Generalized Modigliani–Miller Theory: Applications in Corporate Finance, Investments, Taxation and Ratings (Contributions to Finance and Accounting) 3030938921, 9783030938925

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Contributions to Finance and Accounting

Peter Brusov Tatiana Filatova Natali Orekhova

Generalized Modigliani–Miller Theory Applications in Corporate Finance, Investments, Taxation and Ratings

Contributions to Finance and Accounting

The book series ‘Contributions to Finance and Accounting’ features the latest research from research areas like financial management, investment, capital markets, financial institutions, FinTech and financial innovation, accounting methods and standards, reporting, and corporate governance, among others. Books published in this series are primarily monographs and edited volumes that present new research results, both theoretical and empirical, on a clearly defined topic. All books are published in print and digital formats and disseminated globally.

More information about this series at https://link.springer.com/bookseries/16616

Peter Brusov • Tatiana Filatova • Natali Orekhova

Generalized Modigliani– Miller Theory Applications in Corporate Finance, Investments, Taxation and Ratings

Peter Brusov Financial University under the Government of Russian Federation Moscow, Russia

Tatiana Filatova Financial University under the Government of Russian Federation Moscow, Russia

National Chung Cheng University Jiayi City, Taiwan Natali Orekhova Southern Federal University Rostov-on-Don, Russia

ISSN 2730-6038 ISSN 2730-6046 (electronic) Contributions to Finance and Accounting ISBN 978-3-030-93892-5 ISBN 978-3-030-93893-2 (eBook) https://doi.org/10.1007/978-3-030-93893-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

With nostalgia for pre-covid time, when we been young and happy and with hope that normal life will return Dedicated to our children and nice grandchildren and their future

Preface

In modern conditions, the requirements increase for the quality of the company’s financial management, for the efficiency of corporate finance management, for improving the quality of assessing the effectiveness of investments, for improving the taxation and tax control system, for developing an adequate system of business assessment, and for increasing the objectivity of ratings. In the financial management of a company, the management of the cost of raising capital and the structure of the company’s capital play a fundamental role. Historically, the first quantitative theory of the capital structure of a company–the theory of Nobel laureates Modigliani and Miller, due to the large number of limitations of this theory, had a very distant relationship to economic practice. Since the creation of the theory of Modigliani and Miller, numerous attempts have been made to modify it, the analysis of which is given in the monograph. Of all the modifications, we mention the two most important ones that brought the theory closer to economic practice: accounting for corporate and individual taxes (carried out by the authors Modigliani and Miller themselves) and generalization to the case of companies of arbitrary age and arbitrary lifetime, performed by the authors of this monograph, who created the Brusov–Filatova–Orekhova (BFO theory). The rest of the modifications, although some of them are interesting from a theoretical point of view, have little effect on the possibility of the practical application of the Modigliani–Miller theory. Note that the importance of the Modigliani–Miller theory is determined by the fact that, despite its many limitations (the most significant of which is the perpetuity of companies), it is still widely used in practice, and also by the fact that it is, due to its simplicity and that it is the perpetuity limit of the BFO theory, serves as a good testing ground for new modifications, which, after verifying their significance for practical application, are then used in the Brusov–Filatova– Orekhova theory (BFO theory). In this monograph, the Modigliani–Miller theory is generalized and modified taking into account the conditions of the real functioning of companies: for the case of variable income of companies, for the case of payment of income tax with an arbitrary frequency (monthly, quarterly, semiannual or annual payments), both for vii

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advance payments of income tax and for payments at the end of the period, for the case of payment of interest on a loan with an arbitrary frequency, to simultaneously take into account the above conditions, as well as other conditions of the actual functioning of companies. These generalizations lead to very important consequences: all statements and all formulas by Modigliani and Miller change significantly. For example, in the case of variable profit: • Discount rate changes from the weighted average cost of capital, WACC, to WACC–g (where g is growing rate). WACC grows with g, while real discount rates WACC–g and k0–g decrease with g. This leads to an increase of company capitalization with g. • The tilt angle of the equity cost ke(L) grows with g. This should change the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. • The qualitatively new effect in corporate finance has been discovered: at rate g < g* the slope of the curve ke(L) turns out to be negative that could significantly alter the principles of the company’s dividend policy. Similar important consequences take place for all considering cases of generalization of Modigliani–Miller theory. Accounting them leads to a lot of quite important results, which allows developing a new approach to financial policy and financial strategy of the company. We investigate the applications of obtained theoretical results in corporate finance, investments, taxation, and ratings, where using of generalized Modigliani– Miller theory will be quite useful. In business valuation, it allows to determine the correct value of discount rate, company capitalization, and other financial indicators of the company. New modern investment models have been created, as close as possible to real investment conditions, both long-term and arbitrary duration, with various debt repayment schemes (at the end of the term, a few payments per period, advance payments, etc.) and interest on debt (a few payments per period, advance payments, etc.), with variable income from investments, as well as taking into account the various tax payment options adopted in different countries. Their verification will lead to the creation of a comprehensive system of adequate and correct assessment of the effectiveness of the company’s investment program and its investment strategy. One of the most important elements of calculating the effectiveness of investment projects is the assessment of the discount rate, the calculation methods of which have been generalized for the real conditions of the implementation of investment projects. The development of modern methods of studying the impact of taxation on business and investment in different countries has been carried out using the generalized Modigliani–Miller theory and modern investment models created within the monograph. This will make it possible to develop recommendations to the Regulators of different countries on the amount of corporate income tax, and to the Central Banks of different countries on loan rates.

Preface

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New innovative methodologies for rating nonfinancial issuers and project rating have been developed based on the application of the generalized Modigliani–Miller theory, as well as new investment models created by the authors within the monograph. For this, modifications of the generalized Modigliani–Miller theory, as well as modern investment models for rating needs, have been carried out. The financial ratios used in the rating methodology have been incorporated into the generalized Modigliani–Miller theory, as well as into modern investment models. A complete and detailed study of the dependence of the weighted average cost of capital of the company WACC, used as the discount rate for discounting financial flows, on the financial ratios used in the rating, on the level of debt financing, the level of profitability, the profit tax rate, the frequency of payment of income tax and interest on a loan, on the growth rate of profit in a wide range of values of the cost of equity and debt capital has been done. This will allow for a correct assessment of discount rates taking into account the values of financial ratios. For project rating within the framework of the created new modern investment models of arbitrary duration, a complete and detailed study of the dependence of the main efficiency indicator, NPV, on the level of debt financing, the level of profitability, the profit tax rate, the frequency of payment of income tax and interest on a loan, the growth rate of profit has been carried out in a wide range of values of the cost of equity and debt capital. This creates a new methodological basis for modern project rating. The monograph uses a fundamental approach associated with the generalization, further development and application of the Modigliani–Miller theory, with the generalization and further development of modern investment models created by the authors and well tested in the real sector of the economy, with the creation within the monograph of innovative modified investment models, taking into account the payment of income tax with an arbitrary frequency, various options for payment of debt and interest on debt. The use of a fundamental approach will allow a deeper and more comprehensive study of the studied problems of the theory of corporate finance and corporate governance, as well as problems in the field of investments, business valuation, taxation and tax control, and rating. The relevance of solving the problems discussed is associated with the increase in modern conditions of requirements for the quality of financial management of the company, for the efficiency of corporate finance management, for improving the quality of assessing the effectiveness of investments, for improving the taxation and tax control system, for developing an adequate business assessment system, requirements for increasing the objectivity of ratings. The implementation of the results of the monograph will greatly contribute to the solution of the discussed problems, which determines the practical significance of these results. The scientific novelty of the research is associated with the world’s first generalization, taking into account the conditions of the real functioning of companies, the Modigliani–Miller theory. This book is intended for students, both undergraduate and postgraduate, students of MBA program, teachers of economic and financial Universities, students of MBA

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program, scientists, financial analysts, financial directors of company, managers of insurance companies, managers and analysts of rating agencies, officials of regional and federal ministries and departments, and for ministers responsible for economic and financial management. Moscow, Russia 23 November, 2021

Peter Brusov

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

1 6

Modigliani–Miller Theory in Corporate Finance

2

Capital Structure: Modigliani–Miller Theory . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Traditional Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Modigliani–Miller Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Modigliani–Miller Theory Without Taxes . . . . . . . . 2.3.2 Modigliani–Miller Theory with Taxes . . . . . . . . . . . 2.3.3 Main Assumptions of Modigliani–Miller Theory . . . 2.3.4 Modifications of Modigliani–Miller Theory . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .

3

Modern Theory of Capital Cost and Capital Structure: Brusov–Filatova–Orekhova Theory (BFO Theory) . . . . . . . . . . . . . 3.1 Companies of Arbitrary Age and Companies with Arbitrary Lifetime: Brusov–Filatova–Orekhova Equation . . . . . . . . . . . . 3.2 Comparison of Modigliani–Miller Results (Perpetuity Company) with Myers Results (1-Year Company) and Brusov–Filatova– Orekhova Ones (Company of Arbitrary Age) . . . . . . . . . . . . . 3.3 Brusov–Filatova–Orekhova Theorem . . . . . . . . . . . . . . . . . . . 3.4 From Modigliani–Miller to General Theory of Capital Cost and Capital Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 BFO Theory in the Case, When the Company Ceased to Exist at the Time Moment n (BFO-2 Theory) . . . . . . . . . . . 3.5.1 Application of Formula BFO-2 . . . . . . . . . . . . . . . . .

9 9 10 11 11 13 16 17 30 33 33

37 39 45 48 49

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3.5.2

Comparison of Results Obtained from Formulas BFO and BFO-2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Optimal Capital Structure of the Company: Its Absence in Modigliani–Miller Theory with Risky Debt Capital . . . . . . . . . . . . 4.1 Optimal Capital Structure of the Company . . . . . . . . . . . . . . 4.2 Absence of the Optimal Capital Structure in Modified Modigliani–Miller Theory (MMM Theory) . . . . . . . . . . . . . . 4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51 53 55

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57 57

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60 61 62

5

The Equity Cost in the Modigliani–Miller Theory . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65 65 69

6

The Role of Taxing and Leverage in Evaluation of Capital Cost and Capitalization of the Company . . . . . . . . . . . . . . . . . . . . . 6.1 The Role of Taxes in Modigliani–Miller Theory . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 71 74

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Inflation in Modigliani–Miller Theory . . . . . . . . . . . . . . . . . . . . . . . 7.1 Accounting of Inflation in Modigliani–Miller Theory without Taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Second Original MM Statement . . . . . . . . . . . . . . . . . 7.1.2 Second Modified MM-BFO Statement . . . . . . . . . . . . 7.2 Accounting of Inflation in Modigliani–Miller Theory with Corporate Taxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Fourth Original MM Statement . . . . . . . . . . . . . . . . . 7.2.2 Fourth Modified MM-BFO Statement . . . . . . . . . . . . 7.3 Irregular Inflation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Inflation Rate for a Few Periods . . . . . . . . . . . . . . . . . . . . . . . 7.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modification of the Modigliani–Miller Theory for the Case of Advance Tax on Profit Payments . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Modified Modigliani–Miller Theory in Case of Advance Tax Payments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Tax Shield in Case of Advance Tax Payments . . . . . 8.2.2 Capitalization of the Company . . . . . . . . . . . . . . . . 8.2.3 Equity Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level in “Classical” Modigliani–Miller

77 77 81 81 81 83 83 84 84 85 85

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88 88 89 90

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Theory (MM Theory) and in Modified Modigliani–Miller Theory (MMM Theory) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

10

The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax on Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Focus and Objective of the Chapter . . . . . . . . . . . . . . . . . . . . 9.3 Modification of the Modigliani–Miller Theory for the Case of Arbitrary Frequency of Payments of Tax on Profit . . . . 9.3.1 Tax Shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 The Weighted Average Cost of Capital, WACC . . . . . 9.3.3 Company Value, V . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4 Equity Cost, ke . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4 Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1 Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L at Different p and Fixed kd . . . . . . . . . . . . . . . . . . . . . . 9.5 Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L at Different kd and Fixed p . . . . . 9.6 Dependence of the Company Capitalization, V, on Leverage Level L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.6.1 Dependence of the Company Capitalization, V, on Leverage Level L at Different p and Fixed kd . . . . . 9.6.2 Dependence of the Company Capitalization, V, on Leverage Level L at Different kd and Fixed p . . . . . 9.7 Dependence of the Equity Cost, ke, on Leverage Level L . . . . . 9.7.1 Dependence of the Equity Cost, ke, on Leverage Level L at Different p and Fixed kd ¼ 0.16 . . . . . . . . . 9.7.2 Dependence of the Equity Cost, ke, on Leverage Level L at Different kd and Fixed p ¼ 2 . . . . . . . . . . . 9.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Frequently Should Companies Pay Tax on Profit . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Focus and Objective of the Chapter . . . . . . . . . . . . . . . . . . . 10.4 Modification of the Modigliani–Miller Theory for the Case of Arbitrary Frequency of Advanced Payments of Tax on Profit . 10.4.1 Tax Shield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 The Weighted Average Cost of Capital, WACC . . . . 10.4.3 Company Value, V . . . . . . . . . . . . . . . . . . . . . . . . .

91 93 94 97 97 97 99 100 100 101 102 103 104

104 106 109 112 115 120 120 124 130 131

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133 133 134 137

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138 138 139 140

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10.4.4 Equity Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1 Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L at Different p and Fixed kd . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Dependence of the Weighted Average Cost of Capital, WACC, and of Company Value, V, on Debt Cost Value kd at Fixed Value of Leverage Level L and Fixed p . . . 10.6 Dependence of the Company Capitalization, V, on Leverage Level L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6.1 Dependence of the Company Capitalization, V, on Leverage Level L at Different p and Fixed kd . . . . . . . 10.7 Dependence of the Equity Cost, ke, on Leverage Level L . . . . . 10.8 Dependence of the Equity Cost, ke, and the Weighted Average Cost of Capital, WACC, on Tax on Profit Value t . . . . . . . . . . 10.9 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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10.5

11

Generalization of the Modigliani–Miller Theory for the Case of Variable Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The Traditional Approach . . . . . . . . . . . . . . . . . . . . . 11.1.2 Modigliani–Miller Theory Without Taxes . . . . . . . . . 11.2 Some Modifications of Modigliani–Miller Theory . . . . . . . . . . 11.2.1 Modigliani–Miller Theory with Taxes . . . . . . . . . . . . 11.2.2 Taking into Account Market Risk: Hamada Model . . . 11.2.3 The Account of Corporate and Individual Taxes (Miller Model) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.4 Brusov–Filatova–Orekhova (BFO) Theory . . . . . . . . . 11.2.5 The General WACC Formula . . . . . . . . . . . . . . . . . . 11.2.6 Trade-Off Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Generalization of the Modigliani–Miller Theory for the Case of Variable Profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 Modigliani–Miller Theory Without Taxes . . . . . . . . . 11.3.2 Modigliani–Miller Theory with Taxes . . . . . . . . . . . . 11.4 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.1 Dependence of WACC on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.2 and Different Values of g (0.4; 0.3; 0.2; 0.0; 0.2; 0.3; 0.4) . . . . . . . . . . . . 11.4.2 Dependence of the Weighted Average Cost of Capital WACC on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.3 and Different Values of g . . . . . . . . . . . . . . . . . . . . .

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145 148 148 150 154 157 159 159 163 163 165 166 168 168 169 170 170 171 172 174 174 176 179

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11.4.3

Dependence of Discount Rate i on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.2 and Different Values of g . . . . . . . . . . . . . 11.4.4 Dependence of Discount Rate i on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.3 and Different Values of g . . . . . . . . . . . . . 11.4.5 Dependence of Company Value V on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.2 and Different Values of g . . . . . . 11.4.6 Dependence of Company Value V on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.3 and Different Values of g . . . . . . 11.4.7 Dependence of Equity Cost ke on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.2 and Different Values of g (0; 0.2; 0.3; 0.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4.8 Dependence of Equity Cost ke on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 ¼ 0.3 and Different Values of g . . . . . . . . . . . . . 11.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II 12

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Applications of the Modigliani–Miller Theory in Investments

Investment Models with Debt Repayment at the End of the Project and their Application . . . . . . . . . . . . . . . . . . . . . . . 12.1 Investment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2 The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only . . . . . . . . . . . . . . . . 12.2.1 With the Division of Credit and Investment Flows . . 12.3 Without Flows Separation . . . . . . . . . . . . . . . . . . . . . . . . . . 12.4 Modigliani–Miller Limit (Perpetuity Projects) . . . . . . . . . . . . 12.4.1 With Flows Separation . . . . . . . . . . . . . . . . . . . . . . 12.4.2 Without Flows Separation . . . . . . . . . . . . . . . . . . . . 12.5 The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt . . . . . . . . . . . . 12.5.1 With Flows Separation . . . . . . . . . . . . . . . . . . . . . . 12.5.2 Without Flows Separation . . . . . . . . . . . . . . . . . . . . 12.6 Modigliani–Miller Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 With Flows Separation . . . . . . . . . . . . . . . . . . . . . . 12.6.2 Without Flows Separation . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 201 . 201 . . . . . .

202 202 204 205 205 206

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Investment Models with Uniform Debt Repayment and Their Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1 Investment Models with Uniform Debt Repayment . . . . . . . . 13.2 The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only . . . . . . . . . . . . . . . . 13.2.1 With the Division of Credit and Investment Flows . . 13.2.2 Without Flows Separation . . . . . . . . . . . . . . . . . . . . 13.3 The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt . . . . . . . . . . . . 13.3.1 With Flows Separation . . . . . . . . . . . . . . . . . . . . . . 13.3.2 Without Flows Separation . . . . . . . . . . . . . . . . . . . . 13.4 Example of the Application of the Derived Formulas . . . . . . . 13.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Innovative Investment Models with Debt Repayment at the End of the Project . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1.1 The Literature Review . . . . . . . . . . . . . . . . . . . . . . 14.2 Modern Investment Models . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.1 The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only . . . . . . . . . . 14.2.2 Without Flows Separation . . . . . . . . . . . . . . . . . . . . 14.3 The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt . . . . . . . . . . . . 14.3.1 With Flows Separation . . . . . . . . . . . . . . . . . . . . . . 14.3.2 Without Flows Separation . . . . . . . . . . . . . . . . . . . . 14.4 Discount Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.1 Numerical Calculation of the Discount Rates . . . . . . 14.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Investment Models with Advance Frequent Payments of Tax on Profit and of Interest on Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 The Literature Review . . . . . . . . . . . . . . . . . . . . . . 15.1.2 The Discount Rates . . . . . . . . . . . . . . . . . . . . . . . . 15.1.3 The Structure of the Paper . . . . . . . . . . . . . . . . . . . . 15.2 The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only . . . . . . . . . . . . . . . . 15.2.1 With Flow Separation . . . . . . . . . . . . . . . . . . . . . . . 15.2.2 Without Flow Separation . . . . . . . . . . . . . . . . . . . . 15.3 The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt . . . . . . . . . . . .

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15.3.1 With Flow Separation . . . . . . . . . . . . . . . . . . . . . . . . 15.3.2 Without Flow Separation . . . . . . . . . . . . . . . . . . . . . 15.4 Discount Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.1 Numerical Calculation of the Discount Rates . . . . . . . 15.5.2 The Effectiveness of the Long-Term Investment Project from the Perspective of the Owners of Equity Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.3 The Effectiveness of the Long-Term Investment Project from the Perspective of the Owners of Equity and Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.4 The Effectiveness of the Arbitrary Duration Investment Projects from the Perspective of the Owners of Equity Capital . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.5 The Effectiveness of the Arbitrary Duration Investment Project from the Perspective of the Owners of Equity and Debt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5.6 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part III 16

260 261 262 263 264

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Applications of the Modigliani–Miller Theory Ratings and Rating Methodologies

Application of the Modigliani–Miller Theory in Rating Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 The Closeness of the Rating Agencies . . . . . . . . . . . . . . . . . . 16.3 The Use of Discounting in the Rating . . . . . . . . . . . . . . . . . . . 16.4 Incorporation of Parameters, Using in Ratings, into Perpetuity Limit of Modern Theory of Capital Structure by Brusov– Filatova–Orekhova . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.1 One-Period Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5.2 Multi-period Model . . . . . . . . . . . . . . . . . . . . . . . . . 16.6 Theory of Incorporation of Parameters, Using in Ratings, into Perpetuity Limit of Modern Theory of Capital Structure by Brusov–Filatova–Orekhova . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.1 Coverage Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6.2 More Detailed Consideration . . . . . . . . . . . . . . . . . . . 16.6.3 Leverage Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7 Equity Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8 How to Evaluate the Discount Rate? . . . . . . . . . . . . . . . . . . . . 16.8.1 Using One Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8.2 Using a Few Ratios . . . . . . . . . . . . . . . . . . . . . . . . .

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16.9

Influence of Leverage Level . . . . . . . . . . . . . . . . . . . . . . . . . 16.9.1 The Dependence of Equity Cost ke on Leverage Level at Two Coverage Ratio Values ij ¼ 1 and ij ¼ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.9.2 The Dependence of Equity Cost ke on Leverage Level at Two Leverage Ratio Values lj ¼ 1 and lj ¼ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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. 312 . 313 . 317

Application of the Modigliani–Miller Theory, Modified for the Case of Advance Payments of Tax on Profit, in Rating Methodologies . . . 17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2 Мodified Modigliani–Miller Theory . . . . . . . . . . . . . . . . . . . . 17.3 Application of Modified of Modigliani–Miller Theory for Rating Needs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.1 Coverage Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2 Leverage Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A New Approach to Project Ratings . . . . . . . . . . . . . . . . . . . . . . . 18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2 Investment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.2.1 The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only (Without Flows Separation) . . . . . . . . . . . . . . . . . . . 18.2.2 Modigliani–Miller Limit (Long-term (Perpetuity) Projects) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3 Incorporation of Financial Coefficients, Using in Project Rating, into Modern Investment Models . . . . . . . . . . . . . . . . 18.3.1 Coverage Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.3.2 Leverage Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Dependence of NPV on Coverage Ratios . . . . . . . . . . . . . . . 18.4.1 Coverage Ratio on Debt . . . . . . . . . . . . . . . . . . . . . 18.5 Dependence of NPV on Leverage Ratios . . . . . . . . . . . . . . . 18.5.1 Leverage Ratio of Debt . . . . . . . . . . . . . . . . . . . . . . 18.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

319 319 320 322 323 327 331 332

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Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

About the Authors

Peter Brusov is a professor at the Financial University under the Government of the Russian Federation (Moscow). Originally a physicist, he was the co-founder of (together with Victor Popov) the theory of collective properties of superfluids and superconductors. In the area of finance and economy, Peter Brusov has created a modern theory of capital cost and capital structure, the Brusov–Filatova–Orekhova theory, together with Tatiana Filatova and Natali Orekhova and modified the Modigliani-Miller theory in several directions: for an arbitrary company age (BFO theory), for an arbitrary life-time (BFO-2 theory), for rating needs (BFO-3), to account for variable profits, to account for frequent income tax payments with payment at the end of the periods and in advance. Peter Brusov has been visiting Professor of Northwestern University (USA), Cornell University (USA), Osaka City University (Japan), Chung–Cheng National University (Taiwan) among other places. He is the author of over 500 research publications including eight monographs, numerous textbooks and articles.

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About the Authors

Tatiana Filatova is a professor at the Financial University under the Government of the Russian Federation (Moscow). In the last 20 years, she has been a Dean of the faculties of financial management, management, state and municipal government among others at the Financial University. Tatiana Filatova is co-author of a modern theory of capital cost and capital structure, the Brusov–Filatova–Orekhova theory, and the author of over 260 research publications including seven monographs, numerous textbooks and articles.

Natali Orekhova is a scientist of the Southern Federal University, Russia. Natali Orekhova has been the leading scientist of the Financial University under the Government of the Russian Federation. She is the co-author of a modern theory of capital cost and capital structure, the Brusov–Filatova–Orekhova theory, and author of over 110 research publications including five monographs, numerous textbooks and articles.

Chapter 1

Introduction

In modern conditions, as we mentioned in Preface, the requirements for the quality of the company’s financial management, for the efficiency of corporate finance management, for improving the quality of assessing the effectiveness of investments, for improving the taxation and tax control system, for developing an adequate system of business assessment, for increasing the objectivity of ratings are increasing. In the financial management of a company, the management of the cost of raising capital and the structure of the company’s capital plays a fundamental role. Historically, the first quantitative theory of the capital structure of a company—the theory of Nobel laureates Modigliani and Miller, due to a large number of limitations of this theory, had a very distant relationship to economic practice. Since the creation of the theory of Modigliani and Miller, numerous attempts have been made to modify it, the analysis of which is given in the monograph. Of all the modifications, we mention the two most important ones that brought the theory closer to economic practice: accounting for corporate and individual taxes (carried out by the authors Modigliani and Miller themselves) and generalization to the case of companies of arbitrary age and arbitrary lifetime, performed by the authors of this monograph, who created the Brusov–Filatova–Orekhova (BFO) theory. The rest of the modifications, although some of them are interesting from a theoretical point of view, have little effect on the possibility of the practical application of the Modigliani–Miller theory. Note that the importance of the Modigliani–Miller theory is determined by the fact that, despite its many limitations (the most significant of which is the perpetuity of companies), it is still widely used in practice, and also by the fact that it is, due to its simplicity and that it is the perpetuity limit of the BFO theory, serves as a good testing ground for new modifications, which, after verifying their significance for practical application, are then used in the BFO theory. In monograph the Modigliani–Miller theory, which is the perpetuity limit of the BFO theory, is generalized taking into account the conditions for the actual functioning of companies: for the case of variable company income, for the case of income tax payments with an arbitrary frequency (monthly, quarterly, semiannual or annual payments), for advance tax for profit payments, as well as for payments at the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_1

1

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1 Introduction

end of the period, and for other conditions (Brusov et al. 2020a, 2020b; Brusov et al. 2021a, 2021b; Brusov and Filatova 2021; Filatova et al. 2022). The applications of the generalized Modigliani–Miller theory in corporate finance, investments, taxation, and ratings are developed. New modern investment models have been created that are as close as possible to real investment conditions, with various schemes for repaying debt and interest on it (frequent payments, advance payments, etc.), with variable income from investments, as well as for taking into account various options for paying taxes, adopted in various countries. Consideration is carried out both from the point of view of owners of equity capital, and from the point of view of owners of equity and debt capital. Within the framework of the created new modern investment models, a complete and detailed study of the dependence of the main efficiency indicator, NPV, on the financial ratios used in the rating (coverage ratios and leverage ratios), on the level of debt financing, on the profit tax rate, on the frequency of payment of income tax and interest on a loan, on the growth rate of profit in a wide range of values of equity and debt capital, and the level of profitability was carried out. New modern methodologies for rating nonfinancial issuers and project rating based on the application of the generalized Modigliani–Miller theory, as well as investment models created by the authors of the monograph, have been developed. For this, the modification of the generalized Modigliani–Miller theory, as well as long-term investment models for the needs of rating, has been carried out. The incorporation of financial indicators used in the rating methodology into the generalized theory by Modigliani–Miller, as well as into modern investment models, has been done. Within this theory, a complete and detailed study of the dependence of the weighted average cost of capital of the WACC company, used as the discount rate for discounting financial flows, on the financial ratios used in the rating, on the level of debt financing and the level of taxation, on the frequency of payments of income tax and interest on a loan, on the growth rate of profit in a wide range of values of equity and debt capital costs was carried out for perpetual companies. This will make it possible to carry out a correct assessment of discount rates taking into account the values of financial ratios and to issue correct ratings for nonfinancial issuers. When developing a project rating methodology, a complete and detailed study of the dependence of the main performance indicator, NPV, on the financial ratios used in the rating (coverage and leverage ratios), on the level of debt financing, the level of profitability, income tax rate, frequency of payment of tax on profit and interest on a loan, the growth rate of investment profit in a wide range of values of the cost of equity and debt capital has been done. The influence of taxation on business and investment has been investigated within the framework of the generalized theory of Modigliani–Miller. The monograph has the following structure. Chapter 2 is devoted to a detailed description of the classical version of the Modigliani–Miller theory and some of its known modifications. In Chap. 3, we describe the most important generalization of the Modigliani–Miller theory for the company of arbitrary age—the modern theory of capital cost and capital structure— the Brusov–Filatova–Orekhova theory. Chapter 4 is devoted to discussion of the

1 Introduction

3

problem of the optimal capital structure and it is shown that the optimal capital structure is absent in modified Modigliani–Miller theory (MMM theory) (modified by taking off the suggestion about riskless of debt capital). In Chap. 5 the equity cost in the Modigliani–Miller theory is discussed in detail, including its dependence on tax on profit rate, on leverage level, and other parameters. In Chap. 6 the role of tax shield, taxes, and leverage in the Modigliani–Miller theory is investigated. Chapter 7 is devoted to the investigation of the influence of inflation on capital cost and capitalization of the company within Modigliani–Miller theory. By direct incorporation of inflation into Modigliani–Miller theory, it is shown that inflation not only increases the equity cost and the weighted average cost of capital but as well it changes their dependence on leverage. In particular, it increases the growing rate of equity cost with leverage. Capitalization of the company is decreased under the accounting of inflation. Chapters 8–11 are devoted to the description of the generalization of the Modigliani–Miller theory taking into account the conditions for the actual functioning of companies: for the case of variable company income, for the case of income tax payments with an arbitrary frequency (monthly, quarterly, semiannual, or annual payments), as for advance tax payments for profit, and for payments at the end of the period, as well as other conditions. In Chap. 8 we generalize the Modigliani–Miller theory for the case of advance tax on profit payments, which is widely used in practice, and show that this leads to some important consequences, which change seriously all the main statements by Modigliani and Miller. In Chap. 9 we modify the Modigliani–Miller theory for the case of arbitrary frequency of payment of tax on profit. Combining the theoretical consideration with numerical calculations within MS Excel we show that: 1. All Modigliani–Miller theorems, statements, and all formulas change. 2. All main financial indicators, such as the weighted average cost of capital, WACC, company value, V, and equity cost, ke, depend on the frequency of tax on profit payments. This allows to company manage WACC, V, ke, etc. by choosing the number of payments of tax of profit p per year; 3. in case of income tax payments more than once per year (at p 6¼ 1), as it takes place in practice, the weighted average cost of capital, WACC, company value, V and equity cost, ke start depend on kd, while in ordinary (classical) Modigliani– Miller theory all these values DO NOT depend on kd; 4. the tilt angle of the curve of equity cost, ke (L), decreases with the number of payments of tax of profit p, this modifies the dividend policy of the company, because the economically justified value of dividends is equal to equity cost; 5. obtained results allow to company choose the number of payments of tax of profit per year, as many, as it is profitable to it (of course, within actual tax legislation): more frequent payments of income tax are beneficial for both parties: for the company and for the tax regulator. Chapter 10 studies the influence of two effects (frequent payments of tax on profit and the method of its payment: at the end of periods or by advanced payments) to main financial indicators of the company, such as the weighted average cost of

4

1 Introduction

capital, WACC, company value, V, and equity cost, ke, we modify the Modigliani– Miller theory for the case of arbitrary frequency of payments of tax on profit: for payments at the end of periods as well as for advanced payments. Account of these two effects leads to very important consequences. In Chap. 11, for the first time, we have generalized the world-famous theory by Nobel Prize winners Modigliani and Miller for the case of variable profit, which significantly extends the application of the theory in practice, specifically in business valuation, ratings, corporate finance, etc. We demonstrate that all the theorems, statements, and formulae of Modigliani and Miller are changed significantly. We combine theoretical and numerical (by MS Excel) considerations. The following results are obtained: 1. Discount rate for leverage company changes from the weighted average cost of capital, WACC, to WACC–g (where g is growing rate), for a financially independent company from k0 to k0–g. This means that WACC and k0 are no longer the discount rates as it takes place in the case of classical Modigliani–Miller theory with constant profit. WACC grows with g, while real discount rates WACC–g and k0–g decrease with g. This leads to an increase of company capitalization with g. 2. The tilt angle of the equity cost ke(L) grows with g. This should change the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. 3. A qualitatively new effect in corporate finance has been discovered: at rate g < g* the slope of the curve ke(L) turns out to be negative, which could significantly alter the principles of the company’s dividend policy. Chapters 12–15 are devoted to the description of the innovative investment models, created by the authors and accounting for some features of real investments, such as frequent payments of tax on profit and the method of its payment: at the end of periods or by advanced payments. The modern investment models with debt repayment at the end of the project are described in Chap. 12, while Chap. 13 is devoted to a description of the investment models with uniform debt repayment and their application. Both types of investment models are well tested in the real economy. These models are used by us for the investigation of different problems of investments, such as influence of debt financing, leverage level, taxing, project duration, method of financing, and some other parameters on efficiency of investments and other problems. In Chaps. 14–15 we create eight innovative investment models, considering the long-term as well as arbitrary duration models with payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly), which could be applied in real economic practice. Their verification will lead to the creation of a comprehensive system of adequate and correct assessment of the effectiveness of the company’s investment program and its investment strategy. In Chap. 14 payments of interest on debt and of tax on income are made at the ends of periods, while In Chap. 15 these payments are made in advance. Numerical calculations showed that in the case of advance payments of income tax and interest on debt, all

1 Introduction

5

the results related to the effect of the number of payments of income tax and interest on debt on the effectiveness of the investment projects are opposite to the results in the case of payments at the end of the periods. Results obtained in Chaps. 14–15 help tax regulator (Finance Ministry) to understand the influence of the number of payments of tax on income per period and credit regulator (Central Bank) to understand the influence of the number of payments of interest on debt per period on the effectiveness of investment projects. These allow both Regulators to modify and improve tax legislation and credit policy, respectively. Chapters 16–18 are devoted to the application of the Modigliani–Miller theory in ratings of nonfinancial issues and of project ratings. Chapter 16 describes the rating of nonfinancial issuers, while Chapter 18 describes the long-term project rating. In these chapters, the Modigliani–Miller theory has been modified for rating needs. The key factors of a new approach are: (1) The adequate use of discounting of financial flows virtually not used in existing rating methodologies, (2) The incorporation of rating parameters (financial “ratios”) into the Modigliani–Miller theory. This, on the one hand, allows the use of the powerful tools of this theory in the rating, and on the other hand, it ensures the correct discount rates when discounting financial flows. We discuss also the interplay between rating ratios and leverage level which can be quite important in rating. In Chap. 17 the Modigliani–Miller theory generalized for the case of advance payments of tax on profit (which is widely used in practice) (MMM theory) (see Chap. 8) has been modified for rating needs. A serious modification of MMM theory in order to use it in rating procedure has been required. The financial “ratios” (main rating parameters) were introduced into MMM theory. The necessity of an appropriate use of financial flows discounting in rating methodologies is discussed. The dependence of the weighted average cost of capital (WACC), which plays the role of discount rate, on coverage and leverage ratios is analyzed. All these create a new base for rating methodologies. New approach to ratings and rating methodologies allows to issue more correct ratings of issuers, makes the rating methodologies more understandable and transparent. Obtained results show that the properties of the generalized Modigliani–Miller theory are quite different from ones of the classical Modigliani–Miller theory. This leads to the fact that the generalized Modigliani–Miller theory, as well as modern investment models modified for real investment conditions, created in monograph, are much more applicable in real economy, finance and practice. Since the Modigliani–Miller theory is still widely used in practice, its generalized modification will allow increasing the quality of the company’s financial management, the efficiency of corporate finance management, improving the quality of assessing the effectiveness of investments, improve the taxation and tax control system, to develop an adequate system of business assessment, and increase the objectivity of rating issues.

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1 Introduction

References Brusov P, Filatova T (2021) The Modigliani–miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020a) Application of the Modigliani–miller theory, modified for the case of advance payments of tax on profit, in rating methodologies. Journal of Reviews on Global Economics 9:282–292 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Modification of the Modigliani–miller theory for the case of advance payments of tax on profit. Journal of Reviews on Global Economics 9:257–267 Brusov P, Filatova T, Chang S-I, Lin G (2021a) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9(13):1491 Brusov P, Filatova T, Orekhova N, Kulik V, Chang S-I, Lin G (2021b) Generalization of the Modigliani–miller theory for the case of variable profit. Mathematics 9(11):1286 Filatova T, Brusov P, Orekhova N (2022) Impact of advance payments of tax on profit on effectiveness of investments. Mathematics 10(4):666. https://doi.org/10.3390/math10040666

Part I

Modigliani–Miller Theory in Corporate Finance

Chapter 2

Capital Structure: Modigliani–Miller Theory

2.1

Introduction

One of the two main theories of capital cost and capital structure is the theory of Nobel Prize winners Modigliani and Miller (1958, 1963, 1966). In this chapter, we describe the main results of this theory. Under the capital structure, one understands the relationship between equity and debt capital of the company. Does capital structure affect the company’s main settings, such as the cost of capital, profit, value of the company, and the others, and, if it affects, how? Choice of an optimal capital structure, i.e., a capital structure, which minimizes the weighted average cost of capital, WACC, and maximizes the value of the company, V, is one of the most important tasks solved by a financial manager and by the management of a company. The first serious study (and first quantitative study) of the influence of capital structure of the company on its indicators of activities was the work by Modigliani and Miller (1958). Until this study, the approach existed (let us call it traditional), which was based on empirical data analysis. One of the most important assumptions of the Modigliani–Miller theory is that all financial flows are perpetuity. This limitation was lifted out by Brusov–Filatova– Orekhova in 2008 (Filatova et al. 2008), who have created BFO theory—modern theory of capital cost and capital structure for companies of arbitrary age (BFO–1 theory) and for companies of an arbitrary lifetime (BFO–2 theory) (Brusov et al. 2015). In Fig. 2.1 the historical development of capital structure theory from the traditional (empirical) approach, through perpetuity Modigliani–Miller approach to general capital structure theory—Brusov–Filatova–Orekhova (BFO) theory is shown. Steve Myers (2001) has considered the case of one-year company and showed that in this case the weighted average cost of capital, WACC, is higher than in Modigliani–Miller case, and the capitalization of the company, V, is less than in Modigliani–Miller case.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_2

9

10

2

Capital Structure: Modigliani–Miller Theory

Fig. 2.1 Historical development of capital structure theory (here TA traditional (empirical) approach, MM Modigliani–Miller approach, BFO Brusov–Filatova–Orekhova theory)

Fig. 2.2 MM theory describes perpetuity limit, Myers paper describes one-year company while BFO theory fills the whole numeric axis (from n ¼ 1 up to perpetuity limit n ¼ 1)

So, before 2008 only two results for capital structure of the company were available: Modigliani–Miller for perpetuity company and Myers for one-year company (see Fig. 2.2). BFO theory has filled out the whole interval between t ¼ 1 and t¼1. It gives the possibility to calculate the capitalization V, the weighted average cost of capital, WACC, equity cost ke, and other financial parameters for companies of arbitrary age and for companies of arbitrary lifetime. BFO theory has led to a lot of new meaningful effects in modern capital structure theory, discussed in this monograph.

2.2

The Traditional Approach

The traditional (empirical) approach told that weighted average cost of capital, WACC, and the associated company capitalization, V ¼ CF/WACC, depend on the capital structure, the level of leverage, L. Debt cost always turns out to be lower than equity cost because the first one has lower risk, via the fact that in the event of bankruptcy creditor claims are met prior to shareholders claims. As a result, an increase in the proportion of lower-cost debt capital in the overall capital structure up to the limit which does not cause violation of financial sustainability and growth of risk of bankruptcy leads to a lower weighted average cost of capital, WACC. The profitability required by investors (the equity cost) is growing; however, its growth has not led to compensation of benefits from the use of lower-cost debt capital. Therefore, the traditional approach welcomes the increased leverage L ¼ D/S and the associated increase of company capitalization. The traditional (empirical) approach has existed up to the appearance of the first quantitative theory by Modigliani and Miller (1958).

2.3 Modigliani–Miller Theory

2.3 2.3.1

11

Modigliani–Miller Theory Modigliani–Miller Theory Without Taxes

Modigliani and Miller (ММ) in their first paper (Мodigliani and Мiller 1958) have come to the conclusions which were fundamentally different from the conclusions of the traditional approach. Under assumptions (see Sect. 2.3.3 for details) that there are no taxes, no transaction costs, no bankruptcy costs, perfect financial markets exist with symmetry information, equivalence in borrowing costs for both companies and investors, etc., they have shown that choosing of the ratio between the debt and equity capital does not affect company value as well as capital costs (Fig. 2.3). Under the above assumptions, Modigliani and Miller have analyzed the impact of financial leverage, supposing the absence of any taxes (on corporate profit as well as individual one). They have formulated and proven two following statements: Without taxes, the total cost of any company is determined by the value of its EBIT – Earnings Before Interest and Taxes, discounted with fixed rate k0, corresponding to group of business risk of this company. VL ¼ VU ¼

EBIT : k0

ð2:1Þ

Index L means a financially dependent company (using debt financing), while index U means a financially independent company. Fig. 2.3 Dependence of company capitalization, UL, equity cost, ke, debt cost, kd, and weighted average cost of capital, WACC, in traditional (empirical) approach

U U*

UL

U0

U0

Ke WACC Kd

K0

Kd

0

L*

L

12

2

Capital Structure: Modigliani–Miller Theory

Authors supposed that both companies belong to the same group of business risk, and k0 corresponds to required profitability of the financially independent company, having the same business risk. Because, as it follows from Eq. 2.1, value of the company does not depend on the value of debt; then according to Modigliani–Miller theorem (Мodigliani and Мiller 1958), in the absence of taxes, value of the company is independent of the method of its funding. This means as well that weighted average cost of capital, WАСС, of this company does not depend on its capital structure and is equal to the capital cost, which this company will have under the funding by equity capital only. V 0 ¼ V L;

CF=k0 ¼ CF=WACC,

and thus WACC ¼ k 0 :

Note that the first Modigliani–Miller theorem is based on suggestions about independence of weighted average cost of capital and debt cost on leverage level. From the first Modigliani–Miller theorem (Мodigliani and Мiller 1958), it is easy to derive an expression for the equity capital cost WACC ¼ k 0 ¼ k e we þ k d wd :

ð2:2Þ

Finding from here ke, one gets ke ¼

k ðS þ D Þ k0 w D D  k d ¼ k 0 þ ðk 0  k d Þ  kd d ¼ 0 S S S we we

¼ k0 þ ðk0  kd ÞL

ð2:3Þ

Here, D S D kd , wd ¼ DþS S ke , we ¼ DþS L ¼ D/S

value of debt capital of the company value of equity capital of the company cost and fraction of debt capital of the company cost and fraction of equity capital of the company financial leverage

Thus, we come to the second statement (theorem) of Modigliani–Miller theory about the equity cost of financially dependent (leverage) company (Мodigliani and Мiller 1958). Equity cost of leverage company ke could be found as equity cost of financially independent company k0 of the same group of risk, plus premium for risk, the value which is equal to production of difference (k0  kd) on leverage level L: ke ¼ k0 þ ðk 0  kd ÞL:

ð2:4Þ

Eq. 2.4 shows that equity cost of the company increases linearly with leverage level (Fig. 2.3).

2.3 Modigliani–Miller Theory

13

The combination of these two Modigliani–Miller statements implies that the increasing level of debt in the capital structure of the company does not lead to an increased value of firms, because the benefits gained from the use of more low-cost debt capital markets will be exactly offset by an increase in risk (we are speaking about the financial risk, the risk of bankruptcy) and, therefore, by an increase in cost of equity capital of firms: investors will increase the required level of profitability under increased risk, by which a higher level of debt in the capital structure is accompanied. In this way, the Modigliani–Miller theorem argues that in the absence of taxes, the capital structure of the company does not affect the value of the company and its weighted average cost of capital, WACC, and equity cost increases linearly with the increase of financial leverage. Explanations, given by Modigliani and Miller under receiving of their conclusions, are the following (Мodigliani and Мiller 1958). Value of the company depends on profitability and risk only and does not depend on the capital structure. Based on the principle of preservation of the value, they postulated that the value of the company, which is equal to the sum of the equity and debt funds, is not changed when the ratio between its parts is changed. An important role in the justification of Modigliani–Miller statements is an existence of arbitral awards opportunities for the committed markets plays. Two identical companies, differing only by the leverage level, must have the same value. If this is not the case, the arbitration aligns business cost: investors of less cost company can invest capital in a company of more value. Selling of shares of the first company and buying of stock of the second company will continue until the values of both companies are not equalized. Most of Modigliani and Miller assumptions (Мodigliani and Мiller 1958), of course, are unrealistic. Some assumptions can be removed without changing the conclusions of the model. However, assuming no costs of bankruptcy and the absence of taxes (or the presence of corporate taxes only) are crucial—the change of these assumptions alters conclusions. The last two assumptions rule out the possibility of signaling theory and agency costs theory and, thus, also constitute a critical prerequisite (Fig. 2.4).

2.3.2

Modigliani–Miller Theory with Taxes

In the real situation, taxes on profit of companies always exist. Since the interest paid on debt is excluded from the tax base, it leads to the so-called effect of “tax shield”: the value of the company that used the borrowed capital (leverage company) is higher than the value of the company that financed entirely by the equity (non-leverage company). The value of the “tax shield” for 1 year is equal to kd D T, where D—the value of debt, T—the income tax rate, and kd—the interest on the debt (or debt capital cost) (Мodigliani and Мiller 1963). The value of the “tax shield” for perpetuity company for all time of its existence is equal to (we used the formula for the sum of terms of an infinitely decreasing geometric progression).

14

2

Fig. 2.4 Dependence of equity cost ke and WACC on leverage level L within Modigliani–Miller theory without taxes

Capital Structure: Modigliani–Miller Theory

CC ke =k0+L(k0-kd )

WACC

k0

_ L= D S

ðPVÞTS ¼ kd DT

1 X

ð1 þ k d Þt ¼ DT

ð2:5Þ

t¼1

and the cost of leverage company is equal to V ¼ V 0 þ DT,

ð2:6Þ

where V0 is the value of financially independent company. Thus, we obtain the following result obtained by Мodigliani and Мiller (1963): The value of financially dependent company is equal to the value of the company of the same risk group used no leverage, increased by the value of tax shield arising from financial leverage, and equal to the product of rate of corporate income tax T and the value of debt D. Let us now get the expression for the equity capital cost of the company under the existence of corporate taxes. Accounting that V0 ¼ CF/k0 and that the ratio of debt capital wd ¼ D/V, one gets V ¼ CF=k 0 þ wd VT:

ð2:7Þ

Because the value of leverage company is V ¼ CF/WACC, for weighted average cost of capital, WACC, we get WACC ¼ k0 ð1  wd T Þ:

ð2:8Þ

From here the dependence of WACC on leverage L ¼ D/S becomes the following:

2.3 Modigliani–Miller Theory

WACC ¼ k0 ð1  LT=ð1 þ LÞÞ:

15

ð2:9Þ

On the other hand, on definition of the weighted average cost of capital with “tax shield” accounting, we have WACC ¼ k0 we þ k d wd ð1  T Þ:

ð2:10Þ

Equating Eqs. (2.9) and (2.11), one gets k 0 ð1  wd T Þ ¼ k 0 we þ kd wd ð1  T Þ

ð2:11Þ

and from here, for equity cost, we get the following expression: ð1  w d T Þ w 1 w D  k d d ð1  T Þ ¼ k 0  k 0 d T  k d ð1  T Þ we we S we we DþS D D  k 0 T  kd ð1  T Þ ¼ k0 þ Lð1  T Þðk0  k d Þ: ¼ k0 S S S

ke ¼ k0

ð2:12Þ

So, we get the following statement obtained by Мodigliani and Мiller (1963): Equity cost of leverage company ke paying tax on profit could be found as equity cost of financially independent company k0 of the same group of risk, plus premium for risk, the value which is equal to production of difference (k0  kd) on leverage level L and on tax shield (1 – T). It should be noted that Eq. 2.12 is different from Eq. 2.4 without tax only by the multiplier (1 – T ) in term, indicating a premium for risk. As the multiplier is less than unit, the corporate tax on profits leads to the fact that capital is growing with the increase of financial leverage, slower than it would have been without them. Analysis of formulas (Eqs. 2.4, 2.9, and 2.12) leads to following conclusions. When leverage grows: 1. Value of company increases. 2. Weighted average cost of capital WACC decreases from k0 (at L ¼ 0) up to k0(1  T ) (at L ¼ 1) (when the company is funded solely by borrowed funds). 3. Equity cost increases linearly from k0 (at L ¼ 0) up to 1 (at L ¼ 1). Within their theory, Мodigliani and Мiller (1963) had come to the following conclusions. With the growth of financial leverage (Fig. 2.5): 1. The company value increases. 2. The weighted average cost of capital decreases from k0 (for L ¼ 0) up to k0(1  T ) (for L ¼ 1, when the company is financed entirely with borrowed funds). 3. The cost of equity capital increases linearly from k0 (for L ¼ 0) up to 1 (for L ¼ 1).

16

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Capital Structure: Modigliani–Miller Theory

Fig. 2.5 Dependence of equity capital cost, debt cost, and WACC on leverage in Modigliani– Miller theory without taxes (t ¼ 0) and with taxes (t 6¼ 0)

2.3.3

Main Assumptions of Modigliani–Miller Theory

The most important assumptions of the Modigliani–Miller theory are as follows: 1. Investors are behaving rationally and instantaneously, seeing profit opportunity, inadequate investment risk. Therefore, the possibility of a stable situation of the arbitration, i.e., obtaining the risk-free profit on the difference in prices for the same asset cannot be kept any length of time—reasonable investors quickly take advantage of it for their own purposes and equalize conditions in the market. This means that in a developed financial market capital, the same risk should be rewarded by the same rate of return. 2. Investment and financial market opportunities should be equally accessible to all categories of investors—whether institutional or individual investors, large or small, rapidly growing or stable, or experienced or relatively inexperienced. 3. Transaction costs associated with funding are very small. In practice, the magnitude of transaction costs is inversely proportional to the amount of finance involved, so this assumption is more consistent with reality than the large sums involved: i.e., in attracting small amounts, the transaction costs can be high, while, as in attracting large loans, as well as during placement of shares at a significant amount, the transaction costs can be ignored. 4. Investors get money and provide funds to borrowers at risk-free rate. In all probability, this assumption is due to the fact that the lender seeks to protect himself by using one or other guarantees, pledge of assets, the right to pay claims on third parties, and the treaty provisions restricting the freedom of the borrower to act to the detriment of the creditor. Lender’s risk is really small, but its position can be considered risk-free with respect to the position of the borrower and, accordingly, should be rewarded by a risk-free rate of return.

2.3 Modigliani–Miller Theory

17

5. Companies have only two types of assets: risk-free debt capital and risky equity capital. 6. There is no possibility of bankruptcy, i.e., irrespective of what the level of financial leverage of the company—borrowers are reached—bankruptcy is not threatening them. Thus, bankruptcy costs are absent. 7. There are no corporate taxes and taxes on personal income of investors. If the personal income tax can indeed be neglected, because the assets of the company separated from the assets of shareholders, the corporate income taxes should be considered in the development of more realistic theories (which was done by Modigliani and Miller in their second paper devoted to the capital structure (Мodigliani and Мiller 1963). 8. Companies are in the same class of risky companies. 9. All financial flows are perpetuity. 10. Companies have the same information. 11. Management of the company maximizes the capitalization of the company.

2.3.4

Modifications of Modigliani–Miller Theory

Taking into Account Market Risk: Hamada Model Robert Hаmаdа (1969) unites Capital Asset Pricing Model (CAPM) with Modigliani–Miller model taxation. As a result, he derived the following formula for calculation of the equity cost of financially dependent company, including both financial and business risk of company: k e ¼ kF þ ðk M  kF ÞbU þ ðkM  kF ÞbU

D ð1  T Þ, S

ð2:13Þ

where bU is the β–coefficient of the company of the same group of business risk, that the company under consideration, but with zero financial leverage. Eq. 2.13 represents the desired profitability of equity capital ke as a sum of three components: riskfree profitability kF, compensating to shareholders a temporary value of their money, premium for business risk (kM  kF)bU, and premium for financial risk ðkM  kF ÞbU DS ð1  T Þ. If the company does not have borrowing (D ¼ 0), the financial risk factor will be equal to zero (the third term is drawn to zero), and its owners will only receive the premium for business risk. To apply the Hamada equation, specialists in practice, in most cases, use book value of equity capital as its approach to market value. Nevertheless, the Hamada formula implies the use of market value of the assets. It should be noted also that Eq. 2.13 can be used to derive other equations, using which you can analyze the impact of financial leverage on β–factor of company shares.

18

2

Capital Structure: Modigliani–Miller Theory

Equating CAPM formula to equity cost, we get: kF þ ðkM  kF ÞbU ¼ kF þ ðkM  kF ÞbU þ ðkM  kF ÞbU

D ð1  T Þ S

ð2:14Þ

or   D b ¼ bU 1 þ ð 1  T Þ : S

ð2:15Þ

In this way, the assumptions on which Modigliani–Miller theory and CAPM are based, β–factor of equity capital of financially dependent company is equal to β– factor of financially independent company, corrected on tax on profit rate and applied leverage level. Consequently, market risk of the company, measured by a factor b, depends on both the business risk of the company, a measure of which is bU, and on the financial risk b, which is calculated by the formula in Eq. 2.15. In conclusion, here are the formulas for calculating the capital costs within the CAPM model [in parenthesis, there are formulas within the Modigliani–Miller theory (Modigliani and Miller 1958, 1963, 1966)]. The equity cost for company without debt capital:   ke ¼ kF þ kM  kF βU , ðke ¼ k 0 Þ:

ð2:16Þ

The equity cost for company with debt capital:   k e ¼ kF þ k M  kF βe , ðke ¼ k 0 þ ð1  T Þðk0  k d ÞLÞ:

ð2:17Þ

The debt cost:   kd ¼ kF þ kM  kF βd , ðkd ¼ kF ; βd ¼ 0Þ:

ð2:18Þ

The weighted average cost of capital WACC WACC ¼ ke we þ kd wd ð1  T Þ,

ðWACC ¼ k 0 ð1  Twd ÞÞ:

ð2:19Þ

The Cost of Capital Under Risky Debt Another hypothesis of Modigliani and Miller was the suggestion about free of risk debt (in their theory, there are two types of assets: risky equity and free of risk debt). However, if we assume the risk of bankruptcy of company (and, accordingly, the ability to nonpayment of loans), the situation may change. Stiglitz (1969) and Rubinstein (1973) have shown that the conclusions concerning the total value of company do not change as compared to the findings derived by Modigliani and Miller under assumptions about free of risk debt (Modigliani and Miller 1958, 1963, 1966). However, the debt cost is changed. If previously, under assumption about the

2.3 Modigliani–Miller Theory

19

free of risk debt, it (debt cost) was regarded as a constant kd ¼ kF, now it is not a constant. This claim is based on the work by Hsia (1981), where based on the models of pricing options, Modigliani–Miller and CAPM, it was shown that if one uses the formula for the net discount income, a term, reflecting tax protection on debt, should be discounted at the rate 1 , wd

ð2:20Þ

 ln wd þ kF t 1 pffi pffi kF þ σ t, 2 σ t

ð2:21Þ

k 0d ¼ k F þ ðk 0  kF ÞN ðd 1 Þ where d1 ¼

here t–a moment of payment a credit, N(d1)–cumulative normal distribution of probability of random value d1. The Account of Corporate and Individual Taxes (Miller Model) In the second article, Modigliani and Miller (1963) considered taxation of corporate profits but did not take into account the presence in the economy of individual taxes of investors. Merton Miller (1997) has introduced the model, demonstrating the impact of leverage on the company value with account of the corporate and individual taxes (Miller 1976). To describe his model, we will enter the following legends: TC—tax on corporate profits rate, TS—the tax rate on income of an individual investor from his ownership by stock of corporation, TD—tax rate on interest income from the provision of investor—individuals of credits to other investors and companies. Income from shares partly comes in the form of a dividend and, in part, as capital profits, so that TS is a weighted average value of effective rates of tax on dividends and capital profits on shares, while the income from the provision of loans usually comes in the form of the interests. The last are usually taxed at a higher rate. In the light of the individual taxes, and with the same assumptions that have been made for Modigliani–Miller models previously, the financially independent company value can be determined as follows: VU ¼

EBITð1  T C Þð1  T S Þ : k0

ð2:22Þ

A term (1  TS) allows to take into account the individual taxes in the formula. In this way, numerator indicates which part of the operating company’s profit remains in the possession of the investors, after the company will pay taxes on their profits, and its shareholders then will pay individual taxes on income from stock ownership. Since individual taxes reduce profits, remaining in the disposal of investors, the last,

20

2

Capital Structure: Modigliani–Miller Theory

at other things being equal circumstances, also reduce and an overall assessment of the financially independent company value. We will assess the financially dependent company under condition of a double taxation of income investors. To start, let us divide the annual cash flows of financially dependent company CFL into flows sent to its shareholders CFe and the flows belonging to debt owners CFd, with account of both corporation tax on profits and on the income of individuals: CFL ¼ CFe þ CFd ¼ ðEBIT  I Þð1  T C Þð1  T S Þ þ I ð1  T D Þ,

ð2:23Þ

where I is the annual interest payments on debt. Eq. 2.23 can also be rewritten as follows: CFL ¼ CFe þ CFd ¼ EBITð1  T C Þð1  T S Þ  I ð1  T C Þð1  T S Þ þ I ð1  T D Þ:

ð2:24Þ

The first term of Eq. 2.24 corresponds to cash flow after taxes for financially independent company, shown in Eq. 2.22, which shows its present value. The second and the third terms of the equation, reflecting the financial dependence, correspond to cash flows related to the debt financing, which, as previously, is considered as free of risk. Their present values are obtained by discounting by risk-free nominal rate on debt kd. By combining the present values of all three terms, we get the company value under using the debt financing and in the presence of all types of taxation: VU ¼

EBITð1  T C Þð1  T S Þ I ð1  T C Þð1  T S Þ I ð1  T d Þ  þ : k0 kd kd

ð2:25Þ

First term in Eq. 2.25 is identical to VU in Eq. 2.22. Accounting this and combining two last terms, we get the following formula: VL ¼ VU þ

  I ð1  T d Þ ð1  T C Þð1  T S Þ 1 : kd ð1  T d Þ

ð2:26Þ

The amount of paid interests with taking into account the taxation, divided by the desired profitability of debt capital, I ð1  T d Þ kd

ð2:27Þ

is equal to market value of the debt D. Substituting D in the previous formula, we get the final expression, which is known as a formula of a Miller model:

2.3 Modigliani–Miller Theory

21

  ð1  T C Þð1  T S Þ VL ¼ VU þ 1  D: ð1  T d Þ

ð2:28Þ

The Miller model allows you to obtain an estimate of the value of financially dependent company, taking into account the corporate tax, as well as tax on individuals. The Miller formula in Eq. 2.28 has several important consequences: 1. Second term of sum,  1

 ð1  T C Þð1  T S Þ D, ð1  T d Þ

ð2:29Þ

represents the gains from use of debt capital. This term replaces the tax on profit of corporation rate in the Modigliani–Miller model with corporate taxes: V L ¼ V U þ TD:

ð2:30Þ

2. If we ignore taxes, a term (Eq. 2.29) will be equal to zero. Thus, in this case, the formula in Eq. 2.28 is transformed into the original version of the Modigliani– Miller model without taxes. 3. If we neglect taxes on individuals, the considering term becomes 1  (1  TC) ¼ TC, so, in this case, Eq. 2.28 becomes a Modigliani–Miller model with corporate taxes (Eq. 2.30). 4. If the shareholder receives profit only in the form of dividend, and if effective tax rates on income from shares and bonds are equal (TS ¼ TD), the terms 1  TS and 1  TD are shrinking, and the factor for D in Eq. 2.29 again is equal to TC. 5. If the shareholder receives dividends, and income from capital, the situation is changed. In this case, effective tax rates on income from shares and bonds are not equal. Let us take a look at common case, when individual taxes on income for the company shares are less than individual taxes of creditors. This encourages investors to purchase the shares of the company compared to purchasing the bonds of the company. In this case, TS < TD. Then factor in D in Eq. 2.28 β has a look ð1  T C Þð1  T S Þ ¼ 1  ð1  T C Þα ¼ 1  ð1  T C Þð1 þ γ Þ ð1  T d Þ ¼ T C þ γ ð T C  1Þ < T C :

β¼1

ð2:31Þ

It is less than TC, because γ > 0, TC < 1; therefore, in this case, the effect of using debt financing, although there is, but it is less, than in the absence of individual taxes. In other words, the effect of tax shields for the company in this case

22

2

Capital Structure: Modigliani–Miller Theory

decreases, and it becomes less than the above individual taxes of creditors (individual taxes for the obligations of the company) in comparison with the individual income tax on shares. 6. Let us take a look at case TS > TD, when individual income taxes on shares are bigger than individual taxes creditors. The factor β takes view ð1  T C Þð1  T S Þ ¼ 1  ð1  T C Þα ¼ 1  ð1  T C Þð1  γ Þ ð1  T d Þ ¼ T C þ γ ð1  T C Þ > T C :

β¼1

ð2:32Þ

It is bigger than TC, because γ > 0, TC < 1; therefore, in this case, the effect of use of debt financing is increased compared with the case of the absence of individual taxes. 7. If (1  TC)(1  TS) ¼ 1  TD, then this term is zero, and the effect of using debt financing will also be zero. This means that the benefits of the use of tax shields as a result of the application of debt financing will be fully offset by additional losses of investors, associated with a higher tax rate on interest on income of individuals. In this case, the capital structure will not affect the company value and its capital cost—in other words, you can apply Modigliani–Miller theory without tax (Мodigliani and Мiller 1958). In his paper, Miller (1976) claimed that companies on average will use issuance of shares and debt securities in such a way as to result in taxation of investors’ income to be optimal. In such an equilibrium state will occur equality ð1  T C Þð1  T S Þ ¼ 1  T D ,

ð2:33Þ

and thus, as we have pointed out above, capital structure will not affect the market company value and its capital cost. Thus, by Miller, the conclusions on the irrelevance of the capital structure, made on the base of the original Modigliani–Miller model with zero taxes, remain in force. Subsequently, researchers adapted and checked the Miller results. Their works, as a rule, have been devoted to Miller’s conclusion concerning the absence of the gains from the use of the debt capital by the company. In the United States, an effective tax rate on the income of shareholders is lower than the one on the income of creditors, but, nevertheless, the product (1  TC)(1  TS) is less than 1  TD. Consequently, the companies may receive the benefit from use of debt financing. However, iMiller’s work, in fact, has been shown that the distinction of rates of individual taxes on income of shareholders and creditors to some extent compensates the advantages of use of debt financing, and, in this way, the tax benefits of debt are less than anticipated at a more earlier Modigliani–Miller model, where only corporate taxes have been taken into account.

2.3 Modigliani–Miller Theory

23

The General WACC Formula A more general formula for WACC, than in the famous Modigliani–Miller theory (MM) has been derived and discussed by a few authors in 2006–2007 (Farber et al. 2006; Fernandez 2006; Berk and DeMarzo 2007; Harris and Pringle 1985) . It takes the following form (Equations (18) in (Farber et al. 2006) WACC ¼ k0 ð1  wd t Þ  k d twd þ kTS twd ,

ð2:34Þ

where k0 is the required return on unleveraged company, kd is the required return on its debt, kTS is the expected return on the tax shield, and t is the corporate taxes rate. This formula is derived from the definition of the weighted average cost of capital and the balance sheet identity (for a similar presentation, see Berk and DeMarzo 2007). At any point in time, it should therefore be verified, regardless of whether returns are annually or continuously compounded. Practical applicability of Eq. (2.34) (while it is fairly general) requires additional conditions. Indeed, when the WACC is constant over time, the value of a leveraged company can be computed by discounting with the WACC of the unlevered free cash flows. Therefore, it is interesting to consider the special cases when WACC is constant. The resulting formulas can also be found in textbooks (Brealey et al. 2005; Ross et al. 2005). It was assumed by Modigliani and Miller in 1963 that the debt value D is constant. As the expected after-tax cash flow of the unleveraged company is fixed, V0 is also constant. By this assumption, kTS ¼ kD and the value of the tax shield is TS ¼ tD. Therefore, the capitalization of the leveraged company V is a constant and the general WACC Formula (2.34) simplifies to a constant WACC: WACC ¼ k0 ð1  wd T Þ

ð2:35Þ

However, our opinion is that “classical” Modigliani–Miller (MM) theory, which suggests that the expected returns on the debt kd and the tax shield kTS are equals (because both of them have debt nature), is much more reasonable and in our paper we modify the “classical” Modigliani–Miller (MM) theory, which is still widely used in practice. Trade-off Theory The world-famous trade-off theory has been considered the cornerstone in the solution of the problem of optimal capital structure for a company for many decades and is still used today for decision analysis on capital structure. Below we give two examples. Frank and Goyal (2009), “examines the relative importance of many factors in the capital structure decisions of publicly traded American firms from 1950 to 2003. The most reliable factors for explaining market leverage are: median industry leverage (+ effect on leverage), market-to-book assets ratio (), tangibility (+), profits (), log of assets (+), and expected inflation (+).” In addition, the authors have found that “dividend-paying firms tend to have lower leverage. When considering book

24

2

Capital Structure: Modigliani–Miller Theory

leverage, somewhat similar effects are found. However, for book leverage, the impact of firm size, the market-to-book ratio, and the effect of inflation are not reliable.” The empirical evidence seems to be reasonably consistent with some versions of the trade-off theory of capital structure. Serrasqueiro and Caetano (2015), analyzed “to what extent decisions on the capital structure of small and medium-sized enterprises (SMEs) are closer to the assumptions of trade-off theory or to the assumptions of hierarchy theory. They used a sample of small and medium-sized enterprises located in the Portuguese hinterland, using dynamic LSDVC as the valuation method, and the empirical evidence suggests that the most profitable and oldest SMEs are less leveraged, confirming Pecking Order Theory’s forecasts. Larger SMEs are leveraging more borrowing, confirming the predictions of trade-off theory and hierarchy theory. In addition, SMEs are significantly adjusting their current debt levels towards the optimal debt ratio, which is consistent with the predictions of the compromise theory. It was concluded that theories of compromise and hierarchy are not mutually exclusive in explaining capital structure decisions of small and medium-sized enterprises.” However, the bankruptcy of trade-off theory has been proven by Brusov et al. in 2013 (Brusov et al. 2013a, b). They have shown that risky debt financing (and growing credit rate near the bankruptcy) in contrast to waiting results does not lead to the growing of weighted average cost of capital, WACC, which still decreases with leverage. This means the absence of minimum in the dependence of WACC on leverage level as well as the absence of maximum in the dependence of company capitalization on leverage. Thus, the well-known trade-off theory lacks an optimal capital structure. The explanation for this fact was made by Brusov et al. in 2013 by analyzing the dependence of the cost of equity capital on the leverage level on the assumption that debt capital is risky. Modigliani and Miller have considered tax shields from the interest on debt can increase the value of companies. In 1980, De Angelo and Masulis moved further in the theoretical examination of tax shields. They have noted that there are tax deductibles for companies other than debt to reduce their corporate tax burden and debt and non-debt tax shields should be accounted for. Depreciation, investment tax credits, or net-loss carryforwards could represent examples of such kinds of non-debt tax shields. The first to test for these tax effects (suggested by Harry and Masulis (1980)) has been carried out by Bradley et al. (1984). In contrast to the prediction in Harry and Masulis (1980), by regressing company-specific debt-to-value ratios on non-debt tax shields, they have shown that debt is positively related to non-debt tax shields as measured by depreciation and investment tax credits. Sheridan and Wessels (1988) found that “their results do not provide support for an effect on debt ratios arising from non-debt tax shields. . .” It was pointed out by Graham (2003), if a company invests heavily and uses debt financing to invest, a positive relation between such proxies for non-debt tax shield and debt may result. A mechanical positive relation of this type overwhelms and renders any substitution effects between debt and non-debt tax shields. The original theory by Nobel Prize Winners Modigliani and Miller (1958, 1963, 1966) has been modified by many authors, and above we shortly discussed some of

2.3 Modigliani–Miller Theory

25

Table 2.1 Classification and summary of main theories of capital structures of company Theory Traditional theory

Modigliani–Miller theory (ММ)

Without taxes

With taxes

Brusov–Filatova– Orekhova theory (BFO–1)

For arbitrary age Without inflation

Main thesis Empirical theory, existing before appearance of the first quantitative theory of capital structures (Modigliani–Miller theory) in 1958 (Modigliani and Miller 1958, 1963, 1966). Weighted average cost of capital depends on capital structures of company. There is an optimal dependence on capital structures of company Capital cost and capitalization of the company are irrelevant on the capital structures of company Weighted average cost of capital is decreased with leverage level, equity cost is increased linearly with leverage level, and capitalization of the company is increased with leverage level continuously BFO theory (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b, 2018a, b, c, d, e; Filatova et al. 2008; Brusova 2011) has replaced the famous theory of capital cost and capital structure by Nobel laureates, Modigliani and Miller (1958, 1963, 1966). The authors have moved from the assumption of Modigliani–Miller concerning the perpetuity (infinite time of life) of companies and further elaborated quantitative theory of valuation of core parameters of financial activities of companies with arbitrary age. Results of modern BFO theory turn out to be quite different from that of Modigliani–Miller theory. It shows that later, via its perpetuity, underestimates the assessment of weighted average cost of capital and the equity cost of the company and substantially overestimates the assessment of the capitalization of the company. Such an incorrect assessment of key performance indicators of financial activities of companies has led to an underestimation of risks involved, and impossibility, or serious difficulties in adequate managerial decision-making, which was one of the implicit reasons for the global financial crisis of 2008. In the BFO theory, in investments at certain values of return-oninvestment, there is an optimum investment structure. As well authors have developed a new mechanism of formation of the (continued)

26

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Capital Structure: Modigliani–Miller Theory

Table 2.1 (continued) Theory

Main thesis

For arbitrary age With inflation

Brusov–Filatova– Orekhova theory (BFO–2)

For arbitrary age With increased financial distress costs and risk of bankruptcy For arbitrary lifetime

company optimal capital structure, different from suggested by trade-off theory Inflation not only increases the equity cost and the weighted average cost of capital, but as well it changes their dependence on leverage. In particular, it increases the growing rate of equity cost with leverage. Capitalization of the company is decreased under accounting of inflation In BFO theory, with increased financial distress costs and risk of bankruptcy, the optimal capital structure is absent, which means that trade-off theory does NOT work In perpetuity limit (Modigliani–Miller theory) time of life of company and company age turns out to be the same: both of them are infinite. When we move from the assumption of Modigliani–Miller concerning the perpetuity of companies these concepts (time of life of company and company age) become different and one should distinct them under generalization of Modigliani–Miller theory with respect to finite n. Thus we have developed two kinds of finite n—theories: BFO–1 and BFO–2. BFO–1 theory is related to companies with arbitrary age and BFO–2 theory is related to companies with arbitrary lifetime companies. In other words, BFO–1 is applicable for most interesting case of companies that reached the age of n-years and continue to exist on the market, and allows to analyze the financial condition of the operating companies. BFO–2 theory allows examining the financial status of the companies which ceased to exist, i.e., of those for which n means not age, but a lifetime, i.e., the time of existence. A lot of schemes of termination of activities of the company can exist: Bankruptcy, merger, acquisition, etc. one of those schemes, when the value of the debt capital D becomes zero at the time of termination of activity of company n, is considered in Chapter 3 and comparison of results of BFO–1 and BFO–2 has been done. (continued)

2.3 Modigliani–Miller Theory

27

Table 2.1 (continued) Theory Brusov–Filatova– Orekhova theory (BFO–3)

For rating needs

Trade-off theory

Static

Main thesis A new approach to rating methodology has been developed in Brusov et al. (2018e) (Chapters 21–23). Chapters 21 and 22 in Brusov et al. (2018e) are devoted to rating of nonfinancial issuers, while Chapter 23 is devoted to long-term project rating. The key factors of a new approach are: 1) The adequate use of discounting of financial flows virtually not used in existing rating methodologies, 2) The incorporation of rating parameters (financial “ratios”) into the modern theory of capital structure (Brusov– Filatova–Orekhova (BFO–3 theory)) (in Chapters 21 into its perpetuity limit). This, on the one hand, allows use of the powerful tools of this theory in the rating, and on the other hand, it ensures the correct discount rates when discounting of financial flows. The interplay between rating ratios and leverage level which can be quite important in rating is discussed as well. All these create a new base for rating methodologies. New approach to ratings and rating methodologies allows to issue more correct ratings of issuers, makes the rating methodologies more understandable and transparent. The static trade-off theory is developed with accounting of tax on profit and bankruptcy cost. It attempts to explain the optimal capital structure in terms of the balancing act between the benefits of debt (tax shield from interest deduction) and the disadvantage of debt (from the increased financial distress and expected bankruptcy costs). The tax shield benefit is the corporate income tax rate multiplied by the market value of debt and the expected bankruptcy costs are the probability of bankruptcy multiplied by the estimated bankruptcy costs Does not take into account the costs of the adaptation of financial capital structure to the optimal one, economic behavior of managers, owners, and other participants of economical process, as well as a number of other factors Some main existing principles of financial management have been destroyed by BFO theory: including world-known trade-off (continued)

28

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Capital Structure: Modigliani–Miller Theory

Table 2.1 (continued) Theory

Main thesis

Dynamic

Accounting of transaction cost

Accounting of asymmetry of information

Signaling theory

theory. During many decades it has been considered as keystone of the capital structure theories and has described the formation of optimal capital structure of the company. Bankruptcy of the world famous trade-off theory has been proven be BFO authors (Brusov et al. 2013a, b). As it has been shown, the optimal capital structure is absent in trade-off theory. The reasons for such insolvency were investigated and found within BFO theory The dynamic trade-off models assume that costs of constant capital adjustment are high and thus firms will change capital structure only if benefits exceed costs. Therefore, there is an optimal range, outside of each leverage change but remains unchanged inside. Companies try to adjust their leverage when it reaches the boundary of the optimal range. Subject to types of adjustment costs firms reach target ratio faster or slower. Proportional changes imply slight correction, whereas fixed changes imply considerable costs. In the dynamic model, correct decision on financial structure capital of the company in this period depends on the profit, which the company hopes to receive in the next period In BFO theory, with increased financial distress costs and risk of bankruptcy, the optimal capital structure is absent, which means that trade-off theory does NOT work: In static version as well as in dynamic one Accounting of the recapitalization transaction costs for the company, in which these costs are high, leads to the conclusion that a more cost-effective is not to modify financial capital structure, even if it is not optimal, during a certain period of time. The actual and target capital structure may vary because of the tool costs At the real financial markets, information is asymmetric (managers of the companies have owned more reliable information than investors and creditors), and rationality of economic subjects is limited Information asymmetry may be reduced on the basis of certain signals for creditors and (continued)

2.3 Modigliani–Miller Theory

29

Table 2.1 (continued) Theory

Main thesis

Pecking order theory

Theories of conflict of interests

Theory of agency costs

Theory of corporate control and costs monitoring

Theory of stakeholders

investors, related to the behavior of managers on the capital market. It should take into account the previous development of the company and the current and projected cost-effectiveness of activities The pecking order theory is the preferred, and empirically observed, sequence of financing type to raise capital. That is, firms first tap retained earnings (internal equity) finance, second source is debt, and the last source is issuing new common stock shares (external equity). The empirical evidence of nonfinancial firm debt ratios coupled with the decision-making process of top management and the board of directors point to greater adherence to the pecking order theory Management of the company may take decisions that are contrary to the interests of the shareholders or creditors, respectively; the costs are necessary to monitor its actions. An effective tool for resolving agent problem is the correct selection of compensation package (the share of participation of agent in property, bonus, stock options), allowing to link revenue of managers with the dynamics of equity capital and to provide motivation for managers to its (equity capital) conservation and growth If asymmetries of information exist, creditors, providing the capital, are interested in the possibility of the implementation of the self-monitoring of the effectiveness of its use and return. Costs for monitoring, as a rule, put on the company owners by their inclusion into credit rate. The level of monitoring costs depends on the scale of the business; therefore, with the increase of the business scale, the weighted average cost of capital of the company grows and company market value is reduced Stakeholder theory is a theory that identifies and models the groups that are stakeholders of a corporation or project. The diversity and the intersection of stakeholders’ interests and different assessments by them of acceptable risk generate conditions for conflict of their interest, that is, making (continued)

30

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Capital Structure: Modigliani–Miller Theory

Table 2.1 (continued) Theory

Behavioral theories

Main thesis

Manager investment autonomy

The equity market timing theory

Information cascades

some corrections into the process of optimizing financial capital structure Managers implement those decisions, which, from their point of view, will be positively perceived by investors and, respectively, positively affect the market value of companies: When the market value of shares of a company and the degree of consensus of expectations of managers and investors are high, the company has an additional issue of shares, and in the opposite situation, it uses debt instruments. In this way, the financial capital structure is more influenced by investors, the expectations of which are taken into account by managers Leverage level is determined by market dynamics. Equity market timing theory means that company should issue shares at high price and repurchase them at low price. The idea is to exploit temporary fluctuations in the equity cost relative to the cost of other forms of capital In order to save costs and to avoid errors, financial capital structure can be formed not on the basis of the calculations of optimal capital structure or depending on available in different periods of company life funding sources, but borrow from other companies that have successful, reputable managers (companies’ leaders), as well as using (in the wake of the majority) the most popular methods of management of capital structure

them. In the next paragraph, we will generalize for the first time the Modigliani– Miller theory for the case of variable profit. In conclusion, we present in Table 2.1 a classification and summary of main theories of capital structures of company.

