Investment Valuation and Appraisal: Theory and Practice (Springer Texts in Business and Economics) 3030624390, 9783030624392

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Springer Texts in Business and Economics

Kay Poggensee Jannis Poggensee

Investment Valuation and Appraisal Theory and Practice

Springer Texts in Business and Economics

Springer Texts in Business and Economics (STBE) delivers high-quality instructional content for undergraduates and graduates in all areas of Business/Management Science and Economics. The series is comprised of self-contained books with a broad and comprehensive coverage that are suitable for class as well as for individual self-study. All texts are authored by established experts in their fields and offer a solid methodological background, often accompanied by problems and exercises.

More information about this series at http://www.springer.com/series/10099

Kay Poggensee • Jannis Poggensee

Investment Valuation and Appraisal Theory and Practice

Kay Poggensee School of Business Management University of Applied Sciences Kiel, Germany

Jannis Poggensee Kremperheide, Germany

ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-030-62439-2 ISBN 978-3-030-62440-8 (eBook) https://doi.org/10.1007/978-3-030-62440-8 Translation from the German language edition: Invesitionsrechnung by Kay Poggensee # Springer FachmedienWiesbaden 2015. All Rights Reserved. # The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 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

Preface

Writing a preface is certainly one of the finest activities of an author of an academic textbook, since it is said that a large part of the work on the book is completed, for a preface is probably always written last and thus shortly before publication. This book was not written under time pressure. The author Kay Poggensee can already look back on over 60 semesters of teaching in this subject area and about 200 courses on the subject of investment valuation held at various universities, further education institutions, and practical training courses. The time has hopefully been sufficient to identify the relevant aspects of the theoretical field for the needs in academic teaching and practical application in companies and in particular to identify the difficulties of learners and readers in individual subject areas and to meet these in the text in a didactically meaningful way. The other author Jannis Poggensee gave fresh impetus to the book due to his experience in his various BA, MA, and PhD studies and his work for a bank. In any case, there was enough time for Kay Poggensee to give up his resistance to write a textbook on the subject of investment valuation. Certainly, in the relatively static field of knowledge of investment calculation and investment theory, there are various excellent texts by very qualified colleagues from renowned universities who explain the field of knowledge very well. Some of them have been active for a long time, some of them for more than 60 years with almost unchanged content, and some of them have already had a two-figure print run, successfully supporting the dissemination of the field of knowledge among students and practitioners. Nevertheless, after many years of experience in the field of teaching and in support of practical questions in companies, the author Kay Poggensee felt that the production of a new textbook on the subject of investment valuation would be useful for three reasons. On the one hand, this is due to the fact that the use of information technology and, in this case, standard software is currently increasingly penetrating practice and teaching. While many established textbooks still contain sections in which programming routines for small programs on investment calculation issues are presented, spreadsheet analysis with Microsoft Excel has certainly proved itself today in practice and teaching due to a much higher market penetration and because of the ever-increasing implementation of mathematically relevant functionalities for this subject area in the package. Therefore, this book focuses more than the other

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common textbooks on the solution of investment calculation problems in this software. In addition, students often request a textbook that is specifically tailored to the relevant academic courses on this subject at the universities where the authors hold courses on the subject. This book now takes this into account. Last but not least, especially practitioners in companies and application-oriented students are constantly confronted with the question of how an academic or company problem is to be solved in concrete terms and which problems can arise in the academic solution of a problem in practice. The book should also take this problem into account. Thus, great importance is attached to a thorough elaboration of the academic theories on which the individual methods of investment calculation are based, and their feasibility in practice is critically examined in connection with the assumptions made, as is usual in good academic textbooks on this field of knowledge. In addition, the concrete implementation of the models in business practice is also intensively demonstrated using many examples. This will hopefully put students and practitioners in a position to use the examples to gain access to the investment calculation models and operational problems and to solve them appropriately. Calculation results for the many sample calculations and tasks created with Excel are available for download on the homepage of Springer under the respective chapter of the book (https://link.springer.com, search term “Kay Poggensee”). In this way, all calculations and the Excel formulas stored in the figures can be traced. The Excel files are stored according to their figure numbers in the book. There are all the figures that are marked with an asterisk in the book. Another strong reason to publish this book is the fact that it is written in English. More and more courses worldwide are taught in English. Thus, this title gives academics around the world the opportunity to follow this academic field in English by studying this book. At the end of a preface, it is good academic custom to express thanks. Here the authors have taken a lot of time to write this text, because it is a challenge for a university teacher at the University of Applied Sciences and a PhD student to find the time and other necessary resources to write such a text. The teaching activity at the University of Applied Sciences alone almost covers the time required for a full-time professional activity in other areas of work. In addition, for a committed university lecturer, there is the time-consuming transfer of knowledge from the university to the companies and participation in academic selfadministration and in various strategically active groups. This leaves little time for research and publication. Compared to traditional universities, working at the University of Applied Sciences also means the lack of an academic middle class, i.e., unfortunately the lack of doctoral students, research assistants, and a secretariat of its own, who usually provide extensive support in research, teaching, and the preparation of textbooks. Although this makes acknowledgments superfluous in this area, it means additional complex tasks for the authors in addition to teaching in the production of books. This then had to be done over a long period of time in the evenings, on weekends, and during holidays.

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So our first thanks go to our family, who have accepted the absence in the family activities during these times, who have renounced free and family time with us, and who have partly taken over our obligations in the family during this time. My second thanks go to the students of Kay Poggensee. With their many questions and discussions in the courses in the past, they have helped to know what content, thought processes, exercise steps, and presentation techniques are required for the lecturer to understand this field of study in order to support the students in their learning. We hope we have taken up the suggestions sensibly and developed a text that helps students and practitioners to understand this subject area and to apply the techniques correctly to practical problems. Our special thanks go to Mrs. Cyra-Helena Schmucker, BA. Without her this book would not have been possible. Due to her experience as a translator and her knowledge on investment valuation and her immense work, she made it possible to develop this already existing book in German into a title that can be used for modules taught in English and by students studying in English around the world. Of course, any errors that may still exist are entirely at the authors’ expense. Here we ask the readers, as far as they discover mistakes, to inform Kay Poggensee (Kay. [email protected]), so that we can improve this for following editions. In conclusion, we would like to emphasize that we enjoyed writing this book very much. It was done voluntarily. So the joy must have outweighed the aforementioned renunciation. So, to conclude our activities as the authors of this book, we would like to grant ourselves three wishes which may accompany the reader in reading this book: 1. We hope that this text will help readers to understand the subject of investment evaluation more easily and that they will be satisfied with the text. 2. We hope that the knowledge disseminated by reading this book will help to increase the number of qualified investment decisions in practice. 3. We hope that we will soon be able to write the preface to the second edition, as this would certainly be one of the best proofs of the usefulness of this book. Kiel, Germany Kremperheide, Germany Autumn 2020

Kay Poggensee Jannis Poggensee

Contents

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2

Introduction to Investment Evaluation . . . . . . . . . . . . . . . . . . . . 1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Significance and Relevance of the Investment Evaluation . . . . 1.3 Aim and Definition of Investment Calculation . . . . . . . . . . . . 1.4 Differentiation of Investment Calculation from Other Business Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Investment Calculation Procedures at a Glance . . . . . . . . . . . 1.6 Historical Development of Investment Calculation . . . . . . . . . 1.7 The Organisational Structure for Investment Analysis . . . . . . 1.8 Process Organisation of Investment Calculation . . . . . . . . . . . 1.9 The Problem of Collecting Data for Investment Calculation . . 1.10 Necessity and Limits of Investment Calculation . . . . . . . . . . . 1.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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11 14 17 18 20 22 28 29 30

Static Investment Calculation Methods . . . . . . . . . . . . . . . . . . . . 2.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Fundamental Aspects of Static Investment Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 A Modular System for the Creation of Static Investment Calculation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 The Components of Static Investment Calculation Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 The Combinations for the Creation of Static Investment Calculation Formulas . . . . . . . . . . . . . . . 2.3.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Cost Comparison Calculation . . . . . . . . . . . . . . . . . . . . 2.4.1 Presentation and Criticism of the Cost Comparison Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Formulas of the Cost Comparison Calculation . . . . . 2.4.3 Application of the Cost Comparison Calculation . . . . 2.4.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Profit Comparison Calculation . . . . . . . . . . . . . . . . . . . .

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2.5.1

Presentation and Criticism of the Profit Comparison Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Formulas of the Profit Comparison Calculation . . . . . 2.5.3 Application of the Profit Comparison Calculation . . . 2.5.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 The Profitability Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Presentation and Criticism of the Profitability Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Formulas of the Profitability Calculation . . . . . . . . . 2.6.3 Application of the Profitability Calculation . . . . . . . . 2.6.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 The Static Amortisation Calculation . . . . . . . . . . . . . . . . . . . 2.7.1 Presentation and Criticism of the Static Amortisation Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Formulas of the Static Amortisation Calculation . . . . 2.7.3 Application of the Static Amortisation Calculation . . 2.7.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Dynamic Investment Calculation Methods . . . . . . . . . . . . . . . . . . 3.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Model Assumptions of Dynamic Investment Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Objective of the Dynamic Investment Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Assumptions of the Dynamic Investment Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 Calculation Elements of Dynamic Investment Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Fundamentals of Financial Mathematics . . . . . . . . . . . . . . . . 3.3.1 The One-Time Factors . . . . . . . . . . . . . . . . . . . . . . 3.3.2 The Summation Factors . . . . . . . . . . . . . . . . . . . . . 3.3.3 The Distribution Factors . . . . . . . . . . . . . . . . . . . . . 3.4 The Net Present Value Method . . . . . . . . . . . . . . . . . . . . . . . 3.4.1 Net Present Value with Individual Discounting . . . . . 3.4.2 Net Present Value If DSF Can Be Used . . . . . . . . . . 3.4.3 Net Present Value with Infinite Useful Life . . . . . . . 3.4.4 Case Study Net Present Value Method . . . . . . . . . . . 3.4.5 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 The Horizon Value Method . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The Annuity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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62 66 67 70 70

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70 73 75 78 78 78 79 83 84

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3.7

The Internal Rate of Return Method . . . . . . . . . . . . . . . . . . . 3.7.1 Determination of the Yield with the Regula Falsi . . . 3.7.2 Special Cases in Determining the Return . . . . . . . . . 3.7.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 The Dynamic Amortisation Calculation . . . . . . . . . . . . . . . . 3.9 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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122 124 126 130 131 136 138 140

Selection of Alternatives and Investment Programme Planning . . 4.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Selection of Alternatives as a Problem in Investment Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 An Example of the Ambiguity in the Selection of Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Causes for Ambiguity in the Selection of Alternatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Removal of the Reinvestment Premise . . . . . . . . . . . . . . . . . 4.3.1 Use of Capital in the Dynamics and Reality . . . . . . . 4.3.2 Net Present Value Formula with Removed Reinvestment Premise . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Consequence of the Net Present Value Formula for the Selection of Alternatives After Removing the Reinvestment Premise . . . . . . . . . . . . . . . . . . . . 4.3.4 Application Example . . . . . . . . . . . . . . . . . . . . . . . 4.3.5 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fictitious Investments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 Graphic Form and to an Account Assigned Form of the Fictitious Investment . . . . . . . . . . . . . . . . . . . 4.4.2 Graphical Form of the Fictitious Investment . . . . . . . 4.4.3 To an Account Assigned Form of the Fictitious Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.4 Application Example . . . . . . . . . . . . . . . . . . . . . . . 4.4.5 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Ambiguity of the Internal Rate of Return . . . . . . . . . . . . . . . 4.5.1 Special Net Present Value Functions in Determining the Return . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Examples of Ambiguous Returns . . . . . . . . . . . . . . . 4.5.3 Check Routines for Checking the Economic Validity of Determined Returns . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 The Utility Value Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Procedure of the Utility Value Analysis . . . . . . . . . . 4.6.2 Application Example . . . . . . . . . . . . . . . . . . . . . . .

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4.6.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Account Development Planning . . . . . . . . . . . . . . . . . . . 4.7.1 Presentation of the Account Development Planning . 4.7.2 Application Example for Account Development Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8 The Dean Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.8.1 Representation of the Dean Model . . . . . . . . . . . . . . 4.8.2 Comparison of the Programme Decision According to Dean Model and Account Development Planning . 4.8.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 The Linear Optimisation . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9.1 Linear Optimisation Technique . . . . . . . . . . . . . . . . 4.9.2 Application Example . . . . . . . . . . . . . . . . . . . . . . . 4.9.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.10.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7

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Optimum Useful Life and Optimum Replacement Time . . . . . . . 5.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Useful Life Optimisation as an Economic Problem . . . . . . . . 5.3 Model Assumptions for the Calculation of Useful Life . . . . . . 5.4 Determination of the Optimum Useful Life . . . . . . . . . . . . . . 5.4.1 Optimum Useful Life for a One-Time Investment . . . 5.4.2 Optimum Useful Life with Repeated Investment . . . . 5.4.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Determination of the Optimal Replacement Time . . . . . . . . . 5.5.1 Optimal Replacement Time with Annual Replacement Possibility . . . . . . . . . . . . . . . . . . . . . 5.5.2 Optimum Replacement Time for Replacement After a Multi-Year Period . . . . . . . . . . . . . . . . . . . . 5.5.3 Application Example . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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229 231 233 234 234 234 237

Investment Decisions in Uncertainty . . . . . . . . . . . . . . . . . . . . . . 6.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Data Uncertainty as a Decision-Making Problem . . . . . . . . . . 6.2.1 The Concept of Risk . . . . . . . . . . . . . . . . . . . . . . . .

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xiii

6.2.2 6.2.3

Reasons for Risk in the Investment Decision . . . . . . . The Importance of Considering Risk in the Investment Decision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Correction Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.1 Correction Procedure in Detail . . . . . . . . . . . . . . . . . . 6.3.2 Application Example for the Correction Methods . . . . 6.3.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Sensitivity Analyses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 The Critical Value Calculation . . . . . . . . . . . . . . . . . . 6.4.2 The Triple Calculation . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 The Target Value Change Calculation . . . . . . . . . . . . 6.4.4 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Sequential Investment Decisions . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Procedure for Sequential Planning . . . . . . . . . . . . . . . 6.5.2 Application Example for Sequential Planning . . . . . . . 6.5.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Investment Decision in Uncertainty . . . . . . . . . . . . . . . . . . . . 6.6.1 Principles of Dominance . . . . . . . . . . . . . . . . . . . . . . 6.6.2 The Maximax Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.3 The Minimax Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.4 The Hurwicz Rule . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.5 The Laplace Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.6 The Savage–Niehans Rule . . . . . . . . . . . . . . . . . . . . . 6.6.7 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 The Risk Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7.1 Procedure for Risk Analysis . . . . . . . . . . . . . . . . . . . 6.7.2 Application Example for Risk Analysis . . . . . . . . . . . 6.7.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Portfolio Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Procedure for the Portfolio Selection Model According to Markowitz . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Application Example for Portfolio Selection . . . . . . . . 6.8.3 Section Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Case Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

242 243 244 244 245 246 248 249 249 258 260 264 264 265 268 269 272 274 275 276 276 277 278 279 279 280 284 286 286 287 288 296 296 309 310

Tables of Financial Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

311

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

327

1

Introduction to Investment Evaluation

1.1

Objectives

In this chapter, the reader will deal with the basics of investment evaluation. The aim is to make the reader aware of what exactly the business administration discipline’s theoretical area of “investment theory” covers and on which assumptions the academic model simplifies the complex reality in order to obtain a decision. The goal of investment evaluation will be elaborated and the knowledge should be gained that it is an academic model, whose results are not identical to reality, but are to be interpreted in consideration of the assumptions made. In detail, the following goals should be achieved in this section: • The relevance of the investment calculation is to be documented and evaluated with empirical information from the point of view of the national economy, companies and private households. • The goal of investment calculation should be worked out. Thereby, different possible questions and asset concepts, which can belong to the aim of an investment calculator, are presented. • The investment calculation is to be differentiated from other business management studies of internal accounting. • The different groups of investment calculation methods are distinguished from each other, the individual methods are assigned to the groups. • The reader should be made aware of the importance of the various investment calculation methods, taking into account their chronological development and the computing capacities available at that time. • Ideal-typical organisational systematisation for the allocation of investmentcalculating instances in a company are discussed depending on the size of the company and the capital commitment through investments. • The process organisation of the investment calculation is discussed in detail. • The problems of data collection and the consequences of the obtained results for realistic representation are discussed and. # The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. Poggensee, J. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-62440-8_1

1

2

1 Introduction to Investment Evaluation

• The general purpose and the operational benefit of the investment calculation are discussed in principle. After reading the chapter, the reader should be able to define what investment calculation is, what it means from different perspectives, what its goal is, what different procedures exist and what its limitations are.

1.2

Significance and Relevance of the Investment Evaluation

The implementation of investment evaluation is of great importance for society, individual companies and private households, both from a strategic and an operational point of view. From a strategic point of view, it is important because it is the basis for long-term capital planning, usually on a larger scale. From an operational point of view, it is important because it allows you to evaluate individual investment objects or investment programmes in concrete and quantitative terms. Investment calculation is a technique that can be used for economic issues, for operational problems and for private projects. Since this book is a textbook of business administration, the operational focus is certainly in the foreground. However, the calculation techniques themselves do not differ for all three areas, so the procedures are of course identically applicable in all three areas. First of all, the importance of investment evaluation for the national economy will be demonstrated using empirical data from large economies of the world and additionally from Germany over the past years. The investment ratio is used as an indicator, which in economic terms is defined as the quotient of gross fixed capital formation and gross domestic product. Certainly, a criticism of this indicator is admissible, since in a payment-oriented or value-oriented economy a separation into fixed and current assets is not particularly meaningful, and the use of precisely these two variables as numerator and denominator is also arbitrary, but as empirical support for the above-stated importance of investment calculation for economic issues, the value may be acceptable. In Fig. 1.1 we see the level of the investment ratio of large economies in the world as a percentage. Figure 1.1 illustrates that investments in all major economies of the world have accounted for a high proportion of the use of capital in the economy in recent decades. The values range from 11.6% (Greece, 2015) as the lower limit and 36.8% (ROK (Republic of Korea), 1990) as the upper limit. This means that the total gross domestic product of 1 year, for this small calculation, assuming that the gross domestic product remains constant, will be spent on investments in only approximately 3–10 years. This shows the great importance of investments for the national economy. It is, therefore, all the more important to make qualified investment decisions. The techniques of investment calculation help in this respect. Figure 1.1 also shows that the investment ratio of individual nations generally declines over time. This may be due to the extended durability of investment objects, e.g. automobiles, and a

1.2 Significance and Relevance of the Investment Evaluation

Country

3

Investment ratio (%) 1980

1990

2000

2010

2015

2017/2018

A

27.4

25.4

25.6

21.6

22.7

23.9

B

24.9

24.1

22.7

21.9

22.8

23.1

CZ

..

24.5

30.6

26.9

26.5

25.5

D

25.8

24.3

23.1

19.5

20.0

21.2

DK

20.5

20.4

21.5

18.1

19.9

22.0

ESP

22.8

25.8

25.9

21.8

18.0

19.4

FR

24.4

23.4

21.5

22.1

21.5

22.9

FIN

27.3

29.7

23.1

21.8

20.0

22.4

GR

31.5

25.5

24.6

17.6

11.6

12.9

H

..

23.3

25.5

20.2

22.3

25.2

I

25.3

21.9

20.7

20.0

16.9

17.2

IRL

28.3

19.2

23.8

17.5

24.1

23.4

LUX

21.2

20.4

20.2

17.6

18.2

16.8

NL

22.8

23.1

22.5

19.7

22.1

20.3

P

30.6

26.7

28.0

20.6

15.5

17.6

PL

..

19.3

23.7

20.3

20.1

..

S

25.3

28.7

22.1

22.0

23.3

24.7

RUS

..

26.7

15.4

19.4

19.7

..

UK

21.5

22.2

17.9

15.3

16.7

16.9

CH

28.8

31.0

25.3

22.8

23.8

24.6

N

27.9

22.8

19.8

20.7

23.8

24.1

CDN

22.9

21.7

19.6

23.5

23.9

22.5

J

33.6

34.2

27.4

21.3

23.8

22.5

ROK

32.2

36.8

30.8

29.2

27.6

29.3

USA

23.3

21.2

23.0

18.3

20.1

20.2

Fig. 1.1 Historical investment ratios in large economies in percentage (Source: Deutschland in Zahlen 2019)

4

Year

1 Introduction to Investment Evaluation

Gross domestic product Gross fixed capital formation Investment ratio In current prices, in billion euro

Percent

1991

1,585.80

394.7

24.89

1995

1,894.61

446.0

23.54

2000

2,109.09

487.5

23.11

2005

2,288.31

436.5

19.08

2010

2,564.40

501.1

19.54

2015

3,030.07

605.9

20.00

2018

3,344.37

707.7

21.16

2019

3,435.76

..

..

Fig. 1.2 Historical development of gross domestic product, gross fixed capital formation and investment ratio in Germany (Source: Deutschland in Zahlen 2019)

longer optimum useful life. In individual cases, this effect is masked by economic developments or government incentive programmes (e.g. tax incentives). In developed economies, the growth potential is also lower, so that the investment ratio there is lower in absolute terms than in growing economies. The fact that even if the investment ratio falls, investment activity can rise in absolute terms as shown in Fig. 1.2 using data from Germany over time. In Germany, around 450–700 billion euros have been invested annually in gross fixed capital formation over the past two decades. Despite the tendency for the investment ratio to decline, these are not always declining absolute investment amounts. The importance of investment calculation, therefore, constantly remains high for the best possible use of capital and thus for the competitiveness of the national economy. An improved investment decision through more precise investment calculations could mobilise large efficiency reserves. For example, a 10% saving in gross fixed capital formation through cost savings due to improved data collection and more accurate investment calculations or through the acquisition of economically reasonable assets with a longer optimum useful life would generate an annual savings volume of approximately 45–70 billion euros. Thus, it would be more successful than many government economic stimulus programmes or tax relief programmes. The same applies to the economic importance of investments in companies. Here, too, the significance is first shown empirically. From an operational point of view, the share of investments is interesting for various parameters in the denominator, e.g. number of companies, number of employees and amount of revenue. Just as in the previous economic discussion of the importance of investment calculation, criticism of the use of these indicators is easily possible. Again, it may be said that it is helpful as empirical information for the above-stated importance of investments for companies. Figure 1.3 presents these variables for mining and manufacturing in a

1.2 Significance and Relevance of the Investment Evaluation

Year

5

Companies

Employees

Units

Mill. euro

Investments per Investments per Net Investments return on revenue company employee sales Mill. euro 1.000 euro Percent Percent

Revenue

Investments

1995

47,919

1.000 people Billion euro 6,779 1,060

47,100

0.98

6.95

1.6

4.4

2000

48,913

6,375

1,307

53,287

1.09

8.36

2.7

4.1

2005

45,140

5,785

1,446

45,740

1.01

7.91

3.4

3.2

2010

44,687

5,716

1,576

46,474

1.04

8.13

3.6

2.9

2015

37,150

6,294

1,859

59,773

1.61

9.50

2.9

3.2

2018

38,660

6,640

2,055

68,434

1.77

10.31

..

3.3

Fig. 1.3 Key figures for companies in the mining and manufacturing industries (Source: Statistisches Jahrbuch 2019)

time series. This area was chosen because it is covered continuously by the Federal Statistical Office of Germany. Figure 1.3 shows the importance of investment for these companies. For example, around 50 billion euros are invested in the sector every year. This corresponds to about 1–2 million euros per company and about 8000–10,000 euros per employee. The share of investments in turnover is about 3% and is thus about the same as the net return on sales, i.e. the annual net profit after corporate taxes in relation to turnover. The amount of invested capital each year is, therefore, as high as the annual surplus, to which companies naturally attach great importance. Here, too, the special importance of the investment calculation for companies is evident. An improvement in the ratio of investments in relation to turnover will result in a disproportionately high improvement in the annual surplus, at least in the analysed sector in the years analysed. With a given level of approximately 450–700 billion euros gross fixed capital formation per year, as shown in Fig. 1.2, of which a large part is carried out by companies, the importance of investment calculation for the competitiveness of companies is impressively underpinned. For private households, the importance of the investment calculation can be demonstrated in a similar way, even if it is much less common here than in companies. The consumption expenditure of private households in 2019 in Germany may be used as an indicator here. Criticism of this indicator is easily possible, as with those on the importance of the investment calculation for the national economy and companies. Here, too, it may be said that the indicator is helpful as empirical information for the above-stated importance of investment for private households. Figure 1.4 shows household consumption expenditure broken down by intended use from 2019, again illustrating the importance of cost estimation using investment calculation methods for private households from an economic perspective. A hypothetical more efficient use of financial resources by private households in their

6

1 Introduction to Investment Evaluation

Intended use

billion euro percent

housing, energy, water

401.8

23.56

other

317.3

18.60

Traffic, News

277.2

16.25

Food, beverages, tobacco

237.1

13.90

Leisure, Enterprise, Culture

188.3

11.04

Furnishings, appliances for the household

110.6

6.48

Accommodation, Restaurants

95.8

5.62

Clothing, shoes

77.5

4.55

Domestic consumption

1705.6

100

Fig. 1.4 Private household consumption expenditure in 2019 (Source: Deutschland in Zahlen 2019)

consumption expenditure by means of investment calculation of 10% would have made an additional 170 billion euros available for private consumption in Germany in 2019. In principle, investments carry technical and social progress into the areas in which they are made. For example, expansion investments in wind power plants in wind farms lead to an increased supply of electricity from this sector, while the decline in replacement investments in coal-fired power plants leads to a reduction in the supply of electricity from this sector. This results in a change that is probably desired by society as a whole at the moment. The know-how in this area created in connection with the financial investments will enable the relevant industry and society to maintain or improve its international competitiveness. The investment activity is, therefore, of great relevance. Companies that invest in this market segment and thus improve their competitiveness and know-how can achieve high surpluses in a growing market while at the same time increasing their market share due to the competitive advantages gained through investments. The investment evaluation is, therefore, also quite significant for the continued existence of the companies. Investments by private households in the area of thermal insulation of private property can lead to savings in heating costs, so that these private units will have more private capital available in the future and will thus be able to either consume at a higher level than they would have done without investment or to save more capital. The term “investment” is not clearly defined, even in business terminology. It means different things depending on the situation and author. Therefore, the term concerns • The expenditure of money for an investment object, that is, a financial transaction. • The procurement of a fixed asset, that is, a transaction forming a fixed asset and.

1.2 Significance and Relevance of the Investment Evaluation

7

• The calculation of the advantageousness of an object. Within this book, the definition listed first is rarely used, the second term is referred to as investment object and the last is called investment calculation. Apart from the importance of continuous investment at all levels to implement technical and social progress, the other necessary condition is the success of the investment. An investment should, therefore, at least always create an improved situation compared to the initial situation. The probability of this improved situation increases with the application of investment calculation methods. The description of these techniques in their theoretical derivation and practical application is the subject of this book. The success of investments for the future success of companies is particularly important because • Investments often tie up a high proportion of capital. • Investments often tie up capital for a long time. Thus, an investment greatly reduces operational flexibility. The capital tied up can usually only be mobilised again before the end of the planned useful life but at greater losses, and the acquired investment often also determines the relatively limited direction in which a company is active, as the acquired equipment can only be used for a few applications. For example, a constructed cold store can be used to store food as long as it meets certain legal requirements, but it is unlikely suitable for the use as a car repair shop. In times of ever faster progress, ever higher capital commitment in companies relative to other production factors, and ever more fluctuating expectations of future success, successful investment of capital becomes increasingly vital. This can for example be measured by the development of share prices over time as an indicator of the valuation of listed stock corporations, here using the example of the DAX, by the capital market, as shown in Fig. 1.5, and a corporate management that is increasingly oriented towards maximising returns as a corporate management concept. Figure 1.5 shows the development of the DAX Performance Index from 1997 to 2019, measured at the year-end level of the index. The fluctuations of the index during the year, which are not shown in the figure, were of course much greater than the figure and the adjacent year-end levels of the DAX and the percentage change values indicate, otherwise the index would be constant throughout the year, which is of course not the case. Of course, this indicator is again open to criticism, but it underpins the strongly fluctuating profit expectations of companies and thus also the need for careful planning of the use of capital through investment calculation. It is important that the investment decision is always made by the investor and never by the calculation procedure itself, since the investment calculation procedure is always an academic model, i.e. a simplified representation of reality that cannot always reflect all the criteria relevant to a decision in a practical case. Therefore, the

8

1 Introduction to Investment Evaluation Year

14000

12000

DAX-Points

10000

8000

6000

4000

2000

0 1997 1999 2001 2003 2005 2007 2009 2011 2013 2015 2017 2019 Year

Dax Change Points Percent 1997 4250 1998 5002 17.69 1999 6958 39.10 2000 6433 -7.55 2001 5160 -19.79 2002 2893 -43.93 2003 3965 37.05 2004 4256 7.34 2005 5408 27.07 2006 6597 21.99 2007 8067 22.28 2008 4810 -40.37 2009 5957 23.85 2010 6914 16.07 2011 5898 -14.69 2012 7612 29.06 2013 9552 25.49 2014 9806 2.66 2015 10743 9.56 2016 11481 6.87 2017 12918 12.52 2018 10559 -18.26 2019 13249 25.48

Fig. 1.5 Historical development of the DAX (Source: Statistisches Jahrbuch 2019)

responsibility for the investment decision remains in the hands of the human investor and not the academic model. The conclusion of this section is that investments are of great importance for the future of society, businesses and private households. Therefore, their careful planning, using academic models which are based on mathematical methods, is of particular importance.

1.3

Aim and Definition of Investment Calculation

The purpose of investment calculation is to determine a quantitative result using a defined procedure, usually mathematical academic model, on the basis of given concrete economic data of one or more investment objects. If necessary this can also be done by means of expected values and known statistical distribution of defined data, which normally serves as a basis for an investment decision. As already mentioned in the previous section, the result is not the investment decision as such, but must be interpreted on the premises of the model, and it must be verified whether the premises correspond to the practical decision situation. For example, one possible investment calculation method is the net present value or capital value method. We will take a closer look at it in Chap. 3. It is also a

1.3 Aim and Definition of Investment Calculation

9

mathematical procedure for which the economic data of the investment object under consideration, the so-called calculation elements, have already been determined in some way. In the sense of this model, a positive net present value indicates an advantageous investment. However, this method only considers the quantitative aspects. Thus, despite a positive net present value, an investor could reject a corresponding investment decision because he has other non-monetary target criteria in his specific case, e.g. prestige. In this book, only quantitative methods of investment calculation are presented, including those methods which assign a subjective numerical value to a utility value of an investment object. This is the case, for example, with the utility value analysis. Here, for example the shapeliness of a handbag can be assigned a subjective integer value from a scale of, e.g. 1–5, whereby the value 5 means a handbag that is perceived as shapely by the investor. An exact systematisation is given in Sect. 1.5. With few exceptions, these quantitative methods are based on a company’s performance data. The measurement of success in companies can be done by different concepts. For example, the profit of a company or a project can be calculated from the difference between sales and costs. However, using profit in a cash flow-oriented economy is problematic, • As these figures can be imprecisely determined in external accounting due to disclosure or valuation options when determining these figures. • Since actual payments do not necessarily have to be based on these figures in full in all cases and. • Since the dates of payment are not always recorded exactly. Therefore, as in the predominant modern literature on the subject of investment calculation, this book assumes a cash flow-based concept for the dynamic investment calculation methods. Only by means of these calculation variables is the time of payment of the entrepreneurial activity precisely recorded. This is important because, due to the time preference of an economically active individual, an amount of money today has a higher value than the same amount of money in the future. Moreover, only this variable records the amount that is actually based on financial instruments that are not distorted by imputed elements such as depreciation. Thus, for this book,—in congruence with the definition commonly found in the relevant literature—we define an investment as: An investment is a cash flow that starts with a disbursement, i.e. a pay-out. By this definition, we understand an investment object as a multi-year activity that leads to payments in each year. Due to the complexity of such a series of payments, it is not possible to make a direct comparison of investment activities which certainly may be useful, as the decision-making problem is too complex. To reduce the complexity, we then use an investment calculation procedure as an academic model, which allows a structured preparation of decision-making. Only in Chap. 2 we can fall back on a simpler definition than one based on cash flow, since the static procedures presented there do not or only incompletely take an investor’s time preference into account.

10

1 Introduction to Investment Evaluation object A

object B

A

A NEk

34.5 0

NEk 10 1

NEk

NEk

11

11 2

NEk NEk 34.5

12 3

4

n

0

NEk 10

11

12 1

2

3

NEk 10 4

n

Fig. 1.6 Complexity of investment objects (Source: Author)

In order to give the reader an idea of the complexity of a practical decision problem, the following example from Fig. 1.6 may serve as a guide. It concerns two investment objects (object A, object B), both have a useful life of 4 years and both have an initial net investment, also referred to as acquisition payment, (A) of 34.5 million euros. The net payments, also referred to as net earnings, (NEk, million euros) of the objects are different in amount and time distribution. Which of the two investment opportunities is more attractive is difficult to judge because of the multiyear nature of the investment objects. Therefore, investment calculation procedures are necessary to reduce the complexity of the decision-making problem and to reach the decision-making stage. How the problem in Fig. 1.6 can be brought to a decision is discussed in Sect. 3.4. The aim of the investment calculation is to provide a recommendation for various possible questions. Not all possible investment calculation methods are suitable for these questions. The relevant questions are the following: • Determination of the absolute advantageousness of individual investments. The calculation techniques of statics and dynamics from Chaps. 2 and 3 of this book are suitable for this question. • Determination of the relative advantageousness of individual investments. Some of the calculation techniques from Chap. 4 are suitable for this question. • Determination of questions regarding the optimum useful life and the optimum replacement time. The calculation techniques from Chap. 5 are suitable for this question. • Determination of the ideal investment and financing programme. Some of the calculation techniques from Chap. 4 are suitable for this question. • Determination of investment decisions under data uncertainty. The calculation techniques from Chap. 6 are suitable for this question. The decision for an investment object after the application of an investment calculation method by an investment calculator can only be made for a defined objective of the investor. In principle, the investor as “homo oeconomicus” maximises benefits. However, which quantitative criteria maximise the benefit of an investor can differ. Similarly, instead of a single goal, there may be a canon of goal with similar or competing goals, not all of which need to be quantitative in nature, some of which may also be qualitative in nature (imponderables), aspects which for various reasons cannot be quantified and which need to be taken into

1.4 Differentiation of Investment Calculation from Other Business Studies

11

account in the investment decision outside of the pure investment calculation. In the case of quantitative one-dimensional goals, we distinguish • Asset concepts • Withdrawal concepts and • Yield concepts Mixed forms are possible. Each of these concepts is operationalised by other investment calculation methods. In the asset concept, either a present value (BW) or a terminal value (EW) surplus of the investment is considered, assuming that in extreme cases there are no consumer withdrawals during the life of the investment object. Suitable dynamic investment calculation methods here would be the net present value method and the horizon value method. The withdrawal concept ensures that at the end of the investment the capital invested and its interest is regained. Surpluses available over and above this are withdrawn in equal periodic amounts during the term of the investment object. The annuity method is a method corresponding to this concept. In the yield concept, importance is attached not to the absolute amount of the present value or terminal value of the surpluses or the withdrawals, but to the permanent return on the capital invested. The application of this criterion leads to a different valuation of investment objects. A dynamic method that supports this concept is the internal rate of return method. All methods mentioned in this paragraph are presented in Chap. 3. Their specific calculation features are presented and evaluated in Chap. 4.

1.4

Differentiation of Investment Calculation from Other Business Studies

The separation of investment calculation as academic business administration from financing as academic business administration is rather limited to the Germanspeaking area in contemporary literature. Both areas are often combined internationally to form “Corporate Finance,” “Managerial Finance” or even “Accounting” together with cost accounting and areas of other business administration disciplines. The more intensively value-based management as an academic model becomes the focus of attention in business administration, the less a classical separation of the individual business management disciplines of internal accounting, which includes investment calculation, finance, cost accounting, controlling and, in part, corporate management, makes sense, since value-based management is concerned with determining and improving the value of the company (enterprise valuation), which is determined using the techniques of investment calculation. However, the simpler methods of investment calculation, which simply implicitly reflect the practical financing of an investment object in the adequate target rate, must not be used here. We will look at the consequences later in Chap. 4.

12

1 Introduction to Investment Evaluation

Aspect

Investment calculation

Financing theory

Cost accounting Controlling

Regularity

No

Yes

Yes

Planning period multi-periodic

Yes

single-periodic, single-periodic multi-periodic

single-periodic, multi-periodic Single object, business

Reference object

Single object, business

Single object, business

Purpose of the accounts

long-term plan

ShortRather short-term Short-term/longterm/long-term plan term plan plan

Elements of calculation

Payments

Payments

Costs and benefits

all

Liquidity

Mostly no

Yes/no

Mostly no

Yes/no

Realism

Rather not

Yes/no

Yes/no

Yes/no

Business

Fig. 1.7 Differences between the individual business management disciplines of internal accounting (Source: Author)

The differences between these individual business management theories of internal accounting are illustrated in Fig. 1.7 by means of a few aspects. However, this presentation is very subjective because of its reduction and is discussed later in the text. The delimitation in Fig. 1.7 is, therefore, very subjective, as the mentioned business management theories each consider some issues, so that an allocation is not always clear. The aspects addressed are also an arbitrary selection. The financing theory may serve as an example for the classification. Liquidity, for example, may or may not be taken into account in finance, depending on the subject under consideration. Liquidity is, therefore, naturally taken into account in liquidity planning. When determining the instalments of an annuity loan, for example, which is also the object of finance, it is only implicitly assumed that liquidity is available to serve the instalments, but this is not checked directly when determining the annuities. In the same way, not only payments are relevant for the calculation elements in investment calculation. There are also other options that are available in the static procedures. In the case of the aspect of regularity, the investment calculation is occasionrelated. If a new investment object is to be acquired, or an existing one is replanned, the system carries out investment calculation. Regular applications are common in the other three business management disciplines, e.g. liquidity planning in finance, annual planning in cost accounting and the control of quarterly figures in controlling. However, event-related planning is also possible in the financing, e.g. planning the repayment instalments for external financing to be taken up. The same applies to other business management disciplines. This is identical to the aspect of the planning period. Investment planning is usually multi-periodic. In the case of static procedures, which we will discuss in

1.4 Differentiation of Investment Calculation from Other Business Studies

Term Pay-out Disbursement In-payments Cash inflow Expenditure

Definition

Revenue

Monetary value of sales of goods and services per period

Costs

Valued consumption of material goods and services in the production process during a period, to the extent necessary to produce and maintain operational readiness

Performance

Tangible goods and services resulting from the production process of an enterprise and valued in cash during a period

Expenses

Expenditure, distributed into periods for profit determination (=each equity reduction that does not represent a repayment of capital)

Income

Revenue, distributed into periods for profit determination (= each equity increase that does not represent a capital contribution)

13

Disposal of liquid funds (cash and cash equivalents) per period Acquisition of liquid funds (cash and cash equivalents) per period Monetary value of purchases of goods and services per period

Fig. 1.8 Calculation elements of the investment calculation methods (Source: Däumler and Grabe 2007, p. 24)

Sect. 1.2, you can also consider one-periodic planning. The other business administration disciplines can be carried out on a single-period basis. Multi-periodic aspects are of course possible and are not uncommon. The reference object can be the individual object in all the business management disciplines mentioned, i.e. an investment object that is evaluated using a dynamic investment calculation method, but also the entire operation that is evaluated in the investment calculation using the simultaneous models of the capital budget. In other business studies, both views are usually possible and common. In the finance theory, for example, the external financing for a single machine can be optimised, as can the external financing of the entire operation. The purpose of investment calculation is rather long-term, even though it can also be short-term when using static investment calculation methods; in other business management disciplines both aspects are possible and common. Finance may serve as an example here again. While liquidity planning is rather short-term, equity planning would be more long-term in nature. These are questions that are treated similarly in other business management disciplines, cost accounting and controlling. When considering the calculation elements, payments are usually used in investment calculation. Other calculation elements are only possible in the static investment calculation procedures. In other business studies, different elements of calculation are used depending on the issue at hand. For example, costs can usually be used in cost accounting, depreciation would be a cost item per period. However, payments are required for the calculation of depreciation allowances. A structuring of the various possible calculation elements is found in Däumler and Grabe (2007) and is shown here in slightly modified form in Fig. 1.8.

14

1 Introduction to Investment Evaluation

The consideration of liquidity is possible or not in the individual business management disciplines mentioned, depending on the issue at hand. For example, it is usually ignored in investment calculations, but can be taken into account when applying simultaneous models of the capital budget, e.g. by drawing up account development plans. The situation is similar in other business management disciplines. The closeness to reality of the models also varies depending on the research question. In principle, these academic models, which are used in investment calculation, for example, in this case, the net present value method, are simplified illustrations of reality. And the more complex the assumptions required, the greater the loss of reality. Since the mathematical models of investment calculation sometimes have to work with many assumptions, the closeness to reality here is rather low. In principle, the less complex the procedures are, and the fewer assumptions required, the closer the obtained results are to reality. Therefore, many results of cost accounting are also more realistic than those of investment calculation.

1.5

Investment Calculation Procedures at a Glance

The many different existing investment calculation methods can be partly combined into groups of methods and can be systematised according to different aspects. In principle, the following can be distinguished here • • • •

Procedures with and without consideration of risk Qualitative and quantitative methods Methods with one-dimensional and multidimensional target functions Procedures for the assessment of individual investments or investment programmes

The possibility of combining individual procedures into groups of procedures depends on various aspects. For example, the group of • • • •

The static investment calculation methods The dynamic investment calculation methods The simultaneous models of the capital budget and The procedures for taking risk into account

The assignment of individual methods to the individual branches of Figs. 1.9 and 1.10 is not always clear, since individual methods can be applied to different problems with slight modifications. All the methods mentioned are dealt with in this book. This is done in Chaps. 2, 3, 4 and 6, where the working methods of the procedures are also explained in detail. Figure 1.9 lists the investment calculation methods which assume the certainty of the calculation elements, Fig. 1.10 contains the methods which assume data uncertainty.

Net present value method Horizon value method Annuity method Internal rate of return method Dynamic amortisation calculation

Cost comparison calculation Profit comparison calculation Profitability calculation Static amortisation calculation

After removing the reinvestment premise and fictitious investment: Net present value method Horizon value method Annuity method Internal rate of return method Dynamic amortisation calculation

Account development planning regardless of liquidity

Investment programme

Utility value analysis

Utility value analysis

Account development planning with regard to liquidity

Linear optimization Dean Modell

Investment programme

Account development planning with regard to liquidity

Individual investment

Multi-dimensional target function

Utility value analysis

Individual investment

Utility value analysis

Investment programme

One-dimensional target function

Fig. 1.9 Overview of investment calculation procedures on the assumption of data certainty (Source: Author)

Dynamic methods

Static methods

Individual investment

One-dimensional target function

Quantitative target criteria

Assumption of data certainty

Qualitative target criteria

Utility value analysis

Individual investment

Utility value analysis

Investment programme

Multi-dimensional target function

1.5 Investment Calculation Procedures at a Glance 15

Dominance rule Maximax rule Minimax rule Hurwicz rule Laplace rule Savage-Niehans rule

Correction procedure Critical-value calculation Triple Calculation Target value change calculation

Investment programme

Risk analysis

Individual investment

Sequential investment decisions Risk analysis Portfolio selection

Investment programme

Multi-dimensional target function

No standard procedures

Individual investment

No standard procedures

Investment programme

One-dimensional target function

Fig. 1.10 Overview of investment calculation procedures on the assumption of data uncertainty (Source: Author)

Correction procedure Critical-value calculation Triple Calculation Target value change calculation

Individual investment

One-dimensional target function

Quantitative target criteria

Assumption of data uncertainty

Individual investment

No standard procedures

Investment programme

Multi-dimensional target function

No standard procedures

Qualitative target criteria

16 1 Introduction to Investment Evaluation

1.6 Historical Development of Investment Calculation

1.6

17

Historical Development of Investment Calculation

At this point, no historical review of the investment calculation should be carried out. Basically, investment calculation is a rather static field of knowledge in which no decisive research findings have been achieved in the past 20–40 years. This is due to the fact that the mathematical models which form the basis of investment calculation methods have theoretically been mature for a long time. The problem lies primarily in obtaining reliable data. Since an economically oriented investment decision must always use forecast values of future economic success figures, forecast models must be used to estimate these values. Trivial models or techniques of academic decision theory are used for this purpose. However, since the future is still not exactly predictable in most cases, this is the real weakness of the realism of the calculation results. What has changed in recent decades is primarily the vigour and the operational know-how in the application of information technology. Thus, purely from the point of view of the consideration on the timeline, the scientific development and the application of the relatively simple procedures of the static investment calculation methods can be justified by the lack of operational computing capacity. In a time without the operational spread of computers and pocket calculators, the use of the static investment calculation methods was a useful tool to support operational decisions. With the spread of pocket calculators, the use of dynamic investment calculation methods, accompanied by a higher computational effort, was also possible in companies with a reasonable amount of work. The simultaneous models of the capital budget, which can represent decision situations much more realistically than possible in the other models, did not find their broader operational application until the introduction of personal computers in companies in the 1980s. In addition to the consideration of the timeline, the complexity of the decision problem naturally also has an influence on the used investment calculation method. For the evaluation of an acquisition with an initial net investment (acquisition payment) of 50 euros a static investment calculation method is certainly justified because of the presumably small importance of this investment, if a mathematical method is used at all. The third relevant aspect for the use of investment calculation methods, apart from the existing IT and the importance of the investment object, is probably the size of the company. While in large corporations the know-how about complex investment calculation methods is certainly available, this is not necessarily a matter of course for a sole proprietorship and is unfortunately often not supplemented there by hiring consulting services. As a result, significant investments are often made with too trivial investment calculation methods or without them at all due to a lack of know-how.

18

1.7

1 Introduction to Investment Evaluation

The Organisational Structure for Investment Analysis

The organisational structure of the investment calculation units in a company cannot be systematised in an ideal-typical way. This is strongly dependent on the size of the company. The industry also has an influence. So companies that invest a particularly large amount of capital in a short period of time, e.g. investment banks, are more intensively equipped with investment-calculating departments, even at the upper hierarchical levels of a company’s organisational structure. In very large companies, a separate department is responsible for investment planning. The smaller the companies become, the more likely it is that the organisational structure will initially be assigned to a controlling department, then to an accounting department and then to a general department. In very small companies, this is then the direct responsibility of the management. Implementation in the organisational plan is also strongly dependent on whether the enterprise has a line organisation, a matrix or a project organisation. Mixed forms are of course possible. Possible alternative systematisation of an investment calculation department are shown in Fig. 1.11 and commented on here. In Fig. 1.11 a line organisation for a company is assumed and a part of it is described. Separate considerations for other forms of organisation such as the matrix or project organisation form are not made here. The 12 different concretisations of the investment calculation unit (Inv x) formulated in Fig. 1.11 are to be understood alternatively; in larger companies an organisational structure is certainly anchored at different points. In very small companies, investment calculation, where it exists, is usually only carried out in a very simple form (Inv 1). From the point of view of this specialist

Management Inv 1

Staff Inv 2

Investment planning

Specialist department

Inv 4

Inv 5

Inv 6

Inv 7

Investment planning

Investment planning

Investment planning

Investment planning

Investment planning

Inv 8

Inv 9

Inv 10

Inv 11

Inv 12

General department Inv 3

Accounng

Controlling

……………………………………….

Fig. 1.11 Possible anchoring of investment calculation units in the organisational plan (Source: Author)

1.7 The Organisational Structure for Investment Analysis

19

area, managers should be advised to consult external experts for investment calculations in the case of more significant investments. However, the same organisational structure can also be chosen in very large units with extensive investment activities for very high amounts in absolute terms, e.g. in investment banks. Since the activity of buying and selling companies and shareholdings is very important there and accounts for a very high proportion of turnover, profit and capital commitment, this is often the responsibility of a member of the management, depending on how the management may be organised by the legal form of the company. In such an organisational installation, there is of course always a further operative area in the form of a department, for example, which is responsible for the work of investment calculation and the necessary data collection. This could be organised by a separate investment department (Inv 6) or a department with various departments (Inv 6 with Inv 11). As the size of the company increases compared to a very small company, the investment calculation could then initially be carried out by a specialist department (Inv 7), e.g. production, a general department (Inv 3) or an accounting department (Inv 4) or a controlling department (Inv 5), in each case by the managing body. With the exception of investment calculation in a controlling department, it is always advisable to consult a qualified external consultant for larger capital expenditures from an academic perspective. If, within the sequence of steps described in this paragraph, the heads of the departments do not carry out the investment calculation, but delegate it to a subordinate body in these departments, the job specifications Inv 8, Inv 9, Inv 10 and Inv 12 are shown in Fig. 1.11. If a full-time position is exclusively concerned with questions of investment calculation and the job holder has received appropriate training and further education in this field, this form is preferable to the variants of implementation in the respective heads of departments due to the higher priority and the exclusivity of this job. Of course, the knowledge about investment calculation in a company is manifested much better by a separate department for investment planning in the organisational structure of the presented line organisation. This form of organisation certainly best represents knowledge and competence in the preparation of investment decisions by carrying out investment calculations and by data collection. However, this form of organisation requires a certain minimum size of a company and corresponding activities on a larger scale, so that this form of organisation is probably unfortunately not useful for about 80–90% of all the companies. At this point, it should be emphasised that such an organisational structure can only be successful if suitable internal corporate communications are in place. As soon as the investment planning unit in the organisational plan sees itself as an independent group without exchanging knowledge with other units, it will not be possible to collect enough realistic data. Especially in production companies, the departments responsible for investment calculation depend on information about technical necessities and difficulties from producing departments, since this knowledge is rarely strongly represented in investment planning departments.

20

1 Introduction to Investment Evaluation

More important than the organisational structure of investment planning is a structured and transparent process organisation of investment planning, which is described in the following section.

1.8

Process Organisation of Investment Calculation

The actual decision-making process for an investment can be divided into several phases. Only a small part of these phases of the decision-making process, the actual investment calculation by means of an investment calculation method, is the subject of this book. The individual phases are shown in Fig. 1.12. The First Phase Is the Excitation Phase Here it is assumed that a company has clear corporate goals derived from its corporate vision, corporate mission and corporate strategy. The strengths, weaknesses, opportunities and risks are derived from these corporate goals. On the one hand, the corporate environment is thus observed with regard to the emergence of new information. Such new information can, for example, concern technical progress. The emergence of new processes or more cost-effective machines may make replacement investment necessary to maintain competitive strength. We will discuss this in detail in Chap. 5. Similarly, information on the behaviour of competitors may make it necessary to take action accompanied by investment. This information can be obtained by automated monitoring of new information on the Internet, reading industry services and print media, attending training courses or industry meetings, or by employees approaching the company. On the other hand, information coming from inside the company is included regarding the necessity of expansion or rationalisation or replacement of investment objects. If these suggestions are in line with the goals of the company and a demand is possible, the investment calculator enters the next phase, the planning phase. The Second Phase Is the Planning Phase The planning phase itself is divided into Phase diagram of the investment calculation

Phase of decision-making

Excitation phase

Collection of information

Planning phase

Formulation of objectives and alternatives Data collection Investment calculation Coordination with other business areas

Phase of implementation

Decision phase

Implementation phase

Most favourable alternative

Monitoring of project implementation

Fig. 1.12 Phases of the investment calculation (Source: Author)

Control phase

Target-actual control Deviation analysis Correction decisions

1.8 Process Organisation of Investment Calculation

• • • • •

21

The definition of the target criteria. The collection of the possible alternatives. The focus on the alternatives to be considered more closely. The calculation of the results of the investment calculation procedures and. The coordination with other divisions of the company.

First of all, a concrete definition of objectives is made for the planning problem. Thus it is to be determined whether there are non-monetary criteria in addition to monetary ones. This can be taken into account either by later determining the investment calculation method or by formulating knock-out criteria. Non-monetary criteria could, for example, be a minimum production capacity in terms of quantity of a production plant to be purchased in units per day of a defined intermediate product. A non-monetary criterion could also be the exclusion of a particular supplier while looking for an investment object. Once the criteria have been defined, the most labour-intensive part of investment planning, data collection, takes place. First, all known alternatives are determined. From these alternatives, those that do not meet the knock-out criteria are then eliminated. For the remaining alternatives, you may then have to make a detailed analysis of the planned data, which is quite labour-intensive. The problem of data collection is discussed in more detail in the following section. After the data collection for the possible alternatives, the actual investment calculation is carried out, which is dealt with in all the following main chapters of this book. It is important to ensure that defined target criteria and the selected investment calculation procedure do not contradict each other. With the result of the investment calculation, the planning phase is concluded and moves on to the decision phase. In coordination with other divisions of the company, finance and capacity planning, which are not considered in the investment calculation itself, must be carried out. In financial planning, the origin of funds and liquidity must be planned. Borrowed capital can be used to pay for the acquisition, for example, in the form of a loan, if necessary with a collateral security. Mixed financing from several sources and leasing agreements are also possible, but, except in the simultaneous models of the capital budget, never the subject of the actual investment calculation. This also applies to liquidity. The money required for the investment object must be planned in terms of origin and use, as must the future surpluses generated by the investment, the so-called disinvestments. This is also not the subject of investment calculation, except in the simultaneous models of the capital budget. In capacity planning, the entire supply chain may have to be adjusted for a new investment object. For example, the procurement, funding and, if necessary, storage of intermediate products must be replanned, if the investment object is a production line. Storage capacities, staff capacities and bottlenecks in the production process must be analysed and the sales of products with the necessary organisation and logistics must be determined. This capacity planning is also not part of the actual investment calculation.

22

1 Introduction to Investment Evaluation

The Third Phase Is the Decision Phase This should be the shortest phase in terms of time and organisation. In this phase, the investor, which in the case of larger investments should always be the company management, including the investment calculation unit and the unit operating the investment, must make the investment decision, because the investment calculation cannot make the investment decision itself due to its character as an academic model and due to the often poor quality of the forecast of the planning data. The investment analysis itself has only the task of structuring the decision problem to the maximum possible extent and reducing the complexity of the decision problem by determining an investment calculation key figure. The Fourth Phase Is the Implementation Phase Here, the investment calculation unit must ensure that the plan is implemented identically in reality. Any negative deviation from the plan data, for example, higher set-up disbursements for the set-up of an asset, can make the investment opportunity that was identified as economically viable change to economically unviable due to other plan data. This task is a cooperative task in which the investment-calculating agency, in cooperation with the executing agency, looks for inefficiencies in order to improve them and thus become even more successful. It is by no means a mere control task, but rather a coaching. The Fifth Phase Is the Control Phase However, the control has already regularly accompanied the other phases and completes the process. Here, recalculations and deviation analyses are carried out. If necessary, corrective or adjustment measures are proposed. This activity is of course also carried out in cooperation with the department carrying out the investment.

1.9

The Problem of Collecting Data for Investment Calculation

Probably the main problem of a successful investment calculation lies in the collection of data for the investment calculation, which, however, does not allow a more intensive theoretical analysis specific to the theoretical field of investment calculation. Since investment calculation is usually future-oriented, data on future events must be obtained. This is the case with the current state of the art in forecasting, and this has changed only marginally over the past few decades, but only with incomplete realism. Since the future is not predictable, data for investment objects cannot be predicted with certainty in most cases. There are, of course, gradual differences in the level of certainty of data forecasting. For example, the calculation elements, what the relevant calculation data for an investment calculation is usually called, e.g. for a savings bond with a defined term and fixed interest rate, are usually known with certainty. However, the adequate target rate, i.e. the calculation interest rate, which is also necessary for discounting, is a rather subjective variable even in such a situation, as it should be measured against opportunities. In

1.9 The Problem of Collecting Data for Investment Calculation

23

Forecasting methods

Qualitative

Surveys

Quantitative

Creativity techniques

Univariate methods

Multivariate methods

Fig. 1.13 Overview of possible forecasting methods (Source: Kruschwitz, Lutz, p. 18)

contrast, the data forecast for the establishment of a new company in a young industry, e.g. the operation of a biogas plant, is likely to be much more uncertain. In general, the old wisdom of the builders of academic empirical models applies here as well: “Garbage in, garbage out,” i.e. the realisation that even the most beautiful academic model produces waste as a result when waste is processed as data. With the exception of this section, the possibilities of data collection for investment calculation will not be the subject of this book and, as with most of the existing literature on investment calculation, will be regarded as having already been done when the investment calculation begins, whose actual calculation models are presented intensively in this text. Of course, this leads to a loss of realism of the calculation results of the investment calculation methods, if the calculation elements cannot be predicted with certainty. On the other hand, these prediction models are not specific scientific findings of investment theory, but are researched and taught in academic decision theory, stochastics or econometrics. Moreover, this is the fundamental forecasting problem of future-oriented economics. Thus, although the results of the techniques used will not occur with certainty, they are a useful way of structuring a planning problem. Consequently, the reader may deal with these models elsewhere, but here only a small structuring of possible forecast procedures for generating the necessary calculation elements will be presented, which are very closely oriented to the remarks of Kruschwitz (2005) on this topic. These are presented in Fig. 1.13 and then commented on in the text. In principle, it is possible to distinguish • Qualitative and • Quantitative forecasting methods Qualitative methods are based on subjective assessments of individuals, which may also be aggregated and are also based on experience. These qualitative methods can be divided into • Surveys and • Creativity techniques

24

1 Introduction to Investment Evaluation

Surveys can be structured in many different ways, basically all stakeholders can be interviewed, i.e. suppliers, employees, customers, experts, etc. Other structures are also possible. Creativity techniques are methods of structured reflection, the aim of which is to produce ideas and results on a discussed topic. These include brainstorming, brainwriting, morphological methods, the Collective Notebook method, the Delphi method, synectics, pro and contra, black box, lateral thinking and many other methods. They will not be dealt with in detail here but can be found in other textbooks on decision theory or general business administration. The results of qualitative methods can also be represented by quantitative values. For example, in a customer survey in which customers are asked to rate the attractiveness of the design of the body of a passenger car with values from 1 to 5, where 1 stands for a particularly unattractive design and 5 for a particularly attractive design, this qualitative method can lead to the result 4.2 on this scale, i.e. a quantitative value can be calculated as the mean value of all survey results. The result of the Delphi method as a representative of the creativity techniques and thus of the qualitative techniques on the question of the annual expected increase in electricity prices for the next 10 years can, for example, be 15% and thus also determine a quantitative value. The suitability of these procedures should not be criticised on the basis that the results were determined subjectively. The quality of the calculation elements thus determined depends very much on the reliability of the data sources. For example, the self-assessment of an employee of a company who has submitted an investment application in his company to get a new notebook may be much less positive about the utility value of the old device than it is in reality, so that he will actually receive a new device. On the other hand, the estimation of an experienced master craftsman in production about the set-up times when attaching new tools to machines can be a very qualified estimate based on extensive experience, which is the best source of information available. The quantitative methods are divided into • Univariate and • Multivariate methods With univariate methods (trend methods, exponential smoothing, autoregressive methods), an attempt is made to find a rule for future development from the historical development of a time series of a variable under consideration using the various models. Exogenous factors (structural breaks in the temporal development of the explanatory variables) are ignored. In multivariate methods, (single regression, multiple regression), the development of endogenous variables from several exogenous variables is explained. Statistical techniques can be used to check the significance of the exogenous variables used and to verify the significance over time. How well a future development can be predicted from historical data can only be determined after a few more years in individual cases by comparing the forecast data

1.9 The Problem of Collecting Data for Investment Calculation A B year premium petrol price (cent/ litre) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31

1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019

60.7 67.6 71.1 71.3 79.6 79.3 82.7 85.2 81.2 86.3 101.6 102.3 104.6 109.2 113.2 121.7 128 133.7 138.9 127.3 140.5 152.2 159.8 154.9 149.8 136.9 128.1 134.7 142.8 140.5

C regression 19902019

D regression 19902000

E deviation absolute 19902019

F deviation absolute 19902000

66.6 69.8 73.0 76.2 79.3 82.5 85.7 88.9 92.1 95.3 98.5 101.7 104.9 108.1 111.3 114.5 117.6 120.8 124.0 127.2 130.4 133.6 136.8 140.0 143.2 146.4 149.6 152.8 155.9 159.1

63.3 66.4 69.5 72.6 75.7 78.8 81.9 85.0 88.1 91.2 94.3 97.4 100.5 103.5 106.6 109.7 112.8 115.9 119.0 122.1 125.2 128.3 131.4 134.5 137.6 140.7 143.8 146.9 150.0 153.1

-5.9 -2.2 -1.9 -4.9 0.3 -3.2 -3.0 -3.7 -10.9 -9.0 3.1 0.6 -0.3 1.1 1.9 7.2 10.4 12.9 14.9 0.1 10.1 18.6 23.0 14.9 6.6 -9.5 -21.5 -18.1 -13.1 -18.6

-2.6 1.2 1.6 -1.3 3.9 0.5 0.8 0.2 -6.9 -4.9 7.3 4.9 4.2 5.7 6.6 12.0 15.2 17.8 19.9 5.2 15.3 23.9 28.4 20.4 12.2 -3.8 -15.7 -12.2 -7.2 -12.6

G deviation percent 19902019

-9.7 -3.2 -2.6 -6.8 0.3 -4.1 -3.7 -4.4 -13.4 -10.4 3.1 0.6 -0.3 1.0 1.7 6.0 8.1 9.6 10.7 0.1 7.2 12.2 14.4 9.6 4.4 -6.9 -16.8 -13.4 -9.2 -13.3

H I deviation trend percent 19902000

-4.3 1.8 2.3 -1.8 4.9 0.7 1.0 0.3 -8.5 -5.6 7.2 4.8 4.0 5.2 5.8 9.8 11.9 13.3 14.3 4.1 10.9 15.7 17.8 13.2 8.1 -2.8 -12.2 -9.0 -5.0 -8.9

60.7 67.6 71.1 71.3 79.6 79.3 82.7 85.2 81.2 86.3 101.6 105.7 109.8 113.9 118.0 122.1 126.1 130.2 134.3 138.4 142.5 146.6 150.7 154.8 158.9 163.0 167.0 171.1 175.2 179.3

25 J K deviation updating percent 2000 trend

3.3 5.0 4.3 4.2 0.3 -1.5 -2.6 -3.3 8.7 1.4 -3.7 -5.7 -0.1 6.0 19.0 30.4 27.0 22.7 27.6

101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6 101.6

L deviation percent updating

-0.7 -2.9 -7.0 -10.2 -16.5 -20.6 -24.0 -26.9 -20.2 -27.7 -33.2 -36.4 -34.4 -32.2 -25.8 -20.7 -24.6 -28.9 -27.7

Fig. 1.14 Historical development of premium petrol prices in euro cents/litre and forecasts (Source: ADAC Benzinpreisstatistik, own calculation)

with the real values that have occurred. It is not clear which forecasting method is suitable for the forecast in each individual case. The suitability is to be documented at this point by a small comparison in the forecast according to various methods and the then occurring development of the premium petrol prices measured against the annual average prices in euro cents per litre in Germany. The initial data and forecasts are first shown in Fig. 1.14 and graphically represented in Fig. 1.15. Columns A and B in Fig. 1.14 show the annual average prices for premium petrol in euro at German petrol stations according to ADAC data for the years 1990–2019 (ADAC 2019); data before 2001 were converted into euro cents by the author. In Fig. 1.15, the values are shown as a solid black line. These are, therefore, values that have actually occurred in the past, at least in this type of recording. These data are processed with different prognosis techniques in order to show how individual forecasting techniques work and how forecasting and real occurring data can diverge for individual techniques in this very subjective example.

26

1 Introduction to Investment Evaluation

200 180 160 140 120 100 80 60 40 20 0

premium petrol prices (cent/litre)

regression 1990 ‐ 2019

trend

updating 2000

regression 1990 ‐ 2000

Fig. 1.15 Graphic representation of the historical premium petrol prices in euro cents/litre and forecasts (source: own representation with data from Fig. 1.14)

First, a multivariate method, known as linear regression is applied. In column C in Fig. 1.14 the full-time series has been subjected to linear regression, according to the formula Premium petrol prices ðcents=litreÞ ¼ a  year þ b:

ð1:1Þ

a indicates the slope of the linear function and b the intersection with the ordinate. The results were determined with Microsoft Excel using “slope” and “axis intercept” functions of the spreadsheet. The results are shown in Fig. 1.14. The deviation of the regression from reality in cents was added in column E and the corresponding percentage deviation in column G. The curve is shown in Fig. 1.15 as an irregular dotted line. This regression is not suitable for a forecast that would allow the predicted values to be compared with the reality that has already occurred, since no figures other than those for the period up to 2019 are available. Therefore, in a second step, the regression was carried out using only data from 1990 to 2000, and the values for the years 2001–2019 were forecast using the result of the estimation equation. Figure 1.14 shows the results in column D. The deviation of the regression from reality in cents was added in column F and the corresponding deviation in percentage in column H. The curve is shown in Fig. 1.15 as an evenly dashed short line. As an alternative to multivariate methods, univariate quantitative methods can also be used. One possibility here is trend extrapolation. Two different methods have been applied here, the results of which are documented in columns I and K in Fig. 1.14. To enable a comparison of the forecast result with historical reality, the

1.9 The Problem of Collecting Data for Investment Calculation

27

years 1990 and 2000 were used as the basis for the trend determination according to the following formula for the procedure documented in column I: b ¼

Δ price ðcent=litreÞ Δ time ðyearsÞ

ð1:2Þ

The values from 2001 onwards in column I are, therefore, forecasts according to this trend assessment. The curve is shown in Fig. 1.15 as an evenly dashed long line. In column K a naive forecast was made and the values for 2000 were left unchanged for the coming years. This curve is shown as a dotted line in Fig. 1.15. In addition to quantitative forecasting methods, qualitative methods can also be applied, e.g. expert surveys. Two statements on the development of petrol prices may be mentioned here as examples: In the Berliner Morgenpost of March 4, 2000, the following statement can be read: “According to experts from the Hamburg Institute for Economic Research (HWWA), petrol prices will drop again by about 20 Pfennig in the coming months. Klaus Matthies, the raw materials expert of the HWWA, told the Saarland Broadcasting Corporation, that he expects the price for unleaded premium petrol of about 1.80 DM per litre (equivalent to about 0.92 euro, the author).” In the Wirtschaftswoche of June 22, 2000, the following expert statement can be found: “‘If this continues, we will have reached a price per litre of 2.50 marks (equivalent to approximately 1.28 euro, the author), by the end of 2002’, predicts Ferdinand Dudenhöffer, an expert in automotive economics at the Gelsenkirchen University of Applied Sciences.” A glance at Figs. 1.14 and 1.15 show relatively clearly that the suitability of these forecasting methods was not always ideal in this example. In the case of the quantitative methods, the percentage deviations of the forecast values from the actually occurring values between 2001 and 2019 can be seen in Fig. 1.14. For this data set, ex post the regression 1990–2019 and the trend forecast according to the method chosen here has proven to be favourable. Here, deviations between 0 and 13 for the regression 1990–2019 and deviations between 0 and 17% for the trend forecast of the forecast values from reality. For the naive forecast, the worst result for 2012 was a deviation of 36%. The expert forecasts showed deviations of about 10% and about 20%. However, this evaluation is biased, because the statements of the experts are certainly taken out of context and are compared with annual average data of the corresponding years, which were not the focus of their forecasts. As a conclusion of this empirical observation, the reader should now be aware that in general cases it is very difficult to make a good prediction of the data required for investment calculations. For the general case, it is also not possible to determine ex ante which is the appropriate forecasting method. If this were certain, the future would no longer be uncertain, but predictable. The determination of the adequate target rate, i.e. the calculation interest rate is of particular importance when forecasting the calculation elements. The calculation interest rate is the investor’s subjective minimum interest requirement, that is, the

28

1 Introduction to Investment Evaluation

interest rate that the investor wants to achieve with at least one investment or investment programme so that he considers the investment to be worthwhile. There are various procedures for setting this interest rate, such as setting it by reference to the debit interest rate if only equity is used. Alternatively, if only borrowed capital is used, it is possible to determine the rate of interest by reference to the borrowing interest rate. In the case of mixed financing, the calculation interest rate is weighted with these interest rates and capital shares. Surcharges and discounts on these cost estimates are possible. In principle, it makes sense to determine the interest rate according to the opportunity cost principle. In this case, the total benefit of the best alternative not chosen is determined as a measure, i.e. the return on the best investment just discarded. Determinations according to the Capital Asset Pricing Model (CAPM), in which the risk attitude of the investor is also taken into account, are also possible and are common and widespread when determining the value of a company within the framework of a company valuation according to the shareholder value approach. Basically, the collection of the calculation elements for the calculation of the value of an investment object by means of an investment calculation procedure is thus a major problem, from which this book is abstracted. Poor planning data makes the results of investment analysis only very limited utility for the investment decision. Nevertheless, structured and reflected data collection is the best way to create a transparent basis for decision-making.

1.10

Necessity and Limits of Investment Calculation

The remarks so far on investment calculation in this first chapter of the book may not have seemed like a praise of investment calculation to some readers. In an academic textbook, it is also the author’s task to adopt a critical basic attitude and point out the difficulties and risks involved in the topic dealt with. Cheering on the covered subject would certainly not be appropriate. In general, the investment calculation has two problems. One problem is data collection. Since the data used is a forecast of future values, there is a lack of clarity of whether there is a discrepancy between current planning data and actual values that will occur in the future. On the other hand, is the fact that the investment calculation methods are academic mathematical models and as such, are simplified representations of reality and cannot always accurately reflect complex practical decision-making situations. Nevertheless, carrying out an investment calculation is the best available method for preparing a decision. One advantage is the structured planning of an investment object in the planning phase. By collecting planning data, the planning process is processed in a structured way. This alone increases the realism of the planning process and the knowledge gained. On the other hand, the procedure on the way to the investment decision, the investment calculation, is documented and made available to all those involved. This

1.11

Summary

29

is the usual scientific theoretical procedure in the social sciences, “critical rationalism” according to Sir Karl Popper (1994). In critical rationalism, a researcher has to disclose his assumptions and approach in determining statements, to make them intersubjectively comprehensible in order to call the results scientific. This is exactly what is done by applying an investment calculation method and is the advantage overacting on instinct. The great advantages of using investment calculation methods are the reduction of the complexity of a decision situation and the transparency of the planning path. Only in this way can complex operational decision-making situations be brought to rational decision-making maturity, the consequence is that, due to the complexity, a management body would not be able to understand the situation of the decision without the application of a model. If the investment calculator then pays attention to careful and realistic data collection, applies the investment calculation method suitable for the decision situation and interprets the results taking into account the assumptions of the used investment calculation method, then the best possible and available basis for decision-making is created. There is, therefore, no alternative to investment calculation for the meaningful solution of operational planning problems in the area of longer-term or more extensive use of capital.

1.11

Summary

In this section, the reader was introduced to the subject and problems of investment calculation. After reading this chapter the reader should now be able • To evaluate the relevance of investment calculation from the point of view of the national economy, the companies and the private households. • To interpret the goal of an investment calculation as well as different possible questions and asset concepts which can be among the goals of an investment calculator. • To differentiate the investment calculation from other business management disciplines of internal accounting. • To name the different groups of investment calculation methods and assign the individual methods to these groups. • To reflect the importance of the various investment calculation methods, taking into account their chronological development and the computing capacities available at the time. • To discuss ideal-typical organisational systematisation for the allocation of investment calculation instances depending on the size of the company and the capital commitment in companies. • To propose the ideal-typical process organisation of the investment calculation. • To interpret the problems of data collection and the consequences for the closeness to reality of the obtained results and.

30

1 Introduction to Investment Evaluation

• To know consciously the purpose and the operational benefit of the investment calculation. The reader should now be able to define what investment calculation is, what it means from different perspectives, what its goal is, what different methods there are and what its limitations are. It should also be clear to the reader that, despite the existing shortcomings with regard to the complete transferability of the calculation results to reality, the investment calculation is the best available procedure for the structured evaluation of investment objects.

References ADAC. (Hrsg.). (2019). Benzinpreisstatistik. München. Retrieved from July 5, 2020, from https:// www.adac.de/verkehr/tanken-kraftstoff-antrieb/deutschland/kraftstoffpreisentwicklung/. Däumler, K-D., & Grabe, J. (2007). Grundlagen der Investitions- und Wirtschaftlichkeitsrechnung (12. Aufl.). Herne/Berlin: nwb Verlag Neue Wirtschafts-Briefe. Institut der Deutschen Wirtschaft. (2019). Deutschland in Zahlen 2019. Köln. Retrieved from July 5, 2020, from https://www.deutschlandinzahlen.de/no_cache/tab/welt/oeffentliche-haushalte/ einnahmen-und-ausgaben-des-staates/investitionsquote?tx_diztables_pi1%5BsortBy% 5D¼col_13&tx_diztables_pi1%5BsortDirection%5D¼asc&tx_diztables_pi1%5Bstart% 5D¼0; Retrieved from July 5, 2020, from https://www.deutschlandinzahlen.de/tab/deutschland/ volkswirtschaft/bruttoinlandsprodukt/bruttoinlandsprodukt-nominal; https://www. deutschlandinzahlen.de/tab/deutschland/volkswirtschaft/verwendung/ bruttoanlageinvestitionen-nach-guetergruppen; Retrieved from July 5, 2020, from https://www. deutschlandinzahlen.de/tab/deutschland/volkswirtschaft/verwendung/konsumausgabenprivater-haushalte. Kruschwitz, L. (2005). Investitionsrechnung (10. Aufl.). München: Oldenburg Verlag. Popper, K. R. (1994). Logik der Forschung (10. Aufl.). Tübingen: Mohr Verlag. Statistisches Bundesamt. (Hrsg.). (2019). Statistisches Jahrbuch 2019. Wiesbaden. Retrieved from July 5, 2020, from https://www.destatis.de/DE/Service/Bibliothek/_publikationenfachserienliste-4.html; Retrieved from July 5, 2020, from https://www.statistischebibliothek. de/mir/receive/DESerie_mods_00000442; Retrieved from July 5, 2020, from https://www. deutsch landinzahlen.de/tab/deutschland /b ranchen-u nternehmen/un ternehmen/ erfolgskennziffern-deutscher-unternehmen; Retrieved from July 5, 2020, from https://de. statista.com/statistik/daten/studie/199158/umfrage/jaehrliche-entwicklung-des-dax-seit-1987/.

2

Static Investment Calculation Methods

2.1

Objectives

In this chapter, the reader will deal with static investment calculation methods. The aim is to make the reader aware of the criticism that can be levelled at the static methods and of the risks involved in their application in terms of transferring the results into practice as an investment decision. The general assumptions of the static investment calculation methods are presented, as well as the four most well-known static investment calculation methods, their criteria, their formulas, their risks in detail and their application to practical problems. Also, their quick applicability due to simple data collection and calculation techniques is presented. In detail, the following sub-goals shall be achieved: • The working method of statics in general and the assumptions of this investment calculation method group should be learned. • The criticism of the static procedures should be worked out extensively. • The risks of transferring the calculation results of the static investment calculation procedures as an investment decision into practice should be disclosed and documented very clearly. • The reader should be able to assemble the static formulas from a set of relevant calculation elements according to the problem and the relevant static investment calculation method. • The cost comparison calculation as a static investment calculation method should be known in detail, defined and criticised as a single method, the possible calculation formulas should be presented and applied to practical cases. • The profit comparison calculation as a static investment calculation method should be known in detail, defined and criticised as a single method, the possible

Supplementary Information The online version of this chapter (https://doi.org/10.1007/978-3030-62440-8_2) contains supplementary material, which is available to authorized users. # The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. Poggensee, J. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-62440-8_2

31

32

2

Static Investment Calculation Methods

calculation formulas should be presented and applied to practical cases, the profitability calculation as a static investment calculation method should be known in detail, defined and criticised as a single method, the possible calculation formulas should be presented and applied to practical cases. • The profitability calculation as a static investment calculation method should be known in detail, defined and criticised as a single method, the possible calculation formulas should be presented and applied to practical cases. • The static amortisation calculation as a static investment calculation method should be known in detail, defined and criticised as a single method, the possible calculation formulas should be presented and applied to practical cases and. • All procedures should be applied to a practical problem in a case study. After reading the chapter, the reader should be able to define what static investment calculation methods are, what value they have for practical application, what criticisms and dangers there are in transferring the calculation results into practice and how the methods work in detail. The reader should be able to set up the corresponding formulas independently after reading the chapter. In order to be able to achieve these goals, it is necessary to follow the offered exercise calculations independently doing the mental calculation, with the pocket calculator or with the spreadsheet. Enjoy your work!

2.2

Fundamental Aspects of Static Investment Calculation Methods

Static investment calculation methods should no longer be used today for significant investments. They are too trivial for that. As already described in Sect. 1.6, the methods were eligible in times when electronic calculators were not available to businesses on a large scale, i.e. until a good 60 years ago, when one considers pocket calculators to be electronic calculators. The presentation of these procedures in a modern textbook is, therefore, surprising and hence in need of explanation. The main reason for this presentation is the fact that many companies still use static investment calculation methods for investment calculations. Däumler mentions survey results from the mid-1990s of studies cited by him or studies he himself conducted, according to which almost every second German company still uses static investment calculation methods, at least in addition to other methods (Däumler and Grabe 2007, p. 32 ff). This is of course in relation to the total number of companies. Since large companies probably tend to calculate more dynamically because of the importance of the topic and because of the competence of specialists available for it, this means that the large number of smaller companies, especially the large number of companies in the legal form of sole proprietorships, still calculate statically or at least have done so in the near past. The techniques are presented here so that the reader of this book knows the weaknesses of these methods and can accordingly denounce and criticise them in practice when confronted with such calculations.

2.2 Fundamental Aspects of Static Investment Calculation Methods

33

An obvious advantage of the static methods over the other techniques of investment calculation clearly lies in the simplicity of the calculation. An entrepreneur as well as a private person can make an initial calculation by quickly producing a result with estimated data through the mental calculation for an idea that has just arisen. This works just as well as in negotiations with other economic partners without them noticing the calculation, comparable to as when driving a car very quickly as a driver without endangering those in traffic. This is the advantage of the method. Results can be produced quickly without much effort in data collection and calculation. Since the static procedures do not or only incompletely consider an interest rate, as we will work out in the course of this chapter, the data collection of the calculation elements is also considerably less complex than for the dynamic investment calculation procedures. While only the incoming and outgoing payments as calculation elements of the investment calculation take the timing of the payments exactly into account, as we have worked out in Fig. 1.8, the calculation elements of income/ expenditure and performance/costs, which are easier to determine, can also be used here, due to the lack of consideration of the time of payment in the statics and due to the missing appropriate interest rate. All static procedures are periodic procedures, i.e. they only consider the economic situation in a single period, i.e. in a single year. The decision criteria of the static procedures are, therefore, always based on a single-periodic consideration and an implicitly maximised withdrawal of these amounts from the investment object. Asset accumulation criteria are, therefore, not used for valuation in the static analysis. Static amortisation calculation, which is also a static procedure, involves searching for the number of periods after which the initial payment is recovered. If this procedure is multi-periodic, however, you can assume that the payments are constant in all periods. The data collection is, as already mentioned, less complex for the static procedures, because different calculation variables can be used. Furthermore, only data for a planning period of 1 year needs to be procured. A distinction is made here between the primitive and the improved procedure for data collection. In the primitive method, the planning data of the first year is assumed to be representative and, therefore, valid for all planning periods. In the improved procedure, the data of a representative period are selected and used for the calculation. This procedure of the static methods causes somewhat serious disadvantages with regard to the transferability of the calculation results to reality: • The useful life of an investment object is not fully taken into account. • The time difference of the payment incidence and the associated interest payments are not or only incompletely considered. • Interdependencies with other investment objects or with other years of the useful life of the object under consideration are ignored. • Constant capacity utilisation is assumed over the years of the useful life. • Revenue, costs and profits are assumed to remain constant over the years of the useful life and.

34

2

Static Investment Calculation Methods

• Data security is assumed. In the example in Fig. 1.6, the decision of the static procedures can, therefore, be derived from the data structure alone, object A is preferred over object B, because with the same useful life and acquisition payment (payment for the acquisition of an object), the total net payments, for which no interest is taken into account, are 1 million euros higher than for object B. The annual net incoming payments for object A are, therefore, 250,000 euros higher than for object B. In principle, when comparing investment objects in the static analysis, only those objects should be compared that are mutually exclusive and have the same useful lives and the same acquisition payment. This makes sense in order to avoid different capital budgets (different costs of acquisition payment, which are tied up in the investment object over the term) over different periods (different useful lives of the investment objects). Practice often deviates from that. This is not dramatic because the opportunity costs for capital, the interest rate, are not always fully taken into account in the static model. If it is assumed that capital has no opportunity costs, it is not dramatic to compare different amounts of capital over different periods of time. However, it does not make sense either, but it is generally justifiable in other areas due to the low accuracy of static methods. For dynamic methods, this would have to be assessed much more critically; this will be the subject of Chap. 4. In this chapter, four different static procedures will be considered: • • • •

The cost comparison calculation. The profit comparison calculation. The profitability calculation and. The static amortisation calculation.

Especially in older academic literature, various other static methods were presented in large numbers, which were then often developed by well-known authors of thematically related teaching texts and represented a marginal change compared to one of the known methods. These are not to be presented here. No consideration of tax aspects shall take place neither in this chapter nor in the whole book. The different questions of the static procedures can be based on • A single investment, but this does not make sense in all procedures, or on. • An alternative comparison of investment objects, i.e. the question of whether investment object A or investment object B should be acquired if none exists or. • The replacement problem, where an existing old investment object is to be replaced by a new investment object, if necessary. The necessary calculation formulas differ. Overall, there is a high number of different static formulas. Therefore, in the following section, before introducing the individual static investment calculation methods, a toolkit is to be set up from which each static investment calculation formula can be determined. Thus, it is not necessary to learn

2.3 A Modular System for the Creation of Static Investment Calculation Formulas

35

the formulas by heart if they can be put together again and again from the relevant components. However, the formulas in Sects. 2.4–2.7 are also presented for the individual four static investment calculation procedures. Section Results In this chapter, you: • Got to know the fundamental points of criticism of the static investment calculation methods. • Learned to distinguish the primitive from the improved method of data acquisition. • Learned that there are four common static investment calculation methods and. • Identified the issues that can be addressed with static methods.

2.3

A Modular System for the Creation of Static Investment Calculation Formulas

While, as we will see in Chap. 3, the dynamic investment calculation methods generally manage with 1–3 mathematical formulas for determining the dynamic target value, the static investment calculation methods have considerably more formulas per method, since there are various case distinctions that can be made in the areas of • Comparison of alternatives/replacement problem or. • Residual value available/residual value not available or with. • Notion of capital commitment. What exactly these terms mean will also be discussed in this section. The consequence of these distinctions is that in each defined case of a practical investment question, which is to be calculated with a static investment calculation formula, only exactly one formula correctly reflects the decision situation. Unfortunately, there are about 60 different investment calculation formulas for the four static investment calculation procedures to be dealt with. This opens up two possibilities for an academic textbook: either the presentation of all formulas for the individual static investment calculation procedures or the presentation of a general instruction for static investment calculation formulas, from which the correct investment calculation formula can be created, depending on the case, without having to learn all formulas individually for reproduction. The latter path will be taken first in this section. In addition, the most common static investment calculation formulas are presented in detail in the following four sections, i.e. in Sects. 2.4–2.7, when presenting the individual four static investment calculation methods. For these described procedures, first, the maximum of three different components or summands that can occur in a static investment calculation formula is presented, and then the three different cases or combinations when the components can be combined to a certain formula are presented. As a result, each of the more than

36

2

Static Investment Calculation Methods

60 static investment calculation formulas of the four different investment calculation procedures can be clearly produced from the components and combinations presented.

2.3.1

The Components of Static Investment Calculation Formulas

Three components occur at most in the static formulas: • The revenue (U). This is the revenue generated by the investment object in the year under consideration. • The operating costs (B). These are the annual operating costs of the investment object. • The debt service (KD). This is the distribution of the acquisition payment of the investment object, possibly reduced by a share of the residual value, over an annual period, taking into account an interest component. The debt service is divided into two areas, – The recovery share An , A 2n R . A stands for acquisition payment (initial net investment) of the investment object, R for the residual value and n for the useful life of the investment object, measured in years. Here the amount of the acquisition payment, reduced by a share of the residual value if necessary, is transformed into an annual value. – The interest component (average fixed capital (d. geb. Kap)
i). i is the calculation interest rate in decimal notation. What exactly the average fixed capital represents is clarified in the combinations in this section. The interest portion, if it exists, pays interest on part of the acquisition payment. These are the three components that can occur in a static investment calculation formula which can occur at most. The terms “revenue” and “costs” can be replaced by the terms of the calculation elements mentioned in the previous section. Yet, due to the lack of consideration of the time preference by the statics, this is irrelevant.

2.3.2

The Combinations for the Creation of Static Investment Calculation Formulas

In this sub-section, three situations or combinations that influence the creation of static investment calculation formulas are specified: • The type of calculation • The distinction between “comparison of alternatives” and “replacement problem” and • The notion of capital commitment

2.3 A Modular System for the Creation of Static Investment Calculation Formulas

37

The consideration of the type of calculation, i.e. the question of which static investment calculation method is used, is important, since not all static methods include all components. The distinction between questions for comparing alternatives, i.e. investment decisions where no investment object has been available to the considered department of the company so far and where one object should be selected from several possible investment objects, and the replacement problem is important because the corresponding static investment calculation formulas contain different components. The replacement problem is the question of whether an investment object that already exists in the company should be used further on, or whether it should be replaced by another new investment object. When determining the notion of capital commitment, the second part of the debt service, the interest portion, is assigned to an interest-bearing capital, the average fixed capital, which is to be paid interest at the given interest rate. With the determination of the notion of capital commitment, the average fixed capital is determined, i.e. a determination is made of how the commitment of the acquisition payment in the investment object develops over time, possibly modified by a residual value. Here eight different forms are known.