References Berk J, DeMarzo P (2007) Corporate finance. Pearson–Addison Wesley, Boston, MA Bradley M, Jarrell GA, Kim EH (1984) On the existence of an optimal capital structure: theory and evidence. J Financ 39:857–878

References

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Brealey R, Myers S, Allen F (2005) Principle of corporate finance, 7th edn. McGraw Hill, New York Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 435:2–8 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11): 815–824 Brusov P, Filatova T, Orehova N, Brusov PP, Brusova N (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova P, Orekhova N (2013a) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova T, Orehova N (2013b) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Global Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit–Modigliani–Miller theory. J Rev Global Econ 3:175–185 Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investment and taxation, 1st edn. Springer International Publishing, Cham, pp 1–368 Brusov P, Filatova T, Orehova N, Kulk WI (2018a) New meaningful effects in modern capital structure theory. J Rev Global Econ 7:104–122 Brusov P, Filatova T, Orehova N, Kulk V (2018b) A “Golden Age” of the companies: conditions of its existence. J Rev Global Econ 7:88–103 Brusov P, Filatova T, Orehova N, Kulk V (2018c) Rating methodology: new look and new horizons. J Rev Global Econ 7:63–87 Brusov P, Filatova T, Orehova N, Kulk V (2018d) Rating: new approach. J Rev Global Econ 7:37– 62 Brusov P, Filatova T, Orehova N, Eskindarov M (2018e) Modern corporate finance, investment, taxation and rating, 2nd edn. Springer Nature Publishing, Cham, pp 1–576 Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Prob Sol 34(76):36–42 Farber A, Gillet R, Szafarz A (2006) A general formula for the WACC. Int J Bus 11(2):211–218 Fernandez P (2006) A general formula for the WACC: a comment. Int J Bus 11(2):219 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Frank M, Goyal V (2009) Capital structure decisions: which factors are reliably important? Financ Manag 38(1):1–37. https://doi.org/10.1111/j.1755-053X.2009.01026.x Graham JR (2003) Taxes and corporate finance: a review. Rev Financ Stud 16:1075–1129 Hamada R (1969) Portfolio analysis, market equilibrium, and corporate finance. J Financ 24(1): 13–31 Harris R, Pringle J (1985) Risk–adjusted discount rates – extension form the average–risk case. J Financ Res 8(3):237–244 Harry DA, Masulis RW (1980) Optimal capital structure under corporate and personal taxation. J Financ Econ 8:3–29 Hsia С (1981) Coherence of the modern theories of finance. Financ Rev 16(1):27–42 Miller M (1976) Debt and taxes. J Financ 32(2):261–275

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Miller M (1997) Merton Miller on Derivatives. Wiley, New York Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Myers S (2001) Capital structure. J Econ Perspect 15(2):81–102 Ross S, Westerfiel R, Jaffee D (2005) Corporate finance, 7th edn. McGraw Hill, New York Rubinstein M (1973) A mean–variance synthesis of corporate financial theory. J Financ 28:167– 181 Serrasqueiro Z, Caetano A (2015) Trade-off theory versus pecking order theory: capital structure decisions in a peripheral region of Portugal. J Bus Econ Manag 16(2):445–466 Sheridan T, Wessels R (1988) The determinants of capital structure choice. J Financ 43:1–19 Stiglitz J (1969) A re–examination of the Modigliani–Miller theorem. Am Econ Rev 59(5):784–793

Chapter 3

Modern Theory of Capital Cost and Capital Structure: Brusov–Filatova–Orekhova Theory (BFO Theory)

3.1

Companies of Arbitrary Age and Companies with Arbitrary Lifetime: Brusov–Filatova–Orekhova Equation

The most significant generalization of Modigliani–Miller theory has been done by Brusov, Filatova, and Orekhova in 2008 (Filatova et al. 2008). The Brusov, Filatova, and Orekhova theory (BFO theory) has replaced the famous theory of capital cost and capital structure by Nobel laureates Modigliani and Miller, making latter one the perpetuity limit of BFO theory. The authors have moved from the assumption of Modigliani–Miller concerning the perpetuity (infinite time of life) of companies and further elaborated quantitative theory of valuation of key parameters of financial activities of companies with arbitrary time of life (of arbitrary age). Results of modern BFO theory turn out to be quite different from those of Modigliani–Miller theory. They show that the latter, via its perpetuity, underestimates the assessment of weighted average cost of capital, WACC, and the equity cost of the company and substantially overestimates the assessment of the capitalization of the company. Such an incorrect assessment of key performance indicators of financial activities of companies has led to an underestimation of risks involved, and impossibility, or serious difficulties in adequate managerial decision-making, which was one of the implicit reasons of global financial crisis in 2008. As we discussed in Chap. 2, one of the serious limitations of the Modigliani–Miller theory is the suggestion about perpetuity of the companies. In 2008, Brusov–Filatova–Orekhova (Filatova et al. 2008) have lifted up this limitation and have shown that the accounting of the finite lifetime (or finite age) of the company leads to significant changes of all Modigliani– Miller results (Modigliani and Miller 1958, 1963, 1966): capitalization of the company is changed, as well as the equity cost, ke, and the weighted average cost of capital, WACC, in the presence of corporative taxes. Besides, a number of qualitatively new effects in corporate finance, obtained in Brusov–Filatova– Orekhova theory (Brusov 2018a, b; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_3

33

34

3 Modern Theory of Capital Cost and Capital Structure:. . .

2014a, b, 2015, 2018a, b, c, e, f, g, 2019, 2020; Filatova et al. 2018; Brusov and Filatova 2011), are absent in Modigliani–Miller theory. Only in the absence of corporate taxes, we give a rigorous proof of the Brusov–Filatova–Orekhova theorem that equity cost, ke, as well as its weighted average cost, WACC, does not depend on the lifetime (or age) of the company, so the Modigliani–Miller theory could be generalized for arbitrary lifetime (arbitrary age) companies. Until recently (before 2008, when the first paper by Brusov–Filatova–Orekhova [Filatova et al. 2008] has appeared), the basic theory (and the first quantitative one) of the cost of capital and capital structure of companies was the theory by Nobel Prize winners Modigliani and Miller (Modigliani and Miller 1958, 1963, 1966). One of the serious limitations of the Modigliani–Miller theory is the suggestion about perpetuity of the companies. We lift up this limitation and show that the accounting of the finite lifetime (finite age) of the company leads to change in the equity cost, ke, as well as of the weighted average cost of capital, WACC, in the presence of corporate taxes. The effect of leverage on the cost of equity capital of the company, ke, with an arbitrary lifetime, and its weighted average cost of WACC is investigated. We give a rigorous proof of the Brusov–Filatova–Orekhova theorem that in the absence of corporate taxes, cost of company equity, ke, as well as its weighted average cost, WACC, does not depend on the age (lifetime) of the company. Let us consider companies of arbitrary age and companies with arbitrary lifetime. In perpetuity limit (Modigliani–Miller theory) time of life of company and company age turn out to be the same: both of them are infinite. When we move from the assumption of Modigliani–Miller concerning the perpetuity of companies these concepts (time of life of company and company age) become different and one should distinct them under generalization of Modigliani–Miller theory with respect to finite n. Thus, we have developed two kinds of finite n-theories: BFO-1 and BFO-2. BFO-1 theory is related to companies with arbitrary age and BFO-2 theory is related to companies with arbitrary lifetime companies. In other words, BFO-1 is applicable for the most interesting case of companies that reached the age of n-years and continue to exist on the market and allows to analyze the financial state of the operating companies. BFO-2 theory allows examining the financial status of the companies which ceased to exist, i.e., of those companies for which n means not age, but a lifetime, i.e., the time of existence. A lot of schemes of termination of activities of the company can exist: bankruptcy, merger, acquisition, etc. One of those schemes, when the value of the debt capital D becomes zero at the time of termination of activity of company n, is considered in this chapter below (Sect. 3.5) and comparison of results of BFO-1 and BFO-2 has been done as well. To start with the case of finite n let us first of all, find the value of tax shield, TS, of the company for n years

3.1 Companies of Arbitrary Age and Companies with Arbitrary Lifetime:. . .

TS ¼ kd DT

n X

ð1 þ k d Þt ¼ DT ½1  ð1 þ kd Þn :

35

ð3:1Þ

t¼1

(We used the formula for the sum of n terms of a geometric progression). Here, D is the value of debt capital; kd the cost of debt capital; and T the tax on profit rate. Next, we use the Modigliani–Miller theorem (Modigliani and Miller 1958, 1963, 1966): The value of financially dependent company is equal to the value of the company of the same risk group used no leverage, increased by the value of tax shield arising from financial leverage, and equal to the product of rate of corporate income tax T and the value of debt D.

V ¼ V 0 þ DT:

ð3:2Þ

This theorem was formulated by Modigliani and Miller for perpetuity companies, but we modify it for a company of arbitrary age. V ¼ V 0 þ TS ¼ V 0 þ kd DT

1 X

ð1 þ k d Þt

t¼1 n

ð3:3Þ

¼ V 0 þ wd VT ½1  ð1 þ kd Þ , V ð1  wd VT ½1  ð1 þ kd Þn Þ ¼ V 0 :

ð3:4Þ

There is a common use of the following two formulas for the cost of the financially independent and financially dependent companies (Modigliani and Miller 1958, 1963, 1966): V 0 ¼ CF=k 0 and

V ¼ CF=WACC:

ð3:5Þ

However, these almost always used formulas were derived for perpetuity company, and in the case of a company of finite age (or with a finite lifetime), they must be modified in the same manner as the value of tax shields (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b; Filatova et al. 2008): V 0 ¼ CF½1  ð1 þ k 0 Þn =k0 ;

V ¼ CF½1  ð1 þ WACCÞn =WACC:

ð3:6Þ

From Eq. (3.4), we get Brusov–Filatova–Orekhova equation for WACC (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b; Filatova et al. 2008):

3 Modern Theory of Capital Cost and Capital Structure:. . .

36

1  ð1 þ WACCÞn 1  ð1 þ k0 Þn : ¼ WACC k0 ½1  ωd T ð1  ð1 þ k d Þn Þ

ð3:7Þ

D Here, S—the value of equity capital of the company, wd ¼ DþS —the share of debt S capital, ke , we ¼ DþS —the cost and the share of the equity capital of the company, and L ¼ D/S—financial leverage. At n ¼ 1, we get Myers (2001) formula for 1-year company

WACC ¼ k0 

ð1 þ k0 Þkd wd T 1 þ kd

ð3:8Þ

For n ¼ 2, one has 1  ð1 þ WACCÞ2 1  ð1 þ k0 Þ2  i : ¼ h WACC k0 1  ωd T 1  ð1 þ k d Þ2

ð3:9Þ

This equation can be solved for WACC analytically: WACC ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2α  4α þ 1 , 2α

ð3:10Þ

where α¼

2 þ k0 h i: 2k d þk2d ð1 þ k0 Þ 1  ωd T ð1þk 2 Þ 2

ð3:11Þ

d

For n ¼ 3 and n ¼ 4, equation for the WACC becomes more complicated, but it still can be solved analytically, while for n > 4, it can be solved only numerically. We would like to make an important methodological notice: taking into account the finite lifetime of the company, all formulas, without exception, should be received with use Eq. (3.6) instead of their perpetuity limits (Eq. 3.5). Below, we will describe the algorithm for the numerical solution of Eq. (3.7). Algorithm for Finding WACC in Case of Companies of Arbitrary Age Let us return back to n-year project (n-year company). We have the following equation for WACC in n-year case: 1  ð1 þ WACCÞn  AðnÞ ¼ 0, WACC

ð3:12Þ

3.2 Comparison of Modigliani–Miller Results (Perpetuity Company) with Myers. . .

37

where, AðnÞ ¼

1  ð1 þ k0 Þn : k0 ½1  ωd T ð1  ð1 þ kd Þn Þ

ð3:13Þ

The algorithm of the solving of Eq. (3.12) should be as follows: 1. Putting the values of parameters k0, ωd, T and given n, we calculate A(n). 2. We determine two WACC values, for which the left part of the Eq. (3.12) has opposite signs. It is obvious that as these two values we can use WACC1 and WACC1, because WACC1 > WACCn > WACC1 for finite n  2. 3. Using, for example, the bisection method, we can solve Eq. (3.12) numerically. In MS Excel, it is possible to solve Eq. (3.7) much easily by using the option “matching of parameter”: we will use it through the monograph.

3.2

Comparison of Modigliani–Miller Results (Perpetuity Company) with Myers Results (1-Year Company) and Brusov–Filatova–Orekhova Ones (Company of Arbitrary Age)

Myers (2001) has compared his result for 1-year company (project) (Eq. 3.8) with Modigliani and Miller’s result for perpetuity limits (Eq. 2.8). He has used the following values of parameters: k 0 ¼ 8%  24%;

kd ¼ 7%;

T ¼ 50%;

wd ¼ 0%  60%

and estimated the difference in the WACC values following from the formulas in Eqs. (3.8) and (2.8). We did make similar calculations for 2-, 3-, 5-, and 10-year projects for the same set of parameters, and we have gotten the following results, shown in Table 3.1 [second line (bulk)], Table 3.2 [second line (bulk)], and Table 3.3 and corresponding figures in Figs. 3.1, 3.2, and 3.3. Note that data for equity cost k0 ¼ 8% turns out to be a little bit uncertain: this could be related to the fact that this value of equity cost is quite close to the value of interest rate of debt k0 ¼ 7%. For all other values of equity cost, the results are reproducible and very informative and are discussed below. For a graphic illustration of the results, we use data for n ¼ 1, 2, 1, which adequately reflect the results we have obtained. Discussion of Results 1. From Table 3.1 and Fig. 3.1, it is obvious that WACC is maximum for 1-year company (project) and decreases with the lifetime (age) of the company (project) and reaches the minimum in the Modigliani–Miller perpetuity case. Dependence

3 Modern Theory of Capital Cost and Capital Structure:. . .

38

Table 3.1 WACC dependence on debt share wd for different values of equity cost k0 for companies with different lifetime n k0 k0 ¼ 8%

k0 ¼ 10%

k0 ¼ 12%

k0 ¼ 16%

k0 ¼ 20%

k0 ¼ 24%

n n¼1 n¼2 n¼1 n¼1 n¼2 n¼1 n¼1 n¼2 n¼3 n¼5 n ¼ 10 n¼1 n¼1 n¼2 n¼3 n¼5 n ¼ 10 n¼1 n¼1 n¼2 n¼3 n¼5 n ¼ 10 n¼1 n¼1 n¼2 n¼3 n¼5 n ¼ 10 n¼1

wd ¼ 10% 7.6 7.52 7.6 9.7 9.51 9.5 11.6 11.51 11.46 11.42 11.396 11.4 15.62 15.52 15.44 15.38 15.34 15.2 19.6 19.45 19.41 19.35 19.27 19.0 23.6 23.46 23.39 23.31 23.21 22.8

20% 7.3 7.08 7.2 9.3 9.05 9.0 11.3 11.02 10.93 10.83 10.786 10.8 15.2 14.99 14.88 14.76 14.67 14.4 19.2 18.97 18.82 18.69 18.54 18.0 23.2 22.94 22.77 22.61 22.40 21.6

30% 6.9 6.6 6.8 8.9 8.59 8.5 10.9 10.54 10.39 10.25 10.1695 10.2 14.9 14.5 14.31 14.14 13.99 13.6 18.8 18.45 18.23 18.03 17.80 17.0 22.8 22.37 22.15 21.91 21.60 20.4

40% 6.6 6.17 6.4 8.6 8.13 8.0 10.5 10.07 9.85 9.66 9.5455 9.6 14.5 13.98 13.75 13.51 13.31 12.8 18.4 17.93 17.64 17.36 17.05 16.0 22.4 21.80 21.54 21.21 20.78 19.2

50% 6.2 5.67 6.0 8.2 7.64 7.5 10.2 9.6 9.31 9.06 8.914 9.0 14.1 13.47 13.18 12.88 12.62 12.0 18.1 17.37 17.05 16.70 16.30 15.0 22.0 21.30 20.91 20.51 19.96 18.0

60% 5.9 5.21 5.6 7.8 7.16 7.0 9.8 9.09 8.77 8.46 8.2745 8.4 13.7 12.96 12.61 12.24 11.92 11.2 17.7 16.86 16.45 16.03 15.54 14.0 21.6 20.75 20.29 19.80 19.13 16.8

The bold values correspond to case n = 2, calculated within BFO theory

of all WACC values on debt share wd turns out to be linear at any equity cost k0 for all considered durations of the project (lifetime values of the companies). It is natural for 1-year project because it is described by Myers linear formula (3.8) as well as, in the Modigliani–Miller perpetuity case, described by the formula (2.8), which is linear too, but it is a surprise for 2-year project, where formula for WACC (Eq. 3.7) is obviously nonlinear. The negative slope in WACC increases with the equity cost k0. 2. As it follows from Table 3.2 and Fig. 3.3, the dependence of the average ratios r ¼ hΔ1 =Δ2 i on debt share wd is quite weak and can be considered as almost

3.3 Brusov–Filatova–Orekhova Theorem Table 3.2 Dependence of the differences Δ1 ¼ WACC1  WACC1 (first line), Δ2 ¼ WACC1  WACC2 [second line (bulk)], and their ratio r ¼ Δ1/Δ2 (third line) on debt share wd for different values of equity cost k0

k0 ¼ 10%

k0 ¼ 12%

k0 ¼ 16%

k0 ¼ 20%

k0 ¼ 24%

39 wd ¼ 10% 0.20 0.19 1.05 0.2 0.09 2.22 0.4 0.08 5.0 0.6 0.15 4.0 0.8 0.14 5.7

20% 0.30 0.25 1.2 0.5 0.28 1.76 0.8 0.21 3.81 1.2 0.23 5.22 1.6 0.26 6.15

30% 0.4 0.31 1.29 0.7 0.36 1.94 1.3 0.4 3.25 1.8 0.35 5.14 2.4 0.43 5.58

40% 0.60 0.47 1.28 0.9 0.43 2.09 1.7 0.52 3.27 2.4 0.47 5.11 3.2 0.6 5.33

50% 0.7 0.56 1.25 1.2 0.6 2 2.1 0.63 3.33 3.1 0.73 4.25 4.0 0.7 5.71

60% 0.8 0.64 1.25 1.4 0.71 1.97 2.5 0.74 3.38 3.7 0.84 4.4 4.8 0.85 5.65

The bold values use the results for the case n = 2, calculated within BFO theory Table 3.3 Average (by debt share wd) values of ratios r ¼ hΔ1 =Δ2 i for k0 ¼ 10%; 12%; 16%; 20%; and 24% k0 r ¼ hΔ1 =Δ2 i

10% 1.22

12% 2.00

16% 3.67

20% 4.69

24% 3.69

constant. The value of this constant increases practically linear with the equity cost k0 from 1.22 at k0 ¼ 10% up to 3.69 at k0 ¼ 24% (see Fig. 3.4). 3. The relative difference between 1-year and 2-year projects increases when the equity cost k0 decreases. At the same time, the relative difference between 2-year project and perpetuity MM project increases with the equity cost k0. We have also shown in Table 3.4 and Fig. 3.5 the dependence of the WACC on leverage level L for n ¼ 1, n ¼ 3, and n ¼ 1. From Table 3.4 and Fig. 3.5 it is obvious that WACC has a maximum for 1-year company and decreases with the age (lifetime) of the company, reaching the minimum in the Modigliani–Miller perpetuity case. (Note, however, that this not always be so via the effect of the “golden age” of the company) (Brusov et al. 2018d).

3.3

Brusov–Filatova–Orekhova Theorem

Case of Absence of Corporate Taxes Modigliani–Miller theory in case of absence of corporate taxes gives the following results for dependence of WACC and equity cost ke on leverage:

3 Modern Theory of Capital Cost and Capital Structure:. . .

40 25.00 24.00 23.00 22.00 21.00 20.00 19.00 18.00

WACC

17.00 16.00 15.00 14.00 13.00 12.00 11.00 10.00 9.00 8.00 7.00 6.00 5.00 0

10

20

30

40

50

60

Wd

Fig. 3.1 The dependence of the WACC on debt share wd for companies with different lifetime n for different cost of equity, k0 (from Table 3.1)

1.

V 0 ¼ V L;

CF=k 0 ¼ CF=WACC,

and thus WACC ¼ k0 :

ð3:14Þ

2. WACC ¼ we  ke + wd  kd; and thus L WACC  wd  k d k 0  1 þ L k d ke ¼ ¼ 1 we 1þL ¼ k0 þ Lðk 0  kd Þ:

ð3:15Þ

3.3 Brusov–Filatova–Orekhova Theorem

41

Fig. 3.2 Dependence of the ratio r ¼ Δ1/Δ2 of differencesΔ1 ¼ WACC1  WACC1 and Δ2 ¼ WACC1  WACC2 on debt share wd for different values of equity cost k0 (from Table 3.2) Fig. 3.3 Dependence of the average values of ratio r ¼ hΔ1 =Δ2 i on the equity cost, k0

7.00 6.00 5.00 4.00 – r 3.00 2.00 1.00 0.00 10.00

12.00

16.00

20.00

24.00

Ko

For the finite lifetime (finite age) companies, Modigliani–Miller theorem about equality of value of financially independent and financially dependent companies (V0 ¼ VL) has the following view (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b):

3 Modern Theory of Capital Cost and Capital Structure:. . .

42

Fig. 3.4 The dependence of the WACC on leverage in the absence of corporate taxes [the horizontal line (t ¼ 0)], as well as in the presence of corporate taxes [for 1-year (n ¼ 1) and perpetuity companies (n ¼ 1)]. Curves for the WACC of companies with an intermediate lifetime (age) (1 < n < 1) lie within the shaded region Table 3.4 The dependence of the WACC on leverage level L for n ¼ 1, n ¼ 3, and n ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

wd 0.00 0.50 0.67 0.75 0.80 0.83 0.86 0.88 0.89 0.90 0.91

WACC n¼1 20.00% 18.91% 18.55% 18.36% 18.25% 18.18% 18.13% 18.09% 18.06% 18.04% 18.02%

WACC n¼3 20.00% 18.41% 17.87% 17.61% 17.44% 17.34% 17.26% 17.20% 17.16% 17.12% 17.09%

WACC (MM) 20.00% 18.00% 17.33% 17.00% 16.80% 16.67% 16.57% 16.50% 16.44% 16.40% 16.36%

V 0 ¼ V L; CF 

n

½1  ð1 þ k 0 Þ  ½1  ð1 þ WACCÞn  : ¼ CF  k0 WACC

ð3:16Þ

Using this relation, we prove an important Brusov–Filatova–Orekhova theorem: Under the absence of corporate taxes, the equity cost of the company, ke, as well as its weighted average cost of capital, WACC, does not depend on the lifetime (age) of the company and is equal, respectively, to

3.3 Brusov–Filatova–Orekhova Theorem

43

Fig. 3.5 Dependence of WACC on leverage level for n ¼ 1, n ¼ 3, and n ¼ 1

ke ¼ k 0 þ Lðk0  kd Þ;

WACC ¼ k0 :

ð3:17Þ

Let us consider first the 1- and 2-year companies (a) For 1-year company, one has from Eq. (3.15) 1  ð1 þ k 0 Þ1 1  ð1 þ WACCÞ1 ¼ , k0 WACC

ð3:18Þ

1 1 : ¼ 1 þ k 0 1 þ WACC

ð3:19Þ

WACC ¼ k0 :

ð3:20Þ

and thus

Hence,

Formula for equity cost ke ¼ k0 + L(k0  kd) now obtained by substituting WACC ¼ k0 into Eq. (3.14). (b) For 2-year company, one has from Eq. (3.15) h i 1  ð1 þ k0 Þ2 k0

¼

h i 1  ð1 þ WACCÞ2 WACC

,

3 Modern Theory of Capital Cost and Capital Structure:. . .

44

and thus, 2 þ k0 2 þ WACC ¼ : ð1 þ k0 Þ2 ð1 þ WACCÞ2

ð3:21Þ

2þk0 Denoting α ¼ ð1þk , we get the following quadratic equation for WACC: Þ2 0

α  WACC2 þ ð2α  1Þ  WACC þ ðα  2Þ ¼ 0:

ð3:22Þ

It has two solutions WACC1,2 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  2α  4α þ 1 : 2α

ð3:23Þ

2þk 0 Substituting α ¼ ð1þk , we get Þ2 0



WACC1,2

 k 20  3  ðk 0 þ 3Þð1 þ k0 Þ ¼ : 2ð 2 þ k 0 Þ

WACC1 ¼ k0 ;

WACC2 ¼ 

2k0 þ 3 < 0: k0 þ 2

ð3:24Þ ð3:25Þ

The second root is negative, but the weighted average cost of capital can only be positive, so only one value remains WACC1 ¼ k0 : (c) For company with arbitrary lifetime, n, Brusov–Filatova–Orekhova formula (3.15) gives 1  ð1 þ k0 Þn 1  ð1 þ WACCÞn : ¼ k0 WACC

ð3:26Þ

For a fixed k0, Eq. (3.25) is an equation of (n + 1)-degree relative to WACC. It has n + 1 roots (in general complex). One of the roots, as a direct substitution shows, is always WACC ¼ k0. Investigation of the remaining roots is difficult and not a part of our problem. Formula for equity cost ke ¼ k0 + L(k0  kd) is now obtained by substituting WACC ¼ k0 into Eq. (3.14). Thus, we have proved the Brusov–Filatova–Orekhova theorem.

3.4 From Modigliani–Miller to General Theory of Capital Cost and Capital. . .

45

Case of the Presence of Corporate Taxes Modigliani–Miller theory in case of presence of corporate taxes gives the following results for dependence of WACC and equity cost ke on leverage: 1. WACC V L ¼ V 0 þ Dt; D ¼ wd V L ;

ð3:27Þ

CF=WACC ¼ CF=k0 þ Dt ¼ CF=k0 þ wd tCF=WACC,

ð3:28Þ

1  wd t 1 ¼ ; WACC k0  WACC ¼ k 0 ð1  wd t Þ ¼ k0 1 

ð3:29Þ  L t : 1þL

ð3:30Þ

Thus, WACC decreases with leverage from k0 [in the absence of debt financing (L ¼ 0)] up to k0(1  t) (at L ¼ 1). 2. The equity cost ke WACC ¼ k 0 ð1  wd t Þ ¼ we  ke þ wd  kd ð1  t Þ; and thus WACC  wd  kd  ð1  t Þ we L k ð1  t Þ k0 ð1  wd t Þ  1 þ L d ¼ k0 þ Lðk0  k d Þð1  t Þ: ¼ 1 1þL ke ¼

3.4

ð3:31Þ

From Modigliani–Miller to General Theory of Capital Cost and Capital Structure

Let us consider how the weighted average cost of capital, WACC, and the cost of equity capital, ke, will be changed when taking into account the finite age of the company. (a) One-year company

3 Modern Theory of Capital Cost and Capital Structure:. . .

46

From Eq. (3.7), one has 1  ð1 þ WACCÞn 1  ð1 þ k0 Þn : ¼ WACC k0 ½1  wd t ð1  ð1 þ k d Þn Þ

ð3:32Þ

For 1-year company, we get 1  ð1 þ WACCÞ1 1  ð1 þ k0 Þ1  i ¼ h WACC k0 1  wd t 1  ð1 þ kd Þ1

ð3:33Þ

From Eq. (3.33), we obtain the well-known Myers formula (Eq. 3.8), which is the particular case of Brusov–Filatova–Orekhova formula (Eq. 3.7). WACC ¼ k0 

1 þ k0 k w t: 1 þ kd d d

Thus,   ð1 þ k 0 Þ  k d L t :  WACC ¼ k 0 1  ð1 þ k d Þ  k 0 1 þ L

ð3:34Þ

Thus, WACC decreases with leverage from k0 [in the absence of debt financing 0 Þk d (L ¼ 0)] up to k 0 1  ðð1þk ðat L ¼ 1Þ: 1þkd Þk 0 t Equating the right part of Eq. (3.34) to general expression for WACC WACC ¼ we  k e þ wd  kd ð1  t Þ,

ð3:35Þ

1 þ k0 k w t ¼ we  ke þ wd  k d ð1  t Þ: 1 þ kd d d

ð3:36Þ

one gets k0  Thus,

3.4 From Modigliani–Miller to General Theory of Capital Cost and Capital. . .



1 1 þ k0 ke ¼ k  k w t  k d w d ð1  t Þ we 0 1 þ k d d d k ¼ ð1 þ LÞk 0  L d ½ð1 þ k0 Þt þ ð1 þ kd Þð1  t Þ 1 þ kd   kd ¼ k 0 þ Lð k 0  k d Þ 1  t : 1 þ kd   kd k e ¼ k 0 þ Lð k 0  k d Þ 1  t : 1 þ kd

47

ð3:37Þ

So we see that in case of 1-year company, the perpetuity limit ke ¼ k0 + L(k0  kd)(1  t) is replaced by Eq. (3.37). Difference is due to different values of the tax shield for a 1-year company and perpetuity one (Fig. 3.6). Let us investigate the question of the tax shield value for companies with different lifetime (age) in more detail. Tax Shield General expression for the tax shield for n-year company has the form (Brusov– Filatova–Orekhova) TS ¼

n X i¼1

kd Dt ½1  ð1 þ k d Þn  kd Dt   ¼ Dt ½1  ð1 þ kd Þn : ð3:38Þ ¼ i 1 ð1 þ k d Þ ð1 þ kd Þ 1  ð1 þ k d Þ

1. In perpetuity limit (n ! 1), tax shield is equal to TS1 ¼ Dt, which leads to the so-called effect of the tax shield associated with the appearance of a factor (tax corrector) (1  t) in the equity cost ke ¼ k0 + L(k0  kd)(1  t). 2. For the 1-year company, tax shield value is equal to

Fig. 3.6 Dependence of the equity cost, ke, on leverage in the absence of corporate taxes [the upper line (t ¼ 0)], as well as in the presence of corporate taxes [for 1-year (n ¼ 1) and perpetuity companies (n ¼ 1)]. Dependences of the cost of equity capital of companies, ke, of an intermediate age (1 < n < 1) lie within the shaded region

48

3 Modern Theory of Capital Cost and Capital Structure:. . .

  TS1 ¼ Dt 1  ð1 þ kd Þ1 ¼ Dtk d =ð1 þ kd Þ:

ð3:39Þ

  kd This leads to appearance of a factor 1  1þk t in the equity cost (Eq. 3.36) k e ¼ d   kd k0 þ Lðk0  kd Þ 1  1þk t : d 3. Tax shield for a 2-year company is equal to   TS2 ¼ Dt 1  ð1 þ kd Þ2 ¼ Dtkd ð2 þ kd Þ=ð1 þ kd Þ2

ð3:40Þ

and if the analogy with 1-year company will keep, then factor (1  t) in the Modigliani–Miller theory would be replaced by the factor ! k d ð2 þ k d Þ t : 1 ð1 þ k d Þ2

ð3:41Þ

However, due to a nonlinear relation between WACC and k0 and kd in Brusov– Filatova–Orekhova formula (Eqs. 3.9, 3.10, and 3.11) for 2-year company (and companies of bigger age), such a simple analogy is no longer observed, and the calculations become more complex.

3.5

BFO Theory in the Case, When the Company Ceased to Exist at the Time Moment n (BFO-2 Theory)

From the output of the BFO formula, it follows that developed ideology is applied to companies that have reached the age of n-years and continue to exist on the market, while the theory of MM is only applicable to infinitely old (perpetuity) companies. In other words, BFO is applicable for most interesting case of companies that reached the age of n-years and continue to exist on the market and allows to analyze the financial condition of the operating companies. However, the BFO theory allows also to examine the financial status of the companies which ceased to exist, i.e., of those for which n means not age, but a lifetime, i.e., the time of existence. A lot of schemes of termination of activities of the company can exist: bankruptcy, merger, acquisition, etc. Below we consider one of those schemes when the value of the debt capital D becomes zero at the time of termination of activity of company n: in this case, the BFO theory requires minimal upgrades, as shown below. From the formula for the capitalization of the company (Eq. 3.1), it is easy to get an estimation for the “residual capitalization” of the company, discounted to the time moment k:

3.5 BFO Theory in the Case, When the Company Ceased to Exist at the Time. . .

49

h i CF CF ðnk Þ 1  ð 1 þ WACC Þ ¼ : ð1 þ WACCÞt WACC

ð3:42Þ

Vk ¼

n X t¼kþ1

Using the formula V k ¼ wd D,

ð3:43Þ

we obtain an expression for the tax shield for n years subject to the termination of the activities of the company at the moment n: n X

n ðnkþ1Þ V k1 tk w CF X 1  ð1 þ WACCÞ ¼ d d k k WACC k¼1 ð1 þ k d Þ k¼1 ð1 þ k d Þ

n n ð1 þ kd Þ  ð1 þ WACCÞn tk w 1  ð1 þ kd Þ ¼ d d  : WACC kd WACC  kd

TSn ¼ tk d wd

ð3:44Þ

Substituting this expression into Eq. (3.3) V L ¼ V 0 þ ðTSÞn one gets the equation (let us call it BFO-2) 1  ð1 þ WACCÞn 1  ð1 þ k0 Þn tk w ¼ þ d d WACC k0 WACC

1  ð1 þ kd Þn ð1 þ kd Þn  ð1 þ WACCÞn   , kd WACC  kd ð3:45Þ from which one can find the WACC for companies with arbitrary lifetime n, provided that the company ceases to function at the time moment n. Below in the monograph, we investigate the companies that have reached the age of n-years and continue to exist on the market, i.e., we will use formula BFO (Eq. 3.7), but in this paragraph, we present some results obtained from the formula BFO-2 (Eq. 3.45).

3.5.1

Application of Formula BFO-2

Formula BFO-2 (Eq. 3.45) in MS Excel takes the following form:

3 Modern Theory of Capital Cost and Capital Structure:. . .

50 Fig. 3.7 The dependence of the WACC on leverage level L for n ¼ 3 and n ¼ 5; k0 ¼ 0.2; kd ¼ 0.15 WACC

WACC(L) 0.2050 0.2000 0.1950 0.1900 0.1850 0.1800 0.1750 0.1700 0.1650 0

2

4

6

8

10

12

n n=3

Fig. 3.8 The dependence of the WACC on lifetime n at different leverage level L

n=5 WACC(n)

0.1900 0.1850 WACC

0.1800 0.1750 0.1700 0.1650 0.1600 0

10

20

30

40

50

n L=1

L=2

L=3

L=5

     1  ð1 þ C4ÞðH4Þ =C4  1  ð1 þ D4ÞðH4Þ =D4    þ ðððG4 E4 F4Þ=C4Þ 1  ð1 þ E4ÞðH4Þ =E4     ð1 þ E4ÞðH4Þ  ð1 þ C4ÞðH4Þ =ðC4  E4Þ ¼ 0:

L=7



ð3:46Þ

Using it we get the following results for dependence of WACC on leverage level L, lifetime n, and on tax on profit rate t (Figs. 3.7, 3.8, and 3.9).

3.5 BFO Theory in the Case, When the Company Ceased to Exist at the Time. . . Fig. 3.9 The dependence of the WACC on tax on profit rate t for n ¼ 3 and n ¼ 5; k0 ¼ 0.2; kd ¼ 0.15

51

WACC(t) 0.2500

WACC

0.2000 0.1500 0.1000 0.0500 0.0000 0

0.2

0.4

0.6

0.8

1

1.2

n n=3

3.5.2

n=5

Comparison of Results Obtained from Formulas BFO and BFO-2

Let us compare results obtained from formulas BFO and BFO-2 (Figs. 3.10, 3.11, 3.12, and 3.13). Comparison of results obtained from formulas BFO and BFO-2 shows that WACC values (at the same values of other parameters) turn out to be higher for the companies which ceased to exist at the time moment n, than for companies that have reached the age of n-years and continue to exist on the market. In other words, the companies which ceased to exist at the time moment n can attract a capital at higher rate, than for companies that have reached the age of n-years and continue to exist on the market. We will develop the detailed investigation of the companies which ceased to exist at the time moment n (described by formula BFO-2) somewhere also and in this monograph we will limit ourselves by consideration of the companies which have Fig. 3.10 Comparison of the dependence of the WACC on leverage level L for n ¼ 3 and n ¼ 5 from formulas BFO and BFO-2

3 Modern Theory of Capital Cost and Capital Structure:. . .

52

WACC(n)

0.0760 0.0740 0.0720 WACC

Fig. 3.11 Comparison of the dependence of the WACC on lifetime n from formulas BFO (lower curve) and BFO-2 (upper curve); k0 ¼ 0.08; kd ¼ 0.04

0.0700 0.0680 0.0660 0.0640 0.0620 0.0600 0

10

20

30

40

50

n k0=0.08, kd=0.04

WACC(n) 0.1800 0.1750 0.1700 WACC

Fig. 3.12 Comparison of the dependence of the WACC on lifetime n from formulas BFO (lower curve) and BFO-2 (upper curve); k0 ¼ 0.2; kd ¼ 0.15; L ¼ 3

k0=0.08, kd=0.04

0.1650 0.1600 0.1550 0.1500 0

10

20

30

40

50

n k0=0.2, kd=0.15

WACC(t) 0.2500 0.2000 WACC

Fig. 3.13 Comparison of the dependence of the WACC on tax on profit rate t from formulas BFO (lower curve) and BFO-2 (upper curve); k0 ¼ 0.2; kd ¼ 0.15; n¼5

k0=0.2, kd=0.15

0.1500 0.1000 0.0500 0.0000 0

0.2 n=5

0.4 n=5(2)

0.6 n

0.8

1

1.2

3.6 Conclusions

53

reached the age of n-years and continue to exist on the market (described by formula BFO).

3.6

Conclusions

In this chapter, an important step toward a general theory of capital cost and capital structure of the company has been done. For this, perpetuity theory of Nobel Prize winners Modigliani and Miller, which was until recently (until 2008) the main and the basic theory of capital cost and capital structure of companies, has been extended to the case of companies of arbitrary age and of companies with an arbitrary age (lifetime), as well as for projects of arbitrary duration. We show that taking into account the finite age of the company or the finite lifetime of the company in the presence of corporate taxes leads to a change in the equity cost of the company, ke, as well as in its weighted average cost, WACC, and company capitalization, V. Thus, we have removed one of the most serious limitations of the theory of Modigliani–Miller, connected with the assumption of perpetuity of the companies. The effect of leverage on the cost of equity capital, ke, of the company of an arbitrary age or with arbitrary lifetime and its weighted average cost, WACC, is investigated. We give a rigorous proof of an important Brusov–Filatova–Orekhova theorem that in the absence of corporate tax, equity cost of companies, ke, as well as its weighted average cost, WACC, does not depend on the lifetime (age) of the company. We summarize the difference in results obtained within modern Brusov– Filatova–Orekhova theory and within classical Modigliani–Miller one in Table 3.5. The first four formulas from the right-hand column are sometimes used in practice, but there are several significant nuances. First, these formulas do not take account of the residual value of the company, and only take into account the operating flows and this must be borne in mind. Second, these formulas contain the weighted average cost of capital of the company, WACC. If it is estimated within the traditional approach or the theory of Modigliani–Miller, it gives a lower WACC value, than the real value, and, therefore, overestimates the capitalization of both financially dependent and financially independent companies. Therefore, in order to assess a company’s capitalization by the first two formulas, one needs to use Brusov–Filatova–Orekhova formulas for weighted average cost of capital, WACC, and equity cost, ke. To calculate the equity cost in BFO approximation (last line in Table 3.5), one first needs to use Brusov–Filatova–Orekhova formula for weighted average cost of capital, WACC (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b, 2015, 2018a, b, c, e, f, g, 2019, 2020; Brusov 2018a, b; Filatova et al. 2008, 2018).

ke ¼ k0 þLðk 0  k d Þð1  t Þ

Equity cost, ke

BFO-1: TSn ¼ DT[1  (1 + kd)n] BFO-2: n n n T  k d  wd 1  ð1 þ k d Þ ð1 þ k d Þ  ð1 þ WACCÞ TSn ¼   WACC kd WACC  kd V ¼ V0 + DT[1  (1 + kd)n] BFO-1: 1  ð1 þ WACCÞn WACC 1  ð1 þ k 0 Þn ¼ k 0 ½1  ωd T ð1  ð1 þ k d Þn Þ BFO-2: 1  ð1 þ WACCÞn 1  ð1 þ k 0 Þn Tk w þ d d  ¼ WACC k0 WACC

1  ð1 þ k d Þn ð1 þ k d Þn  ð1 þ WACCÞn   kd WACC  k d k e ¼ ð1 þ LÞ  WACC k d Lð1  T Þ

(TS)1 ¼ DT

V ¼ V0 + DT WACC ¼ k 0 ð1  wd t Þ

CF ½1  ð1 þ WACCÞn  V ¼ WACC

n V 0 ¼ CF k 0 ½1  ð1 þ k 0 Þ 

Brusov–Filatova–Orekhova (BFO) results

V ¼ CF/WACC

Modigliani–Miller (MM) results V0 ¼ CF/k0

Modigliani–Miller theorem with taxes Weighted average cost of capital, WACC

Capitalization of leverage (financially dependent) company Tax shield

Financial parameter Capitalization of financially independent company

Table 3.5 Comparison of results, obtained within Modigliani–Miller theory and within general Brusov–Filatova–Orekhova theory

54 3 Modern Theory of Capital Cost and Capital Structure:. . .

References

55

References Brusov P (2018a) Editorial: Introduction on special issue on the Banking System and financial markets of Russia and other countries: problems and prospects. J Rev Glob Econ 7:i–vi. SCOPUS Brusov P (2018b) Editorial. J Rev Glob Econ 7:i–v. SCOPUS Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 435:2–8 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11):815– 824 Brusov P, Filatova T, Orehova N et al (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova P, Orekhova N (2013a) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova T, Orehova N (2013b) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Global Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit—Modigliani–Miller theory. J Rev Global Econ 3:175–185 Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International, Switzerland, 373 p. Monograph, SCOPUS. https://www. springer.com/gp/book/9783319147314 Brusov P, Filatova T, Orehova N, Eskindarov M (2018a) Modern corporate finance, investments, taxation and ratings. Springer Nature Publishing, Switzerland, 571 p. Monograph Brusov PN, Filatova TV, Orekhova NP (2018b) Modern corporate finance and investments. Monograph. Knorus Publishing House, Moscow, 517 p Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018c) Rating: new approach. J Rev Glob Econ 7:37–62. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018d) A “golden age” of the companies: conditions of its existence. J Rev Glob Econ 7:88–103. SCOPUS. https://doi.org/10.6000/ 1929-7092.2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018e) Rating methodology: new look and new horizons. J Rev Glob Econ 7:63–87. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.06 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2018f) New meaningful effects in modern capital structure theory. J Rev Glob Econ 7:104–122. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.08 Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018g) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Glob Econ 7:360–376. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Glob Econ 8:437–448. SCOPUS. https://doi. org/10.6000/1929-7092.2019.08.37

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3 Modern Theory of Capital Cost and Capital Structure:. . .

Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Glob Econ 9: 257–268 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018) Ratings of the long-term projects: new approach. J Rev Glob Econ 7:645–661. https://doi.org/10.6000/1929-7092.2018.07.59 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Myers S (2001) Capital structure. J Econ Perspect 15(2):81–102

Chapter 4

Optimal Capital Structure of the Company: Its Absence in Modigliani–Miller Theory with Risky Debt Capital

4.1

Optimal Capital Structure of the Company

Choosing of optimal capital structure of the company, i.e., proportion of debt and equity, which minimizes weighted average cost of capital and maximizes the company capitalization, is one of the most important tasks of financial manager and the management of a company. The search for an optimal capital structure, like the search for a “golden fleece,” attracts the attention of economists and financiers for many tens of years. And it is clear why: one can, nothing making, but only by changing the proportion between the values of equity capital and debt one of the company, significantly enhance the company capitalization, in other words to fulfill the primary task, to reach critical goal of the business management. Spend a little less of your own, loan slightly more (or vice versa), and company capitalization reaches a maximum. In the classical Modigliani–Miller theory (Modigliani and Miller 1958), there is no optimal capital structure: an increase in the share of borrowed capital reduces the cost of raising capital, WACC, and increases the capitalization of company, V, up to infinite leverage level L values. We show here that in Modigliani–Miller theory (Modigliani and Miller 1958, 1963) modified by us by taking off the suggestion about riskless of debt capital, the optimal capital structure is still absent as well as in the famous trade-off theory. Note that the problem of capital structure is studied very intensively. There are theories, which consider the perfect market (Brusov and Filatova 2011; Brusov et al. 2011a,b,c, 2012a,b, 2013a,b, 2014a,b; Filatova et al. 2008; Brusova 2011; Modigliani and Miller 1958, 1963, 1966) and other ones, considering the imperfect market (Brennan and Schwartz 1978, 1984; Leland 1994; Kane et al. 1984; Dittmar and Thakor 2007; Bikhchandani et al. 1998; Post et al. 2002; Filbeck et al. 1996; Jenter 2005; Baker and Wurgler 2002; Graham and Harvey 2001; Hovakimian et al. 2001; Myers and Majluf 1984; Myers 1984; Fama and French 2004; Jensen and Meckling 1976). © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_4

57

58

4 Optimal Capital Structure of the Company: Its Absence in Modigliani–Miller. . .

Among former one’s agent cost theory (Jensen and Meckling 1976), stakeholders theory (Post et al. 2002), manager investment autonomy (Dittmar and Thakor 2007), information cascades (Bikhchandani et al. 1998), behavioral theories (Filbeck et al. 1996; Jenter 2005; Baker and Wurgler 2002; Graham and Harvey 2001), signaling theory (Myers and Majluf 1984), pecking order theory (Myers 1984, Hovakimian et al. 2001; Fama and French 2004). Historically the conceptions of the influence of capital structure on the well-being of shareholders have developed not monotonically. We consider the traditional (empirical) approach (Brusov and Filatova 2011; Brusov et al. 2011a,b,c, 2012a,b, 2013a,b, 2014a,b; Filatova et al. 2008), the Modigliani and Miller theory (Modigliani and Miller 1958, 1963, 1966), trade-off theory (Brennan and Schwartz 1978, 1984; Leland 1994; Kane et al. 1984), and modern Brusov–Filatova–Orekhova theory of capital cost and capital structure (Brusov and Filatova 2011; Brusov et al. 2011a,b,c, 2012a,b, 2013a,b, 2014a,b; Filatova et al. 2008). The traditional approach The traditional (empirical) approach told to businessmen, that weighted average cost of capital, WACC, and the associated company capitalization, V ¼ CF/WACC depend on the capital structure, the level of leverage. Debt cost always turns out to be lower, than equity cost, because, first, one has lower risk, because in the event of bankruptcy creditor claims are met prior to shareholders’ claims. As a result, an increase in the proportion of lower-cost debt capital in the overall capital structure up to the limit which does not cause violation of financial sustainability and growth in risk of bankruptcy leads to a lower weighted average cost of capital, WACC. The required by investors profitability (the equity cost) is growing; however, its growth has not led to compensation benefits from the use of more low-cost debt capital. Therefore, the traditional approach welcomes the increased leverage L ¼ D/ S, and the associated increase of company capitalization. The traditional (empirical) approach has existed up to the appearance of the first quantitative theory by Мodigliani and Мiller (1958). Modigliani–Miller theory Modigliani and Miller (ММ) in their first paper (without taxes) (Мodigliani and Мiller 1958) come to the conclusion that under assumptions, that there is no taxes, no transaction costs, no bankruptcy costs, perfect market exists with symmetry information, equivalence in borrowing costs for both companies and investors, etc. the choosing of proportion of debt and equity does not affect WACC and company value as well (Fig. 2.2). Most of Modigliani and Miller assumptions (Мodigliani and Мiller 1958), of course, are unrealistic. Some assumptions can be removed without changing the conclusions of the model. However, assuming no costs of bankruptcy and the absence of taxes (or the presence of only corporate taxes) are crucial—the change of these assumptions alters conclusions. The last two assumptions rule out the possibility of signaling and agency costs and, thus, also constitute a critical prerequisite.

4.1 Optimal Capital Structure of the Company

59

Modigliani–Miller theory with taxes (see Chap. 2) leads to the conclusion that in accordance to obtained by the formula WACC ¼ k0 ð1  wd T Þ

ð4:1Þ

weighted average cost of capital WАСС decreases continuously (see Fig. 2.5) (WACC decreases from k0 (for L ¼ 0) up to k0(1  T ) (for L ¼ 1, when the company is financed entirely with borrowed funds). So, there is no optimal capital structure within this theory. Below we modify the Modigliani–Miller theory with taxes by taking off the suggestion about riskless of debt capital, modeling the growth in risk of bankruptcy by increase of a credit rate, and show that optimal capital structure of the company is still absent. Trade-off theory Reduction in financial sustainability companies and increase of bankruptcy risk, which relate to the use of different forms of borrowing in the formation of financial capital structure of the company, is increased with the increase of debt. Modigliani–Miller theory did not take into account the bankruptcy risk and related costs. From its version with the tax on profit it follows that debt financing brings only some benefits associated with tax benefits (tax shield). Because company capitalization grows with leverage and there is no compensating increase in the debt cost, increasing the capitalization requires the use of debt financing only. This obvious contradiction with the real economy has created many theories, which had tried to find a balance between the advantages and disadvantages of using by the companies of debt financing. The advantage is a reduction of weighted average cost of capital, WACC, and the corresponding increase of capitalization of the companies, V, and the drawback—reduce with the increase of debt financing of financial sustainability of the companies and increased financial distress costs and risk of bankruptcy. One of these theories is trade-off theory (Brennan and Schwartz 1978; Leland 1994). There are two versions of this theory: static and dynamic. The former one is based on the fact that at the low leverage level the benefits of debt financing are manifested: WACC drops with leverage, a company capitalization is growing. Starting with a certain leverage level financial distress costs and risk of bankruptcy are growing, the WACC begins to growth and the value of the company begins to fall. The leverage level, at which the value of tax benefits is approximately equal to the cost of bankruptcy, determines the optimal (objective) capital structure. While the static trade-off theory is a single-period model (Brennan and Schwartz 1978; Leland 1994), in the dynamic trade-off theory (Brennan and Schwartz 1984) the financing decision depends on what the company anticipates in the next periods, which will be a capital structure. Brusov, Filatova, and Orekhova within created by them modern theory of capital structure and capital cost (BFO theory) (Brusov and Filatova 2011; Brusov et al. 2011a,b,c, 2012a,b, 2013a,b, 2014a,b; Filatova et al. 2008) have made the analyses of wide-known trade-off theory. It has been shown that the suggestion of risky debt

60

4 Optimal Capital Structure of the Company: Its Absence in Modigliani–Miller. . .

financing (and growing credit rate near the bankruptcy) in opposite to waiting result does not lead to the growing of weighted average cost of capital, WACC, which still decreases with leverage. This means the absence of minimum in the dependence of WACC on leverage as well as the absence of maximum in the dependence of company capitalization on leverage. Thus, it means that the optimal capital structure is absent in the famous trade-off theory. The explanation for this fact has been done. Under the condition of proved by us insolvency of well-known classical trade-off theory question of finding of new mechanisms of formation of the company optimal capital structure, different from one, suggested by the trade-off theory, becomes very important. One of the real such mechanisms has been suggested by us (Brusov et al. 2014a). Below we show that in Modigliani–Miller theory (Modigliani and Miller 1958, 1963) modified by us by taking off the suggestion about riskless of debt capital, the optimal capital structure is still absent.

4.2

Absence of the Optimal Capital Structure in Modified Modigliani–Miller Theory (MMM Theory)

Let us show that in Modigliani–Miller theory (Modigliani and Miller 1958, 1963) modified by us by taking off the suggestion about riskless of debt capital, the optimal capital structure is still absent. Consider the case of arbitrary dependence of debt cost on leverage f(L). Suppose, that debt cost kd is described by the following function  kd ¼

kd0 ¼ const;

atL  L0

k d0 þ f ðLÞ;

atL > L0

 ð4:2Þ

Here, f(L ) is arbitrary (growing or decreasing) function of leverage level L. We are interested in leverage levels L > L0, because at L < L0 the standard Modigliani– Miller theory works and weighted average cost of capital, WACC, is decreased with leverage WACC ¼ k 0 ð1  wd t Þ

ð4:3Þ

while an equity cost grows linearly with leverage k e ¼ k0 þ Lðk 0  kd Þð1  t Þ:

ð4:4Þ

Here, ke is an equity cost; k0 is an equity cost of financially independent company; kd is debt cost; t is tax on profit rate; WACC is a weighted average cost of capital. In this case for WACC one has

4.3 Conclusion

61

WACC ¼ k e we þ k d wd ð1  t Þ ¼ ke ¼

1 L þ kd ð1  t Þ 1þL 1þL

1 ½k þ k d Lð1  t Þ 1þL e

ð4:5Þ

Substituting Eqs. (4.2) and (4.4) into Eq. (4.5), one has finally for weighted average cost of capital, WACC, 1 ½k þ Lðk0  kd Þð1  t Þ þ Lkd ð1  t Þ 1þL 0 1 ¼ ½k þ k0 Lð1  t Þ 1þL 0 k ½1 þ Lð1  t Þ ¼ 0 ¼ k0 ½we þ wd ð1  t Þ 1þL ¼ k 0 ð1  wd t Þ

WACC ¼

ð4:6Þ

One can see that weighted average cost of capital, WACC, does not depend on f(L). Moreover, it is described by the same expression (Eq. 4.3), as in the case of riskless debt capital. Note that obtained result consistent with conclusions of Rubinstein (1973) and Stiglitz (1969) that cost of company within Modigliani–Miller theory is not changed upon the introduction of debt riskiness. In our approximation, as well as at Hsia (1981) debt cost is not already constant. For derivative from weighted average cost of capital, WACC, on leverage level one has ðWACCÞ0L ¼ k0

½ð1  t Þð1 þ LÞ  1  Lð1  t Þ t ¼ k0 < 0: ð 1 þ LÞ 2 ð1 þ LÞ2

ð4:7Þ

We have proved the following theorem: In modified Modigliani–Miller theory (allowing riskiness debt capital) under arbitrary change of debt cost with leverage (growing, as well as decrease) weighted average cost of capital, WACC always fall down with leverage. This means the absence of the company optimal capital structure.

4.3

Conclusion

In this chapter, we show that in modified Modigliani–Miller theory (allowing riskiness debt capital) under arbitrary change of debt cost with leverage (growing, as well as decrease) weighted average cost of capital, WACC, always fall down with leverage. This means the absence of the company optimal capital structure. In Brusov et al. (2013a), we have proved insolvency well-known classical trade-off theory in its original formulation with the help of the modern theory of capital structure and capital cost by Brusov–Filatova–Orekhova. Thus, the company

62

4 Optimal Capital Structure of the Company: Its Absence in Modigliani–Miller. . .

optimal capital structure is absent as in the famous trade-off theory and in modified Modigliani–Miller theory (allowing riskiness debt capital). The absence of the optimal capital structure in two these theories questioned the existence of an optimal capital structure of the company [but as authors have shown, the optimal capital structure for the investment still exists (Brusov et al. 2011b,c)]. In the search for the “golden fleece” one needs to switch to the study of other mechanisms for the formation of the capital structure of the company, different from ones considering in trade-off theory. Some possible approach to solve this problem, different from trade-off theory and modified Modigliani–Miller theory has been suggested by us in Brusov et al. (2014a).

References Baker M, Wurgler J (2002) Market timing and capital structure. J Financ 57(1):1–32 Bikhchandani S, Hirshleifer D, Welch I (1998) Learning from the behavior of others: conformity, fads, and informational cascades. J Econ Perspect 12(3):151–170 Brennan M, Schwartz E (1978) Corporate income taxes, valuation, and the problem of optimal capital structure. J Bus 51:103–114 Brennan M, Schwartz E (1984) Optimal financial policy and firm valuation. J Financ 39:593–607 Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance and credit 435:2–8 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11): 815–824 Brusov P, Filatova T, Orehova N, Brusova N (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Glob Econ 1:106–111 Brusov P, Filatova T, Orehova N (2013a) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Glob Econ 2:183–193 Brusov P, Filatova P, Orehova N (2013b) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Glob Econ 2:94–116 Brusov P, Filatova T, Orehova N (2014a) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit – Modigliani–Miller theory. J Rev Glob Econ 3:175–185 Brusov P, Filatova P, Orehova N (2014b) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Financ 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Prob Sol 34(76):36–42 Dittmar A, Thakor A (2007) Why do firms issue equity? J Financ 62(1):1–54 Fama E, French K (2004) Financing decisions: who issues stock? Working Paper Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filbeck G, Raymond F, Preece D (1996) Behavioral aspects of the intra-industry capital structure decision. J Financ Strateg Decis 9(2):55–66

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Graham J, Harvey C (2001) The theory and practice of corporate finance: evidence from the field. J Financ Econ 60:187–243 Hovakimian A, Opler T, Titman S (2001) The debt-equity choice. J Financ Quant Anal 36(1):1–24 Hsia С (1981) Coherence of the modern theories of finance. Financ Rev 16(1):27–42 Jensen M, Meckling W (1976) Theory of the firm: managerial behavior, agency costs, and ownership structure. J Financ Econ 3(4):305–360 Jenter D (2005) Market timing and managerial portfolio decisions. J Financ 60(4):1903–1949 Kane A, Marcus A, McDonald R (1984) How big is the tax advantage to debt? J Financ 39:841–853 Leland H (1994) Corporate debt value, bond covenants, and optimal capital structure. J Financ 49(4):1213–1252 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Myers S (1984) The capital structure pussle. J Financ 39(3):574–592 Myers S, Majluf N (1984) Corporate financing and investment decisions when firms have information that investors do not have. J Financ Econ 13:187–221 Post J, Preston L, Sachs S (2002) Redefining the corporation: stakeholder management and organizational wealth. Stanford University Press, Stanford Rubinstein M (1973) A mean–variance synthesis of corporate financial theory. J Financ 28:167– 181 Stiglitz J (1969) A re–examination of the Modigliani–Miller Theorem. Am Econ Rev 59(5): 784–793

Chapter 5

The Equity Cost in the Modigliani–Miller Theory

5.1

Introduction

The cost of equity is very important in finance because the economically justified cost of dividends is equal to the cost of equity. Thus, knowledge of the cost of equity capital helps the company’s management to formulate an adequate dividend policy for the company. But calculation of the cost of equity capital is one of the most difficult tasks of company management. This could be done correctly only within the modern capital structure theory—Brusov–Filatova–Orekhova theory (BFO theory) (Filatova et al. 2008; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b; Brusov and Filatova 2011; Brusova 2011) and within its perpetuity limit Modigliani–Miller theory (Modigliani and Miller 1958, 1963, 1966) (For one year company see Myers 2001). In this chapter, we consider the value of the cost of equity ke in the theory of Modigliani and Miller, its dependence on leverage level L and tax on profit rate T. We study the dependence of cost of equity on leverage level L at different tax on profit rates T and the dependence of cost of equity on tax on profit rates T at different leverage level L and show that, in this perpetuity limit the cost of equity ke is always growing with leverage (for any tax on profit rate T). Note, that within the modern Brusоv–Filаtоvа–Orekhоvа theory for companies of arbitrary (finite) age or with finite lifetime a qualitatively new effect takes place: decreasing of the cost of equity with the leverage. The effect takes place at tax on profit rate T, exceeding a cutoff value T*. In Chaps. 9 and 10 we will study the influence of frequency of payments of tax on profit rate T on equity cost ke and on its dependence on leverage level L and on tax on profit rate T. For weighted average cost of capital WACC in the Modigliani–Miller theory the following expression has been obtained (Мodigliani and Мiller 1958, 1963, 1966)

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_5

65

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5 The Equity Cost in the Modigliani–Miller Theory

WACC ¼ k0 ð1  wd T Þ:

ð5:1Þ

Dependence of WACC on financial leverage L ¼ D/S is described by the formula WACC ¼ k0 ð1  LT=ð1 þ LÞÞ

ð5:2Þ

In accordance with the definition of the weighted average cost of capital accounting the tax shield one has WACC ¼ k0 we þ k d wd ð1  T Þ:

ð5:3Þ

Equating (Eq. 5.1) to (Eq. 5.3), we get k0 ð1  wd T Þ ¼ k0 we þ k d wd ð1  T Þ,

ð5:4Þ

From where, for cost of equity, one has ke ¼ k 0 þ Lð1  T Þðk 0  kd Þ:

ð5:5Þ

Note that Eq. 5.5 is different from the corresponding formula without tax only by multiplier (1 – T ) in the term, indicating premium for risk. As the multiplier is less than unit, the appearance of corporate tax on profits leads to the fact that cost of equity increases with leverage slowly comparing to the case of taxes absence. Analysis of formulas (Eq. 5.1) and (Eq. 5.5) leads to the following conclusions. With the increasing of financial leverage: 1. Value of the company increases. 2. Weighted average cost of capital decreases from k0 (at L ¼ 0) up to k0(1  T ) (at L ¼ 1, when the company is funded solely by borrowing, or its own capital is negligible. 3. Cost of equity increases linearly from k0 (at L ¼ 0) up to 1 (at L ¼ 1). Let us analyze now the influence of taxes on cost of equity in Modigliani– Miller theory by study of the dependence of cost of equity on tax on profit rate (Fig. 5.1). For this we will analyze the formula ke ¼ k 0 þ Lð1  T Þðk 0  kd Þ:

ð5:6Þ

It is seen that dependence is linear: cost of equity decreases linearly with tax on profit rate. The module of negative tilt angle tangent tgγ ¼  L(k0  kd) increases with leverage, and besides all dependences at different leverage level Li, coming from different points ke ¼ k0 + Li(k0  kd) at T ¼ 0, at T ¼ 1 converge at the point k0 (Fig. 5.2).

5.1 Introduction

67

Fig. 5.1 Dependence of cost of equity, cost of debt and WACC on leverage without taxes (t ¼ 0) and with taxes (t 6¼ 0)

Fig. 5.2 Dependence of cost of equity on tax on profit rate T at different leverage level Li

This means that the difference in cost of equity at different leverage level Li decreases with tax on profit rate T, disappearing at T ¼ 1. Let us illustrate these general considerations by the example k0 ¼ 10 % ; kd ¼ 8% (Figs. 5.3, 5.4, and 5.5). From Fig. 5.2 it is seen that dependence is linear: cost of equity decreases linearly with tax on profit rate. The module of negative tilt angle tangent tgγ ¼  L(k0  kd) increases with leverage, and besides all dependences at different leverage level Li,

68 Fig. 5.3 Dependence of cost of equity on tax on profit rate T at different leverage level Li for the case k0 ¼ 10 % ; kd ¼ 8%. (1— L ¼ 0; 2—L ¼ 2; 3—L ¼ 4; 4—L ¼ 6; 5—L ¼ 8)

5 The Equity Cost in the Modigliani–Miller Theory Ke (T), at fix L

Ke 0.3000

5

0.2500

4

0.2000

3

0.1500

2 1

0.1000 0.0500

0

Fig. 5.4 Dependence of cost of equity on leverage L at different tax on profit rates T for the case k0 ¼ 10 % ; kd ¼ 8% (1— T ¼ 0; 2—T ¼ 0.1; 3— T ¼ 0.2; 4—T ¼ 0.3; 5— T ¼ 0.4; 6—T ¼ 0.5; 7— T ¼ 0.6; 8—T ¼ 0.7; 9— 10—T ¼ 0.9; 11—T ¼ 1)

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 T

Ke 0.3000

1

0.0000 1.1

Ke (L) at fix T 1 2 3 4 5 6 7 8 9 10 11

0.2500 0.2000 0.1500 0.1000 0.0500 0.0000 0.0

1.0

2.0

3.0

4.0

L

5.0

6.0

7.0

8.0

9.0

coming from different points ke ¼ k0 + Li(k0  kd) at T ¼ 0, at T ¼ 1 converge at the point k0 (Fig. 5.2). From Fig. 5.4 it is seen that cost of equity increases linearly from k0 (at L ¼ 0) up to 1 (at L ¼ 1), and besides tilt angle tangent decreases with tax on profit rate T, becoming zero at T ¼ 100%. By other words, with an increase of tax on profit rate T, dependence of cost of equity on leverage L becomes smaller, disappearing at T ¼ 100%, i.e. within perpetuity Modigliani–Miller theory there is no any anomaly—effect, announced in the title of paper is absent. In conclusion, here is a three-dimensional graph of dependence of cost of equity on leverage L and on tax on profit rate T for the case k0 ¼ 10 % ; kd ¼ 8%.

References Fig. 5.5 Dependence of cost of equity on leverage L and on tax on profit rate T for the case k0 ¼ 10 % ; kd ¼ 8%

69 Ke

Ke (L,T)

0.3000 0.2500 0.2000 0.1500 0.1000 0.0500 0.0000 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 T

0.9

0,0 1

2,0

6,0 4,0 L

8,0

References Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 435:2–5 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11): 815–824 Brusov P, Filatova T, Orehova N et al (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Glob Econ 1:106–111 Brusov P, Filatova P, Orekhova N (2013a) Absence of an optimal capital structure in the famous Tradeoff theory! J Rev Glob Econ 2:94–116 Brusov P, Filatova T, Orehova N (2013b) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Glob Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit – Modigliani – Miller theory. J Rev Glob Econ 3:175–185 Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Probl Solutions 34(76):36–42 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bullet FU 48:68–77

70

5 The Equity Cost in the Modigliani–Miller Theory

Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Myers S (2001) Capital Structure. J Econ Perspect 15(2):81–102

Chapter 6

The Role of Taxing and Leverage in Evaluation of Capital Cost and Capitalization of the Company

6.1

The Role of Taxes in Modigliani–Miller Theory

In Chap. 2 it has been noted that Modigliani and Miller in their paper in 1958 (Мodigliani and Мiller 1958) have come to conclusions which are fundamentally different from the conclusions of the traditional approach. They have shown that in the framework of assumptions made by them the ratio between equity and debt capital in the company does not affect either the cost of capital or company value. In the context of the studies of the impact tax on the cost of capital and the company’s capitalization we raised among the numerous assumptions Modigliani and Miller two of the most important: 1. Corporate taxes and taxes on personal income of investors are absent. 2. All financial flows are perpetuity ones. From the first of these assumptions, Modigliani and Miller subsequently refused themselves and have modified their theory to the case of the presence of corporate taxes and taxes on personal income of investors that have significantly altered the conclusions of their theory (Мodigliani and Мiller 1963, 1966). The failure of the second assumption has led to the creation of modern theory of capital cost and capital structure by Brusov–Filatova–Orekhova (Brusov et al. 2011a, b, c, d; 2012a, b; 2013a, b; 2014a, b; Brusov and Filatova 2011; Brusova 2011; Filatova et al. 2008). We analyze now the role of taxes in the Modigliani–Miller theory, studying the dependence of weighted average cost of capital WACC and the equity cost ke of tax on profit rate T. With this purpose we analyze the following formulas:

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_6

71

72

6 The Role of Taxing and Leverage in Evaluation of Capital Cost and. . .

Fig. 6.1 The dependence of weighted average cost of capital WACC on tax on profit rate T at different fixed leverage level L

1. For weighted average cost of capital WACC one has WACC ¼ k 0 ð1  wd T Þ,

ð6:1Þ

WACC ¼ k 0 ð1  LT=ð1 þ LÞÞ; 2. For the equity cost ke one has ke ¼ k 0 þ Lð1  T Þðk 0  kd Þ:

ð6:2Þ

Both dependences are linear: both costs of capital decrease linearly with the increase of tax on profit rate T. For dependence of weighted average cost of capital WACC on tax on profit rate T negative tangent of tilt angle in tgβ¼k0L/(1+L ) is growing in the module with the increase of the leverage level L, achieving maximum, equal k0 at an infinite leverage level L ¼ 1 (share of equity capital is insignificantly small compared with the fraction of debt funds) (Fig. 6.1). Let us give a few examples: 1. In accordance to expression tgβ ¼  k0L/(1 + L ) one gets, that at k0 ¼ 10% and L ¼ 1, i.e. D ¼ S increase of tax on profit rate T on 10% leads to decrease of weighted average cost of capital WACC on 0.5%. 2. This dependence of weighted average cost of capital WACC on tax on profit rate T will be even more significant at a higher leverage level L and higher value k0. For example, at k0 ¼ 20% and L ¼ 2, the increase in T on 10% leads to a decrease in WACC on 1.33%.

6.1 The Role of Taxes in Modigliani–Miller Theory

73

For dependence of the equity cost ke on tax on profit rate T from the analysis of formula ke ¼ k0 + L(1  T)(k0  kd) it is seen that negative tangent of tilt angle tgγ ¼  L(k0  kd) is also increased in the module with the increase of the leverage level, in which connection all dependences at the different leverage levels Li, based on the different points ke ¼ k0 + Li(k0  kd) when T ¼ 0, at T ¼ 1 converge at the point k0. 3. In accordance with the formula tgγ ¼  L(k0  kd) we get that, when k0  kd ¼ 6% and L ¼ 1, i.е., D ¼ S, the increase of tax on profit rate T on 10% leads to a reduction in the equity capital cost ke on 0.6%. 4. This dependence of the equity cost ke on tax on profit rate T will be even more significant at a higher leverage level L and higher value k0  kd. For example, at k0  kd ¼ 10% and L ¼ 2, the increase in T on 10% leads to a decrease in ke on 2%. It should be noted that with the rising of tax on profit rate T the difference in the equity cost ke at various levels leverage decreases, disappearing at T ¼ 1. This procedure recalls operational analysis, which examined dependence of financial results of the activities of the company on the costs and volumes of production and the implementation of the products, goods, services. The key elements of operational analysis of any enterprise are: operating lever; the threshold of cost-effectiveness; stock financial strength of enterprise. The operational arm is reflected in the fact that any change proceeds from the disposal always gives rise to a more severe change in earnings. In the present case, as the effects of tax operational lever can be taken the ratio of change of weighted average cost of capital WACC to the change of tax on profit rate T, and the ratio of change of equity capital cost ke to the change of tax on profit rate T, i.е., we can introduce for the first time two tax operating levers: – For weighted average cost of capital WACC: LWACC ¼ ΔWACC=ΔT; – For equity capital cost ke: Lke ¼ Δke =ΔT: For the earlier examples the power of the lever is: 1. 2. 3. 4.

LWACC ¼ 0.05. LWACC ¼ 0.133. Lke ¼ 0:06. Lke ¼ 0:2.

The higher value of the tax operational lever causes the greater change in capital cost of the company at fixed change of tax on profit rate T (see Fig. 6.2).

74

6 The Role of Taxing and Leverage in Evaluation of Capital Cost and. . .

Fig. 6.2 The dependence of equity cost on tax on profit rate T at different fixed leverage level L

Conclusions In this chapter, the role of tax shields, taxes, and leverage is investigated within the Modigliani–Miller theory. It is shown that equity cost of the company as well as weighted average cost of capital decrease with the growth of tax on profits rates. A detailed study of the dependence of weighted average cost of capital WACC and equity cost of the company ke on tax on profit rates at fixed leverage level (fixed debt capital fraction wd) as well as on leverage level (debt capital fraction wd) at fixed tax on profits rate has been done. The very useful concept “tax operating lever” has been introduced.

References Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 435:2–8 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11): 815–824 Brusov P, Filatova T, Orehova N et al (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Orehova N, Brusov PP, Brusova N (2011d) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21

References

75

Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova P, Orekhova N (2013a) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova T, Orehova N (2013b) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Global Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit – Modigliani – Miller theory. J Rev Global Econ 3:175–185 Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal: Probl Solut 34(76):36–42 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bulletin FU 48:68–77 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Мodigliani F, Мiller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391

Chapter 7

Inflation in Modigliani–Miller Theory

7.1

Accounting of Inflation in Modigliani–Miller Theory without Taxes

Introduction Created more than half a century ago by Nobel Prize winners Modigliani and Miller theory of capital cost and capital structure (Modigliani and Мiller 1958, 1963, 1966) did not account for a lot of factors of a real economy, such as taxing, bankruptcy, unperfected capital markets, inflation, and many others. But while taxing has been included into consideration by authors themselves and some other limitations have been taken off by their followers, direct incorporation of inflation to Modigliani–Miller theory was absent still now. The influence of inflation on valuation of capital cost of company and its capitalization is investigated within Modigliani–Miller theory (ММ) (Modigliani and Мiller 1958, 1963, 1966), which is now outdated but still widely used in the West. It is shown (Brusov et al. 2014b) that inflation not only increases the equity cost and the weighted average cost of capital but as well it changes their dependence on leverage. In particular, it increases the growing rate of equity cost with leverage. Capitalization of the company is decreased under the accounting of inflation. We start from the study of inflation within Modigliani–Miller theory without taxing (Мodigliani and Мiller 1958), than with taxing (Мodigliani and Мiller 1963). Note that any modification of Modigliani–Miller theory, as well as of any other one, requires going behind the frame of modified theory. Thus, in the current case, we should go behind the frame of perpetuity of the company (reminder to reader that Modigliani–Miller theory describes only perpetuity companies—companies with infinite lifetime), come to the companies of finite age (or of finite lifetime) (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, d, 2012a, b, 2013a, b, c, 2014a; Brusova 2011; Filatova et al. 2008), make necessary calculations and then use the perpetuity limit. As known, in profit approach capitalization of the company is equal to discounted sum of profits of the company. Suppose that profit is constant for all periods and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_7

77

78

7 Inflation in Modigliani–Miller Theory

equal to CF, one gets for capitalization of the financially independent company V0, existing n years at market V0 ¼

CF CF CF þ þ ... þ : 1 þ k0 ð1 þ k 0 Þ2 ð1 þ k 0 Þn

ð7:1Þ

Here, k0 capital cost of the financially independent company. Under inflation with rate α the capitalization of the financially independent company V 0 becomes equal to V 0 ¼

CF CF CF þ ... þ : þ ð1 þ k 0 Þð1 þ αÞ ½ð1 þ k 0 Þð1 þ αÞ2 ½ð1 þ k0 Þð1 þ αÞn

ð7:2Þ

Using the formula for sum of the terms of indefinitely diminishing geometrical progression with the first term a1 ¼

CF ð1 þ k0 Þð1 þ αÞ

ð7:3Þ

q¼

1 ð1 þ k0 Þð1 þ αÞ

ð7:4Þ

and denominator

one gets for capitalization of the financially independent company V 0 the following expression V 0 ¼ ¼

a1 CF h i ¼ 1  q ð1 þ k Þð1 þ αÞ 1  ðð1 þ k Þð1 þ αÞÞ1 0 0

CF CF ¼ : ð1 þ k 0 Þð1 þ α Þ  1 k 0 ð1 þ α Þ þ α V 0 ¼

CF : k 0 ð1 þ αÞ þ α

ð7:5Þ

It is seen that under accounting of inflation the capitalization of the company decreases. At discount rate k0 ¼ 10% and inflation rate α ¼ 3% the decrease is equal to 5.7%, and at discount rate k0 ¼ 15% and inflation rate α ¼ 7% the decrease is equal to 35%. One can see that influence of inflation on the company capitalization could be significant enough and negative. For leverage company, using debt capital one has without inflation

7.1 Accounting of Inflation in Modigliani–Miller Theory without Taxes

VL ¼

79

CF CF CF þ þ ... þ , 1 þ WACC ð1 þ WACCÞ2 ð1 þ WACCÞn

ð7:6Þ

and in perpetuity limit VL ¼

CF : WACC

ð7:7Þ

Under accounting of inflation the capitalization of the company is equal to V L ¼

CF CF þ þ ... ð1 þ WACCÞð1 þ αÞ ½ð1 þ WACCÞð1 þ αÞ2

CF þ : ½ð1 þ WACCÞð1 þ αÞn

ð7:8Þ

Summing the infinite set, we get for leverage company capitalization under accounting of inflation in Modigliani–Miller limit V L ¼ ¼

a1 CF h i ¼ 1  q ð1 þ WACCÞð1 þ αÞ 1  ðð1 þ WACCÞð1 þ αÞÞ1

CF CF ¼ ð1 þ WACCÞð1 þ αÞ  1 WACCð1 þ αÞ þ α V L ¼

CF : WACCð1 þ αÞ þ α

ð7:9Þ

It is seen that similar to the case of the financially independent company inflation decreases the company capitalization and the decrease could be significant. From (Eqs. 7.7 and 7.9), it follows that effective values of capital costs (equity and WACC) are equal to k0 ¼ k0 ð1 þ αÞ þ α 

WACC ¼ WACC  ð1 þ αÞ þ α

ð7:10Þ ð7:11Þ

Note that both capital costs increase under inflation. We can compare obtained results with the Fisher formula for inflation. i ¼

iα : 1þα

ð7:12Þ

Solving this equation with respect to nominal rate i, one gets an equation similar to (Eqs. 7.10 and 7.11)

80

7 Inflation in Modigliani–Miller Theory

Fig. 7.1 Dependence of the equity cost and the weighted average cost of capital on leverage in the Modigliani– Miller theory without taxing under accounting of inflation. It is seen that growing rate of equity cost increases with leverage. Axis y means capital costs— C.C

i ¼ i  ð1 þ αÞ þ α:

ð7:13Þ

Thus, effective capital costs in our case have the meaning of nominal ones, accounting inflation. From the Modigliani–Miller theorem, that the weighted average cost of capital WACC does not depend on leverage level (without taxing), formulating under accounting of inflation, it is easy to get expression for the equity cost: WACC ¼ k0 ¼ ke we þ k d wd :

ð7:14Þ

Finding from here ke , one gets:  D k0 k  ðS þ D Þ w D  k d d ¼ 0  k d ¼ k 0 þ k0  k d S S we we S      ¼ k0 þ k0  kd L

k e ¼

ð7:15Þ

Putting instead of k0 , k d their expressions, one gets finally   ke ¼ k0 þ k 0  kd L ¼ k0 ð1 þ αÞ þ α þLðk 0  kd Þð1 þ αÞ ¼ ð1 þ αÞ½k0 þ α þ Lðk 0  kd Þ k e ¼ k 0 ð1 þ αÞ þ α þ Lðk0  kd Þð1 þ αÞ:

ð7:16Þ

7.2 Accounting of Inflation in Modigliani–Miller Theory with Corporate Taxes

81

It is seen that inflation not only increases the equity cost but as well it changes its dependence on leverage. In particular, it increases the growing rate of equity cost with leverage by the multiplier (1 + α). The growing rate of equity cost with leverage, which is equal to (k0  kd) without inflation becomes equal to (k0  kd) (1 + α) under accounting of inflation (Fig. 7.1). Thus, we come to the conclusion that it is necessary to modify the second statement of the Modigliani–Miller theory (Мodigliani and Мiller 1958) concerning the equity cost of leverage company.

7.1.1

Second Original MM Statement

Equity cost of leverage company ke could be found as equity cost of financially independent company k0 of the same group of risk, plus premium for risk, which value is equal to production of difference (k0  kd) on leverage level L.

7.1.2

Second Modified MM-BFO Statement

Under existing of inflation with rate α equity cost of leverage company ke could be found as equity cost of financially independent company k0 of the same group of risk, multiplied by (1 + α), plus inflation rate α and plus premium for risk, which value is equal to production of difference (k0  kd) on leverage level L and on multiplier (1 + α).

7.2

Accounting of Inflation in Modigliani–Miller Theory with Corporate Taxes

Let us calculate first the tax shield for perpetuity company under accounting of inflation ðPVÞTS ¼ kd DT

1  X

1 þ k d

t

¼ DT

ð7:17Þ

t¼1

It is interesting to note that in spite of dependence of each term of set on effective credit rate. kd tax shield turns out to be independent of it and equal to “inflationless” value DТ and Modigliani–Miller theorem under accounting of inflation takes the form (Мodigliani and Мiller 1963)

82

7 Inflation in Modigliani–Miller Theory

V L ¼ V 0 þ DT:

ð7:18Þ

Substituting D ¼ wd V L , one gets V L ¼ CF=k0 þ wd V L T

ð7:19Þ

V L ð1  wd T Þ ¼ CF=k 0 :

ð7:20Þ

or

Because leverage company capitalization is equal to V L ¼ CF=WACC for the weighted average cost of capital one has WACC ¼ k 0 ð1  wd T Þ:

ð7:21Þ

From (Eq. 7.21) we get the dependence of WACC* on leverage level L ¼ D/S: WACC ¼ k 0 ð1  LT=ð1 þ LÞÞ: WACC ¼ ½k 0 ð1 þ αÞ þ α  ð1  wd T Þ

ð7:22Þ

On definition of the weighted average cost of capital with accounting of the tax shield one has WACC ¼ k0 we þ kd wd ð1  T Þ:

ð7:23Þ

Equating right-hand parts of (Eqs. 7.21 and 7.23), we get k0 ð1  wd T Þ ¼ k0 we þ k d wd ð1  T Þ,

ð7:24Þ

from where one obtains the following expression for equity cost: ð1  wd T Þ w 1 w D  kd d ð1  T Þ ¼ ke  k 0 d T  kd ð1  T Þ we we S we we   DþS D D   k 0 T  kd ð1  T Þ ¼ k0 þ Lð1  T Þ k0  k d ¼ k0 S S S      ke ¼ k0 þ Lð1  T Þ k0  k d ð7:25Þ ¼ ½k 0 ð1 þ αÞ þ α þ Lð1  T Þðk 0  kd Þð1 þ αÞ

ke ¼ k0

It is seen that similar to the case without taxes inflation not only increases the equity cost, but as well it changes its dependence on leverage (Fig. 7.2). In particular, it increases growing rate of equity cost with leverage by multiplier (1 + α). The growing rate of equity cost with leverage, which is equal to (k0  kd)(1  T) without inflation becomes equal to (k0  kd)(1 + α)(1  T ) under accounting of inflation.

7.2 Accounting of Inflation in Modigliani–Miller Theory with Corporate Taxes

83

Fig. 7.2 Dependence of the equity cost and the weighted average cost of capital on leverage in the Modigliani– Miller theory with taxing under accounting of inflation. It is seen that growing rate of equity cost increases with leverage. Axis y means capital costs— C.C

We can now reformulate the fourth statement of the Modigliani–Miller theory (Мodigliani and Мiller 1963) concerning the equity cost of leverage company for case of accounting of inflation.

7.2.1

Fourth Original MM Statement

Equity cost of leverage company ke paying tax on profit could be found as equity cost of financially independent company k0 of the same group of risk, plus premium for risk, which value is equal to production of difference (k0  kd) on leverage level L and on tax shield (1-T) and on multiplier (1 + α).

7.2.2

Fourth Modified MM-BFO Statement

Equity cost of leverage company ke paying tax on profit under existing of inflation with rate α could be found as equity cost of financially independent company k0 of the same group of risk, multiplied by (1 + α), plus inflation rate α and plus premium for risk, which value is equal to production of difference (k0  kd) on leverage level L, on tax shield (1T) and on multiplier (1 + α).

84

7.3

7 Inflation in Modigliani–Miller Theory

Irregular Inflation

Above we considered inflation rate as constant. Really, as a rule, the inflation rate is a variable. It is possible to generalize all above considerations for the case of non-homogeneous inflation, introducing effective inflation for a few periods. The effective inflation rate for a few periods t ¼ t1 + t2 + . . . + tn is equal to α ¼ ð1 þ α1 Þð1 þ α2 Þ  . . .  ð1 þ αn Þ  1,

ð7:26Þ

where α1, α2, . . ., αn are inflation rates for periods t1, t2, . . ., tn. The proof of the formula (7.26) will be done below in 7.4. In the case of non-homogeneous inflation it could be accounted in both theories: Modigliani–Miller and Brusov–Filatova–Orekhova theory (BFO theory) either through effective inflation rate or directly upon discounting of financial flow.

7.4

Inflation Rate for a Few Periods

Suppose that the inflation rate for the consistent time periods t1, t2, . . ., tn is equal to α1, α2, . . ., αn consequently. Let us find the inflation rate α for total time period t ¼ t1 + t2 + . . . + tn. Common sense dictates that inflation rate is an additive value, so that α, at least approximately, is equal to the sum of the inflation rates α1, α2, . . ., αn α  α1 þ α2 þ . . . þ αn :

ð7:27Þ

Below we will get an exact expression for inflation rate for the total period of time t and will see how it is different from an intuitive result (7.27). At the end of the first commitment period, the gained sum will be equal to the amount S1 ¼ S0(1 + i), and with the account of inflation S1α ¼ S0 ð1 þ iÞt1 =ð1 þ α1 Þ. At the end of the second commitment period, the gained sum will be equal to the amount S2 ¼ S0 ð1 þ iÞt1 þt2 , and with the account of inflation S2α ¼ S0 ð1 þ iÞt1 þt2 =ð1 þ α1 Þð1 þ α2 Þ . At the end of the n–th commitment period, the gained sum will be equal to the amount Sn ¼ S0 ð1 þ iÞt1 þt2 þ...þtn , and with the account of inflation Snα ¼ S0 ð1 þ iÞt1 þt2 þ...þtn =ð1 þ α1 Þð1 þ α2 Þ  . . .  ð1 þ αn Þ:

ð7:28Þ

On the other hand, at inflation rate α for the total period at t¼t1+t2+. . .+tn at the end of this period t gained sum will be equal to

References

85

Snα ¼ S0 ð1 þ iÞt =ð1 þ αÞ:

ð7:29Þ

Equating the right-hand part of (7.28) and (7.29), we get ð1 þ α1 Þð1 þ α2 Þ  . . .  ð1 þ αn Þ ¼ 1 þ α:

ð7:30Þ

α ¼ ð1 þ α1 Þð1 þ α2 Þ  . . .  ð1 þ αn Þ  1:

ð7:31Þ

From where

It is easy to get a strict proof of this formula by the method of mathematical induction. Note that inflation rate for the n–periods does not depend on both the length of constituting periods and on the period t. For equal inflation rates α1¼α2¼. . .¼αn (it is interesting to note, that herewith the time intervals t1,t2,. . .,tn can be arbitrary and do not equal each other) one has α ¼ ð1 þ α1 Þn  1:

7.5

ð7:32Þ

Conclusions

In this chapter, the influence of inflation on capital cost and capitalization of the company within Modigliani–Miller theory (Modigliani and Мiller 1958, 1963, 1966), which is now outdated but still widely used in the West, is investigated. All basic results of Modigliani–Miller theory were modified. It is shown that inflation not only increases the equity cost and the weighted average cost of capital but as well it changes their dependence on leverage. In particular, it increases the growing rate of equity cost with leverage. Capitalization of the company is decreased under the accounting of inflation.

References Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 435:2–8 Brusov P, Filatova T, Orehova N, Brusov PP, Brusova N (2011a) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N, Brusova A (2011b) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11): 815–824 Brusov P, Filatova T, Orehova N et al (2011c) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21

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Brusov P, Filatova T, Orehova N et al (2011d) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova T, Orehova N (2013a) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Global Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2013b) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova T, Orekhova N (2013c) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322037.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit – Modigliani – Miller theory. J Rev Global Econ 3:175–185 Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Probl Solut 34(76):36–42 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bulletin FU 48:68–77 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Мodigliani F, Мiller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391

Chapter 8

Modification of the Modigliani–Miller Theory for the Case of Advance Tax on Profit Payments

8.1

Introduction

The first serious study (and first quantitative study) of the influence of capital structure of the company on its indicators of activities was the work by Nobel Prize Winners Modigliani and Miller (Modigliani and Miller 1958, 1963, 1966). Their theory has a lot of limitations. One of the most important and serious assumptions of the Modigliani–Miller theory is that all financial flows as well as all companies are perpetuity. This limitation was lifted out by Brusov–Filatova– Orekhova in 2008 (Filatova et al. 2008), who have created BFO theory—modern theory of capital cost and capital structure for companies of arbitrary age (Brusov 2018a, b; Brusov et al. 2015, 2018a, b, c, d, e, f, g, 2019, 2020; Filatova et al. 2018). Despite the fact that the Modigliani–Miller theory is currently a particular case of the general theory of capital cost and capital structure—Brusov–Filatova–Orekhova (BFO) theory—it is still widely used in the West. In the current chapter we discuss one more limitation of Modigliani–Miller theory: a method of tax on profit payments. Modigliani–Miller theory accounts for these payments as annuity-immediate while in practice these payments are made in advance and thus should be accounting as annuity-due. We generalize the Modigliani–Miller theory for the case of advance tax on profit payments, which is widely used in practice and show that this leads to some important consequences, which change seriously all the main statements by Modigliani and Miller. These consequences are as follows: WACC starts depend on debt cost kd, WACC turns out to be lower than in the case of classical Modigliani–Miller theory and thus company capitalization becomes higher than in ordinary Modigliani–Miller theory. We show that equity dependence on leverage level L is still linear, but the tilt angle with respect to L-axis turns out to be smaller: this could lead to modification of the divident policy of the company.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_8

87

8 Modification of the Modigliani–Miller Theory for the Case of Advance Tax. . .

88

Correct account of a method of tax on profit payments demonstrates that the shortcomings of Modigliani–Miller theory are dipper than everybody suggested: the underestimation of WACC really turns out to be bigger, as well as overestimation of the capitalization of the company. This means that systematic risks arising from the use of modified Modigliani–Miller theory (which is more correct than ‘classical’ one) in practice is higher than it was suggested by the ‘classical’ version of this theory.

8.2

Modified Modigliani–Miller Theory in Case of Advance Tax Payments

8.2.1

Tax Shield in Case of Advance Tax Payments

Modigliani–Miller theory accounts for these payments as annuity-immediate while in practice these payments are making in advance and thus should be accounting as annuity-due. To calculate tax shield TS in case of advance tax payments one should use annuity-due (Fig. 8.1). TS ¼ k d Dt þ

k d Dt kd Dt k d Dt  þ ... ¼  þ ð1 þ kd Þ ð1 þ k d Þ2 1  ð1 þ k d Þ1

¼ Dt ð1 þ kd Þ

ð8:1Þ

This expression is different from the case of classical Modigliani–Miller theory (which used annuity–immediate) (Fig. 8.2). TS ¼

kd Dt k d Dt k Dt  d  ¼ Dt þ ... ¼ þ 2 ð1 þ k d Þ ð1 þ k d Þ ð1 þ k d Þ 1  ð1 þ kd Þ1

ð8:2Þ

Thus, in the former case tax shield, TS is bigger by multiplier (1 + kd). This is Fig. 8.1 Annuity-due

Fig. 8.2 Annuityimmediate

connected with the time value of money: money today is more expensive than money tomorrow due to the possibility of their alternative investment.

8.2 Modified Modigliani–Miller Theory in Case of Advance Tax Payments

8.2.2

89

Capitalization of the Company

Modigliani–Miller theorem for capitalization of the company V ¼ V 0 þ TS

ð8:3Þ

V ¼ V 0 þ Dt ð1 þ kd Þ:

ð8:4Þ

takes the following form

Thus, we arrive to the following statement, which modifies the original Modigliani and Miller. The value of financially dependent company making tax on profit payments in advance is equal to the value of the company of the same risk group used no leverage, increased by the value of tax shield arising from financial leverage, and equal to the product of rate of corporate income tax t, the value of debt D and multiplier (1 + kd). Substituting D ¼ wd  V one has
V 1  w d t ð1 þ k d Þ ¼ V 0
CF CF  1  w d t ð1 þ k d Þ ¼ WACC k0 And for WACC we have the following formula:
WACC ¼ k 0 1  wd t ð1 þ k d Þ :

ð8:5Þ

At L ! 1WACC ¼ k0 ð1  t ð1 þ kd ÞÞ:

ð8:6Þ

This expression is different from the similar one in classical Modigliani–Miller theory WACC ¼ k 0 1  wd t




At L ! 1WACC ¼ k 0 ð1  t Þ: From these expressions it is seen that WACC decreases with L, achieving lower value WACC ¼ k0(1  t(1 + kd)) at L ! 1 in considering case comparing with the classical Modigliani–Miller theory WACC ¼ k0(1  t). This means also, that company capitalization becomes higher than in ordinary Modigliani–Miller theory (Fig. 8.3).

8 Modification of the Modigliani–Miller Theory for the Case of Advance Tax. . .

90

Fig. 8.3 Dependence of company capitalization, V and WACC on leverage level in “classical” Modigliani–Miller theory (curve 10 and curve 1) and in “modified” Modigliani– Miller theory (curve 20 and curve 2)

8.2.3

Equity Cost

Let us find equity cost WACC ¼ ke we þ kd wd ð1  t Þ:

ð8:7Þ

Equating (8.5) and (8.7), we obtain k0 ð1  wd t ð1 þ kd ÞÞ ¼ ke we þ kd wd ð1  t Þ,

ð8:8Þ

whence we get the following expression for the equity cost: ð1  w d t ð1 þ k d ÞÞ w  k d d ð1  t Þ we we 1 wd D ¼ k0  k 0 ð1 þ k d Þ  t  kd ð1  t Þ we S we DþS D D ¼ k0  k0 ð1 þ k d Þ  t  kd ð1  t Þ S S S ¼ k 0 þ L½ð1  t Þðk0  kd Þ  k0 kd t : ke ¼ k0

ð8:9Þ

Finally, we have for the equity cost k e ¼ k0 þ L½ð1  t Þðk0  k d Þ  k0 k d t :

ð8:10Þ

Thus, we arrive to the following statement, which modifies the original Modigliani and Miller. Equity cost of leverage company ke making tax on profit payments in advance could be found as equity cost of financially independent company k0 of the same group of risk, plus premium for risk, the value which is equal to production of leverage level L on production of difference (k0  kd) and tax shield (1 – t), decreasing by the value k0kdt (Fig. 8.4).

8.3 The Dependence of the Weighted Average Cost of Capital, WACC, on Leverage. . . Fig. 8.4 Dependence of equity cost of the company, ke, on leverage level in “classical”. Modigliani– Miller theory (curve 2), in “modified” Modigliani– Miller theory (curve 3) and for financially independent company (t ¼ 0) (curve 1)

91 1

ke

2

3

k0

L

This means that equity cost dependence on leverage level L is still linear, but the tilt angle with respect to L-axis turns out to be smaller tgα ¼ (k0  kd)  (1  t)  k0kdt. This could lead to modification of the dividend policy of the company, because the equity cost represents itself economically sound value of dividends. Thus, company could decrease the value of dividends.

8.3

The Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level in “Classical” Modigliani–Miller Theory (MM Theory) and in Modified Modigliani–Miller Theory (MMM Theory)

Let us compare the dependence of the weighted average cost of capital, WACC, on leverage level in “classical” Modigliani–Miller theory (MM theory) and in modified Modigliani–Miller theory (MMM theory). Study of such dependence is very important, because, the weighted average cost of capital, WACC, plays the role of discount rate in operating financial flows discounting as well as of financial flows in rating methodologies. WACC’s value determines as well the capitalization of the company V ¼ CF/WACC. We use Microsoft Excel for calculations. From Tables 8.1, 8.2, 8.3 and 8.4 and from Fig. 8.5 it is seen that WACC in MMM theory turns out to be lower than in the case of classical Modigliani–Miller theory and thus company capitalization becomes higher than in ordinary Modigliani–Miller theory. It is seen that WACC decreases with debt cost kd. Correct account of a method of tax on profit payments demonstrates that the shortcomings of Modigliani–Miller theory are dipper, than everybody suggested: the underestimation of WACC really turns out to be bigger, as well as overestimation of

92

8 Modification of the Modigliani–Miller Theory for the Case of Advance Tax. . .

Table 8.1 Dependence of WACC on leverage level in “classical” Modigliani–Miller theory

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

Table 8.2 Dependence of WACC on leverage level in modified Modigliani–Miller theory (MMM theory) at kd ¼ 0.18

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

WACC 0.2000 0.1764 0.1685 0.1646 0.1622 0.1607 0.1595 0.1587 0.1580 0.1575 0.1571

Table 8.3 Dependence of WACC on leverage level in modified Modigliani–Miller theory (MMM theory) at kd ¼ 0.14

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

WACC 0.2000 0.1772 0.1696 0.1658 0.1635 0.1620 0.1609 0.1601 0.1595 0.1590 0.1585

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

WACC 0.2000 0.1800 0.1733 0.1700 0.1680 0.1667 0.1657 0.1650 0.1644 0.1640 0.1636

the capitalization of the company. This means that systematic risks arising from the use of modified Modigliani–Miller theory (MMM theory) (which is more correct than “classical’ one) in practice are higher than it was suggested by the “classical” version of this theory.

8.4 Conclusions

93

Table 8.4 Dependence of WACC on leverage level in modified Modigliani–Miller theory (MMM theory) at kd ¼ 0.1

L 0 1 2 3 4 5 6 7 8 9 10

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

WACC 0.2000 0.1780 0.1707 0.1670 0.1648 0.1633 0.1623 0.1615 0.1609 0.1604 0.1600

WACC(L) 0.2100 0.2000

WACC

0.1900 WACC (0) WACC (0') kd=0,18 WACC (0') kd=0,14 WACC (0') kd=0,1

0.1800 0.1700 0.1600 0.1500

0

1

2

3

4

5

6

7

8

9 10

Fig. 8.5 Dependence of WACC on leverage level L: – in “classical” Modigliani–Miller theory (curve WACC(0)) and – in modified Modigliani–Miller theory (MMM theory) at different values of debt cost: kd ¼ 0.18; kd ¼ 0.14; kd ¼ 0.1 (curves WACC(00 ))

8.4

Conclusions

Despite the fact that the Modigliani–Miller theory is currently a particular case of the general theory of capital cost and capital structure—Brusov–Filatova–Orekhova (BFO) theory—it is still widely used in the West. We generalize the Modigliani– Miller theory for the case of advance tax on profit payments, which is widely used in practice, and show that this leads to some important consequences, which change seriously all the main statements by Modigliani and Miller. These consequences are as follows: WACC starts depend on debt cost kd, WACC turns out to be lower than in case of classical Modigliani–Miller theory and thus company capitalization becomes higher than in ordinary Modigliani–Miller theory.

94

8 Modification of the Modigliani–Miller Theory for the Case of Advance Tax. . .

We study as well the equity cost dependence on leverage level L and show that it is still linear, but the tilt angle with respect to L-axis turns out to be smaller in MMM theory tgα ¼ (k0  kd)  (1  t)  k0kdt, than in classical Modigliani–Miller theory tgα ¼ (k0  kd)  (1  t). This could lead to modification of the dividend policy of the company, because the equity cost represents itself economically sound value of dividends. Thus, company could decrease the value of dividends, which they should pay to shareholders. Correct account of a method of tax on profit payments demonstrates that the shortcomings of Modigliani–Miller theory are dipper, than everybody suggested: the underestimation of WACC really turns out to be bigger, as well as overestimation of the capitalization of the company. This means that systematic risks arising from the use of modified Modigliani–Miller theory (which is more correct than “classical’ one) in practice is higher than it was suggested by the “classical” version of this theory.

References Brusov P (2018a) Editorial: introduction on special issue on the banking system and financial markets of Russia and other countries: problems and prospects. J Rev Global Econ 7:i–vi. SCOPUS Brusov P (2018b) Editorial. J Rev Global Econ 7:i–v. SCOPUS Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International Publishing, Cham, p 373. Monograph, SCOPUS. https://www. springer.com/gp/book/9783319147314 Brusov P, Filatova T, Orehova N, Eskindarov M (2018a) Modern corporate finance, investments, taxation and ratings. Springer Nature Publishing, Cham, p 571. monograph Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018b) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Global Econ 7:360–376. https://doi.org/10.6000/ 1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018c) Rating: new approach. J Rev Global Econ 7:37–62. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018d) A “golden age” of the companies: conditions of its existence. J Rev Global Econ 7:88–103. https://doi.org/10.6000/1929-7092. 2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018e) Rating methodology: new look and new horizons. J Rev Global Econ 7:63–87. https://doi.org/10.6000/1929-7092.2018.07.06 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2018f) New meaningful effects in modern capital structure theory. J Rev Global Econ 7:104–122. https://doi.org/10.6000/ 1929-7092.2018.07.08 Brusov PN, Filatova TV, Orekhova NP (2018g) Modern corporate finance and investments, monograph. Knorus Publishing House, Moscow, p 517 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Global Econ 8:437–448. https://doi.org/10. 6000/1929-7092.2019.08.37 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Global Econ 9:257–268

References

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Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018) Ratings of the long–term projects: new approach. J Rev Global Econ 7:645–661. https://doi.org/10.6000/1929-7092.2018.07.59 Modigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Modigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Мiller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391

Chapter 9

The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax on Profit

9.1 9.1.1

Introduction Literature Review

In their first paper, Modigliani and Miller (1958) consider the problem of capital cost and capital structure for perpetuity companies without any taxes. In the second article, Modigliani and Miller (1963) considered taxation of corporate profits, but did not take into account the presence in the economy of individual taxes of investors. Merton Miller (1977) has introduced the model, demonstrating impact of leverage on the company value with account of the corporate and individual taxes. To describe his model, we will enter the following legends: TC—tax on corporate profits rate, TS—the tax rate on income of an individual investor from his ownership by stock of corporation, TD—tax rate on interest income from the provision of investor–individuals of credits to other investors and companies. Income from shares partly comes in the form of a dividend and, in part, as capital profits, so that TS is a weighted average value of effective rates of tax on dividends and capital profits on shares, while the income from the provision of loans usually comes in the form of the interests. The last are usually taxed at a higher rate. In the light of the individual taxes, and with the same assumptions that have been made for Modigliani–Miller models previously, the financially independent company value can be determined as follows: VU ¼

EBITð1  T C Þð1  T S Þ : k0

ð9:1Þ

A term (1  TS) allows to take into account the individual taxes in the formula. In this way, numerator indicates which part of the operating company’s profit remains in the possession of the investors after the company will pay taxes on their profits, and its shareholders then will pay individual taxes on income from stock ownership. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_9

97

98

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Since individual taxes reduce profits, remaining at the disposal of investors, the last, at other things being equal circumstances, also reduce and an overall assessment of the financially independent company value. Robert Hаmаdа (1969) unites Capital Asset Pricing Model (CAPM) with Modigliani–Miller model taxation. As a result, he derived the following formula for calculation of the equity cost of financially dependent company, including both financial and business risk of company: k e ¼ kF þ ðk M  kF ÞbU þ ðkM  kF ÞbU

D ð1  T Þ, S

ð9:2Þ

where bU is the β–coefficient of the company of the same group of business risk that the company under consideration, but with zero financial leverage. The formula (9.2) represents the desired profitability of equity capital ke as a sum of three components: risk-free profitability kF, compensating to shareholders a temporary value of their money, premium for business risk (kM  kF)bU, and premium for financial risk ðkM  kF ÞbU DS ð1  T Þ. If the company does not have borrowing (D ¼ 0), the financial risk factor will be equal to zero (the third term is drawn to zero), and its owners will only receive the premium for business risk. More general formula for the WACC, then famous Modigliani–Miller (MM) one has been derived and discussed by a few authors (Farber et al. 2006; Fernandez 2007; Harris, R., and J. Pringle, Miles, J.A., and J.R. Ezzel, Peter MDeMarzo et al.). It takes the following form (Eq. (18) in FGS, 2006) WACC ¼ k0 ð1  wd T Þ  kd twd þ kTS twd

ð9:3Þ

where k0, kd, and kTS are the expected returns respectively on the unlevered firm, the debt and the tax shield, V is the value of the levered firm, VTS is the value of the tax shield, D is the value of the debt and TC is the corporate tax rate. This formula is derived from the balance sheet identity and the definition of the weighted average cost of capital (for a similar presentation, see Berk and DeMarzo 2007). It should therefore be verified at any point in time, whatever returns are annually or continuously compounded. Although Eq. (9.3) is fully general, its practical applicability requires additional conditions. Indeed, when the WACC is constant over time, the value of a levered firm can be computed by discounting with the WACC the unlevered free cash flows. Therefore, we pay a special attention to the special cases that make the WACC constant. The resulting particular formulas can also be found in textbooks (Brealey et al. 2006; Ross et al. 2005). First, Modigliani and Miller (1963) assume that the level of debt D is constant. Then, as the expected after-tax cash flow of the unlevered firm is fixed, V0 is also constant. By assumption, kTS ¼ kD and the value of the tax shield is TS ¼ tD. Therefore, the value of the firm V is a constant, and the general WACC formula (9.3) simplifies to a constant WACC:

9.2 Focus and Objective of the Chapter

WACC ¼ k0 ð1  wd T Þ

99

ð9:4Þ

But our opinion is that “classical” Modigliani–Miller (MM) theory, which coming from the suggestion that the expected returns on the debt kd and the tax shield kTS are equals (because both of them have debt nature), is much more reasonable and in our paper we modify namely “classical” Modigliani–Miller (MM) theory, which is still widely used in practice. One of the most serious limitations of the Modigliani–Miller theory is the suggestion about the perpetuity of the companies. In 2008, Brusov–Filatova– Orekhova (Filatova et al. 2008) have lifted up this limitation and have shown that the accounting of the finite age (or finite lifetime) of the company leads to significant changes of all Modigliani–Miller results (Modigliani and Miller 1958, 1963, 1966): capitalization of the company is changed, as well as the equity cost, ke, and the weighted average cost of capital, WACC, in the presence of corporative taxes. Besides, a number of qualitatively new effects in corporate finance, obtained in Brusov–Filatova–Orekhova theory (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b, 2018a, b, c, d, 2019, 2020) are absent in Modigliani–Miller theory (Modigliani and Miller 1958, 1963, 1966). Some recent modifications of Modigliani–Miller theory and Brusov–Filatova–Orekhova theory could be found in (Brusov and Filatova 2021a, b; Brusov et al. 2021; Filatova et al. 2022). The famous Brusov–Filatova–Orekhova formula for weighted average cost of capital, WACC, has the following form 1  ð1 þ WACCÞn 1  ð1 þ k0 Þn : ¼ WACC k0 ½1  ωd T ð1  ð1 þ k d Þn Þ

ð9:5Þ

D Here, S–—the value of own (equity) capital of the company, wd ¼ DþS —the share S of debt capital; k e , we ¼ DþS–—the cost and the share of the equity of the company, L ¼ D/S—financial leverage. A lot of meaningful effects have been discovered within BFO theory: these effects are absent within MM theory. BFO theory has destroyed some main existing principles of financial management: among them trade-off theory, which was considered as keystone of formation of optimal capital structure of the company during many decades and bankruptcy of which has been proven by Brusov–Filatova– Orekhova (Brusov et al. 2013a).

9.2

Focus and Objective of the Chapter

The tax shield plays a crucial role in both main capital structure theories: Brusov– Filatova–Orekhova (BFO theory) and its perpetuity limit Modigliani–Miller theory. How it is formed influences the results of both theories. Both of them describe the

100

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

case of annual payments of tax of profit, but in practice, these payments are made more frequent: semiannually, quarterly, monthly. While a return is not required more than once a year, businesses may be responsible for filing estimated taxes based on profits earned. This requirement is dependent on showing a profit. For example, sole proprietors must file estimated taxes on profits quarterly, on the 15th day of April, June, September and January. In Russia, tax on profit payments could be made annually, quarterly, monthly. In the current chapter, we show that the account of frequency of payment of tax on profit within the Modigliani–Miller theory turns out to be very important and leads to significant consequences. To study this problem, we modify the Modigliani–Miller theory for the case of arbitrary frequency of payment of tax on profit and show that: – All Modigliani–Miller theorems, statements and all formulas change. – All main financial indicators, such as the weighted average cost of capital, WACC, company value, V and equity cost, ke depend on the frequency of tax on profit payments. This allows to company manage WACC, V, ke, etc. by choosing the number of payments of tax of profit p per year. – Increase the number of payments of tax of profit per year leads to decrease of the cost of attracting capital and increase of the company value. – Obtained results allow to company choose the number of payments of tax of profit per year, as many, as it is profitable to it (of course, within actual tax legislation). – In case of income tax payments more than once per year (at p 6¼ 1), as it takes place in practice, the weighted average cost of capital, WACC, company value, V and equity cost, ke start depend on kd, while in ordinary (classical) Modigliani– Miller theory all these values DO NOT depend on kd. – The tilt angle of the curve of equity cost, ke (L ), decreases with the number of payments of tax of profit p, this modifies the dividend policy of the company, because the economically justified value of dividends is equal to equity cost.

9.3

9.3.1

Modification of the Modigliani–Miller Theory for the Case of Arbitrary Frequency of Payments of Tax on Profit Tax Shield

Let us calculate first the tax shield in the Modigliani–Miller theory for the case of p payments of tax on profit per year TS ¼

kd Dt kd Dt k d Dt þ þ þ ... 1 2 3 pð1 þ kd Þ =p pð1 þ kd Þ =p pð1 þ k d Þ =p

TS represents a geometric progression with denominator q ¼

1 1 . ð1þk d Þ =p

ð9:6Þ

9.3 Modification of the Modigliani–Miller Theory for the Case of Arbitrary. . .

101

Summing the progression, one gets TS ¼

k Dt k d Dt :
d ¼
1 1=p pð 1 þ k d Þ 1  ð 1 þ k d Þ p ð1 þ k d Þ =p  1 1=p

ð9:7Þ

Note, that in the case of classical Modigliani–Miller theory TS ¼ Dt:

ð9:8Þ

It is easy to get this result from (9.7), putting the frequency of payments of tax on profit p ¼ 1.

9.3.2

The Weighted Average Cost of Capital, WACC

Let us now calculate the weighted average cost of capital, WACC, in the Modigliani–Miller theory for the case of p payments of tax on profit per year. From the Modigliani–Miller theorem we have for the company value V V ¼ V 0 þ TS:

ð9:9Þ

Putting expression (9.7) for TS, we get kd Dt : V ¼ V0 þ
1 p ð1 þ kd Þ =p  1

ð9:10Þ

Substituting D ¼ wdV, one gets 0

1 k d wd t

A ¼ V 0 : V @1 
1 p ð1 þ kd Þ =p  1

ð9:11Þ

Accounting, that V¼ we have

CF CF , ;V ¼ WACC 0 k0

ð9:12Þ

102

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

0

1 kd wd t

CF CF A ¼  @1 
: 1=p WACC k0 p ð1 þ k d Þ  1

ð9:13Þ

From here, we arrive to the expression for the weighted average cost of capital, WACC, for the case of p payments of tax on profit per year. 0

1 k d wd t

 A: WACC ¼ k 0  @1 
1 p ð1 þ kd Þ =p  1

ð9:14Þ

Note, that in case of classical Modigliani–Miller theory WACC ¼ k0  ð1  wd t Þ:

ð9:15Þ

It is easy to get this result from (9.14), putting the frequency of payments of tax on profit per year p ¼ 1.

9.3.3

Company Value, V

Let us now calculate the company value, V, in the Modigliani–Miller theory for the case of p payments of tax on profit per year. Putting WACC value into the first formula of (9.12) one gets for company value, V V¼

CF ¼ WACC

CF  : k0  1   kd wd1t=p 

ð9:16Þ

p ð1þk d Þ 1

In case of “classical” Modigliani–Miller theory one has V¼

CF CF ¼ : WACC k0  ð1  wd t Þ

ð9:17Þ

It is easy to get this result from (9.16), putting the frequency of payments of tax on profit per year p ¼ 1.

9.3 Modification of the Modigliani–Miller Theory for the Case of Arbitrary. . .

9.3.4

103

Equity Cost, ke

Let us now calculate the equity cost, ke, in the Modigliani–Miller theory for the case of p payments of tax on profit per year. General expression for the weighted average cost of capital, WACC, upon definition, has the following form WACC ¼ ke we þ kd wd ð1  t Þ:

ð9:18Þ

We get from (9.18) the following formula for the equity cost, ke, ke ¼

WACC k d wd ð1  t Þ  : we we

ð9:19Þ

Accounting from (9.14) the expression for the weighted average cost of capital, WACC, 0

1 k d wd t

 A, WACC ¼ k0  @1 
1 p ð1 þ k d Þ =p  1

ð9:20Þ

one gets 0 1 k w ð1  t Þ k0 @ kd wd t A  d d  1
ke ¼ 1=p we we p ð1 þ k Þ  1 d

0

1

kd Lt

A  k d Lð1  t Þ ¼ k 0  @1 þ L 
1 p ð1 þ kd Þ =p  1 0 1 k k t 0 d A: ¼ k 0 þ L @k 0  k d ð 1  t Þ 
1 p ð1 þ kd Þ =p  1

ð9:21Þ

Thus, we have the following formula for the equity cost, ke, in case of arbitrary frequency of payment of tax on profit 0

1 k0 kd t

A k e ¼ k 0 þ L@ k 0  k d ð 1  t Þ 
1 p ð1 þ kd Þ =p  1

ð9:22Þ

From this formula, it is easy to get “classical” Modigliani–Miller expression for the equity cost, ke, in case of annual payment of tax on profit, putting р ¼ 1

104

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

k e ¼ k 0 þ Lð k 0  k d Þ ð 1  t Þ

ð9:23Þ

Thus, from 9.3.1 to 9.3.4 we come to conclusion, that in case of income tax payments more than once per year (at p 6¼ 1), as it takes place in practice, the weighted average cost of capital, WACC, company value, V and equity cost, ke start depend on kd, while in ordinary (classical) Modigliani–Miller theory all these values DO NOT depend on kd. In subsequent paragraphs (9.4–9.6), we will study numerically with the use of Microsoft Excel the dependence of the main financial indicators of the company (the weighted average cost of capital, WACC, the company value, V, the cost of equity, kd, on leverage level L for cases of practical interest with respect to equity and debt costs and frequency of payment of tax on profit p.

9.4 9.4.1

Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L at Different p and Fixed kd

Let us start from the study numerically with the use of Microsoft Excel of the dependence of the weighted average cost of capital, WACC, on leverage level L at different p and fixed kd, using obtained by us formula (9.14) 0

1

k d wd t  A: WACC ¼ k 0  @1 
1 p ð1 þ kd Þ =p  1

Table 9.1 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 1 1 1 1 1 1 1 1 1 1 1

WACC 0.200000 0.180000 0.173333 0.170000 0.168000 0.166667 0.165714 0.165000 0.164444 0.164000 0.163636

9.4 Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L

105

Table 9.2 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 2

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 2 2 2 2 2 2 2 2 2 2 2

WACC 0.200000 0.179230 0.172306 0.168845 0.166767 0.165383 0.164394 0.163652 0.163075 0.162613 0.162236

Table 9.3 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 4 4 4 4 4 4 4 4 4 4 4

WACC 0.200000 0.178837 0.171783 0.168256 0.166139 0.164728 0.163721 0.162965 0.162377 0.161907 0.161522

Table 9.4 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

WACC 0.200000 0.178573 0.171430 0.167859 0.165716 0.164288 0.163267 0.162502 0.161907 0.161431 0.161041

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

From Tables 9.1, 9.2, 9.3 and 9.4 and Figs. 9.1 and 9.2 it is seen that the weighted average cost of capital, WACC, decreases with leverage level L at all values of frequency of payments of tax on profit p. With increase of p WACC decreases. As we will see below, this will lead to increase of company value, V, with p.

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

106

WACC (L) at different p 0.198000

0.193000

0.188000

0.183000

0.178000

0.173000

0.168000

0.163000

0.158000 0

1

2

3 P=1

4 p=2

5

6 p=4

7

8

9

10

p=12

Fig. 9.1 Dependence of the weighted average cost of capital, WACC, on leverage level L at different p and fixed kd ¼ 0.16

To demonstrate that above conclusion takes place for all values of equity cost (at zero leverage level L ) k0, we consider one more example: k0 ¼ 0.22 and different p ¼ 2; 4; 12 (Fig. 9.3 and Table 9.5).

9.5

Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L at Different kd and Fixed p

In this paragraph, we investigate the dependence of the weighted average cost of capital, WACC, on leverage level L at different values of debt cost kd and fixed p (Figs. 9.4 and 9.5). From Tables 9.6, 9.7, 9.8, 9.9, 9.10, 9.11, 9.12 and 9.13 and Figs. 9.6 and 9.7 it is seen that in case of income tax payments more than once per year (at p 6¼ 1), as it takes place in practice, the weighted average cost of capital, WACC, (and as we will see below, company value, V and equity cost, ke start depend on debt cost, kd, while in ordinary (classical) Modigliani–Miller theory all these values DO NOT depend on kd). The weighted average cost of capital, WACC, decreases with leverage level L at all values of debt cost kd and fixed frequency of payments of tax on profit p. With

9.5 Dependence of the Weighted Average Cost of Capital, WACC, on Leverage. . .

107

WACC (L) 0.168000 0.167000 0.166000 0.165000 0.164000 0.163000

0.162000 0.161000 0.160000 0.159000 0.158000 5

6

7 P=1

8

p=2

p=4

9

10

p=12

Fig. 9.2 Dependence of the weighted average cost of capital, WACC, on leverage level L at different p and fixed kd ¼ 0.16 (larger scale)

WACC(L) at different p 0.2250

0.2150

0.2050

p=2 p=4

0.1950

p = 12

0.1850

0.1750 0

1

2

3

4

5

6

7

8

9

10

Fig. 9.3 Dependence of the weighted average cost of capital, WACC, on leverage level L at different p and fixed k0 ¼ 0.22

108

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.5 Dependence of WACC on leverage level L at k0 ¼ 0.22 and different p ¼ 2; 4; 12

WACC p¼2 0.2200 0.1970 0.1893 0.1854 0.1831 0.1816 0.1805 0.1797 0.1790 0.1785 0.1781

L 0 1 2 3 4 5 6 7 8 9 10

p¼4 0.2200 0.1964 0.1886 0.1846 0.1823 0.1807 0.1796 0.1787 0.1781 0.1775 0.1771

p ¼ 12 0.2200 0.1960 0.1881 0.1841 0.1817 0.1801 0.1789 0.1781 0.1774 0.1769 0.1765

WACC(L); p=12 0.20000

0.19500

0.19000

WACC

0.18500 Kd=0,18

0.18000

Kd=0,16 Kd=0,14

0.17500

Kd=0,12 0.17000

0.16500

0.16000 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 9.4 Dependence of the weighted average cost of capital, WACC, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 12

increase of kd WACC decreases. This is connected to the fact that the increase of kd increases the value of tax shield in case of income tax payments more than once per year (at p 6¼ 1). Note, that in “classical” Modigliani–Miller theory with one income tax payment per year, all financial indicators do not depend on debt cost, kd.

9.6 Dependence of the Company Capitalization, V, on Leverage Level L

109

WACC(L); p=12 0.16600

0.16500

WACC

0.16400

Kd=0,18

0.16300

Kd=0,16 Kd=0,14 Kd=0,12

0.16200

0.16100

0.16000 5

6

7

8

9

10

L

Fig. 9.5 Dependence of the weighted average cost of capital, WACC, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 12 (larger scale) Table 9.6 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.18 and p ¼ 1

9.6

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

p 12 12 12 12 12 12 12 12 12 12 12

WACC 0.20000 0.17840 0.17120 0.16760 0.16544 0.16400 0.16297 0.16220 0.16160 0.16112 0.16073

Dependence of the Company Capitalization, V, on Leverage Level L

In this section, we study the dependence of the company capitalization, V, on leverage level L for the following situations: (1) at different p ¼ 1; 2; 4; 12 and fixed kd ¼ 0.16; (2) for fixed p and at different values of debt cost kd ¼ 0.18; 0.16; 0.14; 012.

110

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.7 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 12 12 12 12 12 12 12 12 12 12 12

WACC 0.20000 0.17857 0.17143 0.16786 0.16572 0.16429 0.16327 0.16250 0.16191 0.16143 0.16104

Table 9.8 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.14 and p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

p 12 12 12 12 12 12 12 12 12 12 12

WACC 0.20000 0.17875 0.17166 0.16812 0.16600 0.16458 0.16357 0.16281 0.16222 0.16174 0.16136

Table 9.9 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.12 and p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

p 12 12 12 12 12 12 12 12 12 12 12

WACC 0.20000 0.17892 0.17190 0.16838 0.16628 0.16487 0.16387 0.16311 0.16253 0.16206 0.16168

9.6 Dependence of the Company Capitalization, V, on Leverage Level L

111

Table 9.10 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.18 and p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

p 4 4 4 4 4 4 4 4 4 4 4

WACC 0.20000 0.17870 0.17160 0.16804 0.16591 0.16449 0.16348 0.16272 0.16213 0.16165 0.16127

Table 9.11 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 4 4 4 4 4 4 4 4 4 4 4

WACC 0.20000 0.17884 0.17178 0.16826 0.16614 0.16473 0.16372 0.16296 0.16238 0.16191 0.16152

Table 9.12 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.14 and p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

p 4 4 4 4 4 4 4 4 4 4 4

WACC 0.20000 0.17898 0.17197 0.16847 0.16637 0.16496 0.16396 0.16321 0.16263 0.16216 0.16178

112

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.13 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.12 and p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 4 4 4 4 4 4 4 4 4 4 4

WACC 0.20000 0.17912 0.17216 0.16868 0.16659 0.16520 0.16421 0.16346 0.16288 0.16242 0.16204

WACC(L) at p=4 0.20000

0.19500

0.19000

WACC

0.18500 Kd=0,18

0.18000

Kd=0,16 Kd=0,14

0.17500

Kd=0,12 0.17000

0.16500

0.16000 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 9.6 Dependence of the weighted average cost of capital, WACC, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 4

9.6.1

Dependence of the Company Capitalization, V, on Leverage Level L at Different p and Fixed kd

Obtained results on the dependence of the company capitalization, V, on leverage level L at different p ¼ 1; 2; 4; 12 and fixed kd ¼ 0.16 (Tables 9.14, 9.15, 9.16 and 9.17 and Figs. 9.8 and 9.9) show that the company capitalization, V, increases with p

9.6 Dependence of the Company Capitalization, V, on Leverage Level L

113

WACC(L); p=4 0.16700

0.16600

0.16500

WACC

0.16400 Kd=0,18 Kd=0,16 0.16300

Kd=0,14 Kd=0,12

0.16200

0.16100

0.16000 5

6

7

8

9

10

L

Fig. 9.7 Dependence of the weighted average cost of capital, WACC, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 4 (larger scale) Table 9.14 Dependence of the company capitalization, V, on leverage level L at p ¼ 1, kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 1 1 1 1 1 1 1 1 1 1 1

V 5000.00 5555.56 5769.23 5882.35 5952.38 6000.00 6034.48 6060.61 6081.08 6097.56 6111.11

and reach maximum at p ¼ 12. This means that for company is more profitable to pay tax on profit as much more frequently as tax legislation allows.

114

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.15 Dependence of the company capitalization, V, on leverage level L at p ¼ 2, kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 2 2 2 2 2 2 2 2 2 2 2

V 5000.00 5579.43 5809.62 5922.61 5996.37 6046.58 6082.96 6110.53 6132.15 6149.55 6169.87

Table 9.16 Dependence of the company capitalization, V, on leverage level L at p ¼ 4, kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 4 4 4 4 4 4 4 4 4 4 4

V 5000.00 5591.68 5821.31 5949.34 6019.04 6070.60 6107.96 6136.29 6158.51 6176.39 6191.11

Table 9.17 Dependence of the company capitalization, V, on leverage level L at p ¼ 12, kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 12 12 12 12 12 12 12 12 12 12 12

V 5000.00 5599.96 5839.28 5957.38 6034.41 6086.88 6124.92 6159.77 6176.39 6194.61 6209.59

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

9.6 Dependence of the Company Capitalization, V, on Leverage Level L

115

V(L) at different p 6400.00 6200.00 6000.00

5800.00

V(L)

p=1 p=2

5600.00

p=4 5400.00

p=12

5200.00 5000.00 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 9.8 Dependence of the company capitalization, V, on leverage level L at difference p (1; 2; 4; 12) and fixed kd ¼ 0.16

V(L) at different p 6300.00

6250.00

V(L)

6200.00 p=1

6150.00

p=2

p=4

6100.00

p=12 6050.00

6000.00 5

6

7

8

9

10

L

Fig. 9.9 Dependence of the company capitalization, V, on leverage level L at difference p (1; 2; 4; 12) and fixed kd ¼ 0.16 (larger scale)

9.6.2

Dependence of the Company Capitalization, V, on Leverage Level L at Different kd and Fixed p

Let us now study the dependence of the company capitalization, V, on leverage level L for fixed p and at different values of debt cost kd ¼ 0.18; 0.16; 0.14; 012.

116

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.18 Dependence of the company capitalization, V, on leverage level L at p ¼ 2 and kd ¼ 0.18

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 2 2 2 2 2 2 2 2 2 2 2

V 5000.00 5582.31 5807.78 5927.48 6001.70 6052.22 6088.83 6116.58 6138.34 6155.85 6170.26

Table 9.19 Dependence of the company capitalization, V, on leverage level L at p ¼ 2 and kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 2 2 2 2 2 2 2 2 2 2 2

V 5000.00 5579.43 5809.62 5922.61 5996.37 6046.58 6082.96 6110.53 6132.15 6149.55 6169.87

Table 9.20 Dependence of the company capitalization, V, on leverage level L at p ¼ 2 and kd ¼ 0.14

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 2 2 2 2 2 2 2 2 2 2 2

V 5000.00 5576.53 5799.44 5917.71 5991.01 6040.90 6077.05 6104.44 6125.92 6149.21 6157.43

Obtained results on the dependence of the company capitalization, V, on leverage level L at fixed p (but for p ¼ 1; 2; 4; 12) and different kd ¼ 12%; 14%; 16%; 18% (Tables 9.18, 9.19, 9.20, 9.21, 9.22, 9.23, 9.24, 9.25, 9.26, 9.27, 9.28 and 9.29 and Figs. 9.10, 9.11, 9.12, 9.13, 9.14 and 9.15) show that the company capitalization, V,

9.6 Dependence of the Company Capitalization, V, on Leverage Level L

117

Table 9.21 Dependence of the company capitalization, V, on leverage level L at p ¼ 2 and kd ¼ 0.12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 2 2 2 2 2 2 2 2 2 2 2

V 5000.00 5579.61 5795.22 5912.77 5985.62 6035.19 6071.10 6098.31 6119.65 6136.83 6150.96

Table 9.22 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.18 and fixed p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 4 4 4 4 4 4 4 4 4 4 4

V 5000.00 5596.08 5827.66 5950.79 6027.20 6079.24 6116.97 6145.57 6168.00 6186.06 6200.92

Table 9.23 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.16 and fixed p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 4 4 4 4 4 4 4 4 4 4 4

V 5000.00 5591.68 5821.31 5949.34 6019.04 6070.60 6107.96 6136.29 6158.51 6176.39 6191.11

increases with kd at each value of fixed p and reach maximum at kd ¼ 18%. This corresponds to decrease of the weighted average cost of capital, WACC, with kd at fixed p, obtained by us in Sect. 9.5. This is connected to the fact that the increase of kd increases the value of tax shield in case of income tax payments more than once

118

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.24 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.14 and fixed p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 4 4 4 4 4 4 4 4 4 4 4

V 5000.00 5587.26 5814.92 5935.85 6010.85 6061.91 6098.92 6126.98 6148.98 6166.69 6181.26

Table 9.25 Dependence of the company capitalization, V, on leverage level L at different kd ¼ 0.12 and fixed p ¼ 4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 4 4 4 4 4 4 4 4 4 4 4

V 5000.00 5582.81 5808.49 5928.32 6002.62 6059.19 6089.84 6117.62 6139.41 6156.94 6171.37

Table 9.26 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.18 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

p 12 12 12 12 12 12 12 12 12 12 12

V 5000.00 5605.40 5841.15 5966.63 6044.53 6097.61 6136.09 6165.28 6188.17 6206.60 6221.77

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

per year (at p 6¼ 1). Note that in “classical” Modigliani–Miller theory with one income tax payment per year all financial indicators, including company capitalization, V, do not depend on debt cost, kd.

9.6 Dependence of the Company Capitalization, V, on Leverage Level L

119

Table 9.27 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.16 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 12 12 12 12 12 12 12 12 12 12 12

V 5000.00 5599.96 5839.28 5957.38 6034.41 6086.88 6124.92 6159.77 6176.39 6194.61 6209.59

Table 9.28 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.14 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 12 12 12 12 12 12 12 12 12 12 12

V 5000.00 5594.50 5825.38 5948.11 6024.27 6076.13 6119.73 6142.23 6164.58 6182.59 6197.39

Table 9.29 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.12 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

p 12 12 12 12 12 12 12 12 12 12 12

V 5000.00 5589.01 5817.45 5938.82 6014.10 6065.35 6102.50 6130.67 6152.75 6170.53 6185.16

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

120

V(L) at different kd and p=2 6400.00 6200.00

V(L)

6000.00 5800.00

Kd=0,18

5600.00

kd=0,16

5400.00

kd=0,14

5200.00

kd=0,12

5000.00 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 9.10 Dependence of the company capitalization, V, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 2

V(L), p=2 6200.00 6180.00 6160.00

V(L)

6140.00 6120.00

Kd=0,18

6100.00

kd=0,16

6080.00

kd=0,14

6060.00

kd=0,12

6040.00 6020.00 5

6

7

8

9

10

L

Fig. 9.11 Dependence of the company capitalization, V, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 2 (larger scale)

9.7

Dependence of the Equity Cost, ke, on Leverage Level L

In this section, we study the dependence of the equity cost, ke, on leverage level L for the following situations: (1) at different p ¼ 1; 2; 4; 12 and fixed kd ¼ 0.16; (2) for fixed p and at different values of debt cost kd ¼ 0.18; 0.16; 0.14; 012.

9.7.1

Dependence of the Equity Cost, ke, on Leverage Level L at Different p and Fixed kd = 0.16

Let us start from the study of the dependence of the equity cost, ke, on leverage level L at different p and fixed kd ¼ 0.16.

9.7 Dependence of the Equity Cost, ke, on Leverage Level L

121

V(L) at p=4 6400.00 6200.00

V(L)

6000.00 5800.00

kd=0,18

5600.00

kd=0,16

5400.00

kd=0,14 kd=0,12

5200.00 5000.00 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 9.12 Dependence of the company capitalization, V, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 4

V(L) at p=4 6210.00 6190.00 6170.00

V(L)

6150.00

kd=0,18

6130.00

kd=0,16

6110.00

kd=0,14

6090.00

kd=0,12

6070.00 6050.00 5

6

7

8

9

10

L

Fig. 9.13 Dependence of the company capitalization, V, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 4 (larger scale)

Obtained results on the dependence of the equity cost, ke, on leverage level L at fixed kd ¼ 16% and different p ¼ 1; 2; 4; 12 (Tables 9.30, 9.31, 9.32 and 9.33 and Figs. 9.16 and 9.17) show that the tilt angle of the curve of equity cost, ke (L ), decreases with the number of payments of tax of profit p, this modifies the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. Note that in “classical” Modigliani–Miller theory with one income tax payment per year all financial indicators, including the equity cost, ke, do not depend on debt cost, kd.

122

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

V(L) at p=12 6400.00 6200.00

V(L)

6000.00 5800.00

kd=0,18

5600.00

kd=0,16

5400.00

kd=0,14 kd=0,12

5200.00 5000.00 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 9.14 Dependence of the company capitalization, V, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 12

V(L)

V(L) at p=12 6250.00 6230.00 6210.00 6190.00 6170.00 6150.00 6130.00 6110.00 6090.00 6070.00 6050.00

kd=0,18 kd=0,16 kd=0,14 kd=0,12

5

6

7

8

9

10

L

Fig. 9.15 Dependence of the company capitalization, V, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 12 (larger scale) Table 9.30 Dependence of the equity cost, ke, on leverage level L at p ¼ 1 and fixed kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 1 1 1 1 1 1 1 1 1 1 1

ke 0.2000 0.2320 0.2640 0.2960 0.3280 0.3600 0.3920 0.4240 0.4560 0.4880 0.5200

9.7 Dependence of the Equity Cost, ke, on Leverage Level L

123

Table 9.31 Dependence of the equity cost, ke, on leverage level L at p ¼ 2 and fixed kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 2 2 2 2 2 2 2 2 2 2 2

ke 0.2000 0.2305 0.2609 0.2914 0.3218 0.3523 0.3828 0.4132 0.4437 0.4741 0.5046

Table 9.32 Dependence of the equity cost, ke, on leverage level L at p ¼ 4 and fixed kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 4 4 4 4 4 4 4 4 4 4 4

ke 0.2000 0.2297 0.2593 0.2890 0.3187 0.3484 0.3780 0.4077 0.4374 0.4671 0.4967

Table 9.33 Dependence of the equity cost, ke, on leverage level L at p ¼ 12 and fixed kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

ke 0.2000 0.2291 0.2583 0.2874 0.3166 0.3457 0.3749 0.4040 0.4332 0.4623 0.4915

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

124 0.5500 0.5000 0.4500 0.4000 0.3500 0.3000 0.2500 0.2000 0

1

2

3 p=1

4

5 p=2

6 p=4

7

8

9

10

p=12

Fig. 9.16 Dependence of the equity cost, ke, on leverage level L at different p (1; 2; 4; 12) and fixed kd ¼ 0.16 0.5200

0.4700

0.4200

0.3700

0.3200 5

6

7 p=1

p=2

8 p=4

9

10

p=12

Fig. 9.17 Dependence of the equity cost, ke, on leverage level L at different p (1; 2; 4; 12) and fixed kd ¼ 0.16 (lager scale)

9.7.2

Dependence of the Equity Cost, ke, on Leverage Level L at Different kd and Fixed p = 2

In this section, we study the dependence of the equity cost, ke, on leverage level L at different kd ¼ 18%; 16%; 14%; 12% and fixed number of income tax payments per year p ¼ 2.

9.7 Dependence of the Equity Cost, ke, on Leverage Level L

125

Table 9.34 Dependence of the equity cost, ke, on leverage level L at p ¼ 2 and fixed kd ¼ 0.18

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

p 2 2 2 2 2 2 2 2 2 2 2

ke 0.2000 0.2143 0.2285 0.2428 0.2571 0.2714 0.2856 0.2999 0.3142 0.3285 0.3427

Table 9.35 Dependence of the equity cost, ke, on leverage level L at p ¼ 2 and fixed kd ¼ 0.16

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 2 2 2 2 2 2 2 2 2 2 2

ke 0.2000 0.2305 0.2609 0.2914 0.3218 0.3523 0.3828 0.4132 0.4437 0.4741 0.5046

Table 9.36 Dependence of the equity cost, ke, on leverage level L at p ¼ 2 and fixed kd ¼ 0.14

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

p 2 2 2 2 2 2 2 2 2 2 2

ke 0.2000 0.2466 0.2933 0.3399 0.3866 0.4332 0.4799 0.5265 0.5732 0.6198 0.6665

Conclusions Obtained results on the dependence of the equity cost, ke, on leverage level L at fixed but different p ¼ 1; 2; 4; 12 and different values of debt cost kd (Tables 9.34, 9.35, 9.36, 9.37, 9.38, 9.39, 9.40, 9.41, 9.42, 9.43, 9.44 and 9.45 and Figs. 9.18, 9.19 and 9.20) show that the tilt angle of the curve of equity cost, ke (L ),

126

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.37 Dependence of the equity cost, ke, on leverage level L at p ¼ 2 and fixed kd ¼ 0.12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

p 2 2 2 2 2 2 2 2 2 2 2

ke 0.2000 0.2628 0.3257 0.3885 0.4513 0.5142 0.5770 0.6398 0.7027 0.7655 0.8283

Table 9.38 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.18 and fixed p¼4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

p 4 4 4 4 4 4 4 4 4 4 4

ke 0.2000 0.2134 0.2268 0.2402 0.2536 0.2670 0.2804 0.2938 0.3071 0.3205 0.3339

Table 9.39 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.16 and fixed p¼4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 4 4 4 4 4 4 4 4 4 4 4

ke 0.2000 0.2297 0.2593 0.2890 0.3187 0.3484 0.3780 0.4077 0.4374 0.4671 0.4967

decreases with the increase of values of debt cost kd: this modifies the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. Note, that in “classical” Modigliani–Miller theory with one

9.7 Dependence of the Equity Cost, ke, on Leverage Level L

127

Table 9.40 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.14 and fixed p¼4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

p 4 4 4 4 4 4 4 4 4 4 4

ke 0.2000 0.2460 0.2919 0.3379 0.3838 0.4298 0.4757 0.5217 0.5677 0.6136 0.6596

Table 9.41 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.12 and fixed p¼4

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

p 4 4 4 4 4 4 4 4 4 4 4

ke 0.2000 0.2622 0.3245 0.3867 0.4490 0.5112 0.5735 0.6357 0.6979 0.7602 0.8224

Table 9.42 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.18 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

ke 0.2000 0.2128 0.2256 0.2384 0.2512 0.2640 0.2768 0.2896 0.3024 0.3152 0.3280

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

income tax payment per year all financial indicators, including the equity cost, ke, do not depend on debt cost, kd.