2.3.2.1 The Consideration of the Type of Calculation In this area, it is clarified whether all components are considered in all four static investment calculation methods. The component revenue is not relevant in cost comparison calculation. As a result of this finding, it is possible to dispense with the consideration of the component revenue when creating the calculation formulas for the cost comparison calculation as a static investment calculation procedure. 2.3.2.2 The Distinction Between “Comparison of Alternatives” and “Replacement Problem” While the determination of the type of calculation is self-explanatory, the investor determines which of the four static investment calculation methods he wants to apply to the valuation of an investment object. The distinction between the comparison of alternatives and the replacement problem is now worked out. First of all, Fig. 2.1 shows a graphic representation of the decision problem in a comparison of alternatives. Two investment objects, object 1 and object 2, are to be valued as alternatives. Both objects have revenue (U ), operating costs (B) and debt services (KD). Both have a planned 5-year useful life, as documented by the timeline under the two objects. Thus, the numerical indexing of the variables refers to the year in which they occur. In accordance with the assumptions of the statics, these values are the same in all years, since we make a one-period analysis and, therefore, assume that the payments are the same in all periods. An absolute level of payments is not shown in this figure, as it is irrelevant for explaining the decision-making situation. Nor is it specified

38

2

Object 1

Object 2

0

Static Investment Calculation Methods

U1

U2

U3

U4

U5

B1

B2

B3

B4

B5

KD 1

KD 2

KD 3

KD 4

KD 5

U1

U2

U3

U4

U5

B1

B2

B3

B4

B5

KD 1

KD 2

KD 3

KD 4

KD 5

1

2

3

4

5

Fig. 2.1 Decision problem when comparing alternatives (Source: Author)

which static investment calculation procedure should be used for a decision, this is also irrelevant, since it applies identically to all procedures. The comparison of alternatives was defined as a decision-making situation in which the company currently has no investment object in the consideration department—and one of the two investment objects under consideration is to be acquired. Thus, it quickly becomes clear from the picture what the relevant calculation elements for an alternative comparison are. If object 1 would be acquired, then revenue, operating costs and debt services of object 1 would arise. If object 2 was to be purchased, then revenue, operating costs and debt services of object 2 would arise. Depending on the more favourable target values of the objects in a particular static investment calculation procedure, one or the other investment objects would be acquired. A general equation for the decision would be U 1  B1  KD1

< U  B2  KD2 > 2

ð2:1Þ

In the case of a replacement problem, the decision situation is different. This is first presented in Fig. 2.2. It shows that an existing old object can be replaced by a new object after 2 years if necessary. This is also an alternative decision. Either the old object is kept or it is replaced by a new object. Which payments are relevant for the decision concerning the new object can then first be clarified quickly and equivalently to Fig. 2.1 and the decision situation when comparing alternatives. Revenue, operating costs and the debt service of the new

2.3 A Modular System for the Creation of Static Investment Calculation Formulas

Object old

U 1 old

U 2 old

U 3 old

U 4 old

U 5 old

B 1 old

B 2 old

B 3 old

B 4 old

B 5 old

KD 1 old

KD 2 old

KD 3 old

KD 4 old

KD 5 old

U 1 new

U 2 new

U 3 new

U 4 new

U 5 new

B 1 new

B 2 new

B 3 new

B 4 new

B 5 new

KD 1 new

KD 2 new

KD 3 new

KD 4 new

KD 5 new

Object new

old new

0

1

2

3

4

5

0

1

2

3

4

39

5

Fig. 2.2 Decision problem with the replacement problem (Source: Author)

object only arise if it is actually purchased, i.e. they are fully attributable to it. The situation is different for the old object. If the old object is abolished at the end of the second year, then there will certainly be no revenue and no operating costs for the old object in the following three actually planned years of the useful life, because the object has been abolished. It is different with the debt service of the old object. The debt service is the distribution of the acquisition costs of the investment object over the period of the useful life and should regain the acquisition payment in the years of the useful life, if necessary including a share of interest and less a share of the residual value. If the useful life is shortened during the useful life, then the capital invested has not yet been earned back. The debt services of the old object for the remaining years must also be earned by the new object, because the old object no longer exists and the economic results of the years that have elapsed cannot be changed, since these are plan values from the time before the acquisition of the old object, which led to the acquisition of exactly this amount. Figure 2.3 shows the new planning situation. The debt services of the old object continue until the end of the actually planned useful life of the old object. In practical terms, the reader can perhaps best imagine this with leasing rates, at least on impractical assumptions that a leasing contract for the financing of an investment object always runs until the end of the planned useful life of an investment object, even if this has already been abolished. In practice, this would also lead to an immediate termination of a leasing contract. In Fig. 2.2, five debt service instalments, e.g. the amount of 20,000 euros each, would incur for the old object. After the abolition of the old object after 2 years of the planned 5-year useful life, three debt services of 20,000 euros each would thus remain open. The creditor of the instalments would certainly insist on these payments. An immediate replacement by a special payment or an insurance, which

40

2

Static Investment Calculation Methods

Object old

Object new

old new

0

1

KD 3 old

KD 4 old

KD 5 old

U 1 new

U 2 new

U 3 new

U 4 new

U 5 new

B 1 new

B 2 new

B 3 new

B 4 new

B 5 new

KD 1 new

KD 2 new

KD 3 new

KD 4 new

KD 5 new

2

3

4

5

0

1

2

3

4

5

Fig. 2.3 Obtaining debt service in the replacement problem (Source: Author)

is usual in practice, should not be assumed here. In such a situation, the new object would, therefore, have to earn the same amount of debt service as the old object in the first 3 years of its useful life, as Fig. 2.3 suggests. The here general equation for the decision would now be U old  Bold  KDold

< U  Bnew  KDnew  KDold : > new

ð2:2Þ

In fact, the debt service of the old object would have to be adjusted to the different number of years of the useful life of the new object, as the calculation elements of a planning period must be the same, but this is not the case with the rather imprecise approach of the static investment calculation methods. Static investment calculation methods are decision-making techniques for determining the advantages of investment objects, not like cost accounting methods which are intended to determine cost accounting variables precisely, nor like financial planning methods which are intended to determine the liquidity of alternatives precisely. Since only the advantageousness of old and new objects is to be determined in comparison, the summand KDold can be omitted on each side of the Eq. (2.2) because if the same summand is eliminated on each side of an equation, the advantageousness cannot change. Considering the amounts of this example, this means that it is not necessary to deduct 20,000 euros on each side of the equation for the debt service of the old object. This reduces the calculation effort, but it does not change the relative advantages of the alternatives. U old  Bold

< U  Bnew  KDnew > new

ð2:3Þ

2.3 A Modular System for the Creation of Static Investment Calculation Formulas

41

This leads to the necessity of the distinction in this second combination, the distinction between • The comparison of alternatives, that is, the situation in which there is currently no investment object in the considered business area and one of several possible objects is to be acquired and. • The replacement problem, where an investment object already exists in the company and is to be replaced by a new object from this area, if necessary. Because the debt service of the old system is not relevant for the replacement problem it does not have to be considered on either side of the equation. When comparing alternatives, however, the debt service of the alternative investment objects must be considered on each side of the equation.

2.3.2.3 The Notion of Capital Commitment The third combination deals with the idea of the development of the fixed capital in the investment object over time. It thus operationalises the second part of the third component in a mathematical formula, so that static investment calculation formulas can then be set up in a mathematically clear and unambiguous manner. The mathematical formula for determining the interest portion of the debt service is thus determined. How the interest portion of the average fixed capital (d. geb. Kap.
i) is to be determined depends on the investor’s idea about the course of the fixed capital in the investment object over time. There is rarely an impartial value, since there are reliable market prices at every age and level of use for only a few investment objects. Popular car models for which the data are available would be an example here. In order to use this data in a static investment calculation, it would also have to be assumed that the investor periodically withdraws depreciations from the investment by the amount of the value reduction. If this is not the case, idealtypical capital commitment patterns overtime during the useful life of an investment object can be used to determine the interest portion of the average capital commitment. Here we distinguish between four cases of the idea of a possible capital commitment, each with the presence of a residual value and the absence of a residual value. The ideas of capital commitment with a residual value are shown in Fig. 2.4. The average fixed capital is derived from the graphs to a formula which is also shown in Fig. 2.4. After multiplying it by the interest rate, it then represents the interest-bearing portion of the debt service in its various forms for the static investment calculation formula. In principle, the classic forms of capital commitment are • The linear reduction of capital (part (1) in Fig. 2.4) • The discrete reduction of capital (part (2) and (3) in Fig. 2.4) and • The constant commitment of capital (part (4) in Fig. 2.4)

42

2

Static Investment Calculation Methods

Fig. 2.4 Development of capital commitment for investments with residual value (Source: Author)

These different notions of capital commitment then lead to a different idea of the average capital commitment as the second summand of debt service for each investment object. • If we take the simple notion of capital commitment with continuous capital reduction in (1), the result is an average fixed capital of AþR 2 . • For the same investment object, the presentation of a discrete reduction in capital, i.e. a reduction in capital ad hoc at the end of the period, as would be the case for

2.3 A Modular System for the Creation of Static Investment Calculation Formulas

43

ARþAR

n depreciations in accounting, results in an average fixed capital of þ R. 2 This makes sense if the capital actually is withdrawn from the investment object at the end of the period through the formation of corresponding depreciation equivalents. Non-linear depreciation can be taken into account in this approach but does not fit in with the general assumption in the static procedures that the calculation elements are the same in each year. Þ
• Mathematically identical to the case (2) in Fig. 2.4, is the formula ðAR 2 nþ1 þ R which is shown in case (3). In the case of straight-line depreciation, the n calculation results of the capital commitment ideas for the average fixed capital coincide in (2) and (3). Non-linear depreciation cannot be taken into account with this approach. • The capital commitment concept in (4) assumes that the capital is fully committed over the term, i.e. that there is no reduction in the capital during the term. This means that A is tied up on average.

The ideas of capital commitment in the absence of a residual value are shown in Fig. 2.5. Likewise in Fig. 2.5, as in Fig. 2.4, the average fixed capital is derived from the figures and developed into formulas in order to represent the interest portion of the debt service in its various forms concerning the static investment calculation formulas Basically, the classic forms assume • The linear reduction of capital (part (1) in Fig. 2.5), • The discrete reduction of capital (part (2) and (3) in Fig. 2.5) and • The constant commitment of capital (part (4) in Fig. 2.5)

2.3.3

Section Results

In this section, you: • Learned about a modular system for creating static investment calculation formulas. • Differentiated the three case distinctions for determining static investment calculation formulas. • Worked out the three components of the static investment calculation formulas. • Learned about the three combinations for combining the components of the static investment calculation formulas. • Focused, in particular, on the difference between comparing alternatives and the replacement problem and. • Analysed the different forms of capital commitment.

44

2

Static Investment Calculation Methods

graphic representation

average fixed capital (d. geb. Kap) residual value = 0

(1) linear capital reduction fixed capital (€) A

(1): d. geb. Kap. =

n

A 2

years (n)

(2), (3) discrete capital reduction fixed capital (€) A

A n

(2): d. geb. Kap. =

A+

A n

2

A n (3): d. geb. Kap. =

A n +1 * 2 n

(4): d. geb. Kap. =

A

A n A n n

years (n)

(4) constant capital commitment fixed capital (€) A

A

n

years (n)

Fig. 2.5 Development of capital commitment for investments without residual value (Source: Author)

2.4 The Cost Comparison Calculation

2.4

The Cost Comparison Calculation

2.4.1

Presentation and Criticism of the Cost Comparison Calculation

45

The cost comparison calculation compares costs of investment objects at a given capacity. Unfortunately, this procedure has several disadvantages due to its simplicity. First, these are the disadvantages common to all static methods, which have already been mentioned in Sect. 2.2. Therefore they will not be repeated here. Additional disadvantages, which are based on the cost comparison calculation, are • The missing of consideration of the revenue. Of course, from an economic perspective it is not the costs of investment objects that are relevant, but profits, i.e. a difference between a revenue figure and a cost figure. The part of the revenue is ignored when applying the cost comparison calculation. This only makes sense if the amount and temporal distribution of revenue are identical or nearly identical for all alternatives to be compared. If this is not the case, you should not use cost comparison accounting, but instead use profit comparison accounting, which is discussed in the following Sect. 2.5. • Furthermore, the cost comparison calculation does not determine any actual or realistically calculated costs, as would be the case in cost accounting as other business management disciplines, so that the application of the cost comparison calculation for a single investment is not meaningful, since the cost calculation only determines a comparison figure and no actual costs. • In addition, only investment objects whose costs are quite similar in their amount over time should be compared with the cost comparison calculation. Since the cost comparison calculation is a periodic calculation, averages of different operating costs over time are calculated. Thus, the timing of payments and interest differences, which are relevant in practice, are ignored. • For the same reason of periodically identical calculation elements, the cost comparison calculation also assumes constant capacity utilisation of the investment object. This is a decisive factor for selecting the most favourable investment, especially when comparing investment alternatives, which is done in all applications of cost comparison calculation. Even small changes in utilisation can lead to a different investment decision, but this is not taken into account by the cost comparison calculation. It is possible to take different capacity utilisation rates into account when using cost comparison accounting, not on the total costs of an investment distributed throughout the useful life, that is, per period, but on the costs per unit, which are determined by dividing the total costs by the output quantity, that is the costs per performance unit. This question will not be addressed at this point. The question of the minimum output quantity from which an investment object can be preferred to another is, in the final analysis, identical to the question of critical-value calculation,

46

2

Static Investment Calculation Methods

which will be discussed in Chap. 6. There, however, the calculation will be somewhat more demanding and more realistic, namely dynamic. The presentation in this chapter also forgoes the division into fixed and variable costs. The costs shown are, therefore, always the sum of fixed and variable costs. In accordance with the combinations presented in the previous section, there are now a total of 16 different formulas for cost comparison calculation. This results from the four possible capital commitment concepts, which can be combined once with and once without residual value to eight different possible debt services. Due to the differentiation in the comparison of alternatives and the replacement problem, the number of possible formulas doubles to 16. These formulas are presented hereafter.

2.4.2

Formulas of the Cost Comparison Calculation

In general, the following formula applies to the comparison of alternatives: K1

< K > 2

ð2:4Þ

Of course more than two investment objects can be compared. For the replacement problem the following formula applies: K old

< K > new

ð2:5Þ

The cost components of the operating costs (B) are wages, material, energy, maintenance, occupancy costs, distribution costs, etc. The cost components of the debt service are the depreciation for the recovery of capital and the interest component. First, the eight formulas for the comparison of alternatives are presented, and from these the first four with consideration of a residual value. They then differ only in the different capital commitment concepts. B1 þ

A1  R1 A1 þ R1 < A  R2 A2 þ R2 þ þ
i B2 þ 2
i > n1 2 n2 2

ð2:6Þ

This formula 2.6 is the best-known formula of cost comparison calculation, the so-called engineering formula. ! 1 A1  R1 þ A1nR A1  R1 < A  R2 1 þ B1 þ þ R1
i B2 þ 2 > n1 2 n2 ! 2 A2  R2 þ A2nR 2 þ þ R2
i ð2:7Þ 2

2.4 The Cost Comparison Calculation

47

Formula 2.7 is a cost comparison calculation formula for comparing alternatives with a residual value and assuming a discrete notion of capital commitment.   A  R1 A1  R1 n1 þ 1 < A  R2 B1 þ 1 þ þ R1
i B2 þ 2
n1 > n1 2 n2   A2  R2 n2 þ 1 þ þ R2
i
ð2:8Þ n2 2 Formula 2.8 is a cost comparison calculation formula in the comparison of alternatives with a residual value and also assuming a discrete notion of capital commitment. B1 þ

A1  R1 < A  R2 þ A1
i B2 þ 2 þ A2
i > n1 n2

ð2:9Þ

Formula 2.9 is a cost comparison calculation formula in the comparison of alternatives with a residual value and assuming a constant notion of capital commitment. Since the amount of the acquisition payment corresponds to the residual value (A ¼ R), (A  R)/n must also be zero. B1 þ

A1 A1 < A A þ
i B2 þ 2 þ 2
i > n1 2 n2 2

ð2:10Þ

This formula 2.10 is also the well-known engineering formula, but without a residual value. It is also missing in the following three formulas. ! ! A1 þ An11 A2 þ An22 A1 < A2 B1 þ þ
i B2 þ þ
i ð2:11Þ > n1 2 n2 2 Formula 2.11 is a cost comparison calculation formula in the comparison of alternatives without a residual value and assuming a discrete notion of capital commitment.     A1 A 1 n1 þ 1 < A2 A2 n2 þ 1 B1 þ þ


i B2 þ þ
i ð2:12Þ n1 > n2 n1 2 n2 2 Formula 2.12 is a cost comparison calculation formula in the comparison of alternatives without a residual value and assuming an equally discrete notion of capital commitment. B1 þ

A1 < A þ A1
i B2 þ 2 þ A2
i > n1 n2

ð2:13Þ

Formula 2.13 is a cost comparison calculation formula in the comparison of alternatives without a residual value and assuming a constant notion of capital commitment.

48

2

Static Investment Calculation Methods

Now the eight formulas for the replacement problem are presented. The first four formulas consider a residual value. They only differ in the notion of capital commitment. Bnew þ

Anew  Rnew Anew þ Rnew < þ
i Bold > nnew 2

ð2:14Þ

Formula 2.14 is a cost comparison calculation formula in the replacement problem with a residual value and assuming a continuous notion of capital commitment. ! new Anew  Rnew þ AnewnR Anew  Rnew < new Bnew þ þ ð2:15Þ þ Rnew
i Bold > nnew 2 Formula 2.15 is a cost comparison calculation formula in the replacement problem with a residual value and assuming a discrete notion of capital commitment.   Anew  Rnew Anew  Rnew nnew þ 1 < Bnew þ þ þ Rnew
i Bold ð2:16Þ
nnew > nnew 2 Formula 2.16 is a cost comparison calculation formula in the replacement problem with a residual value and also assuming a discrete notion of capital commitment. Bnew þ

Anew  Rnew < þ Anew
i Bold > nnew

ð2:17Þ

Formula 2.17 is a cost comparison calculation formula in the replacement problem with a residual value, which however does not affect the capital commitment, and assuming a constant notion of capital commitment. Bnew þ

Anew Anew < þ
i Bold > nnew 2

ð2:18Þ

Formula 2.18 is a cost comparison calculation formula in the replacement problem without a residual value, as it is in the following three formulas, and assuming a continuous notion of capital commitment. ! new Anew þ Annew Anew < Bnew þ þ ð2:19Þ
i Bold > nnew 2 Formula 2.19 is a cost comparison calculation formula in the replacement problem without a residual value and assuming a discrete notion of capital commitment.   Anew Anew nnew þ 1 < Bnew þ
þ ð2:20Þ
i Bold nnew > nnew 2 Formula 2.20 is a cost comparison calculation formula in the replacement problem without a residual value and also assuming a discrete notion of capital commitment.

2.4 The Cost Comparison Calculation

Bnew þ

49

Anew < þ Anew
i Bold > nnew

ð2:21Þ

Formula 2.21 is a cost comparison calculation formula in the replacement problem without a residual value and assuming a constant notion of capital commitment.

2.4.3

Application of the Cost Comparison Calculation

The cost comparison calculation is now applied in different ways to the example data set in Fig. 2.6. The corresponding exercises can be found below Fig. 2.6.

2.4.3.1 Exercises

Exercise a) Determine the costs of objects 1 and 2 with the cost comparison calculation in the alternative comparison, and recommend an investment decision based on the results of the calculation. For the notion of capital commitment choose the simplest, i.e. linear capital reduction and apply once the primitive method of data collection and once the improved method. For the improved method of data collection, please calculate the averages of the annual operating costs and consider them as representative. Exercise b) Determine the costs of objects 1 and 2 using the cost comparison calculation in the alternative comparison, and recommend an investment decision based on the results of the calculation. For the capital commitment, use the discrete capital reduction and apply the improved data determination procedure. For the improved method of data determination, calculate the average annual operating costs and consider them as representative. Exercise c) Determine the costs of objects 1 and 2 with the cost comparison calculation in the replacement problem and recommend an investment decision Fig. 2.6 Data set for the application of the cost comparison calculation (Source: Author)

A A

B Object 1 12000

C Object 2 20000

i n

0.1 4

0.1 4

5

B k=1

18000

16000

6

B k=2

15000

20000

7

B k=3

14000

22000

8

B k=4

13000

22000

9

R

2000

5000

1 2 3 4

50

2

Static Investment Calculation Methods

based on the calculation result. Assume a simple and a discrete capital reduction scenario for the notion of capital commitment and use the improved method of data determination. For the improved method of data determination, please calculate the average annual operating costs and consider them as representative. For this exercise, please assume that object 1 is the new object and object 2 is the old object. Exercise d) Determine the costs for objects 1 and 2 with the cost comparison calculation in the replacement problem and recommend an investment decision based on the calculation result. For the capital commitment, use the simple capital reduction concept and use the improved data determination procedure. For the improved method of data determination, calculate the average annual operating costs and consider them as representative. For this exercise, please assume that object 1 is the new object and object 2 is the old object and that, unlike the data in Fig. 2.6, there are no residual values for the old object and the new object.

2.4.3.2 Solutions Exercise a) First, the relevant formula must be identified. In this case, this is formula 2.6, into which the relevant data is substituted. In the case of the primitive method of data acquisition, the first-year data of the operating costs are used, i.e. 18,000 euros for object 1 and 16,000 euros for object 2 from Fig. 2.6. For the improved procedure of data acquisition, the operating costs of each investment object must be summed up individually and divided by the years of useful life. For object 1, the result is (18,000 + 15,000 + 14,000 + 13,000) / 4 ¼ 15,000. For object 2, the result is (16,000 + 20,000 + 22,000 + 22,000) / 4 ¼ 20,000. (2.22) = (2.6) B1 þ

A1  R1 A1 þ R1 < A  R2 A2 þ R2
i B2 þ 2
i þ þ > n1 2 n2 2

ð2:22Þ

For the primitive method of data determination, 18,000 euros are, therefore, used as operating costs for object 1, and 16,000 euros for object 2. This is visible in Eq. (2.23). 12, 000  2000 12, 000 þ 2000 < þ
0:1 16, 000 4 2 > 20, 000  5000 20, 000 þ 5000 þ þ
0:1 4 2

18, 000 þ

ð2:23Þ

In this situation, the costs for object 1 would then sum up to 21,200 euros, and to 21,000 euros for object 2. K Object 1 ð21, 200 eurosÞ > K Object 2 ð21, 000 eurosÞ Thus object 2 would be preferable to object 1 in this combination.

ð2:24Þ

2.4 The Cost Comparison Calculation

51

For the improved method of data determination, 15,000 euros are, therefore, required for object 1 as operating costs and 20,000 euros for object 2. Again formula 2.6 is relevant. (2.25) = (2.6) B1 þ

A1  R1 A1 þ R1 < A  R2 A2 þ R2 þ þ
i B2 þ 2
i > n1 2 n2 2

ð2:25Þ

This is visible in Eq. (2.26). 12, 000  2000 12, 000 þ 2000 < þ
0:1 20, 000 4 2 > 20, 000  5000 20, 000 þ 5000 þ þ
0:1 4 2

15, 000 þ

ð2:26Þ

In this situation, the costs for object 1 would be 18,200 euros, and 25,000 euros for object 2. K Object 1 ð18, 200 eurosÞ < K Object 2 ð25, 000 eurosÞ

ð2:27Þ

Thus object 1 would be preferable to object 2 in this combination. Exercise b) First, the relevant formula must be identified. In this case, these are formulas 2.7 and 2.8, both represent the discrete capital commitment concept, only with different formulas that lead to the same results for a steady reduction of capital. The relevant data must be entered into these formulas, as shown in formulas 2.30 and 2.31. (2.28) = (2.7) A  R1 B1 þ 1 þ n1 þ

A2  R2 þ 2

1 A1  R1 þ A1nR 1 þ R1 2 ! A R 2

n2

!
i

< A  R2 B þ 2 > 2 n2

2

þ R2


i

ð2:28Þ

(2.29) = (2.8)   A1  R1 A1  R1 n1 þ 1 < A  R2 þ þ R1
i B2 þ 2
n1 > n1 2 n2   A2  R2 n2 þ 1 þ þ R2
i
n2 2

B1 þ

ð2:29Þ

52

2

Static Investment Calculation Methods

  12, 000  2000 þ 12, 0002000 12, 000  2000 4 þ þ 2000 15, 000 þ 4 2
0:1 þ

< 20, 000  5000 20, 000 þ > 4

20, 000  5000 þ 20, 0005000 4 þ 5000 2

!
0:1

ð2:30Þ

  12, 000  2000 12, 000  2000 4 þ 1 þ
þ 2000 4 2 4 < 20, 000  5000
0:1 20, 000 þ > 4   20, 000  5000 4 þ 1
þ 5000
0:1 þ 2 4

15, 000 þ

ð2:31Þ

For object 1, this situation will result in costs of 18,325 euros. For object 2 the costs are 25,187.50 euros. K Object 1 ð18, 325 eurosÞ < K Object 2 ð25, 187:50 eurosÞ

ð2:32Þ

Thus object 1 would be preferable to object 2 here. Exercise c) First of all, the relevant formulas have to be identified. In this case, this is 2.14 as well as 2.15 and 2.16. The first formula is used for the simple capital reduction, the others represent the discrete capital commitment concept, only with different formulas that lead to the same results for a steady reduction of capital. Therefore, only one formula is used here for the discrete capital commitment concept. The relevant data must be entered into these formulas, as shown in formulas 2.35 and 2.36. (2.33) = (2.14) Bnew þ

Anew  Rnew Anew þ Rnew <
i Bold þ > nnew 2

ð2:33Þ

(2.34) = (2.15) A  Rnew Bnew þ new þ nnew

Rnew Anew  Rnew þ Anewnnew þ Rnew 2

!
i

< B > old

ð2:34Þ

The following equation results for the simplest notion of capital commitment: 15, 000 þ

12, 000  2000 12, 000 þ 2000 < þ
0:1 20, 000 4 2 >

ð2:35Þ

2.4 The Cost Comparison Calculation

53

In this situation, the costs for the new object would then amount to 18,200 euros, and 20,000 euros for the old object. K Object

new ð18, 200

eurosÞ < K Object

old ð20, 000

eurosÞ

ð2:36Þ

Thus, the new object in this combination would be preferable to the old object. There should be an immediate replacement. The following equation results for the discrete notion of capital commitment:   12, 000  2000 þ 12, 0002000 12, 000  2000 4 15, 000 þ þ þ 2000 4 2 <
0:1 20, 000 ð2:37Þ > In this situation, the costs for the new object would then amount to 18,325 euros, and 20,000 euros for the old object. K Object

new ð18, 325

eurosÞ < K Object

old ð20, 000

eurosÞ

ð2:38Þ

Thus, the new object in this combination would be preferable to the old object. There should be an immediate replacement. Exercise d) First, the relevant formula must be identified. In this case, this is formula 2.18. The relevant data must be entered into this formula. This becomes visible in the formula 2.40. (2.39) = (2.18) Bnew þ

Anew Anew <
i Bold þ > nnew 2

ð2:39Þ

The following equation results for the simplest notion of capital commitment: 15, 000 þ

12, 000 12, 000 < þ
0:1 20, 000 4 2 >

ð2:40Þ

In this situation, the costs for the new object would then sum up to 18,600 euros, and to 20,000 euros for the old object. K Object

new ð18, 600

eurosÞ < K Object

old ð20, 000

eurosÞ

ð2:41Þ

Thus, the new object would be preferable to the old object. There should be an immediate replacement.

54

2

2.4.4

Static Investment Calculation Methods

Section Results

In this section, you: • • • • •

Learned how cost comparison calculation works. Understood the criticism of the cost comparison calculation. Learned about the criteria for the cost comparison calculation. Developed the formulas for the cost comparison calculation and. Applied the cost comparison calculation to an example.

2.5

The Profit Comparison Calculation

2.5.1

Presentation and Criticism of the Profit Comparison Calculation

Profit comparison calculation compares profits of investment objects at a given capacity.This approach has the same disadvantages as the cost comparison calculation, except that revenues are now taken into account. The structure of the static investment calculation formulas is identical to that of the cost comparison calculation, as is the number, except that the formulas now consistently include revenue as an additional summand. The formulas of the corresponding cost comparison calculation formulas are all based on revenue (U). The disadvantages are in detail: • The profits determined in the profit comparison calculation are also not actual or realistic profits, as it would be the case in cost accounting as a different business administration discipline. • Unlike the cost comparison calculation, it is now useful to use the profit comparison calculation for a single investment, even though the profit comparison calculation actually only determines a comparison figure and no actual profits, since the result of the profit comparison calculation is an absolute value and given the simplicity of the procedure, the result can give an indication, whether the investment can be advantageous when the planned data is met. • Furthermore, only investment objects should be compared with the profit comparison statement if their profits are similar in amount over time. Since the comparison of profits is a periodic calculation, averages are formed over time from revenue and costs that vary in amount. • For the same reason of periodically identical calculation elements, the profit comparison calculation also assumes that the capacity utilisation of the investment object remains the same. This is a decisive factor for the selection of the most favourable investment, especially when comparing investment alternatives, which is done in all applications of the profit comparison calculation except when considering a single investment. Even small changes in capacity utilisation can

2.5 The Profit Comparison Calculation

55

lead to a different investment decision. However, this is not taken into account by the profit comparison calculation. In accordance with the combinations presented in Sect. 2.3, there are now a total of 16 different formulas for the profit comparison calculation. This results from the four possible capital commitment concepts, which can be combined once with and once without residual value to eight different possible debt services. Due to the difference in the comparison of alternatives and the replacement problem, the number of possible formulas doubles to 16, exactly as in the cost comparison calculation. The formulas for considering a single investment with the profit comparison calculation correspond to one side of the formulas for the alternative comparison. The formulas are presented in the next chapter.

2.5.2

Formulas of the Profit Comparison Calculation

In general, the following formulas apply to the single investment (2.42) and for the comparison of alternatives (2.43): G0 G1

< G > 2

ð2:42Þ ð2:43Þ

Of course more than two investment objects can be compared. The following formula applies to the replacement problem. Gnew

< G > old

ð2:44Þ

First, the eight formulas for the comparison of alternatives are presented, and from these the first four consider a residual value. They then only differ in the capital commitment concepts. U 1  B1 

A1  R1 A1 þ R1 < A  R2 A2 þ R2  
i U 2  B2  2
i > n1 2 n2 2

ð2:45Þ

Formula 2.45 is a profit comparison formula in the comparison of alternatives with a residual value and assuming a continuous or linear capital commitment. ! 1 A1  R1 þ A1nR A1  R1 < A  R2 1  U 1  B1  þ R1
i U 2  B2  2 > n1 2 n2 ! 2 A2  R2 þ A2nR 2  þ R2
i ð2:46Þ 2

56

2

Static Investment Calculation Methods

Formula 2.46 is a profit comparison formula in the comparison of alternatives with a residual value and assuming a discrete notion of capital commitment.   A  R1 A1  R1 n1 þ 1 < U 1  B1  1  þ R1
i U 2  B2
n1 > n1 2   A  R2 A2  R2 n2 þ 1  2  þ R2
i
ð2:47Þ n2 n2 2 Formula 2.47 is a profit comparison formula in the comparison of alternatives with a residual value and also assuming a discrete notion of capital commitment. U 1  B1 

A1  R1 < A  R2  A1
i U 2  B2  2  A2
i > n1 n2

ð2:48Þ

Formula 2.48 is a profit comparison formula in the comparison of alternatives with a residual value and assuming a constant notion of capital commitment. Since the amount of the acquisition payment corresponds to the residual value (A ¼ R), (A  R)/n must also be zero. U 1  B1 

A1 A1 < A A 
i U 2  B2  2  2
i > n1 2 n2 2

ð2:49Þ

Formula 2.49 is a profit comparison formula in the comparison of alternatives without a residual value and assuming a continuous or linear notion of capital commitment. ! ! A1 þ An11 A2 þ An22 A1 < A2 U 1  B1  
i U 2  B2  
i ð2:50Þ > n1 2 n2 2 Formula 2.50 is a profit comparison formula in the comparison of alternatives without a residual value and assuming a discrete notion of capital commitment.     A1 A1 n1 þ 1 < A2 A 2 n2 þ 1 U 1  B1  


i U 2  B2   n1 > n2 n1 2 n2 2
i

ð2:51Þ

Formula 2.51 is a profit comparison calculation formula in the comparison of alternatives without a residual value and also assuming a discrete notion of capital commitment. U 1  B1 

A1 < A  A1
i U 2  B2  2  A2
i > n1 n2

ð2:52Þ

Formula 2.52 is a profit comparison formula in the comparison of alternatives without a residual value and assuming a constant notion of capital commitment.

2.5 The Profit Comparison Calculation

57

Now the eight formulas for the replacement problem are presented. The first four formulas consider a residual value. They only differ in the notion of capital commitment. U new  Bnew 

Anew  Rnew Anew þ Rnew < 
i U old  Bold > nnew 2

ð2:53Þ

Formula 2.53 is a profit comparison formula in the replacement problem with a residual value and assuming a continuous or linear notion of capital commitment. ! new Anew  Rnew þ AnewnR Anew  Rnew new U new  Bnew   þ Rnew nnew 2
i

< U  Bold > old

ð2:54Þ

Formula 2.54 is a profit comparison calculation formula in the replacement problem with a residual value and assuming a discrete notion of capital commitment.   Anew  Rnew Anew  Rnew nnew þ 1 U new  Bnew   þ Rnew
nnew nnew 2 <
i U old  Bold ð2:55Þ > Formula 2.55 is a profit comparison calculation formula in the replacement problem with a residual value and also assuming a discrete notion of capital commitment. U new  Bnew 

Anew  Rnew <  Anew
i U old  Bold > nnew

ð2:56Þ

Formula 2.56 is a profit comparison calculation formula in the replacement problem with a residual value and assuming a constant capital commitment. Since the amount of the acquisition payment corresponds to the residual value (A ¼ R), (A  R)/n must also be zero. U new  Bnew 

Anew Anew < 
i U old  Bold > nnew 2

ð2:57Þ

Formula 2.57 is a profit comparison calculation formula in the replacement problem without a residual value and assuming a continuous or linear capital commitment. ! new Anew þ Annew Anew < U new  Bnew   ð2:58Þ
i U old  Bold > nnew 2 Formula 2.58 is a profit comparison calculation formula in the replacement problem without a residual value and assuming a discrete capital commitment.

58

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Static Investment Calculation Methods

  Anew Anew nnew þ 1 <
U new  Bnew  
i U old  Bold nnew > nnew 2

ð2:59Þ

Formula 2.59 is a profit comparison calculation formula in the replacement problem without a residual value and assuming an equally discrete notion of capital commitment. U new  Bnew 

Anew <  Anew
i U old  Bold > nnew

ð2:60Þ

Formula 2.60 is a profit comparison calculation formula in the replacement problem without a residual value and assuming a constant capital commitment.

2.5.3

Application of the Profit Comparison Calculation

The profit comparison calculation is now applied to the example data set in Fig. 2.6. In addition to the example data set, it must be added that the annual revenue of object 1 are 25,000 euros and the annual revenue of object 2 are 30,000 euros. The improved procedure of data collection should be assumed.

2.5.3.1 Exercises Exercise a) Determine the profits for objects 1 and 2 using the profit comparison calculation in an alternative comparison and recommend an investment decision based on the results of the calculation. For the capital commitment, use the simplest (or linear) capital reduction. Exercise b) Determine the profits for objects 1 and 2 using the profit comparison calculation in an alternative comparison, and recommend an investment decision based on the calculation result. Use the discrete notion of capital commitment. Exercise c) Determine the profits for objects 1 and 2 using the profit comparison calculation in the replacement problem and recommend an investment decision based on the calculation result. For the notion of capital commitment, use once the simple and once the discrete notion of capital commitment. For this exercise, assume that object 1 is the new object and object 2 is the old object. Exercise d) Determine the profits for objects 1 and 2 using the profit comparison calculation in the replacement problem and recommend an investment decision based on the calculation result. For the capital commitment calculation, assume that the notion of capital commitment is constant. For this exercise, please assume that object 1 is the new object and object 2 is the old object and that, unlike the data record in Fig. 2.6, there are no residual values for both the old and new object.

2.5 The Profit Comparison Calculation

59

2.5.3.2 Solutions The procedure for solving these exercises is almost identical to the procedure for solving the cost comparison exercise, except that the result of the cost comparison must be subtracted from the revenue now given in the exercise to determine the profit. Exercise a) First, the relevant formula must be identified. In this case, this is formula 2.45. (2.61) = (2.45) U 1  B1 

A1  R1 A1 þ R1 < A  R2 A2 þ R2  
i U 2  B2  2
i > n1 2 n2 2

ð2:61Þ

The relevant data must be entered in these. This is visible in Eq. (2.62). 12, 000  2000 12, 000 þ 2000 < 
0:1 30, 000 4 2 > 20, 000  5000 20, 000 þ 5000  20, 000  
0:1 4 2

25, 000  15, 000 

ð2:62Þ

In this situation, object 1 would then make a profit of 6800 euros, object 2 would make 5000 euros. GObject 1 ð6800 eurosÞ > GObject 2 ð5000 eurosÞ

ð2:63Þ

Object 1 would, therefore, be preferable to object 2 under this combination. Exercise b) First, the relevant formula must be identified. In this case, these are formulas 2.46 and 2.47. Both represent the discrete notion of capital commitment, only with different formulas that lead to the same results for a uniform capital reduction. Therefore, only formula 2.46 is presented here. (2.64)=(2.46) A  R1 U 1  B1  1  n1 

1 A1  R1 þ A1nR 1 þ R1 2 !

2 A2  R2 þ A2nR 2 þ R2 2


i

!
i

< A  R2 U  B2  2 > 2 n2 ð2:64Þ

The relevant data must be entered into this formula, as shown in formula 2.65.

60

2

Static Investment Calculation Methods

0 12,0002000 B12,0002000þ @ 25,00015,000 4 2

1 12,0002000 < C 4 þ2000A
0:1 >

0

1 20,0005000 20,0005000 B20,0005000þ C 4 30,00020,000 @ þ5000A
0:1 4 2 ð2:65Þ For object 1, this situation would result in a profit of 6675 euros, for object 2 it is 4812.50 euros. GObject 1 ð6675 eurosÞ > GObject 2 ð4812:50 eurosÞ

ð2:66Þ

Thus object 1 would be preferable to object 2 here. Exercise c) First of all, the relevant formulas must be identified. In this case, these are formulas 2.53, 2.54 and 2.55. The first formula is used for the simple notion of capital commitment, the others represent the discrete notion of capital commitment, only with different formulas that lead to the same results for a uniform reduction of capital. Therefore, only formula 2.54 is used here for the discrete notion of capital commitment. The relevant data must be entered in these formulas, as shown in formulas 2.69 and 2.71. (2.67)=(2.53) U new  Bnew 

Anew  Rnew Anew þ Rnew < 
i U old  Bold > nnew 2

ð2:67Þ

(2.68)=(2.54) A  Rnew  U new  Bnew  new nnew
i

new Anew  Rnew þ AnewnR new þ Rnew 2

!

< U  Bold > old

ð2:68Þ

The following equation results for the simplest notion of capital commitment: 25, 000  15, 000   20, 000

12, 000  2000 12, 000 þ 2000 < 
0:1 30, 000 4 2 > ð2:69Þ

In this situation, the new object will then generate profits of 6800 euros, while the old object will generate 10,000 euros. For the new object, there is of course no change compared to object 1 in exercise a).

2.5 The Profit Comparison Calculation

GObject

new ð6800

eurosÞ < GObject

61

old ð10, 000

eurosÞ

ð2:70Þ

Thus, the old object in this combination would be preferable to the new object. There would be no immediate replacement. The following equation results for the discrete notion of capital commitment: 12, 000  2000 25, 000  15, 000  4   12, 000  2000 þ 12, 0002000 < 4  þ 2000
0:1 30, 000  20, 000 > 2

ð2:71Þ

In this situation, the new object would then generate profits of 6675 euros, while the old object would generate 10,000 euros. For the new object there is of course no change compared to object 1 in exercise a). GObject

new ð6675

eurosÞ < GObject

old ð10, 000

eurosÞ

ð2:72Þ

Thus, the old object in this combination would be preferable to the new object. There would be no immediate replacement. Exercise d) First, the relevant formula must be identified. In this case, this is formula 2.60. (2.73)=(2.60) U new  Bnew 

Anew <  Anew
i U old  Bold > nnew

ð2:73Þ

The relevant data must be entered into this formula, as shown in formula 2.74. 25, 000  15, 000 

12, 000 <  12, 000
0:1 30, 000  20, 000 4 >

ð2:74Þ

In this situation, the new object will then generate profits of 5800 euros, while the old object will generate 10,000 euros. GObject

new ð5800

eurosÞ < GObject

old ð10, 000

eurosÞ

ð2:75Þ

Thus, the old object in this combination would be preferable to the new object. There would be no immediate replacement.

2.5.4

Section Results

In this section, you: • Learned how the method of profit comparison calculation works. • Understood the criticism of the profit comparison calculation. • Got to know the criteria of the profit comparison calculation. • Developed and applied the formulas of the profit comparison calculation.

62

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2.6

The Profitability Calculation

2.6.1

Presentation and Criticism of the Profitability Calculation

In the profitability calculation a ratio is determined. Here a periodic profit value is set in relation to the average fixed capital. Accordingly the equation is: Profitability ðRentÞ ¼

profit ðGÞ average fixed capital ðd:geb:Kap:Þ

ð2:76Þ

In this procedure, a single-period variable is determined again. The numerator and denominator are one-periodic values, and the result is, unlike the return, which is the result of the internal rate of return method and is presented in more detail in Chap. 3, also a one-periodic value that determines the return on capital in 1 year, but on the assumption that all values in the numerator and denominator and, therefore, also the calculation result, remain constant in the planned useful life. The internal rate of return method would also assume compound interest. Profitability is, therefore, only an extremely short-term indicator that is not suitable for strategic decisions. The values in numerator and denominator can be defined and determined in a wide variety of ways, so that there is a wide range of profitability. The profitability calculation can be used for the same questions as the profit comparison calculation, that is, the determination of • Profitability of a single investment • Profitability in comparison with alternatives and • Profitability in the replacement problem The following decision criteria apply: For the single investment: Rentsingle investment  Rentexpected minimum

ð2:77Þ

The profitability of the individual investment must not be below a subjectively defined minimum return. The terms of the procedure correspond to the determination of a calculation interest rate as the subjective minimum return requirement of the investor. When comparing alternatives: RentObject 1

< RentObject 2 >

ð2:78Þ

In the case of mutually exclusive investment alternatives with the same capital commitment and the same useful life, the investment with the higher profitability is the better one. Otherwise, of course, both are advantageous as long as they are both above the minimum rate of return set. For the replacement problem:

2.6 The Profitability Calculation

RentObject new

63

< RentObject old >

ð2:79Þ

In the case of mutually exclusive investment alternatives with the same capital commitment and the same useful life, the investment with the higher profitability is the better one. Otherwise, of course, both are advantageous as long as they are both above the minimum rate of return set. For the periodic success variable in the numerator of Eq. (2.76) the profit has already been named. This is neither uncontroversial nor exactly correct for investment calculation in the literature. Before discussing this, it should first be repeated that the academic value of the static procedures as a whole is quite low, i.e. that the static investment calculation procedures as a whole use quite simplifying and thus unrealistic procedures, so that the use of a profit value in the numerator of the profitability calculation formula must be seen primarily regarding this aspect. The aim of the static profitability calculation is to determine a percentage value for a considered planning year, which relates the economic success of the period to the capital invested in the year. Since the value of static methods is precisely that they make use of easy-todetermine calculation elements, in most sources the profit is defined as a success variable in the numerator. Depending on the profit determination procedure and the form of financing, this means that in the usual determination procedure and with complete external financing, interest payments are already deducted from the sales before the profit is determined. From this point of view, they are already the interest claim. From this perspective, using a profit determined in this way in the numerator of the quotient would only determine the “over profitability,” that is, the interest paid on the invested capital that is achieved in addition to the planned interest. Therefore, the deducted interest payments should be added back to the profit in order to relate this profit value as the numerator to the invested capital in the denominator, in order to obtain the total interest on invested capital in the period. Various methods can be selected for determining the numerator, • Simply entering the profit determined in the profit and loss account, if it concerns the consideration of an entire enterprise. • The profit determination of an individual project in the company, i.e. for reasons of simplicity, ignoring the problems just described. • The attribution of interest payments to profit, which, if not only debt capital is used, can also lead to errors depending on the accounting treatment. • The simple determination according to the procedure of the profit comparison calculation known from Sect. 2.5. The profit was determined according to the formula in Eq. (2.80), however the debt service was always formulated according to the formula.