128

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

Table 9.43 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.16 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

p 12 12 12 12 12 12 12 12 12 12 12

ke 0.2000 0.2291 0.2583 0.2874 0.3166 0.3457 0.3749 0.4040 0.4332 0.4623 0.4915

Table 9.44 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.12 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

p 12 12 12 12 12 12 12 12 12 12 12

ke 0.2000 0.2455 0.2910 0.3365 0.3820 0.4275 0.4730 0.5185 0.5640 0.6094 0.6549

Table 9.45 Dependence of the equity cost, ke, on leverage level L at i ¼ 0.12 and fixed p ¼ 12

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

p 12 12 12 12 12 12 12 12 12 12 12

Ke 0.2000 0.2618 0.3237 0.3855 0.4474 0.5092 0.5711 0.6329 0.6948 0.7566 0.8185

9.7 Dependence of the Equity Cost, ke, on Leverage Level L

129

Ke(L) at p=2 and different kd 0.9000 0.8000 0.7000

Ke

0.6000 0.5000

Kd=0,18

0.4000

Kd=0,16

0.3000

Kd=0,14

0.2000

Kd=0,12

0.1000 0.0000 0

1

2

3

4

5

6

7

8

9

10

L Fig. 9.18 Dependence of the equity cost, ke, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 2

Ke(L) at different kd and p=4 0.9000

0.8000 0.7000

Ke

0.6000 0.5000

Kd=0,18

0.4000

Kd=0,16

0.3000

Kd=0,14

0.2000

Kd=0,12

0.1000 0.0000 0

1

2

3

4

5

6

7

8

9

10

L Fig. 9.19 Dependence of the equity cost, ke, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 4

9 The Modigliani–Miller Theory with Arbitrary Frequency of Payment of Tax. . .

130

Ke(L) at different kd and p=12 0.9000 0.8000 0.7000

Ke

0.6000 0.5000

Kd=0,18

0.4000

Kd=0,16

0.3000

Kd=0,14

0.2000

Kd=0,12

0.1000 0.0000 0

1

2

3

4

5

6

7

8

9

10

L Fig. 9.20 Dependence of the equity cost, ke, on leverage level L at different kd (12%; 14%; 16%; 18%) and fixed p ¼ 12

9.8

Conclusions

The tax shield plays a crucial role in both main capital structure theories: Brusov– Filatova–Orekhova (BFO theory) and its perpetuity limit Modigliani–Miller theory. The way it is formed affects the results of the theories. Both of them describe the case of annual payments of tax of profit, but in practice, these payments are made more frequent: semiannually, quarterly, monthly. While a return is not required more than once a year, businesses may be responsible for filing estimated taxes based on profits earned. This requirement is dependent on showing a profit. For example, sole proprietors must file estimated taxes on profits quarterly, on the 15th day of April, June, September and January. In Russia tax on profit payments could be made annually, quarterly, monthly. We show in the current paper that the account of frequency of payment of tax on profit turns out to be very important and leads to significant consequences. To study this problem, we modify the Modigliani–Miller theory for the case of arbitrary frequency of payment of tax on profit and show that: – All Modigliani–Miller statements and all formulas change. – In case of income tax payments more than once per year (at p 6¼ 1), as it takes place in practice, the weighted average cost of capital, WACC, company value, V and equity cost, ke start depend on kd, while in ordinary (classical) Modigliani– Miller theory all these values DO NOT depend on kd.

References

131

– Obtained results allow to company choose the number of payments of tax of profit per year, as many, as it is profitable to it (of course, within actual tax legislation). – The tilt angle of the curve of equity cost, ke(L ), decreases with p, this modifies the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. – One more influence of taxes within Modigliani–Miller theory is found: under accounting the number of payments of the tax of profit influent on all main indicators of the company: the weighted average cost of capital, WACC, company value, V, equity cost, ke, etc. This allows to company manage WACC, V, ke, etc. by choosing the number of payments of tax of profit p. It turns out that increase the number of payments of tax of profit per year leads to decrease of the cost of attracting capital and increase of the company value. More frequent payments of income tax are beneficial for both parties: for the company and for the tax regulator: for the company, this leads to an increase in the value of the company, and for the tax regulator, earlier payments are beneficial due to the time value of money.

References Berk J, DeMarzo P (2007) Corporate finance. Pearson–Addison Wesley, Boston Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 3(435):2–8 Brusov P, Filatova T (2021a) The Modigliani–Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9:1198. https://doi.org/10.3390/math9111198 Brusov P, Filatova T (2021b) The Modigliani–\Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198. https://doi.org/10.3390/math9111198 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11):815– 824 Brusov P, Filatova T, Orehova N, Brusov PP, Brusova N (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova P, Orekhova N (2013a) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova T, Orehova N (2013b) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Global Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit—Modigliani–Miller theory. J Rev Global Econ 3:175–185

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Brusov P, Filatova T, Orehova N, Kulk WI (2018a) New meaningful effects in modern capital structure theory. J Rev Global Econ 7:104–122 Brusov P, Filatova T, Orehova N, Kulk V (2018b) A “Golden Age” of the companies: conditions of its existence. J Rev Global Econ 7:88–103 Brusov P, Filatova T, Orehova N, Kulk V (2018c) Rating methodology: new look and new horizons. J Rev Global Econ 7:63–87 Brusov P, Filatova T, Orehova N, Kulk V (2018d) Rating: new approach. J Rev Global Econ 7:37– 62 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Global Econ 8:437–448. https://doi.org/10. 6000/1929-7092.2019.08.37 Brusov P, Filatova T, Orehova N (2020) Ratings: critical analysis and new approaches of quantitative and qualitative methodology, Springer Nature Publishing, Cham, Monograph, https:// www.springer.com/de/book/9783030562427 Brusov P, Filatova T, Chang S-I, Lin G (2021) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9:1491. https://doi.org/10. 3390/math9131491 Farber A, Gillet R, Szafarz A (2006) A general formula for the WACC. Int J Bus 11(2):211–218 Fernandez P (2007) A general formula for the WACC: a comment. Int J Bus 12(3):399–403 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP (2022) Impact of advance payments of tax on profit on effectiveness of investments. Mathematics 10(4):666. https://doi.org/10.3390/math10040666 Hamada R (1969) Portfolio analysis, market equilibrium, and corporate finance. J Financ 24(1): 13–31 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Мodigliani F, Мiller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Miller M (1977) Debt and taxes. J Finance 32(2):261–275. https://doi.org/10.1111/j.1540-6261. 1977.tb03267.x Ross S, Westerfiel R, Jaffee D (2005) Corporate finance, 7th edn. McGraw Hill, New York

Chapter 10

How Frequently Should Companies Pay Tax on Profit

10.1

Introduction

Brusov and Filatova (2021) have modified the Modigliani–Miller theory for the case of arbitrary frequency of payment of tax on profit and have shown that the account of frequency of payment of tax on profit within the Modigliani–Miller theory turns out to be very important and leads to significant consequences: • All Modigliani–Miller theorems, statements, and all formulas change. • All main financial indicators, such as the weighted average cost of capital, WACC, company value, V and equity cost, ke depend on the frequency of tax on profit payments. This allows to company manage WACC, V, ke, etc. by choosing the number of payments of tax of profit p per year. • Obtained results allow to company choose the number of payments of tax of profit per year, as many, as it is profitable to it (of course, within actual tax legislation). • In case of income tax payments more than once per year (at p 6¼ 1), as it takes place in practice, the weighted average cost of capital, WACC, company value, V and equity cost, ke start depend on kd, while in ordinary (classical) Modigliani– Miller theory all these values DO NOT depend on kd. • The tilt angle of the curve of equity cost, ke (L), decreases with the number of payments of tax of profit p, this modifies the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. As it was shown in that paper (Brusov and Filatova 2021), more frequent payments of income tax are beneficial for both parties: for the company and for the tax regulator: for the company, this leads to decrease of the cost of attracting capital and thus to an increase in the value of the company, and for the tax regulator, earlier payments are beneficial due to the time value of money. But companies could pay tax of profit in advance and as we show in current paper, this fact changes drastically the main conclusion: more frequent payments of © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_10

133

134

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How Frequently Should Companies Pay Tax on Profit

income tax are NOT beneficial for the company: this leads to increase of the cost of attracting capital and thus to decrease in the value of the company, but for the tax regulator it remains beneficial: earlier payments are beneficial due to the time value of money. The structure of the chapter is as following: 1. We give a literature review for Modigliani–Miller theory and for its modifications. 2. We modify the Modigliani–Miller theory for the case of frequent payment of tax on profit (in case of advanced payments) and get modified Modigliani–Miller theorems as well as new formulas for the weighted average cost of capital, WACC, for the equity cost, ke, and for the capitalization of the company, V. 3. Within new Modified Modigliani–Miller theory (MMM theory), we study the dependence of the main financial indicators of the company (WACC, ke, V ) on leverage level L, on debt cost kd and on tax on profit t at different frequency of payments of tax on profit (at the end of period and in advance). 4. Based on obtained results we come to some very important conclusions, which allow company choose the method of payments of tax of profit (at the end of period or in advance) and number of payments of tax of profit per year, as many, as it is profitable to it (of course, within actual tax legislation).

10.2

Literature Review

In their first paper, Modigliani and Miller (1958) considered the problem of capital cost and capital structure for perpetuity companies without any taxes. In the second article, Modigliani and Miller (1963) considered taxation of corporate profits but did not take into account the presence in the economy of individual taxes of investors. Merton Miller (1977) has introduced the model, demonstrating impact of leverage on the company value with account of the corporate and individual taxes. To describe his model, we will enter the following legends: TC—tax on corporate profits rate, TS—the tax rate on income of an individual investor from his ownership by stock of corporation, TD—tax rate on interest income from the provision of investor– individuals of credits to other investors and companies. Income from shares partly comes in the form of a dividend and, in part, as capital profits, so that TS is a weighted average value of effective rates of tax on dividends and capital profits on shares, while the income from the provision of loans usually comes in the form of the interests. The last are usually taxed at a higher rate. In the light of the individual taxes, and with the same assumptions that have been made for Modigliani–Miller models previously, the financially independent company value can be determined as follows:

10.2

Literature Review

135

VU ¼

EBITð1  T C Þð1  T S Þ : k0

ð10:1Þ

A term (1  TS) allows to take into account the individual taxes in the formula. In this way, numerator indicates which part of the operating company’s profit remains in the possession of the investors, after the company will pay taxes on their profits, and its shareholders then will pay individual taxes on income from stock ownership. Since individual taxes reduce profits, remaining at the disposal of investors, the last, at other things being equal circumstances, also reduce and an overall assessment of the financially independent company value. Robert Hаmаdа (1969) unites Capital Asset Pricing Model (CAPM) with Modigliani–Miller model taxation. As a result, he derived the following formula for calculation of the equity cost of financially dependent company, including both financial and business risk of company: k e ¼ kF þ ðk M  kF ÞbU þ ðkM  kF ÞbU

D ð1  T Þ, S

ð10:2Þ

where bU is the β-coefficient of the company of the same group of business risk that the company under consideration, but with zero financial leverage. The formula (10.2) represents the desired profitability of equity capital ke as a sum of three components: risk-free profitability kF, compensating to shareholders a temporary value of their money, premium for business risk,(kM  kF)bU, and premium for financial risk ðk M  kF ÞbU DS ð1  T Þ. If the company does not have borrowing (D ¼ 0), the financial risk factor will be equal to zero (the third term is drawn to zero), and its owners will only receive the premium for business risk. More general formula for the WACC, then famous Modigliani–Miller (MM) one has been derived and discussed by a few authors (Farber et al. 2007; Fernandez 2007; Harris and Pringle 1985; Miles and Ezzel 1980; DeMarzo 1988). It takes the following form (Equation (18) in FGS, 2006) WACC ¼ k0 ð1  wd T Þ  kd twd þ kTS twd

ð10:3Þ

where k0, kd, and kTS are the expected returns respectively on the unlevered firm, the debt and the tax shield, V is the value of the levered firm, VTS is the value of the tax shield, D is the value of the debt, and TC is the corporate tax rate. This formula is derived from the balance sheet identity and the definition of the weighted average cost of capital (for a similar presentation, see Berk and DeMarzo 2007). It should therefore be verified at any point in time, whatever returns are annually or continuously compounded. Although Eq. (10.3) is fully general, its practical applicability requires additional conditions. Indeed, when the WACC is constant over time, the value of a levered firm can be computed by discounting with the WACC the unlevered free cash flows. Therefore, we pay a special attention to the special cases that make the WACC

136

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How Frequently Should Companies Pay Tax on Profit

constant. The resulting particular formulas can also be found in textbooks (Brealey et al. 2005; Ross et al. 2005). First, Modigliani and Miller (1963) assume that the level of debt D is constant. Then, as the expected after-tax cash flow of the unlevered firm is fixed, V0 is also constant. By assumption, kTS ¼ kD and the value of the tax shield is TS ¼ tD. Therefore, the value of the firm V is a constant, and the general WACC formula (10.3) simplifies to a constant WACC: WACC ¼ k0 ð1  wd T Þ

ð10:4Þ

But our opinion is that “classical” Modigliani–Miller (MM) theory, which coming from the suggestion that the expected returns on the debt kd and the tax shield kTS are equals (because both of them have debt nature), is much more reasonable and in our paper, we modify namely “classical” Modigliani–Miller (MM) theory, which is still widely used in practice. Recently the Modigliani–Miller theory has been modified by Brusov et al. (2019, 2021) into two ways: it was applied for rating methodologies needs and it was generalized for the case of advance payments of tax on profit, which are widely used in practice (MMM theory). As well Brusov et al. (2019) used the Modified Modigliani–Miller theory (MMM theory) and applied it for rating methodologies needs. The financial “ratios” (main rating parameters) were introduced into MMM theory. The dependence of the weighted average cost of capital (WACC), which plays the role of discount rate in financial flows discounting in rating methodologies, on coverage and leverage ratios, is analyzed. Obtained results will help improve the existing rating methodologies. One of the most serious limitations of the Modigliani–Miller theory is the suggestion about the perpetuity of the companies. In 2008, Brusov–Filatova– Orekhova (Filatova et al. 2008) have lifted up this limitation and have shown that the accounting of the finite age (or finite lifetime) of the company leads to significant changes of all Modigliani–Miller results (Modigliani and Miller 1958, 1963, 1966): capitalization of the company is changed, as well as the equity cost, ke, and the weighted average cost of capital, WACC, in the presence of corporative taxes. Besides, a number of qualitatively new effects in corporate finance, obtained within Brusov–Filatova–Orekhova theory (BFO theory) (Brusov 2018a, b; Brusov et al. 2018a, b, c, d, e, f, g, 2019, 2020a, b, c; Filatova et al. 2018), are absent in Modigliani–Miller theory (1958, 1963). The famous Brusov–Filatova–Orekhova formula for weighted average cost of capital, WACC, has the following form 1  ð1 þ WACCÞn 1  ð1 þ k0 Þn : ¼ WACC k0 ½1  ωd T ð1  ð1 þ k d Þn Þ

ð10:5Þ

10.3

Focus and Objective of the Chapter

137

D Here, S—the value of own (equity) capital of the company, wd ¼ DþS —the share S of debt capital; k e , we ¼ DþS—the cost and the share of the equity of the company, L ¼ D/S—financial leverage. A lot of meaningful effects have been discovered within BFO theory: these effects are absent within Modigliani–Miller theory. BFO theory has destroyed some main existing principles of financial management: among them trade-off theory, which was considered as keystone of formation of optimal capital structure of the company during many decades and bankruptcy of which has been proven by Brusov–Filatova–Orekhova.

10.3

Focus and Objective of the Chapter

The tax shield plays a crucial role in both main capital structure theories: Brusov– Filatova–Orekhova (BFO theory) and its perpetuity limit Modigliani–Miller theory. How it is formed influences the results of both theories. Both of them describe the case of annual payments of tax of profit, but in practice, these payments are made more frequent: semiannually, quarterly, monthly. While a return is not required more than once a year, businesses may be responsible for filing estimated taxes based on profits earned. This requirement is dependent on showing a profit. For example, sole proprietors must file estimated taxes on profits quarterly, on the 15th day of April, June, September, and January. In Russia, tax on profit payments could be made annually, quarterly, monthly. Brusov and Filatova (2021) have shown that the account of frequency of payment of tax on profit within the Modigliani–Miller theory turns out to be very important and leads to significant consequences. To study this problem, they have modified the Modigliani–Miller theory for the case of arbitrary frequency of payment of tax on profit and show that: • All Modigliani–Miller theorems, statements, and all formulas change. • All main financial indicators, such as the weighted average cost of capital, WACC, company value, V and equity cost, ke, depend on the frequency of tax on profit payments. This allows to company manage WACC, V, ke, etc. by choosing the number of payments of tax of profit p per year, • Increase the number of payments of tax of profit per year leads to decrease of the cost of attracting capital and increase of the company value. • Obtained results allow to company choose the number of payments of tax of profit per year, as many, as it is profitable to it (of course, within actual tax legislation). • In case of income tax payments more than once per year (at p 6¼ 1), as it takes place in practice, the weighted average cost of capital, WACC, company value, V and equity cost, ke start depend on kd, while in ordinary (classical) Modigliani– Miller theory all these values DO NOT depend on kd.

138

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How Frequently Should Companies Pay Tax on Profit

• The tilt angle of the curve of equity cost, ke (L), decreases with the number of payments of tax of profit p, this modifies the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. As it was shown in that paper (Brusov and Filatova 2021), more frequent payments of income tax are beneficial for both parties: for the company and for the tax regulator: for the company, this leads to decrease of the cost of attracting capital and thus to an increase in the value of the company, and for the tax regulator, earlier payments are beneficial due to the time value of money. But companies could pay tax of profit in advance and as we show in current paper, this fact changes drastically the main conclusion: more frequent payments of income tax are NOT beneficial for the company: this leads to increase of the cost of attracting capital and thus to decrease in the value of the company, but for the tax regulator it remains beneficial: earlier payments are beneficial due to the time value of money.

10.4

Modification of the Modigliani–Miller Theory for the Case of Arbitrary Frequency of Advanced Payments of Tax on Profit

10.4.1 Tax Shield Let us calculate first the tax shield TS in the Modigliani–Miller theory in the case of arbitrary frequency ( p per year) of advanced payments of tax on profit TS ¼

kd Dt kd Dt k d Dt þ þ þ ... 1=p 2 p pð 1 þ k d Þ pð1 þ k d Þ =p

TS represents a geometric progression with denominator q ¼

ð10:6Þ 1 1 . ð1þk d Þ =p

Here D is debt value, kd is debt cost, t is tax on profit rate, p is the frequency of payments of tax on profit. Summing the progression, one gets k Dt  ð1 þ k d Þ =p kd Dt :  ¼
d TS ¼
1 1 p ð1 þ k d Þ =p  1 p 1  ð1 þ k d Þ =p 1

ð10:7Þ

Note, that in case of classical Modigliani–Miller theory, modified by us for the case of one advanced payment of tax on profit per year

10.4

Modification of the Modigliani–Miller Theory for the Case of Arbitrary. . .

TS ¼ Dt ð1 þ kd Þ:

139

ð10:8Þ

It is easy to get this result from Eq. (10.7), putting p ¼ 1.

10.4.2 The Weighted Average Cost of Capital, WACC Let us calculate now the weighted average cost of capital, WACC in the Modigliani– Miller theory in the case of arbitrary frequency ( p per year) of advanced payments of tax on profit. From the Modigliani–Miller theorem we have for the company value V V ¼ V 0 þ TS:

ð10:9Þ

Here V0 is the financially independent company value. Putting expression for tax shield TS, we get k Dt  ð1 þ kd Þ =p : V ¼ V 0 þ
d 1 p ð1 þ kd Þ =p  1 1

ð10:10Þ

Substituting D ¼ wdV, one gets 0

1 1=p k w t  ð 1 þ k Þ d d A ¼ V 0 : V @1 
d 1=p p ð1 þ k d Þ  1

ð10:11Þ

Accounting, that V¼

CF CF ;V ¼ , WACC 0 k0

ð10:12Þ

we have 0

1 1=p k w t  ð 1 þ k Þ CF CF d d A ¼  @1 
d : 1=p WACC k0 p ð1 þ k d Þ  1

ð10:13Þ

From here we arrive to the expression for the weighted average cost of capital, WACC,

140

10

How Frequently Should Companies Pay Tax on Profit

0

1=p

1

k w t  ð1 þ k d Þ A  : WACC ¼ k 0  @1 
d d 1 p ð1 þ kd Þ =p  1

ð10:14Þ

Note, that in case of classical Modigliani–Miller theory, modified by us for the case of one advanced payment of tax on profit per year WACC ¼ k 0  ð1  wd t  ð1 þ kd ÞÞ:

ð10:15Þ

It is easy to get this result from Eq. (10.14), putting p ¼ 1.

10.4.3 Company Value, V Let us calculate now the company value, V, in the Modigliani–Miller theory in the case of arbitrary frequency ( p per year) of advanced payments of tax on profit. Putting WACC value into first formula of (10.12) one gets for company value, V V¼

CF ¼ WACC

CF  : 1 kd wd tð1þkd Þ =p k0  1  1=p

ð10:16Þ

p ð1þk d Þ 1

Here CF is the profit of the company per year, wd is the debt share. In case of classical Modigliani–Miller theory, modified by us for the case of one advanced payment of tax on profit per year (Brusov et al. 2020a, b, c), one has V¼

CF CF ¼ : WACC k0  ð1  wd t ð1 þ kd ÞÞ

ð10:17Þ

It is easy to get this result from Eq. (10.16), putting p ¼ 1.

10.4.4 Equity Cost Let us calculate now the equity cost, kd, in the Modigliani–Miller theory in the case of arbitrary frequency ( p per year) of advanced payments of tax on profit. For the weighted average cost of capital, WACC, we have from formula (10.14)

10.4

Modification of the Modigliani–Miller Theory for the Case of Arbitrary. . .

0

1=p

141

1

k w t  ð1 þ k d Þ A  : WACC ¼ k 0  @1 
d d 1 p ð1 þ kd Þ =p  1

ð10:18Þ

From other side, upon definition one has WACC ¼ ke we þ kd wd ð1  t Þ:

ð10:19Þ

From Eq. (10.19) we get ke ¼

WACC kd wd ð1  t Þ  we we

ð10:20Þ

Putting Eq. (10.18) into Eq. (10.20), we arrive to expression for equity cost ke 0 1 1 kd wd t  ð1 þ kd Þ =p A kd wd ð1  t Þ k0 @   ke ¼  1
1 we we p ð1 þ k Þ =p  1 d

0

1 1=p k Lt  ð 1 þ k Þ d d A  kd Lð1  t Þ ¼ k 0  @1 þ L 
1 p ð1 þ kd Þ =p  1 0 1 1=p k k t  ð 1 þ k Þ d A ¼ k 0 þ L@k 0  kd ð1  t Þ 
0 d 1 p ð1 þ k d Þ =p  1

ð10:21Þ

Finally, one has 0

1 1=p k k t  ð 1 þ k Þ 0 d d A k e ¼ k 0 þ L@ k 0  k d ð 1  t Þ 
1 p ð1 þ kd Þ =p  1

ð10:22Þ

In case of classical Modigliani–Miller theory, modified by us for the case of one advanced payment of tax on profit per year (Brusov et al. 2020a, b, c), one has k e ¼ k0 þ Lðk 0  kd  ð1  t Þ  k0 t ð1 þ k d ÞÞ: Putting in formula (10.22) p ¼ 1, it is easy to get formula (10.23).

ð10:23Þ

142

10.5

10

How Frequently Should Companies Pay Tax on Profit

Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L

10.5.1 Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L at Different p and Fixed kd Let us start from the study numerically with the use of Microsoft Excel of the dependence of the weighted average cost of capital, WACC, on leverage level L at different frequency of payments p and fixed debt cost kd, using obtained by us formulas 0

1 kd wd t

A WACC ¼ k0  @1 
1 p ð1 þ k d Þ =p  1 0 1 1=p k w t  ð1 þ k d Þ A  ¼ k0  @1 
d d 1 p ð1 þ k d Þ =p  1

and

WACC

From Tables 10.1, 10.2, 10.3, 10.4 and Figs. 10.1, 10.2, it is seen that the weighted average cost of capital, WACC, decreases with leverage level L at all values of frequency of payments of tax on profit p. With increase of the frequency of payments of tax on profit per year p WACC (1) decreases in the case of payments at the ends of the periods, and WACC (2) increases in the case of advanced payments of tax on profit. As we will see below, this will lead to increase of company value, V, with increase of p in the case (1) and to decrease of company value, V, with increase of p in the case (2). Note that the weighted average cost of capital, WACC, in the case of advanced payments of tax on profit for all values of frequency of payments turns out to be low

Table 10.1 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 1

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.18 0.173333 0.17 0.168 0.166667 0.165714 0.165 0.164444 0.164 0.163636

10.5

Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L 143

Table 10.2 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 12 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

Table 10.3 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 1 (advanced payment of tax of profit)

L 0 1 2 3 4 5 6 7 8 9 10

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

p 12 12 12 12 12 12 12 12 12 12 12

L 0 1 2 3 4 5 6 7 8 9 10

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.178573 0.17143 0.167859 0.165716 0.164288 0.163267 0.162502 0.161907 0.161431 0.161041

WACC 0.2 0.1768 0.169067 0.1652 0.16288 0.161333 0.160229 0.1594 0.158756 0.15824 0.157818

Table 10.4 Dependence of WACC on leverage level L at k0 ¼ 0.2, kd ¼ 0.16 and p ¼ 12 (advanced payment of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

WACC 0.2 0.178306 0.171075 0.167459 0.16529 0.163843 0.16281 0.162035 0.161433 0.160951 0.160556

144

10

How Frequently Should Companies Pay Tax on Profit

WACC(L) 0.21

WACC

0.2 0.19

p=1

0.18

p=12 p=1'

0.17

p=12'

0.16 0.15 0

1

2

3

4

5

6

7

8

9

10

Fig. 10.1 Dependence of the weighted average cost of capital, WACC, on leverage level L at different p and fixed kd ¼ 0.16

WACC

WACC(L) 0.167 0.166 0.165 0.164 0.163 0.162 0.161 0.16 0.159 0.158 0.157

p=1 p=12 p=1' p=12'

5

6

7

8

9

Fig. 10.2 Dependence of the weighted average cost of capital, WACC, on leverage level L at different p and fixed kd ¼ 0.16 (larger scale)

than for any values of frequency of payments in the case of payments at the ends of the periods.

10.5

Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L 145

10.5.2 Dependence of the Weighted Average Cost of Capital, WACC, and of Company Value, V, on Debt Cost Value kd at Fixed Value of Leverage Level L and Fixed p In this paragraph, we investigate the dependence of the weighted average cost of capital, WACC, and of company value, V on debt cost value kd at fixed value of leverage level L and fixed p. From Tables 10.5, 10.6, 10.7, 10.8 and Fig. 10.3, it is seen that the weighted average cost of capital, WACC, decreases with increase of debt cost value kd, except the classical MM case p ¼ 1 and payment at the end of the period. It supports obtained by us result that account of each of two effects (either advanced payments of tax on profit, or more frequent than once per year payments) leads to dependence of all main financial indicators, such as the weighted average cost of capital, WACC, company value, V, equity cost, ke, etc. on debt cost value kd, while in the classical MM case these financial indicators DO NOT depend on debt cost value kd. Table 10.5 Dependence of WACC on debt cost value kd at k0 ¼ 0.2, at L ¼ 1 and p ¼ 1 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

WACC 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5555.556 5555.556 5555.556 5555.556 5555.556 5555.556 5555.556 5555.556 5555.556 5555.556 5555.556

ke 0.296 0.288 0.28 0.272 0.264 0.256 0.248 0.24 0.232 0.224 0.216

Table 10.6 Dependence of WACC on debt cost value kd at k0 ¼ 0.2, at L ¼ 1 and p ¼ 12 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

i 1.006434 1.007207 1.007974 1.008735 1.009489 1.010237 1.010979 1.011715 1.012445 1.01317 1.013888

WACC 0.179277 0.179188 0.179099 0.179011 0.178923 0.178835 0.178747 0.17866 0.178573 0.178486 0.178399

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5577.965 5580.736 5583.501 5586.259 5589.012 5591.758 5594.499 5597.234 5599.963 5602.686 5605.404

ke 0.294554 0.286376 0.278198 0.270021 0.261845 0.253669 0.245494 0.237319 0.229145 0.220972 0.212799

146

10

How Frequently Should Companies Pay Tax on Profit

Table 10.7 Dependence of WACC on debt cost value kd at k0 ¼ 0.2, at L ¼ 1 and p ¼ 1 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.08 1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

WACC 0.1784 0.1782 0.178 0.1778 0.1776 0.1774 0.1772 0.177 0.1768 0.1766 0.1764

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5605.381 5611.672 5617.978 5624.297 5630.631 5636.979 5643.341 5649.718 5656.109 5662.514 5668.934

ke 0.2928 0.2844 0.276 0.2676 0.2592 0.2508 0.2424 0.234 0.2256 0.2172 0.2088

Table 10.8 Dependence of WACC on debt cost value kd at k0 ¼ 0.2, at L ¼ 1 and p ¼ 12 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

i 1.006434 1.007207 1.007974 1.008735 1.009489 1.010237 1.010979 1.011715 1.012445 1.01317 1.013888

WACC 0.179144 0.179038 0.178932 0.178827 0.178723 0.178618 0.178514 0.17841 0.178306 0.178202 0.178099

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5582.116 5585.412 5588.702 5591.986 5595.266 5598.541 5601.811 5605.077 5608.338 5611.594 5614.846

ke 0.294287 0.286076 0.277865 0.269655 0.261445 0.253236 0.245027 0.236819 0.228612 0.220405 0.212199

Similar to previous paragraph, it is seen that with increase of the frequency of payments of tax on profit per year p WACC (1) decreases in the case of payments at the ends of the periods and WACC (2) increases in the case of advanced payments of tax on profit. As we will see below, this will lead to increase of company value, V, with increase of p in the case (1) and to decrease of company value, V, with increase of p in the case (2). Note that the weighted average cost of capital, WACC, in the case of advanced payments of tax on profit for all values of frequency of payments turns out to be low than for any values of frequency of payments in the case of payments at the ends of the periods. From Tables 10.5–10.8 and Fig. 10.4, it is seen, that the company value, V, increases with increase of debt cost value kd, except the classical Modigliani–Miller theory case p ¼ 1 and payment at the end of the period, when V remains constant. It supports obtained by us result that account of each of two effects (either advanced

10.5

Dependence of the Weighted Average Cost of Capital, WACC, on Leverage Level L 147

WACC(kd) 0.1805

0.18 0.1795 WACC

0.179

Wacc12'

0.1785

Wacc1'

0.178

Wacc12

0.1775

Wacc1

0.177 0.1765 0.176 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

Fig. 10.3 Dependence of WACC on debt cost value kd at k0 ¼ 0.2, at L ¼ 1 and p ¼ 1 and p ¼ 12 (means advanced payments of tax of profit)

V(kd) 5675

V

5655 5635

V12'

5615

V1' V1

5595

V12

5575 5555 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18

Fig. 10.4 Dependence of company value V on debt cost value kd at k0 ¼ 0.2, at L ¼ 1 and p ¼ 1 and p ¼ 12 (means advanced payments of tax of profit)

payments of tax on profit, or more frequent than once per year) leads to dependence of all main financial indicators, such as the weighted average cost of capital, WACC, company value, V, equity cost, ke, etc. on debt cost value kd, while in the classical Modigliani–Miller theory case these financial indicators DO NOT depend on debt cost value kd. Similar to previous paragraph, it is seen that with increase of the frequency of payments of tax on profit per year p the company value, V, (1) increases in the case of payments at the ends of the periods and WACC (2) decreases in the case of advanced payments of tax on profit.

148

10

How Frequently Should Companies Pay Tax on Profit

Note that the company value, V, in the case of advanced payments of tax on profit for all values of frequency of payments turns out to be higher than for any values of frequency of payments in the case of payments at the ends of the periods.

10.6

Dependence of the Company Capitalization, V, on Leverage Level L

10.6.1 Dependence of the Company Capitalization, V, on Leverage Level L at Different p and Fixed kd Let us now calculate the dependence of the company capitalization, V, on leverage level L at different p and fixed kd. From Tables 10.9, 10.10, 10.11, 10.12 and Figs. 10.5, 10.6, it is seen, that the company value, V, increases with the increase of leverage level L for all frequencies of payment of tax on profit per year p. While in case of payments at the end of the periods the company value, V, increases with the increase of the frequency of payment of tax on profit per year p, in case of advanced payments of tax on profit, the company value, V, decreases with the increase of the frequency of payment of tax on profit per year p. These facts consistent with the dependence of the weighted average cost of capital, WACC, on the leverage level L. All the company values, V, in case of advanced payments of tax on profit turns out to be bigger than in case of payments at the end of the periods. Thus if the companies tend to increase their capitalization, they should pay tax on profit (1) in advance; (2) as seldom as tax legislation allows. Table 10.9 Dependence of the company capitalization on leverage level L at kd ¼ 0.16 and at p ¼ 1 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.18 0.173333 0.17 0.168 0.166667 0.165714 0.165 0.164444 0.164 0.163636

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5555.556 5769.231 5882.353 5952.381 6000 6034.483 6060.606 6081.081 6097.561 6111.111

10.6

Dependence of the Company Capitalization, V, on Leverage Level L

149

Table 10.10 Dependence of the company capitalization on leverage level L at kd ¼ 0.16 and at p ¼ 12 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

WACC 0.2 0.178573 0.17143 0.167859 0.165716 0.164288 0.163267 0.162502 0.161907 0.161431 0.161041

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5599.963 5833.279 5957.383 6034.413 6086.883 6124.923 6153.767 6176.39 6194.608 6209.594

Table 10.11 Dependence of the company capitalization on leverage level L at kd ¼ 0.16 and at p ¼ 1 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.1768 0.169067 0.1652 0.16288 0.161333 0.160229 0.1594 0.158756 0.15824 0.157818

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5656.109 5914.826 6053.269 6139.489 6198.347 6241.084 6273.526 6298.992 6319.515 6336.406

Table 10.12 Dependence of the company capitalization on leverage level L at kd ¼ 0.16 and at p ¼ 12 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

WACC 0.2 0.178306 0.171075 0.167459 0.16529 0.163843 0.16281 0.162035 0.161433 0.160951 0.160556

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5608.338 5845.403 5971.613 6049.99 6103.394 6142.121 6171.49 6194.528 6213.082 6228.345

150

10

How Frequently Should Companies Pay Tax on Profit

V(L) 6400 6200 6000

p=1

5800 V

p=12

5600

p=1'

5400

p=12'

5200 5000 0

1

2

3

4

5

6

7

8

9

10

Fig. 10.5 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.16 and p ¼ 1 and 12 (means advanced payment of tax of profit)

V(L) 6400

V

6300 6200

p=1

6100

p=12 p=1'

6000

p=12'

5900 5800 5

6

7

8

9

10

Fig. 10.6 Dependence of the company capitalization, V, on leverage level L at kd ¼ 0.16 and p ¼ 1 and 12 (means advanced payment of tax of profit) (larger scale)

10.7

Dependence of the Equity Cost, ke, on Leverage Level L

Let us now calculate the dependence of the equity cost, ke, on leverage level L at different p and fixed kd. From Tables 10.13, 10.14, 10.15, 10.16 and Figs. 10.7, 10.8, 10.9, it is seen that the equity cost, ke, linearly increases with the increase of leverage level L for all frequencies of payment of tax on profit per year p.

10.7

Dependence of the Equity Cost, ke, on Leverage Level L

151

Table 10.13 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.16 and at p ¼ 1 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.18 0.173333 0.17 0.168 0.166667 0.165714 0.165 0.164444 0.164 0.163636

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5555.556 5769.231 5882.353 5952.381 6000 6034.483 6060.606 6081.081 6097.561 6111.111

ke 0.2 0.232 0.264 0.296 0.328 0.36 0.392 0.424 0.456 0.488 0.52

Table 10.14 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.16 and at p ¼ 12 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

WACC 0.2 0.178573 0.17143 0.167859 0.165716 0.164288 0.163267 0.162502 0.161907 0.161431 0.161041

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5599.963 5833.279 5957.383 6034.413 6086.883 6124.923 6153.767 6176.39 6194.608 6209.594

ke 0.2 0.229145 0.25829 0.287436 0.316581 0.345726 0.374871 0.404017 0.433162 0.462307 0.491452

Table 10.15 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.16 and at p ¼ 1 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.1768 0.169067 0.1652 0.16288 0.161333 0.160229 0.1594 0.158756 0.15824 0.157818

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5656.109 5914.826 6053.269 6139.489 6198.347 6241.084 6273.526 6298.992 6319.515 6336.406

ke 0.2 0.2256 0.2512 0.2768 0.3024 0.328 0.3536 0.3792 0.4048 0.4304 0.456

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Table 10.16 Dependence of the equity cost, ke, on leverage level L at kd ¼ 0.16 and at p ¼ 12 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0 0.5 0.666667 0.75 0.8 0.833333 0.857143 0.875 0.888889 0.9 0.909091

L 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

p 12 12 12 12 12 12 12 12 12 12 12

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

WACC 0.2 0.178306 0.171075 0.167459 0.16529 0.163843 0.16281 0.162035 0.161433 0.160951 0.160556

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5608.338 5845.403 5971.613 6049.99 6103.394 6142.121 6171.49 6194.528 6213.082 6228.345

ke 0.2 0.228612 0.257224 0.285836 0.314448 0.34306 0.371671 0.400283 0.428895 0.457507 0.486119

ke(L) 0.55 0.5

ke

0.45

p=1

0.4

p=12

0.35

p=1'

0.3

p=12'

0.25 0.2 0

1

2

3

4

5

6

7

8

9

10

Fig. 10.7 Dependence of the equity cost, ke, on leverage level L at different p and fixed kd ¼ 0.16 (means advanced payment of tax of profit)

While in the case of tax on profit payments at the end of the periods, the tilt of the curve of the equity cost, ke, decreases with the increase of the frequency of payment of tax on profit per year p, in case of the advanced payments of tax on profit, the tilt of the curve of the equity cost, ke, increases with the increase of the frequency of payment of tax on profit per year p. These facts should be accounted when developing the company’s dividend policy, because the value of the equity cost, ke, is the economically justified amount of dividends. The company’s dividend policy should be as following: 1. If the company would like to minimize the dividend payments: (a) In case of payments of tax on profit at the ends of periods, they should be as frequently as tax legislation allows.

10.7

Dependence of the Equity Cost, ke, on Leverage Level L

153

ke(L) 0.34 0.32

ke

0.3

p=1

0.28

p=12

0.26

p=12'

0.24

p=1'

0.22 0.2 0

1

2

3

4

Fig. 10.8 Dependence of the equity cost, ke, on leverage level L at different p and fixed kd ¼ 0.16 (means advanced payment of tax of profit) (larger scale)

ke(L) 0.53 0.51

ke

0.49 0.47

p=1

0.45

p=12

0.43

p=1'

0.41

p=12'

0.39 0.37 7

8

9

10

Fig. 10.9 Dependence of the equity cost, ke, on leverage level L at different p and fixed kd ¼ 0.16 (means advanced payment of tax of profit) (larger scale)

(b) In case of advanced payments of tax on profit, they should be as seldom as tax legislation allows. 2. If the company would like to maximize the dividend payments: (a) In case of payments of tax on profit at the ends of periods, they should be as seldom as tax legislation allows. (b) In case of advanced payments of tax on profit, they should be as frequently as tax legislation allows.

154

10.8

10

How Frequently Should Companies Pay Tax on Profit

Dependence of the Equity Cost, ke, and the Weighted Average Cost of Capital, WACC, on Tax on Profit Value t

In this paragraph, we study the dependence of the equity cost, ke, and the weighted average cost of capital, WACC, on tax on profit value. Let us first analyze the dependence of the equity cost, ke, on tax on profit value t at different p and fixed debt cost kd. From Tables 10.17, 10.18, 10.19, 10.20 and Figs. 10.10, 10.11, it is seen that the equity cost, ke, decreases with the increase of on tax on profit value t for all frequencies of payment of tax on profit per year p. While in case of payments at the end of the periods the equity cost, ke, decreases with the increase of the frequency of payments of tax on profit per year p (negative Table 10.17 Dependence of the equity cost, ke, and the weighted average cost of capital, WACC, on tax on profit value at p ¼ 1 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.19 0.18 0.17 0.16 0.15 0.14 0.13 0.12 0.11 0.1

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5263.158 5555.556 5882.353 6250 6666.667 7142.857 7692.308 8333.333 9090.909 10,000

ke 0.24 0.236 0.232 0.228 0.224 0.22 0.216 0.212 0.208 0.204 0.2

Table 10.18 Dependence of the equity cost, ke, and the weighted average cost of capital, WACC, on tax on profit value at p ¼ 12 k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p 12 12 12 12 12 12 12 12 12 12 12

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

WACC 0.2 0.189286 0.178573 0.167859 0.157145 0.146432 0.135718 0.125004 0.11429 0.103577 0.092863

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5283.002 5599.963 5957.383 6363.54 6829.129 7368.227 7999.733 8749.634 9654.672 10,768.54

ke 0.24 0.234573 0.229145 0.223718 0.21829 0.212863 0.207436 0.202008 0.196581 0.191154 0.185726

10.8

Dependence of the Equity Cost, ke, and the Weighted Average Cost. . .

155

Table 10.19 Dependence of the equity cost, ke, and the weighted average cost of capital, WACC, on tax on profit value at p ¼ 1 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p 1 1 1 1 1 1 1 1 1 1 1

i 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16 1.16

WACC 0.2 0.1884 0.1768 0.1652 0.1536 0.142 0.1304 0.1188 0.1072 0.0956 0.084

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5307.856 5656.109 6053.269 6510.417 7042.254 7668.712 8417.508 9328.358 10,460.25 11,904.76

ke 0.24 0.2328 0.2256 0.2184 0.2112 0.204 0.1968 0.1896 0.1824 0.1752 0.168

Table 10.20 Dependence of the equity cost, ke, and the weighted average cost of capital, WACC, on tax on profit value at p ¼ 12 (for advanced payments of tax of profit) k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

wd 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

L 1 1 1 1 1 1 1 1 1 1 1

t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

p 12 12 12 12 12 12 12 12 12 12 12

i 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445 1.012445

WACC 0.2 0.189153 0.178306 0.167459 0.156612 0.145765 0.134918 0.124071 0.113224 0.102377 0.09153

CF 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000 1000

V 5000 5286.726 5608.338 5971.613 6385.21 6860.363 7411.917 8059.911 8832.063 9767.838 10,925.41

ke 0.24 0.234306 0.228612 0.222918 0.217224 0.21153 0.205836 0.200142 0.194448 0.188754 0.18306

tilt of the curve ke(t) increases on modulus), in case of advanced payments of tax on profit t the equity cost, ke, increases with the increase of the frequency of payment of tax on profit per year p (negative tilt of the curve ke(t) increases on modulus). All the equity cost, ke, values in case of advanced payments of tax on profit turns out to be lower than in case of payments at the end of the periods. Let us now analyze the dependence of the weighted average cost of capital, WACC, on tax on profit value at different p and fixed debt cost kd. From Tables 10.17–10.20 and Figs. 10.10, 10.11, it is seen that the weighted average cost of capital, WACC, decreases with the increase of on tax on profit value t for all frequencies of payment of tax on profit per year p. While in case of payments at the end of the periods, the weighted average cost of capital, WACC, decreases with the increase of the frequency of payments of tax on profit per year p (negative tilt of the curve WACC(t) increases on modulus), in case of advanced payments of tax on profit t the weighted average cost of capital, WACC,

156

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How Frequently Should Companies Pay Tax on Profit

ke(t); WACC(t)

ke; WACC

0.24 0.22

WACC1

0.2

WACC12

0.18

ke1

0.16

ke12

0.14

WACC1'

0.12

WACC12' ke1'

0.1

ke12'

0.08 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Fig. 10.10 Dependence of the equity cost, ke, and the weighted average cost of capital, WACC, on tax on profit value at different p and fixed kd ¼ 0.16 (means advanced payment of tax of profit)

ke(t); WACC(t) 0.22 WACC1'

0.2

WACC12'

ke; WACC

0.18

ke1'

0.16

ke12'

0.14

WACC1

0.12

WACC12 ke1

0.1

ke12

0.08 0.5

0.6

0.7

0.8

0.9

1

Fig. 10.11 Dependence of the equity cost, ke, and the weighted average cost of capital, WACC, on tax on profit value at different p and fixed kd ¼ 0.16 (means advanced payment of tax of profit) (larger scale)

increases with the increase of the frequency of payment of tax on profit per year p (negative tilt of the curve WACC(t) increases on modulus). All the weighted average cost of capital, WACC, values in case of advanced payments of tax on profit turns out to be lower than in case of payments at the end of the periods.

10.9

10.9

Discussions

157

Discussions

Let us discuss the main obtained results. 1. The weighted average cost of capital, WACC, decreases with leverage level L at all values of frequency of payments of tax on profit p. With increase of the frequency of payments of tax on profit per year p the weighted average cost of capital, WACC, (1) decreases in the case of payments at the ends of the periods and WACC (2) increases in the case of advanced payments of tax on profit. Note that the weighted average cost of capital, WACC, in the case of advanced payments of tax on profit for all values of frequency of payments turns out to be low than for any values of frequency of payments in the case of payments at the ends of the periods. 2. The weighted average cost of capital, WACC, decreases with increase of debt cost value kd, except the classical Modigliani–Miller case p ¼ 1 and payment at the end of the period. It supports obtained by us result that account of each of two effects (either advanced payments of tax on profit, or more frequent than once per year) leads to dependence of all main financial indicators, such as the weighted average cost of capital, WACC, company value, V, equity cost, ke, etc. on debt cost value kd, while in the classical Modigliani–Miller case these financial indicators DO NOT depend on debt cost value kd. Similar to previous paragraph, it is seen that with increase of the frequency of payments of tax on profit per year p WACC (1) decreases in the case of payments at the ends of the periods and WACC (2) increases in the case of advanced payments of tax on profit. As we will see below, this will lead to increase of company value, V, with increase of p in the case (1) and to decrease of company value, V, with increase of p in the case (2). Note that the weighted average cost of capital, WACC, in the case of advanced payments of tax on profit for all values of frequency of payments turns out to be low than for any values of frequency of payments in the case of payments at the ends of the periods. 3. The company value, V, increases with the increase of leverage level L for all frequencies of payment of tax on profit per year p. While in case of payments at the end of the periods the company value, V, increases with the increase of the frequency of payment of tax on profit per year p, in case of advanced payments of tax on profit, the company value, V, decreases with the increase of the frequency of payment of tax on profit per year p. These facts consistent with the dependence of the weighted average cost of capital, WACC, on the leverage level L. All the company values, V, in case of advanced payments of tax on profit turns out to be bigger than in case of payments at the end of the periods. Thus if the companies tend to increase their capitalization, they should pay tax on profit (1) in advance; (2) as seldom as tax legislation allows. 4. The equity cost, ke, linearly increases with the increase of leverage level L for all frequencies of payment of tax on profit per year p.

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How Frequently Should Companies Pay Tax on Profit

While in the case of tax on profit payments at the end of the periods the tilt of the curve of the equity cost, ke, decreases with the increase of the frequency of payment of tax on profit per year p, in case of the advanced payments of tax on profit, the tilt of the curve of the equity cost, ke, increases with the increase of the frequency of payment of tax on profit per year p. These facts should be accounted when developing the company’s dividend policy, because the value of the equity cost, ke, is the economically justified amount of dividends. The company’s dividend policy should be as following: (a) If the company would like to minimize the dividend payments: In case of payments of tax on profit at the ends of periods, they should be as frequently as tax legislation allows. In case of advanced payments of tax on profit, they should be as seldom as tax legislation allows. (b) If the company would like to maximize the dividend payments: In case of payments of tax on profit at the ends of periods, they should be as seldom as tax legislation allows. In case of advanced payments of tax on profit, they should be as frequently as tax legislation allows. 5. The equity cost, ke, decreases with the increase of on tax on profit value t for all frequencies of payment of tax on profit per year p. While in case of payments at the end of the periods the equity cost, ke, decreases with the increase of the frequency of payments of tax on profit per year p (negative tilt of the curve ke(t) increases on modulus), in case of advanced payments of tax on profit t the equity cost, ke, increases with the increase of the frequency of payment of tax on profit per year p (negative tilt of the curve ke(t) increases on modulus). All the equity cost, ke, values in case of advanced payments of tax on profit turns out to be lower than in case of payments at the end of the periods. The weighted average cost of capital, WACC, decreases with the increase of on tax on profit value t for all frequencies of payment of tax on profit per year p. While in case of payments at the end of the periods, the weighted average cost of capital, WACC, decreases with the increase of the frequency of payments of tax on profit per year p (negative tilt of the curve WACC(t) increases on modulus), in case of advanced payments of tax on profit t the weighted average cost of capital, WACC, increases with the increase of the frequency of payment of tax on profit per year p (negative tilt of the curve WACC(t) increases on modulus). All the weighted average cost of capital, WACC, values in case of advanced payments of tax on profit turns out to be lower than in case of payments at the end of the periods.

References

10.10

159

Conclusions

Brusov and Filatova (2021) have modified the Modigliani–Miller theory for the case of arbitrary frequency of payment of tax on profit t and show that the account of frequency of payment of tax on profit within the Modigliani–Miller theory turns out to be very important and leads to significant consequences: (1) all Modigliani–Miller theorems, statements, and all formulas change; (2) all main financial indicators, such as the weighted average cost of capital, WACC, company value, V and equity cost, ke depend on the frequency of tax on profit payments p; (3) this allows to company choose the number of payments of tax of profit per year, as many, as it is profitable to it (of course, within actual tax legislation). As it was shown in that paper (Brusov and Filatova 2021), more frequent payments of income tax are beneficial for both parties: for the company and for the tax regulator: for the company, this leads to decrease of the cost of attracting capital and thus to an increase in the value of the company, and for the tax regulator, earlier payments are beneficial due to the time value of money. But companies could (and sometimes should) pay tax of profit in advance. In current paper, we use two modifications of the Modigliani–Miller theory simultaneously: (1) frequent payments of income tax; (2) advance payments of income tax and show that this fact changes drastically the main conclusions of the paper (Brusov and Filatova 2021): more frequent payments of income tax are NOT beneficial for the company: this leads to increase of the cost of attracting capital (the weighted average cost of capital, WACC, and the equity cost, ke), and thus to decrease in the value of the company, but for the tax regulator it remains beneficial: earlier payments are beneficial due to the time value of money. Thus, the main conclusions of current chapter are as follows: 1. For payments of tax of profit at the end of periods more frequent payments of income tax are beneficial for both parties: for the company and for the tax regulator: (a) for the company, this leads to decrease of the cost of attracting capital, WACC, and thus to an increase in the value of the company, V, and (b) for the tax regulator, more frequent payments are beneficial due to the time value of money. 2. For payments of tax of profit in advance more frequent payments of income tax are NOT beneficial for the company: this leads to increase of the cost of attracting capital, WACC, and thus to decrease in the value of the company, V, but for the tax regulator it remains beneficial: earlier payments are beneficial due to the time value of money.

References Berk J, DeMarzo P (2007) Corporate finance. Pearson–Addison Wesley, Boston Brealey R, Myers S, Allen F (2005) Principle of corporate finance, 7th edn. McGraw Hill, New York

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Brusov P (2018a) Editorial: Introduction on special issue on the banking system and financial markets of Russia and other countries: problems and prospects. J Rev Glob Econ 7:i–vi Brusov P (2018b) Editorial. J Rev Glob Econ 7:i–v Brusov P, Filatova T (2021) The Modigliani – Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198. https://doi.org/10.3390/math9111198 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2018a) New meaningful effects in modern capital structure theory. J Rev Glob Econ 7:104–122. https://doi.org/10.6000/ 1929-7092.2018.07.08 Brusov P, Filatova T, Orehova N, Eskindarov M (2018b) Modern corporate finance, investments, taxation and ratings. Springer Nature, Switzerland, 571 p. monograph Brusov PN, Filatova TV, Orekhova NP (2018c) Modern corporate finance and investments. Monograph, Knorus Publishing House, Moscow, 517 p Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018d) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Glob Econ 7:360–376. https://doi.org/10.6000/ 1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018e) Rating: new approach. J Rev Glob Econ 7:37–62. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018f) A “golden age” of the companies: conditions of its existence. J Rev Glob Econ 7:88–103. https://doi.org/10.6000/1929-7092. 2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018g) Rating methodology: new look and new horizons. J Rev Glob Econ 7:63–87. https://doi.org/10.6000/1929-7092.2018.07.06 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Glob Econ 8:437–448. https://doi.org/10.6000/ 1929-7092.2019.08.37 Brusov P, Filatova T, Orehova N (2020a) Ratings: critical analysis and new approaches of quantitative and qualitative methodology. Springer Nature, Switzerland. Monograph. https:// www.springer.com/de/book/9783030562427 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Application of the Modigliani–Miller Theory, modified for the case of advance payments of tax on profit, in rating methodologies. J Rev Glob Econ 9:282–292. https://doi.org/10.6000/1929-7092.2020.09.28 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020c) Modification of the Modigliani–Miller Theory for the case of advance payments of tax on profit. J Rev Glob Econ 9:257–267. https://doi.org/10.6000/1929-7092.2020.09.25 De Marzo PM (1988) An extension of the Modigliani–Miller theorem to stochastic economies with incomplete markets and interdependent securities. J Econ Theory 45(2):353–369 Farber A, Gillet R, Szafarz A (2007) A general formula for the WACC. Int J Bus 11(2):211–218 Fernandez P (2007) A general formula for the WACC: a comment. Int J Bus (this issue) Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018) Ratings of the long-term projects: new approach. J Rev Glob Econ 7:645–661. https://doi.org/10.6000/1929-7092.2018.07.59 Hamada R (1969) Portfolio analysis, market equilibrium, and corporate finance. J Financ:13–31 Harris R, Pringle J (1985) Risk–adjusted discount rates – extension form the average–risk case. J Financ Res 8(3):237–244 Miles JA, Ezzel JR (1980) The weighted average cost of capital, perfect capital markets and project life: a clarification. J Financ Quant Anal 15(3):719–730 Miller M (1977) Debt and taxes. J Financ 32(2):261–275. https://doi.org/10.1111/j.1540-6261. 1977.tb03267.x

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Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Ross S, Westerfiel R, Jaffee D (2005) Corporate finance, 7th edn. McGraw Hill, New York

Chapter 11

Generalization of the Modigliani–Miller Theory for the Case of Variable Profit

For the first time, we have generalized the world-famous theory by Nobel Prize winners Modigliani and Miller for the case of variable profit, which significantly extends the application of the theory in practice, specifically in business valuation, ratings, corporate finance, etc. We demonstrate that all the theorems, statements and formulae of Modigliani and Miller are changed significantly. We combine theoretical and numerical (by MS Excel) considerations. The following results are obtained: 1. Discount rate for leverage company changes from the weighted average cost of capital, WACC, to WACC
g (where g is growing rate), for a financially independent company from k0 to k0
g. This means that WACC and k0 are no longer the discount rates as it takes place in case of classical Modigliani–Miller theory with constant profit. WACC grows with g, while real discount rates WACC
g and k0
g decrease with g. This leads to an increase of company capitalization with g. 2. The tilt angle of the equity cost ke(L ) grows with g. This should change the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. 3. A qualitatively new effect in corporate finance has been discovered: at rate g < g* the slope of the curve ke(L) turns out to be negative, which could significantly alter the principles of the company’s dividend policy.

11.1

Introduction

The original theory by Nobel Prize Winners Modigliani and Miller (1958, 1963, 1966) has been modified by many authors, and we shortly discuss some of these. A few important modifications have been done by the authors of this monograph (Brusov et al. 2015, 2018, 2020a, 2021a; Filatova et al. 2008; Brusov and Filatova 2021), who created the general theory of capital cost and capital structure, the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_11

163

164

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

Brusov–Filatova–Orekhova (BFO) theory, which generalized the Modigliani–Miller theory for the case of companies of arbitrary age (and arbitrary lifetime), as well as for the case of advance payments of tax on profit (Brusov et al. 2020b), for rating needs (Brusov et al. 2020c), as well as for variable debt cost (Brusov et al. 2021b). Note that a stochastic extension of the Miller–Modigliani theory has been created by some authors (Sethi et al. 1991; DeMarzo 1988). In the current chapter, for the first time, we have generalized the world-famous theory by Nobel Prize winners Modigliani and Miller for the case of variable profit (Brusov et al. 2021b), which significantly extends the application of the theory in practice, specifically in business valuation, ratings, investments and in other areas of the economy and of finance. We consider the case of growing profit as well as decreasing profit and show that all theorems, all statements by Modigliani and Miller (and all their main formulas) are changed significantly. Within the new Generalized Modigliani–Miller theory (GMM theory), we study the dependence of the weighted average cost of capital, WACC, the equity cost, ke, the discount rate, i, and the capitalization of the company, V, on leverage level L. Some important results have been obtained, which allows for the development of a new approach to financial policy and financial strategy for the company. Some of these are as follows: • The discount rate for a leveraged company changes from the weighted average cost of capital, WACC, to WACC
g (where g is growth rate), for an unleveraged company from k0 to k0
g. This means that WACC and k0 are no longer the discount rates they are in the case of classical Modigliani–Miller theory with constant profit. • All curves WACC(L ) for different values of g start from one point, k0. They decrease with L at g < k0 and increase at g > k0. It turns out that WACC grows with g, while real discount rates WACC
g and k0
g decrease with g. This leads to an increase of company capitalization along with g: V ¼ CF/WACC
g. Knowing the correct value of the discount rate allows for management of companies’ financial flows. • The equity cost, ke, which grows linearly with the leverage level, increases with g: the tilt angle ke(L ) grows along with the growth rate g. A qualitatively new effect in corporate finance has been discovered: at rate g < g* the slope of the curve ke(L ) turns out to be negative. Two these effects, which are absent in classical Modigliani–Miller theory, could significantly alter the principles behind the company’s dividend policy, because the economically justified value of dividends is equal to equity cost. The final effect is similar to the qualitatively new effect in corporate finance that was discovered by Brusov–Filatova–Orekhova within the BFO theory: the abnormal dependence of equity cost on leverage level at tax on profit rate T, which exceeds rate value T*. This discovery also significantly alters the principles behind the company’s dividend policy. The current paper, contrary to papers on (stochastically) growing cash flows, can be directly applied for the calculation of all the main indicators for companies. The

11.1

Introduction

165

results obtained will have applications in corporate finance, business valuation, ratings, etc. The structure of the chapter is as follows: 1. We give an introduction to the traditional approach of the Modigliani–Miller theory and to its modifications. (a) We generalize the Modigliani–Miller theory to the case of variable profit and obtain generalized Modigliani–Miller theorems, as well as new formulae for the weighted average cost of capital, WACC, equity cost, ke, discount rate, i, and capitalization of the company, V. (b) Within the new Generalized Modigliani–Miller theory (GMM theory), we numerically study (with MS Excel) the dependence of the main financial indicators of the company (WACC, ke, i, V ) on leverage level L. (c) We discuss the results obtained and based on these, we arrive at some important conclusions. Below we discuss the problem of capital structure. Capital structure is the relationship between the debt and the equity capital of the company. Does capital structure impact the main financial indicators of the company, such as the cost of capital, profit, value of the company, etc., and, if so, how? The choice of an optimal capital structure, i.e., a capital structure that maximizes the company’s capitalization, V, and minimizes the weighted average cost of capital, WACC, is one of the most important problems to be solved by the financial manager and senior management of a company. The first quantitative study of the impact of company capital structure on its financial indicators was performed by Modigliani and Miller (1958). Before 1958, the traditional approach, based on empirical data analysis, was used.

11.1.1 The Traditional Approach The traditional approach supposes that the weighted average cost of capital, WACC, and the associated company capitalization value, V ¼ CF/WACC, depend on the capital structure [the level of leverage (L )]. The reason for this is that the debt cost always turns out to be lower than the equity cost because the first one has lower risk, due to the fact that, in the event of bankruptcy, creditor claims are met prior to shareholders’ claims. Thus, if the firm increases the share of lower-cost debt capital in the overall capital structure, this will lead to a lower weighted average cost of capital and WACC up to the limit, which does not cause violation of financial sustainability or result in increase of bankruptcy risk. Investor profitability is required, and the cost of equity grows with the leverage level; however, its growth has not lead to compensation of benefits from the increasing use of lower cost debt capital. Therefore, at a low leverage level

166

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

WACC decreases with the increase of leverage L ¼ D/S and company capitalization increases. At a high leverage level, when the risk of bankruptcy becomes higher, WACC may increase with the increase of leverage L and company capitalization decreases. Thus, the trade-off between advantages of debt financing at a low leverage level and its shortcomings at a high leverage level forms an optimal capital structure, which maximizes the company capitalization, V, and minimizes the weighted average cost of capital, WACC. The traditional approach has existed up to 1958, when the first quantitative theory by Modigliani and Miller has appeared (Мodigliani and Мiller 1958).

11.1.2 Modigliani–Miller Theory Without Taxes In their first paper, Modigliani and Miller (ММ) (1958) under a lot of assumptions (there are no taxes, no bankruptcy costs, no transaction costs, perfect financial markets exist with symmetry information, equivalence in borrowing costs for both companies and investors, etc.), came to the conclusions that choosing of the ratio between the debt and equity capital does not affect company value as well as capital costs (Fig. 11.1). These conclusions were fundamentally different from the conclusions of the traditional approach.

Fig. 11.1 Dependence of equity capital cost, debt cost and WACC on leverage in Modigliani– Miller theory without taxes (t ¼ 0) and with taxes (t 6¼ 0)

11.1

Introduction

167

Modigliani and Miller, under the above assumptions, have analyzed the impact of financial leverage, assuming the absence of any taxes (on corporate profit as well as individual one). They have formulated and proven two following theorems. Without taxes, the total cost of any company is determined by the value of its EBIT (Earnings Before Interest and Taxes), discounted with fixed rate k0, corresponding to group of business risk of this company: V ¼ V0 ¼

EBIT k0

ð11:1Þ

This leads to the following expressions for WACC: WACC ¼ k 0

ð11:2Þ

Note that k0 here and below is the weighted average cost of capital, WACC for an unleveraged company. For a leveraged company, k0 is the equity cost (and weighted average cost of capital, WACC) at zero leverage level (L ¼ 0). From the first Modigliani–Miller theorem (Modigliani and Miller 1958), it is easy to derive an expression for the equity capital cost: WACC ¼ k 0 ¼ k e we þ k d wd :

ð11:3Þ

Finding from here ke, one gets ke ¼

k ðS þ D Þ k0 w D D
k d ¼ k 0 þ ðk 0
k d Þ
kd d ¼ 0 S S S we we

¼ k0 þ ðk0
kd ÞL

ð11:4Þ

Here, D S D kd , wd ¼ DþS S ke , we ¼ DþS L ¼ D/S WACC

value of debt capital of the company; value of equity capital of the company; cost and fraction of debt capital of the company; cost and fraction of equity capital of the company; financial leverage weighted average cost of capital.

Thus, we come to second theorem of Modigliani–Miller theory regarding the equity cost of a leveraged company (Мodigliani and Miller 1958). The equity cost of a leveraged company (ke) could be found as k0 (the cost of equity of an unleveraged company with the same asset risk), plus premium for risk, which is equal to the product of difference (k0
kd) and leverage level L:

168

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

k e ¼ k0 þ Lðk 0
kd Þ

ð11:5Þ

Formula (11.5) shows that equity cost of the company increases linearly with is leverage level.

11.2

Some Modifications of Modigliani–Miller Theory

11.2.1 Modigliani–Miller Theory with Taxes In 1963 Мodigliani and Miller (1963) accounted for the effect of corporate taxes and obtained the following result for the capitalization value of a leveraged company, V: V ¼ V 0 þ DT,

ð11:6Þ

where V0 is the value of unleveraged company, D is the debt value and T is the corporate taxes on profit rate. The value of a leveraged company is equal to the value of the unleveraged company with the same asset risk, plus the value of tax shield arising from financial leverage, which is equal to the product of corporate income tax rate (T ) and the debt value (D). The following section presents the formal derivation result for WACC and the equity capital cost (ke) of the company with consideration of corporate taxes.

11.2.1.1

Weighted Average Cost of Capital, WACC V ¼ V 0 þ DT V ð1
wd T Þ ¼ V 0

ð11:7Þ

CF CF ð1
wd T Þ ¼ WACC k0

ð11:8Þ

WACC ¼ k 0  ð1
wd t Þ

ð11:9Þ

The above is a very important formula for weighted average cost of capital (WACC) and is one of the main results of the Modigliani–Miller theory with taxes.

11.2.1.2

Equity Cost

Let us derive formula for equity cost. On definition of the weighted average cost of capital with “tax shield,” we have:

11.2

Some Modifications of Modigliani–Miller Theory

WACC ¼ k0 we þ k d wd ð1
T Þ:

169

ð11:10Þ

Equating Eqs. (11.9) and (11.10), one gets: k 0 ð1
wd T Þ ¼ k 0 we þ kd wd ð1
T Þ

ð11:11Þ

and from here, for equity cost, we develop the following expression: ð1
w d T Þ w 1 w D
k d d ð1
T Þ ¼ k 0
k 0 d T
k d ð1
T Þ we we S we we DþS D D ¼ k0
k 0 T
kd ð1
T Þ ¼ k0 þ Lð1
T Þðk0
k d Þ: S S S

ke ¼ k0

ð11:12Þ

Therefore, we see the following theorem obtained by Мodigliani and Miller in 1963: Equity cost of leveraged company (ke) could be found as equity cost of unleveraged company (k0) with the same business risk, plus premium for risk, the value of which is equal to the triple product of difference between the cost of capital for an unleveraged firm and cost of debt (k0
kd), leverage level (L ¼ DSÞ and tax corrector (1 – T ). It should be noted that Formula (11.12) is different from Formula (11.5) without tax only by the multiplier (1 – T ), suggesting the tax benefit of debt will lower the cost of equity as well. Analysis of Formulas (11.5), (11.9) and (11.12) leads to the following conclusions. When leverage grows: 1. The value of company increases. (a) The weighted average cost of capital WACC decreases from k0 (at L ¼ 0) up to k0(1
T ) (at L ¼ 1) (when the company is funded solely by borrowed funds). (b) Equity cost increases linearly from k0 (at L ¼ 0) up to 1 (at L ¼ 1). Within their theory, Мodigliani and Miller (1963) came to the following conclusions regarding the growth of financial leverage (Fig. 11.1).

11.2.2 Taking into Account Market Risk: Hamada Model In 1969 Hаmаdа united CAPM (Capital Asset Pricing Model) and Modigliani– Miller theory with taxes. For the equity cost of a leveraged company, the following formula has been derived, which includes financial and business risk of company:

170

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

k e ¼ kF þ ðk M
kF ÞbU þ ðkM
kF ÞbU

D ð1
T Þ, S

ð11:13Þ

here bU is the CAPM βeta of the unleveraged company with the same asset risk as the leveraged company under consideration. Formula (11.13) represents the cost of equity (ke) for a leveraged firm as a sum of three components: risk-free rate (kF), risk premium for business/asset risk (kM
kF)bU and risk premium for financial risk ðkM
kF ÞbU DS ð1
T Þ. If the company does not have borrowing (D ¼ 0), the financial risk factor will be equal to zero (the third term is drawn to zero), and equity holder will only require the premium for business/asset risk.

11.2.3 The Account of Corporate and Individual Taxes (Miller Model) In their second article, Modigliani and Miller (1963) only considered corporate taxes benefit of debt but did not take into account the effect of individual investors’ income taxes. In 1977 Merton Miller (1977) have developed such a model, showing the influence of financial leverage on the capitalization of the company when the effects of both corporate and individual taxes are accounted for, i.e., there is double taxation on corporate earnings. We will use the following definitions: TC—corporate taxes rate and TS—a weighted average value of effective taxes rates on dividends and capital gains on shares. With the same other assumptions that have been made for Modigliani–Miller models previously, the unleveraged company value can be determined as follows: VU ¼

EBITð1
T C Þð1
T S Þ : k0

A term (1
TS) accounts for the individual taxes. The numerator in (1) indicates which part of the operating company’s profit remains in the possession of the investors, after the company earnings are taxed twice: by corporate taxes and individual taxes. Since individual taxes reduce investors’ residual income, it also reduces the overall assessment of the unleveraged company value.

11.2.4 Brusov–Filatova–Orekhova (BFO) Theory One of the most serious limitations of the Modigliani–Miller theory is the suggestion about the perpetuity of the companies. In 2008, Brusov–Filatova–Orekhova (Filatova et al. 2008) have considered this limitation and showed that accounting

11.2

Some Modifications of Modigliani–Miller Theory

171

for the finite lifetime length (or arbitrary age n) of the company leads to significant changes of all Modigliani–Miller results (Мodigliani and Мiller 1958, 1963, 1966): capitalization of the company V is changed, as well as the equity cost, ke, and the weighted average cost of capital, WACC, in the presence of corporative taxes. Moreover, a number of new findings for corporate finance, in the Brusov–Filatova– Orekhova theory (Brusov et al. 2015, 2018; Filatova et al. 2008), are absent in Modigliani–Miller theory. The formula for weighted average cost of capital (WACC) for the company with arbitrary age n, as derived by Brusov–Filatova–Orekhova, has the following form (Brusov et al. 2015, 2018; Filatova et al. 2008): 1
ð1 þ WACCÞ
n 1
ð1 þ k0 Þ
n ¼ WACC k0 ½1
wd T ð1
ð1 þ kd Þ
n Þ

ð11:14Þ


D Here, wd ¼ DþS ) is the debt share; k0 is the cost of capital for an unleveraged firm with the same asset risk; kd the cost of debt; T denotes the corporate taxes rate; and n stands for the firm’s lifetime length (age). A perpetuity (Modigliani–Miller) limit could be easily obtained from Eq. (11.14), substituting n ! 1. A lot of meaningful effects have been discovered within the BFO theory: these effects are absent within MM theory. BFO theory has challenged some main existing principles of financial management: among them the trade-off theory, which has been prevailing for many decades and established the foundation to claim the existence of an optimal capital structure for firms. However, BFO theory has proven the bankruptcy of trade-off theory (Brusov et al. 2015, 2018).

11.2.5 The General WACC Formula A more general formula for WACC than in the famous Modigliani–Miller theory (MM) has been derived and discussed by a few authors in 2006–2007 (Farber et al. 2006; Fernandez 2006; Berk and DeMarzo 2007; Harris and Pringle 1985). It takes the following form (Eq. 18 in Farber et al. 2006) WACC ¼ k0 ð1
wd t Þ
kd twd þ kTS twd

ð11:15Þ

where k0 is the required return on unleveraged company, kd is the required return on its debt, kTS is the expected return on the tax shield and T is the corporate taxes rate. This formula is derived from the definition of the weighted average cost of capital and the balance sheet identity (for a similar presentation, see Berk and DeMarzo 2007). At any point in time, it should therefore be verified, regardless of whether returns are annually or continuously compounded.

172

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

Practical applicability of Eq. (11.15) (while it is fairly general) requires additional conditions. Indeed, when the WACC is constant over time, the value of a leveraged company can be computed by discounting with the WACC of the unleveraged free cash flows. Therefore, it is interesting to consider the special cases when WACC is constant. The resulting formulas can also be found in textbooks (Brealey et al. 2005; Ross et al. 2005). It was assumed by Modigliani and Miller in 1963 that the debt value D is constant. As the expected after-tax cash flow of the unleveraged company is fixed, V0 is also constant. By this assumption, kTS ¼ kD and the value of the tax shield is TS ¼ tD. Therefore, the capitalization of the leveraged company V is a constant, and the general WACC Formula (11.15) simplifies to a constant WACC: WACC ¼ k0 ð1
wd T Þ However, our opinion is that “classical” Modigliani–Miller (MM) theory, which suggests that the expected returns on the debt kd and the tax shield kTS are equals (because both of them have debt nature), is much more reasonable and in our paper, we modify the “classical” Modigliani–Miller (MM) theory, which is still widely used in practice.