64

2

Static Investment Calculation Methods

ProfitðGÞ ¼ revenue ðU Þ  operating costs ðBÞ  debt service ðKDÞ

ð2:80Þ

Since in this approach the debt service, i.e. the actual or estimated interest on the capital and the recovery of the capital, has already been deducted, a common procedure for determining the numerator of the static profitability calculation is the approach according to the following formula: Profit ðGÞ ¼ revenue ðU Þ  operating costs ðBÞ

ð2:81Þ

Only formula 2.81 will be used in this section to determine the numerator due to reasons of simplicity. Of course, profit values from other operational sources can be used, such as financial statements or a business plan. All of this is evidence of the subjectivity of the numerator. The value can be taken from different sources and be distorted by the possible use of disclosure and measurement options in accounting. However, the calculation of profit without deduction of debt service as given in the formula 2.81 is also seen critically in the literature. Depending on the perspective from which a question is dealt with, some sources also demand the deduction of debt service when calculating profit by the difference between revenue, operating costs and debt service, because at least the recovery share in debt service ensures that the financial substance of the investment is recovered at the end of the term, i.e. the profitability in this form of calculation shows the return while maintaining the substance. Thus, if profit is calculated on the basis of formula (2.81) as the profitability determination, no recovery share of the fixed capital for the consumption of the value of the investment object over time is deducted from turnover, since debt service is not taken into account, the denominator of the profitability value should take into account the entire initial capital and not the average fixed capital, since no recovery of capital is planned in this form of determination of the numerator value. However, this is not assumed in the following text, however, in order to present all the methods of determining the profitability analysis for investment calculations mentioned in the literature. The denominator is, therefore, the average fixed capital and can be taken from the second summand of the third component from Sect. 2.3.2.3. Here too, other sources and corresponding subjective forms are possible, simply by selecting the formula used for the average fixed capital. There is an even more substantial subjectivity in the denominator. The reader will quickly become aware of this when looking at Figs. 2.4 and 2.5 in this chapter. Various capital commitment concepts are possible for an investment object, which the reader may imagine when purchasing a work of art in the form of an oil painting that is to remain with the purchaser for 5 years. How can the price development and thus the development of the fixed capital over time be estimated? • Is it to be depreciated on a straight-line basis because it has no market value at the end of its planned useful life and is worn down by environmental influences?

2.6 The Profitability Calculation

65

• Or must a special depreciation be made in a particular year because the relatively young offspring of the purchaser of the artwork has irreparably “embellished” the work in a creative moment? • Or is no depreciation to be made, as the value does not change or? • Is an attribution in value to be made? All these procedures are certainly possible and possibly equally likely. Now, it is not so difficult to forecast the price development of every investment object or company within this term, but the thoughts show very clearly the subjectivity of this procedure, which is especially in the denominator and less in the so criticised numerator value. The reader is on the basis of Figs. 2.4 and 2.5 aware that in the simple notion of capital commitment, A2 would be the average amount of fixed capital, whereas with constant capital commitment, it would be A. This very subjective decision then leads to a doubling of the calculation result, namely profitability. This clearly shows the subjectivity of the static profitability calculation method. Despite their immense subjectivity, which has the consequence that details of the formulas of the static investment calculation method do not seem to be appropriate in practice, the profitability calculation formulas directly derived from the components derived in Sect. 2.3.2.3 are presented here for reasons of academic consistency. There are seven possible formulas for the profitability analysis, as far as they are derived directly from the components used in the static investment calculation method. Of course, there are other profitability formulas known in general business administration, such as return on equity, return on debt and return on sales. As already mentioned, a differentiation according to the different calculation methods of the static components is actually worthless in practice, since the rough assumptions of the method do not justify a detailed procedure with marginal differences in other areas. Ultimately, the formulas repeatedly show that profitability is the quotient of a periodic performance indicator by the average fixed capital. When applying this method in practice, the user should primarily think about what capital commitment he considers appropriate as the denominator for the problem and use the appropriate formula. In addition to the known problems of the calculation results of the statics in their transferability into practice due to the restrictive assumptions of these methods, there are further problems in the application of the profitability calculation: • A problem is the netting of payments made at different points in time (A-R), as neither of them incur in the considered year. • The comparison of investment alternatives with different terms and different capital requirements is also problematic, since capital has opportunity costs. This problem could be successfully countered with the technique of differential investment, which is presented in Chap. 4. However, from the author’s point of view, it is not justified to enhance these simple static procedures, such as the

66

2

Static Investment Calculation Methods

profitability calculation, by further techniques. This is not the point of these methods. Thus, in this textbook techniques such as differential investment are only presented in connection with the more complex dynamic investment calculation methods.

2.6.2

Formulas of the Profitability Calculation

The seven profitability formulas are Rent ¼

UB AþR 2

ð2:82Þ

Formula 2.82 is a profitability calculation formula with a residual value and assuming a continuous or linear notion of capital commitment. UB Rent ¼ ARþAR n þR 2

ð2:83Þ

Formula 2.83 is a profitability calculation formula with a residual value and assuming a discrete notion of capital commitment. Rent ¼ AR 2

UB
nþ1 n þR

ð2:84Þ

Formula 2.84 is a profitability calculation formula with a residual value and also assuming a discrete notion of capital commitment. Rent ¼

UB A

ð2:85Þ

Formula 2.85 is a profitability calculation formula based on the assumption of constant notion of capital commitment. Rent ¼

UB A 2

ð2:86Þ

Formula 2.86 is a profitability calculation formula without a residual value and assuming a continuous or linear notion of capital commitment. Rent ¼

UB AþAn 2

ð2:87Þ

Formula 2.87 is a profitability calculation formula without a residual value and assuming a discrete notion of capital commitment.

2.6 The Profitability Calculation

67

UB Rent ¼ A nþ1 2
n

ð2:88Þ

Formula 2.88 is a profitability calculation formula without a residual value and also assuming a discrete notion of capital commitment. All calculation results of these formulas are available as percentage values in decimal form. To specify them as percentages, multiply the results by 100.

2.6.3

Application of the Profitability Calculation

2.6.3.1 Exercises Determine for objects 1 and 2 with their data from Fig. 2.6 and the further assumption that the annual revenues for object 1 are 25,000 euros and for object 2 30,000 euros, calculation results with the static profitability calculation by assuming that basically the improved procedure of data collection is applied and that: Exercise a) Assume the simple notion of capital commitment, Exercise b) Assume the discrete notion of capital commitment, Exercise c) Assume the constant notion of capital commitment, Exercise d) Assume the simple notion of capital commitment and, in contrast to the exercise, assume that the residual values of both objects are zero, Exercise e) Assume the discrete notion of capital commitment and deviate from the exercise by assuming that the residual values of both objects are zero. Evaluate the results for all questions in the form of an investment decision.

2.6.3.2 Solutions Exercise a) First, the relevant formula must be identified. In this case, this is formula 2.82. (2.89)=(2.82) Rent ¼

UB AþR 2

ð2:89Þ

The relevant data must be entered in this formula. In the case of investment object 1, the result is

68

2

RentObject

1

¼

Static Investment Calculation Methods

25, 000  15, 000 12, 000þ2000 2

¼ 142:86%

ð2:90Þ

¼ 80:00%

ð2:91Þ

In the case of investment object 2, the result is RentObject

2

¼

30, 000  20, 000 20, 000þ5000 2

RentObject 1 ð142:86%Þ > RentObject 2 ð80:00%Þ

ð2:92Þ

Thus, object 1 would be preferable to object 2 in this combination. Exercise b) First, the relevant formula must be identified. In this case, this would be formulas 2.83 and 2.84. Since they lead to the same result, 2.84 is not applied here. (2.93)=(2.83) UB Rent ¼ ARþAR n þR 2

ð2:93Þ

The relevant data must be entered into the formula. In the case of investment object 1, the result is RentObject

1

¼

25, 000  15, 000 12, 0002000þ12, 0002000 4 2

þ 2000

¼ 121:21%

ð2:94Þ

¼ 69:57%

ð2:95Þ

In the case of investment object 2, the result is RentObject

2

¼

30, 000  20, 000 20, 0005000þ20, 0005000 4 2

þ 5000

RentObject 1 ð121:21%Þ > RentObject 2 ð69:57%Þ

ð2:96Þ

Thus, object 1 would be preferred to object 2 in this combination. Exercise c) First, the relevant formula must be identified. In this case, this is formula 2.85. (2.97)=(2.85) Rent ¼

UB A

ð2:97Þ

The relevant data must be entered into the formula. In the case of investment object 1, the result is

2.6 The Profitability Calculation

RentObject

69

1

¼

25, 000  15, 000 ¼ 83:33% 12, 000

ð2:98Þ

In the case of investment object 2, this results in RentObject

2

¼

30, 000  20, 000 ¼ 50:00% 20, 000

RentObject 1 ð83:33%Þ > RentObject 2 ð50:00%Þ

ð2:99Þ ð2:100Þ

Thus object 1 would be preferable to object 2 in this combination. Exercise d) First, the relevant formula must be identified. In this case, this is formula 2.86 (2.101)=(2.86) Rent ¼

UB

ð2:101Þ

A 2

The relevant data must be entered into this formula. In the case of investment object 1, the result is RentObject

1

¼

25, 000  15, 000 12, 000 2

¼ 166:67%

ð2:102Þ

¼ 100:00%

ð2:103Þ

In the case of investment object 2, this results in RentObject

2

¼

30, 000  20, 000 20, 000 2

RentObject 1 ð166:67%Þ > RentObject 2 ð100:00%Þ

ð2:104Þ

Thus, object 1 would be preferable to object 2 here. Exercise e) First, the relevant formula must be identified. In this case, these are formulas 2.87 and 2.88. Since they lead to the same result, 2.88 is not applied here. (2.105)=(2.87) Rent ¼

UB AþAn 2

ð2:105Þ

The relevant data must be entered in this formula. In the case of investment object 1, the result is

70

2

RentObject

1

¼

Static Investment Calculation Methods

25, 000  15, 000 12, 000þ12,4000 2

¼ 133:33%

ð2:106Þ

¼ 80:00%

ð2:107Þ

In the case of investment object 2, this results in RentObject

2

¼

30, 000  20, 000 20, 000þ20,4000 2

RentObject 1 ð133:33%Þ > RentObject 2 ð80:00%Þ

ð2:108Þ

Thus, object 1 would be preferred to object 2, here.

2.6.4

Section Results

In this section, you: • Learned how the profitability calculation functions. • Understood the criticism of the profitability calculation. • Got to know the criteria of the profitability calculation. • Developed the formulas of the profitability calculation and. • Applied the profitability calculation to an example.

2.7

The Static Amortisation Calculation

2.7.1

Presentation and Criticism of the Static Amortisation Calculation

The static amortisation calculation also belongs to the static investment calculation procedures, but unlike the other static procedures it does not follow the economic principle, i.e. it does not attempt to make an investment decision in which a maximum output is achieved with a given input or in which a given output is achieved with a minimum input. The other static procedures follow this principle, two of the procedures, namely the profit comparison calculation and the profitability calculation (the profitability calculation, at least on special assumptions), have the goal to maximise output when properly applied and the cost comparison calculation has the goal to minimise input when properly applied. The static amortisation calculation implicitly assumes a safety thinking on the part of the investor and elevates this to a decision maxim, but without the actual techniques of the investment decision in uncertainty, which we will discuss in Chap. 6, and without considering the economic principle.

2.7 The Static Amortisation Calculation

71

The static amortisation calculation determines the number of periods, measured in years, after which the initially invested capital is recovered from the returns of the individual years without taking interest into account. The decision criterion is a comparison of the determined number of periods with a predefined number or an alternative number of periods. This allows the static amortisation calculation for the same questions as the profit comparison calculation and the profitability calculation, i.e. the determination of the • Static amortisation period of a single investment. • Static amortisation times in comparison to alternatives and the. • Static amortisation time in the replacement problem. The following criteria apply for the decision: For the single investment: t single investment  t maximum expectation

ð2:109Þ

t stands for the number of periods in years of the determined static amortisation period. The static payback period of the individual investment must not exceed a subjectively determined maximum specified time. For alternative comparisons: t Object

1

< t > Object

2

ð2:110Þ

In the case of mutually exclusive investment alternatives with the same capital commitment and the same useful life, the investment with the shorter static payback period is the better one, according to this method. Otherwise, of course, both are advantageous, as long as they are below the specified maximum time. For the replacement problem: t Object

new

< t > Object

old

ð2:111Þ

In the case of mutually exclusive investment alternatives with the same capital commitment and the same useful life, the investment with the shorter static payback period is the better one. To this extent, the static amortisation calculation is thus not a decision-making procedure based on the economic principle, but rather a simple analysis of a safety issue, without considering by any chance the theoretical aspects of the investment decision under risk. An approximate liquidity analysis in such a way that it becomes clear when the invested capital is recovered from planned returns is also possible. You can use all the calculation elements used in the static methods, since there are no exact date-related interest calculation requirements. Since the static amortisation calculation also answers questions about liquidity, it is of course useful here to

72

2 Object A

Object B

NEk

A 10

A 10

10 0

1

Static Investment Calculation Methods

2

3

4

n

0

1

2

NEk

NEk

10

10

3

4

n

Fig. 2.7 Example of the static amortisation calculation (Source: Author)

process payments instead of other profit and loss figures. Thus, in the formulas in this chapter, only payments should be used for the calculations. The static amortisation calculation implicitly assumes that the fast recovery of the invested money is advantageous, since the further future is associated with higher risks than the near future. It is assumed that the remaining time of the remaining useful life after recovery of the invested capital can be used to generate the desired return on the capital invested and then additional surpluses. Long-term investments are discriminated in this way. Thus, out of two investment alternatives that are assessed with the premise of a fixed maximum payback period of 2 years, object A in Fig. 2.7 is the only advantageous one. It is obvious that object A must be an unattractive investment if an interest rate exists, because an equally high recovery of the invested capital after a period of time, here of 2 years, can only be a loss due to the lack of interest. Investment object B, however, would fulfil this criterion. It is, therefore, the better investment alternative, but is classified as unfavourable by the static amortisation calculation at the specified maximum amortisation time. Like that, the static amortisation calculation can lead to wrong economic decisions and should, therefore, never be used as a separate investment calculation method, but only as a small, non-binding additional calculation that gives a rough overview of the time of the return of capital. The static amortisation calculation assumes a different amortisation pattern for fixed capital than the other static methods. Whereas the profitability calculation looks at the development of the average fixed capital, the development of the invested capital with regular reduction through depreciation, the static amortisation calculation assumes that the net payments, i.e. higher amounts than the debt service in the case of worthwhile investments, are used for repayment. Unfortunately, despite this flawed procedure, the method has been quite popular in practice, at least in the past, and is also used as an independent procedure, as analyses by Däumler show (Däumler and Grabe 2007, p. 211). The static amortisation time can be calculated in two different ways in the static amortisation calculation, with the cumulative calculation and with the average calculation. In the accumulation calculation, the net payments or possible other annual performance figures are added up until they reach the amount of the initial payment.

2.7 The Static Amortisation Calculation

73

After the elapsed time, measured in years, after which this is the case, the static payback period, measured in this number of years, is reached. In the average calculation, the initial payment is divided by the average net payments or an equivalent profit figure. The resulting value is the static payback period, measured in years. This method is of course much less accurate. Therefore the problems of the static amortisation calculation lie in particular in the fact that • • • •

The decision criterion does not follow the economic principle. The risk is not considered in the data set, but only in a very abstract way. The useful life of the investment object is not taken into account. The acquisition payment of the investment object is not taken into account in comparisons and. • The returns after the payback period are not taken into account. The latter may result in • Investments which pay off statically before the maximum defined payback period are unfavourable overall if they do not earn at least the interest on the capital invested during the remaining term. • Investments with the same static payback period have different advantages depending on the returns that occur after the static payback period. • Investments that do not pay off in the maximum static payback period are overall worthwhile investments because they still provide substantial returns after the maximum tolerated static payback period.

2.7.2

Formulas of the Static Amortisation Calculation

The formula for determining the static amortisation time for the accumulation calculation is t m ðyearsÞ for A ¼

m X

ðet  at Þ

ð2:112Þ

t

The static amortisation time t is reached when the undiscounted sum of the net payments, net earnings NEK (et  at), starting from time 1 has reached the amount of the acquisition payment A. This is the case after m years. If the amount of the acquisition payment is between two values of the total net payments, an interpolation would be possible to determine the point in time when both values are exactly the same. However, this does not seem to make sense to the author for several reasons: • On the one hand, this presupposes an even (linear) progression of the net payments over the relevant years. In practice, however, it is rather unlikely that the payments will be distributed over the year in this way.

74

2

Static Investment Calculation Methods

• On the other hand, this ignores the actually common assumption of arrears in investment payments, which is rather irrelevant here due to the lack of an appropriate interest rate in the static investment calculation methods, but which is assumed to be dynamic, as we will see in Chap. 3. If payments are actually only made at the end of the period, a static payback period cannot be reached during the year. Actually, the year when the criterion is met for the first time should be declared the static amortisation time. • Furthermore, an interpolation creates an apparent accuracy that is not appropriate for the imprecise method. Besides the presented Eq. (2.112), another equation for the determination of the static amortisation time according to the accumulation calculation is often found in the literature for the case of an existing residual value, but the author rejects it. It is described in Eq. (2.113). t m ðyearsÞ for A  R ¼

m X

ðet  at Þ

ð2:113Þ

t

In this approach, it is assumed that it is not the acquisition payment that has to be recovered, but the actual capital invested without taking interest into account due to the different timing of the payments, i.e. the difference between the acquisition payment and the residual value. The idea so far is a good one, as the investment is statically amortised when the capital actually used is recovered from the returns. However, this approach is worthless under the implicit question of the static amortisation calculation, the calculus of risk. The acquisition payment is due at the beginning of the investment and is paid by the investor. In the future, the net cash inflow of funds should then compensate for the outflow. Under the calculus of risk, the number of periods is measured until the outflow of money is repaid. And in the time further in the future, the investment object will still have a residual value if the data occurs as planned. Thus, an inflow of money, the residual value, whose time of inflow is after the time of risk equalisation, i.e. the time at which inflow and outflow are equal in amount, is included in the calculation. This should be omitted if there is a risk. This also applies to the following average calculation. In addition to the use of payment parameters, as described in this section, all the profit and loss variables commonly used in static investment calculation methods can of course also be used for the calculation both according to the accumulative calculation and the average calculation. The average calculation is carried out according to the following formula 2.114. t m ðyearsÞ ¼

A ð e  aÞ

ð2:114Þ

The prerequisite for this is that the net payments (e  a) for the investment object are the same in each period, or if this is not the case, an average is formed for the

2.7 The Static Amortisation Calculation

75

calculation. Here, too, the other performance indicators commonly used in static investment calculation methods can of course be used. For the replacement problem, a further calculation formula is often found in the literature for the average calculation: t m ðyearsÞ ¼

Anew ðaold  anew Þ

ð2:115Þ

Here it is, therefore, formulated that if, through cost savings (aold  anew) the acquisition payment of the new object (Anew) is recovered in a reasonable number of periods tm, a replacement is worthwhile. However, this formula says nothing about a direct comparison of the static amortisation times of the new and old object. The author does not consider the following formula 2.116 to be appropriate for the same reasons as those just argued for the accumulative calculation: t m ðyearsÞ ¼

2.7.3

AR ð e  aÞ

ð2:116Þ

Application of the Static Amortisation Calculation

2.7.3.1 Exercises In addition to their data from Fig. 2.6 for objects 1 and 2, the annual revenues for object 1 are 25,000 euros and for object 2 30,000 euros. Calculate the results using the static amortisation calculation by assuming that Exercise a) the average calculation is applied, Exercise b) the accumulative calculation is applied, Exercise c) the accumulative calculation is applied and, contrary to the author’s opinion, the residual value from the exercise is included in the calculation. Evaluate the results for all questions in the form of an investment decision.

2.7.3.2 Solutions Exercise a) First, the relevant formula must be identified. In this case, this is formula 2.114 (2.117)=(2.114) t m ðyearsÞ ¼

A ð e  aÞ

ð2:117Þ

The relevant data must be entered into this formula, as in the formulas 2.118 and 2.119.

76

2

years

in-payments euro/year 0 1 2 3 4

25000 25000 25000 25000

pay-out euro/year

Static Investment Calculation Methods

net payments euro/year

18000 15000 14000 13000

accumulated returns euro 7000 10000 11000 12000

balance euro -12000 7000 -5000 17000 5000 28000 16000 40000 28000

Fig. 2.8 Static amortisation time after the accumulation calculation for object 1* (Source: Author)

t m ðyearsÞ, Object 1 ¼

12, 000 ¼ 1:2 years ð25, 000  15, 000Þ

ð2:118Þ

20, 000 ¼ 2 years ð30, 000  20, 000Þ

ð2:119Þ

t m ðyearsÞ, Object 2 ¼

In this situation, if no amortisation during the year is allowed for object 1 because of the assumption of payments in arrears, the static payback period is 2 years, for object 2 it is also 2 years. Therefore, object 1 would be preferred to object 2 in this situation, since object 1 has a shorter amortisation period, even if only arithmetically, which coincides with the static amortisation period of object 2 due to the assumption of payments in arrears. A further prerequisite is that this was an alternative selection that fulfilled the usual requirements and that the calculated static amortisation time is less than the maximum useful life specified but not mentioned in this exercise. Exercise b) First, the relevant formula must be determined. In this case, this is formula 2.112. (2.120)=(2.112) t m ðyearsÞ for A ¼

m X

ðet  at Þ

ð2:120Þ

t

The relevant data must be entered in this formula. This is visible in Fig. 2.8 for object 1 and Fig. 2.9 for object 2. For reasons of clarity, a solution for the cumulative calculation is always in tabular form. For object 1, this situation results in a static payback period of 2 years, for object 2 it is also 2 years. Therefore, object 1 would be preferred to object 2 here, since object 1 has the shorter amortisation period, even if only arithmetically, which coincides with the static amortisation period of object 2 due to the assumption of the payment in arrears. A further prerequisite is that this was an alternative selection that meets the usual requirements and that the calculated static amortisation time is less than the specified maximum useful life, which is not mentioned in this exercise.

2.7 The Static Amortisation Calculation years

in-payments euro/year 0 1 2 3 4

30000 30000 30000 30000

pay-out euro/year

77 net payments euro/year

16000 20000 22000 22000

accumulated returns euro

balance euro -20000 14000 -6000 24000 4000 32000 12000 40000 20000

14000 10000 8000 8000

Fig. 2.9 Static amortisation time after the accumulation calculation for object 2* (Source: Author)

years

in-payments euro/year 0 1 2 3 4

25000 25000 25000 25000

pay-out euro/year

net payments euro/year

accumulated returns euro

18000 15000 14000 13000

balance euro -10000 7000 -3000 17000 7000 28000 18000 40000 30000

7000 10000 11000 12000

Fig. 2.10 Static amortisation time after the accumulation calculation for object 1* (Source: Author)

years

in-payments euro/year 0 1 2 3 4

30000 30000 30000 30000

pay-out euro/year

net payments euro/year

16000 20000 22000 22000

accumulated returns euro 14000 10000 8000 8000

balance euro -15000 14000 -1000 24000 9000 32000 17000 40000 25000

Fig. 2.11 Static amortisation time after the accumulation calculation for object 2* (Source: Author)

Exercise c) First, the relevant formula must be identified. In this case, this is formula 2.121 (2.121)=(2.113) t m ðyearsÞ for A  R ¼

m X

ðet  at Þ

ð2:121Þ

t

The relevant data must be entered into this formula, as shown in Fig. 2.10 for object 1 and Fig. 2.11 for object 2. For reasons of clarity, a solution for the accumulative calculation is always in tabular form. For object 1, this situation results in a static payback period of 2 years, for object 2 it is also 2 years.

78

2

Static Investment Calculation Methods

In this combination, object 2 would, therefore, be preferable to object 1, since object 2 has the shorter payback period, even if only arithmetically, which coincides with the static payback period of object 1 due to the assumption of the payment in arrears. A further prerequisite is that this was an alternative selection that meets the usual requirements and that the calculated static amortisation time is less than the specified maximum useful life, which is not mentioned in this exercise.

2.7.4

Section Results

In this section, you have: • • • • • •

Learned how the static amortisation calculation works. Understood the criticism of the static amortisation calculation. Got to know the criteria of static amortisation calculation. Learned the difference between the average and accumulative calculation. Developed the formulas of the static amortisation calculation and. Applied the static amortisation calculation to an example.

2.8

Case Study

In addition to your current job, you would like to increase your income by working part-time. You are planning a restaurant for students on the eastern bank of Kiel. Your calculation interest rate is 10%. Two concepts have been shortlisted, which are characterised by the calculation elements in Fig. 2.12:

2.8.1

Exercises

Exercise a) Determine the costs of both investment objects 1 and 2 according to the cost comparison calculation in the alternative comparison. Start from the simplest notion of capital commitment. Based on the results of the calculation, make an investment decision and document it. Calculation element n (years)

Investment object 1

Investment object 2

Old object

4

4

4

A (T€)

150

120

80

ek (T€)

100

80

50

ak (T€)

25

15

10

R (T€)

30

20

15

Fig. 2.12 Calculation elements for the case study of statics (Source: Author)

2.8 Case Study

79

Exercise b) Comment on the suitability of the cost comparison calculation for the investment decision in this case. Exercise c) Determine the profit of investment object 1 using the profit comparison calculation for the replacement problem. Assume that the old object in the table in Fig. 2.12 has already been in operation for 2 years. Continue to assume a discrete capital commitment. Exercise d) Determine the profitability according to the profitability calculation of investment object 1. Assume that • The simplest notion of capital commitment applies for the determination of the first profitability. • For the determination of the second profitability, the discrete notion of capital commitment applies. • For the determination of the third profitability, it applies that the fixed capital commitment in the investment object does not change over the term.

Exercise e) Determine the static amortisation period of investment object 1 according to the accumulation calculation and the static payback period of object 2 according to the average calculation.

2.8.2

Solution

Exercise a) First, the relevant formula must be identified. In this case, this is Eq. (2.6). (2.122)=(2.6) B1 þ

A1  R1 A1 þ R1 < A  R2 A2 þ R2 þ þ
i B2 þ 2
i > n1 2 n2 2

ð2:122Þ

The relevant data are to be entered into this formula, as can be seen in Eq. (2.123). 150, 000  30, 000 150, 000 þ 30, 000 < þ
0:1 15, 000 4 2 > 120, 000  20, 000 120, 000 þ 20, 000 þ þ
0:1 4 2

25, 000 þ

ð2:123Þ

In this situation, the costs for object 1 would then amount to 64,000 euros, and for object 2 to 47,000 euros. K Object 1 ð64, 000 eurosÞ > K Object 2 ð47, 000 eurosÞ Thus object 2 would be preferable to object 1 here.

ð2:124Þ

80

2

Static Investment Calculation Methods

Exercise b) First of all, the general criticism of the static procedures should be mentioned here: • The useful life of an investment object is not fully taken into account; only individual years are considered. • The time difference of the payment incidence and the associated interest on payments are not or only incompletely considered. • Interdependencies with other investment objects or with other years of the useful life of the considered objects are ignored. • Constant capacity utilisation is assumed over the years of the useful life. • Profits and costs are assumed to remain constant over the years of the useful life. • Data security is assumed. In addition, the following applies to the cost comparison calculation • The cost comparison calculation only makes sense if the sales sides of the alternatives are comparable. • The cost comparison calculation only makes sense if the output quantities and qualities of the alternatives are comparable. • The cost comparison calculation only makes sense if the acquisition payment of the alternatives are comparable. • The cost comparison calculation only makes sense if the useful lives of the alternative are comparable. In general, as in this case, comparative cost accounting is, therefore, less suitable for a qualified investment decision. Exercise c) First, the relevant formula must be identified. In this case, these are formulas 2.54 and 2.55. The formulas represent the discrete notion of capital commitment concept, only with different formulas that lead to the same results for a uniform capital reduction. Therefore, only formula 2.54 is used here for the discrete notion of capital commitment. (2.125)=(2.54) A  Rnew U new  Bnew  new  nnew
i

< U  Bold > old

new Anew  Rnew þ AnewnR new þ Rnew 2

!

ð2:125Þ

The relevant data must be entered into this formula, as shown in the formula 2.126.

2.8 Case Study

81

100, 000  25, 000  

150, 000  30, 000 4

000 150, 000  30, 000 þ 150, 00030, 4 þ 30, 000 2

!
0:1

< 50, 000 >

 10, 000

ð2:126Þ

In this situation, the new object would then generate profits of 34,500 euros, while the old object would generate 40,000 euros. GObject

new ð34, 500

eurosÞ < GObject

old ð40, 000

eurosÞ

ð2:127Þ

Thus, the old object in this combination would be preferable to the new object. There would be no immediate replacement. Exercise d) First identify the relevant formulas, and then insert the data. This is done one after the other according to the indents in the exercise. The formula 2.82 applies to the first bullet point. (2.128)=(2.82) Rent ¼

UB

ð2:128Þ

AþR 2

The relevant data must be entered in the formula. In the case of investment object 1, the result is RentObject

1

¼

100, 000  25, 000 150, 000þ30, 000 2

¼ 83:33%

ð2:129Þ

Object 1 thus has a profitability of 83.33%. If this is above the desired minimum profitability, which was not specified in the exercise, object 1 is worthwhile with this notion of capital commitment. The formula 2.83 applies to the bullet point. (2.130)=(2.83) UB Rent ¼ ARþAR n þR 2

ð2:130Þ

The relevant data must be entered into this formula. In the case of investment object 1, the result is RentObject

1

¼

100, 000  25, 000 000 150, 00030, 000þ150, 00030, 4 2

þ 30, 000

¼ 71:43%

ð2:131Þ

82

2

years

in-payments euro/year 0 1 2 3 4

pay-out euro/year

100000 100000 100000 100000

Static Investment Calculation Methods

net payments euro/year

25000 25000 25000 25000

accumulated returns euro 75000 75000 75000 75000

balance euro -150000 75000 -75000 150000 0 225000 75000 300000 150000

Fig. 2.13 Static amortisation time after the accumulation calculation for object 1* (Source: Author)

Object 1, therefore, has a profitability of 71.43%. If this is above the desired minimum profitability, which was not specified in the exercise, object 1 is worthwhile with this notion of capital commitment. For the third bullet point, the formula 2.85 applies. (2.132)=(2.85) Rent ¼

UB A

ð2:132Þ

The relevant data must be entered into this formula. In the case of investment object 1, the result is RentObject

1

¼

100, 000  25, 000 ¼ 50:00% 150, 000

ð2:133Þ

Thus, object 1 has a profitability of 50.00%. If this is above the desired minimum profitability, which was not specified in the exercise, object 1 is worthwhile with this notion of capital commitment. Exercise e) Solution to the first bullet point: First of all the right formula has to be found. In this case, this is formula 2.112. (2.134)=(2.112) t m ðyearsÞ for A ¼

m X

ðet  at Þ

ð2:134Þ

t

The relevant data must be entered into this formula. This is shown in Fig. 2.13 for object 1. For reasons of clarity, a solution for the accumulation calculation is always in tabular form. For object 1, this situation results in a static amortisation period of 2 years. Thus, object 1 would be a sensible investment if the calculated static amortisation period is less than the specified maximum useful life, which is not defined in this exercise. Solution to the second bullet point: First, the relevant formula must be found. In this case, this is formula 2.114. (2.135)=(2.114)

2.9 Summary

83

t m ðyearsÞ ¼

A ð e  aÞ

ð2:135Þ

In this formula, the data are to be inserted as seen in Eq. (2.136). 1:85 years ¼

120 ð80  15Þ

ð2:136Þ

In this situation, the average calculation of the static amortisation period for object 2 is 2 years. The result for this data record would coincide with that of the accumulation calculation, since the net payments in the data record are the same for each period, i.e. they are already average. If the maximum static amortisation time, not specified here, is at least 2 years, investment object 2 would be worthwhile according to this method.

2.9

Summary

In this chapter, the static investment calculation methods were presented. The aim was to make the reader aware of the criticism that can be levelled at the static methods and of the risks involved in their application in terms of transferring the results into practice as an investment decision. The general assumptions of the static investment calculation methods were presented, as well as the four best-known static investment calculation methods, their criteria, their formulas, their risks in detail and their application to practical issues. Also, their quick applicability due to the easy data collection and calculation technique was presented. In detail, the following sub-goals have been achieved: • The working method of statics in general and the assumptions of this investment calculation method group were learned. • The criticism of the static procedures was elaborated extensively. • The risks of transferring the calculation results of the static investment calculation procedures as an investment decision into practice were very clearly disclosed and documented. • The reader was put in the position to assemble the static formulas from a set of relevant calculation elements according to the problem and the relevant static investment calculation method. • The cost comparison calculation as a static investment calculation method was presented in detail, defined and criticised as a single method, the possible calculation formulas were presented and applied to practical cases. • The profit comparison calculation as a static investment calculation method was presented in detail, defined and criticised as a single method, the possible calculation formulas were presented and applied to practical cases. • The profitability calculation as a static investment calculation method was presented in detail, defined and criticised as a single method, the possible calculation formulas were presented and applied to practical cases.

84

2

Static Investment Calculation Methods

• The static amortisation calculation as a static investment calculation method was presented in detail, defined and criticised as a single method, the possible calculation formulas were presented and applied to practical cases. • All procedures were applied to a practical problem in a case study. After reading the chapter, the reader is able to define what static investment calculation methods are, what value they have for practical application, what criticism and what dangers there are in transferring the calculation results into practice and how the methods work in detail. The reader should be able to set up the corresponding formulas independently after reading the chapter. The reader should be aware that the static methods are investment calculation methods that are still very common in practice, especially in small companies or investment calculation procedures with low capital commitment. The methods are particularly suitable for small and quick calculations, as the relevant data are relatively easy to obtain and the computing effort required to generate the results is quite low. However, the reader must be aware that a transfer of the calculation results into practice as an investment decision is very dangerous in itself, as the calculations work with very unrealistic simplifying assumptions, e.g. the lack of proper consideration of the interest claim of the used capital. Therefore, the static investment calculation methods should never be used to decide on major investment projects or those with a relatively high capital investment.

Reference Däumler, K-D., & Grabe, J. (2007). Grundlagen der Investitions- und Wirtschaftlichkeitsrechnung (12. Aufl.). Herne/Berlin: nwb Verlag Neue Wirtschafts-Briefe.

3

Dynamic Investment Calculation Methods

3.1

Objectives

After you have already become acquainted with the basics and thus the terminology of investment calculation and the static investment calculation methods in the previous two chapters, the aim of this section is to present the dynamic investment calculation methods and their application. As a result of this chapter you should be able to • • • • • • •

Know the five dynamic investment calculation methods. Know the assumptions of dynamic investment calculation methods. Know the mathematical way to determine the methods. Know the decision criteria of the methods. Apply the dynamic investment calculation methods to practical cases. Interpret the calculation results of the dynamic investment calculation methods. Apply the dynamic investment calculation methods appropriately to your practical operational investment problems and. • Use the results of the dynamic investment calculation methods appropriately for your practical operational decision problems. In order to achieve these goals, it is necessary to follow the offered exercise calculations independently with the pocket calculator or spreadsheet. Enjoy your work!

Supplementary Information The online version of this chapter (https://doi.org/10.1007/978-3030-62440-8_3) contains supplementary material, which is available to authorized users. # The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. Poggensee, J. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-62440-8_3

85

86

3.2

3 Dynamic Investment Calculation Methods

Model Assumptions of Dynamic Investment Calculation Methods

The differentiation of the dynamic investment calculation methods from the other investment calculation method groups is already familiar with the first chapter of this book. Because of its significance for the characterisation of the dynamic investment calculation methods, this will be briefly repeated at this point. A characteristic of the dynamic procedures is that the different timing of payments is taken into account for the evaluation of investment projects and these payments are valued with a fixed interest rate. For example all dynamic procedures stand out positively from static procedures, since, in contrast to the approach of static procedures, in the case of dynamics all cash-effective activities of the future are worth less in absolute amounts in the present than activities that trigger the same payment effects in the present, since individual discounting is carried out. In a dynamic view, receiving 100 euros today is, therefore, more valuable than receiving 100 euros in, say, 5 years. In a static view, both activities would usually be equally valuable. To this extent, the dynamic view due to discounting is much closer to reality than the static view. The fact that there is a fixed interest rate in the dynamic procedures for the evaluation of all activities is also positive compared to the static procedures, since in the static procedures an interest rate is not used for all activities. Compared to the currently most developed group of scientific investment calculation methods, the simultaneous models of the capital budget, which are covered in Chap. 4, there is also a weakness of the dynamic methods with regard to the transferability of their calculation results to reality. In reality, different activities have different interest claims. Debt capital, for example usually carries a different interest rate than equity capital, and within these groups, there are different interest claims depending on maturities, securities and creditor structure. While the simultaneous models of the capital budget can realistically depict this by means of payment-specific interest rates, the use of only one fixed interest rate in the dynamic investment calculation procedures creates a clear discrepancy in today’s operational reality. Whether this elaborate planning technique of simultaneous models of the capital budget always makes sense for operational practice has to be decided individually, as the planning process is very complex but the data basis is very uncertain in many companies. Thus, dynamic investment calculation methods are probably still the most frequently used investment calculation methods in practice, measured by the number of companies using investment calculation.

3.2.1

Objective of the Dynamic Investment Calculation Methods

The five existing dynamic investment calculation methods are: • Net present value method

3.2 Model Assumptions of Dynamic Investment Calculation Methods

• • • •

87

Horizon value method Annuity method Internal rate of return method and Dynamic amortisation calculation

These are methods that all evaluate the absolute advantage of an investment accounting object using a singular criterion. A single criterion means that only one cash relevant, i.e. cash-based criterion (a monetary amount), or an interest rate, or a point in time is considered, at which the invested capital has flowed back, taking interest into account. Multidimensional target functions, for example securing of market shares or stakeholder contacts, such as long-term supplier relationships, customer loyalty or incentive programmes for employees, which do not have any direct cash relevant consequences in the period under review, are not considered. For multidimensional target functions, for example the utility value analysis or linear optimisation with multidimensional target function must be used. The techniques are also presented in Chap. 4. The pure consideration of the absolute advantageousness also deserves special attention. Due to the mathematical construct of the formulas of the dynamic investment calculation methods and the underlying assumptions, it is only possible to assess whether a planned investment object is worthwhile in reality when all planning data is available. Whether the profitable investment object would be preferable to another profitable investment object (relative advantageousness) can no longer be decided in a qualified manner on the basis of the dynamic investment calculation methods. The absolute level of the target criterion is of very little importance. A selection from several worthwhile investment objects is methodically impossible. With regard to these phenomena, dynamic procedures are often applied incorrectly in practice. We will take a closer look at this in Chap. 4. In order to achieve the goal of the dynamic investment calculation methods of evaluating the absolute advantageousness of an investment calculation object on the basis of a singular criterion, a considerable abstraction from reality is necessary so that every conceivable investment object can be expressed in a single key figure. The necessary assumptions are presented in the following section.

3.2.2

Assumptions of the Dynamic Investment Calculation Methods

In order to depict complex reality in a simplified image of reality, i.e. in a model, at least six assumptions must be made for the functionality of the dynamic investment calculation methods: • All calculation elements are known with certainty. • All calculation elements are incurred in arrears, i.e. at the end of a period.

88

• • • •

3 Dynamic Investment Calculation Methods

Payments can be deferred over time. There is only one interest rate. Only payments are considered as calculation elements and. The assumption of profit maximisation and polypoly.

3.2.2.1 The Safety Assumption All calculation elements are known with certainty. Through this assumption, dynamic investment calculation reflects a lack of objectivity, since calculation elements are often estimated with different probabilities of occurrence, i.e. with different standard deviations. Deterministic calculation results are thus subject to different levels of objectivity. Evaluation The addition of decision-theoretical models could remove this limitation by carrying out risk analyses and modifying the target criteria, but this goes well beyond the approach of dynamic investment calculation methods. This will be discussed in Chap. 6.

3.2.2.2 The Assumption of Payments in Arrears All calculation elements are incurred in arrears. The calculation results of the dynamic investment calculation methods are thus based on incorrect payment dates. In general, payments in a capital investment object generally accrue continuously in practice, customers pay several times within the period of a year, and wages and purchases of equipment are paid several times a year. The assumption made shifts all payments of a 1-year-period to a year-end-date. The advantage of this assumption is that in the dynamic investment calculation formulas you can work with sum signs, otherwise you would have to calculate with integrals or additionally with interest rates during the year, which is much more complex. This assumption leads to the fact that the interest rates for payments during the year are generally ignored. Depending on the time and amount of the payment, this has a different distorting effect on the calculation result of the dynamic investment calculation methods. Evaluation This could be remedied relatively easily by introducing interest factors during the year or by integral calculation, but is not done in classical dynamic investment calculation.

3.2.2.3 The Assumption of a Payment Deferral Payments can be deferred over time. This assumption differs significantly from most cases of operational reality, in which payment dates for payments to employees (wages), suppliers (advance payments), creditors (interest and redemption payments) must be adhered to in the same way as customers should adhere to payment dates. Dynamic capital budgeting thus completely ignores liquidity planning. An investment that is worthwhile according to a dynamic investment calculation, therefore, says nothing about the liquidity status of the project, which can certainly lead to insolvency. This assumption is necessary in order to be able to express

3.2 Model Assumptions of Dynamic Investment Calculation Methods

89

complex investment projects in a key figure, otherwise no present value could be determined for a series of payments over several years, for example. Evaluation This discrepancy with reality can be healed if, in addition to investment planning, complete financial plans are drawn up which can monitor liquidity.

3.2.2.4 The Interest Assumption There is only one interest rate. This assumption is divided into the fundraising premise: • Unlimited amounts can be raised for an unlimited time at a constant interest rate. and the reinvestment premise: • Unlimited amounts can be invested for an unlimited period of time at a constant and equal interest rate. This assumption also contains a clear discrepancy in reality. It was made in order to assess the value of an investment and not the influence of the financing. Without this assumption, the identical data set of an investment project would possibly show a positive target value of a dynamic calculation method in a case of a relatively low-interest financing option, whereas a rejection of the project would be recommended for a financing with higher interest rates. Evaluation This assumption was made in order to assess the investment object itself and not its form of financing. However, in reality this means that the actual financing costs must always be taken into account. One possibility for this is to use complete financial plans.

3.2.2.5 The Calculation Element Assumption Only payments are considered as calculation elements. Since the results of the dynamic investment calculation procedures are one-dimensional cash relevant criteria, costs that do not lead to payments during the period under review and corresponding benefit aspects are not taken into account. If, for example the marginal social costs of society differ from the marginal private costs of the company, which is the case, for example with legal consumption of environmental resources (emissions) that is free of charge or if society uses services provided by companies without paying the full market price (e.g. apprenticeship), these aspects are not taken into account in the dynamic investment calculation method. Evaluation Consideration would be possible, e.g. via a cost-benefit analysis, but this would necessarily require a multidimensional target function.