11.2.6 Trade-Off Theory The world-famous trade-off theory has been considered the cornerstone in the solution of the problem of optimal capital structure for a company for many decades and is still used today for decision analysis on capital structure. Below we give two examples. Frank, M., & Goyal, V., in their 2009 paper, “examines the relative importance of many factors in the capital structure decisions of publicly traded American firms from 1950 to 2003. The most reliable factors for explaining market leverage are: median industry leverage (+ effect on leverage), market–to–book assets ratio (
), tangibility (+), profits (
), log of assets (+), and expected inflation (+).” In addition, the authors have found that “dividend–paying firms tend to have lower leverage. When considering book leverage, somewhat similar effects are found. However, for book leverage, the impact of firm size, the market–to–book ratio, and the effect of inflation are not reliable.” The empirical evidence seems to be reasonably consistent with some versions of the trade-off theory of capital structure. Serrasqueiro, Z., & Caetano, A., in 2015, analyzed “to what extent decisions on the capital structure of small and medium–sized enterprises (SMEs) are closer to the assumptions of trade–off theory or to the assumptions of hierarchy theory. They used a sample of small and medium–sized enterprises located in the Portuguese hinterland, using dynamic LSDVC as the valuation method, and the empirical evidence suggests that the most profitable and oldest SMEs are less leveraged, confirming Pecking Order Theory ‘s forecasts. Larger SMEs are leveraging more borrowing,

11.2

Some Modifications of Modigliani–Miller Theory

173

confirming the predictions of trade–off theory and hierarchy theory. In addition, SMEs are significantly adjusting their current debt levels towards the optimal debt ratio, which is consistent with the predictions of the compromise theory. It was concluded that theories of compromise and hierarchy are not mutually exclusive in explaining capital structure decisions of small and medium–sized enterprises.” However, the bankruptcy of trade-off theory has been proven by Brusov et al. in 2013 (Brusov et al. 2018). They have shown that risky debt financing (and growing credit rate near the bankruptcy) in contrast to waiting results does not lead to growing of weighted average cost of capital, WACC, which still decreases with leverage. This means the absence of minimum in the dependence of WACC on leverage level as well as the absence of maximum in the dependence of company capitalization on leverage. Thus, the well-known trade-off theory lacks an optimal capital structure. The explanation for this fact was made by Brusov et al. in 2013 (Brusov et al. 2018) by analyzing the dependence of the cost of equity capital on the leverage level on the assumption that debt capital is risky. Modigliani–Miller have considered tax shields from the interest on debt can increase the value of companies. In 1980, De Angelo and Masulis moved further in the theoretical examination of tax shields. They have noted that there are tax deductibles for companies other than debt to reduce their corporate tax burden, and debt and non-debt tax shields should be accounted for. Depreciation, investment tax credits, or net loss carryforwards could represent examples of such kind of non-debt tax shields. The first to test for these tax effects (suggested by DeAngelo and Masulis 1980) has been carried out by Bradley et al. (1984). In contrast to the prediction in De Angelo and Masulis (1980), by regressing company-specific debt-to-value ratios on non-debt tax shields, they have shown that debt is positively related to non-debt tax shields as measured by depreciation and investment tax credits. Titman and Wessels in 1988 found that “their results do not provide support for an effect on debt ratios arising from non–debt tax shields. . .” It was pointed out in 2003 by Graham, if a company invests heavily and uses debt financing to invest, a positive relation between such proxies for non-debt tax shield and debt may result. A mechanical positive relation of this type overwhelms and renders any substitution effects between debt and non-debt tax shields. The original theory by Nobel Prize Winners Modigliani and Miller (1958, 1963, 1966) has been modified by many authors and above, we shortly discussed some of them. In next paragraph, we will generalize for the first time the Modigliani–Miller theory for the case of variable profit.

174

11.3

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

Generalization of the Modigliani–Miller Theory for the Case of Variable Profit

11.3.1 Modigliani–Miller Theory Without Taxes 11.3.1.1

Company Value, V

In this part, we generalize for the first time the world-famous theory by Nobel Prize winners Modigliani and Miller for the case of variable profit. Let us calculate capitalization for leverage company, assuming profit per period grows with growth rate g. V¼

CFð1 þ gÞ CFð1 þ gÞ2 CF þ þ þ 2 1 þ WACC ð1 þ WACCÞ ð1 þ WACCÞ3

ð11:16Þ

Here, CF is the annual profit of company, WACC is the weighted average cost of capital. This is geometric progression with denominator b g¼

1þg 1 þ WACC

ð11:17Þ

Summing Eq. (11.16), one gets V¼

CF 1  1 þ WACC 1
1þg

¼

1þWACC

CF WACC
g

ð11:18Þ

Let us calculate the capitalization for a financially independent company 2

V0 ¼

CFð1 þ gÞ CFð1 þ gÞ CF þ þ þ 1 þ k 0 ð1 þ k 0 Þ2 ð1 þ k 0 Þ3

ð11:19Þ

This is geometric progression with denominator b g¼

1þg 1 þ k0

ð11:20Þ

Summing Eq. (11.19), one gets V0 ¼

CF 1 CF  ¼ 1 þ k 0 1
1þg k 0
g

ð11:21Þ

1þk 0

The original first theorem by Modigliani–Miller changes by the following one:

11.3

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

175

Without taxes, the total cost of any company with growing profit with rate g is determined by the value of its EBIT—Earnings Before Interest and Taxes— discounted with fixed rate k0
g, corresponding to group of business risk of this company: V ¼ V0 ¼

EBIT k0
g

ð11:22Þ

Note, that this formula for company capitalization is similar to one of the Gordon Growth Model (GGM), which assumes a company is perpetuity and pays dividends per share that increase at a constant rate. To estimate the value of a stock, the model takes the infinite series of dividends per share and discounts them back into the present using the required rate of return. P¼

DPS1 ke
g

ð11:23Þ

where p—value of stock;DPS1—expected Dividends 1 year from now (next period); ke—required rate of return for equity investors (equity cost);g—growth rate in dividends forever. Note that, despite some formal similarities, these are not the same model. In our model, CF is the company’s annual income (not dividends), and we study all the financial indicators of the company (WACC, V, ke, i, etc.), all of them (except V ) are absent in Gordon’s growth model.

11.3.1.2

The Weighted Average Cost of Capital, WACC

Below we calculate the WACC value. V ¼ V0 CF CF ¼ WACC
g k 0
g WACC ¼ k 0

ð11:24Þ

It is seen that the WACC value turns out to be the same as in case of classical Modigliani–Miller theory with constant profit.

11.3.1.3

Equity Cost, ke

It is easy to show that the expression for equity cost turns out to be the same as in case of classical Modigliani–Miller theory with constant profit

176

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

ke ¼ k0 þ ðk 0
kd ÞL:

ð11:25Þ

Therefore, as we show in the case of the Modigliani–Miller theory without taxes, accounting of variable profit changes the first of three Modigliani–Miller theorems only. As we show below, in the case of the Modigliani–Miller theory with taxes (practically interesting case), all three Modigliani–Miller theorems (concerning value of the company, V, its WACC and its equity cost, ke) change significantly.

11.3.2 Modigliani–Miller Theory with Taxes 11.3.2.1

The Weighted Average Cost of Capital, WACC

The following result is obtained by Мodigliani and Miller (1963): The value of leveraged company is equal to the value of the company of the same risk group used no leverage, increased by the value of tax shield arising from financial leverage, and equal to the product of rate of corporate income tax T and the value of debt D. V ¼ V 0 þ DT

ð11:26Þ

From here it is easy to get expression for WACC (see above). V ð1
wd T Þ ¼ V 0

ð11:27Þ

CF CF ð1
wd T Þ ¼ WACC k0

ð11:28Þ

WACC ¼ k 0  ð1
wd t Þ

ð11:29Þ

Under variable profit, as we have seen above in the case of Modigliani–Miller theory without taxes, one has the same statement for capitalization, but discount rates are different: WACC ! WACC
g k0 ! k0
g

ð11:30Þ

CF CF ð1
wd T Þ ¼ WACC
g k0
g

ð11:31Þ

WACC
g ¼ ðk0
gÞ  ð1
wd T Þ

ð11:32Þ

WACC ¼ ðk 0
gÞ  ð1
wd T Þ þ g

ð11:33Þ

This expression for WACC is different from the original result for WACC by Мodigliani and Miller (Eq. 11.29):

11.3

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

177

WACC ¼ k0  ð1
wd T Þ In order to understand the character of dependence of WACC(L ), let us calculate the derivative of WACC with respect to L using Eq. (11.33): WACC ¼ ðk 0
gÞ  ð1
wd T Þ þ g ¼ k0
ðk0
gÞ  T  WACC0L ¼


ð k 0
gÞ  T ð1 þ LÞ2

L 1þL

ð11:34Þ ð11:35Þ

At g < k0 the derivative WACC0L is negative, thus WACC(L) decreases with L, while at the g > k0 derivative WACC0L is positive and WACC(L ) increases with L. We will see these effects in Sect. 11.4 under numerical calculations.

11.3.2.2

The Company Value, V

For the company value from the Formulas (11.30) and (11.32), we get the following formula V¼

CF CF ¼ WACC
g ðk 0
gÞ  ð1
wd t Þ

ð11:36Þ

We should note that in case of growing rate g equals to equity cost (at L ¼ 0) k0, the company value V becomes infinite. This is the limitation of the perpetuity Modigliani–Miller theory. As we will see in our future publications, such kind of restriction is absent for companies of finite ages when we consider the generalization of BFO theory for the case of variable profit.

11.3.2.3

Equity Cost, ke

Let us calculate the equity cost ke. From Eq. (11.33), we have for the weighted average cost of capital, WACC, WACC ¼ ðk 0
gÞ  ð1
wd T Þ þ g

ð11:37Þ

As well according to the definition of the weighted average cost of capital, considering the “tax shield” we have WACC ¼ ke we þ k d wd ð1
T Þ Equating Eqs. (11.37) and (11.38), we obtain

ð11:38Þ

178

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

ðk 0
gÞ  ð1
wd T Þ þ g ¼ ke we þ kd wd ð1
T Þ

ð11:39Þ

whence we get the following expression for the cost of equity: k e ¼ WACCð1 þ LÞ
Lk d ð1
T Þ ¼ k 0 þ L  ½ðk 0
kd Þ  ð1
T Þ þ gT 

ð11:40Þ

This expression is different from Formula (11.12) in case of the classical Modigliani–Miller theory with tax. Thus, we have the following modified statement for equity cost within Generalized Modigliani–Miller theory. Equity cost of leverage company ke could be found as equity cost of unleveraged company k0 of the same group of risk, plus premium for risk, the value which is equal to production of leverage level L on sum of production of difference (k0
kd) on tax corrector and gT: k e ¼ k0 þ L½ðk0
kd Þð1
T Þ þ gT 

ð11:41Þ

The Qualitatively New Effect in Corporate Finance From the Formula (11.41), it is seen that with decrease of g tilt angle of straight ke(L ) decreases and at some value of g ¼ g* ke ¼ k0

ð11:42Þ

Let us find g* value. Equating ketok0 in Formula (11.41), one has L½ðk0
k d Þð1
T Þ þ gT  ¼ 0 and from here, we find g* g ¼


ðk 0
k d Þð1
T Þ T

ð11:43Þ

It is clear that at with growth rate g < g* the slope of the straight ke(L ) turns out to be negative. Thus, the qualitatively new effect in corporate finance has been discovered: at growth rate g < g* the slope of the straight ke(L ) turns out to be negative. This effect, which is absent in the classical Modigliani–Miller theory, could significantly alter the principles of the company’s dividend policy. Rating methodology takes into account the company’s dividend policy, thus the results obtained could change the credit rating of an issuer. We have shown in Sect. 11.3, that in case of the Modigliani–Miller theory with taxes, generalized for the case of variable profit all three Modigliani–Miller statements (concerning value of the company, V, the weighted average cost of capital,

11.4

Results and Discussions

179

WACC, and its equity cost, ke) change significantly. The consequences of such changes will be investigated numerically in the next part.

11.4

Results and Discussions

In this section, we study numerically (within Microsoft Excel) the dependence of the weighted average cost of capital, WACC, discount rate, i, company value, V, and equity cost, ke, on leverage level L in the Generalized Modigliani–Miller theory (GMM theory) at two values of equity cost k0 (0.2 and 0.3) and different values of growth rate, g. The obtained results are discussed.

11.4.1 Dependence of WACC on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.2 and Different Values of g (0.4; 0.3; 0.2; 0.0; 20.2; 20.3; 20.4) We study first the dependence of the weighted average cost of capital, WACC, on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and different values of g (0.4; 0.3; 0.2; 0.0;
0.2;
0.3;
0.4). The results of Table 11.1 are shown in Fig. 11.2. From Fig. 11.2, it is seen that all curves WACC(L ) for different g start from one point k0, in this case from point (0; 0.2). They decrease with leverage level L at g < 0.2 (at g ¼ 0; 0.2;
0.3;
0.4) and increase at g > 0.2 (at g ¼ 0.3; 0.4). The curves WACC(L) increase with growth rate, g. Note that cutoff value of g, which Table 11.1 Dependence of WACC on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and g ¼ 0.4; 0.3; 0.2; 0.0;
0.2;
0.3;
0.4 L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

WACC g ¼ 0.4 0.20000 0.22000 0.22667 0.23000 0.23200 0.23333 0.23429 0.23500 0.23556 0.23600 0.23636

g ¼ 0.3 0.20000 0.21000 0.21333 0.21500 0.21600 0.21667 0.21714 0.21750 0.21778 0.21800 0.21818

g ¼ 0.2 0.20000 0.20000 0.20000 0.20000 0.20000 0.20000 0.20000 0.20000 0.20000 0.20000 0.20000

g¼0 0.20000 0.18000 0.17333 0.17000 0.16800 0.16667 0.16571 0.16500 0.16444 0.16400 0.16364

g ¼
0.2 0.20000 0.16000 0.14667 0.14000 0.13600 0.13333 0.13143 0.13000 0.12889 0.12800 0.12727

g ¼
0.3 0.20000 0.15000 0.13333 0.12500 0.12000 0.11667 0.11429 0.11250 0.11111 0.11000 0.10909

g ¼
0.4 0.20000 0.14000 0.12000 0.11000 0.10400 0.10000 0.09714 0.09500 0.09333 0.09200 0.09091

180

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

WACC(L),ko=0.2 0.25 0.23 0.21

WACC

0.19

g=0.4

0.17

g=0.3

0.15

g=0.2

0.13

g=0 g=-0.2

0.11

g=-0.3

0.09

g=-0.4

0.07 0.05 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 11.2 Dependence of WACC on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and g ¼ 0; 0.2; 0.3; 0.4

separate increasing curves WACC(L) from decreasing ones, is equal to k0 ¼ 0.2, and WACC is constant at g ¼ k0 and equal to k0. Below we check this observation at different value of k0 (k0 ¼ 0.3).

11.4.2 Dependence of the Weighted Average Cost of Capital WACC on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.3 and Different Values of g Let us study the dependence of WACC on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and different values of g. The results of Table 11.2 are shown in Fig 11.3. From Fig. 11.3, it is seen that all curves WACC(L ) for different g start from one point, k0, in this case from (0; 0.3). They decrease with L at g < 0.3 (at g ¼ 0; 0.2;0.3;
0.4) and increase at g > 0.3 (at g ¼ 0.4). Note that the cutoff value of g, which separate increasing curves WACC(L ) from decreasing ones as in above case, is equal to k0 (k0 ¼ 0.3), and WACC is constant at g ¼ k0 and equal to k0. Therefore, our observation that this conclusion is valid at different value of k0 is right. From Figs. 11.2 and 11.3, it is seen that all curves WACC(L ) for different g start from one point, k0. They decrease with L at g < k0 and increase at g > k0. It turns out that WACC grows with g, while, as we will see below, real discount rates WACC – g

11.4

Results and Discussions

181

Table 11.2 Dependence of WACC on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and g ¼ 0; 0.2;
0.3;
0.4 L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

WACC g ¼ 0.4 0.300 0.310 0.313 0.315 0.316 0.317 0.317 0.318 0.318 0.318 0.318

g ¼ 0.3 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300 0.300

g ¼ 0.2 0.300 0.290 0.287 0.285 0.284 0.283 0.283 0.283 0.282 0.282 0.282

g¼0 0.300 0.270 0.260 0.255 0.252 0.250 0.249 0.248 0.247 0.246 0.245

g ¼
0.2 0.300 0.250 0.233 0.225 0.220 0.217 0.214 0.213 0.211 0.210 0.209

g ¼
0.3 0.300 0.240 0.220 0.210 0.204 0.200 0.197 0.195 0.193 0.192 0.191

g ¼
0.4 0.300 0.230 0.207 0.195 0.188 0.183 0.180 0.178 0.176 0.174 0.173

WACC(L), ko=0.3 0.33 0.31 0.29 g=0.4

WACC

0.27

g=0.3

0.25

g=0.2

0.23

g=0

0.21

g=-0.2

0.19

g=-0.3 g=-0.4

0.17 0.15 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 11.3 Dependence of WACC on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and g ¼ 0; 0.2; 0.3; 0.4

and k0 – g decrease with g. This leads to increase of company capitalization with g: V ¼ CF/WACC
g.

182

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

11.4.3 Dependence of Discount Rate i on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.2 and Different Values of g Discount rate for the leverage company change from the weighted average cost of capital, WACC, to WACC
g (where g is growth rate), for a financially independent company from k0 to k0
g. This means that WACC and k0 are no longer the discount rates as it takes place in case of the classical Modigliani–Miller theory with constant profit. Below we study the dependence of discount rate i on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at two values of k0 (0.2; 0.3) and different values of g (
0.4;
0.3;
0.2; 0.0; 0.1; 0.15; 0.2; 0.3; 0.4). Let us start from k0 ¼ 0.2. The results of Table 11.3 are shown in Fig. 11.4. From Fig. 11.4 it is seen that discount rate i decreases with leverage level L at growth values g < k0 (at g ¼ 0; 0.1; 0.15; 0.2;
0.3;
0.4). Discount rate i in contrast to WACC decreases with g: this provides the increase of company value V with g. At g > k0 discount rate i increases with L, but being negative it is not shown in Fig. 11.4.

11.4.4 Dependence of Discount Rate i on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.3 and Different Values of g Let us study the dependence of discount rate i on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and different values of g. The results of Table 11.4 are shown in Fig. 11.5. From Fig. 11.5 it is seen that discount rate i decreases with leverage level L at growth values at g < k0 (at g ¼ 0; 0.1; 0.15; 0.2;
0.3;
0.4). Discount rate i in opposite to WACC decreases with g: this provides the increase of company value V with g. At g > k0 discount rate i increases with L, but being negative it is not shown in Fig. 11.5. From Figs. 11.4 and 11.5 it is seen that for k0 ¼ 0.2 and 0.3 discount rate i decreases with g: this provides the increase of company value V with g. Discount rate i decreases with L at g < k0 (at g ¼ 0; 0.1; 0.15; 0.2;
0.3;
0.4). At g > k0 discount rate i increases with L, but being negative it is not shown in Figs. 11.4 and 11.5.

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

i g ¼ 0.4
0.200
0.180
0.173
0.170
0.168
0.167
0.166
0.165
0.164
0.164
0.164

g ¼ 0.3
0.100
0.090
0.087
0.085
0.084
0.083
0.083
0.083
0.082
0.082
0.082

g ¼ 0.2 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

g ¼ 0.15 0.050 0.045 0.043 0.043 0.042 0.042 0.041 0.041 0.041 0.041 0.041

g ¼ 0.1 0.100 0.090 0.087 0.085 0.084 0.083 0.083 0.083 0.082 0.082 0.082

g¼0 0.200 0.180 0.173 0.170 0.168 0.167 0.166 0.165 0.164 0.164 0.164

g ¼
0.2 0.400 0.360 0.347 0.340 0.336 0.333 0.331 0.330 0.329 0.328 0.327

g ¼
0.3 0.500 0.450 0.433 0.425 0.420 0.417 0.414 0.413 0.411 0.410 0.409

g ¼
0.4 0.600 0.540 0.520 0.510 0.504 0.500 0.497 0.495 0.493 0.492 0.491

Table 11.3 Dependence of discount rate i on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and g ¼
0.4;
0.3;
0.2; 0.0; 0.1; 0.15; 0.2; 0.3; 0.4

11.4 Results and Discussions 183

184

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

i(L), k0=0.2 0.7

0.6

0.5 g=0.2

0.4

g=-0.2

i

g=-0.3 g=-0.4

0.3

g=0 g=0.1

0.2

g=0.15 0.1

0 0

1

2

3

4

5

6

7

8

9

10

L Fig. 11.4 Dependence of discount rate i on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and g ¼ 0; 0.1; 0.15; 0.2;
0.3;
0.4

11.4.5 Dependence of Company Value V on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.2 and Different Values of g Let us study the dependence of company value V on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and different values of g. The results of Table 11.5 are shown in Fig. 11.6. From Fig. 11.6 it is seen, that at k0 ¼ 0.2 and g ¼ 0; 0.1; 0.15;
0.2;
0.3;
0.4 the company value V at fixed growth rate g increases with leverage level L in Generalized Modigliani–Miller theory (GMM theory). The company value V also increases with growth rate g.

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

i g ¼ 0.4
0.100
0.090
0.087
0.085
0.084
0.083
0.083
0.083
0.082
0.082
0.082

g ¼ 0.3 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000

g ¼ 0.2 0.100 0.090 0.087 0.085 0.084 0.083 0.083 0.083 0.082 0.082 0.082

g ¼ 0.15 0.150 0.135 0.130 0.128 0.126 0.125 0.124 0.124 0.123 0.123 0.123

g ¼ 0.1 0.200 0.180 0.173 0.170 0.168 0.167 0.166 0.165 0.164 0.164 0.164

g¼0 0.300 0.270 0.260 0.255 0.252 0.250 0.249 0.248 0.247 0.246 0.245

g ¼
0.2 0.500 0.450 0.433 0.425 0.420 0.417 0.414 0.413 0.411 0.410 0.409

g ¼
0.3 0.600 0.540 0.520 0.510 0.504 0.500 0.497 0.495 0.493 0.492 0.491

g ¼
0.4 0.700 0.630 0.607 0.595 0.588 0.583 0.580 0.578 0.576 0.574 0.573

Table 11.4 Dependence of discount rate i on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and g ¼ 0; 0.1; 0.15; 0.2; 0.3; 0.4

11.4 Results and Discussions 185

186

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

11

i(L), k0=0.3 0.800 0.700 0.600 g=0.2

0.500

i

g=0.15 g=0.1

0.400

g=0 0.300

g=-0.2

g=-0.3

0.200

g=-0.4

0.100 0.000 0

1

2

3

4

5

6

7

8

9

10

L Fig. 11.5 Dependence of discount rate i on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and g ¼ 0; 0.1; 0.15; 0.2;
0.3;
0.4

11.4.6 Dependence of Company Value V on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.3 and Different Values of g Let us study the dependence of company value V on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and different values of g. The results of Table 11.6 are shown in Fig. 11.7. From Fig. 11.7 it is seen, that at k0 ¼ 0.3 and g ¼ 0; 0.1; 0.15;
0.2;
0.3;
0.4 the company value V at fixed growth rate g increases with leverage level L in Generalized Modigliani–Miller theory (GMM theory). The company value V as well increases with growth rate g. From Figs. 11.6 and 11.7, it is seen that at two values of k0 (0.2; 0.3) and g ¼ 0; 0.1; 0.15;
0.2;
0.3;
0.4, the company value V at fixed growth rate g increases with leverage level L in Generalized Modigliani–Miller theory (GMM theory). The company value V as well increases with growth rate g. We should note that in case of growing rate g equals to equity cost (at L ¼ 0) k0, the company value V becomes infinite. This is the limitation of the perpetuity Modigliani–Miller theory. As one can see in our future publications, where we

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 100 100 100 100 100 100 100 100 100 100 100

V g ¼ 0.1 1000 1111.111 1153.846 1176.471 1190.476 1200 1206.897 1212.121 1216.216 1219.512 1222.222 g ¼ 0.15 2000 2222.222 2307.692 2352.941 2380.952 2400 2413.793 2424.242 2432.432 2439.024 2444.444

g¼0 500 555.5556 576.9231 588.2353 595.2381 600 603.4483 606.0606 608.1081 609.7561 611.1111

g ¼
0.2 250 277.7778 288.4615 294.1176 297.619 300 301.7241 303.0303 304.0541 304.878 305.5556

g ¼
0.3 200 222.2222 230.7692 235.2941 238.0952 240 241.3793 242.4242 243.2432 243.9024 244.4444

g ¼
0.4 166.6667 185.1852 192.3077 196.0784 198.4127 200 201.1494 202.0202 202.7027 203.252 203.7037

Table 11.5 Dependence of company value V on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and different values of g ¼ 0.0; 0.1; 0.15;
0.2;
0.3;
0.4

11.4 Results and Discussions 187

188

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

11

V(L),k0=0.2 3000 2500

V

2000

g=0.15 g=0.1

1500

g=-0.2 g=-0.3

1000

g=-0.4 g=0

500 0

0

1

2

3

4

5

6

7

8

9

10

L Fig. 11.6 Dependence of company value V on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and g ¼ 0; 0.1; 0.15;
0.2;
0.3;
0.4

will consider the generalization of BFO theory for the case of variable profit such kind of restriction is absent for companies of finite ages.

11.4.7 Dependence of Equity Cost ke on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.2 and Different Values of g (0; 0.2; 0.3; 0.4) The economically justified value of dividends is equal to equity cost. Thus, knowing the dependence of equity cost of company on leverage level L, on growth rate g is very important, because it could impact the dividend policy of the company and help to company management to develop reasonable dividend policy. Below we study the dependence of equity cost ke on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and different values of g (0; 0.2; 0.3; 0.4). The results of Table 11.7 are shown in Fig. 11.8. We have investigated the dependence of equity cost ke on leverage level L in the Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and g ¼ 0; 0.2;

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

CF 100 100 100 100 100 100 100 100 100 100 100

V g ¼ 0.1 500 555.5556 576.9231 588.2353 595.2381 600 603.4483 606.0606 608.1081 609.7561 611.1111 g ¼ 0.15 666.6667 740.7407 769.2308 784.3137 793.6508 800 804.5977 808.0808 810.8108 813.0081 814.8148

g ¼ 0.25 2000 2222.222 2307.692 2352.941 2380.952 2400 2413.793 2424.242 2432.432 2439.024 2444.444

g¼0 333.3333 370.3704 384.6154 392.1569 396.8254 400 402.2989 404.0404 405.4054 406.5041 407.4074

g ¼
0.2 200 222.2222 230.7692 235.2941 238.0952 240 241.3793 242.4242 243.2432 243.9024 244.4444

g ¼
0.3 166.6667 185.1852 192.3077 196.0784 198.4127 200 201.1494 202.0202 202.7027 203.252 203.7037

g ¼
0.4 142.8571 158.7302 164.8352 168.0672 170.068 171.4286 172.4138 173.1602 173.7452 174.216 174.6032

Table 11.6 Dependence of company value V on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and g ¼ 0.25

11.4 Results and Discussions 189

190

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

V(L),k0=0.3 2500

2000

g=0.25

1500

g=0.1

V

g=-0.2 g=-0.3

1000

g=-0.4 g=0 g=0.15

500

0 0

1

2

3

4

5

6

7

8

9

10

L

Fig. 11.7 Dependence of company value V on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and g ¼ 0; 0.1; 0.15; 0.25;
0.2;
0.3;
0.4

0.3; 0.4. From Fig. 11.8 it is seen that the equity cost, ke, which linearly grows with leverage level L increases with g: the tilt angle ke(L) grows with g. It is interesting that at k0 ¼ 0.2; kd ¼ 0.16 and at g* ¼
0.16 in accordance with formula g ¼
ðk0
kdTÞð1
T Þ the equity cost ke turns out to be equal to k0 and does not change with leverage level L. This should change the dividend policy of the company, because the economically justified value of dividends is equal to equity cost.

11.4.8 Dependence of Equity Cost ke on Leverage Level L in Generalized Modigliani–Miller Theory (GMM Theory) at k0 = 0.3 and Different Values of g Presented below is the dependence of equity cost ke on leverage level L in the Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and different values of g (0; 0.2; 0.3; 0.4). The results of Table 11.8 are shown in Fig. 11.9. We have investigated the dependence of equity cost ke on leverage level L in the Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and g ¼ 0; 0.2; 0.3; 0.4. From Fig. 11.9, it is seen that the equity cost, ke, which linearly grows

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

ke g ¼ 0.4 0.2 0.312 0.424 0.536 0.648 0.76 0.872 0.984 1.096 1.208 1.320 g ¼ 0.3 0.2 0.292 0.384 0.476 0.568 0.66 0.752 0.844 0.936 1.028 1.120

g ¼ 0.2 0.2 0.272 0.344 0.416 0.488 0.560 0.632 0.704 0.776 0.848 0.920

g¼0 0.2 0.232 0.264 0.296 0.328 0.360 0.392 0.424 0.456 0.488 0.520

g ¼
0.2 0.2 0.192 0.184 0.176 0.168 0.16 0.152 0.144 0.136 0.128 0.120

g ¼
0.3 0.2 0.172 0.144 0.116 0.088 0.060 0.032 0.004
0.024
0.052
0.080

g ¼
0.4 0.2 0.152 0.104 0.056 0.008
0.040
0.088
0.136
0.184
0.232
0.280

Table 11.7 Dependence of equity cost ke on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2 and different values of g (0; 0.2; 0.3; 0.4)

11.4 Results and Discussions 191

192

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

ke(L),ko=0.2;kd=0.16 1.4 1.2 1 g=0.4

0.8

g=0.3

0.6

ke

g=0.2 g=-0.2

0.4

g=-0.4 0.2

g=-0.3 g=0

0 0

1

2

3

4

5

6

7

8

9

10

-0.2 -0.4

L

Fig. 11.8 Dependence of equity cost ke on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.2; kd ¼ 0.16 and g ¼ 0; 0.2; 0.3; 0.4

with leverage level increases with g: the tilt angle ke(L ) grows with g. It is interesting, that at k0 ¼ 0.3; kd ¼ 0.22 and at g* ¼
0.32 in accordance with Formula (11.43), the equity cost ke turns out to be equal to k0 and does not change with leverage level L. Here, we have investigated the dependence of equity cost ke on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at two values of k0 (0.2 and 0.3) and different values of g. The equity cost, ke, which linearly grows with leverage level, increases with g: the tilt angle ke(L ) grows with g. This should change the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. It is interesting, that at k0 ¼ 0.2; kd ¼ 0.16 and at g* ¼
0.16 and at k0 ¼ 0.3; kd ¼ 0.22 and at g* ¼
0.32 in accordance with Formula (11.43) the equity cost ke turns out to be equal to k0 and does not change with leverage level L. The qualitatively new effect in corporate finance has been discovered: at rate g < g* the slope of the curve ke(L ) turns out to be negative. This effect, which is absent in classical Modigliani–Miller theory, could significantly alter the principles of the company’s dividend policy. The last effect is similar to the qualitatively new effect in corporate finance, which has been discovered by Brusov–Filatova–Orekhova within the BFO theory: abnormal dependence of equity cost ke on leverage level at tax on profit rate T, which exceed some rate value T*: this discovery also significantly alters the principles of the company’s dividend policy.

L 0 1 2 3 4 5 6 7 8 9 10

T 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3 0.3

kd 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16 0.16

ke g ¼ 0.4 0.3 0.444 0.588 0.732 0.876 1.020 1.164 1.308 1.452 1.596 1.740 g ¼ 0.3 0.3 0.424 0.548 0.672 0.796 0.920 1.044 1.168 1.292 1.416 1.540

g ¼ 0.2 0.3 0.404 0.508 0.612 0.716 0.820 0.924 1.028 1.132 1.236 1.340

g¼0 0.3 0.364 0.428 0.492 0.556 0.620 0.684 0.748 0.812 0.876 0.940

g ¼
0.2 0.3 0.324 0.348 0.372 0.396 0.420 0.444 0.468 0.492 0.516 0.540

g ¼
0.3 0.3 0.304 0.308 0.312 0.316 0.32 0.324 0.328 0.332 0.336 0.340

g ¼
0.4 0.3 0.284 0.268 0.252 0.236 0.220 0.204 0.188 0.172 0.156 0.140

Table 11.8 Dependence of equity cost ke on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3 and different g (0; 0.2; 0.3; 0.4)

11.4 Results and Discussions 193

194

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

ke(L),ko=0.3;kd=0.22 2 1.8 1.6

ke

1.4

g=0.4

1.2

g=0.3

1

g=0.2 g=-0.2

0.8

g=-0.3

0.6

g=-0.4

0.4

g=0

0.2

0 0

1

2

3

4

5

6

7

8

9

10

L Fig. 11.9 Dependence of equity cost ke on leverage level L in Generalized Modigliani–Miller theory (GMM theory) at k0 ¼ 0.3; kd ¼ 0.22 and g ¼ 0; 0.2;0.3;0.4

11.5

Conclusions

The main purpose of the current study is to generalize the Modigliani–Miller theory by taking into account one of the most important conditions of companies real functioning: variable profit. We use analytical and numerical methods: we derive all main formulas of the Generalized Modigliani–Miller theory theoretically and then use them to obtain all the main financial indicators of company and their dependences on different parameters by MS Excel. The generalized Modigliani– Miller theory significantly extends its application in practice, especially in business valuation, in ratings and in other areas of economy and finance. We consider the case of growing profit as well as decreasing of profit and show that all the statements by Modigliani and Miller change significantly. Within the new Generalized Modigliani–Miller theory (GMM theory), we study the dependence of the weighted average cost of capital, WACC, the cost of equity, ke, the discount rate i and the capitalization of the company, V, on leverage level L at different values of growth rate g and obtained the following results: 1. Discount rate for leverage company changes from the weighted average cost of capital, WACC, to WACC
g (where g is growing rate), for a financially independent company from k0 to k0
g. This means that WACC and k0 are no

11.5

Conclusions

195

longer the discount rates as it takes place in case of classical Modigliani–Miller theory with constant profit. 2. All curves WACC(L ) for different g start from one point, k0. They decrease with L at g < k0 and increase at g > k0. It turns out that WACC grows with g, while real discount rates WACC
g and k0
g decrease with g. This leads to increase of company capitalization with g: V ¼ CF/WACC
g. 3. We have investigated the dependence of equity cost ke on leverage level L in the Generalized Modigliani–Miller theory (GMM theory) at two values of k0 (0.2 and 0.3) and different values of g. The equity cost, ke, which linearly grows with leverage level L increases with g: the tilt angle ke(L ) grows with g. This should change the dividend policy of the company, because the economically justified value of dividends is equal to equity cost. It is interesting, that at k0 ¼ 0.2; kd ¼ 0.16 and at g* ¼
0.16 and at k0 ¼ 0.3; kd ¼ 0.22 and at g* ¼
0.32 in accordance with Formula (11.43) the equity cost ke turns out to be equal to k0 and does not change with leverage level L. A new qualitative finding related to corporate finance has been discovered in this study: at rate g < g* the slope of the curve ke(L) turns out to be negative. This effect, which is absent in the classical Modigliani–Miller theory, could significantly alter the principles of the company’s dividend policy. Rating methodology should take into account the company’s dividend policy, thus such results obtained could change the credit rating of an issuer. The last effect found in this study is similar to the qualitatively new effect in corporate finance, discussed above, which has been discovered by Brusov–Filatova– Orekhova within BFO theory: abnormal dependence of equity cost on leverage level when corporate taxes rate T exceeds some rate value T*; this discovery also significantly alters the principles of the company’s dividend policy. Thus, within the new Generalized Modigliani–Miller theory, a lot of significant implications are obtained, allowing us to develop new approach to financial policy and financial strategy of the company. Note that the limitations of the Generalized Modigliani–Miller theory are similar to the limitations of the classical Modigliani–Miller theory and are connected with the numerous assumptions of classical Modigliani–Miller theory (Modigliani and Miller 1958, 1963, 1966). Therefore, the direction of future research is clear: we will consider the generalization of BFO theory, which describes the companies of arbitrary age, for the case of variable profit. The limitations of the Generalized Modigliani–Miller theory will be absent in the generalized BFO theory, which is valid for companies of an arbitrary age. The Generalized Modigliani–Miller theory could and should be used in corporate finance and corporate management, in investments, business valuation, taxation, ratings, etc., for more correct assessments and more qualified management decisions. Authors planning further development of the Modigliani–Miller theory should account for practical conditions of business and its application in all mentioned above areas. In particular, more frequent payments of tax of profit will be considered, combination of this case with advance payments of tax of profit, modification of

196

11

Generalization of the Modigliani–Miller Theory for the Case of Variable. . .

BFO theory (which is valid for the companies of arbitrary ages) for the case of variable profit.

References Berk J, DeMarzo P (2007) Corporate finance. Pearson–Addison Wesley, Boston Bradley M, Jarrell GA, Kim EH (1984) On the existence of an optimal capital structure: theory and evidence. J Financ 39:857–878 Brealey R, Myers S, Allen F (2005) Principle of corporate finance, 7th edn. McGraw Hill, New York Brusov P, Filatova T (2021) The Modigliani–Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198 (WoS Q1) Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International Publishing, Switzerland, pp 1–373. https://www.springer.com/ gp/book/9783319147314 Brusov P, Filatova T, Orehova N, Eskindarov M (2018) Modern corporate finance, investments, taxation and ratings, 2nd edn. Springer Nature, Switzerland, pp 1–571 Brusov P, Filatova T, Orehova N (2020a) Ratings: critical analysis and new approaches of quantitative and qualitative methodology. Springer Nature, Switzerland, pp 1–369. https:// www.springer.com/de/book/9783030562427 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Modification of the Modigliani–Miller Theory for the case of advance payments of tax on profit. J Rev Glob Econ 9:257–267 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020c) Application of the Modigliani–Miller Theory, modified for the case of advance payments of tax on profit, in rating methodologies. J Rev Glob Econ 9:282–292 Brusov P, Filatova T, Chang S-I, Lin G (2021a) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9(13):1491 (WoS Q1) Brusov P, Filatova T, Orekhova N, Kulik V, Chang S-I, Lin G (2021b) Generalization of the Modigliani–Miller theory for the case of variable profit. Mathematics 9(11):1286 (WoS Q1) DeAngelo H, Masulis R (1980) Optimal capital structure under corporate and personal taxation. J Financ Econ 8:3–29 DeMarzo PM (1988) An extension of the Modigliani–Miller theorem to stochastic economies with incomplete markets and interdependent securities. J Econ Theory 45(2):353–369 Farber A, Gillet R, Szafarz A (2006) A general formula for the WACC. Int J Bus 11(2):211–218 Fernandez P (2006) A general formula for the WACC: a comment. Int J Bus 11(2):219 Filatova TV, Orekhova NP, Brusova АP (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Frank M, Goyal V (2009) Capital structure decisions: which factors are reliably important? Financ Manag 38(1):1–37. https://doi.org/10.1111/j.1755-053X.2009.01026.x Hamada R (1969) Portfolio analysis, market equilibrium, and corporate finance. J Financ 24(1): 13–31 Harris R, Pringle J (1985) Risk–adjusted discount rates – extension form the average–risk case. J Financ Res 8(3):237–244 Miller M (1977) Debt and taxes. J Financ 32(2):261–275. https://doi.org/10.1111/j.1540-6261. 1977.tb03267.x Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297

References

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Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Ross S, Westerfiel R, Jaffee D (2005) Corporate finance, 7th edn. McGraw Hill, New York Sethi SP, Derzko NA, Lehoczky JP (1991) A stochastic extension of the Miller–Modigliani framework. Math Financ 1(4):57–76

Part II

Applications of the Modigliani–Miller Theory in Investments

Chapter 12

Investment Models with Debt Repayment at the End of the Project and their Application

In this and next chapter, we describe the modern investment models, created by the authors and well tested in the real economy. These models used by us for investigation of different problems of investments, such as influence of debt financing, leverage level, taxing, project duration, method of financing, and some other parameters on efficiency of investments and other problems. But we use them in Chap. 18 for modification of methodology of project ratings. In this chapter, we consider the investment models with debt repayment at the end of the project and their application, while in Chap. 18 we consider the investment models with uniform debt repayment and their application.

12.1

Investment Models

The effectiveness of the investment project is considered from two perspectives: the owners of equity and debt and the equity holders only. For each of these cases, NPV is calculated in two ways: with the division of credit and investment flows (and thus discounting of the payments using two different rates) and without such a division (in this case, both flows are discounted using the same rate, as which can be, obviously, chosen WACC). For each of the four situations, two cases are considered: (1) a constant value of equity S and (2) a constant value of the total invested capital I ¼ S + D (D is value of debt funds). As it was stated above, the effectiveness of the investment project is considered from two perspectives: the owners of equity and debt and the equity holders only. In the first case, the interest and duty paid by owners of equity (negative flows) returned to the project because they are exactly equal to the flow (positive), obtained by owners of debt capital. The only effect of leverage, in this case, is–the effect of the tax shield, generated from the tax relief: interest on the loan is entirely included into the cost and thus reduces the tax base. After-tax flow of capital for each period in this case is equal to © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_12

201

202

12 Investment Models with Debt Repayment at the End of the Project and their. . .

NOIð1  t Þ þ k d Dt

ð12:1Þ

and the value of investments at the initial time moment T ¼ 0 is equal to –I ¼ –S – D. Here NOI stands for net operating income (before taxes). In the second case, investments at the initial time moment T ¼ 0 are equal to –S and the flow of capital for the period (in addition to the tax shields kd Dt it includes a payment of interest on a loan kdD): ðNOI  kd DÞð1  t Þ:

ð12:2Þ

Here, for simplicity, we suppose that interest on the loan will be paid in equal shares kd D during all periods. Note that principal repayment is made at the end of the last period. Some variety of repayment of long-term loans will be considered below (see in Chap. 14). We will consider two different ways of discounting: 1. Operating and financial flows are not separated and both are discounted, using the general rate (as which, obviously, the weighted average cost of capital (WACC) can be selected). In this case for perpetuity projects, the Modigliani–Miller formula (Мodigliani and Мiller 1958, 1963, 1966) for WACC will be used and for projects of finite (arbitrary) duration Brusov–Filatova–Orekhova formula (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b, 2015, 2018a, b, c, d, e, f, g, 2019, 2020; Filatova et al. 2008; Brusova 2011; Brusov 2018a, b; Filatova et al. 2018). 2. Operating and financial flows are separated and are discounted at different rates: the operating flow at the rate which is equal to the equity cost ke, depending on leverage, and credit flow at the rate which is equal to the debt cost kd, which until fairly large values of leverage remain constant and start to grow only at high values of leverage L, when there is a danger of bankruptcy. Note that loan capital is the least risky, because interest on credit is paid after taxes in the first place. Therefore, the cost of credit will always be less than the equity cost, whether of ordinary or of preference shares ke > kd; kp > kd. Here ke; kp is the equity cost of ordinary or of preference shares consequently.

12.2

The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only

12.2.1 With the Division of Credit and Investment Flows Projects of Finite (Arbitrary) Duration In this case, the expression for NPV has a view

12.2

The Effectiveness of the Investment Project from the Perspective of the. . .

NPV ¼ S þ

n X NOIð1  t Þ i

þ

n X k d Dð1  t Þ i



203

D ð1 þ k d Þn

ð1 þ k e Þ ð1 þ k d Þ i¼1     NOIð1  t Þ 1 1 D ¼ S þ 1 :  Dð1  t Þ 1   ke ð1 þ k e Þn ð1 þ kd Þn ð1 þ kd Þn i¼1

ð12:3Þ The last term in the first line-discounted (present) value of credit, extinguished a one-off payment at the end of the last period n. Below we will look at two cases: 1. A constant value of the invested capital I ¼ S + D (D–debt value). 2. A constant value of equity capital S. We will start with the first case. At a Constant Value of the Invested Capital (I = const) In the case of a constant value of the invested capital (I ¼ const), taking into account D ¼ IL/(1 + L ), S ¼ I/(1 + L), one gets      I 1 1 1 þ L ð1  t Þ 1  þ 1þL ð1 þ k d Þn ð1 þ k d Þn   NOIð1  t Þ 1 þ 1 : ke ð1 þ k e Þn

NPV ¼ 

ð12:4Þ

For 1-Year Project Putting at the Eq. (12.4) n ¼ 1, one gets for NPV NPV ¼ 

   1 þ k d ð1  t Þ NOIð1  t Þ I 1þL : þ 1 þ ke 1þL ð1 þ k d Þ

ð12:5Þ

At a Constant Value of Equity Capital (S = const) Accounting that in the case S ¼ const NOI is proportional to the invested capital, NOI ¼ βI ¼ βS(1 + L ), we get 

    1 1 NPV ¼ S 1 þ L ð1  t Þ 1  þ ð1 þ kd Þn ð1 þ k d Þn   βSð1 þ LÞð1  t Þ 1 þ 1 : ke ð1 þ k e Þn For 1-Year Project Putting at the Eq. (12.6) n ¼ 1, one gets for NPV

ð12:6Þ

12 Investment Models with Debt Repayment at the End of the Project and their. . .

204



  1 þ kd ð1  t Þ βSð1 þ LÞð1  t Þ NPV ¼ S 1 þ L : þ 1 þ kd 1 þ ke

12.3

ð12:7Þ

Without Flows Separation

In this case, operating and financial flows are not separated and are discounted, using the general rate (as which, obviously, WACC can be selected). The credit reimbursable at the end of the project (at the end of the period (n)) can be discounted either at the same rate WACC or at the debt cost rate kd. Now we choose a uniform rate and the first option. n X NOIð1  t Þ  kd Dð1  t Þ

D ð 1 þ WACC Þn ð1 þ WACCÞ i¼1   NOIð1  t Þ  k d Dð1  t Þ 1 D 1 ¼ S þ :  WACC ð1 þ WACCÞn ð1 þ WACCÞn

NPV ¼ S þ

i



ð12:8Þ At a Constant Value of the Invested Capital (I = const) In case of a constant value of the invested capital (I ¼ const), taking into account D ¼ IL/(1 + L ), S ¼ I/(1 + L), one gets     kd ð1  t Þ I 1 L 1 1þL NPV ¼  þ WACC 1þL ð1 þ WACCÞn ð1 þ WACCÞn   NOIð1  t Þ 1 þ 1 : WACC ð1 þ WACCÞn ð12:9Þ For 1-Year Project Putting into Eq. (12.9) n ¼ 1, one gets for NPV   1 þ k d ð1  t Þ NOIð1  t Þ I þ : 1þL NPV ¼  1 þ WACC 1 þ WACC 1þL

ð12:10Þ

At a Constant Value of Equity Capital (S = const) Accounting that in the case S ¼ const NOI is proportional to the invested capital, I, NOI ¼ βI ¼ βS(1 + L ), and substituting D ¼ LS, we get

12.4

Modigliani–Miller Limit (Perpetuity Projects)

205

  NOIð1  t Þ  kd Dð1  t Þ 1 1 NPV ¼ S þ WACC ð1 þ WACCÞn D , ð1 þ WACCÞn     Lk d ð1  t Þ 1 L NPV ¼ S 1 þ 1 þ WACC ð1 þ WACCÞn ð1 þ WACCÞn   βSð1 þ LÞð1  t Þ 1 þ 1 : WACC ð1 þ WACCÞn 

ð12:11Þ

ð12:12Þ

For 1-Year Project Putting into Eq. (12.12) n ¼ 1, one gets for NPV     Lk d ð1  t Þ 1 L 1 þ NPV ¼ S 1 þ WACC ð1 þ WACCÞn ð1 þ WACCÞn   βSð1 þ LÞð1  t Þ 1 þ 1 : WACC ð1 þ WACCÞn NPV ¼ S þ

NOIð1  t Þ  k d Dð1  t Þ  D : 1 þ WACC

ð12:13Þ

Substituting D ¼ LS, NOI ¼ βI ¼ βS(1 + L ), we get   Lðkd ð1  t Þ  1Þ βSð1 þ LÞð1  t Þ þ : NPV ¼ S 1 þ 1 þ WACC 1 þ WACC

12.4

ð12:14Þ

Modigliani–Miller Limit (Perpetuity Projects)

12.4.1 With Flows Separation In perpetuity limit (Modigliani–Miller limit) (turning to the limit n ! 1 in the relevant equations), we have NPV ¼ S þ

NOIð1  t Þ  Dð1  t Þ: ke

ð12:15Þ

At a Constant Value of the Invested Capital (I = const) At a constant value of the invested capital (I ¼ const), accounting D ¼ IL/(1 + L ), S ¼ I/(1 + L ), we get

206

12 Investment Models with Debt Repayment at the End of the Project and their. . .

NPV ¼  NPV ¼ 

NOIð1  t Þ I : ð1 þ Lð1  t ÞÞ þ ke 1þL

NOIð1  t Þ I ð1 þ Lð1  t ÞÞ þ : 1þL k0 þ ðk0  kd ÞLð1  t Þ

ð12:16Þ ð12:17Þ

In order to obtain Eqs. (12.17) from (12.16), we used the Modigliani–Miller formula (Мodigliani and Мiller 1963) for equity cost ke for perpetuity projects: k e ¼ k0 þ ðk0  kd ÞLð1  t Þ:

ð12:18Þ

At a Constant Value of Equity Capital (S = const) Accounting D ¼ LS, we get in perpetuity limit (n ! 1) (Modigliani–Miller limit) NPV ¼ Sð1 þ Lð1  t ÞÞ þ

βSð1 þ LÞð1  t Þ : k0 þ ðk 0  kd ÞLt

ð12:19Þ

12.4.2 Without Flows Separation In perpetuity limit (n ! 1) (Modigliani–Miller limit) (turning to the limit n ! 1 in the relevant equations), we have NPV ¼ S þ

NOIð1  t Þ  k d Dð1  t Þ : WACC

ð12:20Þ

At a Constant Value of the Invested Capital (I = const) At a constant value of the invested capital (I ¼ const), accounting D ¼ IL/(1 + L ), S ¼ I/(1 + L ), we get L k ð1  t Þ NOIð1  t Þ  I 1 1 þ L d NPV ¼ I  þ WACC 1þL   Lkd ð1  t Þ NOIð1  t Þ 1 ¼ I  1þ þ : 1þL k0 ð1  Lt=ð1 þ LÞÞ k0 ð1  Lt=ð1 þ LÞÞ

ð12:21Þ

At a Constant Value of Equity Capital (S = const) NPV ¼ S þ Substituting D ¼ LS, we get

NOIð1  t Þ  kd Dð1  t Þ WACC

ð12:22Þ

The Effectiveness of the Investment Project from the Perspective of the. . .

12.5

 Lkd ð1  t Þ NOIð1  t Þ þ NPV ¼ S 1 þ WACC WACC   Lkd ð1  t Þ βSð1 þ LÞð1  t Þ ¼ S 1 þ þ : k 0 ð1  Lt=ð1 þ LÞÞ k 0 ð1  Lt=ð1 þ LÞÞ

207



12.5

ð12:23Þ

The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt

12.5.1 With Flows Separation Projects of Arbitrary (Finite) Duration In this case, operating and financial flows are separated and are discounted, using different rates: the operating flow at the rate equal to the equity cost ke, depending on leverage, and credit flow at the rate equal to the debt cost kd, which until fairly large values of leverage remain constant and start to grow only at high values of leverage L, when there is a danger of bankruptcy. n X NOIð1  t Þ

n X

kd Dt i i¼1 ð1 þ k e Þ i¼1 ð1 þ k d Þ     NOIð1  t Þ 1 1 ¼ I þ 1 þ Dt 1  : ke ð1 þ k e Þn ð1 þ kd Þn

NPV ¼ I þ

i

þ

ð12:24Þ

Below we will consider two cases: 1. At a constant value of the invested capital (I ¼ S + D (D is the debt value). 2. At a constant value of equity capital S. We will start with the first case. At a Constant Value of the Invested Capital (I = const) At a constant value of the invested capital (I ¼ const), accounting D ¼ IL/(1 + L ), S ¼ I/(1 + L ), we get     NOIð1  t Þ 1 ILt 1 1 NPV ¼ I þ 1 þ ke 1þL ð1 þ k e Þn ð1 þ kd Þn      NOIð1  t Þ Lt 1 1 ¼ I 1  1 1 þ : ke 1þL ð1 þ k d Þn ð1 þ k e Þn ð12:25Þ At a Constant Value of Equity Capital (S = const) Accounting D ¼ LS, I ¼ S(1 + L ), we get

208

12 Investment Models with Debt Repayment at the End of the Project and their. . .

    NOIð1  t Þ 1 1 NPV ¼ S  LS þ 1 þ Dt 1  : ð12:26Þ ke ð1 þ k e Þn ð1 þ k d Þn Accounting that in the case S ¼ const NOI is not a constant, but is proportional to the invested capital, NOI ¼ βI ¼ βS(1 + L ), we get   NPV ¼ S 1 þ L  tL 1    1

 1 : ð1 þ k e Þn

1 ð1 þ k d Þn

 þ

βSð1 þ LÞð1  t Þ ke ð12:27Þ

For 1-Year Project  NPV ¼ S 1 þ L  tL

 βSð1 þ LÞð1  t Þ kd : þ 1 þ ke 1 þ kd

ð12:28Þ

12.5.2 Without Flows Separation In this case, operating and financial flows are not separated and both are discounted, using the general rate (as which, obviously, WACC can be selected): NPV ¼ I þ

n X NOIð1  t Þ þ kd Dt

ð1 þ WACCÞi   NOIð1  t Þ þ kd Dt 1 1 ¼ I þ : WACC ð1 þ WACCÞn i¼1

At a Constant Value of the Invested Capital (I = const) At a constant value of the invested capital (I ¼ const), we have NPV ¼ I þ

  NOIð1  t Þ þ k d Dt 1 1 : WACC ð1 þ WACCÞn

Accounting D ¼ IL/(1 + L ), S ¼ I/(1 + L ), we get

ð12:29Þ

12.6

Modigliani–Miller Limit

209

2

0 13 L 1 6 7 1 þ L  B1    n C NPV ¼ I 41   @ A5 L L 1 þ k 0 1  γ 1þL t k0 1  γ t 1þL 0 1 kd t

þ

ð12:30Þ

NOIð1  t Þ B 1  n C   @1   A: L L 1 þ k 0 1  γ 1þL t k0 1  γ t 1þL

For 1-Year Project Putting at the Eq. (12.30) n ¼ 1, one gets for NPV  NPV ¼ I 1 

 L kd t 1þL NOIð1  t Þ : þ 1 þ WACC 1 þ WACC

ð12:31Þ

At a Constant Value of Equity Capital (S = const)   NOIð1  t Þ þ kd Dt 1 1 NPV ¼ I þ WACC ð1 þ WACCÞn    k Lt 1 ¼ S 1 þ L  d 1 WACC ð1 þ WACCÞn   βSð1 þ LÞð1  t Þ 1 þ 1 : WACC ð1 þ WACCÞn

ð12:32Þ

For 1-Year Project NOIð1  t Þ þ kd Dt NPV ¼ I þ 1 þ WACC   NOIð1  t Þ k d Lt ¼ S 1 þ L  : þ 1 þ WACC 1 þ WACC

12.6

ð12:33Þ

Modigliani–Miller Limit

12.6.1 With Flows Separation In perpetuity limit (n ! 1) (Modigliani–Miller limit), we have NPV ¼ I þ

NOIð1  t Þ þ Dt: ke

ð12:34Þ

210

12 Investment Models with Debt Repayment at the End of the Project and their. . .

At a constant value of the invested capital (I ¼ const), accounting D ¼ IL/(1 + L ), we have  NPV ¼ I 1  t

 NOIð1  t Þ L þ : ke 1þL

ð12:35Þ

For equity cost ke and WACC in Modigliani–Miller theory, we have consequently k e ¼ k0 þ ðk 0  kd ÞLð1  t Þ,

ð12:36Þ

WACC ¼ k0 ð1  wd t Þ ¼ k0 ð1  Lt=ð1 þ LÞÞ:

ð12:37Þ

Putting Eqs. (12.36) into (12.37), we get  NPV ¼ I 1  t

 NOIð1  t Þ L þ : 1þL k 0 þ ðk0  k d ÞLð1  t Þ

ð12:38Þ

At a Constant Value of Equity Capital (S = const) Accounting D ¼ LS, I ¼ S(1 + L), in perpetuity limit (n ! 1) (Modigliani–Miller limit), we have NPV ¼ Sð1 þ Lð1  t ÞÞ þ

NOIð1  t Þ : k0 þ ðk 0  kd ÞLt

ð12:39Þ

Note that in the case S ¼ const NOI is not a constant, but is proportional to the invested capital, NOI ¼ βI ¼ βS(1 + L ). In this case, Eq. (12.38) is replaced by NPV ¼ Sð1 þ Lð1  t ÞÞ þ

βSð1 þ LÞð1  t Þ , k0 þ ðk0  kd ÞLt

ð12:40Þ

12.6.2 Without Flows Separation In perpetuity limit (n ! 1) (Modigliani–Miller limit), we have NPV ¼ I þ

NOIð1  t Þ þ kd Dt : WACC

At a constant value of the invested capital (I ¼ const), we have

ð12:41Þ

References

211

NOIð1  t Þ þ kd Dt WACC 1 L kd t NOIð1  t Þ B C ¼ I @1   1 þ L A þ  : L L k0 1  t t k0 1  1þL 1þL

NPV ¼ I þ 0

ð12:42Þ

At a Constant Value of Equity Capital (S = const) In perpetuity limit (n ! 1) (Modigliani–Miller limit), we have   NOIð1  t Þ kd Lt : þ NPV ¼ S 1 þ L  WACC WACC 2 3 βSð1 þ LÞð1  t Þ k Lt 5 þ   : NPV ¼ S41 þ L   d L L k 0 1  1þL k0 1  1þL t t

ð12:43Þ ð12:44Þ

References Brusov P (2018a) Editorial: introduction on special issue on the banking system and financial markets of Russia and other countries: problems and prospects. J Rev Global Econ 7:i–vi. SCOPUS Brusov P (2018b) Editorial. J Rev Global Econ 7:i–v. SCOPUS Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 3(435):2–8 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11):815– 824 Brusov P, Filatova T, Orehova N, Brusov PP, Brusova N (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova P, Orekhova N (2013a) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova T, Orehova N (2013b) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Global Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit—Modigliani–Miller theory. J Rev Global Econ 3:175–185

212

12 Investment Models with Debt Repayment at the End of the Project and their. . .

Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International Publishing, Cham, p 373. monograph, SCOPUS https://www. springer.com/gp/book/9783319147314 Brusov P, Filatova T, Orehova N, Eskindarov M (2018a) Modern corporate finance, investments, taxation and ratings. Springer Nature Publishing, Cham, p 571. Monograph Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018b) Rating: new approach. J Rev Global Econ 7:37–62. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018c) A “golden age” of the companies: conditions of its existence. J Rev Global Econ 7:88–103. SCOPUS. https://doi.org/10.6000/ 1929-7092.2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018d) Rating methodology: new look and new horizons. J Rev Global Econ 7:63–87. SCOPUS. https://doi.org/10.6000/1929-7092.2018. 07.06 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2018e) New meaningful effects in modern capital structure theory. J Rev Global Econ 7:104–122. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.08 Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018f) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Global Econ 7:360–376. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP (2018g) Modern corporate finance and investments, monograph. Knorus Publishing House, Moscow, p 517 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Global Econ 8:437–448. SCOPUS. https://doi. org/10.6000/1929-7092.20112.08.37 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Global Econ 9:257–268 Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Prob Solut 34(76):36–42 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018) Ratings of the long–term projects: New approach. J Rev Global Econ 7:645–661., SCOPUS. https://doi.org/10.6000/1929-7092.2018. 07.59 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391

Chapter 13

Investment Models with Uniform Debt Repayment and Their Application

In the previous chapter, we have established investment models with debt repayment at the end of the project, well proven in the analysis of real investment projects. In practice, however, a scheme of uniform debt repayment during the duration of the project is more extended. In this chapter, we describe new investment models with uniform debt repayment during the duration of the investment project, quite adequately describing real investment projects. Within these models, it is possible, in particular, to analyze the dependence of effectiveness of investment projects on debt financing and taxation. We will work on the modern theory of capital cost and capital structure developed by Brusov–Filatova–Orekhova (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b, 2015, 2018a, b, c, d, e, f, g, 2019, 2020; Filatova et al. 2008; Brusova 2011; Brusov 2018a, b; Filatova et al. 2018) as well as on perpetuity limit (Мodigliani and Мiller 1958, 1963, 1966). In this chapter, we consider the application of the investment models with uniform debt repayment to rating methodology.

13.1

Investment Models with Uniform Debt Repayment

As in the case of debt repayment at the end of the project, the effectiveness of the investment project is considered from two perspectives: the owners of equity and debt and the equity holders only. In the first case, the interest and duty paid by owners of equity (negative flows) returned to the project because they are exactly equal to the flow (positive), obtained by owners of debt capital. The only effect of leverage, in this case, is the effect of tax shield, generated from the tax relief: interest on the loan is entirely included into the net cost and, thus, reduces the tax base. For each of these cases, NPV is calculated in two ways: with the division of credit and investment flows (and thus discounting the payments, using two different rates) and without such a division (in this case, both flows are discounted at the same rate as © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_13

213

214

13

Investment Models with Uniform Debt Repayment and Their Application

Table 13.1 The sequence of debt and interest values and credit values

Period number Debt

1 D

Interest

kdD

2

3

D  n1 n k d D  n1 n

D  n2 n k d D  n2 n

... ...

n

...

k d D  1n

D  1n

which, obviously, WACC can be chosen). For each of the four situations, two cases are considered: (1) a constant value of equity S and (2) a constant value of the total invested capital I ¼ S + D (D is value of debt funds). The main debt repayment occurs evenly (by equal parts) at the end of each period, and the remaining debt at the end of the each period is an arithmetic progression with the difference D/n n

o n     o D 2D D n1 n2 D D, D  , D  , . . . , ¼ D, D ,D , ..., n n n n n n

ð13:1Þ

Interest constitutes a sequence: n     o n1 n2 D k d D, kd D , kd D , . . . , kd : n n n

ð13:2Þ

In the case of consideration from the point of view of equity owners and debt owners, the after-tax flow of capital for each period is equal to NOIð1  t Þ þ k d Di t,

ð13:3Þ

n  ð i  1Þ , n

ð13:4Þ

where Di ¼ D

and investments at time moment T ¼ 0 are equal to I ¼ S  D. Here NOI stands for net operating income (before tax). In the second case (from the point of view of equity owners only), investments at the initial moment T ¼ 0 are equal to S, and the flow of capital for the ith period (apart from tax shields kdDt it includes payment of interest on the loan kdDi) is equal to ðNOI  kd Di Þð1  t Þ 

Di : n

ð13:5Þ

We suppose that the interest on the loan and the loans itself are paid in tranches kdDi and D  ni n consequently during the all ith periods. We cite in Table 13.1 the sequence of debt and interest values and credit values. As in the case of debt repayment at the end of the project, we will consider two different ways of discounting:

13.2

The Effectiveness of the Investment Project from the Perspective of the. . .

215

1. Operating and financial flows are not separated and both are discounted, using the general rate (as which, obviously, the weighted average cost of capital (WACC) can be selected). For perpetuity projects, the Modigliani–Miller formula (Мodigliani and Мiller 1963) for WACC will be used and for projects of finite duration Brusov–Filatova–Orekhova formula for WACC (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b; Filatova et al. 2008). 2. Operating and financial flows are separated and are discounted at different rates: the operating flow at the rate equal to the equity cost ke, depending on leverage, and credit flow at the rate equal to the debt cost kd, which until fairly large values of leverage remain constant and start to grow only at high values of leverage L, when there is a danger of bankruptcy. Note once again that loan capital is the least risky, because interest on credit is paid after tax in the first place. Therefore, the cost of credit will always be less than the equity cost, whether of ordinary or of preference shares ke > kd; kp > kd. Here ke; kp is equity cost of ordinary or of preference shares consequently. One can show that the present value of interest can be calculated by using the following formula, which we have been able to derive: að1  an Þ 1 2 3 n n þ 2þ 3þ⋯þ n ¼  : 2 a a a ð a  1Þan a ða  1Þ

ð13:6Þ

Here a ¼ 1 + i. We will use this formula in the further calculations.

13.2

The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only

13.2.1 With the Division of Credit and Investment Flows To obtain an expression for NPV, the discounted flow values for one period, given by formulas (Eqs. 13.3 and 13.5), must be summed, using our obtained formula (Eq. 13.6), in which a ¼ 1 + i, where i is the discount rate. Its accurate assessment is one of the most important advantages of BFO theory (Brusov–Filatova–Orekhova) (Brusov and Filatova 2011; Brusov et al. 2011a, b, c, 2012a, b, 2013a, b, 2014a, b; Filatova et al. 2008) over its perpetuity limit Modigliani–Miller theory (Мodigliani and Мiller 1958, 1963, 1966). In this case, the expression for NPV has a view

216

13

NPV ¼ S þ

Investment Models with Uniform Debt Repayment and Their Application

n X NOIð1  t Þ i¼1

ð1 þ k e Þi

þ

n k d D X i¼1

nþ1i D ð1  t Þ  n n ð1 þ k d Þi

NOIð1  t Þð1  ð1 þ ke Þn Þ ¼ S þ ke   1  ð1 þ k Þn D nþ1 d þ kd D ð1  t Þ  n n kd   ð1 þ k d Þ½1  ð1 þ kd Þn  D n þkd ð1  t Þ  n k d ð1 þ k d Þn k2d

ð13:7Þ

In perpetuity limit (let us call it Modigliani–Miller limit), one has NPV ¼ S þ

NOIð1  t Þ  Dð1  t Þ: ke

ð13:8Þ

13.2.2 Without Flows Separation In this case, operating and financial flows are not separated and are discounted, using the general rate (as which, obviously, WACC can be selected). The main debt repayment, which occurs evenly (by equal parts) at the end of each period, can be discounted either at the same rate WACC or at the debt cost rate kd. Now we choose a uniform rate and the first option. We still consider the effectiveness of the investment project from the perspective of the equity holders only. nþ1i D ð1  t Þ  n n NPV ¼ S þ i ð 1 þ WACC Þ i¼1 D nþ1 ð1  t Þ NOIð1  t Þ   kd D n n ¼ S þ   WACC 1  1 ð1 þ WACCÞn  ð1 þ WACCÞ½1  ð1 þ WACCÞn  kd D þ ð1  t Þ n WACC2  n  WACCð1 þ WACCÞn n NOIð1  t Þ  k d D X

ð13:9Þ

In perpetuity limit (Modigliani–Miller limit) (turning to the limit n ! 1 in the relevant equations), we have

13.3

The Effectiveness of the Investment Project from the Perspective of the. . .

NPV ¼ S þ

NOIð1  t Þ  k d Dð1  t Þ : WACC

217

ð13:10Þ

Note that formula (13.10) as well as other formulas for perpetuity limit (13.12) and (13.14) could be applied for analyzing the effectiveness of the long-term investment projects.

13.3

The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt

13.3.1 With Flows Separation Projects of Arbitrary (Finite) Duration In the case of consideration from the perspective of the owners of equity and debt

NPV ¼ I þ

n X NOIð1  t Þ i¼1

ð1 þ k e Þi

þ

nþ1i t n i ð1 þ k d Þ

n kd D X i¼1

NOIð1  t Þð1  ð1 þ k e Þn Þ ¼ I þ ke nþ1 t  ½1  ð1 þ kd Þn  þD n  n D ð1 þ kd Þ½1  ð1 þ kd Þ  n kd t  n k d ð1 þ k d Þn k 2d

ð13:11Þ

In perpetuity limit (Modigliani–Miller limit) (turning to the limit n ! 1 in the relevant equations), we have NOI ¼ I þ

NOIð1  t Þ þ Dt: ke

ð13:12Þ

13.3.2 Without Flows Separation We still consider the effectiveness of the investment project from the perspective of the owners of equity and debt.

218

13

Investment Models with Uniform Debt Repayment and Their Application

nþ1i t n NPV ¼ I þ ð1 þ WACCÞi i¼1 nþ1   t NOIð1  t Þ þ kd D 1 n ¼ I þ 1 WACC ð1 þ WACCÞn  n k D ð1 þ WACCÞ½1  ð1 þ WACCÞ   d t n WACC2  n  WACCð1 þ WACCÞn n NOIð1  t Þ þ k d D X

ð13:13Þ

In perpetuity limit (Modigliani–Miller limit) (turning to the limit n ! 1 in the relevant equations), we have NPV ¼ I þ

13.4

NOIð1  t Þ þ kd Dt : WACC

ð13:14Þ

Example of the Application of the Derived Formulas

As an example of application of the obtained formulas, let us take a look at the dependence of the NPV of project on the leverage level at three values of the tax on profit rates in the case of consideration from the perspective of the equity holders only without flows separation on operating and finance ones. We use formula (Eq. 13.10) and the next parameters values NOI ¼ 800; S ¼ 500; k0 ¼ 22%; kd ¼ 19%; T ¼ 15%; 20%; 25%: Making the calculations in Excel, we get the data, which are shown in Fig. 13.1. From the calculations and Fig. 13.1, one can make the following conclusions: 1. With growth of the tax on profit rate, the NPV of the project decreases and our model makes it possible to assess, for how many percent, with growth of tax on profit rate, for example, by 1%. It should be noted that the possibility of such evaluations is unique. 2. The effect of taxation on the NPV significantly depends on the leverage level: With its increase, the impact of changing of tax on profit rate is greatly reduced. This is valid for increasing of the tax on profit rate and for its reduction. 3. At tax on profit rates 20% (as in Russia) and 25%, there is an optimum in NPV dependence on leverage. Investors should take into account the invested capital structure: in this case, they may, without special effort (only changing this structure), obtain (sometimes very substantial) gains in NPV. Note that at tax

Conclusions

Fig. 13.1 Dependence of NPV of the project on the leverage level at three values of the tax on profit rates NOI ¼ 800; S ¼ 500; k0 ¼ 22 % ; kd ¼ 19 % ; T ¼ 15 % ; 20 % ; 25%

219 NPV(L) 3000.00 2500.00 2000.00 NPV

13.5

1500.00

T=0.15 T=0.2 T=0.25

1000.00 500.00 0.00 0

1

2

3

4

5

L

on profit rate 15%, there is no optimum in NPV dependence on leverage: NPV descends monotonically with leverage.

13.5

Conclusions

New investment models with uniform debt repayment during the duration of the project, quite adequately describing real investment projects, are described. Within these models, it is possible, in particular, to analyze the dependence of the effectiveness of investment projects on debt financing and taxation. We work on the modern theory of capital cost and capital structure developed by Brusov–Filatova– Orekhova as well as on perpetuity limit–MM theory. As in the case of debt repayment at the end of the project, the effectiveness of the investment project is considered from two perspectives: the owners of equity and debt and the equity holders only. For each of these cases, NPV is calculated in two ways: with the division of credit and investment flows (and thus discounting the payments, using two different rates) and without such a division (in this case, both flows are discounted at the same rate as which, obviously, WACC can be chosen). For each of the four situations, two cases are considered: (1) a constant value of equity S and (2) a constant value of the total invested capital I ¼ S + D (D is value of debt funds). As an example of application of the obtained formulas, the dependence of the NPV of project on the leverage level at three values of the tax on profit rate has been investigated in the case of consideration from the perspective of the equity holders only and without flows separation on operating and financial ones. It has been shown that the effect of taxation on the NPV significantly depends on the leverage level: with its increase, the impact of changing of tax on profit rates is greatly reduced. This is valid for increasing of the tax on profit rate and for its reduction. The model allows investigating the dependence of effectiveness of the investment project on leverage level, on the tax on profit rate, on credit rate, on equity cost, etc.

220

13

Investment Models with Uniform Debt Repayment and Their Application

References Brusov P (2018a) Editorial: Introduction on special issue on the Banking System and financial markets of Russia and other countries: problems and prospects. J Rev Glob Econ 7:i–vi. SCOPUS Brusov P (2018b) Editorial. J Rev Glob Econ 7:i–v. SCOPUS Brusov PN, Filatova ТV (2011) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Finance Credit 3(435):2–8 Brusov P, Filatova T, Orehova N, Brusova A (2011a) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11):815– 824 Brusov P, Filatova T, Orehova N, Brusov PP, Brusova N (2011b) From Modigliani–Miller to general theory of capital cost and capital structure of the company. Res J Econ Bus ICT 2:16–21 Brusov P, Filatova T, Orehova N et al (2011c) Influence of debt financing on the effectiveness of the investment project within the Modigliani–Miller theory. Res J Econ Bus ICT (UK) 2:11–15 Brusov P, Filatova T, Eskindarov M, Orehova N (2012a) Influence of debt financing on the effectiveness of the finite duration investment project. Appl Financ Econ 22(13):1043–1052 Brusov P, Filatova T, Eskindarov M, Orehova N (2012b) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova P, Orekhova N (2013a) Absence of an optimal capital structure in the famous tradeoff theory! J Rev Global Econ 2:94–116 Brusov P, Filatova T, Orehova N (2013b) A qualitatively new effect in corporative finance: abnormal dependence of cost of equity of company on leverage. J Rev Global Econ 2:183–193 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit—Modigliani–Miller theory. J Rev Global Econ 3:175–185 Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International, Switzerland, 373 p. Monograph, SCOPUS. https://www. springer.com/gp/book/9783319147314 Brusov PN, Filatova TV, Orekhova NP (2018a) Modern corporate finance and investments. Monograph. Knorus Publishing House, Moscow, 517 p Brusov P, Filatova T, Orehova N, Eskindarov M (2018b) Modern corporate finance, investments, taxation and ratings. Springer Nature, Switzerland, 571 p. Monograph Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018c) Rating: new approach. J Rev Glob Econ 7:37–62. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018d) A “golden age” of the companies: conditions of its existence. J Rev Glob Econ 7:88–103. SCOPUS. https://doi.org/10.6000/ 1929-7092.2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018e) Rating methodology: new look and new horizons. J Rev Glob Econ 7:63–87. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.06 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2018f) New meaningful effects in modern capital structure theory. J Rev Glob Econ 7:104–122. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.08 Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018g) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Glob Econ 7:360–376. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Glob Econ 8:437–448. SCOPUS. https://doi. org/10.6000/1929-7092.2019.08.37

References

221

Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Glob Econ 9: 257–268 Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Prob Solut 34(76):36–42 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018) Ratings of the long-term projects: new approach. J Rev Glob Econ 7:645–661. SCOPUS. https://doi.org/10.6000/1929-7092.2018. 07.59 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391

Chapter 14

Innovative Investment Models with Debt Repayment at the End of the Project

14.1

Introduction

Investments play a crucial role in economy and finance. Investments in tangible and intangible assets are a necessary condition for structural adjustment and economic growth. They provide the enhancement of existing basic funds and industries and the creation of new ones. The role of investment is increased many times at the current stage. With this respect the role of the evaluation of the efficiency of investment projects, which allows the realization of the most effective projects in the context of scarcity and limited investment resources, increases. Since virtually most investment projects use debt financing, the study of the influence of capital structure and debt financing on the efficiency of investment projects, determining the optimal capital structure, is especially actual at the present time. This is why in spite of the fact that a lot of different types of investment models have been developed: stochastic, dynamics (Bond and Meghir 1994), investment banking valuation models (Rosenbaum and Pearl 2013), etc. the main problem, which has been discussed during last decades is the impact of debt financing on the efficiency of investment projects, on the investment decisions of the companies. Below we review some approaches to these problems.

14.1.1 The Literature Review Lang et al. (1996) used so-called Tobin’s q ratio and considered companies with low Tobin’s q ratio as well as with high one. Remind, that the q ratio, or Tobin’s q ratio, equals the market value of a company divided by its assets’ replacement cost. The equilibrium takes place when market value equals replacement cost. The q ratio expresses the relationship between market valuation and intrinsic value. It means

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_14

223

224

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Innovative Investment Models with Debt Repayment at the End of the Project

estimating of the fact whether a given business or market is undervalued or overvalued. The authors have shown that there is a negative correlation between future growth and leverage level at the company level and, for diversified companies, at the level of business segment. This negative correlation between growth and leverage level holds for companies with low Tobin’s q ratio, but not for high-q companies or companies in high-q industries. Therefore, for companies known to have good investment opportunities leverage does not reduce growth, but it is negatively correlated to growth for companies whose growth opportunities are either not recognized by the capital markets or are not enough valuable to overcome the effects of their debt load. Whited (1992) studied the influence of debt financing on company’s investment decision of pharmaceutical firms in India during the 11 years: from 1998 to 2009. To study the impact of debt financing on firm’s investment decision Whited (1992) has used pooling regression, random and fixed-effect models. Leverage level, retained earnings, Tobin’s Q, sales, Return on Asset, cash flow, liquidity have been considered as independent variables and investment as dependent ones. Whited has considered three types of companies, depending on their size: small companies, medium companies and large companies. He has shown, that there is a significant positive correlation between leverage level and investment for large companies, while for medium companies a negative correlation between leverage level and investment took place. Kang (1995) studied the connection between leverage level and investment decision. “Interdependent tax models” tried to explain the specific for company leverage level by analyzing the Interdependency between financing decision and investment. These models account so called investment effect: influence of investment on debt tax benefit and financial risk. One of the questions is how “investment effect” influences on bond financing decision and hence on leverage level. Different “Interdependent tax models” lead to different connection between investment and leverage level. Some authors mentioned a positive connection via the fact that the financial risk and hence the cost of bond financing decrease with an increase of investment at a given leverage level. A negative connection has been mentioned by DeAngelo and Masulis (1980) and Dotan and Ravid (1985). First, authors conclude this since tax benefit of debt compete with one of capital investment. Second, authors refer to the fact that financial risk and thus the cost of bond financing will increase with investment increase. The impact of investment increase on financial risk may depend on companyspecific factors: company-specific technology Dammon and Senbet (1988). This paper provides an analysis of the effect of corporate and personal taxes on the firm’s optimal investment and financing decisions under uncertainty. It extends the DeAngelo and Masulis capital structure model by endogenizing the firm’s investment decision. The authors’ results indicate that, when investment is allowed to adjust optimally, the existing predictions about the relationship between investmentrelated and debt-related tax shields must be modified. In particular, the authors show

14.1

Introduction

225

that increases in investment-related tax shields due to changes in the corporate tax code are not necessarily associated with reductions in leverage at the individual firm level. In cross-sectional analysis, firms with higher investment-related tax shields (normalized by expected earnings) need not have lower debt-related tax shields (normalized by expected earnings) unless all firms utilize the same production technology. Differences in production technologies across firms may thus explain why the empirical results of recent cross-sectional studies have not conformed to the predictions of DeAngelo and Masulis (1980). In this paper, a model of corporate leverage choice is formulated in which corporate and differential personal taxes exist and supply side adjustments by companies enter into the determination of equilibrium prices of debt and equity. The presence of corporate tax shield substitutes for debt such as accounting depreciation, depletion allowances, and investment tax credits is shown to imply a market equilibrium in which each firm has a unique interior optimum leverage decision (with or without leverage-related costs). The optimal leverage model yields a number of interesting predictions regarding crosssectional and time-series properties of firms’ capital structures. Extant evidence bearing on these predictions is examined. Some of the major problems under the evaluation of the effectiveness of the investment projects are the following: 1. Which financial flows and why should be taken into account under calculating parameters of efficiency of the project (NPV, IRR, etc.)? 2. How many discount rates should be used under financial flows associated with investment discounting? 3. How can these discount rates be accurately evaluated? The first two problems are still discussed. Concerning the third issue, we need to note that, in the last decade, a significant progress in the accurate determination of the cost of the equity and company weighted average cost, which just is the discount rates when evaluating the effectiveness of the project, has been achieved. The progress is associated with work performed by Brusov, Filatova, and Orekhova (BFO theory) (Brusov et al. 2015, 2018b; Filatova et al. 2008), in which the general theory of capital cost of the company and its capital structure was established and the dependence of capital cost on leverage level and on the age of company was found for the companies of arbitrary age. The main difference between their theory and Modigliani–Miller theory is that the former one waives from the perpetuity of the companies, which leads to significant differences of a new theory from the theory of Nobel laureates Modigliani and Miller (1958, 1963, 1966). In modern conditions, the requirements for improving the quality of assessing the effectiveness of investments are increased. The modern investment models, which have been well tested in real economic situations have been developed by Brusov, Filatova, and Orekhova (Brusov et al. 2015, 2018b; Filatova et al. 2008). They considered the long-term as well as arbitrary duration models. The effectiveness of the investment project has been considered from two perspectives: the owners of equity and debt and the equity holders only. For each of these cases, NPV (net present value) has been calculated in two ways: with the division of credit and investment flows (and thus discounting of

226

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Innovative Investment Models with Debt Repayment at the End of the Project

the payments using two different rates) and without such a division (in this case, both flows are discounted using the same rate, as which can be, obviously, chosen WACC). Applying their modern investment models on the evaluation of the dependence of the effectiveness of investments on debt financing of one of the telecommunication company in 2010–2012 from the point of view of optimal structure of investment authors shown that in 2012 the company lost USD675 million, because investments structure has been far from optimal one. The ability of calculating of optimal structure of investment is a strong feature created by Brusov, Filatova, and Orekhova’s modern investment models (Brusov et al. 2015, 2018b; Filatova et al. 2008). As we mentioned above, one of the most important elements of calculating the effectiveness of investment projects is the assessment of the discount rate. In the case of the long-term investment models without the division of credit and investment flows the discount rate WACC has been calculated, using the Modigliani–Miller formula (Мodigliani and Мiller 1958, 1963, 1966) WACC ¼ k0  ð1  wd t Þ,

ð14:1Þ

while in case of arbitrary duration models the Brusov–Filatova–Orekhova formula for WACC (Brusov et al. 2015, 2018b; Filatova et al. 2008) 1  ð1 þ WACCÞn 1  ð1 þ k0 Þn ¼ WACC k 0  ð1  wd t ½1  ð1 þ kd Þn Þ

ð14:2Þ

has been used. Here and below WACC is the weighted average cost of capital; k0 is the equity cost at zero leverage (L ¼ 0); kd is the debt cost; wd is the debt share; t is the tax on profit; n is the project duration; ke is the equity cost; L is the leverage level. In case of the long-term investment models with the division of credit and investment flows the discount rate for discounting the investment flows (equity cost ke) has been calculated, using the Modigliani–Miller formula (Мodigliani and Мiller 1958, 1963, 1966) ke ¼ k0 þ L  ðk 0  kd Þð1  t Þ,

ð14:3Þ

while in the case of arbitrary duration models equity cost ke has been calculated from the formula WACC ¼ ke we þ kd wd ð1  t Þ,

ð14:4Þ

using the Brusov–Filatova–Orekhova value for WACC (Brusov et al. 2015, 2018b; Filatova et al. 2008). Recent development of Brusov–Filatova–Orekhova theory see in (Brusov et al. 2020, 2021a, b; Filatova et al. 2022).

14.2

Modern Investment Models

227

The calculation methods of the discount rates (WACC, equity cost ke) have been generalized in (Brusov and Filatova 2021) for the real conditions of the implementation of investment projects: for arbitrary frequency of payment of tax on profit. In this chapter, new modern investment models, both long-term and arbitrary duration, will be created, as close as possible to real investment conditions. They will account for the payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly), which are applied in real economic practice. Their verification will lead to the creation of a comprehensive system of adequate and correct assessment of the effectiveness of the company’s investment program and its investment strategy. The structure of the paper is as follows: 1. Above we presented the literature review. 2. In Sect. 14.2 we consider the modern investment models; In Sect. 14.2.1 the effectiveness of the investment project from the perspective of the equity holders only has been described and we consider the case with flows separation. In Sect. 14.2.2 we consider case without flows separation. 3. In Sect. 14.3 we consider the effectiveness of the investment project from the perspective of the owners of equity and debt In Sect. 14.3.1 we consider case with flows separation. In Sect. 14.3.2 we consider case without flows separation. 4. In Sect. 14.4 the problem of calculation of discount rates is discussed and expressions for modified values are obtained. 5. In Sect. 14.5 we study numerically with use of Microsoft Excel the effectiveness of the four models, created by us in this paper. We consider long-term projects as well as projects of arbitrary duration from two perspectives: the owners of equity and debt and the equity holders only without the division of credit and investment flows. 6. In Conclusions we discussed obtained results and their impact on correctness of valuation of efficiency of investment projects.

14.2

Modern Investment Models

The effectiveness of the investment project could be considered from two perspectives: the equity holders and the owners of equity and debt. For each of these cases, NPV could be calculated in two ways: with the division of credit and investment flows (and thus discounting of the payments using two different rates) and without such a division (in this case, both flows are discounted using the same rate, as which can be, obviously, chosen WACC). The following designations are used below: the equity value S, the investment value I, the net operating income NOI, the leverage level L, the profitability of investments β, the tax on profit t, the project

228

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Innovative Investment Models with Debt Repayment at the End of the Project

duration n, the equity cost k0, the debt cost kd, and the number of payments of interest on debt p1 and of tax on income p2; D is the debt value. In the first case (from the perspective of the equity holders), investments at the initial time moment T ¼ 0 are equal to S and the flow of capital for the period (in addition to the tax shields kd Dt it includes a payment of interest on a loan kdD): CF ¼ ðNOI  kd DÞð1  t Þ:

ð14:5Þ

For simplicity, we suppose that interest on the loan will be paid in equal shares kdD during all periods. Note that principal repayment is made at the end of the last period. In the second case (from the perspective of the equity holders), the interest and duty paid by owners of equity (negative flows) returned to the project because they are exactly equal to the flow (positive), obtained by owners of debt capital. The only effect of leverage in this case is the effect of the tax shield, generated from the tax relief: interest on the loan is entirely included into the cost and thus reduces the tax base. After-tax flow of capital for each period in this case is equal to CF ¼ NOIð1  t Þ þ kd Dt

ð14:6Þ

and the value of investments at the initial time moment T ¼ 0 is equal to I ¼ S  D. We will consider two different ways of discounting: 1. Operating and financial flows are not separated and both are discounted, using the general rate (as which, obviously, the weighted average cost of capital (WACC) can be selected). In this case for long-term projects, the Modigliani–Miller formula for WACC (Мodigliani and Мiller 1958, 1963, 1966), modified by us for the case of payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly) will be used while for projects of finite (arbitrary) duration Brusov–Filatova–Orekhova formula for WACC (Brusov et al. 2015, 2018b; Filatova et al. 2008) modified by us for the case of payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly). 2. Operating and financial flows are separated and are discounted at different rates: the operating flow at the rate which is equal to the equity cost ke, depending on leverage and on the number of payments of interest on debt and of tax on income, and credit flow–at the rate which is equal to the debt cost kd, which until fairly large values of leverage remain constant and start to grow only at high values of leverage L, when there is a danger of bankruptcy. Note that loan capital is the least risky, because interest on credit is paid after taxes in the first place. Therefore, the cost of credit will always be less than the equity cost, whether of ordinary or of preference shares ke > kd; kp > kd. Here ke; kp is the equity cost of ordinary or of preference shares consequently.

14.2

Modern Investment Models

229

14.2.1 The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only In this case, the expression for NPV (net present value) per period has a view NOIð1  t Þ  kd Dð1  t Þ ¼ NOIð1  t Þ þ k d Dt  kd D:

ð14:7Þ

Here, the first term is the value of operating income from the investment project after tax deduction, the second term is the value of the tax shield, the third term is the value of interest on debt for one period. We will need the following auxiliary formulas for summing the reduced values of financial flows (Eq. 14.7) when calculating NPV:for annual payments of interest on debt and of tax on income n X i¼1

1  ð1 þ k d Þ 1 1 ¼  1 1  1þk ð1 þ k d Þi 1 þ k d d

n

¼

1  ð1  ð1 þ k d Þn Þ kd

ð14:8Þ

for more frequent payments ( p times per period) of interest on debt and of tax on income (semiannually, quarterly, monthly) np X

R i

i¼1

pð1 þ k d Þ=p

¼

1  ð1 þ kd Þn R ð1  ð1 þ kd Þn Þ R  ¼  1=p p ð1 þ k d Þ1=p  1 1  1 1=p pð 1 þ k d Þ ð1þk Þ

ð14:9Þ

d

Similar formulas are obtained using the cost of equity ke and WACC as the discount rates. Summing up the given values of financial flows for each period (Eq. 14.7), we obtain for NPV of n-years project in the case with separated flows NPV ¼ S þ 

n n n X X NOIð1  t Þ X k d Dt kd D þ  i i i i¼1 ð1 þ k e Þ i¼1 ð1 þ k d Þ i¼1 ð1 þ k d Þ

D ð1 þ k d Þn

ð14:10Þ

Here, the second term is the reduced value of operating income from the investment project, the second term is the reduced value of the tax shield, the third term is the reduced value of interest paid annually (at the end of the year), the fourth term is the reduced value of the debt paid at the end of the project. After summing we have the following expression for NPV

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Innovative Investment Models with Debt Repayment at the End of the Project

  n  NOIð1  t Þ 1 NPV ¼ S þ  1 þ Dt ð1  ð1 þ kd Þn Þ ke 1 þ ke  Dð1  ð1 þ kd Þn Þ 

D ð1 þ k d Þn

ð14:11Þ

In the case of more frequent ( p-times per year) payment of income taxes ( p1), frequent payments of interest on the debt ( p2) we have np np n X X NOIð1  t Þ X kd Dt kd D þ  i i=p i=p i¼1 ð1 þ k e Þ i¼1 p1 ð1 þ k d Þ 1 i¼1 p2 ð1 þ k d Þ 2 D  ; ð14:12Þ ð1 þ k d Þn

NPV ¼ S þ

After summing we have the following expression for NPV   n  NOIð1  t Þ k Dt ð1  ð1 þ kd Þn Þ 1  NPV ¼ S þ  1 þ d 1 ke 1 þ ke p ð1 þ k Þ =p1  1 1



n

kd Dð1  ð1 þ k d Þ Þ D   1=p ð 1 þ k d Þn 2 p2 ð 1 þ k d Þ  1

d

ð14:13Þ

Long-term investment projects To obtain the expression for NPV of long-term investment projects one should find the limit of Eq. (14.13) n ! 1. NPV ¼ S þ

NOIð1  t Þ kd Dt kd D    ð14:14Þ þ  1=p 1 ke 1 p1 ð 1 þ k d Þ  1 p2 ð1 þ kd Þ =p2  1

14.2.2 Without Flows Separation In this case, operating and financial flows are not separated and are discounted, using the general rate (as which, obviously, WACC can be selected). The credit reimbursable at the end of the project [at the end of the period (n)] can be discounted either at the same rate WACC or at the debt cost rate kd. Now we choose a uniform rate and the first option. Summing up the given values of financial flows for each period (Eq. 14.7), we obtain for NPV of n-years project in the case without separated flows

14.2

Modern Investment Models

NPV ¼ S þ 

231

n n n X X X NOIð1  t Þ kd Dt þ  i i i¼1 ð1 þ WACCÞ i¼1 ð1 þ WACCÞ i¼1

kd D D  n ð1 þ WACCÞi ð1 þ WACCÞ

ð14:15Þ

After summing we have the following expression for NPV NPV ¼ S þ 

  n  k Dt ð1  ð1 þ WACCÞn Þ NOIð1  t Þ 1  1 þ d WACC WACC 1 þ WACC

kd Dð1  ð1 þ WACCÞn Þ D  WACC ð1 þ WACCÞn ð14:16Þ

In the case of more frequent ( p-times per year) payment of income taxes, frequent payments of interest on the debt we have NPV ¼ S þ 

np X

np n X X NOIð1  t Þ kd Dt iþ i=p i¼1 ð1 þ WACCÞ i¼1 p1 ð1 þ WACCÞ 1

kd D i

i¼1

p2 ð1 þ WACCÞ=p2



D ð1 þ WACCÞn

ð14:17Þ

After summing we have the following expression for NPV   n  k Dt ð1  ð1 þ WACCÞn Þ NOIð1  t Þ 1   1 þ d NPV ¼ S þ 1 WACC 1 þ WACC p ð1 þ WACCÞ =p1  1 1



n

kd Dð1  ð1 þ WACCÞ Þ D   1=p ð 1 þ WACC Þn 2 p2 ð1 þ WACCÞ  1 ð14:18Þ

Long-term investment projects To obtain the expression for NPV of long-term investment projects one should find the limit of Eq. (14.18) n ! 1.

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Innovative Investment Models with Debt Repayment at the End of the Project

NPV ¼ S þ 

14.3

NOIð1  t Þ kd Dt  þ  1 WACC p1 ð1 þ WACCÞ =p1  1



kd D

p2 ð1 þ WACCÞ =p2  1 1



ð14:19Þ

The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt

14.3.1 With Flows Separation In this case, operating and financial flows are separated and are discounted, using different rates: the operating flow at the rate equal to the equity cost ke, depending on leverage, and credit flow–at the rate equal to the debt cost kd, which until fairly large values of leverage remain constant and start to grow only at high values of leverage L, when there is a danger of bankruptcy. After-tax flow of capital for each period in this case is equal to Eq. (14.6) NOIð1  t Þ þ kd Dt

ð14:20Þ

After summing over n-periods, one gets the following expression for NPV NPV ¼ I þ ¼ I þ

n n X NOIð1  t Þ X k d Dt i þ i i¼1 ð1 þ k e Þ i¼1 ð1 þ k d Þ

NOIð1  t Þ ð1  ð1 þ ke Þn Þ þ Dt ð1  ð1 þ kd Þn Þ ke

ð14:21Þ

In case of a few ( p) payments of tax on profit per period the expression for NPV will be modified NPV ¼ I þ

np n X NOIð1  t Þ X kd Dt þ i i=p : i¼1 ð1 þ k e Þ i¼1 pð1 þ k d Þ

ð14:22Þ

After summing over n-periods, one gets the following modified expression for NPV

14.3

The Effectiveness of the Investment Project from the Perspective of the. . .

NPV ¼ I þ

NOIð1  t Þ½1  ð1 þ ke Þn  kd Dt ½1  ð1 þ k d Þn  i þ h 1 ke p ð1 þ k Þ =p  1

233

ð14:23Þ

d

To obtain the expression for NPV of long-term investment projects one should find the limit of Eq. (14.23) n ! 1. NPV ¼ I þ

NOIð1  t Þ k d Dt i þ h 1 ke p ð1 þ k Þ =p  1

ð14:24Þ

d

14.3.2 Without Flows Separation In this case, operating and financial flows are not separated and both are discounted, using the general rate (as which, obviously, WACC can be selected). After summing expression (14.6) over n-periods, one gets the following expression for NPV NPV ¼ I þ

NPV ¼ I þ

n n X X NOIð1  t Þ k d Dt iþ i i¼1 ð1 þ WACCÞ i¼1 ð1 þ WACCÞ

ð14:25Þ

NOIð1  t Þ k Dt ð1  ð1 þ WACCÞn Þ þ d ð1  ð1 þ WACCÞn Þ WACC WACC ð14:26Þ

In case of a few ( p) payments of tax on profit per period the expression for NPV will be modified NPV ¼ I þ

np n X X NOIð1  t Þ kd Dt þ i i=p : i¼1 ð1 þ WACCÞ i¼1 pð1 þ WACCÞ

ð14:27Þ

After summing over n-periods, one gets the following modified expression for NPV

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14

Innovative Investment Models with Debt Repayment at the End of the Project

  n  NOIð1  t Þ 1  1 NPV ¼ I þ WACC 1 þ WACC þ

kd Dt ð1  ð1 þ WACCÞn Þ   1 p ð1 þ WACCÞ =p  1

ð14:28Þ

To obtain the expression for NPV of long-term investment projects one should find the limit of Eq. (14.28) n ! 1. NPV ¼ I þ

14.4

NOIð1  t Þ k d Dt  þ  1 WACC p ð1 þ WACCÞ =p  1

ð14:29Þ

Discount Rates

In the case without the division of credit and investment flows (both flows are discounted using the same rate, as which can be, obviously, chosen WACC) WACC is calculated by the following formulas:for arbitrary duration project (Brusov and Filatova 2021) 1  ð1 þ WACCÞn ¼ WACC



1  ð1 þ k0 Þn

k0  1  kd pwd t

;

½1ð1þk d Þn  1 ð1þk d Þ =p 1

ð14:30Þ

for long-term project (Brusov and Filatova 2021) ! WACC ¼ k0 

k wt 1 1 d d p ð1 þ k d Þ1=p  1

ð14:31Þ

In case of the division of credit and investment flows (and thus discounting of the payments using two different rates equity cost ke and debt cost kd) ke should be found from the equation WACC ¼ ke we þ kd wd ð1  t Þ,

ð14:32Þ

where we substitute WACC from the formula (14.30) for arbitrary duration project and from the formula (14.31) for long-term project. Note that formulas (14.30) and (14.31) are quite different from original formulas by Brusov–Filatova–Orekhova (Brusov et al. 2015, 2018b; Filatova et al. 2008) and Modigliani–Miller (Мodigliani and Мiller 1958, 1963, 1966), where payments of

14.5

Results and Discussions

235

interest on debt and of tax on income are made once per year and are turned into them if we put p ¼ 1 1  ð1 þ WACCÞn 1  ð1 þ k 0 Þn ¼ WACC k0  ð1  wd t ½1  ð1 þ kd Þn Þ WACC ¼ k 0  ð1  wd t Þ

14.5

ð14:33Þ

ð14:34Þ

Results and Discussions

In this section, we study numerically with the use of Microsoft Excel the effectiveness of the four models, created above by us. We consider long-term projects as well as projects of arbitrary duration from two perspectives: the owners of equity and debt and the equity holders only. We will study the case without flows separation. In this case, operating and financial flows are not separated and both are discounted, using the general rate, as which we select the weighted average cost of capital, WACC. Let us start from the study numerically of the dependence of the discount rates (weighted average cost of capital, WACC) on leverage level L at different frequencies of payment of tax on profit p.

14.5.1 Numerical Calculation of the Discount Rates 14.5.1.1

The Long-Term Investment Projects

For long-term investment projects (the Modigliani–Miller limit) discount rate (WACC) in case of arbitrary frequency of payment of tax on profit p is described by formula (14.31) ! WACC ¼ k0 

k w t 1 1 d d p ð1 þ kd Þ1=p  1

For WACC calculation we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0;1;2....;10.

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Innovative Investment Models with Debt Repayment at the End of the Project

Table 14.1 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12

L 0 1 2 3 4 5 6 7 8 9 10

WACC p¼1 0.2200 0.1980 0.1907 0.1870 0.1848 0.1833 0.1823 0.1815 0.1809 0.1804 0.1800

p¼6 0.2200 0.1967 0.1890 0.1851 0.1828 0.1812 0.1801 0.1793 0.1787 0.1781 0.1777

p ¼ 12 0.2200 0.1966 0.1888 0.1849 0.1826 0.1810 0.1799 0.1791 0.1784 0.1779 0.1775

Fig. 14.1 Dependence of the weighted average cost of capital, WACC, on leverage level L at different frequency of payments of tax on profit p ¼ 1,6,12

14.5.1.2

The Arbitrary Duration Investment Projects

For arbitrary duration investment projects discount rate (WACC) in case of arbitrary frequency of payment of tax on profit p is described by formula (14.30) [modified BFO formula (Brusov and Filatova 2021)]. For WACC calculation we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0;1;2....;10; p ¼ 1;6;12; n ¼ 3

14.5

Results and Discussions

Table 14.2 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year company

237

L 0 1 2 3 4 5 6 7 8 9 10

WACC p¼1 0.2200 0.1987 0.1915 0.1879 0.1858 0.1843 0.1833 0.1825 0.1819 0.1815 0.1811

p¼6 0.2200 0.1974 0.1899 0.1861 0.1838 0.1823 0.1812 0.1804 0.1798 0.1793 0.1788

p ¼ 12 0.2200 0.1973 0.1897 0.1859 0.1836 0.1821 0.1810 0.1802 0.1795 0.1790 0.1786

Fig. 14.2 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project

1  ð1 þ WACCÞn ¼ WACC



1  ð1 þ k0 Þn

k0  1  kd pwd t



½1ð1þk d Þn  1 ð1þk d Þ =p 1

Note, that obtained above dependences of WACC on leverage level L will be used below under study the effectiveness of the long-term as well of arbitrary duration investment projects, from the perspective of the owners of equity capital and of the owners of equity and debt.

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Fig. 14.3 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project (larger scale)

14.5.1.3

The Effectiveness of the Long-Term Investment Project from the Perspective of the Owners of Equity Capital

For long-term investment projects discount rate (WACC) in case of arbitrary frequency of payment of tax on profit p is described by formula (14.19) and for NPV one has NPV ¼ S þ 



NOIð1  t Þ kd Dt  þ  1 WACC p1 ð1 þ WACCÞ =p1  1 kd D

p2 ð1 þ WACCÞ =p2  1 1



For NPV calculation we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0;1;2....;10; p1 ¼ p2 ¼ p ¼ 1;6;12; S ¼ 1000; D ¼ LS; NOI ¼ 800.

14.5.1.4

The Effectiveness of the Long-Term Investment Project from the Perspective of the Owners of Equity and Debt

For long-term investment projects NPV in case of arbitrary frequency of payment of tax on profit p is described by formula (14.29)

14.5

Results and Discussions

239

Table 14.3 Dependence of WACC and NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12

NPV p¼1 1909 4899 7895 10,893 13,892 16,891 19,890 22,890 25,889 28,889 31,889

L 0 1 2 3 4 5 6 7 8 9 10

p¼6 1909 4891 7883 10,878 13,874 16,871 19,868 22,865 25,863 28,860 31,858

p ¼ 12 1909 4891 7882 10,877 13,873 16,869 19,866 22,863 25,860 28,858 31,855

NPV(L) at different p 35000

30000

NPV

25000 20000 NPV (p=1) 15000

NPV (p=6) NPV (p=12)

10000 5000 0 1

2

3

4

5

6

7

8

9

10

L Fig. 14.4 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for long-term project

NPV ¼ I þ

NOIð1  t Þ k d Dt  þ  1 WACC p ð1 þ WACCÞ =p  1

For NPV calculation we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0;1;2....;10; p1 ¼ p2 ¼ p ¼ 1;6;12; S ¼ 1000; D ¼ LS; NOI ¼ 800.

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NPV(L) at different p 29000 28500

NPV

28000 27500 NPV (p=1) 27000

NPV (p=6) NPV (p=12)

26500

26000 25500 9

10

L Fig. 14.5 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for long-term project (larger scale)

Table 14.4 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12

14.5.1.5

L 0 1 2 3 4 5 6 7 8 9 10

NPV p¼1 1909 4606 7364 10,139 12,922 15,709 18,498 21,289 24,081 26,874 29,667

p¼6 1909 4659 7478 10,316 13,163 16,015 18,870 21,726 24,583 27,441 30,300

p ¼ 12 1909 4665 7489 10,334 13,188 16,047 18,908 21,771 24,635 27,500 30,365

The Effectiveness of the Arbitrary Duration Investment Projects from the Perspective of the Owners of Equity Capital

For the arbitrary duration investment projects NPV in case of arbitrary frequency of payment of tax on profit p is described by formula (14.18)

14.5

Results and Discussions

241

NPV(L) at different p 30000

25000

NPV

20000

NPV (p=1)

15000

NPV (p=6) NPV (p=12)

10000

5000

0 1

2

3

4

5

6

7

8

9

10

L

Fig. 14.6 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for long-term project

NPV(L) (I=const) at different p 28000 27000 26000

NPV

25000 NPV (p=1) 24000

NPV (p=6) NPV (p=12)

23000 22000 21000 8

9

10

L

Fig. 14.7 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for long-term project (larger scale)

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  n  k Dt ð1  ð1 þ WACCÞn Þ NOIð1  t Þ 1   1 NPV ¼ S þ þ d 1 WACC 1 þ WACC p ð1 þ WACCÞ =p1  1 1



n

kd Dð1  ð1 þ WACCÞ Þ D   1=p ð 1 þ WACC Þn p2 ð1 þ WACCÞ 2  1

For NPV calculation we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0;1;2....;10; p1 ¼ p2 ¼ p ¼ 1;6;12; S ¼ 1000; D ¼ LS; NOI ¼ 800; n ¼ 3. Table 14.5 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project

NPV p¼1 307 885 1438 1985 2530 3072 3615 4156 4698 5239 5780

L 0 1 2 3 4 5 6 7 8 9 10

p¼6 307 869 1406 1937 2464 2991 3516 4042 4566 5091 5615

p ¼ 12 307 867 1403 1932 2458 2983 3506 4030 4553 5076 5599

NPV(L) (S=const) at different p 6000

5000

NPV

4000

3000

NPV (p=1) NPV (p=6)

2000

NPV (p=12)

1000

0 1

2

3

4

5

6

7

8

9

10

L Fig. 14.8 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project

14.5

Results and Discussions

243

NPV(L) at different p 5200 5000

NPV

4800 NPV (p=1)

4600

NPV (p=6) NPV (p=12)

4400 4200 4000 8

9

10

L Fig. 14.9 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project (larger scale)

14.5.1.6

The Effectiveness of the Arbitrary Duration Investment Project from the Perspective of the Owners of Equity and Debt

For the arbitrary duration investment projects NPV in case of arbitrary frequency of payment of tax on profit p is described by formula (14.28)   n  k Dt ð1  ð1 þ WACCÞn Þ NOIð1  t Þ 1   1 NPV ¼ I þ þ d 1 WACC 1 þ WACC p ð1 þ WACCÞ =p  1 For NPV calculation we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0;1;2....;10; p1 ¼ p2 ¼ p ¼ 1;6;12; S ¼ 1000; D ¼ LS; NOI ¼ 800; n ¼ 3.

14.5.1.7

Discussions

The analysis of results from Tables 14.1, 14.2, 14.3, 14.4, 14.5, 14.6 and Figs. 14.1, 14.2, 14.3, 14.4, 14.5, 14.6, 14.7, 14.8, 14.9, 14.10, 14.11 shows that: 1. The weighted average cost of capital, WACC, decreases with leverage level L at all values of frequency of payments of tax on profit p (see Tables 14.1, 14.2 and Figs. 14.1–14.3). 2. With an increase of p, WACC decreases: WACC(L ) curves lie lower with an increase of p.

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Table 14.6 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project

NPV p¼1 307 761 1218 1676 2135 2594 3053 3511 3970 4429 4888

L 0 1 2 3 4 5 6 7 8 9 10

p¼6 307 771 1238 1707 2175 2644 3113 3582 4051 4520 4989

p ¼ 12 307 772 1240 1710 2179 2649 3119 3589 4059 4529 4999

NPV(L) (I=const) at different p 5000 4500 4000

3500

NPV

3000 2500

NPV (p=1)

2000

NPV (p=6)

NPV (p=12)

1500 1000 500 0 1

2

3

4

5

6

7

8

9

10

L Fig. 14.10 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project

3. The difference between curves, corresponding to p ¼ 1 and p ¼ 6 is much more than the difference between curves, corresponding to p ¼ 6 and p ¼ 12. This difference decreases with p. It turns out that increasing the number of payments of tax of profit per year p leads to a decrease in the cost of attracting capital (WACC). Whether this decrease in discount rate will increase the effectiveness of investment project? As we see from Tables 14.3–14.6 and Figs. 14.4–14.11, the situation is different for owners of equity capital and for owners of equity and debt capital. 4. NPV practically linear increases with leverage level at all values of frequency of payments of tax on profit p and all frequency of payments of interest on debt.

14.5

Results and Discussions

245

NPV(L) (I=const) at different p 4600

4400

NPV

4200

4000

NPV (p=1) NPV (p=6)

3800

NPV (p=12)

3600

3400 8

9

10

L Fig. 14.11 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1,6,12 for 3-year project (larger scale)

5. In case of considering of the effectiveness of long-term investment project for owners of equity capital or long-term projects NPV is changed with change of p (both p1 and p2 are equal everywhere below) but on very small value (it is seen from the Tables, but not from the Figures). 6. In case of considering of the effectiveness of long-term investment project for owners of equity and debt NPV is changed with change of p more significantly (it is seen from the Tables and from the Figures). 7. For arbitrary duration projects this difference in NPV with change of p is more significant and should be accounted under valuation of the effectiveness of investment project. 8. In case of considering of the effectiveness of investment project for owners of equity capital, we need to note that increase of p leads to decrease of NPV: this means that the effectiveness of investment project decreases with p. 9. In case of considering of the effectiveness of investment project for owners of equity and debt capital, we need to note that situation is opposite and an increase of p leads to an increase of NPV: this means that the effectiveness of investment project increases with p. 10. Above results show that in former case companies should pay tax on profit and interest on debt once per year, while in latter case more frequent payments are profitable for the effectiveness of investment. Thus, while for long-term projects NPV the impact of more frequent payments of both values p1 and p2 is insignificant, for arbitrary duration projects the account of the frequency of both types of payments could be important and could lead to more

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significant influence on the effectiveness of investment project, decreasing it (in former case), or increasing it (in latter case). Note that specific value of the effect depend on the values of parameters of the project (k0, kd, n, t, S, etc).

14.6

Conclusions

We developed here eight innovative investment models: long-term [describing by Eqs. (14.14), (14.19), (14.24), and (14.29)] as well arbitrary duration [describing by Eqs. (14.13), (14.18), (14.23), and (14.28)], which account payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly) as it happened in practice. These investment models allow investigating the impact of all main parameters of investment projects (equity value S, investment value I, net operating income NOI, leverage level L, profitability of investments β, tax on profit t, project duration n, equity cost k0, debt cost kd and number of payments of interest on debt p1 and of tax on income p2) on main indicator effectiveness of investment projects NPV (net present value). They could be used for investigation of different problems of investments, such as influence of debt financing, leverage level, taxing, project duration, method of financing, number of payments of interest on debt and of tax on income, and some other parameters on the efficiency of investments and other problems. In particular, they will improve the issue of project ratings (Brusov et al. 2018a). These new models allow making more correct evaluation of the effectiveness of investment projects long-term as well as of arbitrary duration. Numerical calculations, conducted for four investment models (without flow separation), show that: • In case of considering of the effectiveness of investment project for owners of equity capital, the increase of number of payments of tax on income and of interest on debt p leads to decrease of NPV: this means that the effectiveness of investment project decreases with p. • In the case of considering the effectiveness of investment project for owners of equity and debt capital, the increase of number of payments of tax on income and of interest on debt p leads to increase of NPV: this means that the effectiveness of investment project increases with p. In the former case, companies should pay tax on profit and interest on debt once per year, while in the latter case, more frequent payments are profitable for the effectiveness of investment. Thus, while for long-term projects NPV the impact of more frequent payments of both values p1 and p2 is insignificant, for arbitrary duration projects the account of the frequency of both types of payments could be important and could lead to more significant influence on the effectiveness of investment project, decreasing it (in former case), or increasing it (in latter case). Note that the specific value of the effect depend on the values of parameters of the project (k0, kd, n, t, S, etc). There are some limitations of the applicability of the proposed models:

References

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1. For long-term projects they are connected with the limitations of the Modigliani– Miller theory. 2. For consideration without flows separation they are connected with the wellknown limitations of WACC approach. 3. For arbitrary duration projects (using BFO theory) they are connected with the fact that not all the conditions of real investments are accounted yet.

References Bond S, Meghir C (1994) Dynamic investment models and the firm’s financial policy. Rev Econ Stud 67:197–222 Brusov P, Filatova T (2021) The Modigliani–Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9:1198. https://doi.org/10.3390/math9111198 Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International, Switzerland, pp 1–373. https://www.springer.com/gp/book/ 9783319147314 Brusov P, Filatova T, Orehova N (2018a) Ratings: critical analysis and new approaches of quantitative and qualitative methodology. Springer Nature, Switzerland, pp 1–379. https:// www.springer.com/de/book/9783030562427 Brusov P, Filatova T, Orehova N, Eskindarov M (2018b) Modern corporate finance, investments, taxation and ratings, 2nd edn. Springer Nature, Switzerland, pp 1–571 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Glob Econ 9:257–267. https://doi.org/10.3390/math9111286 Brusov P, Filatova T, Orekhova N, Kulik V, Chang S-I, Lin G (2021a) Generalization of the Modigliani–Miller theory for the case of variable profit. Mathematics 9(11):1286. https://doi. org/10.3390/math9111286 Brusov P, Filatova T, Chang S-I, Lin G (2021b) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9:1491. https://doi.org/10. 3390/math9131491 Dammon R, Senbet L (1988) The effect of taxes and depreciation on corporate investment and financial leverage. J Financ 2:357. https://doi.org/10.1111/j.1540-6261.1988.tb03944.x DeAngelo H, Masulis R (1980) Optimal capital structure under corporate and personal taxation. J Financ Econ 8(1):3 Dotan A, Ravid S (1985) On the interaction of real and financial decisions of the firm under uncertainty. J Financ 40(2):501 Filatova TV, Orekhova NP, Brusova АP (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP (2022) Impact of advance payments of tax on profit on effectiveness of investments. Mathematics 10(4):666. https://doi.org/10.3390/math10040666 Kang J (1995) The conditional relationship between financial leverage and corporate investment: further clarification. J Bus Financ Acc 22(8):1211. https://doi.org/10.1111/j.1468-5957.1995. tb00902.x Lang LE, Ofek E, Stulz R (1996) Leverage, investment, and firm growth. J Financ Econ 40:3–29 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297

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Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Rosenbaum J, Pearl J (2013) Investment banking valuation models. Wiley, USA, pp 1–342 Whited T (1992) Debt, liquidity constraints and corporate investment: evidence from panel data. J Financ 47:1425–1461

Chapter 15

Investment Models with Advance Frequent Payments of Tax on Profit and of Interest on Debt

15.1

Introduction

Investments are quite important in economy and finance of any country. At the current stage, the role of investment is increased many times as well as the role of the evaluation of the efficiency of investment projects, which allows for the realization of the most effective projects in the context of scarcity and limited investment resources, increases. Since virtually most investment projects use debt financing, the study of the influence of capital structure and debt financing on the efficiency of investment projects and determining the optimal capital structure is especially important at the present time. This is why in spite of the fact that a lot of different types of investment models have been developed—stochastic, dynamics (Bond and Meghir 1994), investment banking valuation models (Rosenbaum and Pearl 2013), etc.—the main problem, which has been discussed during the last few decades, is the impact of debt financing on the efficiency of investment projects and on the investment decisions of companies. A comprehensive literature review of investment models has been done in Brusov et al. (2021). Below we shortly summarize it and add a review of a few recent papers devoted to the investigation of frequent payments of tax on income as well as advance payments of tax on income.

15.1.1 The Literature Review Lang et al. (1996) have shown that there is a negative correlation between future growth and leverage level at the company level and, for diversified companies, at the level of business segments. This negative correlation between growth and leverage level holds for companies with low Tobin’s q ratio, but not for high-q companies or companies in high-q industries (the q ratio, or Tobin’s q ratio, equals the market © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_15

249

250

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Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

value of a company divided by its assets’ replacement cost). Therefore, for companies known to have good investment opportunities, leverage does not reduce growth, but it is negatively correlated to growth for companies whose growth opportunities are either not recognized by the capital markets or are not valuable enough to overcome the effects of their debt load. Whited (1992) studied the influence of debt financing on companies’ investment decisions and showed that there is a significant positive correlation between leverage level and investment for large companies, while for medium companies a negative correlation between leverage level and investment took place. Kang (1995) studied the connection between leverage level and investment decisions. “Interdependent tax models” were used to try to explain the specifics of company leverage levels by analyzing the interdependency between financing decisions and investment. Some authors mentioned a positive connection via the fact that the financial risk and hence the cost of bond financing decrease with an increase in investment at a given leverage level. A negative connection has been mentioned by DeAngelo and Masulis (1980) and Dotan and Ravid (1985). The first authors concluded this since the tax benefits of debt compete with those of capital investment. The second authors refer to the fact that financial risk and thus the cost of bond financing will increase with investment increase. The impact of investment increase on financial risk may depend on companyspecific factors, like company-specific technology (Dammon and Senbet 1988). An analysis of the impact of corporate and personal taxes on a firm’s optimal investment and financing decisions under uncertainty is provided in Dammon and Senbet (1988). By endogenizing firms’ investment decisions, it extends the DeAngelo and Masulis capital structure model. The authors’ results indicate that the existing predictions about the relationship between investment-related and debt-related tax shields must be modified in cases where investment is allowed to adjust optimally. Differences in production technologies across companies may thus explain why the empirical results of recent cross-sectional studies have not conformed to the predictions of DeAngelo and Masulis (1980). Below we discuss some portfolio investment models as well as behavioral aspects of investors, which play an important role in investments. Among those portfolio, investment models is the well-known Black–Litterman model, which was created by Fischer Black and Robert Litterman in 1992. They developed the model to address the problems that institutional investors encountered during the application of modern portfolio theory in practice. Starting with an asset allocation based on the equilibrium assumption, the model then modifies that allocation by accounting the investors’ opinions with respect to future asset performance. Anthony Loviscek (2021) has applied the Black–Litterman model of modern portfolio theory to well-known index mutual funds—one guided by the classic 60%/ 40% stock/bond allocation and one based on an all-equity allocation. The period under their study is from 2000 to 2020. Although statistical evidence supports that the efficacy of a precious metal allocation is elusive, the results suggest an average

15.1

Introduction

251

allocation of about 2% for “buy-and-hold” investors who seek one. He has shown that, from 2003 to 2010 and from 2016 to 3Q2020, the allocations were in the range of 5–10%. Other periods, however, register only a little more than 0%. Nusret Cakici and Adam Zaremba (2021) reconsidered the performance of the Fama–French factors in global markets. As their results showed, the value, profitability, and investment factors are far less reliable than what is commonly thought. Their performance depends strongly on the geographical regions and periods examined. Moreover, most factor returns are driven by the smallest companies. Virtually no value or investment effects are present among the big companies representing most of the total market capitalization worldwide. These results cast doubt on the five-factor model’s applicability in international markets, citing that the smallest companies are typically not invested in by major financial institutions. A growing number of investors want to use firm sustainability information in their investment decision processes to avoid risk, satisfy their own asset preference, or find a new alpha-generating factor. Not too many users of environment, social and governance (ESG) data understand how ESG ratings change over time. Bahar Gidwani (2020) used the CSRHub data set to show that ESG ratings regress strongly toward the mean. The observed regression persists during 9 years within the ratings data, for a sample set of more than 8000 firms. Newly rated firms show even more reversion than “seasoned” firms. Firms can only rarely maintain an especially high or low ESG rating. Investors and firm managers should understand that ESG ratings are likely to change toward the mean. This however does not mean that a good firm is getting worse or a bad one is getting better. Keith C. Brown et al. (2021) have developed statistics (holdings-based) to estimate the volatility with time of investment style characteristics of funds. They found that funds with lower levels of style volatility significantly outperform funds with higher levels of style volatility on a risk-adjusted basis. It was concluded that deciding to maintain a less volatile investment style is an important aspect of the portfolio management process. Some behavioral aspects of investors have been considered by Marcos EscobarAnel et al. (2020), who introduced a strategy generalizing the CPPI (Constant Proportion Portfolio Insurance) approach. The target of this strategy is to guarantee the investment goal or floor during participation in the performance of the assets and limiting the downside risk of the portfolio at the same time. The authors show that the strategy accounts for the following behavioral aspects of investors: a risk-averse behavior for gains, distorted probabilities recognition and a risk-seeking behavior for losses. The developed strategy turns out to be optimal within the Cumulative Prospect Theory framework by Kahneman and Tversky (1996). In the current paper, we study the innovative investment models with advance frequent payments of tax on income. Thus, below we add a review of a few recent papers devoted to the investigation of frequency of payments of tax on income as well as advance payments of tax on income. Brusov et al. (2021) discuss one of the additional limitation of Modigliani–Miller theory: a method of tax on profit payments. Modigliani–Miller theory accounts these

252

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payments as annuity-immediate while in practice these payments are made in advance and thus should be accounted as annuity-due. Brusov et al. (2021) have generalized the Modigliani–Miller theory for the case of advance payments of tax on profit, which is widely used in practice, and show that this leads to some important consequences, which change seriously all the main statements by Modigliani and Miller. These consequences are as follows: WACC starts to depend on debt cost kd, WACC turns out to be lower than in case of classical Modigliani–Miller theory and thus company capitalization becomes higher than in ordinary Modigliani–Miller theory. We show that dependence of equity cost on leverage level L is still linear, but the tilt angle with respect to L-axis turns out to be smaller: this could lead to modification of the dividend policy of the company. A correct account of a method of tax on profit payments demonstrates that shortcomings of Modigliani–Miller theory are dipper, than everybody suggested: the underestimation of WACC really turns out to be bigger, as well as overestimation of the capitalization of the company. This means that systematic risks arising from the use of modified Modigliani–Miller theory (MMM theory) (which is more correct than the “classical” one) in practice are higher than it was suggested by the “classical” version of this theory. Brusov et al. (2020b) use the modified Modigliani–Miller theory (MMM theory) and apply it for rating methodologies needs. A serious modification of MMM theory in order to use it in rating procedure has been required. The financial “ratios” were incorporated into MMM theory. The dependence of the weighted average cost of capital (WACC), which plays the role of discount rate, on coverage and leverage ratios is analyzed. Obtained results make it possible to use the power of the MMM theory in rating and create a new base for rating methodologies. Brusov et al. (2021) the Modigliani–Miller theory has been generalized and developed taking into account one of the conditions of the real functioning of companies for the case of paying income tax with an arbitrary frequency (monthly, quarterly, semiannual or annual payments). While a return is not required more than once a year, businesses may be responsible for filing estimated taxes based on profits earned. This requirement is dependent on showing a profit. For example, sole proprietors must file estimated taxes on profits quarterly, on the 15th day of April, June, September, and January. In Russia, tax on profit payments could be made annually, quarterly, or monthly. Authors suppose, that more frequent payment of income tax impacts on all main financial indicators of the company and leads to some important consequences. They used analytical and numerical methods: all main formulas of the modified Modigliani–Miller theory have been derived theoretically and then used to obtain all main financial indicators of company and their dependences on different parameters by MS Excel. Authors show that: (1) all Modigliani–Miller theorems, statements, and formulas change; (2) all main financial indicators, such as the weighted average cost of capital (WACC), company value, V, and equity cost, ke, depend on the frequency of tax on profit payments; (3) in the case of income tax payments more than once per year (at p 6¼ 1), as takes place in practice, the WACC, company value, V and equity cost, and ke start depend on debt cost, kd,

15.1

Introduction

253

while in ordinary (classical) Modigliani–Miller theory all these values do not depend on kd; (4) obtained results allow a company to choose the number of payments of tax on profit per year (of course, within actual tax legislation): more frequent payments of income tax are beneficial for both parties, for the company and for the tax regulator. As we mentioned in Brusov et al. (2021), some of the major problems under the evaluation of the effectiveness of investment projects are suggested as follows: 1. Which financial flows should be taken into account when calculating the parameters of efficiency of a project (NPV, IRR, etc.)? 2. How many discount rates should be used for discounting various cashflows? 3. How can these discount rates be accurately evaluated? The first problem is still under intensive discussion. Concerning the second issue either two different discount rates could be used (equity cost, ke and debt cost, kd) or a general discount rate for operating and financial flows (for which WACC can, obviously, be chosen). Concerning the third issue, we need to note that, in the last decade, significant progress in the accurate determination of the cost of the equity and company weighted average cost, which just are the discount rates when evaluating the effectiveness of the project, has been achieved. The progress is mainly associated with the studies by Brusov, Filatova, and Orekhova (BFO theory) (Brusov et al. 2015, 2018b; Filatova et al. 2008), in which a general theory of capital cost of the company and its capital structure was established, and the dependence of capital cost on leverage level and on the age of a company was found for the companies of arbitrary age. The main difference between their theory and Modigliani–Miller theory is that the former one removes the assumption of perpetuity for the companies under discussion, which leads to a significantly different new theory from the theory established by the Nobel laureates Modigliani and Miller (1958, 1963, 1966). In modern conditions, the requirements for improving the quality of assessing the effectiveness of investments have increased. The modern investment models, which have been well-tested in real economic situations, have been developed by Brusov, Filatova, and Orekhova (Brusov et al. 2015, 2018b; Filatova et al. 2008). They have created long-term as well as arbitrary duration models and have considered the effectiveness of the investment project from two points of view: from the equity holders and from the owners of equity and debt. NPV in each of these cases could be calculated by two different methods: with the division of credit and investment flows and using two different discounting rates (equity cost, ke and debt cost, kd), and without such a division and using a general discounting rate (for which WACC can, obviously, be chosen).

254

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

15.1.2 The Discount Rates As we mentioned above, one of the most important elements of calculating the effectiveness of investment projects is the assessment of the discount rate. In the case of long-term investment models without the division of credit and investment flows, the discount rate WACC has been calculated using the modified for advance payments Modigliani–Miller formula (Modigliani and Miller 1958, 1963, 1966) WACC ¼ k 0  ð1  wd t  ð1 þ kd ÞÞ

ð15:1Þ

while in the case of arbitrary duration models, the modified for advance payments Brusov-Filatova-Orekhova formula for WACC (Brusov et al. 2015, 2018b; Filatova et al. 2008) 1  ð1 þ WACCÞn 1  ð1 þ k 0 Þn ¼ WACC k0  ð1  wd t  ½1  ð1 þ kd Þn   ð1 þ kd ÞÞ

ð15:2Þ

has been used. Note that both formulas for the weighted average cost of capital, WACC, [Modigliani–Miller (Eq. 15.1) and Brusov-Filatova-Orekhova (Eq. 15.2)] in case of advance payments are different from ones in case of payments at the ends of periods (Modigliani and Miller 1958, 1963, 1966; Brusov et al. 2015, 2018b; Filatova et al. 2008). Here and below, WACC is the weighted average cost of capital; k0 is the equity cost at zero leverage (L ¼ 0); kd is the debt cost; wd is the debt share; t is the tax on profit; n is the project duration; ke is the equity cost; and L is the leverage level. In the case of long-term investment models with the division of credit and investment flows, the discount rate for discounting the investment flows (equity cost ke) has been calculated using the Modigliani–Miller formula (Modigliani and Miller 1958, 1963, 1966) k e ¼ k 0 þ L  ðk 0  k d Þð1  t Þ

ð15:3Þ

while in the case of arbitrary duration models, equity cost ke has been calculated from the formula WACC ¼ ke we þ kd wd ð1  t Þ

ð15:4Þ

using the Brusov-Filatova-Orekhova value for WACC (Brusov et al. 2015, 2018b; Filatova et al. 2008). The calculation methods of the discount rates (WACC, equity cost ke) has been generalized for the real conditions of the implementation of investment projects: for arbitrary frequency of payments of tax on profit (Brusov et al. 2021) and for advance payments of tax on profit (Brusov et al. 2020a).

15.1

Introduction

255

In this paper, new modern investment models, both long-term and arbitrary duration, will be created, as close as possible to real investment conditions. They will account for the advance payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly), which are applied in real economic practice. Their verification will lead to the creation of a comprehensive system of adequate and correct assessment of the effectiveness of the company’s investment program and its investment strategy.

15.1.3 The Structure of the Paper The structure of the paper is as follows: 1. In Sect. 15.1 above, we presented: 1.1 The literature review 1.2 Discount rates 2. In Sect. 15.2, we consider the effectiveness of the investment project from the perspective of the equity holders only. In Sect. 15.2.1, we consider a case with flow separation. In Sect. 15.2.2, we consider a case without flow separation. 3. In Sect. 15.3, we consider the effectiveness of the investment project from the perspective of the owners of equity and debt. In Sect. 15.3.1, we consider a case with flow separation. In Sect. 15.3.2, we consider a case without flow separation. 4. In Sect. 15.4, the problem of calculation of discount rates is discussed and expressions for their modified values are obtained. 5. In Sect. 15.5, we study numerically with the use of Microsoft Excel the effectiveness of the four models, created by us in this paper. We consider long-term projects as well as projects of arbitrary duration from two perspectives: the owners of equity and debt and the equity holders only without the division of credit and investment flows. 6. In Conclusions, we discuss obtained results and their impact on the correctness of valuation of efficiency of investment projects. As well we compare obtained in current paper results for advance payments of tax on profit with ones, obtained by Brusov et al. (2021) for payments of tax on profit at the end of periods.

256

15.2

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only

We will consider the effectiveness of the investment project from two points of view: from the equity holders and from the owners of equity and debt. NPV in each of these cases could be calculated by two different methods: with the division of credit and investment flows and using two different discounting rates, and without such a division and using a general discounting rate (for which WACC can, obviously, be chosen). The following designations are used below: The equity value S, the investment value I, the net operating income NOI, the leverage level L, the profitability of investments β, the tax on profit t, the project duration n, the equity cost k0, the debt cost kd, and the number of payments of interest on debt p1 and of tax on income p2; D is the debt value. In the first case (from the perspective of the equity holders), at the initial moment in time, T ¼ 0 investments are equal to S and the flow of capital, CF, for the period is equal to: CF ¼ ðNOI  kd DÞð1  t Þ:

ð15:5Þ

In addition to the tax shields, kdDt includes a payment of interest on a loan, kdD. We suppose below that interests on the loan are paid in equal shares of kdD during all periods and principal repayment is made at the end of the project. In the second case (from the perspective of the equity and debt holders), the negative flows (the interest and duty paid by owners of equity) are returned to the project and they are exactly equal to the positive flows obtained by owners of debt capital. Thus, in this case, the only effect of debt financing is the effect of the tax shield, generated from the tax relief: interest on the loan is entirely included in the cost and thus reduces the tax base. After-tax flow of capital, CF, for each period is equal to CF ¼ NOIð1  t Þ þ kd Dt

ð15:6Þ

and the value of investments at the initial moment in time T ¼ 0 is equal to I ¼ S  D. Below two different ways of discounting will be considered: 1. If operating and financial flows are not separated, both flows are discounted by the general rate. In this case, the weighted average cost of capital (WACC) can be selected as a discounting rate. For long-term projects, we will use the Modigliani– Miller formula for WACC (Мodigliani and Мiller 1958, 1963), modified by us for the case of payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly) and for projects of finite (arbitrary) duration we will use the Brusov–Filatova–Orekhova formula for WACC (Brusov

15.2

The Effectiveness of the Investment Project from the Perspective of the. . .

257

et al. 2018b; Filatova et al. 2008), modified by us for the case of payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly). 2. Operating and financial flows are separated and are discounted at different rates: the operating flow at the rate which is equal to the equity cost ke, depending on leverage and on the number of payments of interest on debt and of tax on income, and credit flow at the rate which is equal to the debt cost kd. Note that loan capital is the least risky because the credit (including the interest on credit) is paid after taxes in the first place. Therefore, the cost of credit will always be less than the equity cost, whether for ordinary or for preference shares ke > kd; kp > kd. Here, ke; kp is the equity cost of ordinary or of preference shares consequently.

15.2.1 With Flow Separation In this case, the expression for NPV (net present value) per period has a view NPV ¼ NOIð1  t Þ  kd Dð1  t Þ ¼ NOIð1  t Þ þ kd Dt  kd D:

ð15:7Þ

Here, the first term is the value of operating income from the investment project after-tax deduction, the second term is the value of the tax shield, the third term is the value of interest on debt for one period. We will need the following auxiliary formulas for summing the reduced values of financial flows (Eq. 15.7) when calculating NPV: For annual advance payments of interest on debt and of tax on income: n1 X i¼0

1  ð1 þ k d Þ 1 i ¼ 1 1  1þk ð1 þ k d Þ d

n

¼

1 þ kd  ð1  ð1 þ k d Þn Þ kd

ð15:8Þ

For more frequent advance payments [p times per period (per year)] of interest on debt and of tax on income (semiannually, quarterly, monthly): np1 X

R i

i¼0

pð1 þ kd Þ=p

¼

R 1  ð1 þ k d Þ  p 1  1 1= ð1þkd Þ

n

p

¼

n

R ð1  ð1 þ k d Þ Þ  ð1 þ k d Þ  1 p ð1 þ kd Þ =p  1

1=p

ð15:9Þ

Similar formulas are obtained using the cost of equity ke and WACC as the discount rates. Summing up the given values of financial flows for each period (Eq. 15.7), we obtain for NPV of n-years project in the case of separated flows:

258

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

NPV ¼ S þ 

n n1 n1 X X NOIð1  t Þ X k d Dt kd D þ  i i 1 þ k 1 þ k d Þi ð Þ ð Þ ð 1 þ k e d i¼1 i¼0 i¼0

D ð1 þ k d Þn

ð15:10Þ

Here, the second term is the reduced value of operating income from the investment project, the third term is the reduced value of the tax shield, the fourth term is the reduced value of interest paid annually (at the beginning of the year), and the fifth term is the reduced value of the debt paid at the end of the project. After summing, we have the following expression for NPV:   n  NOIð1  t Þ 1  1 þ Dt ð1  ð1 þ kd Þn Þ  ð1 þ k d Þ ke 1 þ ke D Dð1  ð1 þ kd Þn Þ  ð1 þ k d Þ  ð1 þ k d Þn ð15:11Þ

NPV ¼ S þ

In the case of more frequent ( p-times per year) advance payment of income taxes ( p1) and frequent advance payments of interest on debt ( p2) we have: np1 np1 n X X NOIð1  t Þ X kd Dt kd D þ  i i=p i=p i¼1 ð1 þ k e Þ i¼0 p1 ð1 þ k d Þ 1 i¼01 p2 ð1 þ k d Þ 2 D  ; ð15:12Þ ð1 þ kd Þn

NPV ¼ S þ

After summing, we have the following expression for NPV:   n  1 NOIð1  t Þ k Dt ð1  ð1 þ kd Þn Þ  ð1 þ kd Þ =p1 1   NPV ¼ S þ  1 þ d 1 ke 1 þ ke p ð1 þ k Þ =p1  1 1



n

1=p

kd Dð1  ð1 þ kd Þ Þ  ð1 þ kd Þ   1 p2 ð1 þ kd Þ =p2  1

2



d

D ð1 þ k d Þn ð15:13Þ

Long-term investment projects To obtain the expression for NPV of long-term investment projects, one should find the limit of Eq. (15.13) n ! 1.

15.2

The Effectiveness of the Investment Project from the Perspective of the. . .

NOIð1  t Þ k Dt  ð1 þ kd Þ =p1 k D  ð1 þ kd Þ =p2   d  ð15:14Þ þ d 1 1 ke p ð1 þ k Þ =p1  1 p ð1 þ k Þ =p2  1 1

NPV ¼ S þ

259

d

1

1

2

d

15.2.2 Without Flow Separation In the case of no separation between operating and financial flows, both flows are discounted by the general rate WACC. The credit reimbursable at the end of the project can be discounted either at the same rate WACC or at the debt cost rate kd. Below, a uniform rate for WACC has been used. Summing up the given values of financial flows for each period (Eq. 15.7), we obtain for NPV of n-years project in the case without separated flows: NPV ¼ S þ 

n n1 n1 X X X NOIð1  t Þ kd Dt þ  i i i¼1 ð1 þ WACCÞ i¼0 ð1 þ WACCÞ i¼0

kd D D  n ð1 þ WACCÞi ð1 þ WACCÞ

ð15:15Þ

After summing, we have the following expression for NPV: NPV ¼ S þ

  n  NOIð1  t Þ 1  1 WACC 1 þ WACC

kd Dt ð1  ð1 þ WACCÞn Þ  ð1 þ WACCÞ WACC kd Dð1  ð1 þ WACCÞn Þ  ð1 þ WACCÞ D   WACC ð1 þ WACCÞn þ

ð15:16Þ

In the case of more frequent ( p-times per year) advance payment of income taxes and frequent payments of interest on the debt we have: NPV ¼ S þ 

np1 n X X NOIð1  t Þ k d Dt þ i i=p ð 1 þ WACC Þ i¼1 i¼0 p1 ð1 þ WACCÞ 1

np1 X

kd D

i¼0

p2 ð1 þ WACCÞ=p2

i

D  ð1 þ WACCÞn

After summing, we have the following expression for NPV:

ð15:17Þ

260

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

  n  NOIð1  t Þ 1  1 NPV ¼ S þ WACC 1 þ WACC þ

k d Dt ð1  ð1 þ WACCÞn Þ  ð1 þ WACCÞ =p1   1 p1 ð1 þ WACCÞ =p1  1



k d Dð1  ð1 þ WACCÞn Þ  ð1 þ WACCÞ =p2 D    1=p ð 1 þ WACC Þn 2 p2 ð1 þ WACCÞ  1

1

1

ð15:18Þ Long-term investment projects To obtain the expression for NPV of long-term investment projects, one should find the limit of Eq. (15.18) n ! 1. NOIð1  t Þ k Dt ð1 þ WACCÞ =p1  þ d 1 WACC p1 ð1 þ WACCÞ =p1  1 1

NPV ¼ S þ

k Dð1 þ WACCÞ =p2 d  1 p2 ð1 þ WACCÞ =p2  1 1



15.3

ð15:19Þ

The Effectiveness of the Investment Project from the Perspective of the Owners of Equity and Debt

15.3.1 With Flow Separation In this case, operating and financial flows are separated and are discounted using different rates: the operating flow at the rate equal to the equity cost ke, depending on leverage, and credit flow at the rate equal to the debt cost kd, which remain constant until fairly large values of leverage and start to grow only at high values of leverage L, when there is a danger of bankruptcy. After-tax flow of capital for each period, in this case, is equal to Eq. (15.6): CF ¼ NOIð1  t Þ þ kd Dt

ð15:20Þ

After summing over n-periods, one gets the following expression for NPV:

15.3

The Effectiveness of the Investment Project from the Perspective of the. . .

NPV ¼ I þ

n X NOIð1  t Þ i¼1

ð1 þ k e Þ

i

þ

n1 X i¼0

261

kd Dt ð1 þ k d Þi

NOIð1  t Þ ¼ I þ ð1  ð1 þ ke Þn Þ þ Dt ð1  ð1 þ kd Þn Þ  ð1 þ k d Þ ke ð15:21Þ In the case of a few ( p) advance payments of tax on profit per period, the expression for NPV will be modified: NPV ¼ I þ

np1 n X NOIð1  t Þ X k d Dt þ i i=p ð Þ 1 þ k e i¼1 i¼0 pð1 þ k d Þ

ð15:22Þ

After summing over n-periods, one gets the following modified expression for NPV: NPV ¼ I þ

NOIð1  t Þ½1  ð1 þ ke Þn  ke

kd Dt ½1  ð1 þ k d Þn ð1 þ kd Þ =p h i 1 p ð1 þ kd Þ =p  1 1

þ

ð15:23Þ

To obtain the expression for NPV of long-term investment projects, one should find the limit of Eq. (15.23) n ! 1. NOIð1  t Þ k Dt ð1 þ kd Þ =p i þ hd 1 ke p ð1 þ k Þ =p  1 1

NPV ¼ I þ

ð15:24Þ

d

15.3.2 Without Flow Separation In the case of no separation between operating and financial flows, both flows are discounted by the general rate WACC. After summing the expression (15.6) over n-periods, one gets the following expression for NPV: NPV ¼ I þ

n n1 X X NOIð1  t Þ kd Dt iþ ð 1 þ WACCÞi ð 1 þ WACC Þ i¼1 i¼0

ð15:25Þ

262

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

NPV ¼ I þ þ

NOIð1  t Þ ð1  ð1 þ WACCÞn Þ WACC

ð15:26Þ

k d Dt ð1  ð1 þ WACCÞn Þð1 þ WACCÞ WACC

In the case of a few ( p) advance payments of tax on profit per period, the expression for NPV will be modified: NPV ¼ I þ

np1 n X X NOIð1  t Þ k d Dt iþ i=p i¼1 ð1 þ WACCÞ i¼0 pð1 þ WACCÞ

ð15:27Þ

After summing over n-periods, one gets the following modified expression for NPV:   n  NOIð1  t Þ 1  1 NPV ¼ I þ WACC 1 þ WACC kd Dt ð1  ð1 þ WACCÞn Þð1 þ WACCÞ =p   1 p ð1 þ WACCÞ =p  1 1

þ

ð15:28Þ

To obtain the expression for NPV of long-term investment projects, one should find the limit of Eq. (15.28) n ! 1. NOIð1  t Þ k Dt ð1 þ WACCÞ =p  þ d 1 WACC p ð1 þ WACCÞ =p  1 1

NPV ¼ I þ

15.4

ð15:29Þ

Discount Rates

In the case without the division of credit and investment flows (both flows are discounted using the same rate, for which WACC can obviously be chosen), WACC is calculated by the following formulas: For arbitrary project duration (Brusov et al. 2018b): 1  ð1 þ WACCÞn ¼ WACC

1  ð1 þ k 0 Þn   1=p n k0  1  kd pwd t ½1ð1þkd Þ 1=pð1þkd Þ ð1þkd Þ 1

For long-term projects (Modigliani and Miller 1958, 1963):

ð15:30Þ

15.5

Results and Discussions

263 = k w t ð1 þ k d Þ p 1 d d 1=p p ð1 þ k d Þ  1 1

WACC ¼ k0 

! ð15:31Þ

Note, that both formulas for the weighted average cost of capital, WACC, [Modigliani–Miller (Eq. 15.31) and Brusov-Filatova-Orekhova (Eq. 15.30)] in case of advance payments, obtained by Brusov et al. (2021) are different from ones in case of payments in the ends of periods (classical Modigliani–Miller formulas and Brusov-Filatova-Orekhova formulas). In the case of the division of credit and investment flows (and thus discounting of the payments using two different rates, equity cost ke and debt cost kd), ke should be found from the equation WACC ¼ ke we þ kd wd ð1  t Þ where we substitute WACC from the formula (15.30) for a project with arbitrary duration and from the formula (15.31) for a long-term project. Note that formulas (15.30) and (15.31) are quite different as well from the original formulas by Brusov-Filatova-Orekhova (Brusov et al. 2015, 2018b; Filatova et al. 2008) and Modigliani–Miller (1958, 1963), where payments of interest on debt and of tax on income are made once per year at the end of the periods. They are turned into an advanced version of them (not original ones, which suppose the payments at the end of periods), if we put p ¼ 1: 1  ð1 þ WACCÞn 1  ð1 þ k 0 Þn ¼ WACC k0  ¼ ð1 k0 wðd1t   ½1w þþkdkÞdn WACC ÞÞ  ð1 þ kd ÞÞ d t ð1ð1

15.5

ð15:32Þ ð15:33Þ

Results and Discussions

In this section, we study numerically with the use of Microsoft Excel the effectiveness of the four models created above by us. We consider long-term projects as well as projects of arbitrary duration from two perspectives: the owners of equity and debt and the equity holders only. We will study the case without flow separation. In this case, operating and financial flows are not separated and both are discounted, using the general rate, for which we select the weighted average cost of capital, WACC. Let us start from the study numerically on the dependence of the discount rates (weighted average cost of capital, WACC) on leverage level L at different frequencies of payment of tax on profit p.