3.2.2.6 The Market Assumption The assumption of profit maximisation and polypoly.

90

3 Dynamic Investment Calculation Methods product price (p)

supply function S

0

supply function S P

1

demand function D

0

q0

product quantity (q)

q1

Fig. 3.1 Growth investments in polypoly markets (Source: Author)

product price (p)

supply function S

0

supply function S P

1

0

P1

demand function D q0 q1

product quantity (q)

Fig. 3.2 Growth investments in non-polypoly markets (Source: Author)

Dynamic investment calculation methods only deliver meaningful results if both criteria, i.e. the goal of maximising profit and the market form of the polypoly, are fulfilled for the company making the investment calculation. It is probably easy to understand that investments that are not worthwhile in the period under consideration can be realised after all if the goal of the company is not profit maximisation but, for example increase of market share at any price. Figures 3.1 and 3.2 show how growth investments must be assessed in relation to the market form. One of the characteristics of a polypoly market, in which a large number of customers and a large number of suppliers are active so that no market participant can dictate the market conditions, is a completely elastic demand function. This means that customers in this market form purchase any quantity of the product at a constant price. This market form is the classical starting point of economic microeconomic theory when considering markets. In the initial situation in Fig. 3.1 the supply function S0 may be present. In a balanced situation, the supply quantity q0 and the price P0 are then set. The turnover rectangle (q0/P0) thus formed is divided into a cost portion of production, the area the part of this rectangle under the supply function, and a profit portion of production,

3.2 Model Assumptions of Dynamic Investment Calculation Methods

91

the area of the small triangle above the supply function up to the demand function at point P0. A growth investment is now characterised by the shift of the supply function to the right, from S0 to S1 as shown in Fig. 3.1. The quantity sold increases to q1, the revenue rectangle increases accordingly (q1/P0) and the profit triangle in Fig. 3.1 also grows. It now extends along line P0 up to the perpendicular to the sales quantity q1. The automatic result in a polypoly market, where demand and thus the market price does not change with changes in supply, is that growth investments are worthwhile if the initial investment was worthwhile. In non-polypoly markets, growth in sales volume is unfortunately not automatically that positive. According to von Stackelberg, one characteristic of the eight non-polypoly market forms is a decreasing demand function D. This means that if there is a higher supply on the market due to an expansion of the production volume through a growth investment from S0 to S1, the market no longer can be emptied at the old price P0, but all goods only achieve the lower price P1. This is visible in Fig. 3.2. An automatic profit growth of an expansion of an investment that was worthwhile in the old state is, therefore, not given. If the capacity of a growth investment is increased from S0 to S1, a loss of revenue due to a price reduction from P0 to P1 in the old investment must also be accepted. This case is usually not taken into account by the dynamic investment calculation, which urgently needs to be taken into account when determining the planning data, otherwise the dynamic investment calculation methods deliver far too positive results. Overall, the profit after growth investment can even be smaller than in the initial situation. This depends on the elasticities of the supply and demand functions. In Fig. 3.2 the triangle at point P0 shows the profit situation in the initial situation. The triangle at point P1 shows the profit after the growth investment. In this illustration, both triangles are approximately the same size. Since more corporate capital is tied up in the activity after a growth investment, profitability has thus fallen significantly in the constellation shown in Fig. 3.2. In this respect, these assumptions, which must necessarily be made for the application of dynamic investment calculation methods, appear to be more likely to discourage the user from using these methods because they are unrealistic. On the other hand, this is the only possible way to reduce a complex investment project mathematically flawlessly to a single key figure and, by reducing its complexity, make it a business problem ready for decision. This procedure alone already increases the decision-making quality of an entrepreneurial decision, because the documentation of the planning data and the mathematically determined procedure for the calculation of the dynamic investment calculation values makes the procedure intersubjectively comprehensible and thus fulfils the criterion of Critical Rationalism for a scientific procedure as demanded by Sir K. R. Popper and widely used in economics: That way, another person thus concludes at the same result when using the identical data and the identical procedure.

92

3 Dynamic Investment Calculation Methods

Evaluation So far these assumptions are suitable to lead to a structured decision in a business situation.

3.2.3

Calculation Elements of Dynamic Investment Calculation Methods

There are three types of calculation elements in the dynamic investment calculation model: • The amount of the payments • The length of the useful life • The level of the interest rate This data set is compiled before the actual application of the dynamic investment calculation methods and requires the largest part of the working time of the investment calculator, the collection of planning data as close to reality as possible. This has already been discussed in Chap. 1. In this chapter, the data is considered as already collected and we concentrate on the application of the dynamic investment calculation methods. Apart from the amount of the payments, mathematically speaking, as already mentioned above, only their timing is of importance. However, we divide the area of payments into four sub-aspects in order to better assess the significance of our planning data from a business perspective. The four aspects of payment size and the labelling of variables, which we use in this book, are • • • •

A: the amount of the initial net investment or acquisition payment R: the amount of the residual value ek: the amount of ongoing in-payments in period k ak: the amount of ongoing payouts or disbursements in period k The labelling of the variables of the remaining calculation elements are

• n: the length of the period of use • i: the level of the interest rate The useful life runs from periods k ¼ 1 to k ¼ n. n is, therefore, the last year, the end of the planned useful life. Five of these six calculation elements are objectively known in the sense of the model, i.e. according to the assumption made, with certainty; the calculation interest rate represents a subjective demand of the investor, i.e. the subjective minimum interest demand of the investor. This means that different investment calculators can come to different investment decisions for an identical data set, since they have used different calculation interest rates.

3.3 Fundamentals of Financial Mathematics

3.3

93

Fundamentals of Financial Mathematics

Financial mathematics operationalise the assumption that payments can be shifted over time, and can thus shift calculation elements on the timeline within the duration of the investment, taking into account interest and compound interest. The example in Fig. 3.3 illustrates that this represents a considerable reduction in the amount of work compared with allocating the value development of a payment over time to an account. But first a note on the presentation of the payments on the timeline: Due to the assumption of payments in arrears, the payments are at the end of the period, and would have to be represented graphically as a line on the timeline at the end of the period. For better comprehensibility, the payments are displayed as bars that extend into the past year. Instead of following the value development of the initial capital of 10 euros at an interest rate of 10% over 3 years, taking into account interest and compound interest over 3 years, as shown in Fig. 3.3 on the timeline, multiplying the initial capital by the interest rate (1 + i) exponentiated by the term in years results in the same arithmetical value in a much faster way, as can be seen beneath the timeline. This is the importance of financial mathematical factors. In general, the financial mathematical factors in dynamic investment calculation shift payments along the timeline. If this is done on the timeline to the right, we speak of accumulating interest, compound interest or adding unaccrued interest, if this is done on the timeline to the left, we speak of discounting.

Fig. 3.3 Capital development with interest and compound interest (Source: Author)

0.01 0.1 0.1 1 0.1

0.1 1

1

10

0 10 * (1 + i) 10 10

1

1

1

10

10

10

1

2 * (1 + i)

* (1 + i) n 11 12.1

3 * (1 + i)

13.31

n

94

3 Dynamic Investment Calculation Methods

Accumulating interest is, therefore, a postponement of a payment on the timeline from the further past into the near past, from the past into the present, from the past into the future, from the near future into the further future or, from the present into the future which is the rule for investment calculation. Discounting is thus a shift of a payment on the timeline from the nearer past to the further past, from the present to the past, from the future to the past, from the further future to the nearer future or, from the future to the present, which is the rule for investment calculation. In this respect, financial mathematics is a single mathematical operation, since discounting represents a strict reciprocal of accumulating. In this respect, the six known factors of financial mathematics can be developed directly from each other. When applying financial mathematics outside of IT, however, it makes sense to know all six factors, as this considerably shortens the calculation time for determining the individual dynamics. Therefore, the individual factors of financial mathematics will be presented in the following chapters. They will be combined into three groups, each of which will be explained in a further section. For the calculations in this chapter, we recommend, as far as you will work with a pocket calculator, to use the tables of financial mathematics in the appendix of this book or to have a financial mathematics table at hand, e.g. the financial mathematics table by Däumler (1998).

3.3.1

The One-Time Factors

First there is the group of one-time factors, they move a single payment on the timeline: • The accumulation factor (Auf), which shifts a one-time payment on the timeline to the right. Auf ¼ ð1 þ iÞn

and

ð3:1Þ

• The discount factor (Abf), which shifts a one-time payment to the left on the timeline. Abf ¼ ð1 þ iÞn :

ð3:2Þ

A numerical application for this can be seen in Fig. 3.4 for the discount factor and in Fig. 3.5 for the accumulation factor.

3.3 Fundamentals of Financial Mathematics

95 DSF: 2.486852

24.86852

0 Abf:

10

10

10

1 0.909091

2 0.826446

3 0.751315

n

Fig. 3.4 Discounting with Abf and DSF (Source: Author) EWF: 3.31 33.10

10

0 Auf:

1 1.21

10

10

2 1.1

3 1

n

Fig. 3.5 Accrued interest with Auf and EWF (Source: Author)

3.3.2

The Summation Factors

The group of sum factors is a summation of one-time factors which, under certain conditions, allows a group of payments to be discounted or accumulated together, making it considerably easier to calculate with a pocket calculator. In order to apply these sum factors to a group of payments, all such payments must be • In arrears, • Equidistant and • Uniform This means that the payments • Are at the end of the period, which is a general assumption of the dynamic investment calculation methods and, therefore, it is always fulfilled. • Must be at the same distance from each other, which is always 1 year, and.

96

3 Dynamic Investment Calculation Methods

• Must have the same height. The discount sum factor (DSF) discounts a group of payments to the point in time before the first payment considered. The DSF is a summation of the individual discount factors. Please check this in Fig. 3.4. The formula for DSF is DSF ¼

ð1 þ i Þn  1 ið1 þ iÞn

ð3:3Þ

In the example in Fig. 3.4, the present value of three payments from years 1 to 3 is to be formed at an interest rate of 10%. The payments amount to 10 euros each. The continuous black arrows illustrate the operation of the discount factor (Abf), which requires three summands for the solution, the dotted arrow shows the path of the discount sum factor (DSF), which requires only one calculation step. On both paths, the present value is 24.86852. Please calculate the solution for both approaches. The solution results from: • 24.86852 ¼ 10  0.909091 + 10  0.826446 + 10  0.751315 with the Abf or • 24.86852 ¼ 10  2.486852 with the DSF. The terminal value factor (EWF) adds interest on a group of payments to the date of the last payment considered. The EWF is a summation of the individual accumulation factors. However, there is always one point in time less to be taken into account when adding interest than when discounting. When discounting, n + 1 points in time are considered in the result, since DSF discounts to the point in time before the first payment taken into account, and also the discount factor in the dynamic investment calculation discounts the first payment to the point in time before that, i.e. to zero, since we assume that there will be payments in arrears. In contrast, we consider only n points in time for accumulation, since the interest is accumulated to the date of the last payment. Thus, the terminal value factor is also a summation of the accumulation factors. However, for the corresponding terminal value factor, the numerical accumulation factors k ¼ 1 to k ¼ n  1 are added together, as the last payment is on the same date as the terminal value. In this case, the accumulation factor is exactly 1, which must be added to the previous sum of the accumulation factors. For this reason, the DSF and EWF summation factors are not reciprocal values, although the one-time factors are reciprocal values, since n + 1 points in time are taken into account when discounting several payments at the same time and thus one point in time more than when adding interest on several payments at the same time. In the example in Fig. 3.5, the terminal value of three payments from years 1 to 3 is to be formed at an interest rate of 10%. The payments amount to 10 euros each. The continuous black arrows illustrate the operation of the accumulation factor

3.3 Fundamentals of Financial Mathematics

97

(Auf), which requires three summands for the solution. The dotted arrow shows the path of the terminal value factor (EWF), which requires only one calculation step. Both ways result in a terminal value of 33.10. Please calculate the solution on both ways. The formula for the EWF is: ð1 þ iÞn  1 i

EWF ¼

3.3.3

ð3:4Þ

The Distribution Factors

The remaining two factors of financial mathematics are the distribution factors: capital recovery factor (KWF) and residual value distribution factor (RVF). The Capital recovery Factor distributes a one-time payment into post-numeric, equidistant and uniform payments for a given period of use, starting on the first date after the payment to be distributed. Figure 3.6 illustrates this procedure. It also shows that the KWF represents exactly the reciprocal value of the DSF. In the example in Fig. 3.6, a one-time payment of 24.86852 at time zero with an interest rate of 10% is to be spread over 3 years. Check the calculation yourself. The formula for the KWF is: KWF ¼

ið1 þ iÞn ð1 þ i Þn  1

ð3:5Þ

The residual value distribution factor (RVF) distributes a one-time payment into post-numeric, equidistant and uniform payments for a given period of use on the timeline to the left, starting at the time of the payment to be distributed. Figure 3.7 illustrates this procedure. It also shows that the RVF represents exactly the reciprocal value of the EWF. Please check this by comparing Figs. 3.5 and 3.7. Because of the different number of points in time that the two distribution factors consider, the distribution factors, as well as the sum factors above, are not reciprocal to each other. In the example in Fig. 3.7, a one-time payment of 33.10 at time point Fig. 3.6 Distributing a payment with the KWF (Source: Author)

KWF: 0.402115

24.86852

10

0

10

1

10

2

3

n

98

3 Dynamic Investment Calculation Methods

Fig. 3.7 Distributing a payment with the RVF (Source: Author)

RVF 0.302115 33.10

10

0

10

1

10

2

3

n

3 with an interest rate of 10% is to be distributed over 3 years. Check the calculation yourself. The solution is 10 ¼ 33.10  0.302115. The formula for the RVF is: RVF ¼

i ð1 þ i Þn  1

ð3:6Þ

So far, you have dealt intensively with financial mathematics. By linking the assumptions of dynamic investment calculation methods, the calculation elements of dynamic investment calculation methods and financial mathematics, all dynamic investment calculation methods can now be easily developed and interpreted in the following sections.

3.4

The Net Present Value Method

In Sect. 3.2.1 we have already pointed out that all dynamic investment calculation methods are methods that all evaluate the absolute advantageousness of an investment object on the basis of a singular criterion in a world that is considered to be safe, i.e. in which all premises occur. The target criterion of the net present value method, the net present value Co, considers the value of an investment at the present time (time today), i.e. it transforms all information of the future into a key figure at the time today, in order to be able to assess the attractiveness of the investment. The following possible definitions of net present value are suggested: The net present value is the difference between an investment object’s incoming and outgoing payments at present time. This is the definition we now want to deal with. In general, it makes sense to reduce the complex payment flows of an investment to one key figure. This may be shown by an initial example, which you can follow in Fig. 3.8. You already know the data from Fig. 1.6.

3.4 The Net Present Value Method

99

Object A

Object B

Co

Co

* (1+i)-k

A

* (1+i)-k

A

NEk 10 0

1

NEk

NEk

11

11 2

3

NEk

NEk

12

12

NEk

4

n

0

1

11 2

NEk 10

NEk 10

3

4

n

Fig. 3.8 Reducing complexity by forming key figures in dynamics (Source: Author)

There are two possible investment objects, object A and object B. Both have a 4-year useful life, both have an acquisition payment of 34.5 million euros, the residual value is zero in both cases. The difference between the ongoing incoming payments ek and the ongoing outgoing payments ak is also called net (incoming) payments or net earnings NEk. The net payments of both objects are shown in the columns in Fig. 3.8. If one adds up the net payments for each object, which is of course not useful for a dynamic investment calculation because of the different payment dates of the individual net payments, object A has net payments of 44 million euros, object B of 43 million euros. In a special situation, which is not desirable from a business point of view, with an interest rate of zero, the net present value of object A is, therefore, 44 million euros and the net present value of object B is 43 million euros, because with an interest rate of zero payments of different points in time can be added. However, the timing of the payments is more favourable for object B, as the relatively higher payments are made relatively early in the useful life and do not have to be discounted to the point of zero. Which of the investments is more advantageous? The formation of a relative advantageousness, i.e. the selection or ranking of investment objects is generally not possible. However, this may still be justified at this point in order to analyze the importance of reducing complexity by reducing a multi-year investment object to one key figure. In order to be able to carry out such a valuation, an interest rate for calculation still has to be determined, which we set at 10% (i ¼ 0.1). Exercise Now decide subjectively which investment is the better one. Solution If you now check your decision arithmetically by discounting the net payments for each object individually with (1 + i)k and then adding up and subtracting the acquisition payment from this, you will find that object A has a net present value of 0.14 million euros when rounded to two decimal places, while object B has a negative net present value of 0.16 million euros. The formula necessary for the solution will be worked out in Sect. 3.4.1. For example the solution for object A is

100

3 Dynamic Investment Calculation Methods

Co ¼ 10  1:11 þ 11  1:12 þ 11  1:13 þ 12  1:14 34:5 ¼ 0:14: Had you decided “right”? In any case, you can see from this example that by reducing the complex timeline according to given rules, an intersubjectively comprehensible decision is now possible. In fact, in this example, a reasonable selection is also possible, since an object has a positive and an object a negative net present value. After having dealt with the first definition of net present value, a different definition is now being developed, with a slightly different perspective on net present value. The net present value is an absolute amount at time zero, equivalent to the value of the investment over the term. First of all, this definition states that the net present value is an absolute amount, i.e. a currency amount, which, since it is not a percentage value, is not related to the capital invested. It is a pure asset value that determines a surplus from the investment. The temporal reference value is the point in time today (zero), the point in time when the investment is to be planned and, if necessary, also implemented. Thus, we determine the immediate asset contribution that an investment project to be carried out would make through payments that incur in the future and are discounted to time zero. This construction only makes sense if, as assumed, all calculation elements are known with certainty. Only then does it make sense economically to “borrow” payments of the future on today. A reference to a later point in time, as is done, for example with the horizon value, which we will discuss in Sect. 3.5, is no more realistic, since this value is also calculated with the same data before the start of the investment. Only the payments are mathematically related to a different point in time. Finally, the relative clause in the definition is significant: The net present value is equivalent to the value of the investment. In other words, it means that it is economically as attractive to maintain the net present value as to make the entire investment. The net present value is, therefore, the fair purchase and sales price of an investment object in the sense of this model. Before this sentence is analysed in more detail, the net present value criterion should first be highlighted. Co  0

ð3:7Þ

This criterion is the only decision criterion existing for the net present value method. An investment is worthwhile if the net present value is not negative. This means that we are dealing with an absolute criterion which, when evaluating investment projects, can only result in the statements “worthwhile,” i.e. Co > 0, or “not worthwhile” with Co < 0. There is the borderline case Co ¼ 0, in which the investment object is just worthwhile. A ranking of investment projects, that is, an

3.4 The Net Present Value Method

101

alternative selection or relative consideration, is not covered by this criterion. We will discuss this in more detail in Chap. 4. Nor is a higher positive net present value better than a lower positive net present value. This results directly from the following points: A positive net present value means • A full recovery of the acquisition payment. • Interest is paid on all outstanding amounts using the calculation interest rate i. • A net present value surplus in the amount of Co. What exactly is meant by “interest on all outstanding amounts at using the calculation interest rate i” is discussed at the end of this subsection. If in an investment project only the acquisition payment is recovered (of course, some projects do not even reach this level), the net present value is negative, because due to the delay in the term of the investment, interest on the acquisition payment is due and because presumably in the current investment project further payments are due, for example wages or preliminary products. If, in addition to the recovery of the acquisition payment, interest can be paid on the ongoing payments at the calculation interest rate, which represents the investor’s subjective minimum interest requirement, then the investor’s expectations have just been met, then the net present value is exactly zero. If, in addition to the recovery of the acquisition payment and the interest on the ongoing payments at the calculation interest rate, a further present value amount remains, the investor’s expectations are exceeded to a certain extent by this present value amount. The net present value is then positive at the level of this present value surplus. The interpretation of the net present value criterion at this point also makes it clear why the net present value represents the fair purchase and sale price for an investment object. This applies at least if the assumptions made are valid. We have specified that all calculation elements are known with certainty, which means that no matter who the owner of an investment project is, the payments always remain the same. Management differences between different owners or synergy effects are, therefore, excluded. A potential buyer of an investment project has the expectation that he will recover the capital invested and receive interest at his subjective minimum interest requirement. Thus, in addition to the acquisition payment for the investment object, a buyer can also pay the cash surplus (Co) as purchase price to the seller of the investment idea. His expectation of interest on his investment at the calculation interest rate is nevertheless fulfilled, the net present value now is zero. The potential seller of an investment project can in turn sell it exactly for Co without deteriorating. He can then invest the unpaid acquisition payment in another project at the assumed (there is only one interest rate!) calculation interest rate. There he will not generate a present value surplus. Since all payments bear interest at the calculation interest rate, the net present value of this investment is exactly zero.

102

3 Dynamic Investment Calculation Methods

However, the seller still has the net present value of the sold investment project, which can also be invested in the calculation interest rate. So far, the seller has neither improved nor worsened his situation. Thus, within the assumptions of the dynamic model, the net present value is the fair sales price of an investment. An example may illustrate this: As a buyer, you have the opportunity to have a financial investment brokered by a financial services provider. Your interest expectation (calculation interest rate) is 10%. It is a zero bond with a 4-year term. The purchase price is 10,000 euros (acquisition payment). The repayment value is 15,735.19 euros (residual value). By discounting the repayment value at 12%, you determine that the capital in the financial asset earns interest at 12%, since the net present value is then exactly 10,000 euros, i.e. corresponds to the purchase price (15,735.19  1.124 ¼ 10,000). The interest rate is, therefore, higher than you would expect with your calculation interest rate of 10%. Discounting the repayment value at 10% would result in a present value of 10,747.35 euros and thus a net present value of 747.35 euros, since the acquisition payment for the savings certificate was 10,000 euros and must be deducted from the present value to determine the net present value. Unfortunately, the financial service provider demands a commission of 747.35 euros for acting as a broker of this savings certificate, payable at the time of the purchase of the savings certificate, i.e. exactly the amount of the net present value. Altogether you have to pay an amount of 10,747.35 euros now, which corresponds to the acquisition payment of the savings certificate plus the net present value. This capital now accrues interest at exactly 10% over 4 years, because the following applies: (10,747.35  1.14 ¼ 15,735.19). So far, the buyer’s expectations are still just about fulfilled if he pays the seller the net present value as a purchase price in addition to the actual acquisition of the investment object. The financial service provider as the seller of the savings certificate is not worse off by selling this financial asset, because he can now invest the 10,000 euros at the calculation interest rate. The financial service provider has received the additional surplus, which cannot be realised in the sold savings certificate, as a commission, so he is not worse off by the sale at net present value. To the extent that the assumptions made are valid, the net present value is the fair purchase and sale price. We still have to consider the following points formulated in the interpretation of the net present value the second bullet point, which reads “interest is paid on all outstanding amounts using the calculation interest rate i.” Whether it is relevant which amounts are outstanding in an investment project is shown in the Fig. 3.9. In practice, it is significant whether a loan is repaid during the term of an investment project or is only serviced at the end of the project. As a rule, this results in different economic successes of a project. In Fig. 3.9 the following example data apply: • • • •

Initial net investment: 40 million euros Useful life: 3 years Residual value: zero Net payments in each period: 20 million euros

3.4 The Net Present Value Method

103

Fig. 3.9 Development of capital commitment in an investment project (Source: Author)

- 53.24

AUF=1.331

+ 66.20 12.96 EWF=3.31

A 40

NEk 20

NEk 20

NEk 20

0

1

2

3

-40

-44 20 -24

-26.4 20 -6.4

-7.04 20 12.96

n

• Calculation interest rate: 10% To elaborate on the difference, the payments are first accumulated and only discounted at the end. This results in an identical net present value as with direct discounting. In the graphical representation above the timeline, the acquisition payment is not repaid. The net payments are invested with the calculation interest rate until the end of year 3. (20  EWF), interest is also calculated on the unredeemed acquisition payment (40  Auf). The difference between the two results in year 3 is 12.96 million euros. The net present value is then 9.737 million euros (12.96  1.13 ¼ 9.737). Below the timeline, an account is kept in which the acquisition payment bears interest at 10% annually. As soon as net payments are made, these are deducted from the interest-bearing acquisition payment, so that only a smaller amount of capital is tied up and interest is paid. After 3 years, this also results in an account balance of 12.96 million euros. As long as only one interest rate is used, as usual in dynamic investment calculation, the development of the capital commitment is irrelevant for the result of the investment calculation. Besides, the concept of the net present value method developed here is not only used in dynamic investment calculation. As a discounted cash flow method (DCF method) with the same technical content, it is used in the valuation of companies for various issues. For example it is the basis of the shareholder value concept and is thus used to determine the value of a company when planning IPOs (e.g. initial public offerings, IPOs) or company merger or sales of companies (mergers and acquisitions, M&A).

104

3 Dynamic Investment Calculation Methods

After we have developed the concept of the net present value method, the corresponding calculation formulas are now to be formulated. This is done in the following three subsections. First, the net present value is developed for individual discounting. It is the general case. If the calculation elements have a special structure, a calculation using a pocket calculator is less calculation-intensive than in the general case. These two special cases are the net present value when DSF can be used and the net present value with infinite useful life, which are presented in the following two sections.

3.4.1

Net Present Value with Individual Discounting

The calculation method for the net present value in the case of individual discounting can be seen in Fig. 3.10. This is followed by the relevant formula. First, the periodic difference between the net payments (ek  ak), is discounted. In the graph, the discounting of each individual payment transaction is shown, but the ongoing payments of each period can of course be discounted together after balancing. They are then totalled. Imagine the graph and all the following ones please in such a way that the column of in-payments ek is located behind the column of disbursements ak and the part of the in-payments column represents only the part that exceeds the disbursements column. The discounted residual value is added. The result is the present value of the investment. This is the total column at time zero, (A + Co). By deducting the acquisition payment from the present value, we obtain the net present value. The formula for this is:

Abf Co

R A BW

0

ek

ek

ek

ek

ak

ak

ak

ak

1

2

3

Fig. 3.10 Net present value for individual discounting (Source: Author)

4

n

3.4 The Net Present Value Method Fig. 3.11 Data set for the net present value calculation (Source: Author)

Co ¼

n X

105

k

ek

ak

1

300

200

2

500

300

3

700

300

4

400

300

ðek  ak Þ  ð1 þ iÞk þ R  ð1 þ iÞn  A

ð3:8Þ

k¼1

Now you are able to calculate the net present value of each investment object with a given data set. You will also be given the opportunity to do this immediately. Exercise Please determine the net present value of given the data. The useful life is 4 years, the acquisition payment 1000 euros. The residual value is 600 euros. The calculation interest rate is 10%. You can see the ongoing payments in Fig. 3.11. Solution The solution can be found in Eq. (3.9). C o ¼ ð300  200Þ  1:11 þ ð500  300Þ  1:12 þ ð700  300Þ  1:13 þð400  300Þ  1:14 þ 600  1:14  1000 ¼ 34:83 ð3:9Þ The net present value is 34.83 euros. In the following two subsections, two special cases are presented for special data sets.

3.4.2

Net Present Value If DSF Can Be Used

If the calculation elements have a special structure, a calculation with a pocket calculator is less calculation-intensive compared to the general case. This is for example if the prerequisites for applying the discount sum factor are met. This is the case if the current annual incoming payments (ek) and the current annual outgoing payments (ak) are the same in each period. Alternatively, it is also sufficient if only the net earnings (NEk) are the same in each period. This case is shown in Fig. 3.12, where all payments in years 1–3 are the same. In year 4, the payments are lower in absolute terms, but the difference (ek  ak), i.e. the net earnings (NEk), is the same as in previous years. Figure 3.12 shows the discounting of net payments with DSF. The corresponding formula for the net present value when DSF is used is:

106

3 Dynamic Investment Calculation Methods Abf Co

R DSF A BW

ek

ak 0

ek

ek

ak 1

ek

ak 2

ak 3

4

n

Fig. 3.12 Net present value if DSF can be used (Source: Author) k

ek

ak

1

500

300

2

500

300

3

500

300

4

500

300

Fig. 3.13 Data set for the net present value calculation with DSF (Source: Author)

C o ¼ ðe  aÞ  DSFni þ R  ð1 þ iÞn  A

ð3:10Þ

You can also apply this formula 3.10 immediately. Exercise Please determine the net present value for the given data. The useful life is 4 years, the acquisition payment 1000 euros. The residual value is 600 euros. The calculation interest rate is 10%. You can see the ongoing payments in Fig. 3.13. Solution You can find the solution in Eq. (3.11). Co ¼ 200  3:169865 þ 600  1:14  1000 ¼ 43:78

ð3:11Þ

The net present value is 43.78 euros. Evaluation The advantage of this formula for using a pocket calculator is obvious. Whereas the net present value formula for individual discounting generally requires n + 2 summations, n discounting of net incoming payments plus discounted residual value minus acquisition payment, this formula requires only three calculation steps, discounting of net earnings with the discount sum factor plus discounted residual value minus acquisition disbursement. If you use the pocket calculator, this method

3.4 The Net Present Value Method

107

is an extreme simplification. This applies all the more, the longer the useful life of the investment object. This is of course irrelevant for use in spreadsheets.

3.4.3

Net Present Value with Infinite Useful Life

There is also a special net present value formula for determining a net present value with an infinite useful life. An infinite useful life (Going-Concern principle, the principle of infinite company continuation) is often assumed in company valuation in order to determine a “future value” of a company after a phase of a concrete planning period with annually planned calculation elements. In other words, this is a resale value of the enterprise after a few years, resulting from the discounted net payments of an infinite future, which is calculated so that not all years of an infinite future are included in the planning. The formula of the net present value with infinite useful life can be easily derived from the general net present value formula. This is based on the assumption that the net payments of the future are constant, an assumption that is not to be criticised due to the extreme data uncertainty of the further future. Since the company is continued infinitely, there is no residual value. The net present value formula is then initially Co ¼ ðe  aÞ  DSFni  A

ð3:12Þ

An analysis of the discount sum factor makes the simplification clear. The formula for DSF is (3.13) = (3.3) DSF ¼

ð1 þ i Þn  1 ið1 þ iÞn

ð3:13Þ

The following notation results from reducing the fraction by the Accumulation factor (1 + i)n: DSF ¼

1 1  ð1þi Þn

i

ð3:14Þ

As the limit value in the numerator of the fraction converges towards zero due to the assumption of infinite useful life, lim

n!1 ð1

1 0 þ iÞn

the DSF remains as 1i . This is the net present value formula for an infinite useful life:

ð3:15Þ

108

3 Dynamic Investment Calculation Methods

1 Co ¼ ðe  aÞ   A: i

ð3:16Þ

Please test this net present value formula on an example. Exercise Please determine the net present value for the following data set. The useful life is infinite, the acquisition payment is 1000 euros. The calculation interest rate is 10%. The ongoing net payments are 300 euros. Solution The solution can be found in Eq. (3.17).

C o ¼ 300 

1  1000 ¼ 2000 0:1

ð3:17Þ

The net present value is 2000 euros. The method has the advantage that it requires very little calculation effort. The assumption of an infinite useful life is not very problematic due to the discounting effect of payments in the further future.

3.4.4

Case Study Net Present Value Method

In this subsection, you will now apply the theory of the net present value method that you have learned. For this purpose, please deal with the following case study. You have the opportunity to participate in a project for renewable energies. To determine whether an investment is attractive for you, determine the net present value of the overall project from the following data. The government subsidises this project, at least you assume this when planning. In addition to the in-payments ek that result from the project, there are also government subsidies that are marked with S or Sk. The acquisition payment (A) amounts to 60 million euros and the initial subsidy, which is paid at point zero, is 15 million euros. There is no residual value (R) of the project, your calculation interest rate (i) is 10%. The useful lifetime (n) is 6 years. The ongoing payments are shown in Fig. 3.14, the payment amount is given in million euros. Fig. 3.14 Data on the net present value case study (Source: Author)

Time, k 1 2 3

Pay-outs, ak 2.0 2.0 3.0

In-payments, ek 15

subsidy, sk 0.5 0.5 0.5

4

3.0

20

0.5

5 6

2.5 2.5

30 45

0.5 0.5

3.4 The Net Present Value Method

k

ek 1 2 3 4 5 6

s 0 0 15 20 30 45

109

ak 0.5 0.5 0.5 0.5 0.5 0.5

total 2.178

NEk 2 2 3 3 2.5 2.5

-1.5 -1.5 12.5 17.5 28 43

BW Co -1.364 -1.240 9.391 11.953 17.386 24.272 60.399 15.399 solution a -1.779 solution b

Fig. 3.15 Excel file with results of the net present value case study* (Source: Author)

Exercises Exercise a) Determine the net present value of the project. Exercise b) Determine the net present value of the project without taking subsidies into account. You can determine the solution in a spreadsheet or with a pocket calculator. We will present both options for all the following exercises, since due to the large group sizes at universities, there is usually no possibility to have the exam “Investment” processed with computers for organisational reasons, but pocket calculators are still used. In practice, such questions will hopefully exclusively be dealt with IT. Solutions First, we present the solution as results in the spreadsheet in Fig. 3.15. Since it is a matter of discounting and summing up the calculation elements, there are various correct procedures for formulating the solution in the spreadsheet. So far the shown solution is only a suggestion from which you can deviate, as long as you come to the same result. Rounding differences may occur between representations in figures and in formulas, as the printed table representation is limited in the decimal places. For the solution of exercise a) with the pocket calculator, we use the net payments NEk already formed in the spreadsheet, discounting them with the discount factor (1 + i)k, i.e. 1.1k, individually, add them up, subtract the acquisition payment and add the initial subsidy. The procedure is explained in Eq. (3.18). C o ¼ 1:5  1:11  1:5  1:12 þ 12:5  1:13 þ 17:5  1:14 þ 28  1:15 þ43  1:16  60 þ 15 ¼ 15:39904073 million euros ð3:18Þ When solving exercise b) with a pocket calculator, it makes sense to take advantage of the discounting sum factor. Only the subsidies need then be deducted from the result of exercise a) without recalculating.

110

3 Dynamic Investment Calculation Methods

C o ¼ 15:39904073  0:5  DSFn¼6 i¼0,1  15 ¼ 1:77858977 million euros

ð3:19Þ

Evaluation of the Results In the case of exercise a), it is a worthwhile investment for which the net present value criterion is met. The interpretation then tells us that • A full recovery of the acquisition payment • Interest payments on all outstanding amounts at the discount rate i and • A net present value surplus in the amount of approximately 15.4 million euros can be achieved. Of course, this interpretation only applies to the assumptions made. This means in particular that the calculation elements occur as planned and that interest can be calculated on all amounts at the calculation interest rate. In the case of exercise (b), the criterion is no longer met; the investment would be disadvantageous at the desired rate of return.

3.4.5

Section Results

In this section, you: • • • • • •

Got to know the net present value method. Developed the net present value criterion. Learned the interpretation of the net present value. Developed the net present value formula. Got to know special cases of the net present value formula. Applied net present value method.

3.5

The Horizon Value Method

Just as with the net present value method, the horizon value method determines the absolute advantageousness of an investment object on the basis of a singular criterion in a world considered to be safe, i.e. a world in which all premises are fulfilled. The target criterion of the horizon value method, the horizon value Cn, considers the value of the capital gain at the time of the end of the useful life of the investment, i.e. it transforms all information of the future into a key figure, at the time of the end of the useful life of the investment project, in order to be able to assess the attractiveness of the investment. The following possible definitions of the horizon value are possible:

3.5 The Horizon Value Method

111

Cn Auf EW

R

A

0

ek

ek

ek

ek

ak

ak

ak

ak

1

2

3

4

n

Fig. 3.16 Horizon value for individual discounting (Source: Author)

• The horizon value is the difference between the terminal value of incoming and outgoing payments. • The horizon value is an absolute amount at time n that is equivalent to the value of the investment over the term. As with the net present value method, the horizon value criterion is presented as a decision-making criterion. Cn  0

ð3:20Þ

This criterion is the only decision criterion existing for the horizon value method. An investment is worthwhile if the horizon value is not negative. A positive horizon value means • A full recovery of the acquisition payment. • Interest is paid on all outstanding amounts using the calculation interest rate i. • A final surplus of Cn. The horizon value of an investment is, therefore, the concentration of the transformed calculation elements of an investment at time n, that is, only at a different time than the net present value. To this extent, the increase in information about the value of the investment project is very small. But first, the formulas for the calculation of the horizon value are to be presented. This is done first again with a graphical representation of the way the horizon value method works in the general case in Fig. 3.16. All payments are now accumulated to time n.

112

3 Dynamic Investment Calculation Methods

The formula for the horizon value for individual discounting is Cn ¼

n X

ðek  ak Þ  ð1 þ iÞnk þ R  A  ð1 þ iÞn

ð3:21Þ

k¼1

Evaluation For the terminology of the exponent at the accumulation factor ()nk, which may initially be somewhat confusing, please consider, for example the payments in year 1 in Fig. 3.16. This payment can obviously still be invested for 3 years, since the total length of the useful life in the illustration is 4 years. In the given terminology the variable n is the total length of the useful life of the investment object, the variable k stands for the year in which the payment is due, here, for example the year 1. nk in this example, therefore, results in 3, that is, the number of periods in which the payment can still be invested. In this way, the exponent is formulated as the accumulation factor. In the graph, the accumulating of each individual payment transaction is shown. However, the ongoing payments of a period can of course be accumulated together after balancing, as the formula also suggests. If the calculation elements have a special structure, a calculation with a pocket calculator is less calculation-intensive compared to the general case. This is for example if the conditions for the application of factors of financial mathematics, here the terminal value factor, are fulfilled. These are the same as for the discount sum factor, which have already been described in Sect. 3.3. The formula for the horizon value Cn when applying the terminal value factor is: C n ¼ ðe  aÞ  EWFni þ R  A  ð1 þ iÞn :

ð3:22Þ

The advantage of this formula for the use of pocket calculators is obvious. While the horizon value formula for individual discounting generally requires n + 2 summations, this formula requires only three calculation steps, a multiplication of the net incoming payments with the terminal value factor plus residual value minus the accrued acquisition payment. This is a great simplification when using the pocket calculator. This is all the more true, the longer the useful life of the investment object. A horizon value with infinite useful life, such as the corresponding special net present value formula, can of course not exist in the horizon value method, since this value would itself be in infinity and would, therefore, automatically be infinite itself. However, there is a much simpler way of calculating the horizon value if the net present value has already been determined. This also applies reciprocally. Figure 3.17 illustrates the case. The net present value in Fig. 3.17 has been determined directly from the payments of the investment, i.e. a transformation of all payments to time zero. This is not visible in Fig. 3.17, but you know it from Fig. 3.10. The horizon value of this investment is a transformation of the same payments to point n. Thus, the horizon

3.5 The Horizon Value Method

113

Cn *(1+i)n

*(1+i)-n

EW Auf

R Co

A

0

ek

ek

ek

ek

ak

ak

ak

ak

1

2

3

4

n

Fig. 3.17 Determining the horizon value from the net present value (Source: Author)

value can also be determined directly from the net present value by simple compounding and the net present value can be determined by discounting from the existing horizon value. This is illustrated by the dotted arrows in Fig. 3.17. The formulas are as follows: C n ¼ C o  ð1 þ i Þn

or

C o ¼ C n  ð1 þ iÞn

ð3:23Þ ð3:24Þ

This presentation also illustrates the close connection between net present value and horizon value. For a given investment project, the multiplier for getting from net present value to horizon value depends only on the specified calculation elements i and n. The horizon value is, therefore, a function of the net present value. The net present value is a function of the horizon value. Exercise Calculate the horizon value of the case study from Sect. 3.4.4. Solution Whether you use a spreadsheet or a pocket calculator, you can quickly find the solution by using the formula (3.25) = (3.23)

114

3 Dynamic Investment Calculation Methods

Fig. 3.18 Excel file with results of the horizon value case study* (Source: Author)

k

ek 1 2 3 4 5 6

s 0 0 15 20 30 45

NEk BW Cn 2 -1.5 -2.416 2 -1.5 -2.196 3 12.5 16.638 3 17.5 21.175 2.5 28 30.8 2.5 43 43 107.001 27.280 solution a -3.151 solution b

ak 0.5 0.5 0.5 0.5 0.5 0.5

total 3.858

C n ¼ C o  ð1 þ i Þn

ð3:25Þ

This leads to the following result: Cn ¼ 15:39904073  1:16 ¼ 27:28033999 million euros

ð3:26Þ

If you would like to find the solution for the purpose of practice on the independent way of the horizon value method, you can follow the same pattern as for the exercise in Sect. 3.4.4 and check it on Fig. 3.18. For the solution with the pocket calculator according to the independent procedure of the horizon value method in exercise a) we use the net payments already calculated in the spreadsheet, add interest individually with the Compounding Factor (1 + i)nk, i.e. 1.1nk, add them up, subtract the compounded acquisition payment and add the compounded initial subsidy. This is documented in Eq. (3.27). Cn ¼ 1:5  1:15  1:5  1:14 þ 12:5  1:13 þ 17:5  1:12 þ 28  1:11 þ 43 60  1:16 þ 15  1:16 ¼ 27:28033999 million euros ð3:27Þ When solving exercise b) with a pocket calculator, it makes sense to take advantage of the terminal value factor. From the result of exercise a) only the subsidies need to be deducted without recalculating. 6 C n ¼ 27:28033999  0:5  EWFn¼6 i¼0:1  15  1:1 ¼ 3:15088000 million euros

ð3:28Þ

The interpretation of the results corresponds to the interpretation of the net present value method. Section results In this section, you: • • • • •

Got to know the horizon value method. Developed the horizon value criterion. Learned the interpretation of the horizon value. Edited the horizon value formula. Got to know the special case of the horizon value formula.

3.6 The Annuity Method

115

• Applied the horizon value method.