264

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

15.5.1 Numerical Calculation of the Discount Rates 15.5.1.1

The Long-Term Investment Projects

For long-term investment projects (the Modigliani–Miller limit), the discount rate (WACC) in the case of arbitrary frequency of advance payment of tax on profit p is described by formula (15.31) (Brusov et al. 2021): = k w t ð1 þ k d Þ p 1 d d 1=p p ð1 þ k d Þ  1 1

WACC ¼ k0 

!

For WACC calculation, we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0; 1; 2...; 10. Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 is shown in Table 15.1 and Figs. 15.1, 15.2.

15.5.1.2

The Arbitrary Duration Investment Projects

For arbitrary duration investment projects, the discount rate (WACC) in the case of arbitrary frequency of payment of tax on profit p is described by formula (15.30) [modified BFO formula (15.22)]. For WACC calculation, we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0; 1; 2...; 10; p ¼ 1; 4; 12; n ¼ 3

Table 15.1 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12

L 0 1 2 3 4 5 6 7 8 9 10

WACC p¼1 0.2200 0.1949 0.1866 0.1824 0.1799 0.1782 0.1770 0.1761 0.1754 0.1749 0.1744

p¼4 0.2200 0.1961 0.1881 0.1842 0.1818 0.1802 0.1790 0.1782 0.1775 0.1770 0.1766

p ¼ 12 0.2200 0.1964 0.1885 0.1845 0.1822 0.1806 0.1795 0.1786 0.1780 0.1775 0.1770

15.5

Results and Discussions

265

WACC(L) at different p 0.2300

0.2200

WACC

0.2100

0.2000

WACC (p=1) WACC (p=4)

0.1900

WACC (p=12)

0.1800

0.1700 0

1

2

3

4

5

6

7

8

9

10

L Fig. 15.1 Dependence of the weighted average cost of capital, WACC, on leverage level L at different frequencies of payments of tax on profit, p ¼ 1; 4; 12

WACC(L) at different p 0.1810 0.1800

WACC

0.1790 0.1780 WACC (p=1) 0.1770

WACC (p=4) WACC (p=12)

0.1760

0.1750 0.1740 5

6

7

8

9

10

L Fig. 15.2 Dependence of the weighted average cost of capital, WACC, on leverage level L at different frequencies of payments of tax on profit, p ¼ 1; 4; 12 (larger scale)

266

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

Table 15.2 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year company

WACC p¼1 0.2199 0.1957 0.1876 0.1836 0.1812 0.1796 0.1784 0.1775 0.1768 0.1763 0.1758

L 0 1 2 3 4 5 6 7 8 9 10

p¼4 0.2199 0.1969 0.1892 0.1853 0.1830 0.1815 0.1804 0.1795 0.1789 0.1784 0.1780

p ¼ 12 0.2199 0.1971 0.1895 0.1857 0.1834 0.1819 0.1808 0.1800 0.1793 0.1788 0.1784

WACC(L) at different p 0.2300

0.2200

WACC

0.2100

0.2000

WACC (p=1) WACC (p=4)

0.1900

WACC (p=12)

0.1800

0.1700 0

1

2

3

4

5

6

7

8

9

10

L Fig. 15.3 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project

1  ð1 þ WACCÞn ¼ WACC

1  ð1 þ k0 Þn   1= n k d wd t ½1ð1þk d Þ ð1þk d Þ p k0  1  p 1=p ð1þk d Þ 1

Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 6, 12 for a 3-year company is shown in Table 15.2 and Figs. 15.3, 15.4. Note that the dependences of WACC on leverage level L obtained above will be used below under a study of the effectiveness of long-term as well arbitrary duration investment projects, from the perspective of the owners of equity capital and of the owners of equity and debt.

15.5

Results and Discussions

267

WACC(L) at different p 0.1830 0.1820 0.1810

WACC

0.1800 0.1790 WACC (p=1)

0.1780

WACC (p=4)

0.1770

WACC (p=12)

0.1760 0.1750 0.1740 5

6

7

8

9

10

L Fig. 15.4 Dependence of WACC on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project (larger scale)

15.5.2 The Effectiveness of the Long-Term Investment Project from the Perspective of the Owners of Equity Capital For long-term investment projects, the discount rate (WACC) in the case of arbitrary frequency of advance payment of tax on profit p is described by formula (15.19) and for NPV one has: NOIð1  t Þ k Dt ð1 þ WACCÞ =p1  þ d 1 WACC p1 ð1 þ WACCÞ =p1  1 1

NPV ¼ S þ

k Dð1 þ WACCÞ =p2 d  1 p2 ð1 þ WACCÞ =p2  1 1



For NPV calculation, we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0; 1; 2...; 10; p1 ¼ p2 ¼ p ¼ 1; 4; 12; S ¼ 1000; D ¼ LS; NOI ¼ 800. It is seen from Figs. 15.5 and 15.6 that in the case of considering the effectiveness of long-term investment projects for owners of equity capital, NPV will vary with the change of p, but with not much variation (it is seen from Table 15.3 and from Fig. 15.6, but not from Fig. 15.5).

268

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

15

NPV(L) at different p 35000 30000

NPV

25000 20000

NPV (p=1) 15000

NPV (p=4) NPV (p=12)

10000 5000 0 0

1

2

3

4

5

6

7

8

9

10

L Fig. 15.5 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a long-term project

NPV(L) at different p 31600 31100

NPV

30600 NPV (p=1)

30100

NPV (p=4) NPV (p=12)

29600 29100

28600 9

10

L Fig. 15.6 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a long-term project (larger scale)

15.5

Results and Discussions

Table 15.3 Dependence of WACC and NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12

Table 15.4 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12

269

L 0 1 2 3 4 5 6 7 8 9 10

NPV p¼1 1909 4880 7867 10,858 13,852 16,846 19,841 22,837 25,833 28,829 31,825

p¼4 1909 4888 7877 10,871 13,866 16,862 19,858 22,854 25,851 28,848 31,844

p ¼ 12 1909 4889 7880 10,874 13,869 16,865 19,862 22,858 25,855 28,852 31,850

L 0 1 2 3 4 5 6 7 8 9 10

NPV p¼1 1909 4738 7648 10,581 13,525 16,474 19,427 22,382 25,338 28,295 31,252

p¼4 1909 4687 7537 10,408 13,289 16,175 19,064 21,954 24,846 27,738 30,631

p ¼ 12 1909 4676 7513 10,371 13,239 16,111 18,985 21,862 24,740 27,618 30,497

15.5.3 The Effectiveness of the Long-Term Investment Project from the Perspective of the Owners of Equity and Debt For long-term investment projects, NPV in the case of arbitrary frequency of payment of tax on profit p is described by formula (15.29): NOIð1  t Þ k Dt ð1 þ WACCÞ =p  þ d 1 WACC p ð1 þ WACCÞ =p  1 1

NPV ¼ I þ

For NPV calculation, we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0; 1; 2...; 10; p1 ¼ p2 ¼ p ¼ 1; 6; 12; S ¼ 1000; D ¼ LS; NOI ¼ 800. Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a long-term project is shown in Table 15.4 and Figs. 15.7, 15.8.

270

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

15

NPV(L) at different p 30000

25000

NPV

20000

NPV (p=1)

15000

NPV (p=4) NPV (p=12)

10000

5000

0 0

1

2

3

4

5

6

7

8

9

L

Fig. 15.7 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a long-term project

NPV(L) at different p 31500 30500

NPV

29500 28500 NPV (p=1) 27500

NPV (p=4) NPV (p=12)

26500 25500 24500 8

9

10

L

Fig. 15.8 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a long-term project (larger scale)

15.5

Results and Discussions

271

15.5.4 The Effectiveness of the Arbitrary Duration Investment Projects from the Perspective of the Owners of Equity Capital For the arbitrary duration investment projects, NPV in the case of arbitrary frequency of payment of tax on profit p is described by formula (15.18):   n  NOIð1  t Þ 1  1 NPV ¼ S þ WACC 1 þ WACC þ

k d Dt ð1  ð1 þ WACCÞn Þ  ð1 þ WACCÞ =p1   1 p1 ð1 þ WACCÞ =p1  1



k d Dð1  ð1 þ WACCÞn Þ  ð1 þ WACCÞ =p2 D    1=p ð 1 þ WACC Þn 2 p2 ð1 þ WACCÞ  1

1

1

For NPV calculation, we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0; 1; 2...; 10; p1 ¼ p2 ¼ p ¼ 1; 6; 12; S ¼ 1000; D ¼ LS; NOI ¼ 800; n ¼ 3. Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project is shown in Table 15.5 and Figs. 15.9, 15.10.

15.5.5 The Effectiveness of the Arbitrary Duration Investment Project from the Perspective of the Owners of Equity and Debt For the arbitrary duration investment projects, NPV in the case of arbitrary frequency of payment of tax on profit p is described by formula (15.28): Table 15.5 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project

L 0 1 2 3 4 5 6 7 8 9 10

NPV p¼1 307 845 1359 1865 2369 2871 3373 3874 4375 4876 5376

p¼4 307 861 1389 1911 2430 2948 3465 3982 4498 5014 5530

p ¼ 12 307 864 1396 1921 2444 2965 3485 4005 4525 5044 5563

272

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

NPV(L) at different p 6000

5000

NPV

4000

3000

NPV (p=1) NPV (p=4)

2000

NPV (p=12)

1000

0 0

1

2

3

4

5

6

7

8

9

10

L Fig. 15.9 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 6, 12 for a 3-year project

NPV(L) at different p 5550 5350

NPV

5150 NPV (p=1)

4950

NPV (p=4) NPV (p=12)

4750 4550

4350 8

9

10

L Fig. 15.10 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project (larger scale)

15.5

Results and Discussions

Table 15.6 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project

273

L 0 1 2 3 4 5 6 7 8 9 10

NPV p¼1 307 785 1267 1749 2232 2714 3197 3680 4164 4648 5132

p¼4 307 776 1248 1721 2194 2667 3140 3614 4087 4561 5034

p ¼ 12 307 774 1244 1714 2185 2657 3128 3599 4071 4542 5013

  n  NOIð1  t Þ 1  1 NPV ¼ I þ WACC 1 þ WACC kd Dt ð1  ð1 þ WACCÞn Þ  ð1 þ WACCÞ =p   1 p ð1 þ WACCÞ =p  1 1

þ

For NPV calculation, we will use the following parameters: k0 ¼ 0.22; kd ¼ 0.14; t ¼ 20%; L ¼ 0; 1; 2...; 10; p1 ¼ p2 ¼ p ¼ 1; 6; 12; S ¼ 1000; D ¼ LS; NOI ¼ 800; n ¼ 3. Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project is shown in Table 15.6 and Figs. 15.11, 15.12.

15.5.6 Discussions The analysis of results from Tables 15.1–15.6 and Figs. 15.1–15.12 shows that: 1. The weighted average cost of capital, WACC, decreases with leverage level L at all values of frequency of payments of tax on profit p (see Tables 15.1, 15.2 and Figs. 15.1–15.4). 2. With an increase in p, WACC increases: WACC(L ) curves lie higher with an increase of p. 3. The difference between curves corresponding to p ¼ 1 and p ¼ 4 is much more than the difference between the curves corresponding to p ¼ 4 and p ¼ 12. This difference decreases with p. It turns out that an increase in the number of payments of tax on profit per year p leads to an increase in the cost of attracting capital (WACC). Will this increase in the discount rate decrease the effectiveness of investment projects? As we see from Tables 15.3–15.6 and Figs. 15.5–

274

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

NPV(L) at different p 6000

5000

NPV

4000

3000

NPV (p=1) NPV (p=4)

2000

NPV (p=12)

1000

0 0

1

2

3

4

5

6

7

8

9

10

L Fig. 15.11 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 4, 12 for a 3-year project

NPV(L) at different p 5150

4950

NPV

4750

4550

NPV (p=1) NPV (p=4)

4350

NPV (p=12)

4150

3950 8

9

10

L Fig. 15.12 Dependence of NPV on leverage level L at k0 ¼ 0.22, kd ¼ 0.14 and different p ¼ 1, 6, 12 for a 3-year project (larger scale)

15.12, the situation is different for owners of equity capital and for owners of equity and debt capital. 4. NPV practically linearly increases with leverage level at all values of frequency of payments of tax on profit p and all frequency of payments of interest on debt.

15.6

Conclusions

275

5. In the case of considering the effectiveness of long-term investment projects for owners of equity capital, NPV is changed with a change of p (both p1 and p2 are equal everywhere below) but by a very small value (it is seen in the Tables, but not in the Figures); NPV is growing with p. 6. In the case of considering the effectiveness of long-term investment projects for owners of equity and debt, NPV is changed with a change of p more significantly (it is seen in the Tables and in the Figures); NPV decreases with p. 7. For arbitrary duration projects, this difference in NPV with a change of p is more significant and should be accounted under valuation of the effectiveness of an investment project; 8. In the case of considering the effectiveness of an investment project for owners of equity capital, we need to note that an increase in p leads to an increase in NPV: this means that the effectiveness of an investment project increases with p. 9. In the case of considering the effectiveness of an investment project for owners of equity and debt capital, we need to note that the situation is opposite and an increase in p leads to decrease in NPV: this means that the effectiveness of an investment project decreases with p. 10. The above results show that in the latter case, companies should pay tax on profit and interest on debt once per year, while in the former case, more frequent payments are profitable for the effectiveness of an investment. 11. Thus, while for long-term projects’ NPV, the impact of more frequent payments of both values p1 and p2 is insignificant, for arbitrary duration projects the account of the frequency of both types of payments could be important and could lead to more significant influence on the effectiveness of an investment project, increasing it (in the former case), or decreasing it (in the latter case). Note that the specific value of the effect depends on the values of the parameters in the project (k0, kd, n, t, S, etc.).

15.6

Conclusions

There are too few investment models which can numerically valuate the effectiveness of investment projects, among them the investment models developed by the authors of this paper. Moreover, investment models are practically absent which account for the conditions of the real functioning of investment projects. This paper covers this gap in the literature and science in the field of investments and develops innovative investment models which are much closer to economic practice. We developed here for the first time eight innovative investment models: longterm [described by Eqs. (15.14), (15.19), (15.24) and (15.29)] as well arbitrary duration [described by Eqs. (15.13), (15.18), (15.23) and (15.28)], which account two effects: (1) payments of interest on debt and of tax on income a few times per year (semiannually, quarterly, monthly); (2) advance payments of taxes and of debt interest, as it happened in practice. Note that no one before had investigated the

276

15

Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

impact of the frequency of payments of taxes and of debt interest and payments of taxes and of debt interest on the effectiveness of investment projects. These investment models allow for investigation of the impact of all main parameters of investment projects (equity value S, investment value I, net operating income NOI, leverage level L, profitability of investments β, tax on profit t, project duration n, equity cost k0, debt cost kd and number of payments of interest on debt p1 and of tax on income p2) on the main indicator of effectiveness of investment projects NPV (net present value). They could be used for investigation of different problems of investments, such as the influence of debt financing, leverage level, taxing, project duration, method of financing, number of payments of interest on debt and of tax on income and some other parameters on efficiency of investments and other problems. In particular, they will improve the issue of project ratings (Brusov et al. 2018a). These new models allow for making more correct evaluations of effectiveness of investment projects long-term as well as of arbitrary duration. Numerical calculations, conducted for four investment models (without flow separation) show that: • In the case of considering the effectiveness of an investment project for owners of equity capital, the increase in the number of payments of tax on income and of interest on debt p leads to an increase in NPV: this means that the effectiveness of an investment project increases with p. • In the case of considering the effectiveness of an investment project for owners of equity and debt capital, the increase in the number of payments of tax on income and of interest on debt p leads to decrease in NPV: this means that the effectiveness of an investment project decreases with p. In the latter case, companies should pay tax on profit and interest on debt once per year, while in the former case, more frequent payments are profitable for the effectiveness of an investment. Thus, while for long-term projects’ NPV, the impact of more frequent payments of both values p1 and p2 is insignificant, for arbitrary duration projects the account of the frequency of both types of payments could be important and could lead to more significant influence on the effectiveness of an investment project, decreasing it (in the latter case), or increasing it (in the former case). Note that the specific value of the effect depends on the values of the parameters in the project (k0, kd, n, t, S, etc.). There are some limitations of the applicability of the proposed models: 1. For long-term projects, they are connected with the limitations of the ModiglianiMiller theory. 2. For consideration of without flow separation, they are connected with the wellknown limitations of the WACC approach. 3. For arbitrary duration projects (using BFO theory), they are connected with the fact that not all the conditions of real investments are accounted yet. We could compare obtained by us results with the ones of our paper Brusov et al. (2021), where we have frequent payments of tax on income and of interest on

15.6

Conclusions

277

debt p at the end of periods. The results turn out to be the opposite. In Brusov et al. (2021) we have: (1) WACC decreases with p; (2) in the case of considering the effectiveness of an investment project for owners of equity capital, the increase in the number of payments of tax on income and of interest on debt p leads to a decrease in NPV: this means that the effectiveness of an investment project decreases with p; (3) in the case of considering the effectiveness of an investment project for owners of equity and debt capital, the increase in the number of payments of tax on income and of interest on debt p leads to an increase in NPV: this means that the effectiveness of an investment project increases with p. In the former case, companies should pay tax on profit and interest on debt once per year, while in the latter case, more frequent payments are profitable for the effectiveness of investment. In the current chapter, where payments of income tax and interest on debt are made in advance, we have: 1. WACC increases with p. 2. In the case of considering the effectiveness of an investment project for owners of equity capital, the increase in the number of payments of tax on income and of interest on debt p leads to an increase in NPV: this means that the effectiveness of an investment project increases with p. 3. In the case of considering the effectiveness of an investment project for owners of equity and debt capital, the increase in the number of payments of tax on income and of interest on debt p leads to the decrease in NPV: this means that the effectiveness of an investment project decreases with p. In the latter case, companies should pay tax on profit and interest on debt once per year, while in the former case, more frequent payments are profitable for the effectiveness of investment. So, we have created and fully investigated eight innovative investment models with frequent payments of income tax and interest on debt in advance (Filatova et al. 2022) and at the end of periods (Brusov et al. 2021). We have shown, that method of payments of tax on income and of interest on debt (in advance or at the ends of periods) changes drastically the effect of the number of payments of income tax and interest on debt on the effectiveness of investments. Depending on the relationship between the investor and the lender, the investor can choose (within the framework of tax legislation): • The frequency of payment of income tax • Scheme for payment of income tax (in advance or at the end of periods) Obtained results help tax regulator (Finance Ministry) to understand the influence of the number of payments of tax on income and credit regulator (Central Bank) the influence of the number of payments of interest on debt on the effectiveness of investment projects. This allows them to modify and improve tax legislation and credit policy, respectively.

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Investment Models with Advance Frequent Payments of Tax on Profit and of. . .

The contribution of this study to finance and economics is mainly related to the goal that new modern investment models have been created to be as close as possible to real investment conditions. Our models, namely, models with advance payments of interest on debt and of tax on income which occur a few times per year (semiannually, quarterly or monthly), could be more successfully applied in real economic practice.

References Black F, Litterman R (1992) Global Portfolio Optimization Financ Anal J 48:28–43 Bond S, Meghir C (1994) Dynamic investment models and the firm’s financial policy. Rev Econ Stud 67:197–222 Brown K, Harlow W, Zhang H (2021) Investment style volatility and mutual fund performance. J Invest Manag 19:78 Brusov P, Filatova T (2021) The Modigliani–Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198. https://doi.org/10.3390/math9111198 Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International, Cham, pp 1–373. https://www.springer.com/gp/book/978331 9147314 Brusov P, Filatova T, Orehova N (2018a) Ratings: critical analysis and new approaches of quantitative and qualitative methodology. Springer Nature, Switzerland, pp 1–379. https:// www.springer.com/de/book/9783030562427 Brusov P, Filatova T, Orehova N, Eskindarov M (2018b) Modern corporate finance, investments, taxation and ratings, 2nd edn. Springer Nature, Cham, Switzerland, pp 1–571 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020a) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Glob Econ 9:257–267 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Application of the Modigliani–Miller theory, modified for the case of advance payments of tax on profit, in rating methodologies. J Rev Glob Econ 9:282–292 Brusov PN, Filatova TV, Chang S-I, Lin YCG (2021) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9(13):1491. https://doi.org/10. 3390/math9131491 Cakici N, Zaremba A (2021) Size, value, profitability, and investment effects in international stock returns: are they really there? J Invest 30:65–86. https://doi.org/10.3905/joi.2021.1.176 Dammon R, Senbet L (1988) The effect of taxes and depreciation on corporate investment and financial leverage. J Financ 43:357–373. https://doi.org/10.1111/j.1540-6261.1988.tb03944.x DeAngelo H, Masulis R (1980) Optimal capital structure under corporate and personal taxation. J Financ Econ 8:3–29 Dotan A, Ravid S (1985) On the interaction of real and financial decisions of the firm under uncertainty. J Financ 40:501–517 Escobar-Anel M, Lichtenstern A, Zagst R (2020) Behavioral portfolio insurance strategies. Financ Mark Portf Manag 34:353–399 Filatova TV, Orekhova NP, Brusova АP (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova T, Brusov P, Orekhova N (2022) Impact of advance payments of tax on profit on effectiveness of investments. Mathematics 10(4):666. https://doi.org/10.3390/math10040666 Gidwani B (2020) Some issues with using ESG ratings in an investment process. J Invest 29:76–84. https://doi.org/10.3905/joi.2020.1.147.

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Kahneman D, Tversky A (1996) On the reality of cognitive illusions. Psychol Rev 103:582–591. https://doi.org/10.1037/0033-295X.103.3.582 Kang J (1995) The conditional relationship between financial leverage and corporate investment: further clarification. J Bus Financ Acc 22:1211–1219. https://doi.org/10.1111/j.1468-5957. 1995.tb00902.x Lang LE, Ofek E, Stulz R (1996) Leverage, investment, and firm growth. J Financ Econ 40:3–29 Loviscek A (2021) how much to invest, if any, in precious metals? A 21st-century perspective. J Invest 30:95–111. https://doi.org/10.3905/joi.2021.1.179 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Rosenbaum J, Pearl J (2013) Investment banking valuation models. Wiley, New York, NY, pp 1– 342 Whited T (1992) Debt, liquidity constraints and corporate investment: Evidence from panel data. J Financ 47:1425–1461

Part III

Applications of the Modigliani–Miller Theory Ratings and Rating Methodologies

Chapter 16

Application of the Modigliani–Miller Theory in Rating Methodology

16.1

Introduction

Rating agencies play a very important role in economics. Their analysis of issuer’s state, generated credit ratings of issuers help investors make a reasonable investment decision, as well as help issuers with good enough ratings get credits on lower rates, etc. But from time to time we listen about scandals involving rating agencies and their credit ratings: let us just remind the situation with sovereign rating of the USA in 2011 and of Russia in 2015 (Brusov et al. 2012). Were these ratings an objective? And how objective could be issued credit ratings in principal? To answer this question, we need to understand how rating agencies (RA) consider, evaluate, analyze. But this is the secret behind the seven seals: rating agencies stand to the death but did not reveal their secrets, even under the threat of multibillion-dollar sanctions. Thus, rating agencies represent some “black boxes,” about which information on the methods of work is almost completely absent.

16.2

The Closeness of the Rating Agencies

The closeness of the rating agencies is caused by multiple causes: 1. The desire to preserve their “know-how.” Rating agencies get big enough money for generated ratings (mostly from issuers) to replicate its methodology. 2. On the other hand closeness of rating agencies is caused by the desire to avoid public discussion of the ratings with anyone, including the issuer. It is a very convenient position—rating agency “a priori” removes himself from beneath any criticism of generated ratings. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_16

283

284

16

Application of the Modigliani–Miller Theory in Rating Methodology

3. The absence of any external control and external analysis of the methodologies resulted in the fact that shortcomings of methodologies are not subjected to serious critical analysis and stored long enough. The illustration of the closeness of rating agencies is the behavior of the S&P (Standard & Poor’s) Director after declining the sovereign rating of the United States, who left his position but has not opened the methodology used. But even in this situation, some analysis of the activities and findings of the rating agencies is still possible, based on knowledge and understanding of existing methods of evaluation. Rating agencies cannot use methods other than developed up to now by leading economists and financiers.

16.3

The Use of Discounting in the Rating

One of the major flaws of all existing rating methodologies is a failure or a very narrow use of discounting. But even in those rare cases where it is used, it is not quite correct, since the discount rate when discounting financial flows is chosen incorrectly. The need to take into account the time factor in terms of discounting is obvious, because it is associated with the time value of money. The financial part of the rating is based on a comparison of generated income with the value of the debt and the interest payable. Because income and disbursement of debt and interest are separated in time, the use of discounting when comparing revenues with the value of debt and interest is absolutely necessary for assigning credit ratings for issuers. This raises the question about the value of discount rate. This question has always been one of the major and extremely difficult in many areas of Finance: corporate finance, investment, it is particularly important in business valuation, where a slight change in the discount rate leads to a significant change in the assessment of company capitalization, that is used by unscrupulous appraisers for artificial bankruptcy of the company. And the value of discount rate is extremely essential as well in rating. The most correct valuation of discount rate could be done within BrusovFilatova-Orekhova (BFO) theory (Brusov 2018a, b; Brusov and Filatova 2021; Brusov et al. 2015, 2018a, b, d, e, f, g, 2019, 2020a, b, c, 2021a, b; Filatova et al. 2008, 2018) and with less accuracy within Modigliani–Miller theory (Modigliani and Miller 1958, 1963, 1966).

16.4

Incorporation of Parameters, Using in Ratings, into Perpetuity Limit of Modern Theory of Capital Structure by Brusov–Filatova–Orekhova

In quantification of the creditworthiness of the issuers the crucial role belongs to the so-called financial “ratios,” which constitute direct and inverse ratios of various generated cash flows to debt values and interest ones. We could mention such ratios

16.5

Models

285

as DCF/Debt, FFO/Debt, CFO/Debt, FOCF/Debt, FFO/cash interest, EBITDA/ interest, Interests/EBITDA, Debt/EBITDA and some others. We incorporate these rating parameters (financial “ratios”) into the modern theory of capital structure—BFO theory (for beginning into its perpetuity limit) (Brusov et al. 2018c). The importance of such incorporation, which has been done by us for the first time, is in using this theory as a powerful tool when discounting financial flows using the correct discounting rate in rating. Only this theory allow to valuate adequately the weighted average cost of capital WACC and equity cost of capital ke used when discounting financial flows. The use of the tools of well-developed theories in rating opens completely new horizons in the rating industry, which could go from mainly the use of qualitative methods of the evaluation of the creditworthiness of issuers to predominantly quantitative evaluation methods that will certainly enhance the quality and correctness of the rating. Currently, rating agencies just directly use financial ratios, while the new methodology will allow [knowing the values of these “relations” (and parameter k0)] determine the correct values of discount rates (WACC and ke) that should be used when discounting the various financial flows, both in terms of their timing and forecasting. This has required the modification of the BFO theory (and its perpetuity limit— Modigliani–Miller theory), as used in financial management the concept of “leverage” as the ratio of debt value to the equity value substantially differs from the concept of “leverage” in the rating, where it is understood as the direct and inverse ratio of the debt value to the generated cash flow values (income, profit, etc.). The authors introduced some additional ratios, allowing more fully to characterize the issuer’s ability to repay debts and to pay interest thereon. Thus the bridge is building between the discount rates (WACC, ke) used when discounting financial flows, and “ratios” in the rating methodology. The algorithm for finding the discount rates for given ratio values is developed.

16.5

Models

Two kinds of models of the evaluation of the creditworthiness of issuers, accounting for the discounting of financial flows could be used in rating: one-period model and multi-period model.

16.5.1 One-Period Model One-period model is described by the following formula (see Fig. 16.1)

286

16

Application of the Modigliani–Miller Theory in Rating Methodology

Fig. 16.1 One-period model

CF

t

kdD

t

1

CFð1 þ iÞt2 t  D þ kd Dð1 þ iÞt2 t1
 CFð1 þ iÞt2 t  D 1 þ k d ð1 þ iÞt2 t1

D

t

2

ð16:1Þ

Here, CF is the value of income for period, D is debt value, t, t1, t2 the moments of income, payment of interest and payment of debt consequently, i is the discount rate, kd is credit rate and kdD is interest on credit.

16.5.2 Multi-period Model One-period model of the evaluation of the creditworthiness of issuers, accounting for the discounting of financial flows could be generalized for more interesting multiperiod cases. Multi-period model is described by the following formula X X h  t t  t t i CF j 1 þ i j 2j j  D j 1 þ kdj 1 þ i j 2j 1j j

ð16:2Þ

j

Here CFj is income for j-th period, Dj is debt value in j-th period, tj, t1j, t2jthe moments of income, payment of interests and payment of debt consequently in j-th period, ij is the discount rate in j-th period, kdj is credit rate in j-th period. There are several options to work with these models: 1. One can check the creditworthiness of the issuer, knowing parameters CFj, Dj, tj, t1j, t2j, kdj and defining discount rate i by the method described below. 2. When the preset Dj, tj, t1j, t2j, kdj, one can determine which income CFj the issuer would require to ensure its creditworthiness. 3. When the preset Dj, tj, t1j, t2j, kdj one can define an acceptable level of debt financing (including the credit value Dj and credit rates kdj) when issuer retains its creditworthiness.

16.6

Theory of Incorporation of Parameters, Using in Ratings, into Perpetuity. . .

16.6

287

Theory of Incorporation of Parameters, Using in Ratings, into Perpetuity Limit of Modern Theory of Capital Structure by Brusov–Filatova–Orekhova

For the first time, we incorporate below the parameters, using in ratings, into perpetuity limit of modern theory of capital structure by Brusov–Filatova–Orekhova (BFO theory). We will consider two kinds of ratios: coverage ratios and leverage ratios. Let us start with the coverage ratios.

16.6.1 Coverage Ratios We will consider three kinds of coverage ratios: coverage ratios of debt, coverage ratios of interest on the credit, and coverage ratios of debt and interest on the credit.

16.6.1.1

Coverage Ratios of Debt Here i1 ¼ CF=D

ð16:3Þ

Modigliani–Miller theorem for case with corporate taxes (Мodigliani and Мiller 1958, 1963) tells that capitalization of leveraged company, VL, is equal to the capitalization of unleveraged company, V0, plus tax shield for perpetuity time, Dt, V L ¼ V 0 þ Dt:

ð16:4Þ

Substituting the expressions for both capitalizations, one has CF CF ¼ þ Dt WACC k0 Dividing both parts by D one gets i1 i ¼ 1 þt WACC k0 i k WACC ¼ 1 0 i1 þ tk 0

ð16:5Þ

This ratio (i1) can be used to assess the following parameters used in rating, DCF/Debt, FFO/Debt, CFO/Debt, FOCF/Debt and some others. We will use the last formula to build a curve of dependence WACC(i1).

288

16.6.1.2

16

Application of the Modigliani–Miller Theory in Rating Methodology

Coverage Ratios of Interest on the Credit Here i2 ¼ CF=kd D

ð16:6Þ

Using the Modigliani–Miller theorem for case with corporate taxes V L ¼ V 0 þ Dt, we derive the expression for WACC(i2) CF CF ¼ þ Dt WACC k0 i2 i i ¼ 2þ 2 WACC k 0 kd i k k WACC ¼ 2 0 d i2 k d þ tk 0

ð16:7Þ

This ratio (i2) can be used to assess the following parameters, used in rating, FFO/cashinterest,EBITDA/interest and some others. We will use the last formula to build a curve of dependence WACC(i2).

16.6.1.3

Coverage Ratios of Debt and Interest on the Credit (New Ratios)

Let us consider the coverage ratios of debt and interest on the credit simultaneously: this is new ratio, introduced by us for the first time here. Here i3 ¼

CF Dð1 þ k d Þ

ð16:8Þ

Using as above the Modigliani–Miller theorem for case with corporate taxes V L ¼ V 0 þ Dt one gets the expression for WACC(i3)

16.6

Theory of Incorporation of Parameters, Using in Ratings, into Perpetuity. . .

289

CF CF ¼ þ Dt WACC k0 i3 i t ¼ 3þ WACC k0 1 þ kd i k ð1 þ k d Þ WACC ¼ 3 0 i3 ð1 þ k d Þ þ tk 0

ð16:9Þ

This ratio (i3) can be used to assess the following parameters used in rating, FFO/Debt + interest, EBITDA/Debt + interest and some others. We will use the last formula to build a curve of dependence WACC(i3). Let us analyze the dependence of company’s weighted average cost of capital (WACC) on the coverage ratios on debt i1, on interest on the credit i2 and on coverage ratios on debt and interest on the credit with the following data: k0 ¼ 12%; kd ¼ 6%; t ¼ 20%; ij run from 0 up to 10. The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on interest on the credit i2 is presented at Fig. 16.3. The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt and interest on the credit i2 is presented in Fig. 16.4. The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i1, on interest on the credit i2, and on debt and interest on the credit i3 is presented at Fig. 16.5. It is seen from Figs. 16.2–16.5 that WACC(ij) is increasing function on ij with saturation around ij value of order 1 for ratios i1 and i3 and of order 4 or 5 for ratios i2. At saturation, WACC reaches the value k0 (equity value at zero leverage level). This means that for high values of ij one can choose k0 as a discount rate with a good accuracy. Thus the role of parameter k0 increases drastically. The method of determination of parameter k0 has been developed by Anastasiya Brusova (2011). So,

WACC

WACC(i1) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

2

4

6

8

10

12

i1 Fig. 16.2 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i1

290

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Application of the Modigliani–Miller Theory in Rating Methodology

WACC

WACC(i2) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0

2

4

6

8

10

12

i2 Fig. 16.3 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on interest on the credit i2

WACC

WACC(i3) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0

2

4

6

8

10

12

i3 Fig. 16.4 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt and interest on the credit i3

parameter k0 is the discount rate for limit case of high values of ij (see however below for more detailed consideration). It is clear from Figs. 16.2–16.5 that case of low values of ij requires more detailed consideration. Let us consider the situation with low values of ij which seems to be the case of the most interest.

16.6

Theory of Incorporation of Parameters, Using in Ratings, into Perpetuity. . .

291

WACC(i) 0.14 0.12

WACC

0.1 0.08 0.06 0.04 0.02 0 0

2

4

6

8

10

12

I WACC 1

WACC 2

WACC 3

Fig. 16.5 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i1, on interest on the credit i2, and on debt and interest on the credit i3

16.6.2 More Detailed Consideration Below we consider the case of low values of ij with more details. ij will vary from zero up to 1 with all other parameters to be the same (Figs. 16.6, 16.7, 16.8 and 16.9). More detailed consideration leads us to the following conclusions: 1. In case of coverage ratio on debt and interest on the credit i3 WACC goes to saturation very fast: with an accuracy of 20% at i3 ¼ 0.15 and with an accuracy of 5% at i3 ¼ 0.5. 2. In case of coverage ratio on debt i1, WACC practically linearly increases with parameter i1 and goes to saturation at i1 ¼ 0.1. 3. In case of coverage ratio on interest on the credit i2 WACC increases with parameter i2 much more slowly than in two previous cases and goes to saturation at high values of i2: with an accuracy of 10% at i2 ¼ 4.

16.6.3 Leverage Ratios Let us consider now the leverage ratios. We will consider three kinds of leverage ratios: leverage ratios of debt, leverage ratios of interest on the credit, and leverage ratios of debt and interest on the credit.

292

16

Application of the Modigliani–Miller Theory in Rating Methodology

WACC(i1) 0.1400 0.1200 0.1000 0.0800 WACC(i1)

0.0600 0.0400 0.0200 0.0000 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Fig. 16.6 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt and interest on the credit i1

WACC(i2) 0.0900 0.0800 0.0700 0.0600 0.0500 WACC(i2)

0.0400 0.0300 0.0200 0.0100 0.0000 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Fig. 16.7 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt and interest on the credit i2

16.6.3.1

Leverage Ratios for Debt Here l1 ¼ D=CF

ð16:10Þ

As above for coverage ratios, we use the Modigliani–Miller theorem for case with corporate taxes

16.6

Theory of Incorporation of Parameters, Using in Ratings, into Perpetuity. . .

293

WACC(i3) 0.1400 0.1200 0.1000

0.0800 WACC(i3)

0.0600 0.0400 0.0200 0.0000 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

Fig. 16.8 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt and interest on the credit i3

Fig. 16.9 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i1, on interest on the credit i2, and on debt and interest on the credit i3

294

16

Application of the Modigliani–Miller Theory in Rating Methodology

V L ¼ V 0 þ Dt, we derive the expression for WACC(l1) CF CF ¼ þ Dt WACC k0 1 1 ¼ þ l1 t WACC k0 k0 WACC ¼ 1 þ tl1 k 0

ð16:11Þ

This ratio (l1) can be used to assess of the following parameters used in rating, Debt/EBITDAand some others. We will use last formula to build a curve of dependence WACC(l1).

16.6.3.2

Leverage Ratios for Interest on Credit Here l2 ¼ k d D=CF

ð16:12Þ

We use again the Modigliani–Miller theorem for case with corporate taxes V L ¼ V 0 þ Dt, we derive the expression for WACC(l2) CF CF ¼ þ Dt WACC k0 1 1 l t ¼ þ 2 WACC k 0 kd k0 kd WACC ¼ k d þ tl2 k0

ð16:13Þ

This ratio (l2) can be used to assess the following parameters used in rating, Interests/EBITDAand some others. We will use the last formula to build a curve of dependence WACC(l2).

16.6.3.3

Leverage Ratios for Debt and Interest on Credit Here l3 ¼ Dð1 þ kd Þ=CF

Using the Modigliani–Miller theorem for case with corporate taxes

ð16:14Þ

16.6

Theory of Incorporation of Parameters, Using in Ratings, into Perpetuity. . .

295

V L ¼ V 0 þ Dt, we derive the expression for WACC(l3) CF CF ¼ þ Dt WACC k0 1 1 l t ¼ þ 3 WACC k 0 1 þ kd k ð1 þ k d Þ WACC ¼ 0 1 þ k d þ tl3 k0

ð16:15Þ

This ratio (l3) can be used to assess the following parameters used in rating, Debt +interest / FFO, Debt+interest / EBIT, Debt+interest / EBITDA(R), and some others. We will use the last formula to build a curve of dependence WACC(l3). Let us analyze the dependence of company’s weighted average cost of capital (WACC) on the leverage ratios with the following data: k0 ¼ 12%; kd ¼ 6%; t ¼ 20%; li runs from 0 up to 10. The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on debt l1 is presented in Fig. 16.10. The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on interest on credit l2 is presented in Fig. 16.11. The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on debt and interest on credit l3 is presented in Fig. 16.12.

WACC

WACC (l1) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0

0

2

4

6

8

10

12

l1 Fig. 16.10 The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on debt l1

296

16

Application of the Modigliani–Miller Theory in Rating Methodology

WACC

WACC (l2) 0.14 0.12 0.1 0.08 0.06 0.04 0.02 0 0

2

4

6

8

10

12

l2 Fig. 16.11 The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on interest on credit l2

WACC(l3) 0.14

0.12

WACC

0.1 0.08 0.06 0.04 0.02 0 0

2

4

6

8

10

12

l3 Fig. 16.12 The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on debt and interest on credit l3

The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on debt, l1, on interest on credit, l2, and on debt and interest on credit, l3 simultaneously is presented in Fig. 16.13. Analysis of the dependences of company’s weighted average cost of capital (WACC) on the leverage ratio on debt, l1, on interest on credit, l2, and on debt and interest on credit, l3 shows the following: for all leverage ratios weighted average cost of capital (WACC) decreases with leverage ratios. For leverage ratio on debt l1 and leverage ratio on debt and interest on credit l3 WACC decreases very similar and practically linearly from k0 ¼ 12% at l1,3 ¼ 0 up to 9.7% at l1,3 ¼ 10. For leverage

16.7

Equity Cost

297

WACC (l) 0.14 0.12

WACC

0.1

0.08 0.06 0.04 0.02 0

0

2

4

6

8

10

12

l WACC1

WACC2

WACC3

Fig. 16.13 The dependence of company’s weighted average cost of capital (WACC) on the leverage ratio on debt, l1, on interest on credit, l2, and on debt and on interest on credit, l3 simultaneously

ratio on interest on credit l2 WACC decreases nonlinearly and much faster from k0 ¼ 12% at l2 ¼ 0 up to 2.4% at l2 ¼ 10.

16.7

Equity Cost

Equity cost plays a very important role in economy and finance because it is the essence of the dividend policy of companies, which should be accounted in rating. A modern approach to the dividend policy of companies, based on the real value of their equity capital cost, compared to its efficiency of planned investment is suggested in the article (Brusov et al. 2012). This allows return to the economic essence of dividends, as the payment to shareholders for the use of equity capital. Equity cost ke determines the economically reasonable dividend value. Rating agencies will be able to compare payable dividend value with economically reasonable dividend level and make a conclusion about the adequacy of the dividend policy of companies and its influence on company’s credit rating. For finding the dependence of equity cost ke on coverage ratios and leverage ratios we consider consistently the dependence of equity cost ke on ratios i1, i2, i3, l1, l2, l3, using the following formula, which couples weighted average cost of capital WACC (calculated by us above: see Tables 16.1, 16.2, 16.3, 16.4, 16.5, 16.6, 16.7, 16.8 and 16.9) and equity cost ke

298

16

Application of the Modigliani–Miller Theory in Rating Methodology

Table 16.1 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i1

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

i1 0 1 2 3 4 5 6 7 8 9 10

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

Table 16.2 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i2

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

i2 0 1 2 3 4 5 6 7 8 9 10

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

Table 16.3 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i3

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

i3 0 1 2 3 4 5 6 7 8 9 10

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

WACC 0 0.1171875 0.1185771 0.1190476 0.1192843 0.1194268 0.1195219 0.11959 0.1196411 0.1196809 0.1197127

WACC 0 0.085714 0.1 0.105882 0.109091 0.111111 0.1125 0.113514 0.114286 0.114894 0.115385

WACC 0 0.1173432 0.1186567 0.1191011 0.1193246 0.1194591 0.1195489 0.1196131 0.1196613 0.1196989 0.1197289

16.7

Equity Cost

299

Table 16.4 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i1

i1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

WACC(i1) 0.0000 0.0117 0.0234 0.0352 0.0469 0.0586 0.0703 0.0820 0.0938 0.1055 0.1172

Table 16.5 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i1

i2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

WACC(i2) 0.0000 0.0240 0.0400 0.0514 0.0600 0.0667 0.0720 0.0764 0.0800 0.0831 0.0857

Table 16.6 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt i3

i3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

WACC(i3) 0.0000 0.0978 0.1078 0.1116 0.1136 0.1148 0.1156 0.1162 0.1167 0.1171 0.1173

300

16

Application of the Modigliani–Miller Theory in Rating Methodology

Table 16.7 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt l1

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

l1 0 1 2 3 4 5 6 7 8 9 10

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

WACC 0.12 0.117188 0.114504 0.11194 0.109489 0.107143 0.104895 0.10274 0.100671 0.098684 0.096774

Table 16.8 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt l2

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

l2 0 1 2 3 4 5 6 7 8 9 10

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

WACC 0.12 0.085714 0.066667 0.054545 0.046154 0.04 0.035294 0.031579 0.028571 0.026087 0.024

Table 16.9 The dependence of company’s weighted average cost of capital (WACC) on the coverage ratio on debt l3

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

l3 0 1 2 3 4 5 6 7 8 9 10

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

WACC 0.12 0.117353 0.114819 0.112393 0.110068 0.107836 0.105693 0.103634 0.101654 0.099747 0.097911

16.7

Equity Cost

301

Table 16.10 The dependence of equity cost ke on coverage ratio i1 at leverage level L ¼ 1

L 1 1 1 1 1 1 1 1 1 1 1

i1 0 1 2 3 4 5 6 7 8 9 10

WACC(i1) 0.00000 0.11719 0.11858 0.11905 0.11928 0.11943 0.11952 0.11959 0.11964 0.11968 0.11971

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke –0.0480 0.1864 0.1892 0.1901 0.1906 0.1909 0.1910 0.1912 0.1913 0.1914 0.1914

Table 16.11 The dependence of equity cost ke on coverage ratio i1 at leverage level L ¼ 2

L 2 2 2 2 2 2 2 2 2 2 2

i1 0 1 2 3 4 5 6 7 8 9 10

WACC(i1) 0.00000 0.11719 0.11858 0.11905 0.11928 0.11943 0.11952 0.11959 0.11964 0.11968 0.11971

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke –0.0960 0.2556 0.2597 0.2611 0.2619 0.2623 0.2626 0.2628 0.2629 0.2630 0.2631

k e ¼ WACCð1 þ LÞ  Lkd ð1  t Þ:

ð16:16Þ

The dependence of equity cost ke on coverage ratios i1, i2, i3. Let us study the dependence of equity cost ke on coverage ratios i1, i2, i3 for the same set of parameters as used above and for leverage levels L ¼ 1 and L ¼ 2 (Tables 16.10, 16.11 and 16.12). 1. L ¼ 1 2. L ¼ 2 We could make some conclusions, based on Tables 16.13, 16.14, 16.15 and Figs. 16.14, 16.15, 16.16. In all three cases, equity cost ke increases with coverage ratios and goes to saturation at high values of coverage ratios. Saturation values increase with leverage level from 19% at L ¼ 1 up to value above 26% at L ¼ 2. Note, that for coverage ratios i1 and i2 the saturation takes place at values i1,2 of order unit, while for coverage ratio i3 the saturation takes place at much higher i3 values of order 6 or 7.

302

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Application of the Modigliani–Miller Theory in Rating Methodology

Table 16.12 The dependence of equity cost ke on coverage ratio i2 at leverage level L ¼ 1

L 1 1 1 1 1 1 1 1 1 1 1

i2 0 1 2 3 4 5 6 7 8 9 10

WACC(i2) 0.00000 0.11734 0.11866 0.11910 0.11932 0.11946 0.11955 0.11961 0.11966 0.11970 0.11973

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke –0.0480 0.1867 0.1893 0.1902 0.1906 0.1909 0.1911 0.1912 0.1913 0.1914 0.1915

Table 16.13 The dependence of equity cost ke on coverage ratio i2 at leverage level L ¼ 2

L 2 2 2 2 2 2 2 2 2 2 2

i2 0 1 2 3 4 5 6 7 8 9 10

WACC(i2) 0.00000 0.11734 0.11866 0.11910 0.11932 0.11946 0.11955 0.11961 0.11966 0.11970 0.11973

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke –0.0960 0.2560 0.2600 0.2613 0.2620 0.2624 0.2626 0.2628 0.2630 0.2631 0.2632

Table 16.14 The dependence of equity cost ke on coverage ratio i3 at leverage level L ¼ 1

L 1 1 1 1 1 1 1 1 1 1 1

i3 0 1 2 3 4 5 6 7 8 9 10

WACC(i3) 0.00000 0.08571 0.10000 0.10588 0.10909 0.11111 0.11250 0.11351 0.11429 0.11489 0.11538

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke –0.0480 0.1234 0.1520 0.1638 0.1702 0.1742 0.1770 0.1790 0.1806 0.1818 0.1828

16.7

Equity Cost

303

Table 16.15 The dependence of equity cost ke on coverage ratio i3 at leverage level L ¼ 2

L 2 2 2 2 2 2 2 2 2 2 2

WACC(i3) 0.00000 0.08571 0.10000 0.10588 0.10909 0.11111 0.11250 0.11351 0.11429 0.11489 0.11538

i3 0 1 2 3 4 5 6 7 8 9 10

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke –0.0960 0.1611 0.2040 0.2216 0.2313 0.2373 0.2415 0.2445 0.2469 0.2487 0.2502

Ke(i1) 0.3000 0.2500 0.2000 0.1500

0.1000

L=1

0.0500

L=2

0.0000 -0.0500

0

1

2

3

4

5

6

7

8

9

10

-0.1000 -0.1500

Fig. 16.14 The dependence of equity cost ke on coverage ratio i1 at two leverage level values L ¼ 1 and L ¼ 2

Equity cost ke should be used as discount rate for unleveraged (financially independent) companies. For coverage ratios i1 and i2 saturation values of equity cost ke could be used as discount rate above unit, while for coverage ratio i3 saturation values of equity cost ke could be used as discount rate at i3 value above 6 or 7. The dependence of equity cost ke on leverage ratios l1, l2, l3. We study below the dependence of equity cost ke on leverage ratios l1, l2, l3 for the same set of parameters as used above and for leverage levels L ¼ 1 and L ¼ 2 (Figs. 16.17, 16.18 and 16.19) (Tables 16.16, 16.17, 16.18, 16.19, 16.20 and 16.21).

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Ke(i2) 0.3000 0.2500 0.2000 0.1500 0.1000

L=1

0.0500

L=2

0.0000 -0.0500

0

1

2

3

4

5

6

7

8

9

10

-0.1000 -0.1500

Fig. 16.15 The dependence of equity cost ke on coverage ratio i2 at two leverage level values L ¼ 1 and L ¼ 2

Ke(i3) 0.3000 0.2500 0.2000 0.1500 0.1000

L=1

0.0500

L=2

0.0000

-0.0500

0

1

2

3

4

5

6

7

8

9

10

-0.1000 -0.1500

Fig. 16.16 The dependence of equity cost ke on coverage ratio i3 at two leverage level values L ¼ 1 and L ¼ 2

16.7

Equity Cost

305

Ke(l1) 0.3000 0.2500 0.2000 L=1

0.1500

L=2 0.1000 0.0500 0.0000 0

1

2

3

4

5

6

7

8

9

10

Fig. 16.17 The dependence of equity cost ke on leverage ratio l1 at two leverage level values L ¼ 1 and L ¼ 2

Ke(l2) 0.3000 0.2500 0.2000 0.1500

L=1

0.1000

L=2

0.0500 0.0000 -0.0500

0

1

2

3

4

5

6

7

8

9

10

Fig. 16.18 The dependence of equity cost ke on leverage ratio l2 at two leverage level values L ¼ 1 and L ¼ 2

The dependence of equity cost ke on leverage ratio l1 1. L ¼ 1 2. L ¼ 2 The dependence of equity cost ke on leverage ratios l2 1. L ¼ 1 2. L ¼ 2

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Ke(l3) 0.3000 0.2500 0.2000 L=1

0.1500

L=2 0.1000 0.0500 0.0000 0

1

2

3

4

5

6

7

8

9

10

Fig. 16.19 The dependence of equity cost ke on leverage ratio l3 at two leverage level values L ¼ 1 and L ¼ 2 Table 16.16 The dependence of equity cost ke on leverage ratio l1 at leverage level L ¼ 1

L 1 1 1 1 1 1 1 1 1 1 1

l1 0 1 2 3 4 5 6 7 8 9 10

WACC(l1) 0.12000 0.11719 0.11450 0.11194 0.10949 0.10714 0.10490 0.10274 0.10067 0.09868 0.09677

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1920 0.1864 0.1810 0.1759 0.1710 0.1663 0.1618 0.1575 0.1533 0.1494 0.1455

Table 16.17 The dependence of equity cost ke on leverage ratio l1 at leverage level L ¼ 2

L 2 2 2 2 2 2 2 2 2 2 2

l1 0 1 2 3 4 5 6 7 8 9 10

WACC(l1) 0.12000 0.11719 0.11450 0.11194 0.10949 0.10714 0.10490 0.10274 0.10067 0.09868 0.09677

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.2640 0.2556 0.2475 0.2398 0.2325 0.2254 0.2187 0.2122 0.2060 0.2001 0.1943

16.7

Equity Cost

307

Table 16.18 The dependence of equity cost ke on leverage ratio l2 at leverage level L ¼ 1

L 1 1 1 1 1 1 1 1 1 1 1

l2 0 1 2 3 4 5 6 7 8 9 10

WACC(l2) 0.12000 0.08571 0.06667 0.05455 0.04615 0.04000 0.03529 0.03158 0.02857 0.02609 0.02400

Table 16.19 The dependence of equity cost ke on leverage ratio l2 at leverage level L ¼ 2

L 2 2 2 2 2 2 2 2 2 2 2

l2 0 1 2 3 4 5 6 7 8 9 10

WACC(l2) 0.12000 0.08571 0.06667 0.05455 0.04615 0.04000 0.03529 0.03158 0.02857 0.02609 0.02400

Table 16.20 The dependence of equity cost ke on leverage ratio l2 at leverage level L ¼ 1

L 1 1 1 1 1 1 1 1 1 1 1

l3 0 1 2 3 4 5 6 7 8 9 10

WACC(l3) 0.12000 0.11735 0.11482 0.11239 0.11007 0.10784 0.10569 0.10363 0.10165 0.09975 0.09791

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1920 0.1234 0.0853 0.0611 0.0443 0.0320 0.0226 0.0152 0.0091 0.0042 0.0000

ke 0.2640 0.1611 0.1040 0.0676 0.0425 0.0240 0.0099 –0.0013 –0.0103 –0.0177 –0.0240

ke 0.1920 0.1867 0.1816 0.1768 0.1721 0.1677 0.1634 0.1593 0.1553 0.1515 0.1478

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Table 16.21 The dependence of equity cost ke on leverage ratio l2 at leverage level L ¼ 2

L 2 2 2 2 2 2 2 2 2 2 2

l3 0 1 2 3 4 5 6 7 8 9 10

WACC(l3) 0.12000 0.11735 0.11482 0.11239 0.11007 0.10784 0.10569 0.10363 0.10165 0.09975 0.09791

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.2640 0.2561 0.2485 0.2412 0.2342 0.2275 0.2211 0.2149 0.2090 0.2032 0.1977

The dependence of equity cost ke on leverage ratios l3 1. L ¼ 1 2. L ¼ 2

16.8

How to Evaluate the Discount Rate?

Let us discuss now the algorithm of valuation of the discount rate, if we know one or a few ratios (coverage or leverage ones). The above developed method allows to estimate discount rate with the best accuracy characteristic for the used theory of capital structure (perpetuity limit).

16.8.1 Using One Ratio If one knows one ratio (coverage or leverage one) the algorithm of valuation of the discount rate is as follows: • Determination of the parameter k0 • Knowing k0, kd and t, one builds the curve of dependence WACC(i) or WACC(l). • Then, using the known value of coverage ratio (i0) or leverage ratio (l0) one finds the value WACC(i0) or WACC(l0), which represents the discount rate.

16.8.2 Using a Few Ratios If we know say m values of coverage ratios (ij) and n values of leverage ratios (lk):

16.9

Influence of Leverage Level

309

• We find by the above algorithm m values of WACC(ij) and n values of WACC(lk) first. • Then we find the average value of WACC by the following formula: " # m n X   X 1 WACCav ¼ WACC i j þ WACCðlk Þ : m þ n j¼1 k¼1 This found value WACCav should be used when discounting the financial flows in rating.

16.9

Influence of Leverage Level

We discuss also the interplay between rating ratios and leverage level which can be quite important in rating.

16.9.1 The Dependence of Equity Cost ke on Leverage Level at Two Coverage Ratio Values ij = 1 and ij = 2 1. i1 ¼ 1 2. i1 ¼ 2 1. i2 ¼ 1 2. i2 ¼ 2 1. i3 ¼ 1 2. i3 ¼ 2 It is seen from Tables 16.22, 16.23, 16.24, 16.25, 16.26, 16.27 and Figs. 16.20, 16.21, 16.22 that equity cost ke increases practically linearly with leverage level for all coverage ratios i1, i2, i3. For each of two coverage ratios i1, i2, curves ke(L) for two values of ij (1 and 2) practically coincide. For coverage ratio i3 curves ke(L ) for value of i3 ¼ 2 lies above one for i3 ¼ 1 and angle of inclination for value of i3 ¼ 2 is bigger.

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Application of the Modigliani–Miller Theory in Rating Methodology

Table 16.22 The dependence of equity cost ke on leverage level L at coverage ratio i1 ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

WACC(i1 ¼ 1) 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1172 0.1864 0.2556 0.3248 0.3939 0.4631 0.5323 0.6015 0.6707 0.7399 0.8091

Table 16.23 The dependence of equity cost ke on leverage level L at coverage ratio i1 ¼ 2

L 0 1 2 3 4 5 6 7 8 9 10

WACC(i1 ¼ 2) 0.11858 0.11858 0.11858 0.11858 0.11858 0.11858 0.11858 0.11858 0.11858 0.11858 0.11858

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1186 0.1892 0.2597 0.3303 0.4009 0.4715 0.5420 0.6126 0.6832 0.7538 0.8243

Table 16.24 The dependence of equity cost ke on leverage level L at coverage ratio i2 ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

WACC(i2 ¼ 1) 0.11734 0.11734 0.11734 0.11734 0.11734 0.11734 0.11734 0.11734 0.11734 0.11734 0.11734

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1173 0.1867 0.2560 0.3254 0.3947 0.4641 0.5334 0.6027 0.6721 0.7414 0.8108

16.9

Influence of Leverage Level

311

Table 16.25 The dependence of equity cost ke on leverage level L at coverage ratio i2 ¼ 2

L 0 1 2 3 4 5 6 7 8 9 10

WACC(i2 ¼ 2) 0.11866 0.11866 0.11866 0.11866 0.11866 0.11866 0.11866 0.11866 0.11866 0.11866 0.11866

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1187 0.1893 0.2600 0.3306 0.4013 0.4719 0.5426 0.6133 0.6839 0.7546 0.8252

Table 16.26 The dependence of equity cost ke on leverage level L at coverage ratio i3 ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

WACC(i3 ¼ 1) 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.0857 0.1234 0.1611 0.1989 0.2366 0.2743 0.3120 0.3497 0.3874 0.4251 0.4629

Table 16.27 The dependence of equity cost ke on leverage level L at coverage ratio i3 ¼ 2

L 0 1 2 3 4 5 6 7 8 9 10

WACC(i3 ¼ 2) 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000 0.10000

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1000 0.1520 0.2040 0.2560 0.3080 0.3600 0.4120 0.4640 0.5160 0.5680 0.6200

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Application of the Modigliani–Miller Theory in Rating Methodology

Ke(L) 0.7800 0.6800 0.5800 0.4800

i1=1

0.3800

i1=2

0.2800

0.1800 0.0800 0

1

2

3

4

5

6

7

8

9

10

Fig. 16.20 The dependence of equity cost ke on leverage level at two coverage ratio values i1 ¼ 1 and i1 ¼ 2

0.9000

Ke(L)

0.8000 0.7000 0.6000 0.5000

0.4000

i2=1

0.3000

i2=2

0.2000 0.1000 0.0000 0

1

2

3

4

5

6

7

8

9

10

Fig. 16.21 The dependence of equity cost ke on leverage level at two coverage ratio values i2 ¼ 1 and i2 ¼ 2

16.9.2 The Dependence of Equity Cost ke on Leverage Level at Two Leverage Ratio Values lj = 1 and lj = 2 Let us now study the dependence of equity cost ke on leverage level at two leverage ratio values lj ¼ 1 and lj ¼ 2 for leverage ratios l1, l2, l3.

16.10

Conclusion

313

Fig. 16.22 The dependence of equity cost ke on leverage level at two coverage ratio values i3 ¼ 1 and i3 ¼ 2

1. l1 ¼ 1 2. l1 ¼ 2 1. l2 ¼ 1 2. l2 ¼ 2 1. l3 ¼ 1 2. l3 ¼ 2 It is seen from Tables 16.28, 16.29, 16.30, 16.31, 16.32, 16.33 and Figs. 16.23, 16.24, 16.25 that equity cost ke increases practically linearly with leverage level for all leverage ratios l1, l2, l3. For each of two leverage ratios l1, l3, curves ke(L ) for two values of lj (1 and 2) practically coincide. For leverage ratio l2 curves ke(L ) for value of l3 ¼ 1 lies above one for l3 ¼ 2 and angle of inclination for value of l3 ¼ 1 is bigger.

16.10

Conclusion

This chapter, Chaps. 17 and 18 suggest a new approach to rating methodology. This chapter as well as the next one are devoted to rating of nonfinancial issuers, while Chap. 18 is devoted to long-term project rating. The key factors of a new

314

16

Application of the Modigliani–Miller Theory in Rating Methodology

Table 16.28 The dependence of equity cost ke on leverage level L at leverage ratio l1 ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

WACC(l1 ¼ 1) 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719 0.11719

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1172 0.1864 0.2556 0.3248 0.3939 0.4631 0.5323 0.6015 0.6707 0.7399 0.8091

Table 16.29 The dependence of equity cost ke on leverage level L at leverage ratio l1 ¼ 2

L 0 1 2 3 4 5 6 7 8 9 10

WACC(l1 ¼ 2) 0.11450 0.11450 0.11450 0.11450 0.11450 0.11450 0.11450 0.11450 0.11450 0.11450 0.11450

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1145 0.1810 0.2475 0.3140 0.3805 0.4470 0.5135 0.5800 0.6465 0.7130 0.7795

Table 16.30 The dependence of equity cost ke on leverage level L at leverage ratio l2 ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

WACC(l2 ¼ 1) 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571 0.08571

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.0857 0.1234 0.1611 0.1989 0.2366 0.2743 0.3120 0.3497 0.3874 0.4251 0.4629

approach are: (1) The adequate use of discounting of financial flows virtually not used in existing rating methodologies, (2) The incorporation of rating parameters (financial “ratios”) into the perpetuity limit of the modern theory of capital structure BFO. This, on the one hand, allows the use of the powerful tool of this theory in the

16.10

Conclusion

315

Table 16.31 The dependence of equity cost ke on leverage level L at leverage ratio l2 ¼ 2

L 0 1 2 3 4 5 6 7 8 9 10

WACC(l2 ¼ 2) 0.06667 0.06667 0.06667 0.06667 0.06667 0.06667 0.06667 0.06667 0.06667 0.06667 0.06667

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.0667 0.0853 0.1040 0.1227 0.1413 0.1600 0.1787 0.1973 0.2160 0.2347 0.2533

Table 16.32 The dependence of equity cost ke on leverage level L at leverage ratio l3 ¼ 1

L 0 1 2 3 4 5 6 7 8 9 10

WACC(l3 ¼ 1) 0.11735 0.11735 0.11735 0.11735 0.11735 0.11735 0.11735 0.11735 0.11735 0.11735 0.11735

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1174 0.1867 0.2561 0.3254 0.3948 0.4641 0.5335 0.6028 0.6722 0.7415 0.8109

Table 16.33 The dependence of equity cost ke on leverage level L at leverage ratio l3 ¼ 2

L 0 1 2 3 4 5 6 7 8 9 10

WACC(l3 ¼ 2) 0.11482 0.11482 0.11482 0.11482 0.11482 0.11482 0.11482 0.11482 0.11482 0.11482 0.11482

kd 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06 0.06

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

ke 0.1148 0.1816 0.2485 0.3153 0.3821 0.4489 0.5157 0.5826 0.6494 0.7162 0.7830

316

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Application of the Modigliani–Miller Theory in Rating Methodology

Ke(L) 0.9000 0.8000 0.7000 0.6000 0.5000

L1=1

0.4000

L1=2

0.3000 0.2000 0.1000 0.0000 0

1

2

3

4

5

6

7

8

9

10

Fig. 16.23 The dependence of equity cost ke on leverage level at two leverage ratio values l1 ¼ 1 and l1 ¼ 2

Ke(L) 0.5000 0.4500 0.4000 0.3500

0.3000 0.2500

L2=1

0.2000

L2=2

0.1500 0.1000 0.0500 0.0000 0

1

2

3

4

5

6

7

8

9

10

Fig. 16.24 The dependence of equity cost ke on leverage level at two leverage ratio values l2 ¼ 1 and l2 ¼ 2

References

317

Ke(L) 0.9000 0.8000 0.7000 0.6000 0.5000

L3=1

0.4000

L3=2

0.3000 0.2000 0.1000 0.0000 0

1

2

3

4

5

6

7

8

9

10

Fig. 16.25 The dependence of equity cost ke on leverage level at two leverage ratio values l3 ¼ 1 and l3 ¼ 2

rating, and on the other hand, it ensures the correct discount rates when discounting financial flows. Two models for accounting of discounting of financial flows—oneperiod and multi-period are discussed. An algorithm of valuation of the correct discount rate, accounting rating ratios is suggested. We discuss also the interplay between rating ratios and leverage level which can be quite important in rating. All above creates a new base for rating methodologies.

References Brusov P (2018a) Editorial: Introduction on special issue on the Banking System and financial markets of Russia and other countries: problems and prospects. J Rev Glob Econ 7:i–vi. SCOPUS Brusov P (2018b) Editorial. J Rev Glob Econ 7:i–v. SCOPUS Brusov P, Filatova T (2021) The Modigliani–Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198 (WoS Q1) Brusov P, Filatova T, Eskindarov M, Orehova N (2012) Hidden global causes of the global financial crisis. J Rev Global Econ 1:106–111 Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International, Switzerland, 373 p. Monograph, SCOPUS. https://www. springer.com/gp/book/9783319147314 Brusov P, Filatova T, Orehova N, Eskindarov M (2018a) Modern corporate finance, investments, taxation and ratings. Springer Nature Publishing, Switzerland, 571 p. Monograph Brusov PN, Filatova TV, Orekhova NP (2018b) Modern corporate finance and investments. Monograph. Knorus Publishing House, Moscow, 517 p

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Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018c) Rating: new approach. J Rev Glob Econ 7:37–62. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018d) A “golden age” of the companies: conditions of its existence. J Rev Glob Econ 7:88–103. SCOPUS. https://doi.org/10.6000/ 1929-7092.2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018e) Rating methodology: new look and new horizons. J Rev Glob Econ 7:63–87. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.06 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2018f) New meaningful effects in modern capital structure theory. J Rev Glob Econ 7:104–122. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.08 Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018g) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Glob Econ 7:360–376. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Glob Econ 8:437–448. SCOPUS. https://doi. org/10.6000/1929-7092.2019.08.37 Brusov P, Filatova T, Orekhova N (2020a) Ratings: critical analysis and new approaches of quantitative and qualitative methodology. Springer Nature, Switzerland, 369 p. Monograph, https://www.springer.com/de/book/9783030562427 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Application of the Modigliani–Miller theory, modified for the case of advance payments of tax on profit, in rating methodologies. J Rev Glob Econ 9:282–292 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020c) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Glob Econ 9:257–267 Brusov P, Filatova T, Chang S-I, Lin G (2021a) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9(13):1491 (WoS Q1) Brusov P, Filatova T, Orekhova N, Kulik V, Chang S-I, Lin G (2021b) Generalization of the Modigliani–Miller theory for the case of variable profit. Mathematics 9(11):1286 (WoS Q1) Brusova A (2011) А comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Prob Solut 34(76):36–42 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018) Ratings of the long-term projects: new approach. J Rev Glob Econ 7:645–661. https://doi.org/10.6000/1929-7092.2018.07.59 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391

Chapter 17

Application of the Modigliani–Miller Theory, Modified for the Case of Advance Payments of Tax on Profit, in Rating Methodologies

17.1

Introduction

In our papers (Brusov et al. 2018a, 2019, 2020b), we have applied the theory of Nobel Prize winners Modigliani and Miller, which is the perpetuity limit of the general theory of capital cost and capital structure—Brusov–Filatova–Orekhova (BFO) theory (Brusov et al. 2011, 2014a, b, 2015, 2018a, 2021a, b; 2018b, c, d, e, f, g; 2020c; Brusov 2018a, b; Brusov and Filatova 2021; Filatova et al. 2008, 2018) for rating needs. It has become a very important step in developed of a qualitatively new rating methodology. Recently we have generalized the Modigliani and Miller theory (Мodigliani and Мiller 1958, 1963, 1966) for a more realistic method of tax on profit payments: for the case of advance payments of tax on profit, which is widely used in practice (Brusov et al. 2020a). Modigliani–Miller theory accounts these tax payments as annuity-immediate, while in practice, these payments are making in advance and thus should be accounting as annuity-due. We have shown that this generalization leads to some important consequences, which change seriously all the main statements by Modigliani and Miller (Мodigliani and Мiller, 1958, 1963, 1966). These consequences are as follows: WACC starts depend on debt cost kd, WACC turns out to be lower than in case of classical Modigliani–Miller theory and thus company capitalization becomes higher than in ordinary Modigliani–Miller theory. We show that equity dependence on leverage level L is still linear, but the tilt angle with respect to L-axis turns out to be smaller: this could lead to modification of the dividend policy of the company. In current chapter, we use modified Modigliani–Miller theory (MMM theory) and apply it for rating methodologies needs. A serious modification of MMM theory in order to use it in rating procedure has been required. The financial “ratios” (main rating parameters) were introduced into MMM theory. The necessity of an appropriate use of financial flows discounting in rating methodologies is discussed. The

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_17

319

320

17

Application of the Modigliani–Miller Theory, Modified for the Case of. . .

dependence of the weighted average cost of capital (WACC), which plays the role of discount rate, on coverage and leverage ratios is analyzed. Obtained results make it possible to use the power of this theory in the rating and create a new base for rating methodologies, by other words, this allows to develop a new approach to methodology of rating, requiring a serious modification of existing rating methodologies.

17.2

Мodified Modigliani–Miller Theory

Let us shortly discuss some main points of Мodified Modigliani–Miller theory (MMM) and its features (see Chap. 4), which are different from ones of “classical” Modigliani–Miller theory (Brusov et al. 2020a). Tax shield To calculate tax shield TS in case of advance tax payments, one should use annuitydue TS ¼ k d Dt þ

kd Dt kd Dt k d Dt  þ ... ¼  þ ð1 þ k d Þ ð1 þ k d Þ2 1  ð1 þ k d Þ1

¼ Dt ð1 þ kd Þ

ð17:1Þ

This expression is different from the case of classical Modigliani–Miller theory (which used annuity-immediate). TS ¼

kd Dt k d Dt k Dt  d  ¼ Dt þ ... ¼ þ ð1 þ kd Þ ð1 þ kd Þ2 ð1 þ k d Þ 1  ð1 þ kd Þ1

ð17:2Þ

Thus in the former case tax shield TS is bigger by multiplier (1+kd). This is connected with the time value of money: money today is more expensive than money tomorrow due to the possibility of their alternative investment. The Weighted Average Cost of Capital, WACC For WACC in MMM theory, we have the following formula:
WACC ¼ k 0 1  wd t ð1 þ k d Þ :

ð17:3Þ

At L ! 1WACC ¼ k0 ð1  t ð1 þ kd ÞÞ:

ð17:4Þ

This expression is different from the similar one in classical Modigliani–Miller theory

Мodified Modigliani–Miller Theory

Fig. 17.1 Dependence of WACC on leverage level L: in “classical” Modigliani– Miller theory [curve WACC (0)] and in modified Modigliani–Miller theory (MMM theory) at different values of debt cost: kd ¼ 0.18; kd ¼ 0.14; kd ¼ 0.1 [curves WACC(00 )]

321 WACC(L)

0.2100 0.2000

WACC

17.2

0.1900 WACC (0) WACC (0') kd=0,18 WACC (0') kd=0,14 WACC (0') kd=0,1

0.1800 0.1700 0.1600 0.1500

0

1

2

3

4

5

WACC ¼ k 0 1  wd t

6

7




8

9 10

ð17:5Þ

At L ! 1WACC ¼ k0 ð1  t Þ: From these expressions, it is seen that WACC decreases with L, achieving lower value WACC ¼ k0(1  t(1 + kd)) at L ! 1 in considering case comparing with classical Modigliani–Miller theory WACC ¼ k0(1  t). This means also that company capitalization becomes higher than in ordinary Modigliani–Miller theory. Let us compare the dependence of the weighted average cost of capital, WACC, on leverage level in “classical” Modigliani–Miller theory (MM theory) and in modified Modigliani–Miller theory (MMM theory). The study of such dependence is very important because the weighted average cost of capital, WACC, plays the role of discount rate in operating financial flows discounting as well as of financial flows in rating methodologies. WACC’s value determines as well the capitalization of the company V ¼ CF/WACC. From Fig. 17.1, it is seen that WACC in MMM theory turns out to be lower than in case of classical Modigliani–Miller theory and thus company capitalization becomes higher than in ordinary Modigliani–Miller theory. It is seen that WACC decreases with debt cost kd. In paper (Brusov et al. 2020a), we study as well the equity cost dependence on leverage level L and show that it is still linear, but the tilt angle with respect to L-axis turns out to be smaller in MMM theory tgα ¼ (k0  kd)  (1  t)  k0kdt, than in classical Modigliani–Miller theory tgα ¼ (k0  kd)  (1  t). This could lead to modification of the dividend policy of the company, because the equity cost represents itself economically sound value of dividends. Thus, company could decrease the value of dividends, which they should pay to shareholders. Correct account of a method of tax on profit payments demonstrates that the shortcomings of Modigliani–Miller theory are dipper than everybody suggested: the underestimation of WACC really turns out to be bigger, as well as the overestimation of the capitalization of the company. This means that systematic risks arising from the use of modified Modigliani–Miller theory (MMM theory) (which is more correct

322

17

Application of the Modigliani–Miller Theory, Modified for the Case of. . .

than “classical” one) in practice is higher than it was suggested by the “classical” version of this theory. Because the advance payments of tax on profit are widely used in practice, modified Modigliani–Miller theory (MMM theory) should be used instead of the classical version of this theory (MM theory). And below, we apply the modified Modigliani–Miller theory for rating needs.

17.3

Application of Modified of Modigliani–Miller Theory for Rating Needs

The financial “ratios” constitute a direct and inverse ratios of various generated cash flows to debt values and interest ones, play quite significant role in quantification of the creditworthiness of the issuers. The examples of such ratios are as follows: DCF/Debt, FFO/Debt, CFO/Debt, FOCF/Debt, FFO/cashinterest, EBITDA/interest, Interests/EBITDA, Debt/EBITDA, and some others. We introduce these financial “ratios” into the modified perpetuity limit of modern theory of capital structure—BFO theory (MMM theory), which is valid for companies of arbitrary age. This is quite important because it allows use of this theory as a powerful tool when discounting of financial flows using the correct discounting rate in rating. This has required the modification of the perpetuity limit of the BFO theory— Modified Modigliani–Miller theory (MMM theory). The need of modification is connected to the fact that used in financial management, the concept of “leverage” as the ratio of debt value to the equity value substantially differs from the concept of “leverage” in the rating, where it is understood as ratio of the debt value to the generated cash flow values (income, profit, etc.). Developed by us recently (Brusov et al. 2020a) modified Modigliani–Miller theory (MMM theory) with corporate taxes (Modigliani and Miller 1966) shows that capitalization of financially dependent (leveraged) company, VL, is equal to the capitalization of financially independent (unleveraged) company, V0, increased by the size of the tax shield for perpetuity time, Dt(1 + kd), V L ¼ V 0 þ Dt ð1 þ kd Þ:

ð17:6Þ

Substituting the expressions for both capitalizations, one has CF CF ¼ þ Dt ð1 þ k d Þ WACC k0

ð17:7Þ

Let us now introduce the parameters, using in ratings (ratios), into Modified Modigliani–Miller theory (MMM theory), which represents a perpetuity limit of the

17.3

Application of Modified of Modigliani–Miller Theory for Rating Needs

323

modern theory of capital structure by Brusov–Filatova–Orekhova theory (BFO theory) (Brusov et al. 2015, 2018a, 2019). Two kinds of financial ratios will be considered: coverage ratios and leverage ratios. We will start from the coverage ratios.

17.3.1 Coverage Ratios We will consider three kinds of coverage ratios: coverage ratio of debt, coverage ratio of interest on the credit and coverage ratio of debt and interest on the credit.

17.3.1.1

Coverage Ratios of Debt

Let us consider first the coverage ratios of debt i1 ¼ CF/D. Dividing both parts of Eq. (17.7) by D one gets i1 i ¼ 1 þ t ð1 þ k d Þ WACC k0 i þ k0 t ð1 þ k d Þ i1 ¼ 1 k0 WACC i1 k0 WACC ¼ i1 þ tk 0 ð1 þ kd Þ

ð17:8Þ

The coverage ratio of debt i1 ¼ CF/D could be used for assessment of the following rating ratios:DCF/Debt, FFO/Debt, CFO/Debt, FOCF/Debt and some others. Formula (17.8) will be used to find a dependence WACC(i1).

17.3.1.2

Coverage Ratios of Interest on the Credit

Consider now coverage ratio of interest on the credit i2 ¼ CF/kdD. By use of the Modified Modigliani–Miller theory (MMM theory) for case with corporate taxes V L ¼ V 0 þ Dt ð1 þ kd Þ, and dividing the both parts by kdD, one could derive the expression for dependence WACC(i2)

324

17

Application of the Modigliani–Miller Theory, Modified for the Case of. . .

CF CF ¼ þ Dt ð1 þ k d Þ WACC k0 t ð1 þ k d Þ i2 i ¼ 2þ kd WACC k0 i2 k 0 k d WACC ¼ i2 k d þ tk 0 ð1 þ kd Þ

ð17:9Þ

This ratio (i2) could be used for the assessment of the following parameters, used in rating, FFO/cashinterest, EBITDA/interest, and some others. Formula (17.9) will be used to find a dependence WACC(i2).

17.3.1.3

Coverage Ratios of Debt and Interest on the Credit

Below we consider the coverage ratios of debt and interest on the credit simultaCF neously i3 ¼ Dð1þk . This is a new value, introduced by us here for the first time. dÞ Using the Modified Modigliani–Miller theory (MMM theory) for case with corporate taxes V L ¼ V 0 þ Dt ð1 þ k d Þ and dividing the both parts by (1+kd)D we get the dependence WACC(i3) CF CF ¼ þ Dt ð1 þ k d Þ WACC k0 i3 i t ¼ 3þ WACC k 0 kd i k k WACC ¼ 3 0 d i3 k d þ tk 0

ð17:10Þ

This ratio (i3) could be used for the assessment of the following rating ratios: FFO/Debt + interest, EBITDA/Debt + interest and some others. Formula (17.10) will be used to find a dependence WACC(i3).

17.3.1.4

Dependence of WACC on Leverage Ratios of Debt in “Classical” Modigliani–Miller Theory (MM Theory) and Modified Modigliani–Miller Theory (MMM Theory)

Below we study the dependences of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory (MM theory) and modified Modigliani–Miller theory (MMM theory). We use Microsoft Excel, and the following parameters (Tables 17.1, 17.2, 17.3 and 17.4):

17.3

Application of Modified of Modigliani–Miller Theory for Rating Needs

325

Table 17.1 Dependence of WACC on coverage ratios of debt in “classical” Modigliani–Miller theory

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

Table 17.2 Dependence of WACC on coverage ratios of debt in modified Modigliani– Miller theory (MMM theory) at kd ¼ 0.18

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

i1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

WACC 0.0000 0.1910 0.1954 0.1969 0.1977 0.1981 0.1984 0.1987 0.1988 0.1990 0.1991

Table 17.3 Dependence of WACC on coverage ratios of debt in modified Modigliani– Miller theory (MMM theory) at kd ¼ 0.14

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

i1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

WACC 0.0000 0.1913 0.1955 0.1970 0.1977 0.1982 0.1985 0.1987 0.1989 0.1990 0.1991

i1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

WACC 0.0000 0.1923 0.1961 0.1974 0.1980 0.1984 0.1987 0.1989 0.1990 0.1991 0.1992

equity cost at L ¼ 0, k0 ¼ 0.2; tax on profit rate t ¼ 0.2; debt cost kd ¼ 0.1;0.14;0.18; leverage ratios of debt l1 ¼ 0;1;2;3;4;5;6;7;8;9;10. Let us analyze the dependence of company’s weighted average cost of capital (WACC) on the coverage ratios on debt i1. It is seen from Fig. 17.2 that the weighted

17

Application of the Modigliani–Miller Theory, Modified for the Case of. . .

Table 17.4 Dependence of WACC on coverage ratios of debt in modified Modigliani– Miller theory (MMM theory) at kd ¼ 0.1

Fig. 17.2 Dependence of WACC on coverage ratios of debt in “classical” Modigliani–Miller theory [curve WACC(1)] and in modified Modigliani–Miller theory (MMM theory) [curves WACC(10 )] at kd ¼ 0.1;0.14;0.18

i1 0 1 2 3 4 5 6 7 8 9 10

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

WACC 0.0000 0.1916 0.1957 0.1971 0.1978 0.1983 0.1985 0.1988 0.1989 0.1990 0.1991

WACC(i1)

0.2000 0.1800 0.1600 0.1400 WACC

326

0.1200

WACC (1) WACC (1') kd=0,18 WACC (1') kd=0,14 WACC (1') kd=0,1

0.1000 0.0800 0.0600 0.0400 0.0200 0.0000

0

1

2

3

4

5

6

7

8

9 10

average cost of capital, WACC, increases with coverage ratio of debt for both versions of Modigliani–Miller theory: for “classical” Modigliani–Miller theory (MM theory) as well as for modified Modigliani–Miller theory (MMM theory). WACC increases very rapidly at L increases from L ¼ 0 to L ¼ 1 and then comes to saturation very fast (after L ¼ 3 WACC changes very weak. At saturation, WACC reaches the value k0 (equity cost at zero leverage level). This means that for high values of ij one can choose k0 as a discount rate with a good accuracy. Thus the role of parameter k0 increases drastically. The method of determination of parameter k0 has been developed by Anastasiya Brusova (2011). So, parameter k0 is the discount rate for limit case of high values of i1. We observe that results for both versions of Modigliani–Miller theory are very closed: only from Tables it is seen that curve of dependence of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory [curve WACC(2)] lies a little bit above all dependences of WACC on coverage ratio of debt in modified Modigliani–Miller theory. The WACC values are practically independent on kd. As we will see below, the situation is quite difference in case of leverage ratios, where the influence of kd on dependences of WACC on leverage ratios is significant.

17.3

Application of Modified of Modigliani–Miller Theory for Rating Needs

327

17.3.2 Leverage Ratios We will consider now the leverage ratios. Three kinds of leverage ratios will be considered: leverage ratios of debt, leverage ratios of interest on the credit and leverage ratios of debt and interest on the credit.

17.3.2.1

Leverage Ratios for Debt Here l1 ¼ D=CF

ð17:11Þ

As above for coverage ratios, we use the modified Modigliani–Miller theorem for case with corporate taxes one has V L ¼ V 0 þ Dt ð1 þ kd Þ, and dividing the both parts by CF, we derive the expression for WACC(l1) CF CF ¼ þ Dt ð1 þ k d Þ WACC k0 1 1 ¼ þ l 1 t ð1 þ k d Þ WACC k 0 k0 WACC ¼ 1 þ tl1 k0 ð1 þ kd Þ

ð17:12Þ

This ratio (l1) can be used to assess of the following parameters used in rating, Debt/EBITDA and some others. We will use the last formula to build a curve of dependence WACC(l1).

17.3.2.2

Leverage Ratios for Interest on Credit Here l2 ¼ k d D=CF

ð17:13Þ

We use again Modified Modigliani–Miller theory (MMM theory) for case with corporate taxes V L ¼ V 0 þ Dt ð1 þ kd Þ, and dividing the both parts by CF/kd, we derive the expression for WACC(l2)

328

17

Application of the Modigliani–Miller Theory, Modified for the Case of. . .

CF CF ¼ þ Dt ð1 þ k d Þ WACC k0 1 1 l t ð1 þ k d Þ ¼ þ 2 kd WACC k0 k0 kd WACC ¼ kd þ tl2 k 0 ð1 þ kd Þ

ð17:14Þ

This ratio (l2) can be used to assess of the following parameters used in rating, Interests/EBITDA and some others. We will use the last formula to build a curve of dependence WACC(l2).

17.3.2.3

Leverage Ratios for Debt and Interest on Credit Here l3 ¼ Dð1 þ kd Þ=CF

ð17:15Þ

Using the Modified Modigliani–Miller theory (MMM theory) for case with corporate taxes V L ¼ V 0 þ Dt ð1 þ kd Þ, and dividing the both parts by CF/(1 + kd), we derive the expression for WACC(l3) CF CF ¼ þ Dt ð1 þ k d Þ WACC k0 1 1 ¼ þ l3 t WACC k0 k0 WACC ¼ 1 þ tl3 k 0

ð17:16Þ

This ratio (l3) can be used to assess of the following parameters used in rating, Debt+interest / FFO, Debt+interest / EBIT, Debt+interest / EBITDA(R), and some others. We will use the last formula to build a curve of dependence WACC(l3).

17.3.2.4

Dependence of WACC on Leverage Ratios of Debt in “Classical” Modigliani–Miller Theory (MM Theory) and Modified Modigliani–Miller Theory (MMM Theory)

Below we study the dependences of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory (MM theory) and modified Modigliani–Miller theory (MMM theory). We use Microsoft Excel, and the following parameters (Tables 17.5, 17.6, 17.7 and 17.8):

17.3

Application of Modified of Modigliani–Miller Theory for Rating Needs

329

Table 17.5 Dependence of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

Table 17.6 Dependence of WACC on leverage ratios of debt in modified Modigliani– Miller theory (MMM theory) at kd ¼ 0.18

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

l1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18 0.18

WACC 0.2000 0.1910 0.1827 0.1752 0.1682 0.1618 0.1559 0.1503 0.1452 0.1404 0.1359

Table 17.7 Dependence of WACC on leverage ratios of debt in modified Modigliani– Miller theory (MMM theory) at kd ¼ 0.14

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

l1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

WACC 0.2000 0.1913 0.1833 0.1759 0.1691 0.1629 0.1570 0.1516 0.1465 0.1418 0.1374

l1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

WACC 0.2000 0.1923 0.1852 0.1786 0.1724 0.1667 0.1613 0.1563 0.1515 0.1471 0.1429

equity cost at L ¼ 0, k0 ¼ 0.2; tax on profit rate t ¼ 0.2; debt cost kd ¼ 0.1;0.14;0.18; leverage ratios of debt l1 ¼ 0;1;2;3;4;5;6;7;8;9;10. Analysis of the dependences of company’s weighted average cost of capital (WACC) on the leverage ratio on debt, l1 shows the following: (see Fig. 17.3) the

17

Application of the Modigliani–Miller Theory, Modified for the Case of. . .

Table 17.8 Dependence of WACC on leverage ratios of debt in modified Modigliani– Miller theory (MMM theory) at kd ¼ 0.1

Fig. 17.3 Dependence of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory [curve WACC(2)] and in modified Modigliani–Miller theory (MMM theory) [curves WACC(20 )] at kd ¼ 0.1;0.14;0.18

l1 0 1 2 3 4 5 6 7 8 9 10

k0 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

kd 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1

WACC 0.2000 0.1916 0.1838 0.1767 0.1701 0.1639 0.1582 0.1529 0.1479 0.1433 0.1389

WACC(l1)

0.2200 0.2000 WACC

330

0.1800

WACC (2) WACC (2') kd=0,18 WACC (2') kd=0,14 WACC (2') kd=0,1

0.1600 0.1400 0.1200 0.1000

0

1

2

3

4

5

6

7

8

9 10

weighted average cost of capital, WACC, decreases with leverage ratios for both versions of Modigliani–Miller theory: for “classical” Modigliani–Miller theory (MM theory) as well as for modified Modigliani–Miller theory (MMM theory). But we observe that curve of dependence of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory [curve WACC(2)] lies above all dependences of WACC on leverage ratios of debt in modified Modigliani–Miller theory. The WACC values decrease with kd. So we are observing the influence of kd on dependences of WACC on leverage ratios. This effect is absent in “classical” Modigliani–Miller theory. Thus the using of the modified Modigliani–Miller theory (MMM theory) leads to more correct valuation of the weighted average cost of capital (WACC), which plays the role of discount rate in financial flows discounting in rating methodologies. Obtained results will help improve the existing rating methodologies.

17.4

17.4

Discussions

331

Discussions

In the current paper, we use the Modified Modigliani–Miller theory (MMM theory) and apply it for rating methodologies needs. The financial “ratios” (main rating parameters) were introduced into MMM theory. The dependence of the weighted average cost of capital (WACC), which plays the role of discount rate in financial flows discounting in rating methodologies, on coverage and leverage ratios is analyzed. Obtained results will help improve the existing rating methodologies. The analysis of the dependence of company’s weighted average cost of capital (WACC) on the coverage ratios on debt i1 shows that WACC increases with coverage ratio of debt for both versions of Modigliani–Miller theory: for “classical” Modigliani–Miller theory (MM theory) as well as for modified Modigliani–Miller theory (MMM theory). WACC increases very rapidly at L increases from L ¼ 0 to L ¼ 1 and then comes to saturation very fast (after L ¼ 3 WACC changes very weak). At saturation, WACC reaches the value k0 (equity cost at zero leverage level). This means that for high values of ij one can choose k0 as a discount rate with a good accuracy. Thus the role of parameter k0 increases drastically. The method of determination of parameter k0 has been developed by Anastasiya Brusova (2011). So, parameter k0 is the discount rate for limit case of high values of i1. We observe that results for both versions of Modigliani–Miller theory are very closed: only from Tables it is seen that curve of dependence of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory lies a little bit above all dependences of WACC on coverage ratio of debt in modified Modigliani–Miller theory. The WACC values are practically independent on kd. The situation is quite difference in case of leverage ratios, where the influence of kd on dependences of WACC on leverage ratios is significant. Analysis of the dependences of company’s weighted average cost of capital (WACC) on the leverage ratio on debt, l1 shows that WACC decreases with leverage ratios for both versions of Modigliani–Miller theory: for “classical” Modigliani– Miller theory (MM theory) as well as for modified Modigliani–Miller theory (MMM theory). We observe that curve of dependence of WACC on leverage ratios of debt in “classical” Modigliani–Miller theory lies above all dependences of WACC on leverage ratios of debt in modified Modigliani–Miller theory. The WACC values decrease with kd. So we are observing the influence of kd on dependences of WACC on leverage ratios. This effect is absent in “classical” Modigliani–Miller theory. Thus the using of the modified Modigliani–Miller theory (MMM theory) leads to more correct valuation of the weighted average cost of capital (WACC), which plays the role of discount rate in financial flows discounting in rating methodologies. Obtained results will help improve the existing rating methodologies, which are used for the valuation of the creditworthiness of companies. Correct account of a method of tax on profit payments demonstrates that the shortcomings of Modigliani–Miller theory are dipper than everybody suggested: the underestimation of WACC really turns out to be bigger, as well as overestimation of the capitalization of the company. This means that systematic risks arising from the

332

17

Application of the Modigliani–Miller Theory, Modified for the Case of. . .

use of modified Modigliani–Miller theory (MMM theory) (which is more correct than “classical” one) in practice is higher than it was suggested by the “classical” version of this theory.

References Brusov P (2018a) Editorial: Introduction on special issue on the Banking System and financial markets of Russia and other countries: problems and prospects. J Rev Glob Econ 7:i–vi. SCOPUS Brusov P (2018b) Editorial. J Rev Glob Econ 7:i–v. SCOPUS Brusov P, Filatova T (2021) The Modigliani–Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198 (WoS Q1) Brusov P, Filatova T, Orehova N, Brusova A (2011) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Appl Financ Econ 21(11):815– 824 Brusov P, Filatova P, Orekhova N (2014a) Mechanism of formation of the company optimal capital structure, different from suggested by trade off theory. Cogent Econ Finance 2:1–13. https://doi. org/10.1080/23322039.2014.946150 Brusov P, Filatova T, Orehova N (2014b) Inflation in Brusov–Filatova–Orekhova theory and in its perpetuity limit—Modigliani–Miller theory. J Rev Global Econ 3:175–185 Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International, Switzerland, 373 p. Monograph, SCOPUS. https://www. springer.com/gp/book/9783319147314 Brusov P, Filatova T, Orehova N, Eskindarov M (2018a) Modern corporate finance, investments, taxation and ratings. Springer Nature, Switzerland, 571 p. Monograph Brusov PN, Filatova TV, Orekhova NP (2018b) Modern corporate finance and investments. Monograph. Knorus Publishing House, Moscow, 517 p Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018c) Rating: new approach. J Rev Glob Econ 7:37–62. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018d) A “golden age” of the companies: conditions of its existence. J Rev Glob Econ 7:88–103. SCOPUS. https://doi.org/10.6000/ 1929-7092.2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018e) Rating methodology: new look and new horizons. J Rev Glob Econ 7:63–87. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.06 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2018f) New meaningful effects in modern capital structure theory. J Rev Glob Econ 7:104–122. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.08 Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018g) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Glob Econ 7:360–376. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Glob Econ 8:437–448. SCOPUS. https://doi. org/10.6000/1929-7092.2019.08.37 Brusov P, Filatova T, Orekhova N (2020a) Ratings: critical analysis and new approaches of quantitative and qualitative methodology. Springer Nature, Switzerland, 369 p. Monograph, https://www.springer.com/de/book/9783030562427 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Application of the Modigliani–Miller theory, modified for the case of advance payments of tax on profit, in rating methodologies. J Rev Glob Econ 9:282–292

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Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020c) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Glob Econ 9:257–267 Brusov P, Filatova T, Chang S-I, Lin G (2021a) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9(13):1491 (WoS Q1) Brusov P, Filatova T, Orekhova N, Kulik V, Chang S-I, Lin G (2021b) Generalization of the Modigliani–Miller theory for the case of variable profit. Mathematics 9(11):1286 (WoS Q1) Brusova A (2011) А Comparison of the three methods of estimation of weighted average cost of capital and equity cost of company. Financ Anal Probl Solut 34(76):36–42 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018) Ratings of the long-term projects: new approach. J Rev Glob Econ 7:645–661. https://doi.org/10.6000/1929-7092.2018.07.59 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–19517. Am Econ Rev 56:333–391

Chapter 18

A New Approach to Project Ratings

18.1

Introduction

Rating agencies play a very important role in economics. Their analysis of issuer’s state, generated credit ratings of issuers help investors make a reasonable investment decision, as well as help issuers with good enough ratings, get credits on lower rates, etc. The chapter continues to create a new approach to rating methodology: in addition to two papers, which have considered the creditworthiness of the nonfinance issuers (Brusov et al. 2018c, d), we develop here a new approach to project rating. We work within investment models, created by authors. One of them describes the effectiveness of investment project from the perspective of equity capital owners, while the other model describes the effectiveness of investment project from the perspective of equity capital and debt capital owners. We work here within Modigliani–Miller (MM) approach, while Brusov–Filatova–Orekhova (BFO) approach (Brusov et al. 2015, 2020a, b; Brusov 2018a, b; Filatova et al. 2018a, b) will be consider somewhere also. The important features of current consideration as well as in previous studies are: (1) The adequate use of discounting of financial flows virtually not used in existing rating methodologies, (2) The incorporation of rating parameters (financial “ratios”), used in project rating, into considered modern investment models. Analyzing within these investment models with incorporated rating parameters the dependence of NPV on rating parameters (financial “ratios”) at different values of equity cost k0, at different values of credit rates kd as well as at different values of on leverage level L we come to a very important conclusion, that NPV (in units of NPV NOI) (NPV NOI ) (as well as NPV (in units of D) ( D )) depends only on equity cost k0, on credit rates kd, on leverage level L as well as on one of the leverage ratios lj (on one of the coverage ratios ij) and does not depend on equity value S, debt value D and NOI. This means that results on the dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratios lj (as well as on the dependence of NPV (in units of D) (NPV D ) on coverage ratios ij) at different equity costs k0, at different credit rates kd, at different © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_18

335

336

18

A New Approach to Project Ratings

leverage levels L carry the universal character: these dependencies remain valid for investment projects with any equity value S, debt value D and NOI.

18.2

Investment Models

We work within investment models, created by authors. One of them describes the effectiveness of investment project from the perspective of equity capital owners, while the other model describes the effectiveness of investment project from the perspective of equity capital and debt capital owners. In the former case, investments at the initial time moment T ¼ 0 are equal to –S and the flow of capital for the period (in addition to the tax shields kdDt it includes a payment of interest on a loan kdD): CF ¼ ðNOI  kd DÞð1  t Þ:

ð18:1Þ

Here, for simplicity, we suppose that interest on the loan will be paid in equal shares kdD during all periods. Note that principal repayment is made at the end of the last period. We will consider the case of discounting, when operating and financial flows are not separated and both are discounted, using the general rate (as which, obviously, the weighted average cost of capital (WACC) can be selected). In this case for longterm (perpetuity) projects, the Modigliani–Miller formula (Мodigliani and Мiller 1958, 1963, 1966) for WACC will be used and for projects of finite (arbitrary) duration Brusov–Filatova–Orekhova formula will be used (Brusov et al. 2018a, b, c, d, e, f, g; 2019; Filatova et al. 2008). For one-year company Myers’ formula (Myers 2001) could be used (this is Brusov–Filatova–Orekhova formula at n ¼ 1). Note that debt capital is the least risky, because interest on credit is paid after taxes in the first place. Therefore, the cost of credit will always be less than the equity cost, whether of ordinary or of preference shares ke > kd; kp > kd. Here ke, kp is the equity cost of ordinary or of preference shares consequently.

18.2.1 The Effectiveness of the Investment Project from the Perspective of the Equity Holders Only (Without Flows Separation) In this case, operating and financial flows are not separated and are discounted using the general rate (as which, obviously, WACC can be selected). The credit reimbursable at the end of the project (at the end of the period (n)) can be discounted either at the same rate WACC or at the debt cost rate kd. Now we choose a uniform rate and the first option.

18.2

Investment Models

337

n X NOIð1  t Þ  kd Dð1  t Þ

D ð1 þ WACCÞn ð1 þ WACCÞ i¼1   NOIð1  t Þ  k d Dð1  t Þ 1 D 1 ¼ S þ :  WACC ð1 þ WACCÞn ð1 þ WACCÞn

NPV ¼ S þ

i



ð18:2Þ At a Constant Value of Equity Capital (S = const) Accounting that in the case S ¼ const NOI is proportional to the invested capital, I, NOI ¼ βI ¼ βS(1 + L ) and substituting D ¼ LS, we get NPV ¼ S þ

  NOIð1  t Þ  kd Dð1  t Þ 1 1 WACC ð1 þ WACCÞn

D , ð1 þ WACCÞn     Lk d ð1  t Þ 1 L NPV ¼ S 1 þ 1 þ WACC ð1 þ WACCÞn ð1 þ WACCÞn   βSð1 þ LÞð1  t Þ 1 þ 1 : WACC ð1 þ WACCÞn 

ð18:3Þ

ð18:4Þ

18.2.2 Modigliani–Miller Limit (Long-term (Perpetuity) Projects) In perpetuity limit (n ! 1) (Modigliani–Miller limit) (turning to the limit n ! 1 in the relevant equations), we have NPV ¼ S þ

NOIð1  t Þ  k d Dð1  t Þ : WACC

ð18:5Þ

At a Constant Value of Equity Capital (S = const) NPV ¼ S þ Substituting D ¼ LS, we get

NOIð1  t Þ  kd Dð1  t Þ WACC

ð18:6Þ

338

18

A New Approach to Project Ratings



 Lkd ð1  t Þ NOIð1  t Þ þ NPV ¼ S 1 þ WACC WACC   Lkd ð1  t Þ βSð1 þ LÞð1  t Þ ¼ S 1 þ þ : k 0 ð1  Lt=ð1 þ LÞÞ k 0 ð1  Lt=ð1 þ LÞÞ

ð18:7Þ

In the last equation, we substituted the perpetuity (Modigliani–Miller) formula for WACC   Lt WACC ¼ k0 1  : 1þL

ð18:8Þ

So, below we consider the long-term (perpetuity) projects and will use the following formula for calculations 2

3 Lk ð 1  t Þ βSð1 þ LÞð1  t Þ d 5 þ   NPV ¼ S41 þ  Lt Lt k0 1  1þL k0 1  1þL

18.3

ð18:9Þ

Incorporation of Financial Coefficients, Using in Project Rating, into Modern Investment Models

Below we incorporate the financial coefficients, used in project rating, into modern investment models, created by authors. We will consider two kinds of financial coefficients: coverage ratios as well as leverage coefficients. In each group of financial coefficients, we incorporate three particular quantities. For coverage ratios we incorporate: (1) coverage ratios of debt, i1 ¼ NOI D ; (2) cov; (3) coverage ratios of debt and interest erage ratios of interest on the credit i2 ¼ NOI kd D NOI on the credit i3 ¼ ð1þkd ÞD. D ; (2) leverFor leverage ratios we incorporate: (1) leverage ratios of debt, l1 ¼ NOI kd D age ratios of interest on the credit l2 ¼ NOI; (3) leverage ratios of debt and interest on d ÞD the credit l3 ¼ ð1þk NOI .

18.3

Incorporation of Financial Coefficients, Using in Project Rating, into. . .

339

18.3.1 Coverage Ratios 18.3.1.1

Coverage Ratios of Debt

Let us first incorporate the coverage ratios, using in project rating, into modern investment models, created by authors. Dividing both parts of Eq. (18.9) by D one gets NPV 1 ðk  i Þð1  t Þ  ¼  d 1 D L k 1  Lt 0

ð18:10Þ

1þL

Here i1 ¼

18.3.1.2

NOI D

ð18:11Þ

Coverage Ratios of Interest on the Credit

Dividing both parts of Eq. (18.9) by kdD one gets ð1  i2 Þð1  t Þ NPV 1    ¼ kd D Lk d k 1  Lt 0

ð18:12Þ

1þL

Here i2 ¼

18.3.1.3

NOI kd D

ð18:13Þ

Coverage Ratios of Debt and Interest on the Credit

Dividing both parts of Eq. (18.9) by (1 + kd)D one gets ½k  i3 ð1 þ kd Þð1  t Þ NPV 1   ¼  d Lð1 þ k d Þ ð1 þ kd ÞD k 1  Lt 0

Here

1þL

ð18:14Þ

340

18

i3 ¼

A New Approach to Project Ratings

NOI ð1 þ kd ÞD

ð18:15Þ

Analyzing the formulas (18.10), (18.12), and (18.14) we come to a very important conclusion, that NPV (in units of D) (NPV D ) depends only on equity cost k0, on credit rates kd, on leverage level L as well as on one of the coverage ratios ij and does not depend on equity value S, debt value D and NOI. This means that results on the dependence of NPV (in units of D) (NPV D ) on coverage ratios ij at different equity costs k0, at different credit rates kd, at different leverage levels L carry the universal character: these dependencies remain valid for investment projects with any equity value S, debt value D and NOI.

18.3.2 Leverage Ratios 18.3.2.1

Leverage Ratios for Debt

Now let us incorporate the leverage ratios, using in project rating, into modern investment models, created by authors. Dividing both parts of Eq. (18.9) by NOI one gets NPV l1 ð1  kd l1 Þð1  t Þ   ¼ þ NOI L k 1  Lt 0

ð18:16Þ

1þL

Here l1 ¼

18.3.2.2

D NOI

ð18:17Þ

Leverage Ratios for Interest on Credit NPV l2 ð1  l2 Þð1  t Þ   ¼ þ NOI kd L k 1  Lt 0

ð18:18Þ

1þL

Here l2 ¼

kd D NOI

ð18:19Þ

18.4

Dependence of NPV on Coverage Ratios

18.3.2.3

341

Leverage Ratios for Debt and Interest on Credit ð1 þ k d  l 3 k d Þð1  t Þ NPV l3   ¼ þ NOI ð1 þ kd ÞL ð1 þ k Þk 1  Lt d

0

ð18:20Þ

1þL

Here l3 ¼

ð1 þ kd ÞD : NOI

ð18:21Þ

Analyzing the formulas (18.16), (18.18), and (18.20) we come to a very important conclusion, that NPV (in units of NOI) (NPV NOI ) depends only on equity cost k0, on credit rates kd, on leverage level L as well as on one of the leverage ratios lj and does not depend on equity value S, debt value D and NOI. This means that results on the dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratios lj at different equity costs k0, at different credit rates kd, at different leverage levels L carry the universal character: these dependencies remain valid for investment projects with any equity value S, debt value D and NOI. We investigate below the effectiveness of long-term investment projects studying the dependence of NPV on coverage ratios and on leverage ratios. We make calculations for coefficients i1 and l1. Calculations for the rest of coefficients (i2, i3 and l2, l3) could be made in a similar way. We start from the calculations of the dependence of NPV on coverage ratios. We consider different values of equity costs k0, of debt costs kd and of leverage level L ¼ D/S. Here t is tax on profit rate, which in our calculations is equal to 20%.

18.4

Dependence of NPV on Coverage Ratios

18.4.1 Coverage Ratio on Debt Below we calculate the dependence of NPV (in units of D) (NPV D ) on coverage ratio on debt i1 at different equity costs k0 (k0 is equity cost at L ¼ 0). We will make calculations for two leverage levels L (L ¼ 1 and L ¼ 3) and for different credit rates k d. For calculation within MM approximation we use the formula (18.10) NPV 1 ðk  i Þð1  t Þ  ¼  d 1 D L k  1  Lt : 0

1þL

342

18

18.4.1.1

A New Approach to Project Ratings

The Dependence of NPV on Coverage Ratio on Debt I1 at Equity Cost k0 = 24%

Below we investigate the dependence of NPV on coverage ratio on debt i1 at different values of equity costs k0, at different values of debt costs kd at a fixed value of equity cost, as well as at different values of leverage levels L. Let us start our calculations from the case of equity cost k0 ¼ 24%. The results of calculations of the dependence of NPV on coverage ratio on debt i1 at equity cost k0 ¼ 24%, at different values of debt costs kd and L ¼ 1 are shown in Table 18.1. The dependence of NPV (in units of D) on coverage ratio on debt i1 at k0 ¼ 24%, kd ¼ 6%, 10%, 14% and 20% and L ¼ 1 is illustrated in Fig. 18.1. Let us calculate the value of i1 above which the investment project remains effective (NPV > 0) kd i1

0.20 0.48

0.14 0.42

0.1 0.38

0.06 0.32

One can see from this table that the value of i1 above which the investment project remains effective (NPV > 0) increases with credit rate kd, which means that the effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd. Let us calculate the dependence of NPV (in units of D) on coverage ratio on debt i1 at k0 ¼ 24%, kd ¼ 6%, 10%, 14% and 20% and L ¼ 3. The dependence of NPV (in units of D) on coverage ratio on debt i1 at k0 ¼ 24%, kd ¼ 6%, 10%, 14% and 20% and L ¼ 3 is illustrated in Fig. 18.2. Let us calculate the value of l1 above which the investment project remains effective (NPV > 0)

Table 18.1 The dependence of NPV on coverage ratio on debt i1 at equity cost k0 ¼ 24%, kd ¼ 6%, 10%, 14%, 20% and L ¼ 1 i1 0 1 2 3 4 5 6 7 8 9 10

L 1 1 1 1 1 1 1 1 1 1 1

k0 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

NPV/D (kd ¼ 0.2) 1.741 1.963 5.667 9.37 13.07 16.78 20.48 24.19 27.89 31.59 35.3

NPV/D (kd ¼ 0.14) 1.519 2.185 5.889 9.593 13.3 17 20.7 24.41 28.11 31.81 35.52

NPV/D (kd ¼ 0.1) 1.37 2.333 6.037 9.741 13.44 17.15 20.85 24.56 28.26 31.96 35.67

NPV/D (kd ¼ 0.06) 1.222 2.481 6.185 9.889 13.59 17.3 21 24.7 28.41 32.11 35.81

18.4

Dependence of NPV on Coverage Ratios

343

NPV/D( i 1 )AT L=1 kd=0,20

kd=0,14

kd=0,10

kd=0,06

40 35 30

NPV/D

25 20 15 10 5 0 -5

0

1

2

3

4

5

6

7

8

9

10

I1

Fig. 18.1 The dependence of NPV (in units of D) on coverage ratio on debt i1 at k0 ¼ 24%, kd ¼ 6%, 10%, 14% and 20% and L ¼ 1

NPV/D ( i 1 ) at L=3 kd=0,20

kd=0,14

kd=0,10

kd=0,06

45 40 35

NPV/D

30 25 20 15

10 5 0 -5

0

1

2

3

4

5 I1

6

7

8

9

10

Fig. 18.2 The dependence of NPV (in units of D) on coverage ratio on debt i1 at k0 ¼ 24%, kd ¼ 6%, 10%, 14% and 20% and L ¼ 3 kd i1

0.20 0.3

0.14 0.23

0.1 0.18

0.06 0.12

One can see from this table that like the case of L ¼ 1 the value of i1 above which the investment project remains effective (NPV > 0) increases with credit rate kd,

344

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A New Approach to Project Ratings

Table 18.2 The dependence of NPV on coverage ratio on debt i1 at equity cost k0 ¼ 24%, kd ¼ 6%, 10%, 14%, 20% and L ¼ 3 i1 0 1 2 3 4 5 6 7 8 9 10

L 3 3 3 3 3 3 3 3 3 3 3

k0 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24 0.24

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

NPV/D (kd ¼ 0.2) 1.118 2.804 6.725 10.65 14.57 18.49 22.41 26.33 30.25 34.18 38.1

NPV/D (kd ¼ 0.14) 0.882 3.039 6.961 10.88 14.8 18.73 22.65 26.57 30.49 34.41 38.33

NPV/D (kd ¼ 0.1) 0.725 3.196 7.118 18.04 14.96 18.88 22.8 26.73 30.65 34.57 38.49

NPV/D (kd ¼ 0.06) 0.569 3.353 7.275 18.2 15.12 19.04 22.96 26.88 30.8 34.73 38.65

which means that effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd. Comparing the case of L ¼ 1 one can see that at bigger leverage level (L ¼ 3) the investment project becomes effective (NPV > 0) starting from smaller coverage ratio i1, so bigger leverage level favors the effectiveness of the investment project as well as its creditworthiness. We see from Tables 18.1 and 18.2 and Figs. 18.1 and 18.2, that NPV D increases with values turn out to be very closed each other at all i values. It is seen as i1 and that NPV 1 D well that NPV increases with decreasing kd. This means that the effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd. Below we investigate the dependence of NPV D on i1 at different values of kd in more (i ) curves at different values of kd, as well as detail and will show the ordering of NPV D 1 at different leverage levels L.

18.4.1.2

The Dependence of NPV on Coverage Ratio on Debt i1 at Equity Cost k0 = 12%

We study here the dependence of NPV D on i1 at fixed equity cost k0 ¼ 12% and at different values of kd in more details and will show the ordering of NPV D (i1) curves at different values of kd, as well as at different leverage levels L. The results of calculations of the dependence of NPV on coverage ratio on debt i1 at equity cost k0 ¼ 12%, at different values of debt costs kd and L ¼ 1 are shown in Table 18.3. The results of calculations of the dependence of NPV on coverage ratio on debt i1 at equity cost k0 ¼ 12%, at different values of debt costs kd and L ¼ 3 are shown in Table 18.4. We see from Tables 18.3 and 18.4 that NPV (in units of D) (NPV D ) increases with i1 and that NPV values turn out to be very closed to each other at all i1 values. D

18.4

Dependence of NPV on Coverage Ratios

345

Table 18.3 The dependence of NPV(in units of D) on coverage ratio on debt i1 at k0 ¼ 12%, kd ¼ 2%, 4%, 6%, 8% and 10% and L ¼ 1 i1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

NPV/D kd ¼ 0.1 1.741 5.667 13.074 20.481 27.889 35.296 42.704 50.111 57.519 64.926 72.333

NPV/D kd ¼ 0.08 1.593 5.815 13.222 20.630 28.037 35.444 42.852 50.259 57.667 65.074 72.481

NPV/D kd ¼ 0.06 1.444 5.963 13.370 20.778 28.185 35.593 43.000 50.407 57.815 65.222 72.630

NPV/D kd ¼ 0.04 1.296 6.111 13.519 20.926 28.333 35.741 43.148 50.556 57.963 65.370 72.778

NPV/D kd ¼ 0.02 1.148 6.259 13.667 21.074 28.481 35.889 43.296 50.704 58.111 65.519 72.926

Table 18.4 The dependence of NPV (in units of D) on coverage ratio on debt i1 at k0 ¼ 12%, kd ¼ 2%, 4%,6%, 8% and 10% and L ¼ 3 i1 0 1 2 3 4 5 6 7 8 9 10

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

k0 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12 0.12

NPV/D kd ¼ 0.1 1.118 6.725 14.569 22.412 30.255 38.098 45.941 53.784 61.627 69.471 77.314

NPV/D kd ¼ 0.08 0.961 6.882 14.725 22.569 30.412 38.255 46.098 53.941 61.784 69.627 77.471

NPV/D kd ¼ 0.06 0.804 7.039 14.882 22.725 30.569 38.412 46.255 54.098 61.941 69.784 77.627

NPV/D kd ¼ 0.04 0.647 7.196 15.039 22.882 30.725 38.569 46.412 54.255 62.098 69.941 77.784

NPV/D kd ¼ 0.02 0.490 7.353 15.196 18.039 30.882 38.725 46.569 54.412 62.255 70.098 77.941

To show the difference in NPV D values in more details we show in Fig. 18.3 the on parameter i1 for range i1 from 1 to 2. dependence of NPV D One can see, that all NPV (i1) curves corresponding to L ¼ 3 lie above the curves corresponding to L ¼ 1. This means that NPV increases with L (with increasing of the debt financing). At fixed value L NPV increases with decreasing the credit rate kd. This means that the effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd. Analyzing the obtained results one should remember, that NPV (in units of D) (NPV D ) depends only on equity cost k0, on credit rates kd, on leverage level L as well as on one of the coverage ratios ij and does not depend on equity value S, debt value D and NOI. This means that obtained results on the dependence of NPV (in units of D) (NPV D ) on coverage ratios ij at different equity costs k0, at different credit rates kd, at

346

18

A New Approach to Project Ratings

NPV/D (i1) (for i1 from 1 to 2) at L=1 and L=3 15.500

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

NPV/D

13.500

NPV/D, L=1, Kd=0,1 (10) NPV/D, L=1, Kd=0,08 (9)

NPV/D, L=1, Kd=0,06 (8)

11.500

NPV/D, L=1, Kd=0,04 (7) NPV/D, L=1, Kd=0,02 (6) NPV/D, L=3, Kd=0,1 (5) NPV/D, L=3, Kd=0,08 (4)

9.500

NPV/D, L=3, Kd=0,06 (3) NPV/D, L=3, Kd=0,04 (2) NPV/D, L=3, Kd=0,02 (1) 7.500

5.500

1

2

Fig. 18.3 The dependence of NPV (in units of D) on coverage ratio on debt i1 at k0 ¼ 12%, kd ¼ 2%, 4%, 6%, 8% and 10% and L ¼ 1 and L ¼ 3

different leverage levels L carry the universal character: these dependencies remain valid for investment projects with any equity value S, debt value D and NOI.

18.5

Dependence of NPV on Leverage Ratios

18.5.1 Leverage Ratio of Debt Below we calculate the dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at different equity costs k0 (k0 is equity cost at L ¼ 0). We make calculations for two leverage levels L (L ¼ 1 and L ¼ 3) and for different credit rates kd. For calculation within MM approximation we use the formula (18.19)

18.5

Dependence of NPV on Leverage Ratios

347

NPV l1 ð1  K d  l1 Þð1  t Þ   : ¼ þ NOI L K  1  Lt 0

18.5.1.1

1þL

The Dependence of NPV (in Units of NOI) (NPV NOI ) on Leverage Ratio on Debt l1 at Equity Cost k0 = 0.12

Results are shown in Tables 18.5 and 18.6 and in Figs. 18.4 and 18.5. Based on the above calculations, we plot the dependences of NPV/NOI on leverage ratio on debt l1 at different leverage levels L. From Tables 18.5 and 18.6 and Figs. 18.4 and 18.5 one can come to the conclusion that the NPV (in units of NOI) (NPV/NOI) decreases with increasing of the leverage ratio on debt l1. With the increasing of the cost of debt capital kd, curves of the dependence of NPV/NOI (l1), outgoing from a single point at a zero value of l1, lie below (i.e., the rate of decrease (or negative slope of curves) grows). Note, that while the dependences of NPV(in units of D) on coverage ratio on debt i1 lie very close to each other (see above), the dependences of NPV(in units of NOI) on leverage ratio on debt l1 are separated significantly more. Also, Figs. 18.6, 18.7, 18.8 and 18.9 of the NPV/NOI dependence on l1 can be plotted for fixed values of the debt cost kd and two values of the leverage level L ¼ 1 and L ¼ 3. One can see, that the rate of decrease of the ratio NPV/NOI decreases with increasing of the leverage level L.

18.5.1.2

The Dependence of NPV (in Units of NOI) (NPV NOI ) on Leverage Ratio on Debt l1 at Equity Cost k0 = 0.14

Based on the obtained data, we plot the dependences of NPV/NOI on l1 at k0 ¼ 14%, at different values of debt cost kd, and at two different leverage levels L ¼ 1 and L ¼ 3 in Figs. 18.10 and 18.11. From Tables 18.7 and 18.8 and Figs. 18.10 and 18.11 one can come to the conclusion that the NPV (in units of NOI) (NPV/NOI) decreases with increasing of the leverage ratio on debt l1. With the increasing of the cost of debt capital kd, curves of the dependence of NPV/NOI (l1), outgoing from a single point at a zero value of l1, fall below (i.e., the rate of decrease grows).

l1 kd ¼ 0.10 kd ¼ 0.08 kd ¼ 0.06 kd ¼ 0.04

0 7.407 7.407 7.407 7.407

1 5.667 5.815 5.963 6.111

2 3.926 4.222 4.519 4.815

3 2.185 2.63 3.074 3.519

4 0.444 1.037 1.63 2.222

5 1.296 0.556 0.185 0.926

6 3.037 2.148 1.259 0.37

7 4.778 3.741 2.704 1.667

8 6.519 5.333 4.148 2.963

9 8.259 6.926 5.593 4.259

10 10 8.519 7.037 5.556

Table 18.5 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.12, kd ¼ 4%, 6%, 8% and 10% and L ¼ 1

348 18 A New Approach to Project Ratings

l1 kd ¼ 0.10 kd ¼ 0.08 kd ¼ 0.06 kd ¼ 0.04

0 7.843 7.843 7.843 7.843

1 6.725 6.882 7.039 7.196

2 5.608 5.922 6.235 6.549

3 4.49 4.961 5.431 5.902

4 3.373 4 4.627 5.255

5 2.255 3.039 3.824 4.608

6 1.137 2.078 3.02 3.961

7 0.02 1.118 2.216 3.314

8 1.098 0.157 1.412 2.667

9 2.216 0.804 0.608 2.02

10 3.333 1.765 0.196 1.373

Table 18.6 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.12, kd ¼ 4%, 6%, 8% and 10% and L ¼ 3

18.5 Dependence of NPV on Leverage Ratios 349

350

18

A New Approach to Project Ratings

NPV/NOI

NPV/NOI (l1) at L=1 10 8 6 4 2 0 -2 -4 -6 -8 -10 -12

Kd=10 0

1

2

3

4

5

6

7

8

9

10

Kd=8

Kd=6 Kd=4

l1

Fig. 18.4 The dependence of NPV (in units of D) on leverage ratio on debt l1 at k0 ¼ 12%, kd ¼ 4%, 6%, 8% and 10% and L ¼ 1

NPV/NOI (l1 ) at L=3 10 8

NPV/NOI

6

Kd=10

4

Kd=8

2

Kd=6

0 -2 -4

0

1

2

3

4

5

6

7

8

9

10

Kd=4

l1

Fig. 18.5 The dependence of NPV (in units of D) on leverage ratio on debt l1 at k0 ¼ 12%, kd ¼ 4%, 6%, 8% and 10% and ¼ 3

18.5.1.3

The Dependence of NPV (in Units of NOI) (NPV NOI ) on Leverage Ratio on Debt l1 at Equity Cost k0 = 0.26

The formula of Modigliani and Miller in Excel will look like: ¼(A3/C3)+(((1–(E3*A3))*(1–B3))/(D3*(1–((C3*B3)/(1+C3)))))

18.5

Dependence of NPV on Leverage Ratios

351

NPV/NOI (l1) at L=1 and L=3 for Kd=0.10 10

NPV/NOI

5 0 -5

0

1

2

3

4

5

6

7

8

9

10

L=1 L=3

-10 -15

l1

Fig. 18.6 The dependence of NPV (in units of NOI) on leverage ratio on debt l1 at k0 ¼ 12%, kd ¼ 10% and L ¼ 1 and L ¼ 3

NPV/NOI

NPV/NOI (l1) at L=1 and L=3 for Kd=0.08 10 8 6 4 2 0 -2 -4 -6 -8 -10

L=1 0

1

2

3

4

5

6

7

8

9

10

L=3

l1

Fig. 18.7 The dependence of NPV (in units of NOI) on leverage ratio on debt l1 at k0 ¼ 12%, kd ¼ 8% and L ¼ 1 and L ¼ 3

Using this formula we calculate the dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, at different values of kd ¼ 22%, 16%, 10%, 6% and at two values of leverage level L ¼ 1 and L ¼ 3. Let us start from the case L ¼ 1 (Table 18.9 and Fig. 18.12). Let us calculate the value of l1 below which the investment project remains effective (NPV > 0)

352

18

A New Approach to Project Ratings

NPV/NOI (l1) at L=1 and L=3 for Kd=0.06 10

NPV/NOI

5 L=1

0 0

1

2

3

4

5

6

7

8

9

10

L=3

-5 -10

l1

Fig. 18.8 The dependence of NPV(in units of NOI) on leverage ratio on debt l1 at k0 ¼ 12%, kd ¼ 6% and L ¼ 1 and L ¼ 3

NPV/NOI(l1) at L=1 and L=3 for Kd=0.04 10 8

NPV/NOI

6 4 2

L=1

0 -2

0

1

2

3

4

5

6

7

8

9

10

L=3

-4 -6 -8

l1

Fig. 18.9 The dependence of NPV(in units of NOI) on leverage ratio on debt l1 at k0 ¼ 12%, kd ¼ 4% and L ¼ 1 and L ¼ 3

kd l1

0.22 1.9

0.16 2.2

0.1 2.5

0.06 2.7

One can see from this table that the value of l1 below which the investment project remains effective (NPV > 0) decreases with credit rate kd, which means that the

18.5

Dependence of NPV on Leverage Ratios

353

NPV/NOI (l1) at L=1 6 4 2 0 1 NPV/NOI

2

3

4

5

6

7

8

9

10

11

-2 -4 -6 1

-8

2

-10

3 4

-12

Fig. 18.10 The dependence of NPV (in units of NOI) on leverage ratio on debt l1 at k0 ¼ 14%, kd ¼ 6%–(1), 8%–(2), 10%–(3), 12%–(4) at L ¼ 1

NPV/NOI (l1) at L=3 7

5

3

NPV/NOI 1

-1

1

2

3

4

5

6

7

8

9

10

11 1

2 -3

-5

3 4

Fig. 18.11 The dependence of NPV (in units of NOI) on leverage ratio on debt l1 at k0 ¼ 14%, kd ¼ 6%–(1), 8%–(2), 10%–(3), 12%–(4) at L ¼ 3

354

18

A New Approach to Project Ratings

Table 18.7 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.14, kd ¼ 6%, 8%, 10%, 12% and L ¼ 1 l1 0 1 2 3 4 5 6 7 8 9 10

L 1 1 1 1 1 1 1 1 1 1 1

k0 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

NPV/NOI (kd ¼ 0.12) 6.349206349 4.587301587 2.825396825 1.063492063 0.698412698 2.46031746 4.222222222 5.984126984 7.746031746 9.507936508 18.26984127

NPV/NOI (kd ¼ 0.1) 6.349206349 4.714285714 3.079365079 1.444444444 0.19047619 1.825396825 3.46031746 5.095238095 6.73015873 8.365079365 10

NPV/NOI (kd ¼ 0.08) 6.349206349 4.841269841 3.333333333 1.825396825 0.317460317 1.19047619 2.698412698 4.206349206 5.714285714 7.222222222 8.73015873

NPV/NOI (kd ¼ 0.06) 6.349206349 4.968253968 3.587301587 2.206349206 0.825396825 0.555555556 1.936507937 3.317460317 4.698412698 6.079365079 7.46031746

Table 18.8 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.14, kd ¼ 6%, 8%, 10%, 12% and L ¼ 3 l1 0 1 2 3 4 5 6 7 8 9 10

L 3 3 3 3 3 3 3 3 3 3 3

k0 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14 0.14

t 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

NPV/NOI kd ¼ 0.12 6.722689 5.582633 4.442577 3.302521 2.162465 1.022409 0.11765 1.2577 2.39776 3.53782 4.67787

NPV/NOI kd ¼ 0.1 6.722689 5.717087 4.711485 3.705882 2.70028 1.694678 0.689076 0.31653 1.32213 2.32773 3.33333

NPV/NOI kd ¼ 0.08 6.722689 5.851541 4.980392 4.109244 3.238095 2.366947 1.495798 0.62465 0.2465 1.11765 1.9888

NPV/NOI kd ¼ 0.06 6.722689 5.985994 5.2493 4.512605 3.77591 3.039216 2.302521 1.565826 0.829132 0.092437 0.64426

effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd (Table 18.10 and Fig. 18.13). Let us calculate the value of l1 below which the investment project remains effective (NPV > 0) kd l1

0.22 3.85

0.16 4

0.1 5.6

0.06 6.6

One can see from this table that like the case of L ¼ 1 the value of l1 below which the investment project remains effective (NPV > 0) decreases with credit rate kd, which means that the effectiveness of the investment project as well as its

18.5

Dependence of NPV on Leverage Ratios

355

Table 18.9 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 22%, 16%, 10%, 6% and L ¼ 1 l1 0 1 2 3 4 5 6 7 8 9 10

NPV/NOI (l1) kd ¼ 0.16 3.4188034 1.8717949 0.3247863 1.2222222 2.7692308 4.3162393 5.8632479 7.4102564 8.957265 10.504274 12.051282

NPV/NOI (l1) kd ¼ 0.22 3.418803419 1.666666667 0.08547009 1.83760684 3.58974359 5.34188034 7.09401709 8.84615385 10.5982906 12.3504274 14.1025641

NPV/NOI (l1) kd ¼ 0.1 3.41880342 2.07692308 0.73504274 0.60683761 1.94871795 3.29059829 4.63247863 5.97435897 7.31623932 8.65811966 10

NPV/NOI (l1) kd ¼ 0.06 3.4188034 2.2136752 1.008547 0.196581 1.401709 2.606838 3.811966 5.017094 6.222222 7.42735 8.632479

2.00 0.00 -2.00

0

1

2

3

4

5

6

7

8

-4.00 -6.00 -8.00 -10.00 -12.00 -14.00 -16.00

Kd = 0,22

Kd = 0,16

Kd = 0,1

Kd = 0,06

Fig. 18.12 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 22%, 16%, 10%, 6% and L ¼ 1

creditworthiness decreases with credit rate kd. Comparing the case of L ¼ 1 one can see that at bigger leverage level (L ¼ 3) the investment project remains effective (NPV > 0) until bigger leverage ratio l1, so bigger leverage level favors to the effectiveness of the investment project as well as its creditworthiness. Let us analyze also the dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26 and each value of kd at two leverage levels L ¼ 1 and L ¼ 3 (Figs. 18.14, 18.15, 18.16 and 18.17). Studying the dependence of NPV (in units of NOI) (NPV NOI) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26 and each value of kd at two leverage levels L ¼ 1 and L ¼ 3 shows that the curve NPV NOI (l1) corresponding to bigger leverage level (L ¼ 3) lies

356

18

A New Approach to Project Ratings

Table 18.10 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 22%, 16%, 10%, 6% and L ¼ 3 NPV/NOI (l1) kd ¼ 0.16 3.6199095 2.7073906 1.7948718 0.8823529 0.0301659 0.9426848 1.8552036 2.7677225 3.6802413 4.5927602 5.505279

NPV/NOI (l1) kd ¼ 0.22 3.619909502 2.490196078 1.360482655 0.230769231 0.89894419 2.02865762 3.15837104 4.28808446 5.41779789 6.54751131 7.67722474

l1 0 1 2 3 4 5 6 7 8 9 10

NPV/NOI (l1) kd ¼ 0.1 3.6199095 2.92458522 2.22926094 1.53393665 0.83861237 0.14328808 0.5520362 1.24736048 1.94268477 2.63800905 3.33333333

NPV/NOI (l1) kd ¼ 0.06 3.6199095 3.0693816 2.5188537 1.9683258 1.4177979 0.86727 0.3167421 0.233786 0.784314 1.334842 1.88537

4.00 2.00

0.00 0

1

2

3

4

5

6

7

8

9

-2.00 -4.00 -6.00 -8.00 -10.00 Kd = 0,22

Kd = 0,16

Kd = 0,1

Kd = 0,06

Fig. 18.13 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 22%, 16%, 10%, 6% and L ¼ 3

above the curve NPV NOI (l1) corresponding to smaller leverage level (L ¼ 1). The curve NPV NOI (l1) corresponding to bigger leverage level (L ¼ 3) has smaller (negative) slope. This means that debt financing of long-term projects favors effectiveness of the investment project as well as its creditworthiness. Analyzing the obtained results one should remember, that NPV (in units of NOI) (NPV NOI ) depends only on equity cost k0, on credit rates kd, on leverage level L as well as on one of the leverage ratios lj and does not depend on equity value S, debt value D and NOI. This means that obtained results on the dependence of NPV (in units of NOI) (NPV NOI) on leverage ratios lj at different equity costs k0, at different credit rates kd,

18.5

Dependence of NPV on Leverage Ratios

357

6.00 4.00 2.00 0.00 -2.00

1

2

3

4

5

6

7

8

9

10

11

-4.00 -6.00 -8.00 -10.00 -12.00 -14.00

L=1

-16.00

L=3

Fig. 18.14 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 22 and L ¼ 1 and L ¼ 3 6.00 4.00 2.00 0.00 -2.00

1

2

3

4

5

6

7

8

9

10

11

-4.00

-6.00 -8.00 -10.00 -12.00

L=1

-14.00

L=3

Fig. 18.15 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 16% and L ¼ 1 and L ¼ 3 6.00 4.00 2.00

0.00 -2.00

1

2

3

4

5

6

7

8

9

10

11

-4.00

-6.00 -8.00 -10.00

-12.00

L=1

L=3

Fig. 18.16 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 10% and L ¼ 1 and L ¼ 3

358

18

A New Approach to Project Ratings

6.00 4.00 2.00 0.00 -2.00

1

2

3

4

5

6

7

8

9

10

11

-4.00

-6.00 -8.00

L=1

-10.00

L=3

Fig. 18.17 The dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratio on debt l1 at equity cost k0 ¼ 0.26, kd ¼ 6% and L ¼ 1 and L ¼ 3

at different leverage levels L carry the universal character: these dependencies remain valid for investment projects with any equity value S, debt value D and NOI.

18.6

Conclusions

This chapter continues to create a new approach to rating methodology: in addition to two previous Chapters (Chaps. 16 and 17), which have considered the creditworthiness of the nonfinance issuers (Brusov et al. 2018c, d), we develop here a new approach to project rating. We work within investment models, created by authors. One of them describes the effectiveness of investment project from the perspective of equity capital owners, while the other model describes the effectiveness of investment project from the perspective of equity capital and debt capital owners. The important features of current consideration as well as in previous studies are: 1. The adequate use of discounting of financial flows virtually not used in existing rating methodologies 2. The incorporation of rating parameters (financial “ratios”), used in project rating, into considered modern investment models Analyzing within these investment models with incorporated rating parameters the dependence of NPV on rating parameters (financial “ratios”) at different values of equity cost k0, at different values of credit rates kd as well as at different values of on leverage level L we come to very important conclusion, that NPV in units of NOI NPV (NPV NOI ) (as well as NPV in units of D ( D )) depends only on equity cost k0, on credit rates kd, on leverage level L as well as on one of the leverage ratios lj (on one of the coverage ratios ij) and does not depend on equity value S, debt value D and NOI. This means that obtained results on the dependence of NPV (in units of NOI) (NPV NOI ) on leverage ratios lj (as well as on the dependence of NPV (in units of D) (NPV D ) on

18.6

Conclusions

359

coverage ratios ij) at different equity costs k0, at different credit rates kd, at different leverage levels L carry the universal character: these dependencies remain valid for investment projects with any equity value S, debt value D and NOI. Calculations on dependence of NPV in units of D (NPV/D) on the coverage ratio NPV on debt i1 show, that NPV D increases with i1 and that D values turn out to be very close to each other at all i1 values. It is seen as well that NPV increases with decreasing kd. This means that the effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd. One can see, that all NPV (i1) curves corresponding to L ¼ 3 lie above the curves corresponding to L ¼ 1. This means that NPV increases with leverage level L (with increasing of the debt financing). Thus, debt financing favors to the effectiveness of the long-term project. At fixed value L, NPV increases with decreasing the credit rate kd. It is shown the value of the coverage ratio on debt i1 above which the investment project remains effective (NPV > 0) increases with credit rate kd, which means that the effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd. Comparing the cases of L ¼ 1 and L ¼ 3 one can see that at bigger leverage level (L ¼ 3) the investment project becomes effective (NPV > 0) starting from smaller coverage ratio i1, so bigger leverage level favors to the effectiveness of the investment project as well as its creditworthiness. Calculations on dependence of NPV in units of NOI (NPV/NOI) on the leverage ratio on debt l1 show that NPV in units of NOI decreases with increasing of the leverage ratio on debt l1. With the increasing of the cost of debt capital kd, curves of the dependence of NPV/NOI (l1), outgoing from a single point at a zero value of l1, lie below (i.e., the rate of decrease (or negative slope of curves) grows). Note that while the dependences of NPV(in units of D) on coverage ratio on debt i1 lie very close to each other, the dependences of NPV(in units of NOI) on leverage ratio on debt l1 are separated significantly more. One can see that the value of l1 below which the investment project remains effective (NPV > 0) decreases with credit rate kd, which means that the effectiveness of the investment project as well as its creditworthiness decreases with credit rate kd. Studying the dependence of NPV (in units of NOI) (NPV NOI) on leverage ratio on debt l1 at fixed equity cost k0 and fixed credit rate kd at two leverage levels L ¼ 1 and L ¼ 3 it was shown that the curve NPV NOI (l1) corresponding to bigger leverage level (L ¼ 3) lies above the curve NPV (l ) corresponding to smaller leverage level (L ¼ 1). NOI 1 The curve NPV (l ) corresponding to bigger leverage level (L ¼ 3) has smaller 1 NOI (negative) slope. This means that debt financing of long-term projects favors effectiveness of the investment project as well as its creditworthiness. Investigations, conducted in the current Chapter, create a new approach to rating methodology with respect to the long-term project rating. And this chapter in combination with two of our previous papers on this topic (Brusov et al. 2018c, d) creates a new base for rating methodology as a whole. In our future papers, we will consider rating methodology for investment projects of arbitrary duration.

360

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A New Approach to Project Ratings

References Brusov P (2018a) Editorial: Introduction on special issue on the Banking System and financial markets of Russia and other countries: problems and prospects. J Rev Global Econ 7:i–vi. SCOPUS Brusov P (2018b) Editorial. J Rev Global Econ 7:i–v. SCOPUS Brusov P, Filatova T, Orehova N, Eskindarov M (2015) Modern corporate finance, investments and taxation. Springer International Publishing, Cham, p 373. monograph, SCOPUS https://www. springer.com/gp/book/9783319147314 Brusov P, Filatova T, Orehova N, Eskindarov M (2018a) Modern corporate finance, investments, taxation and ratings. Springer Nature Publishing, Cham, p 571. Monograph Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018b) Rating: new approach. J Rev Global Econ 7:37–62. SCOPUS. https://doi.org/10.6000/1929-7092.2018.07.05 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018c) A “golden age” of the companies: conditions of its existence. J Rev Global Econ 7:88–103. SCOPUS. https://doi.org/10.6000/ 1929-7092.2018.07.07 Brusov PN, Filatova TV, Orekhova NP, Kulik VL (2018d) Rating methodology: new look and new horizons. J Rev Global Econ 7:63–87. SCOPUS. https://doi.org/10.6000/1929-7092.2018. 07.06 P.N. Brusov, T.V. Filatova, N.P. Orekhova, V.L. Kulik and I. Weil (2018e) New meaningful effects in modern capital structure theory J Rev Global Econ, 7: 104–122. SCOPUS https://doi.org/10. 6000/1929-7092.2018.07.08 Brusov PN, Filatova TV, Orekhova NP (2018f) Modern corporate finance and investments, monograph. Knorus Publishing House, Moscow, p 517 Brusov P, Filatova T, Orekhova N, Kulik V, Weil I, Brailov A (2018g) The impact of the central bank key rate and commercial banks credit rates on creating and maintaining of a favorable investment climate in the country. J Rev Global Econ 7:360–376. SCOPUS. https://doi.org/10. 6000/1929-7092.2018.07.31 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Weil I (2019) Ratings of the investment projects of arbitrary durations: new methodology. J Rev Global Econ 8:437–448. SCOPUS. https://doi. org/10.6000/1929-7092.2019.08.37 Brusov P, Filatova T, Orekhova N (2020a) Ratings: critical analysis and new approaches of quantitative and qualitative methodology. Springer Nature Publishing, Cham, p 369. monograph, https://www.springer.com/de/book/9783030562427 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Global Econ 9:257–268 Filatova Т, Orehova N, Brusova А (2008) Weighted average cost of capital in the theory of Modigliani–Miller, modified for a finite life–time company. Bull FU 48:68–77 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018a) Ratings of the long–term projects: new approach. J Rev Global Econ 7:645–661., SCOPUS. https://doi.org/10.6000/1929-7092.2018. 07.59 Filatova TV, Brusov PN, Orekhova NP, Kulik VL (2018b) Ratings of the long–term projects: new approach. Journal of Reviews on Global Econ 7:645–661., SCOPUS. https://doi.org/10.6000/ 1929-7092.2018.07.59 Мodigliani F, Мiller M (1958) The cost of capital, corporate finance, and the theory of investment. Am Econ Rev 48:261–297 Мodigliani F, Мiller M (1963) Corporate income taxes and the cost of capital: a correction. Am Econ Rev 53:147–175 Modigliani F, Miller M (1966) Some estimates of the cost of capital to the electric utility industry 1954–1957. Am Econ Rev 56:333–391 Myers S (2001) Capital structure. J Econ Perspect 15(2):81–102

Chapter 19

Conclusions

In monograph, the Modigliani–Miller theory, which is the perpetuity limit of the BFO theory and nevertheless widely used in practice, is generalized taking into account the conditions for the actual functioning of companies: for the case of variable company income, for the case of income tax payments with an arbitrary frequency (monthly, quarterly, semiannual, or annual payments), as for advance tax payments for profit, and for payments at the end of the period, as well as other conditions (Brusov et al. 2020a, b; 2021a, b; Brusov and Filatova 2021; Filatova et al. 2022). New modern investment models have been created that are as close as possible to real investment conditions, with various schemes for repaying debt and interest on it (frequent payments of debt, advance payments of debt, etc.), with variable income from investments, as well as taking into account various options for paying taxes, adopted in various countries. Consideration is carried out both from the point of view of owners of equity capital and from the point of view of owners of equity and debt capital. Within the framework of the created new modern investment models, a complete and detailed study of the dependence of the main efficiency indicator, NPV, on the level of debt financing, level of taxation and the level of profitability in a wide range of values of equity and debt capital was carried out. New modern methodologies for rating of nonfinancial issuers and for project rating based on the application of the generalized Modigliani–Miller theory, as well as of investment models created by the authors of the monograph, have been developed. For this, the modification of the generalized Modigliani–Miller theory, as well as modern investment models (modified for real investment conditions) for the needs of rating, has been done. The incorporation of financial indicators used in the rating methodology into the generalized theory of Modigliani–Miller, as well as into modern investment models (modified for real investment conditions), has been carried out. Within the framework of the generalized Modigliani–Miller theory modified for the needs of rating, a complete and detailed study of the dependence of the weighted average cost of capital, WACC, of the company, used as the discount rate for discounting financial flows, on the financial ratios used in the rating, on the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 P. Brusov et al., Generalized Modigliani–Miller Theory, Contributions to Finance and Accounting, https://doi.org/10.1007/978-3-030-93893-2_19

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Conclusions

level of debt financing and the level of taxation in a wide range of values of costs of equity and debt capital for perpetuity companies was carried out. In project rating, a complete and detailed study of the dependence of the main efficiency indicator, NPV, on the financial ratios used in the rating (coverage ratios and leverage ratios), on the level of debt financing and the level of taxation, on frequency of tax on profit and interest on debt payments in a wide range of values of costs of equity and debt capital for long-term projects was carried out. This will make it possible to carry out a correct assessment of discount rates taking into account the values of financial ratios and to issue correct ratings for both nonfinancial issuers and long-term projects. Obtained in monograph results demonstrate that properties of the generalized Modigliani–Miller theory are quite different from ones of the classical Modigliani– Miller theory. This leads to the fact that the generalized Modigliani–Miller theory, as well as modern investment models modified for real investment conditions, created in monograph, and new rating methodologies, developed in monograph and based on the application of the generalized Modigliani–Miller theory are much more applicable in real economy, finance, and practice. Since the Modigliani–Miller theory is still widely used in practice, its generalized modification will allow to increase the quality of the company’s financial management, the efficiency of corporate finance management, to improve the quality of assessing the effectiveness of investments, to improve the taxation and tax control system, to develop an adequate system of business assessment, increase the objectivity of rating issues. Investigations of the generalized Modigliani–Miller theory will be continued, and in Second Edition, we will consider more its applications in corporate finance, investments, taxation, in business valuation and ratings, where using of generalized Modigliani–Miller theory will be quite useful.

References Brusov P, Filatova T (2021) The Modigliani–Miller theory with arbitrary frequency of payment of tax on profit. Mathematics 9(11):1198. (WoS Q1) Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020a) Application of the Modigliani–Miller theory, modified for the case of advance payments of tax on profit, in rating methodologies. J Rev Global Econ 9:282–292 Brusov PN, Filatova TV, Orekhova NP, Kulik VL, Chang S-I, Lin YCG (2020b) Modification of the Modigliani–Miller theory for the case of advance payments of tax on profit. J Rev Global Econ 9:257–267 Brusov P, Filatova T, Chang S-I, Lin G (2021a) Innovative investment models with frequent payments of tax on income and of interest on debt. Mathematics 9(13):1491. (WoS Q1) Brusov P, Filatova T, Orekhova N, Kulik V, Chang S-I, Lin G (2021b) Generalization of the Modigliani–miller theory for the case of variable profit. Mathematics 9(11):1286. (WoS Q1) Filatova TV, Brusov PN, Orekhova NP (2022) Impact of advance payments of tax on profit on effectiveness of investments. Mathematics 10(4):666. https://doi.org/10.3390/math10040666