3.6

The Annuity Method

Just as with the net present value method and the horizon value method, the annuity method determines the absolute advantageousness of an investment accounting object on the basis of a singular criterion in a world considered to be safe, i.e. in which all premises occur. The target criterion of the annuity method, the annuity, this term corresponds to the term average annual surplus (DJÜ), considers the periodically constant surplus of an investment, thus transforming all information of the future to a periodically equally high key figure in order to be able to make an assessment of the attractiveness of the investment. This amount could be withdrawn periodically, i.e. annually on the basis of the assumption of payments in arrears, if all planning data occurs, without the investment becoming disadvantageous. The average annual surplus (DJÜ) is a periodic absolute amount equivalent to the value of the investment for the term. As with the other dynamics, the annuity criterion is presented as a decision criterion. € 0 DJU

ð3:29Þ

This criterion is the only decision criterion existing for the annuity method. An investment is worthwhile if the average annual surplus is not negative. A positive DJÜ means: • A full recovery of the acquisition payment. • Interest is paid on all outstanding amounts using the calculation interest rate i. • A periodic surplus in the amount of DJÜ. The annuity of an investment is thus the uniform distribution of the transformed calculation elements of an investment over time from point 1 to n. Before presenting the independent determination path of the annuity, it should be shown, as concluded in the previous chapter, that the annuity can also be determined from the other dynamics. This is illustrated in Fig. 3.19. Like the horizon value Cn, the net present value Co is determined directly from the payments of the investment, i.e. a transformation of all payments to time zero or n. The average annual surplus of this investment is a uniform transformation of the same payments to time 1 to n. In this way, the annuity can also be determined directly from the net present value or the horizon value. A reciprocal procedure is also possible. € ¼ C o  KWF DJU

ð3:30Þ

116

3 Dynamic Investment Calculation Methods

RVF Cn

KWF

Co

R

A

0

ek

ek

ek

ek

ak

ak

ak

ak

1 DJÜ

2 DJÜ

3 DJÜ

4

n

DJÜ

Fig. 3.19 Determining the annuity from the net present value or horizon value (Source: Author)

€  DSF C o ¼ DJU

ð3:31Þ

€ ¼ C n  RVF DJU

ð3:32Þ

€  EWF C n ¼ DJU

ð3:33Þ

This presentation also makes clear the close connection between annuity, net present value and horizon value. For a given investment project, the multiplier depends only on the specified calculation elements i and n to calculate the annuity from the net present value or the horizon value. This point of view also clarifies that the term annuity is not clearly defined in business administration. In finance, the term annuity refers to the sum of periodic interest and redemption payments. In an annuity loan, the proportion of interest is periodically reduced and the proportion of redemptions is increased. The sum of both activities is constant. Thus, in the terminology of investment theory, a repaid annuity loan re-procures the acquisition payment in the form of repayments and the interest claim of the invested capital in the form of interest payments. From this perspective, the numerical value of the net present value is predetermined, namely zero, since the first two bullet points of the net present value criterion are fulfilled. From the investment calculation definition of the annuity DJÜ ¼ Co  KWF, a positive annuity could never result from the argumentation in the previous paragraph, but it must result from the positive net present value and the mathematical structure of the KWF. Obviously, investment theory defines the term annuity differently. In addition to the definition already given above, we can also define:

3.6 The Annuity Method

k 0 1 2 3 4 Total

117

fixed capital interest repayment annuity 100000.00 78452.90 10000.00 21547.10 31547.10 54751.09 7845.29 23701.81 31547.10 28679.10 5475.11 26071.99 31547.10 -0.09 2867.91 28679.19 31547.10 26188.31 100000.09 126188.40

Fig. 3.20 Financial annuity to the numerical example* (Source: Author)

k 0 1 2 3 4 Total

profit fixed capital interest contribution intrest creditor repayment annuity BW profit DJÜ 100000.00 78452.90 10000.00 6000.00 4000.00 21547.10 31547.10 5454.55 4097.40 54751.09 7845.29 4707.17 3138.12 23701.81 31547.10 3890.23 4097.40 28679.10 5475.11 3285.07 2190.04 26071.99 31547.10 2468.12 4097.40 -0.09 2867.91 1720.75 1147.16 28679.19 31547.10 1175.29 4097.40 26188.31 15712.99 10475.32 100000.09 126188.40 12988.18

Fig. 3.21 Investment mathematical annuity to the numerical example* (Source: Author)

The average annual surplus (DJÜ) is a periodic absolute amount which, in addition to a full recovery of the acquisition payment and, in addition to interest on all outstanding amounts at the calculation interest rate i, is available for withdrawal in periods 1 to n, without the investment becoming unfavourable. The reason for the different definitions should quickly become clear to the reader through the following example. In practical financial management, the assumption of dynamics that there is only one interest rate does not apply, of course. A creditor of a financing usually refinances itself when granting a loan. Thus, the repayments of his debtor are necessary to repay the nominal amount of the refinancing. However, the interest portion is naturally divided into the part that the creditor has to pay to his refinancer as interest and the profit portion for the creditor who had passed on the loan amount to his debtor with an interest surcharge. Both interest components are included together in the financial definition. In the investment calculation definition, these two parts are divided into the interest component (interest on all outstanding amounts with the calculation interest rate i) and the annuity (DJÜ). You can see this in a numerical example in the following text and in Figs. 3.20 and 3.21. In an annuity loan of 100,000 euros, repaid annually in arrears, with a 4-year term, the annual annuity is 31,547.10 [100,000  KWF(n ¼ 4, i ¼ 0.1)] according to the financial definition at an interest rate of 10%. You can check how the payments develop in Fig. 3.20. Assuming that the same redemption payments are made as in the calculation of the financial annuity and that the creditor refinances itself with its refinancer at 4% and has a profit share of 6 percentage points, the investment annuity (Co  KWF

118

3 Dynamic Investment Calculation Methods

DJE

DJE

DJE

DJE

RVF EW

R AUF

0

1

ek

ek

ek

ek

2

3

4

n

Fig. 3.22 DJE formation (Source: Author)

(n ¼ 4, i ¼ 0.1), Co ¼ total BW profit) amounts to 4097.40 euros. This is visible in Fig. 3.21. Note: In the classic annuity method, there is of course only one interest rate, the calculation interest rate. If an interest split as in this example is not common in the method, an interest rate of 10% would result in an investment annuity of zero. You will learn how to work with several interest rates in Chap. 4. If the net present value and horizon value have not been determined, the annuity must be determined using a separate calculation method. Here, the annuity is determined from the difference between the average annual incoming payments (DJE) and the average annual outgoing payments (DJA). € ¼ DJE  DJA DJU

ð3:34Þ

To form the DJE, the components of incoming payments of the investment project, i.e. the ongoing in-payments (ek) and the residual value (R) are transformed to either time zero or time n and then distributed with the corresponding factor of financial mathematics. Figure 3.22 shows this case for the accumulation to time n, where the terminal value of the in-payments is then distributed to the DJE with the RVF. The formulas for the formation of the DJE are then " # n X k n DJE ¼ ek  ð 1 þ i Þ þ R  ð 1 þ i Þ ð3:35Þ  KWFni k¼1

if the present value of the in-payments is formed first, and

3.6 The Annuity Method

119

BW ak

KWF

DJA

DJA

DJA

DJA

ak

ak

BW DJA

Abf

ak

ak

A 0

1

2

3

4

n

Fig. 3.23 DJA formation (Source: Author)

" DJE ¼

n X

# ek  ð1 þ iÞnk þ R  RVFni

ð3:36Þ

k¼1

if the terminal value of the in-payments is formed first, as shown in Fig. 3.22. To form the DJA, the disbursement components of the investment project, i.e. the ongoing payouts (ak) and the acquisition payment (A), are transformed to either time zero or time n and then distributed with the corresponding factor of financial mathematics. Figure 3.23 shows this case for discounting to zero. The result is the present value (BW) of the disbursements, which is distributed to the DJA with the KWF. The formulas for the formation of the DJA are then " DJA ¼

n X

# ak  ð 1 þ i Þ

k

þ A  KWFni

ð3:37Þ

k¼1

if first, as shown in Fig. 3.23, present value (BW) of the disbursements is formed, and " # n X nk n DJA ¼ ak  ð1 þ iÞ þ A  ð1 þ iÞ  RVFni ð3:38Þ k¼1

if the terminal value of the disbursements is formed first. Exercises Calculate the annuity of the case study from Sect. 3.4.4. Exercise a) with subsidies, Exercise b) without subsidies.

120

3 Dynamic Investment Calculation Methods

k

ek 1 2 3 4 5 6

Total

s 0 0 15 20 30 45

ak 0.5 0.5 0.5 0.5 0.5 0.5 2.178

2 2 3 3 2.5 2.5

BW (ek +s) BW ak 0.455 1.818 0.413 1.653 11.645 2.254 14.002 2.049 18.938 1.552 25.684 1.411 71.137 10.738 DJE DJA DJÜ 19.778 16.242 3.536 Solution a 15.833 16.242 -0.408 Solution b

Fig. 3.24 Excel file with results of the annuities case study* (Source: Author)

Solutions No matter whether you use a spreadsheet or a pocket calculator, you can quickly find the solution, e.g. using the formula € ¼ C o  KWFn DJU i

ð3:39Þ

Thus, this is the result for exercise a) € ¼ 15:39904073  0:229607 ¼ 3:53572754 million euros DJU

ð3:40Þ

If you want to practice and solve the problem by using the annuity method, you can follow the same pattern as for the exercise in Sect. 3.4.4. In the presented solution in Fig. 3.24 the present value formation is shown, the representation of the formation of the terminal value, which is also possible, was omitted. For the solution with the pocket calculator for exercise a) we add the ongoing in-payments ek and the ongoing subsidies s for each year separately. Then we discount them individually with the discount factor (1 + i)k, i.e. 1.1k, add them up, add the initial subsidy and distribute the result with the KWF, thus obtaining the DJE. Accordingly, ak and A are treated for the formation of the DJA. The procedure is shown in Eqs. (3.41) and (3.42) for the data set of exercise a). " # 0:5  1:11 þ 0:5  1:12 þ 15:5  1:13 þ DJE ¼ 20:5  1:14 þ 30:5  1:15 þ 45:5  1:16 þ 15  KWFn¼6 i¼0:1

ð3:41Þ

3.6 The Annuity Method

" DJA ¼

121

2  1:11 þ 2  1:12 þ 3  1:13 þ 3  1:14 þ 2:5  1:15 þ2:5  1:16 þ 60

#

 KWFn¼6 i¼0:1 ð3:42Þ (3.43) = (3.34) € ¼ DJE  DJA DJU

ð3:43Þ

When solving exercise b) with a pocket calculator, it makes sense to take advantage of the discount sum factor. Only the subsidies need then to be deducted from the result of exercise a) without recalculating. If you get marginal rounding differences here or elsewhere, this is because in the excel calculations the factors of financial mathematics were saved as formulas and not as numbers. When calculating with the pocket calculator the financial mathematical tables are probably used, which round the factors of financial mathematics to six decimal places. DJE ¼ 19:778  0:5  15  KWFn¼6 i¼0:1

ð3:44Þ

The DJA remains unchanged at 16,242 million euros, and thus the DJÜ is (3.45) = (3.34) € ¼ DJE  DJA ¼ 0:408 million euros DJU

ð3:45Þ

The interpretation of the results corresponds to the interpretation of the net present value method. Evaluation In the case of exercise a), therefore, the investment is worthwhile and fulfils the annuity criterion. The interpretation then tells us that: • A full recovery of the acquisition payment • Interest payments on all outstanding amounts at the discount rate i and • A periodic surplus of 3.53572754 million euros can be achieved. Of course, this interpretation only applies for the assumptions made. This means in particular that the calculation elements occur as planned and that interest is paid on all amounts at the calculation interest rate. In the case of exercise b), the criterion is no longer met; the investment would be disadvantageous at the desired rate of return.

122

3 Dynamic Investment Calculation Methods

Section results In this section, you: • • • • •

Learned about the annuity method. Developed the annuity criterion. Learned how to interpret the annuities. Edited the annuity formula. Applied the annuity method.

3.7

The Internal Rate of Return Method

Just like the other methods discussed so far, the internal rate of return method determines the absolute advantageousness of an investment object on the basis of a singular criterion in a world considered to be safe, i.e. a world in which all premises are fulfilled. While the net present value, horizon value and annuity methods show a positive target value (or zero) for worthwhile investments, the internal rate of return method changes the interest rate, which is now an endogenous, i.e. dependent variable as the result of this method, until the target values of the other dynamics mentioned become zero. While the net present value, horizon value and annuity methods calculate an absolute amount for a given dimensionless criterion (percentage rate), the internal rate of return method calculates an exclusively dimensionless criterion. The internal rate of return method determines the return or yield (r), (internal interest rate, internal rate of return, IRR). The internal rate of return is the rate at which the net present value, horizon value and annuity are zero. The target criterion of the internal rate of return method, the return on investment, considers the interest rate of an investment where all payments transformed to the point in time of the end of the useful life of the investment project or to the point of the beginning of the useful life of the investment project or to the period 1 to n are of such a value that the target values of the related dynamics add up to zero. As with the other dynamics, the internal rate of return criterion is presented as a decision-making criterion. ri

ð3:46Þ

This criterion is the only one existing for the internal rate of return method. An investment is worthwhile if the return (r) is not less than the calculation interest rate (i). Therefore, the fact that the internal rate, unlike the other dynamics, is a dimensionless criterion is also irrelevant when comparing the informative value of the individual methods. All methods offer only one criterion each, which is checked for

3.7 The Internal Rate of Return Method

123

fulfilment or non-fulfilment. For this reason, there are no qualitative differences between the methods. The fact that the internal rate of return method has some major problems in addition to the restrictive assumptions of the other dynamics is ignored at this point and only revisited in Chap. 4. A return (r) above the calculation interest rate (i) means • A full recovery of the acquisition payment. • Interest on all outstanding amounts at the yield. Evaluation Here lies a clear difference between the internal rate of return method and the other dynamics. While all the methods dealt with so far have assumed that all outstanding amounts are invested at the calculation interest rate (i), the internal rate of return method assumes that all outstanding amounts are invested at the higher yield (r) for worthwhile investments. For the same investment projects, the internal rate of return method, therefore, has different ideas about the interest on the amounts still committed than the other dynamics. This phenomenon is also dealt with in more detail in Chap. 4. In order to determine the return in the general case, the net present value, horizon value or annuity equation would have to be set to zero, as defined above and be dissolved according to the interest rate, which would then correspond to the yield. This is illustrated by the net present value equation. In the following figures, too, only the net present value function is used as a representative for the other dynamics. (3.47) = (3.8) Co ¼

n X

ðek  ak Þ  ð1 þ iÞk þ R  ð1 þ iÞn  A

ð3:47Þ

k¼1

After zeroing the net present value, the calculation interest rate (i) is converted into the rate of return (r). 0¼

n X

ðek  ak Þ  ð1 þ r Þk þ R  ð1 þ r Þn  A

ð3:48Þ

k¼1

Since this is generally a higher order equation, it is obvious that solving the equation for r and thus determining a mathematically exact result of the return is not immediately possible. Except in special cases where the data set of the investment project has a special structure, only an approximate solution for determining the return is possible. The approximate solution is treated in Sect. 3.7.1, the special cases will be analysed in Sect. 3.7.2.

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3 Dynamic Investment Calculation Methods

3.7.1

Determination of the Yield with the Regula Falsi

The yield can be determined arithmetically with any degree of accuracy using various approximation methods. The most common method is the regula falsi (Latin: rule of the wrong), which is based on linear interpolation. Figure 3.25 illustrates the procedure. Before we elaborate on the method of determining the return, we would like to discuss the course of the net present value curve Co. The mathematical function C o ¼ f ðiÞ,

 A, ek , ak , R, n

ð3:49Þ

states that the endogenous variable, the net present value Co, is varied by six exogenous variables, but five of the exogenous variables are set constant. Only the change in the net present value is considered when the calculation interest rate changes. You can recapitulate what effect this has by means of Fig. 3.10. If all payment parameters are unchanged, but the interest rate is changed, this has a direct effect on Co Co = f(i), (A, ek ,ak ,R, n)

Co 1

P1 (i1/Co1)

chord

r calculated (r/0) 0 i2

i1 r true

Co 2

P2 (i2/Co 2)

Fig. 3.25 Determining the yield with the regula falsi (Source: Author)

i

3.7 The Internal Rate of Return Method

125

the net present value. An increase in the interest rate means that a larger part of the given payments must be reserved for the interest claim. This means that less present value is left because the discounting is higher. If the less present value remains, the net present value after deducting the given acquisition payment from the present value is also less. This means that the net present value decreases as the interest rate increases, and increases as the interest rate decreases. That the function is convex downwards is due to the available net present value function in exponential form. In any case, it should be assumed at this point. The assumption is generalised in Sect. 4.5. Now, the actual calculation of the return with the regula falsi: Using two test (or trial) interest rates i1 and i2, two net present values Co1 and Co2 are calculated with the net present value formula for individual discounting. This results in the two function points P1 and P2 in Fig. 3.25, through which the intrinsically unknown net present value function passes with certainty. In the spreadsheet calculation, the number of these points can be increased almost arbitrarily, but that does not result in a mathematically estimated function either, only a sequence of many small linear sections that cannot be resolved by the human eye. However, this is irrelevant for the understanding and accuracy of the arithmetic calculation. The equation of the interpolation line (chord) can now be established according to the two-point form of the linear equation. In the investment calculation terminology used here, this is: Co  Co1 C o2  Co1 ¼ i  i1 i2  i1

ð3:50Þ

The point (i/Co) represents the calculated return and has the coordinates (r/0). Taking this variable into account, the following results are obtained 0  C o1 C o2  C o1 ¼ r  i1 i2  i1

ð3:51Þ

After solving the equation according for the searched yield r we obtain the regula falsi: r ¼ i1  C o1 

i2  i1 Co2  Co1

ð3:52Þ

Since this is a downward convex net present value function that is interpolated, the calculated return overestimates the true return. It is not necessary, as shown in Fig. 3.25, for the interpolation to be carried out with a positive and a negative net present value as the result of the test interest rates. Interpolation is also possible with two positive or two negative net present values. The accuracy of the interpolated return is hardly inferior to an exact value. If the two experimental interest rates i1 and i2 are no more than 1 percentage point away from the yield determined that way, the calculated yield overestimates the true yield in most cases by less than one hundredth of a percentage point.

126

3 Dynamic Investment Calculation Methods

Fig. 3.26 Data set for the calculation of the return (Source: Author)

k

ek

ak

1

300

200

2

500

300

3

700

300

4

400

300

Let us take an example of how the internal rate of return method works. The useful life of the investment object is 4 years, the acquisition payment is 1000 euros. The residual value is 600 euros. The calculation interest rate is 10%. The trial interest rates that you need for determining the net present values are 10% and 12%. You can find more data in Fig. 3.26. Exercise Please determine the return for the data set from Fig. 3.26. Solution First, determine the two net present values with the trial interest rates. The net present value at the trial interest rate 10% is 34.83 euros. You can find the solution in Eq. (3.53). C o ¼ ð300  200Þ  1:11 þ ð500  300Þ  1:12 þ ð700  300Þ  1:13 þð400  300Þ  1:14 þ 600  1:14  1000 ¼ 34:83 euros ð3:53Þ The net present value at the 12% test interest rate is 21.70 euros. You can find the solution in Eq. (3.54). Co ¼ ð300  200Þ  1:121 þ ð500  300Þ  1:122 þ ð700  300Þ  1:123 þð400  300Þ  1:124 þ 600  1:124  1000 ¼ 21:70 euros ð3:54Þ These net present values are then entered into the regula falsi. r ¼ 0:1  34:83 

0:12  0:1 ¼ 0:1123 21:70  34:83

ð3:55Þ

This results in a return of 11.23%. The investment is, therefore, advantageous, as the return is higher than the calculation interest rate.

3.7.2

Special Cases in Determining the Return

If the calculation elements of the investment projects are, in contrast to the general case, available in a special structure, then it is possible to determine the return without applying the calculation-intensive regula falsi.

3.7 The Internal Rate of Return Method

127

Four such special cases are known, three of which lead to mathematically exactly determined returns, i.e. they are not approximate solutions. The four special cases are: • • • •

Perpetual annuity Two-payment case Residual value equal to the acquisition payment and Investment without residual value

3.7.2.1 The Perpetual Annuity The formula of the net present value with infinite useful life is already given in Sect. 3.4.3 (3.56) = (3.16) 1 C o ¼ ð e  aÞ   A i

ð3:56Þ

Here it is immediately clear that after zeroing the net present value, whereby the calculation interest rate (i) is transformed into the yield (r), an exact solving for r is possible. The formula for determining the yield then is: r¼

ð e  aÞ A

ð3:57Þ

The calculation of the yield according to the perpetual annuity is of course only possible if there is indeed an infinite useful life and if the net payments are the same in each period. Exercise Determine the return on an investment with an infinite useful life with constant annual net payments of 15,000 euros and an acquisition payment of 100,000 euros. Solution The return is 15%.

r¼

15, 000 100, 000

ð3:58Þ

3.7.2.2 The Two-Payment Case In the two-payment case (zero bond), there is only one acquisition payment and one residual value. No other calculation elements exist. The net present value formula then is: C o ¼ R  ð1 þ iÞn  A

ð3:59Þ

128

3 Dynamic Investment Calculation Methods

Here too, it is immediately clear that after zeroing the net present value, whereby the calculation interest rate (i) is transformed into the yield (r), an exact resolution to r is possible. The formula for determining the yield then is: rffiffiffi n R 1 ð3:60Þ r¼ A The determination of the yield after the two-payment case is of course only possible if only the residual value and the acquisition payment are actually available as relevant payment parameters. Exercise Determine the return on investment for an investment with a useful life of 4 years, an acquisition payment of 100,000 euros and a residual value of 200,000 euros. Solution: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 200, 000 r¼ 1 100, 000

ð3:61Þ

The return is 18.92%.

3.7.2.3 Residual Value Equal to the Acquisition Payment In the case that the acquisition payment is equal to the residual value, one of the variables can be suppressed because the equality of the two amounts. If the net payments are then also the same in each period, the return can be calculated exactly and without applying the regula falsi. The net present value function then is: C o ¼ ðe  aÞ  DSFni þ A  ð1 þ iÞn  A

ð3:62Þ

By zeroing the net present value, the calculation interest rate (i) converted into the yield (r). The equation then is: 0 ¼ ð e  aÞ 

ð1 þ r Þn  1 1 þA A r ð1 þ r Þn ð1 þ r Þn

ð3:63Þ

By transferring the rear two summands to the left side of the equation and dividing by A, the result is 1

ð e  aÞ ð 1 þ r Þ n  1 1  n ¼ A r ð1 þ r Þn ð1 þ r Þ

ð3:64Þ

By multiplying by the KWF (reciprocal value of the DSF) and cancel the KWF by the Accumulation factor (1 + i)n the result is

3.7 The Internal Rate of Return Method

r¼

129

ð e  aÞ A

ð3:65Þ

as the remaining multipliers are cancelled except for the r. Of course, it is only possible to determine the return on an investment with an acquisition payment equal to the residual value if the residual value and the acquisition payment actually reach the same amount and if the net payments are constant in each period. Exercise Determine the return on an investment with constant annual net payments of 15,000 euros and an acquisition payment and a residual value of 100,000 euros. Solution: r¼

15, 000 100, 000

ð3:66Þ

The yield is 15%.

3.7.2.4 The Investment Without Residual Value In the case of an investment without residual value with constant net payments, the net present value function is as follows: Co ¼ ðe  aÞ  DSFni  A

ð3:67Þ

If the net present value is set to zero, the calculation interest rate (i) transforms to the yield (r). After the function is resolved, the formula is: KWFnr ¼

ð e  aÞ A

ð3:68Þ

This is not a mathematically exact solution for the internal interest rate. The interpolation of the net present value function is replaced by the interpolation of the KWF (or in the case of the reciprocal value: DSF) function. However, this interpolation is not performed mathematically, but can be determined using the financial mathematics table already mentioned. Empirically, the formula in Eq. (3.68) is used to determine a capital recovery factor for a given useful life of the investment object. This value is compared with the values stored in the financial mathematics table in the KWF column and in the relevant line n. Where the empirical value and the table value come as close as possible, the return on investment can be read in the header of the table. This procedure is, as already mentioned, not an exact procedure, but so far a special case that the rather complex regula falsi does not have to be applied. Prerequisites for its possible application are the absence of a residual value and constant net payments.

130

3 Dynamic Investment Calculation Methods

Exercise Calculate the return on investment of the case study from Sect. 3.4.4. Solution This is not a special case of the internal rate of return method, so the regula falsi is to be applied. As far as you have determined the net present values with the spreadsheet in Sect. 3.4.4 and have constructed your table according to the solution path given there, you can now simply change the interest rate in the interest cell and thus automatically obtain the necessary net present values for the regula falsi. If you have chosen this procedure, we suggest you work with the calculation interest rates i1 and i2 of 16% and 17%. If you have worked with the pocket calculator, calculate the net present value formula used for the solution in Sect. 3.4.4 with the second interest rate i2 ¼ 0.17. To reduce the effort, we would then choose 10% as the first interest rate. The calculation result of the regula falsi will then of course also differ. You can also use the results of the horizon value or annuity method for this procedure to obtain the same result. The return for exercise a) is then

r ¼ 0:16  1:24564871 

0:17  0:16 ¼ 0:1664 0:7001421  1:24564871

ð3:69Þ

This results in a return of approximately 16.64%. The return for exercise b) is then r ¼ 0:10 þ 1:77858962 

0:09  0:10 ¼ 0:0936 1:00568249 þ 1:77858962

ð3:70Þ

This gives a return of about 9.36%. For the interpretation of the calculated returns, the following applies in both exercises • A full recovery of the acquisition payment. • Interest payments on all outstanding amounts at the yield. In exercise b), however, the return was much lower than in exercise a). Since the criterion of the internal rate of return method is r  i and the fixed calculation interest rate was 10%, the investment from exercise a) is worthwhile, whereas that from exercise b) would not be worthwhile.

3.7.3

Section Results

In this section, you: • Got to know the internal rate of return method. • Developed the internal rate of return criterion. • Learned the interpretation of the internal rate of return criterion.

3.8 The Dynamic Amortisation Calculation

131

• Got to know special cases of the internal rate of return method and. • Applied the internal rate of return method.

3.8

The Dynamic Amortisation Calculation

Just like the other methods discussed so far, dynamic amortisation calculation determines the absolute advantage of an investment object based on a singular criterion in a world that is considered to be safe, i.e. where all premises are fulfilled. While the net present value, horizon value and annuity methods determine the target values of these dynamics for a given useful life, the technical procedure of dynamic amortisation calculation is comparable to that of the internal rate of return method. With the internal rate of return method, the interest rate, which is now an endogenous, i.e. dependent variable as the result of this method’s calculation, is changed until the target values of the other dynamics mentioned above, the net present value method, horizon value method and annuity method become zero. In dynamic amortisation calculation, on the other hand, the useful life, which now represents an endogenous, that is, dependent variable as the result of this method, is changed until the target values of the other dynamics mentioned above, the net present value method, horizon value method and annuity method, become zero. The dynamic amortisation calculation determines the dynamic amortisation time ndyn. The dynamic payback period is the time at which the net present value, horizon value and annuity are zero. The target criterion of the dynamic amortisation calculation considers the time period of an investment after which the acquisition payment and the interest on all outstanding amounts at the calculation interest rate (i) is achieved from the net payments. As with the other dynamics, the criterion of dynamic amortisation calculation is presented as a decision criterion. ndyn  nmax

ð3:71Þ

This is the only decision criterion that exists for dynamic amortisation calculation. An investment is worthwhile if the dynamic amortisation time is not longer than the maximum specified amortisation time (nmax). This criterion is thus consistent with the other dynamics, since at the point of the dynamic payback period, the net present value, horizon value and annuity are just zero. From this point on, an investment according to the other dynamics is worthwhile, since their target values then become positive. If the dynamic amortisation time is less than the maximum specified amortisation time, the object is still worthwhile according to the dynamic amortisation calculation method. Conversely, if an

132

3 Dynamic Investment Calculation Methods

investment object does not reach its dynamic amortisation time within the specified maximum amortisation time, the target values of the other dynamics are still negative after the maximum amortisation time has expired, or do not meet their criteria, and these methods would also reject the investment. All five dynamic methods that have now been developed, therefore, offer only one criterion each, which is checked for fulfilment or non-fulfilment. For this reason, there are no qualitative differences between the methods. A dynamic amortisation time (ndyn) below the maximum permissible amortisation time (nmax) means • A full recovery of the acquisition payment. • Interest paid on all outstanding amounts at the calculation interest rate i and. • A surplus of unknown size. Evaluation The criterion of the dynamic amortisation calculation is only checked for fulfilment or non-fulfilment in the dynamic investment calculation for the assessment of the absolute advantageousness of an investment. To that extent, the criterion is also useful. Consistency with the criteria of other dynamic methods has already been demonstrated. However, this criterion does not follow the economic principle, as no surplus is determined in its amount, but only the point is sought at which the capital invested, including the interest requirement, is recovered. Such a question would still be interesting to analyse under a risk question, but the dynamic excludes the consideration of a risky world due to the assumptions made. To determine the dynamic amortisation time in the general case, the net present value, horizon value or annuity equation would have to be set to zero in accordance with the above definition and would have to be dissolved according to the useful life, which would then correspond to the dynamic amortisation time. The net present value equation provides an example to illustrate this. (3.72) = (3.8) Co ¼

n X

ðek  ak Þ  ð1 þ iÞk þ R  ð1 þ iÞn  A

ð3:72Þ

k¼1

After zeroing the net present value, the useful life (n) changes to the dynamic payback period (ndyn). 0¼

ndyn X

ðek  ak Þ  ð1 þ iÞk þ R  ð1 þ iÞndyn  A

ð3:73Þ

k¼1

Since in the general case, as with the internal rate of return method, this is again a higher order equation, it is obvious that solving the equation for ndyn and thus determining a mathematically exact result of the dynamic payback period is not immediately possible.

3.8 The Dynamic Amortisation Calculation

133

Co

Co = f(n), (A, ek ,ak , R, i) Co2 P2 (n2/Co 2)

chord n true 0 n1

Co1

n calculated

n2

n

P1 (n1/Co 1)

-A

Fig. 3.27 Determining the dynamic payback period (Source: Author)

Except in special cases where the data set of the investment project has a special structure, only an approximate solution for determining the dynamic payback period is possible. This is again done using the regula falsi. The procedure is similar to that of the internal rate of return method, except that the net present value is now considered in relation to the useful life and not the interest rate. You can see this in Fig. 3.27. Before we elaborate on the determination of the dynamic payback period, we would like to discuss the course of the net present value curve Co. The mathematical function  Co ¼ f ðnÞ, A, ek , ak , R, i ð3:74Þ states that the endogenous variable, the net present value Co, is varied by six exogenous variables, but five of the exogenous variables are set constant, only the change in net present value when the useful life changes are considered. You can recapitulate what effect this has in Fig. 3.27. If all payment variables are unchanged, e.g. if the residual value does not fall and ongoing payments do not rise, but the useful life of a worthwhile investment is

134

3 Dynamic Investment Calculation Methods

extended, the net present value as an asset accumulating variable rises further and further, albeit at a much lower rate, since payments further in the future are discounted more and more. Therefore, the function is not linear, but upward convex, the net present value converges towards a limit value. In any case, this is the approach used in dynamic investment calculation to consider the net present value as a function of the useful life. In Chap. 5, which deals with the determination of the optimal useful life, a different functional relationship between net present value and useful life is seen, where all calculation elements are then considered variable over time. Now, the actual calculation of the dynamic amortisation time with the regula falsi, which you can see in Fig. 3.27: Using two test useful lives n1 and n2, two net present values Co1 and Co2 are calculated with the net present value formula for individual discounting. This results in the two function points P1 and P2, through which the intrinsically unknown net present value function passes with certainty. In the spreadsheet calculation, the number of these points can be increased almost arbitrarily, but that does not result in a mathematically estimated function either, only a sequence of many small linear sections that cannot be resolved by the human eye. For the understanding and the accuracy of the arithmetic determination this is however insignificant. The equation of the interpolation line (chord) can now be established according to the two-point form of the linear equation already known from Sect. 3.7.1. In the investment calculation terminology used here it is then: Co  Co1 C o2  Co1 ¼ n  n1 n2  n1

ð3:75Þ

The point (n/Co) represents the calculated dynamic payback time and has the coordinates (ndyn/0). Taking these variable names into account, the following results are obtained 0  C o1 C  Co1 ¼ o2 ndyn  n1 n2  n1

ð3:76Þ

After solving the equation according to the dynamic payback time we are looking for, we get the regula falsi: ndyn ¼ n1  Co1 

n2  n1 C o2  C o1

ð3:77Þ

Since it is an upwardly convex net present value function that is interpolated, the calculated dynamic payback time overestimates the true dynamic payback time. It is not necessary, as shown in Fig. 3.27, for the interpolation to be performed with a positive and a negative net present value as a result of the test useful lives. Interpolation is also possible with two positive or two negative net present values. The accuracy of the dynamic payback period interpolated with the regula falsi can be considered accurate. Since the curved function, shown in Fig. 3.27, actually has a staircase shape due to the assumption of the additional slope, it is possible to

3.8 The Dynamic Amortisation Calculation

135

calculate the payback time by the selection of the corresponding test useful lives, an exact dynamic amortisation time can be determined on the assumption of payments in arrears. Exercise Calculate the dynamic payback time of the case study from Sect. 3.4.4. Solution The regula falsi should be applied here as well. If you have determined the net present values in Sect. 3.4.4 with the spreadsheet calculation, you can obtain the net present values for shorter useful lives for the regula falsi by summing up the in-payments of a shorter useful life. If you have chosen this procedure, we suggest you work with the useful lives n1 and n2 of 5 and 6 years. If you have worked with the pocket calculator, calculate the net present value formula used for the solution in Sect. 3.4.4 with the second useful life n2 ¼ 5. You can also use the results of the horizon value or annuity method for this procedure to obtain the same result. The dynamic amortisation time for exercise a) is then

ndyn ¼ 6  15:39904073 

56 ¼ 5:37 8:87333826  15:39904073

ð3:78Þ

This results in a dynamic payback period of 6 years. Mathematically you get a result of about 5.37 years. However, since we have assumed that the payments will be made in arrears, it makes no sense to show a result during the year. In this case, therefore, you should round up to the next whole numbered year. Therefore, the investment is worthwhile because it just about meets the criterion, as the dynamic amortisation time (ndyn) is equal to the maximum permissible amortisation time (nmax), which is determined by the length of the useful life. Therefore, • A full recovery of the acquisition payment • Interest payments on all outstanding amounts at the calculation interest rate i and • A net present value surplus of approximately 15.399 million euros is achieved. In this special case, in which the dynamic payback period coincides with the maximum useful life, the surplus can actually be indicated, as it is identical to Co. It does not make sense to process exercise b) because the net present value is still negative after 6 years, and the dynamic payback period is, therefore, above the maximum dynamic payback period. Section results In this section, you: • Got to know the dynamic amortisation calculation. • Developed the criterion of dynamic amortisation calculation.

136

3 Dynamic Investment Calculation Methods

• Learned the interpretation of dynamic payback time. • Edited the formula for the dynamic amortisation calculation. • Applied the dynamic amortisation calculation.

3.9

Case Study

Inspired by reading this book, you decided to change your life and want to open up a beach bar offering cocktails. The purchase price is 150,000 euros, your calculation interest rate is 10% and the payments are made in arrears. You plan a useful life of 6 years, the residual value of the bar will then be zero. As further calculation elements, you assume the values shown in Fig. 3.28. Exercises Exercise a) Calculate the net present value, horizon value and annuity of the investment object. Exercise b) Determine the dynamic payback period of the object. Exercise c) Determine the rate of return of the object. If you use the regula falsi, please use the net present value from exercise a). Use 16% as the second test interest rate. Solutions You can calculate the solution in a spreadsheet or with a pocket calculator. Following the familiar pattern, we present the solution in the spreadsheet in Fig. 3.29. Since it is a discounting and summation of the calculation elements, there are several k 1 2 3 4 5 6

ek (euro) 80,000 85,500 51,950 53,020 99,850 96,250

ak (euro) 40,179 45,679 12,129 13,199 60,029 56,429

Fig. 3.28 Data set for the dynamics case study (Source: Author)

k

ek 1 2 3 4 5 6

80 85.5 51.95 53.02 99.85 96.25

ak 40.179 45.679 12.129 13.199 60.029 56.429

NEk BW Co 39.821 36.201 -113.799 39.821 32.910 -80.889 39.821 29.918 -50.971 39.821 27.198 -23.773 39.821 24.726 0.953 39.821 22.478 23.431

Fig. 3.29 Dynamics case study solution* (Source: Author)

Cn DJÜ -201.602 -125.179 -143.300 -46.608 -90.298 -20.496 -42.115 -7.500 1.688 0.251 41.509 5.380

3.9 Case Study

137

correct ways of formulating the spreadsheet. So far, the solution shown is only a suggestion from which you can deviate as long as you come to the same result. For the solution with the pocket calculator, the net present value formula with the possibility of applying the discounting sum factor is a good starting point. Exercise a) C o ¼ 39, 821  DSFn¼6 i¼0:1  150, 000 ¼ 23, 430:85 euros

ð3:79Þ

the horizon value can then be calculated as: C n ¼ Co  AUFn¼6 i¼0:1 ¼ 41, 509:18 euros

ð3:80Þ

The annuity can be calculated as: € ¼ C o  KWFn¼6 ¼ 5379:88 euros DJU i¼0:1

ð3:81Þ

In the case of exercise (a), therefore, it is a worthwhile investment that meets the relevant criteria. The interpretation then tells us that: • A full recovery of the acquisition payment • Interest is paid on all outstanding amounts at the discount rate i and • A present value surplus or terminal value surplus or periodic surplus of Co or Cn or DJÜ is achieved. Of course, this interpretation only applies to the assumptions made. This means in particular that the calculation elements occur as planned and that interest can be calculated on all amounts at the calculation interest rate. Exercise b) Here the solution is calculated using the regula falsi, ndyn ¼ n1  Co1 

n2  n1 ¼ 5 years C o2  C o1

ð3:82Þ

as far as you work with the pocket calculator. In the spreadsheet, the root in connection with the assumption of the arrears of payments can be read directly in Fig. 3.29. The dynamic payback period is 5 years, as the net present value becomes positive for the first time after 5 years. This makes the investment worthwhile according to this criterion as well, since this period is below the maximum dynamic payback period of 6 years. Exercise c) Here the solution is obtained either via the regula falsi or via a special case solution

138

3 Dynamic Investment Calculation Methods

r ¼ i1  C o1 

i2  i1 Co2  Co1

ð3:83Þ

If you have calculated the net present values with the spreadsheet and have built up your table according to the solution described in Sect. 3.4.4, you can now simply change the interest rate in the cell and thereby automatically obtain the necessary net present values for the regula falsi. If you have chosen this procedure, we suggest you work with the calculation interest rates i1 and i2 of 15% and 16%. You can also use the results of the horizon value or annuity method for this procedure to obtain the same result. We chose the net present value method for the documentation. The return for exercise c) is then a percentage of approximately 15.18%. r ¼ 0:15  701:88535 

0:16  0:15 ¼ 0:1518 3270:13139  701:88535

ð3:84Þ

If you work with the pocket calculator, you can use the special case formula of the investment without residual value, KWFnr ¼

39, 821 ¼ 0:265473 150, 000

ð3:85Þ

You compare the calculated empirical value of 0.265473 with the table values in the financial mathematics table in the KWF column and in line n ¼ 6. Where the empirical value comes as close as possible to the value in the table, you can read the return in the table header. i ¼ 0:15 0 Co>0 DJÜ>0 r>i

−−−−−−−−−−−−−

DJÜ = f (i ), ( A, ek , a k , R, n) −−−−−−−−−−−−− Co = f (i ), ( A, e , a , R, n) k k −−−−−−−−−−−−−

Cn = f (i ), ( A, ek , a k , R, n)

Cn=0 Co=0 DJÜ=0 r=i

Cn DJU € new GU

ð5:30Þ

At the end of the year, the old object is then replaced because for the second year it applies: € k,old < DJU € new , GU

ð5:31Þ

as the marginal surplus of the old object in the second extension year is only 16,500 euros. For the investment decision of exercise b), the relevant formula for the old object is " # n X
k € GUold ¼ ðek
ak Þ  ð1 þ iÞ
R0  KWFRND þ Rn  RVFRND ð5:32Þ i

i

k¼1

The solution can be seen in column O in Fig. 5.23 with the results of the old object. The marginal surplus (which corresponds to a DJÜ in terms of the calculation method) of the old object is 17,935.79 euros. The average annual surplus of the new object amounts to 24,018.13 euros each year, as can be seen in cell M4 in Fig. 5.24, since the maximum DJÜ then applies to each year of the useful life. Therefore, the investment decision is to immediately replace the old object because the following applies € old < DJU € new GU

5.5.4

ð5:33Þ

Section Results

In this section, you learned: • To make the distinction between determining the optimum useful life and the optimum replacement date. • To distinguish between the determination of the optimal replacement date in the case of annual possibilities and after a multi-year period. • To determine the optimal time for replacement in case of annual replacement.

234

5 Optimum Useful Life and Optimum Replacement Time

• To determine the optimal time for replacement in the case of a replacement after a multi-year period. • To identify special cases with simplified calculation methods. • To apply the developed criteria to practical cases.

5.6

Case Study

Inspired by reading this chapter, you have decided to change your life. Therefore, you are rethinking your involvement in some potential and already running economic projects. The acquisition payment (A) for a project is 100,000 euros. Your calculation interest rate (i) is 10%. The useful life (n) is a maximum of 6 years. The annual in-payments (ek) from the sale of your products can be considered constant at 41,200 euros per year. You assume the calculation elements from Fig. 5.25 for the future.

5.6.1

Exercises

Exercise a) Determine the optimum useful life of the investment object if it is a one-time investment. Exercise b) Determine the optimum useful life if the following investments are equal in net present value and repeated infinitely. You can include the data from exercise a) for this purpose. Exercise c) At the beginning of k ¼ 5, and only there, you will be offered a successor model with a legally defined useful life of 6 years and a net present value of 10,000 euros as a replacement for the current investment project. Calculate whether an immediate replacement should take place or whether the old object should be carried out until the planned end.

5.6.2

Solutions

You can calculate the solution in a spreadsheet or with a pocket calculator. Following the familiar pattern, we present the solution in the spreadsheet. Since it is a matter of discounting and summing up the calculation elements, there are various Fig. 5.25 Data set for the case study (Source: Author)

k 1

R (euro) 70,000

ak (euro) 8,000

2

50,000

12,000

3

40,000

20,000

4

30,000

30,000

5

24,000

41,000

6

20,000

54,000

5.6 Case Study

235

correct procedures for formulating the solution in the spreadsheet. So far the solution is only a suggestion from which you can deviate, as long as you come to the same result. Due to the structure of the data set, exercises a) and b) also offer a special case solution. Figure 5.26 shows all possible solutions for exercise a) and exercise b). The solution for exercise a) is based on the criteria Co ¼ max! or GE ¼ GA on the secondary condition of increasing marginal payouts. The given data set is in columns A to D of the solution in Fig. 5.26. The special case solution GE ¼ GA on the secondary condition of increasing marginal payouts can be seen in columns C, F-H. The marginal payouts increase again after the third year and are already greater than the marginal in-payments after 4 years, so the optimum useful life is 3 years. The general solution with the criteria Co ¼ max! is shown in columns N to R. The net present value is at a maximum after the third year, so the optimum useful life is 3 years. The solution for exercise b) is based on the criteria DJÜ ¼ max! or DJA ¼ min! The special case solution DJA ¼ min! is shown in columns H to M in Fig. 5.26. The DJA is minimal after the third year, so the optimum useful life is 3 years. The fact that the optimum useful life is shorter for repeated investments than for one-time investments also applies here, but this is not visible in this data set due to the assumption of payments in arrears. The general solution with the criteria DJÜ ¼ max! is shown in columns R to T. The annuity is at a maximum after the third year, so the optimum useful life is 3 years. This was already clear without calculation. Since the data set only contains a positive net present value, as already determined in exercise a), there can only be one positive DJÜ due to the mathematical structure of the KWF. The solution for exercise c) is based on the criterion € old  DJU € new : GU

ð5:34Þ

The marginal payouts cannot be used, since only a net present value is known for the new object, but no in-payments are known. Since only replacements after a multi-year period are possible, this formula applies to the old object: " # n X
k € old ¼ GU ðek
ak Þ  ð1 þ iÞ
R0  KWFRND þ Rn  RVFRND ð5:35Þ i

i

k¼1

Since the GÜ of the old object is negative as seen in Fig. 5.27, immediate replacement is necessary, since the DJÜ of the new object must be positive because its net present value is positive. The DJÜ is € ¼ Co  KWFn ¼ 2296:07: DJU i

ð5:36Þ

Fig. 5.26 Solution to the case study, area of useful life* (Source: Author)

236 5 Optimum Useful Life and Optimum Replacement Time

5.7 Summary

k

ek (GE) 0 1 2

Ro= i=

41200 41200 30000 0.1

ak

237

Rk NEK Abf NEk * Abf ΣNEk * Abf Rn * Abf KWF GÜ/DJÜ 30000 41000 200 0.909091 181.82 181.82 1.1 -13752.38 54000 20000 -12800 0.826446 -10578.51 -10396.69 16528.93 0.576190 -13752.38

Fig. 5.27 Solution to the case study, area substitute time* (Source: Author)

5.7

Summary

In this chapter, you have learned not to take the useful life as a given date, but to optimise it economically. This optimisation is carried out on assumptions that must be made, beyond the general assumptions of dynamics, in order to determine a clear optimum useful life or a clear optimum replacement date. The assumptions mean that the optimisation cannot be used for all questions of interest in practice. The optimisation of the useful life of an investment object is carried out on two different aspects. When determining the optimum useful life, the economic useful life is mathematically optimised before the start of the investment. When determining the optimal replacement date, a check is made to determine whether the useful life should be adjusted for an investment that is already in progress. The determination of the optimum useful life is based on a distinction between one-time and repeated investments. The decision criteria differ. In the case of a one-time investment, the net present value must be maximised while varying the useful life. The optimal economic useful life is reached in the year of the useful life, in which the net present value is at its maximum. In the case of investments that are repeated infinitely and have the same net present value, the annuity must be maximised while varying the useful life. The optimal economic useful life is in the year of the useful life in which the annuity is at a maximum. The determination of the optimal replacement time is based on the case distinction whether an annual replacement is possible or after a multi-year period. The decision criteria then differ. In principle, the replacement of an ongoing investment chain with an ongoing infinitely repeated new investment chain of equal net present value is carried out if the marginal surplus in the current year or in the current period of the ongoing investment object in the ongoing old investment chain is smaller than the average annual surplus of an investment object in a new infinitely repeated investment chain of equal net present value. The case distinctions of the proceedings from Chap. 5 addressed in this summary are shown graphically in Fig. 5.28.

GA = ak + (Rk-1-Rk ) + iRk-1

5.5.1

formula

section

not constant

5.5.1

GÜ = ek -ak - (Rk-1-Rk ) - iRk-1

GAold ≥≤ DJÜnew

n

5.5.2

5.5.2

GÜk = (Σ (ek -ak ) * (1+i)-k - R0 + Rn * Abf) * KWFRND

not constant

general

× (1 + i) – k × KWFi n

GAalt = (Σ ak *(1+i)-k + R0 - Rn Abf ) * KWFRND

k =1

k

Σ GA

GAold ≥≤ DJÜnew

constant

special case

long-term replacement

5.4.2.4

Fig. 5.15

DJA =

DJA=min!

constant

special case

infinitely repeated investment

GAold ≥≤ DJAnew

5.4.2.2

Fig. 5.28 Case distinctions and criteria of useful life problems (Source: Author)

GAold ≥≤ DJAnew

criteria

constant

general

type

payments

annual replacement

5.4.1.1

case

special case

5.4.1.2

section

replacement problem

Fig. 5.5

figure

Fig. 5.12

account development

Co = Σ (ek -ak ) * (1+i)-k + R * (1+i)-n - A

GA = ak + (Rk-1-Rk ) + iRk-1

formula

Fig. 5.3

individual

Co=max!

GE=GA, NB rising GA

not constant

individual

finite repeated investment

criteria

not constant

general

constant

special case

type

payments

singel investment

case

useful life

5.4.2.3

---

DJÜ=Co *KWF

DJÜ=max!

not constant

general

238 5 Optimum Useful Life and Optimum Replacement Time

6

Investment Decisions in Uncertainty

6.1

Objectives

The aim of this chapter is to take risk into account in investment decisions. Up to now, we have assumed that all calculation elements are known with certainty, both when applying the static and dynamic investment calculation methods in Chaps. 2 and 3, and when making investment programme decisions in Chap. 4 and investment duration decisions in Chap. 5. This is, of course, only true in very rare cases. For example, if we consider a savings bond with a fixed interest rate as a form of financial investment and hold it for the entire term, then from a practical point of view all calculation elements are known with certainty for the entire term. Some theoretical concerns about this view should be ignored here. If, in the context of a comparison of alternatives, this financial investment is to be compared with an operational investment, in which, for example, a new branch of industry is to be established in the company, it usually makes no sense to compare the two classically determined net present values considering fictitious investment and the removal of the reinvestment premise. The probability that the data of the planned operational investment will occur exactly as planned is much lower. Even if the investment data were planned very elaborately and are based on industry information and experts, the residual value of a production facility is, for example, much more difficult to determine than the repayment amount of a fixed-rate savings bond. This was already indicated in Sect. 1.9. In the case of two equal net present values or other target values of the dynamic investment calculation methods, the rational investment decision of a cautious businessman would certainly be to invest in the safe savings certificate instead of the investment alternative with the same net present value, which, however, would be more likely to deviate from the planned value. But what should the rational investment decision be if the net present value of the safer

Supplementary Information The online version of this chapter (https://doi.org/10.1007/978-3030-62440-8_6) contains supplementary material, which is available to authorized users. # The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. Poggensee, J. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-62440-8_6

239

240

6 Investment Decisions in Uncertainty

investment is less than the net present value of the more uncertain alternative? This chapter deals with the rational decision of the investor in such a situation. There are generally two approaches to taking risk into account in investment calculations. The theoretically better approach is to consider the risk–benefit of an investment decision. Here, the investor must specify a risk–benefit function or, if the probability of occurrence of the environmental situation is unknown, the investor uses the so-called ad hoc decision rules as defined decision routines. This procedure of determining the risk–benefit is quite complex and, theoretically, requires a high degree of abstraction. Furthermore, risk–benefit functions of investors are not intertemporally stable, i.e. the investor’s attitude at the beginning of the investment does not have to be valid at the end of the investment anymore. We will deal with these techniques in Sects. 6.5–6.8 in this chapter. As an alternative to this procedure, there are the correction procedures and sensitivity analyses, which change the data set of the considered investment object in order to determine the effects on the target values of the investment calculation. These procedures are, theoretically, not very complex and are also strongly criticised in the academic investment theory, but in practice, they are widely used and they are also a good hand to structure decision problems of investment calculation under uncertainty. These techniques are discussed in Sects. 6.3 and 6.4 in this chapter. The consideration of risk is always an important issue for an investor. This will become even more important in the future due to corporate financing reasons. The consideration of risk is particularly important for German companies, as they generally have considerably less equity capital than their international competitors. The regulations of Basel II and III, which aim to achieve an international competition equality of banks, determine how much liable and limited equity capital a credit institution must back for a transaction. Previously, for Basel I, this was a uniform 8%, but today it varies according to the risk of the individual transaction. The riskier a transaction is for a credit institution, the more it must be backed by equity capital, which is then not available for other profitable activities. A debtor or credit-seeking company should, therefore, always give its creditor or credit institution the opportunity to be classified as safe as possible. To this end, credit institutions generally use ratings to assess the creditworthiness of their debtors. This assessment also includes risk analyses that the debtor carries out in his company. This makes the risk analysis of investments with the current regulations of Basel II and III even more important for companies. In order to achieve these goals, it is necessary to follow the offered exercise calculations independently with the pocket calculator or spreadsheet. Enjoy your work!

6.2

Data Uncertainty as a Decision-Making Problem

In this section, we will now look at the topic of uncertainty. When applying dynamic investment calculation methods, we have so far always assumed that the world is safe. On the one hand, this is certainly wrong for most operational investment questions, on the other hand, the proper consideration of uncertainty in the investment decision is difficult to grasp, which has several reasons.

6.2 Data Uncertainty as a Decision-Making Problem

6.2.1

241

The Concept of Risk

First, however, we will deal with the concept of risk. Risk describes the fact that in reality, a value may deviate from the planned value. If the deviation is perceived as positive, this is the area of opportunity; if the deviation is negative, this is the actual risk. Whether the investor perceives the possibility of the deviation as positive is subjective. In general, we distinguish: • Risk seeking • Risk neutrality • Risk aversion The attitude of risk seeking prefers an uncertain situation to a safe situation. The attitude of risk neutrality is an indifferent attitude towards a risk-free and risky situation, each with the same expected value. The risk-averse attitude tries to avoid risk. Thus, a lottery player is willing to take risks because the expected value of a win is less than the stake. The lottery player probably derives his pleasure from the possible batch size transformation, i.e. from the fact that he can achieve a high profit, even if with a very low probability. The expected value of the target value is measured as the product of the target value and the probability of its occurrence. For example, if the prize in a lottery is 1000 euros and the probability of winning is 10%, the expected value of the prize is 100 euros, because 1000
0.1 ¼ 100. In accordance with the principles of commercial caution, we should always imagine a company to have at least a risk-neutral, usually risk-averse investor. A risk-averse investor is prepared to forego profit or pay a premium for transferring a risky situation into a safe situation. This premium or waiver of profit is called the certainty equivalent. A risk-averse investor would, therefore, be prepared to pay a price of exactly 100 euros for our small lottery in the previous paragraph, because his risk neutrality means that he is indifferent to the fact that he can access the safe amount of 100 euros by paying price, because in return he gets the chance to participate in the lottery, with an expected value of 100 euros. If he were to participate in the lottery an infinite number of times, this would have no effect on his financial situation, because if the lottery were repeated infinitely, the expected value of the winnings would be an average of all lottery participation. If he participates only once, he waives a safe 100 euros by paying the price and either receives zero euro back, with a probability of 90%, or he receives 1000 euros back, with a probability of 10%. For these reasons, the risk-averse lottery participant is only prepared to pay less than 100 euros for participating in the lottery. How much less depends on the degree of his risk aversion. Please ask yourself how much you would be willing to pay to participate in this lottery. Please make a note of this value. Then ask your friends how much they would be willing to pay to participate in such a lottery and write that down as well. After reading the entire chapter, ask yourself this question again. Then compare the values. You will probably be surprised at the different assessments that people, with

242

6 Investment Decisions in Uncertainty

whom you may sit on the same decision-making bodies in companies, come to in the same decision-making situation.

6.2.2

Reasons for Risk in the Investment Decision

First of all, one reason for the risk lies in the data collection itself. While we have assumed a secure world, the data is taken for granted and already given for the investment calculation, experts in the departments of investment calculation, controlling or accounting have probably thought thoroughly about the calculation elements used and determined them from old projects, from industry data collections or from experts. So, while these were estimates that we assumed were certain, they may have been quite close to reality. When considering risk, we now no longer work with a deterministic value for the calculation elements, but with an expected value and a probability distribution, for example, measured by a variance. The calculation element is, therefore, understood as a random number whose value fluctuates around the mean value according to the given probability distribution. But whether the expected value we plan and its probability distribution will occur in reality is just as open as the planning of deterministic values. The difference here is necessarily the higher planning effort involved in data planning. The second problem is the investment decision itself. In determining two net present values in the comparison of alternatives, we have chosen the investment alternative with the higher net present value but had to believe in the validity of the assumptions in the dynamic model. When making an investment decision taking risk into account, the investor now has to choose between alternatives in a selection situation where one object has, for example, a higher expected value but also a higher variance. For this purpose, the investment calculator must know his risk– benefit function and this should also be intertemporally stable so that he considers his decision to be reasonable even after some time. Here, the decision-maker is required to have a high degree of abstraction, which easily loses its relation to practical operational activities. The situation becomes complex when a decisionmaking committee is to bring about an investment decision in a collegial manner. As an alternative to using the benefit function, simpler decision routines can be applied or variations can be made to the data set. The third problem is the realisation of the investment decision, at least if the project is a project without frequent implementation. If any kind of investment calculation procedure, taking risk into account, has advised us to carry out a project because it promises a large profit with, for example, an 80% probability, it can, in reality, lead to a loss, because in reality we have a discrete occurrence of a situation and the probability calculation only tells us that if we carry out the project 100 times, we will make a large profit in 80 times, which is not the case in 20 times. Nevertheless, the application of planning techniques under risk makes sense even in this situation, because in the comparison of alternatives the better option is chosen with a higher probability. The actual occurrence of the situation, in reality, is then random

6.2 Data Uncertainty as a Decision-Making Problem

243

in the statistical sense, i.e. cannot be influenced by the entrepreneur, just as the entrepreneur is at the mercy of changes in the law, for example. Since these situations cannot be influenced by the entrepreneur, they are not subject to his rational decision.

6.2.3

The Importance of Considering Risk in the Investment Decision

Despite all these critical aspects, the analysis of investment projects under risk is of course always a valuable structuring of the decision-making situation, which should always be carried out if the effort involved is justified. In addition, the concept of certainty equivalents is of course a very valuable approach for an investor to determine the price for himself to switch from an uncertain situation to a safe situation. In principle, the entire range of instruments of academic decision theory can be used for risk analysis in investment calculation, i.e. with an investment problem as an application example of decision theory. For example, there are techniques for one-dimensional target functions and techniques for multidimensional target functions available. For example, a technique with a one-dimensional target function would be the maximisation of the expected value, possibly taking into account a riskrelevant restriction, i.e. comparable to the net present value method in the safe world. Multidimensional target functions are also possible but can only be processed reasonably in IT and are not considered here. Besides the focus on the target function, the risk can be considered in different ways. In a very simple form, this is possible by changing the data. The degree of uncertainty is also important. If the economic success of an investment depends on the reaction of a competitor, we are in the field of academic gaming situations. This area is comparable to a game of chess. A rationally acting competitor, whose target function is known, and who has relatively few and completely known options available in a given situation, influences the economic success of our previous investment, here our previous move, considerably. In practice, these situations are much more complex than a chess game, since the competitor’s target function and his courses for action are probably not known. For example, the reaction of Boeing to the design of the Airbus A 380 was much more difficult to predict at the start of construction. Such game-theoretical decision situations are not taken into account here. In situations of uncertainty, the economic outcome of an action is not influenced by a systematically acting competitor, it is, therefore, not a game-theoretical situation. The result of an uncertainty situation depends on chance. These situations are divided into uncertainty situations and risk situations. In uncertainty situations, all possible environmental combinations that may occur are known, but no probability of occurrence can be assigned to them. In risk situations, all possible environmental combinations that may occur are known and the probability of their occurrence can be assigned to them.

244

6 Investment Decisions in Uncertainty

For uncertainty situations, decision concepts are presented in Sect. 6.6, and for risk analysis is presented in Sects. 6.7 and 6.8. The degree of uncertainty in investment decisions can often be improved by obtaining information. The procurement of information as an economic problem, with the associated techniques in which additional information is sought until the marginal costs of the information search correspond to the marginal benefit of the advantage thus gained, are not considered here; the degree of uncertainty is taken as given.

6.2.4

Section Results

In this section, you: • • • • • •

Got to know an overview of the concept of uncertainty. Learned to understand risk as an expected value and as a probability distribution. Learned different forms of the investor’s risk attitudes. Recognised that data collection under uncertain information is a problem. Recognised that the rational investment decision under risk is a problem. Learned about the application problems of the decision techniques under risk on singular projects. • Learned to accept the decision techniques under risk as the best available decision routine, knowing that the occurrence of the planning situation, in reality, does not necessarily happen and that ex post the decision may have been suboptimal.

6.3

The Correction Procedures

Correction procedures are a very trivial and non-systematic way of considering uncertainty. Uncertainty is considered via the subjective change of the originally determined calculation elements by either: • Changing the calculation interest rate • Changing the useful life • Changing the considered payment parameters Based on the caution principle, payouts and the calculation interest rate are naturally increased, while in-payments and useful life are shortened. The effects of uncertainty are thus recorded summarily rather than analytically. When all variables are changed, e.g. also the calculation interest rate, which should actually be fixed as the investor’s subjective minimum interest requirement, variables that are not affected by the risk are also varied. By correcting the calculation elements in this way, even a very pessimistic and not probable situation is used as a basis for decision-making. Another problem is that if different operational instances are involved in data collection, e.g. procurement, production or sales, they may have

6.3 The Correction Procedures

245

already applied different deduction for risks or risk premiums when providing the data. Thus, a rational investment decision by the management is finally no longer possible. In spite of all the criticism at the beginning of this chapter, the techniques of the correction procedures are to be presented here, as they are widely used in practice and the consequences of their application are thus clearly visible in some examples.

6.3.1

Correction Procedure in Detail

First of all, we would like to introduce the individual procedures for changing the calculation interest rate: • • • •

The lump-sum procedure The double discounting The different interest rates for incoming and outgoing payments The different interest rates of the payments depending on their distance from the present

In the case of the lump-sum procedure the interest rate is increased to take account of the risk of a higher discounting of payments expected in the future. In the case of double discounting, the payments already discounted at present value at the calculation interest rate are again discounted with a risk interest rate for the corresponding term. The double discounting method yields values quite similar to the lump-sum method if the product of the interest rates of double discounting (1 +i)
(1+i) is the interest rate of the lump-sum method. The procedure for the other two methods of changing the discount rate is probably self-explanatory. We will apply these procedures to an example in the next chapter. When the useful life is changed, the planned useful life is shortened by truncating the planning data of the last year or years. This procedure should take the risk into account in such a way that if a positive target value of a dynamic investment calculation procedure is already achieved with a shortened useful life, it can be assumed that the investment is worthwhile even under risk. In this respect, the procedure is identical to the procedure of the dynamic amortisation calculation. However, it is questionable to simply ignore planning data of the last few years if, for example, positive or strongly negative residual values are not taken into account. If the payment amounts change, incoming payments are reduced by a percentage to be determined and outgoing payments are increased accordingly. This procedure is also problematic if an absolute value as a dynamic target value, e.g. the net present value, is the basis for the decision. This is because a blanket change in incoming and outgoing payments by, for example, 10% changes these variables absolutely in other dimensions if they were available as an output parameter in different dimensions. A 10% reduction on in-payments of 100,000 euros reduces them by 10,000 euros,

246

6 Investment Decisions in Uncertainty

while an increase in payouts of 15,000 euros by 10% increases them by 1500 euros. The effects on the net present value are significant.

6.3.2

Application Example for the Correction Methods

We will look at these aspects in their effects on an example. The following data set applies: The acquisition payment is 100,000 euros. All values in Fig. 6.1 are given in euro. No residual values exist. Exercises: Exercise a) Calculate the net present value at the calculation interest rate of 8%. Exercise b) Calculate the net present value at a calculation interest rate of 10%, increased by the lump-sum procedure. Exercise c) Calculate the net present value with double discounting using the interest rates 8% and 2%. Exercise d) Calculate the net present value by discounting all in-payments with 10% and all payouts with 6%. Exercise e) Calculate the net present value by discounting the net earnings of year 1 with 6% and the net earnings of all subsequent years with an interest rate increased by 1 percentage point. Exercise f) Calculate the net present value by calculating with in-payments (ek) reduced by 5%. The calculation interest rate is 8%. Exercise g) Calculate the net present value with payouts (ak) increased by 5%. The calculation interest rate is 8%. Exercise h) Calculate the net present value by calculating with 5% less in-payments (ek) and 5% more payouts (ak). The calculation interest rate is 8%. Solutions: We will show you the solution in the spreadsheet. Since it is a matter of discounting and summing up the calculation elements, there are various correct procedures for formulating the solution in the spreadsheet. So far, the shown solution is only one suggestion, from which you can deviate as long as you come to the same result. k

ek(euro)

ak(euro)

1

75,000

40,000

2

75,000

40,000

3

75,000

40,000

4

85,000

40,000

Fig. 6.1 Data set for the application of the correction methods (Source: Author)

6.3 The Correction Procedures

k 0 1 2 3 4 A= i1 = i2 = i3 = i4 = i5 = i6 = i7 =

ek ak NEk (TEuro) (TEuro) (TEuro) BW 75 75 75 85 100 0.08 0.1 0.02 0.07 0.06 0.08 0.09

40 40 40 40

35 35 35 45

32.41 30.01 27.78 33.08

247

Co, i=0.08

Co, i=0.1

Co, Co, ek i=0.08; iek=0.1, Co, i (TEuro) 0.02 iak=0.06 increase - 5%

-67.59 -37.59 -9.80 23.27

-68.18 -39.26 -12.96 17.78

-68.23 -39.39 -13.20 17.35

-69.55 -43.17 -20.41 5.97

-66.98 -36.41 -8.63 23.25

% to F6

0.76

0.75

0.26

1.00

ak Co,ek, (TEuro) Co,ek Co, ak ak -/+ + 5% - 5% + 5% 5%

71.25 71.25 71.25 80.75

42 42 42 42

-71.06 -44.27 -19.47 10.49

3.47 1.8045

0.45

-69.44 -72.92 -41.15 -47.84 -14.96 -24.62 16.65 3.86 0.72

0.17

Fig. 6.2 Solution to the correction procedures* (Source: Author)

Exercise

Value (Euro)

Percentage

a

23,274.74

100

b

17,775.43

76.37

c

17,352.66

74.56

d

5,965.82

25.63

e

23,252.49

99.90

f

10,486.75

45.06

g

16,650.48

71.54

h

3,862.49

16.69

Fig. 6.3 Solution to the Exercise Correction procedure, overview of results (Source: Author)

In the presentation of the spreadsheets in the book, the figures are often rounded to two decimal places. This can result in rounding differences between the presentation in a figure and in the text. Of course, a solution with the pocket calculator is also possible. You have learned the corresponding procedures in Chap. 3. The classic net present value is 23,274.74 euros. You can see the calculation method in Fig. 6.2. The deviations caused by the correction methods are shown in Fig. 6.3 as a percentage of the results in relation to the initial solution in exercise a), which was set equal to 100%. This deviation is caused on the one hand by the technique of the correction method and on the other hand by the fixed amount of the deviation. The problem with correction procedures is, as already mentioned above and as is also clear from this example in Fig. 6.3, that no systematic risk assessment is carried out by changing the calculation elements. We would also like to point out a danger of the lump-sum procedure and of the double discounting. The increase in interest rates for reasons of allegedly greater security may well result in an investment that is not worthwhile at the chosen

248

6 Investment Decisions in Uncertainty

Calculation element

Value

n

3 years

i

0.1

A

20 euro

R

- 90 euro

NEk = 1, NEk = 2

50 euro p. a.

NEk = 3

0 euro

Fig. 6.4 Problem of the lump-sum procedure and double discounting (Source: Author)

calculation interest rate becoming advantageous if, for example, it has a negative residual value. The example in Fig. 6.4 may show this. Exercises: Now calculate the net present value for a: Exercise a) Calculation interest rate of 10% Exercise b) Calculation interest rate of 20% Solutions: Exercise a) The net present value is 0.84 euros (0.84 ¼ 50
DSF(n ¼ 2, i ¼ 0.1)  90
Abf (n ¼ 3, i ¼ 0.1)  20), the investment is, therefore, not profitable. Exercise b) The net present value is 4.31 euros, so the investment is worthwhile. We can clearly see that at the actual calculation interest rate of 10%, the net present value is negative and thus the investment is unfavourable. If the calculation interest rate is increased to 20%, which is actually done as a precautionary measure, the net present value becomes positive. The opposite was intended. Lump-sum procedures and double discounting are, therefore, not suitable if there are negative net earnings or negative residual values.

6.3.3

Section Results

In this section, you: • • • • •

Learned the importance of the correction procedures. Learned and applied the correction procedure for the calculation interest rate. Learned correction procedures for the useful life. Learned and applied correction procedures for payments. Evaluated and questioned the application of the application for practical problems.

6.4 Sensitivity Analyses

6.4

249

Sensitivity Analyses

As with the correction procedures, sensitivity analyses are not a theoretically challenging or, in the sense of an academic model, objective form of taking risk into account in investment decisions. In general, one calculation element or several calculation elements are varied and the effects on the target value, i.e. the calculation results of the dynamic investment calculation methods, are analysed. The difference to the correction methods is that the focus is not on changing the calculation elements, but on the effect on the target value. Three methods of sensitivity analysis are known: • The critical value calculation • The triple calculation • The target value change calculation

6.4.1

The Critical Value Calculation

6.4.1.1 Display of the Critical Value Calculation In the critical value calculation only one calculation element is usually varied and the effect on the target value is analysed. In most cases, the target value, such as the capital value, is set to zero and then the possible deviation of the calculation element from the planned value is recorded. This is possible without the investment becoming disadvantageous. Of course, it is also possible to set the target value to a certain level, for example, 1 million euros. It is also possible to determine critical values with regard to two or more investments, if you equate the target values of the considered investment alternatives and solve them for the calculation element you are looking for. The weakness of this technique is the fact that it assumes the independence of calculation elements, i.e. it assumes that the change of one calculation element does not result in changes of other calculation elements. But this is typically the case in practice. For example, if the acquisition payment changes, the residual value as the derived price often changes as well. For various reasons, changes are also expected for almost all other calculation elements. Therefore, the practical significance of this technique as a risk analysis tool is limited. If several calculation elements were to be varied, the functional relationship between the exogenous variables would have to be known so that it is clear to what extent one exogenous variable changes when the other changes by, for example, 10%. As critical value calculation is widely used in practical applications, it is presented here so that the reader can assess its deficits. In principle, a distinction must be made between critical maximum and critical minimum values. Critical maximum values of a calculation element cannot be exceeded without an investment becoming disadvantageous. This is, therefore, a negative correlation between the considered calculation element and the target value. An increase in

250

6 Investment Decisions in Uncertainty

the value of the calculation element leads to a decrease in the target value. Critical minimum values cannot be fallen below without an investment becoming disadvantageous. This is, therefore, a positive correlation between the considered calculation element and the target value. An increase in the value of the calculation element leads to an increase in the target value. Critical maximum values are: • Acquisition payment (A) • Ongoing payouts (ak) • Interest rate (i) Critical minimum values are: • Residual value (R) • Ongoing in-payments (ek) • Useful life (n) The assignment of the individual calculation elements to the group of maximum and minimum values is, first of all, to be shown on the graphic representation of the courses of these calculation elements as a function of the net present value in Fig. 6.5. The representation could also have taken place with the other dynamics as a function of the considered calculation element; therefore, the net present value is only a representative of the group of the dynamic investment calculation procedures. From the representation it becomes obvious that for the six possible calculation elements of the investment calculation, whose variation in their effect on the variation of the net present value is shown in Fig. 6.5, solution methods for the determination of the critical values are already known for two of the calculation elements: • The critical interest rate, which is the rate of return that we learned about in Sect. 3.7. • The critical useful life. It corresponds to the dynamic amortisation period, which we dealt with in Sect. 3.8. For both procedures, therefore, no independent path for determining the solutions is presented here, but reference is made to the corresponding sections in Chap. 3. According to the causality shown above, a predetermined solution path results in the determination of the critical values of the remaining calculation elements. This path generally consists of three steps: • First of all, a dynamic investment calculation formula must be established. • Then this function must be zeroed or set to a fixed target value. This causes the searched calculation element to change to the value you are looking for. • The third step is to solve the function for the calculation element searched for. In principle, this step sequence can be carried out with all five dynamic investment calculation procedures from Chap. 3 for all six calculation elements. However,

6.4 Sensitivity Analyses

251

critical interest rate (i kr)

critical useful life (nkr)

Co

Co

Co = f(i), (A, R, e, a, n)

i kr = r

Co = f(n), (A, R, e, a, i)

i

nkr = ndyn

critical acquisition payment (Akr)

n

critical residual value( Rkr )

Co

Co

Co = f(A), (R, e, a, n, i)

Co = f( R ), (A, e, a, n, i) Akr

A

Rkr

critical ongoing payouts (akr)

R

critical ongoing in-payments ( e kr )

Co

Co

Co = f(a), (A, R, e, n, i) Co = f( e ), (A, R, a, n, i) akr

a

e kr

e

Fig. 6.5 Critical values of the six calculation elements (Source: Author)

due to the higher calculation effort, the internal rate of return method and the dynamic amortisation calculation should only be used if the calculation elements that are congruent with the determination path for this dynamic procedure, i.e. the critical interest rate and the critical useful life, are involved.

252

6 Investment Decisions in Uncertainty

This leaves a maximum of three possible dynamic investment calculation methods, with which the critical value calculation can be operated for the four remaining calculation elements: the net present value method, the horizon value method and the annuity method. In addition to this algebraic form of determining the critical values, a graphical solution is also possible, in which some values (2–4) are determined for the dynamic investment calculation formula or for the dynamic investment calculation formulas for the analysed calculation element, for which the dynamic target values are determined. Thus, for the determination of the critical interest rate, 2–4 different trial interest rates could be defined for which the corresponding 2–4 net present values are then determined. These values would be drawn in a corresponding coordinate system and then connected. The calculation elements interest rate and useful life are convex functions, the functions of the other four calculation elements are linear, as shown in Fig. 6.5. The critical value calculation is now presented in an example.

6.4.1.2 Application Example for the Critical Value Calculation You have the opportunity to acquire commercial real estate in the northern outskirts of Hamburg with good transport connections. The building is currently leased for 10 years at a net rent fixed for the individual years. In the first 3 years the annual net rent, which is paid annually in arrears, is 110 TEuro, in years 4–7 it is 160 TEuro and in years 8–10 it is 185 TEuro. At the end of the 10 years, the tenant has contractually agreed to purchase the building at a residual value of 1 million euros. There are no annual costs. You calculate with a calculation interest rate of 8%. Exercise Determine the critical acquisition payment Akr. Solution In principle, as already mentioned, the solution is to be determined with all five dynamic investment calculation methods. However, the dynamic amortisation calculation and the internal rate of return method are not suitable because of the calculation effort involved. The net present value method is the most suitable method, in the formula the acquisition payment occurs without weighting with a factor of financial mathematics. In addition, due to the definition equation and the three-step sequence mentioned above for the following applies for the solution. Co ¼ BW  A

ð6:1Þ

0 ¼ BW  Akr

ð6:2Þ

Akr ¼ BW

ð6:3Þ

The critical acquisition payment, therefore, always corresponds to the present value of an investment or, in other words, the sum of the acquisition payment given in the data record and the net present value Co.

6.4 Sensitivity Analyses

253

Fig. 6.6 Determining the critical acquisition payment* (Source: Author)

k 1 2 3 4 5 6 7 8 9 10 R (TEuro) = i=

NEk (TEuro) Abf BW Sum BW 110 0.925926 101.852 101.852 110 0.857339 94.307 196.159 110 0.793832 87.322 283.481 160 0.735030 117.605 401.085 160 0.680583 108.893 509.979 160 0.630170 100.827 610.806 160 0.583490 93.358 704.164 185 0.540269 99.950 804.114 185 0.500249 92.546 896.660 185 0.463193 85.691 982.351 1000 0.463193 463.193 1445.54445 0.08

Abf (n = 10)

BW NE 185

Abf (n = 7) R 1000

Abf (n = 3) DSF (n = 3) DSF (n = 4)

BW NE 160

NE 185 DSF

NE 160 DSF

DSF (n = 3) BW NE 110 0

1

2

3

4

5

6

7

8

9

10 years

Fig. 6.7 Determining the critical acquisition payment with the calculator (Source: Author)

Akr ¼ C o þ A

ð6:4Þ

The solution is shown in Fig. 6.6. In principle, a different table structure is also conceivable, unless you arrive at a different result. According to cell E12 in Fig. 6.6, the present value is 1.44554 million euros. This is, therefore, the critical acquisition payment. If this amount is paid as a purchase price and the plan data is fulfilled, a net present value of zero would result at a calculation interest rate of 8%. If you use a pocket calculator to process this data set, you can accelerate the work by applying the discounting sum factor as opposed to individual discounting. This is shown graphically in Fig. 6.7. The following formulaic representation results:

254

6 Investment Decisions in Uncertainty

n¼4 n¼3 n¼3 n¼7 Akr ¼ 110
DSFn¼3 i¼0:08 þ 160
DSFi¼0:08
Abf i¼0:08 þ 185
DSF i¼0:08
Abf i¼0:08

þ1000
Abf n¼10 i¼0:08 ¼ 1:445 million euros ð6:5Þ The second example deals with critical annual payouts. A food manufacturer offers you a silent partnership in a business area that is new to him. He plans to enter the production of lamb salami for 8 years, as he has been offered an exclusive supply contract with a gourmet restaurant chain for this period. The planned useful life is, therefore, 8 years. The machines necessary for the production of this lamb salami, which are not yet installed at the site, have a purchase price of 120,000 euros and can be sold again after 8 years for 50,000 euros. The contractually guaranteed price per delivered lamb salami is 12 euros. The quantity of salami produced, measured in pieces, has the variable labelling q. Your calculation interest rate for this project, in which you would like to participate with 100%, is 10%. You are assuming an annual delivery quantity of the same amount, i.e. constant. The ongoing operating and maintenance payouts consist of a fixed cost component and variable costs for operating resources and raw materials in the amount of: a ¼ 3000 þ 6q

ð6:6Þ

Exercise Determine the annual critical sales quantity. Solution A solution can only be achieved if the annual quantities remain constant and can be determined for a single investment using the horizon value, net present value and annuity methods. When comparing several investment objects, the methods would only all be applicable if the useful lives of all investment objects were the same. If this is not the case, only the annuity method can be used. In this case, you should use the annuity method and the net present value method. C o ¼ ðe  aÞ
DSFni þ R
Abf ni  A

ð6:7Þ

The in-payments are based on the selling price per salami and the quantity sold (q). This results in the following numerical formula: n¼8 Co ¼ ð12q  ð3000 þ 6qÞÞ
DSFn¼8 i¼0:1 þ 50, 000
Abf i¼0:1  120, 000

ð6:8Þ

The formula can now be zeroed, then the quantity changes to the critical quantity: n¼8 0 ¼ ð6qkr  3, 000Þ
DSFn¼8 i¼0:1 þ 50, 000
Abf i¼0:1  120, 000

Solving for qkr results in the following:

ð6:9Þ

6.4 Sensitivity Analyses

qkr ¼

255

120, 000  50, 000
Abf ni 3000 ¼ 3520:18 þ 6 6
DSFni

ð6:10Þ

The critical sales quantity is, therefore, 3521 lamb salami per year. Alternatively, this data set can also be solved using the annuity method. The initial equation here is € ¼ DJE  DJA DJU

ð6:11Þ

Here, the calculation is as follows: € ¼ 12q þ 50, 000
Abf n¼8
KWFn¼8  3000  6q  120, 000 DJU i¼0:1 i¼0:1
KWFn¼8 i¼0:1

ð6:12Þ

n¼8 0 ¼ 6qkr þ 50, 000
Abf n¼8 i¼0:1
KWFi¼0:1  3000  120, 000


KWFn¼8 i¼0:1 qkr ¼

n¼8 n¼8 50, 000
Abf n¼8 i¼0:1
KWFi¼0:1 þ 3000 þ 120, 000
KWFi¼0:1 6

ð6:13Þ ð6:14Þ

Of course, the annuity method results in the same critical sales quantity of 3521 lamb salami as per the net present value method.

6.4.1.3 Presentation of the Critical Value Calculation in Relation to Two Investments In a further part, the critical value calculation will be presented with regard to two investment objects. For this purpose, Fig. 6.8 will first document which critical values are being searched for. When determining the critical value in relation to two investment objects, the corresponding dynamic functions are set up for each investment object and their target values are then equated. So, the three steps to the solution are: • First of all, the dynamic investment calculation formulas must be set up. • Then these functions are equated. This causes the considered calculation element to change to its searched value. • In the third step, the equation set up is to be solved for the calculation element. When analysing the critical values of several investments, there are several critical values in total. First, the absolute critical values already worked out in Fig. 6.5. In these values, the absolute advantageousness of an investment object changes to the absolute disadvantageousness or vice versa, depending on whether the values are critical maximum values or critical minimum values. In the case of relative critical values, whose determination path is described in this paragraph, the relative

256

6 Investment Decisions in Uncertainty critical interest rate (ikr)

critical useful life (nkr) Co

Co

Co = f(i), (A, R, e, a, n)

Co = f(n), (A, R, e, a, i)

nkrI = ndyn ikrII = r

ikrI = r

nkrII = ndyn

i

ikrII/I

n

nkrI/II

critical acquisition payment (Akr)

critical residual value ( Rkr ) Co

Co

Co = f(A), (R, e, a, n, i)

Co = f( R ), (A, e, a, n, i) AkrI

AkrII

RkrI

A

AkrI/II

RkrII

R

RkrI/II

critical ongoing payouts (akr)

critical ongoing in-payments

Co

Co

Co = f( e ), (A, R, a, n, i) Co = f(a), (A, R, e, n, i)

akrI akrI/II

akrII

a

ekrI

ekrII

e ekrI/II

Fig. 6.8 Critical values of the six calculation elements in relation to two investment objects (Source: Author)

advantageousness of an investment object switches to an alternative object. These are the intersections of the functions in Fig. 6.8 where the perpendicular has been dropped. The net present value function of one investment object I is always shown in solid black, the function of the other investment object II is shown in dashed lines.

6.4 Sensitivity Analyses

257

Calculation element

line 1

line 2

Useful life (n, years)

6

7

Calculation interest rate (i)

0.08

0.08

Acquisition payment (A, euro)

120,000

190,000

Residual value (R, euro)

0

20,000

Variable payments (av, euro/unit/year)

120

105

Fixed payments (af, euro/year)

110,000

160,000

Fig. 6.9 Data set for the critical value determination in relation to two investments (Source: Author)

At the intersection points of the functions, the individual investment objects can be absolutely advantageous, as shown in Fig. 6.8, or absolutely disadvantageous. The determination is made according to the procedure described in this section by equating the relevant dynamic investment calculation formulas.

6.4.1.4 Application Example for the Critical Value Calculation in Relation to Two Investment Objects We can see this in an example using the data set shown in Fig. 6.9. A new production line is to be created in the chocolate bar production of your company. Line 1 and line 2 are the alternatives, the data are presented in Fig. 6.9. Assume that the chocolate bars can only be sold in whole units (pieces) as wholesale pallets and that in-payments of 350 euros per unit can be realised. Exercises: Exercise a) Determine the annual sales volume for which line 1 is absolutely advantageous. Exercise b) Determine the annual sales volume where line 2 is absolutely advantageous. Exercise c) Determine the annual sales volume for which line 1 and line 2 have the same relative advantage. Solutions: First of all, we would like to give you some hints for the solution. Exercise c) can only be solved with the annuity method, since the useful lives from line 1 and line 2 differ. Since the results from exercise parts a) and b) can be used to solve exercise c), the annuity method should also be used there. Exercise c) can actually only be solved in a realistic manner after a fictitious investment calculation. At this point, this should be omitted. The solutions are given in integer numbers, since the pallets can only be sold in whole units according to the exercise. The target value must not be rounded commercially (half away from zero), but it must be positive. € 1 ¼ 350q  120q  110, 000  120, 000
KWFn¼6 DJU i¼0:08

ð6:15Þ

258

6 Investment Decisions in Uncertainty

0 ¼ 230qkr1  110, 000  120, 000
KWFn¼6 i¼0:08

ð6:16Þ

110, 000 þ 120, 000
KWFn¼6 i¼0:08 ¼ 592 230

ð6:17Þ

qkr1 ¼

€ 2 ¼ 350q  105q  160, 000  190, 000
KWFn¼7 þ 20, 000 DJU i¼0:08 ð6:18Þ


RVFn¼7 i¼0:08

n¼7 0 ¼ 245qkr2  160, 000  190, 000
KWFn¼7 i¼0:08 þ 20, 000
RVFi¼0:08 ð6:19Þ

qkr2 ¼

n¼7 160, 000 þ 190, 000
KWFn¼7 i¼0:08  20, 000
RVFi¼0:08 ¼ 793 245

ð6:20Þ

€ 1 ¼ DJU €2 DJU

ð6:21Þ

230qkr1  110, 000  120, 000
KWFn¼6 i¼0:08 ¼ n¼7 245qkr2  160, 000  190, 000
KWFn¼7 i¼0:08 þ 20, 000
RVFi¼0:08

qkr2=1 ¼

ð6:22Þ

n¼7 n¼7 50,000120,000
KWFn¼6 i¼0:08 þ190,000
KWFi¼0:08 20,000
RVFi¼0:08 15 ð6:23Þ

qkr2=1 ¼ 3887 pallets

ð6:24Þ

In Eq. (6.17) you see the solution for exercise a), in Eq. (6.20) the solution for exercise b) and in Eq. (6.24) the solution for exercise c). If line 1 is purchased, at least 592 pallets must be sold for the given data set if the company does not want to make a loss with this activity; for line 2, this would be 793 pallets. The relative advantageousness in relation to the sales volume changes from line 1 to line 2 for approximately 3887 pallets, line 2 is advantageous for 3887 pallets and line 1 is still advantageous for 3886 pallets.

6.4.2

The Triple Calculation

6.4.2.1 Presentation of the Triple Calculation With the triple calculation, a classic data set for a dynamic investment calculation is compared once with the planned probable data set and then with a pessimistic and an optimistic data set. The determination of the pessimistic and optimistic data variants is usually done by percentage additions or deductions and is, therefore, again quite subjective.

6.4 Sensitivity Analyses

259

An investment is considered to be reliably worthwhile if it yields a positive target value of a dynamic investment calculation method, e.g. a positive net present value, even in the pessimistic variant. An investment is considered to be unfavourable if it also yields a negative target value in the optimistic variant. However, the same investment decision would also have been made in the classical dynamic, because the probable value must necessarily be negative in a normal investment if the optimistic value is already negative. If, for example, the results of the optimistic assessment and the probable assessment are positive, but the result of the pessimistic assessment is negative, another subjective investment decision is necessary, even in the sense of this model. This then depends on the absolute level of the values determined, the assessment of the probabilities of occurrence of the three states and the subjective risk attitude of the investor. The difference between the triple calculation and the critical value calculation is that in the critical value calculation only one calculation element is usually varied, whereas in the triple calculation all calculation elements are usually varied, except for the acquisition payment, which is usually known with certainty at the beginning of the investment. Just as in the critical value calculation, the triple calculation does not take into account the exact functional relationships between the calculation elements. The triple calculation will now be demonstrated using an example.

6.4.2.2 Application Example of the Triple Calculation The data set in Fig. 6.10 applies. For the optimistic and for the pessimistic assessment, the percentage changes of the calculation elements from Fig. 6.11 apply. Exercise Now calculate the net present values for the optimistic, the probable and the pessimistic situation. In this example, we refrain from changing the calculation interest rate and the useful life. Solution The solution is presented in the spreadsheet. Since it is a matter of discounting and summing up the calculation elements, there are various correct Calculation element

Value

n

4 years

i

0.1

A (TEuro)

80

R (TEuro)

15

(e - a)k = 1 (TEuro)

(75 – 40)

(e - a)k = 2 (TEuro)

(75 – 40)

(e - a)k = 3 (TEuro)

(75 – 40)

(e - a)k = 4 (TEuro)

(85 – 40)

Fig. 6.10 Data set for the triple calculation (Source: Author)

260

6 Investment Decisions in Uncertainty

Calculation element

Optimistic

Pessimistic

ek

+ 10

– 20

ak

– 15

+ 30

R

+ 20

– 20

Fig. 6.11 Percentages of change of payment values for the triple calculation (Source: Author)

ek ak NEk (TEuro) (TEuro) (TEuro) Co

k 0 1 2 3 4 A (TEuro) = i= R (TEuro) =

75 75 75 85 80 0.1 15

40 40 40 40

35 35 35 45

-48.18 -19.26 7.04 48.02

ek ak ek ak (TEuro) + (TEuro) - (TEuro) - (TEuro) + Co opt Co pess 10% 15% 20% 30% 82.5 82.5 82.5 93.5

34 34 34 34

60 60 60 68

52 -35.91 52 4.17 52 40.61 52 93.55

-72.73 -66.12 -60.11 -40.98

Fig. 6.12 Solution to the triple calculus* (Source: Author)

procedures for formulating the solution in the spreadsheet. So far, the shown solution is only a suggestion from which you can deviate, as long as you come to the same result. Of course, a solution with a pocket calculator is also possible. You got to know the corresponding procedures in Chap. 3. For this example, the optimistic situation results in a positive net present value of 93,545.86 euros, the probable situation in a positive net present value of 48,020.63 euros and the pessimistic situation in a negative net present value of 40,980.81 euros. You can see these values in Fig. 6.12. Thus, for this example, no clear investment decision can be made with this technology. Only when the investor determines subjective probabilities of occurrence for the three situations and weights the individual net present values to form a total net present value a clear decision in the sense of this procedure is possible. However, this is very arbitrary. If we assume, for our example, that the optimistic and the pessimistic situation each have a 25% probability of occurrence and the probable situation has a 50% probability of occurrence, the total net present value is 37,151.58 euros (0.25
93,545.86 + 0.5
48,020.63 + 0.25
40,980.81) and the investment would be worthwhile from this perspective.

6.4.3

The Target Value Change Calculation

6.4.3.1 Presentation of the Target Value Change Calculation For the calculation of changes in target values, the causality of the critical value calculation is reversed. In the critical value calculation, the result of one of the

6.4 Sensitivity Analyses

261

dynamic investment calculation procedures was set to a specific target value. For example, the net present value was set to zero, and then the system analysed how far a calculation element, such as the residual value, can deviate from its planned value until this condition is met. Thus, the target value of the dynamic used in the critical value calculation was the exogenous variable of this consideration, while the analysed calculation element was the endogenous variable. In the target value change calculation, this causality is now changed. The analysed calculation element is changed by a certain percentage and is, therefore, the exogenous variable, while the change of the target value of the used dynamic procedure is considered and is, therefore, the endogenous variable in this analysis. In general, the target values of all dynamics are possible as target values. However, the internal rate of return method and the dynamic amortisation calculation are not recommended due to the higher calculation effort involved. This technique allows analysing the influence of the individual calculation elements on the target value. Of course, as with the triple calculation, it is also possible to analyse the change of several calculation elements in one calculation, if the functional relationship between the individual calculation elements, that is, their correlation, is known. However, this approach goes beyond the classic procedure of the target value change calculation. This procedure also assumes that the calculation elements are independent of each other, which is unlikely to be the case in practice. The general procedure for calculating changes in target values is as follows: • Setting up the relevant function of the dynamic investment calculation procedures. • Definition of the deviation of the considered calculation element from the probable initial value to be analysed. • Calculation of the changes to the target value that result from the changes to the calculation element under otherwise identical circumstances. The relative target value change is calculated by relating the difference between the new and old target value, e.g. for the net present value, to the old value, using the formula C onew  Coold Coold

ð6:25Þ

We want to apply the relative target value change calculation to an example. The basis for this example is again the net present value method. To analyse the effects of more than the classical six calculation elements, the net present value formula is defined as follows:
 C o ¼ pq  ðlm þ rxÞ
q  a f
DSFni þ R
Abf  A ð6:26Þ • pq are the in-payments resulting from the product of product price ( p) and sold product quantity (q).

262

6 Investment Decisions in Uncertainty

• rx is the product of input price (r) and input quantity consumed (x) per product quantity (q). • lm is the product of the working price (l) and the amount of consumed labor (m) per product quantity (q). • af are fixed annual payouts.

6.4.3.2 Application Example of the Target Value Change Calculation In order to capture the target value change contribution of the individual input variables without much calculation effort, it was assumed for this example in Fig. 6.13 that all input variables are constant over the term. Of course, a calculation with annually different quantities is also possible. Exercise Determine the results of the target value change calculation for a change of 10% for each calculation element individually. For the useful life, please assume that the change is 1 year. Solution First, the solution is presented in the spreadsheet in Fig. 6.14. Since it is a matter of discounting and summing up the calculation elements, there are various correct procedures for formulating the solution in the spreadsheet. Of course, a solution with the pocket calculator is also possible. You have learned about the corresponding procedures in Chap. 3. The results are summarised in the Excel table in Fig. 6.15. In Fig. 6.15 you can now see the influence of the calculation elements on the net present value. The product price change has a relatively large influence on the net present value for this data set. A change of 10% changes the net present value by 17.82%. The changes in fixed costs and residual value by 10% have a minor influence. They only change the net present value by 0.05%. The net present value functions relating to the change of Calculation element

initial value

p

100

q

100

l

6

m

5

r

4

x

3

af

30

i

0.1

n

6

R

200

A

800

Fig. 6.13 Data set for the target value change calculation (Source: Author)

6.4 Sensitivity Analyses i= n= change

263

0.1 6 p

q

l

m

r

Output p p q q l l m m r r x x af

0 10 -10 10 -10 10 -10 10 -10 10 -10 10 -10 10

100 110 90 100 100 100 100 100 100 100 100 100 100 100

100 6 5 100 6 5 100 6 5 110 6 5 90 6 5 100 6.6 5 100 5.4 5 100 6 5.5 100 6 4.5 100 6 5 100 6 5 100 6 5 100 6 5 100 6 5

af i i n n R R A A

-10 10 -10 1 -1 10 -10 10 -10

100 100 100 100 100 100 100 100 100

100 100 100 100 100 100 100 100 100

6 6 6 6 6 6 6 6 6

5 5 5 5 5 5 5 5 5

x

q

DSF

af

R

Abf

A

Co

rel Co change

4 4 4 4 4 4 4 4 4 4.4 3.6 4 4 4

3 3 3 3 3 3 3 3 3 3 3 3.3 2.7 3

100 100 100 110 90 100 100 100 100 100 100 100 100 100

30 30 30 30 30 30 30 30 30 30 30 30 30 33

4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261 4.355261

200 200 200 200 200 200 200 200 200 200 200 200 200 200

0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474 0.564474

800 800 800 800 800 800 800 800 800 800 800 800 800 800

24442.75 28798.01 20087.49 26968.80 21916.70 23136.17 25749.33 23136.17 25749.33 23920.12 24965.38 23920.12 24965.38 24429.68

0 0.178 -0.178 0.103 -0.103 -0.053 0.053 -0.053 0.053 -0.021 0.021 -0.021 0.021 -0.001

4 4 4 4 4 4 4 4 4

3 3 3 3 3 3 3 3 3

100 100 100 100 100 100 100 100 100

27 30 30 30 30 30 30 30 30

4.355261 4.230538 4.485919 4.868419 3.790787 4.355261 4.355261 4.355261 4.355261

200 200 200 200 200 220 180 200 200

0.564474 0.534641 0.596267 0.513158 0.620921 0.564474 0.564474 0.564474 0.564474

800 800 800 800 800 800 800 880 720

24455.81 23717.13 25203.00 27393.41 21197.02 24454.04 24431.46 24362.75 24522.75

0.001 -0.030 0.031 0.121 -0.133 0.000 0.000 -0.003 0.003

Fig. 6.14 Solution for the calculation of changes in target values* (Source: Author) Calculation element

initial value

change+

change-

Co % change+

Co % change-

p

100

10 %

10 %

17.82

17.82

q

100

10 %

10 %

10.33

10.33

l

6

10 %

10 %

5.35

5.35

m

5

10 %

10 %

5.35

5.35

r

4

10 %

10 %

2.14

2.14

x

3

10 %

10 %

2.14

2.14

af

30

10 %

10 %

0.05

0.05

i

0.1

10 %

10 %

2.97

3.11

n

6

1 year

1 year

12.07

13.28

R

200

10 %

10 %

0.05

0.05

A

800

10 %

10 %

0.33

0.33

Output Co

24,442.75

Fig. 6.15 Summary of the solution for the calculation of changes in target values (Source: Author)

a calculation element are usually linear. If you increase the calculation element or decrease the calculation element, the same changes in percentage occur. A doubling of the change in the calculation elements then also leads to a doubling of the net present value change. The changes in net present value are entered in the table without a sign; whether an increase in the calculation element contributes to an increase in net present value depends on its sign. In the case of interest rate and useful life, the changes when

264

6 Investment Decisions in Uncertainty

increasing or decreasing do not run in the same direction, since the functions are not linear, but of a higher order. The changes in factor price and factor quantity on the one hand and working price and amount of work on the other hand must be the same in percentage terms since they are included as products in the net present value formula. Thus, the target value change calculation provides an overview of the significance of the individual calculation elements for the net present value assuming that the calculation elements are independent of each other. However, this is not a theoretically sophisticated technique for taking risk into account in investment decisions.

6.4.4

Section Results

In this section, you: • • • • • • • •

Got to know the importance of sensitivity analyses. Systematised the critical value calculation as a form of sensitivity analysis. Applied the critical value calculation as a form of sensitivity analysis. Got to know the triple calculation as a form of sensitivity analysis. Applied the triple calculation as a form of sensitivity analysis. Got to know the target value change calculation as a form of sensitivity analysis. Applied the target value change calculation as a form of sensitivity analysis. Evaluated and questioned the usefulness of the application for practical problems.

6.5

Sequential Investment Decisions

The aim of sequential planning is to determine the optimum sequence of investments over time. In sequential investment decisions, the investor must choose between alternative courses of action. This finite number of alternatives can be exposed to different environmental conditions after selection. The number of environmental states is finite. A discrete probability of occurrence can be assigned to the environmental states and these environmental states will then occur with the given probability at all or not at all. The chosen action alternatives have different and measurable economic consequences among the randomly occurring environmental conditions. These economic consequences occur discretely, i.e. without themselves being subject to a probability distribution. The planning can take place in several stages, i.e. an alternative course of action, an environmental condition and a responding alternative course of action, etc. can be formulated. Planning can be rigid, i.e. starting from zero for the entire duration, or continuous, i.e. reacting to environmental conditions that have already occurred. Rigid planning does not necessarily find the optimum combination of actions. In flexible planning, the roll-back procedure is presented as one of many possible planning procedures. It is the most frequently used procedure in flexible planning. It assumes that the

6.5 Sequential Investment Decisions

265

investor can take action not only at the time of planning but also during the investment period. Thus, the entire action strategy is not determined at the beginning of the planning process but can be adjusted over time. The strand of a plot with the best target contribution is selected. Since this contribution is determined recursively, i.e. from the endpoint of the action in the future backwards to the present, it is assumed that the occurrence of the situation in the future is known. The representation of planning is done with the so-called decision tree, which is why the technique is also called the decision tree method. In the decision tree, the action alternatives are symbolised by rectangles, the randomly occurring environmental conditions as circles and the economic consequences as diamonds. Thus, each alternative action and each environmental state can be assigned a probability of occurrence. Conditional probabilities can then be used to determine the occurrence of the economic consequences. In very small planning problems with very few action alternatives and very few environmental states, the economic consequences of each action alternative can be determined completely for all environmental states. In complex planning problems, this can only be done with an algorithm in IT. Recursive planning is the best choice here. It can again be carried out in the spreadsheet, e.g. with Excel. A decision rule then determines the most favourable action. In this section, the risk neutrality of the investor is assumed, the action alternative with the maximum expected value is the most favourable alternative. Risk-averse decision rules, in which an investor is prepared to forego potential profit if he can exchange an uncertain situation for a more secure one, are only much more complex to depict and are not dealt with in this section. Section 6.7 on risk analysis shows which decision-making rules are possible in the risk-averse area. The special feature of sequential planning is, therefore, contingency planning, in which the investment alternatives of the investor are analysed, this is still on the bottom of the classical alternative selection, which environmental conditions, proven with concrete probabilities, can occur, here lies a special feature of the decision tree procedure, and how the investment alternatives can be optimally adapted after the occurrence of certain environmental constellations. This is also a special feature of the procedure. The decision is then made on the basis of planning the economic consequences of the action alternatives on the environmental conditions. Whether a planning of all these constellations is possible for an operational decision situation and whether the planning effort is economically justifiable due to the improvement of results depends on the size and the economic importance of the analysed project.

6.5.1

Procedure for Sequential Planning

We will illustrate the procedure of sequential planning with a graph. To do so, the structure of a decision tree is first illustrated in Fig. 6.16. The starting point of the decision tree is the first decision node (E). In Fig. 6.16 it is marked with (E1). At this starting point, the investor plans an alternative

266

6 Investment Decisions in Uncertainty to

Period 1

Period 2 U3 HA 11

Y1

Z3

U4

Y2

Y10/E2

U3 U1 HA 12

Y3

Z4

U4

Y4

Z1

U5 HA 11 U2

U6 HA 01

Y5

Z3 Y6

Y20/E2

U5 HA 12

Y7

Z4

U6

Y8

E1

U3 HA 21

Y9

Z3

U4

Y10

Y30/E3

HA 02

U3 U1 HA 22

Y11

Z4

U4

Y12

Z2

U5 HA 21 U2

Y13

Z3

U6

Y14

Y40/E3

U5 HA 22

U6

Fig. 6.16 Structure of a decision tree (Source: Author)

Y15

Z4

Y16

6.5 Sequential Investment Decisions

267

investment decision. Several action alternatives (HA) are available. In Fig. 6.16, two action alternatives, HA 01 and HA 02, are shown. However, there must be a finite number of action alternatives, and if the number is very large, the planning technique is not easy to handle. After a decision has been made, environmental conditions (U ) occur that cannot be influenced by the investor, this is symbolised by the node for a random event (Z ). The probability ( p) of the occurrence of the environmental conditions (U ) is known. (U1) and (U2) must add up to 1, because the probabilities of the alternative situations must always add up to one. In general, of course, more than just two environmental states can exist. The environmental states that occur cause a defined economic consequence (Y ) of the action alternative (HA). This economic consequence is exactly known. It is displayed in the result node (Y). The economic consequence (Y ) of the action alternative (HA) depends on the environmental state (U ) that has occurred. In a further step, once the result of flexible planning is known, an alternative decision can be made based on it, for example, in point (E2) the choice of the action alternative (HA 11) or the action alternative (HA 12). This is symbolised by decision nodes (E2). Since these decision nodes coincide with the result nodes (Y ), they are displayed together. Beyond the simple representation in Fig. 6.16, further decision steps can be linked and additional action alternatives can be available. In the last step of the tree, the last decision (E) again leads via a random variable (Z ) to a certain environmental situation (U ), which causes a certain economic consequence (Y ). The decision tree does not have to be symmetrical. For example, the upper branch, according to the choice of HA 01, can have more decision nodes through more action alternatives or more action steps or other action alternatives in a later period than the lower branch, according to the choice of HA 02. Also, nodes for random events do not always have to follow decision nodes; several nodes for random events can also follow each other. The value of an action alternative is then derived from the backward addition of the possible economic consequences (Y) weighted with the probabilities of occurrence of the environmental conditions (U). This is first shown for rigid planning. In Fig. 6.16, for example, the value of the action alternative HA 01 at time E1 is obtained when the action alternative HA 11 is carried out, which in rigid planning must also be determined at the beginning of planning according to the following formula: Y ðHA 01Þ ¼ ðY1
U3 þ Y2
U4Þ
U1 þ ðY5
U5 þ Y6
U6Þ
U2

ð6:27Þ

For flexible planning, the economic consequence of the action alternative is determined differently. In Fig. 6.16, for example, the value of the action alternative HA 01 at time E1 in flexible planning is calculated using the following formula:

268

6 Investment Decisions in Uncertainty

Y ðHA 01Þ ¼ max hðY1
U3 þ Y2
U4Þ, ðY3
U3 þ Y4
U4Þi
U1 þ max hðY5
U5 þ Y6
U6Þ, ðY7
U5 þ Y8
U6Þi
U2 ð6:28Þ

6.5.2

Application Example for Sequential Planning

We will now apply this procedure to the example in Fig. 6.17. In this example, you are planning an expansion. It can be carried out at a time point to by expanding the existing company (HA 1) or by acquiring a company (HA 2). The considered environmental condition, which occurs randomly with a given probability, is the market situation (M growth to). Under favourable conditions, the market grows (high, U1); otherwise, it stagnates (zero, U2). In period 2, you can either leave the company unchanged (HA 12, HA 21) or change it, that is, continue to grow if you started with the expansion in to (HA 11) or resell part of the acquired company (HA 22) if you started with the purchase in to. Again, economic success depends on the environmental situation of market growth as a conditional probability. Therefore, you now know the planning situation. We still need the probabilities of occurrence p (U ) and the economic consequences (Y). Figure 6.18 shows the probabilities of occurrence of the environmental situations you have researched. You have researched the economic consequences described in Fig. 6.19. These are the result values. Therefore, necessary payments have already been balanced. The values are discounted, i.e. related to the time zero. In order to reduce complexity, E1 HA 1

Expansion

HA 2

Purchase

U1,2 M-growth to high (U1) high (U1) zero (U2) zero (U2) high (U1) high (U1) zero (U2) zero (U2)

E2, E3 HA 11 HA 12 HA 11 HA 12 HA 21 HA 22 HA 21 HA 22

growth status quo growth status quo status quo sale status quo sale

U3-U6 M-growth t1 high (U3lU1) high (U4lU1) zero (U5lU2) zero (U6lU2) high (U3lU1) high (U4lU1) zero (U5lU2) zero (U6lU2)

Fig. 6.17 Planning data for the decision tree procedure (Source: Author)

M-growth to high high zero zero

p (U) M-growth t1 p (U1) = 0.6 high p (U1) = 0.6 zero p (U2) = 0.4 high p (U2) = 0.4 zero

p (U) p (U3lU1) = 0.8 p (U4lU1) = 0.2 p (U5lU2) = 0.3 p (U6lU2) = 0.7

Fig. 6.18 Probabilities of occurrence of environmental situations (Source: Author)

6.5 Sequential Investment Decisions Fig. 6.19 Economic consequences (Y ) of the action alternatives for the environmental situations (Source: Author)

269

Y Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8 Y9 Y10 Y11 Y12 Y13 Y14 Y15 Y16

Euro 150 10 120 110 110 20 250 200 200 110 60 50 180 40 50 200

these values have not been generated from calculation elements evaluated for different environmental situations. Exercises Determine the economic consequences of the action alternatives HA 1 and HA 2 and select the optimal alternative: Exercise a) for rigid planning Exercise b) for flexible planning

Solutions: Exercise a) In the case of rigid planning, in which all decisions must be determined at the time of zero, the action alternative HA 12 is to be chosen, the extension with later status quo provides a maximum economic consequence of 156.8 monetary units compared to the other three alternatives. This can be seen in Fig. 6.20. Exercise b) In flexible planning, where adjustments can still be made at time 1, the alternative HA 2 is to be chosen, the acquisition of a company, which provides a maximum economic consequence of 171.2 monetary units compared to the other alternative, as shown in Fig. 6.21.

6.5.3

Section Results

In this section, you: • Learned about the importance of sequential planning. • Learned about the assumptions and prerequisites for sequential planning.

270

6 Investment Decisions in Uncertainty to

Period 1

Period 2 0.8 122 HA 11

HA 11

Z3

92

0.2

150 10

Y10/E2

HA 12

156.8

0.8 0.6 118 HA 12

Z4

0.2

120

110

Z1

0.3 47 HA 11 0.4

Z3

0.7 HA 01

110

20

Y20/E2

0.3 215 HA 12

250

Z4

0.7

200

E1

0.8 182 HA 21

200

Z3

0.2

110

Y30/E3

HA 02

0.8 0.6 58 HA 22

Z4

0.2

60 50

Z2

0.3 82 HA 21 0.4 HA 21

Z3

142

0.7

180 40

Y40/E3

HA 22

96.8

0.3 155 HA 22

Z4

0.7

Fig. 6.20 Solution Exercise a)* (Source: Author)

50 200

6.5 Sequential Investment Decisions to

271

Period 1

Period 2 0.8 122 HA 11

Y10

150

Z3

122

0.2

10

Y10/E2

Y20

215

0.8 118 HA 12

120

Z4

0.2

110

Z1

0.3 0.6

0.4

47 HA 11

159.2 HA 01

110

Z3

0.7

20

Y20/E2

0.3 215 HA 12

250

Z4

0.7

200

E1

0.8 182 HA 21

200

Z3

110

0.2 Y30/E3

HA 02 171.2

0.8

0.4 0.6

58 HA 22

60

Z4 50

0.2 Z2

0.3

82 HA 21 Y30

180

Z3

182

0.7

40

Y40/E3

Y40

155

0.3 155 HA 22

Z4

0.7

Fig. 6.21 Solution Exercise b)* (Source: Author)

50 200

272

• • • •

6 Investment Decisions in Uncertainty

Developed the technique of rigid planning. Developed the technique of flexible planning. Applied the rigid and flexible planning with the roll-back method to an example. Evaluated and questioned the usefulness of the application for practical problems.

6.6

Investment Decision in Uncertainty

Uncertain situations are devided in the decision theory into risk and uncertainty. We have already dealt with this in Sect. 6.2.3. In the case of risk, possible environmental conditions and their probability of occurrence and their economic consequences for the alternatives for action are known. In the case of uncertainty, possible environmental conditions and their economic consequences are known, but their probability of occurrence is not. Situations in which not all possible environmental states (U ) are known cannot be led to a rational solution with decision theory. Section 6.6 considers the techniques for making decisions under uncertainty and Sects. 6.7 and 6.8 consider the techniques for making decisions under risk. In the approaches presented as decision routines in this section, the decisionmaker can, therefore, not specify a probability of occurrence ( p) for the various environmental states. We are in the realm of uncertainty. Here, the number of action alternatives (HA) must be finite. The environmental states must occur discretely, i.e. the environmental states occur completely or not at all. All action alternatives must be assigned an exact economic consequence (Y) for all environmental states, e.g. a certain profit contribution. The planning situation can then be formally represented in a matrix, which is shown in Fig. 6.22. Thus, in Fig. 6.22, an entrepreneur has n action alternatives (HA), e.g. the expansion of the existing enterprise (HAi), the acquisition of another enterprise

U1

U2

Environmental conditions (U) U3 .. Uj .. Um

HA1

Y11

Y12

Y13

..

Y1j

..

Y1m

HA2

Y21

Y22

Y23

..

Y2j

..

Y2m

HA3 Alternatives for action (HA) : : HAi : : HAn

Y31 : : Yi1 : : Yn1

Y32 : : Yi2 : : Yn2

Y33 : : Yi3 : : Yn3

.. .. .. .. .. .. ..

Y3j : : Yij : : Ynj

.. .. .. .. .. .. ..

Y3m : : Yim : : Ynm

Fig. 6.22 Planning matrix in an uncertainty situation (Source: Author)

6.6 Investment Decision in Uncertainty

273

(HA1) or the withdrawal of capital from the enterprise to support the life of the spouse (HAn), between which he or she can actively choose. The value of these alternative courses of action, their economic consequence (Y ), depends on the m possible environmental conditions (U ), with unknown probabilities, that occur and which cannot be influenced by the entrepreneur. Possible environmental conditions could be, for example, economic growth (U1), an increase in taxation on company acquisitions (Uj) or the entry of a rival into your spouse’s life (Um). Depending on the occurrence of the environmental situations (U ), the action alternatives (HA) already chosen then have different economic consequences (Y ), which are compiled in a table as in Fig. 6.22. For the situation of uncertainty, some classical rules of decision-making have been developed. The dominance rule is the most undisputed. The following are still widespread: • • • • •

Maximax rule Minimax rule Hurwicz rule Laplace rule Savage–Niehans rule (Minimum regret rule)

Before the rules are developed and then applied to an example, an example problem is first designed. A German company is considering a major expansion and can imagine seven alternative ways of growth (action alternatives, HA): • • • • • • •

HA 1: Growth at the site through expansion investment HA 2: Reestablishment in a nearby new industrial area HA 3: Acquisition of a company in Germany HA 4: Migration and establishment of production plant in Eastern Europe HA 5: Acquisition of a company in Eastern Europe HA 6: Migration and establishment of production plant in Asia HA 7: Minority shareholding in an owner-managed competitor with a high loss potential

The following six environmental situations (U ), which cannot be influenced by the company itself, may occur, the probability of occurrence is not known: • U 1: the social security contributions to be paid by the company in Germany increase by 5% per year. • U 2: the construction costs for factory buildings will rise by 12% due to material price increases starting next week. • U 3: social security contributions to be paid by the company in Eastern Europe increase by 7% per year. • U 4: transport costs increase by 9% per year due to energy price increases.

274

6 Investment Decisions in Uncertainty

Environmental conditions (U) U3 U4 U5 U6

U2

U1 HA1

5

8

8

9

11

13

HA2

4

5

7

8

10

10

Alternatives for action (HA) HA3

6

10

9

7

12

9

HA4

12

4

2

4

4

5

HA5

10

12

1

5

6

6

HA6

11

3

12

3

4

4

HA7

1

1

2

1

4

1

Fig. 6.23 Economic consequences of the action alternatives under the environmental conditions (Source: Author)

• U 5: due to consumer scandals, consumer acceptance of “Made in Germany” products on the German market is increasing considerably, higher prices can be charged to customers. • U 6: communication costs increase considerably due to taxation. The economic consequences (Y ) of the environmental situations (U ) on the action alternatives (HA) have been planned by the company for a given period and are expressed as net present values in million euro. It is assumed that the economic consequences of the environmental situations for the respective alternative are certain and only these environmental situations can occur. This results in the planning situation in Fig. 6.23.

6.6.1

Principles of Dominance

This decision rule is the most undisputed rule for decisions under uncertainty, because the risk attitude of the decision-maker does not have to be determined, the assumption of rational action is sufficient. As a rule, the dominance rule does not select the optimal action alternative, but eliminates inferior action alternatives from the decision field. A distinction is made between absolute dominance and state dominance. In absolute dominance, an action alternative HAi absolutely dominates an action alternative HAk if the lowest possible economic consequence (Y ) of the action alternative HAi is greater than the highest possible economic consequence (Y ) of the action alternative HAk. Exercise Determine whether absolute dominance is present in our example in Fig. 6.23.

6.6 Investment Decision in Uncertainty

275

Solution An absolute dominance exists, since action alternative 7 is absolutely dominated by action alternative 1, for example. Action alternative 7 has a net present value of 4 million euros as the best economic consequence among all environmental states in environmental state 5. Action alternative 1, as the worst economic consequence of all environmental states in environmental state 1, has a net present value of 5 million euros. Thus, in the worst environmental situation for this action alternative 1, it is still preferable to action alternative 7 under the most favourable environmental condition for action alternative 7. Action alternative 7 can, therefore, be eliminated from further planning. In the dominance of states, one action alternative HAi dominates an action alternative HAk, if in a pairwise comparison of the economic consequences (Y ) under an environmental state (U ) the action alternative HAi is in no case exceeded by the action alternative HAk in the level of economic consequence (Y ). Exercise Determine whether a state dominance exists in our example in Fig. 6.23. Solution A state dominance exists because action alternative 2 is dominated by action alternative 1. Under each individual environmental state (U ), action alternative 2 has no better economic consequence (Y ) than action alternative 1. Action alternative 2 can, therefore, be eliminated from further planning. Further state dominance does not exist, since: • Action alternative 1 under environmental states 4 and 6 has the highest economic consequence of all action alternatives for these environmental states. • Action alternative 3 under environmental state 5 has the highest economic consequence of all action alternatives for this environmental state. • Action alternative 4 under environmental condition 1 has the highest economic consequence of all action alternatives for this environmental state. • Action alternative 5 under environmental condition 2 has the highest economic consequence of all action alternatives for this environmental state. • Action alternative 6 under environmental condition 3 has the highest economic consequence of all action alternatives for this environmental state. The dominance analysis has thus eliminated two of the seven considered alternatives but does not provide any decision criterion as to which of the remaining alternatives should be carried out. This is possible with the following decision techniques.

6.6.2

The Maximax Rule

According to the Maximax rule the environmental situation (U) is determined, for each individual action alternative (HA) with the greatest economic consequence (Y) for this action alternative (HA). This economic consequence (Y) of the action alternative (HA) is used as a basis for the decision and compared with the greatest

276

6 Investment Decisions in Uncertainty

economic consequences (Y) of the other action alternatives (HA). The action alternative (HA) with the greatest overall economic consequence is chosen. The Maximax rule, therefore, has a very optimistic view. Exercise Use our example in Fig. 6.23 to determine which action alternative is selected according to the Maximax rule. Solution According to the Maximax rule, action alternative 1 must be selected. Here, the maximum net present value is 13 million euros, whereas the other remaining alternatives have a maximum net present value of only 12 million euros.

6.6.3

The Minimax Rule

According to the Minimax rule, the environmental situation (U) is determined for each individual action alternative (HA) with the lowest economic consequence (Y) for this action alternative (HA). This economic consequence (Y) of this action alternative (HA) is used as a basis for the decision and compared with the lowest economic consequences (Y) of the other action alternatives (HA). The action alternative (HA) with the highest lowest economic consequences (Y) is chosen. The minimax rule, therefore, has a very pessimistic view. Exercise Use our example in Fig. 6.23 to determine which action alternative is selected according to the minimax rule. Solution According to the minimax rule, action alternative 3 must be selected. There, the minimum net present value under environmental condition 1 is 6 million euros; the minimum net present value is lower for the other remaining alternatives. For action alternative 1, it is 5 million euros under environmental condition 1. For action alternative 4 it is 2 million euros under environmental condition 3. For action alternative 5 it is 1 million euros under environmental condition 3. For action alternative 6 it is, e.g. under environmental condition 2 at 3 million euros.

6.6.4

The Hurwicz Rule

The Hurwicz rule is a compromise between the Minimax and Maximax principles. For each action alternative (HA), the minimax principle is used to determine the environmental situation (U ) for each individual action alternative (HA) with the lowest economic consequence (Y ). The economic consequence (Y ) of this action alternative (HA) is weighted with the pessimism parameter α, which is between 0 and 1. Then the environmental situation (U ) with the greatest economic consequence (Y) for each individual action alternative (HA) is determined according to the Maximax rule. The economic consequence (Y ) of this action alternative (HA) is weighted with the optimism parameter (1α), which is between 0 and 1 and adds up

6.6 Investment Decision in Uncertainty

277

U2

U1 HA1

5

Alternatives for action (HA) HA3

6

HA4

12

HA5 HA6

Environmental conditions (U) Hurwicz U3 U4 U5 U6 13 12 2

12

7.8 5

1 12

7.4

4.3 3

5.7

Fig. 6.24 Solution according to Hurwicz* (Source: Author)

to 1 with the pessimism parameter. Both summands are added together. The action alternative with the highest economic consequence (Y) thus determined is chosen. Exercise For our example in Fig. 6.23, determine which alternative action should be chosen according to the Hurwicz rule. The optimism parameter is 0.3. Solution According to the Hurwicz rule, alternative action 3 must be chosen, since the net present value determined this way is 7.8 million euros. This can be seen in Fig. 6.24. The subjectivity of this approach is particularly evident here in the fact that, according to the planning, none of the environmental conditions that are assumed to occur with unknown probability but discreetly, i.e. not at all, will lead to an economic consequence of 7.8 million euros.

6.6.5

The Laplace Rule

Since there is no information about the probabilities of occurrence of the individual environmental situations (U ) in the situation of uncertainty, the Laplace rule assumes the naive prognosis, all situations are equally probable. On this assumption, all economic consequences (Y) for each environmental state (U ) must be added up for each alternative action (HA) individually (and, if necessary, divided by the number of existing environmental situations in order to arrive at the same order of size as the economic consequences (Y). Of course, this division does not change the ranking of the action alternatives). The action alternative (HA) with the greatest economic consequence (Y ) must be chosen. Exercise From our example in Fig. 6.23, determine which action alternative should be chosen according to the Laplace rule. Solution According to the Laplace rule, alternative 1 must be chosen, since the net present value determined in this way is 9 million euros. You can see this in Fig. 6.25.

278

6 Investment Decisions in Uncertainty Environmental conditions (U) Laplace U3 U4 U5 U6

U2

U1 HA1

5

8

8

9

11

13

9

Alternatives for action (HA) HA3

6

10

9

7

12

9

8.83

HA4

12

4

2

4

4

5

5.17

HA5

10

12

1

5

6

6

6.67

HA6

11

3

12

3

4

4

6.17

Fig. 6.25 Solution to the Laplace rule* (Source: Author)

6.6.6

The Savage–Niehans Rule

The Savage–Niehans rule is also known as the rule of least regret (Minimum regret rule). In this rule, the decision-maker is not guided by the economic consequence (Y) of his chosen action alternative (HA), but rather by minimising the deviation of the economic consequence (Y) of his chosen action alternative (HA) from the maximum economic consequence (Y) of the other possible action alternatives (HA) on each environmental condition (U). The deviations are added up for all environmental states (U ) for each action alternative (HA) individually. The action alternative with the smallest deviation is chosen. Exercise For our example in Fig. 6.23, determine which action alternative should be chosen according to the Savage–Niehans rule. Solution According to the Savage–Niehans rule, action alternative 1 is chosen, since the deviation value determined in this way is 16 million euros and is smaller than the deviation values for the other action alternatives. You can see this in Fig. 6.26. The procedure of these approaches is positive to the extent that the decisionmaker has to structure his decision problem and has to conclude at a formal solution.

U1

SavageEnvironmental conditions (U) Niehans U3 U4 U5 U6

U2

HA1

-7

-4

-4

0

Alternatives for action (HA) HA3

-6

HA4

0

HA5 HA6

-2

-3

-8

-10

-2

0

-1

-9

Fig. 6.26 Savage–Niehans solution* (Source: Author)

-1

0

-16

-2

0

-4

-17

-5

-8

-8

-39

-11

-4

-6

-7

-30

0

-6

-8

-9

-33

6.7 The Risk Analysis

279

However, since the available information for structuring the problem is very limited, the result of the calculation remains highly subjective, as does the exact determination of the economic consequences, the determination of the possible environmental situations and the assumption that these will occur exactly, just as the selection of the decision rule and its procedure is highly subjective. This is also shown by the fact that the decision rules, of which presumably different decision-makers each prefer a different decision rule, can each identify different action alternatives as the most favourable one.

6.6.7

Section Results

In this section, you: • Got to know the planning techniques under uncertainty. • Learned that the following techniques belong to the planning techniques under uncertainty: the dominance rule, the Maximax rule, the Minimax rule, the Hurwicz rule, the Laplace rule and the Savage–Niehans rule. • Got to know the assumptions and prerequisites for planning under uncertainty. • Applied the planning techniques under uncertainty to an example. • Evaluated and questioned the usefulness of the application for practical problems.

6.7

The Risk Analysis

In contrast to the procedures described in the previous sub-chapter, a probability distribution of the data must be known when applying risk analysis. This can mean that for an investment object the expected values and distributions of the individual calculation elements and any covariances are known, but it can also mean that probability distributions of the target values of alternative investment objects are available. Risk analysis is the collective term for procedures of operations research, in which a single distribution of the target criterion is determined as the result of the decision criterion for the advantageousness of an investment. The risk analysis is a statistically formal procedure, the advantage of which is that a decision is made based on a very structured and meaningful model. However, risk analysis requires that the distribution of the data is known or correctly estimated. This is the weak point that is common to all forecasting models. The future is not exactly predictable. For example, errors can occur in the forecast, the initial data can be incorrect, the assumption of the relevant statistical distribution can be wrong, or the distribution for the concrete problem changes in the forecast period. The chosen decision criterion must maximise the benefit to the investor in the considered time horizon. This assumes that the investor knows his benefit criterion, that it is stable over time and that it is consistent with the decision criterion.

280

6.7.1

6 Investment Decisions in Uncertainty

Procedure for Risk Analysis

In principle, the following sequence of steps must be carried out in the risk analysis: 1. The risk attitude of the investor must be known and expressed by the decision criterion. We elaborate on this point in more detail after this step sequence. 2. The data must be collected. Basically, this is the main problem for the practical value of the results. If the data does not come true in the future, the results of the risk analysis are no better than simple calculations using the dynamic methods. 3. Definition of probability distributions and stochastic dependencies between the individual uncertain input variables. 4. Calculation and presentation of the result distribution. This can be done in three ways: • Through full numeration, in which all possible data constellations are considered, which only makes sense with very small amounts of data. • Through analytical approaches, in which the distributions of results are calculated on the basis of the limit theorem of statistics. However, restrictive assumptions about the statistical distribution and the interdependence of the input variables are necessary here. • Through simulation. Here data is generated with random generators, e.g. with Monte Carlo simulation. 5. Interpretation of the results and selection of the best investment alternative. In principle, the risk attitude can be assumed to be that of a risk-seeking, riskneutral or risk-averse investor. The risk attitude is the willingness to change from a safe to an uncertain situation or vice versa. The easiest way to explain this is with the classic example of lottery participation. In this lottery, there is a 90% probability of blank tickets and a 10% probability of winning tickets. Each prize is exactly 100 euros. The expected value for the winnings is, therefore, 10 euros, since 0.9
0 + 0.1
100 ¼ 10. With a ticket price of 15 euros only a risk-seeking investor would participate in the lottery, he would give up his safe 15 euros to win 100 euros with a 10% probability. His stake, the ticket price of 15 euros, is higher than the expected value of the win of 10 euros, so he is willing to take risks, the relatively small chance of a win means more to him than the safe amount. A risk-neutral investor could spend exactly 10 euros on the purchase of a lottery ticket, he is indifferent to security or risk. Since the ticket price corresponds to the expected value of the prize of 10 euros, he behaves rationally when participating in the lottery. The risk-averse lottery participant is only willing to pay ticket prices below 10 euros, i.e. to give up the safe, e.g. 8 euros for the purchase of a ticket in order to have the chance of winning the lottery with an expected value of 10 euros. If only one ticket is purchased, in the reality of this lottery there is either a total loss of money with 90% probability or a win of 100 euros with 10% probability. In contrast to Sect. 6.6, we will start from a risk-averse investor, i.e. a cautious businessman. How great the degree of risk aversion is, i.e. whether the investor is prepared to invest 1 euro or perhaps 2, 3. . . or 9 euros for a lottery ticket,

6.7 The Risk Analysis

281

expected value (μ)

Ф3 Ф2 Ф1

scattering(σ)

Fig. 6.27 Indifference curves of a risk-averse investor (Source: Author)

must be determined for each individual investor in a separate step, the formation of the individual risk–benefit function. Whether this determined risk–benefit function is then intertemporarily stable, i.e. whether it remains unchanged at least for the duration of the investment, is a further decision-theoretical problem that will not be addressed at this point. The determination is often made according to the Bernoulli principle, which is described in the literature in an elaborate manner (Kruschwitz 2005, p. 307 ff.). The risk attitude of an investor can be represented with an indifference curve (Ф), the curve of equal benefit level. This benefit is influenced by the expected value (μ), i.e. the average level of the target variable and its distribution or dispersion measured in the form of the variance or the square root of the variance, the standard deviation (σ). Figure 6.27 shows that for a risk-averse investor with increasing dispersion (σ): the same level of benefit along the indifference curve (Ф) can only be maintained if the expected value (μ), i.e. the average level of the achieved target variable, also increases. The further away an indifference curve is from the origin, the higher the level of benefit. In the case of investment alternatives with the same expected value (EW, μ) as the target figure, e.g. the net present value, the investor prefers the alternative with the lower dispersion. Figure 6.28 illustrates this. Figure 6.28 shows two investment alternatives with their target values and dispersion. According to the risk theory, the graphs show the dispersion of the target values, here the net present values, around the expected value (mean value) if the investment object would be carried out more frequently. The mean value is indicated by the perpendicular in the maximum of the functions. Both alternatives have the same mean value and are, therefore, equally advantageous for a risk-neutral investor.

6 Investment Decisions in Uncertainty

relative frequency of target value

282

EW target value and scattering

Fig. 6.28 Expected value and dispersion of the target values of two investment alternatives (Source: Author)

However, our assumed risk-averse investor will prefer the solid black investment alternative to the dotted black alternative, as the solid black one has a lower spread. However, alternative risky investment objects rarely have the same expected values of the target values, e.g. the net present values. In this case, the individual risk–benefit function determined by the Bernoulli method, for example, is required. The procedure and significance are now developed using an example. The main problem is to measure the investor’s risk attitude, which can then be represented in the investor’s risk–benefit function. The risk–benefit function is not developed empirically at this point, as is possible according to Bernoulli’s theory, but is derived from Gossen’s First law. First of all, there is a functional relationship between income (Y ), the net present value as a representative of the dynamic investment calculation methods and as the surplus of an investment project is an income variable, and benefit (U ). Increasing income creates increasing benefit. With increasing income, however, according to Gossen’s law, the incremental benefit decreases, i.e. we assume a decreasing marginal benefit. This results in a concave curve of the benefit function U ¼ f (Y ), as shown in Fig. 6.29. The curvature of the function is now, of course, drawn subjectively; it would have to be determined by the Bernoulli method. With this given benefit function, the investor’s risk attitude can be determined precisely. Figure 6.29 shows a simple decision situation. An investor is planning an uncertain investment project. In an unfavourable case, he generates an income of 20 million euros (Y1). In the favourable case, he generates an income of 60 million euros (Y2). Both situations are equally probable, i.e. they have a probability ( p) of p ¼ 0.5. Of course, reality is more complex, there may be intermediate stages between these values and the values themselves are not certain. But these are the assumptions made so far.

6.7 The Risk Analysis

283

Benefit U (Y)

U (Y2) U (EW Y)

U = f (Y)

EW (U) U (Y1)

10

Y1Y(EW(U)),SE 20 30

EW (Y) 40

50

Y2 60

Income (Y)

Fig. 6.29 Determining the certainty equivalent (SE) (Source: Author)

The expected value of the income (EW (Y )) is, therefore, 40 million euros (40 ¼ 0.5
20 + 0.5
60). The aim of the entrepreneur is not to maximise income, but to maximise benefits. Thus, he can achieve either the benefit U (Y1) or the benefit U (Y2). Both are assumed to be equally likely. The average benefit is, therefore, EW (U ), exactly halfway between U (Y1) and U (Y2). However, this benefit EW (U ) is much smaller than the benefit U (EW (Y )) of the average income (EW (Y )). This is due to the concave benefit function. Since the investor is benefit-driven and not income-driven, he expects the income Y (EW (U )) that triggers the average benefit (EW (U )). This value is known as the certainty equivalent (SE) and is the value at which the investor switches from the uncertain investment with the expected value of 40 million euros to a safe investment in the event of a course of the benefit function as shown in Fig. 6.29, with the expected value of the income of 10 million euros being waived. In our presentation, this value is around 30 million euros. This risk-averse attitude of the investor can be solved for empirical applications with the so-called μ-σ rule. According to this rule, the investor has three decisionrelevant components in his benefit function, the expected value of the income, e.g. measured by the expected value of the net present value (EW(Co), μ), the dispersion of the income around the mean value measured as standard deviation (σ) and the weighting of the standard deviation in the benefit function (α). For risk-averse investors, alpha is always negative, and in absolute terms even greater, the greater the risk aversion. The target function, therefore, means that the investor is interested

284

6 Investment Decisions in Uncertainty

in a high income at the lowest possible risk. This conflict of targets is resolved by the degree of risk aversion, i.e. α. The utility function of the μ-σ rule is, therefore, U ¼ μ þ α
σ:

ð6:29Þ

Various other decision routines that reflect the benefit function are possible, but will not be considered further at this point. In the literature, different approaches have been suggested for this purpose, e.g. by Eisenführ and Weber (2002, Chap. 9).

6.7.2

Application Example for Risk Analysis

We now want to apply this rule to an example and, given the investor’s risk attitude α, select the more advantageous alternative. The assumptions made in Chap. 4 in Sects. 4.3 and 4.4 about election decisions may all be true and are considered in this example. The example does not start at the level of the calculation elements of an investment object. In this case, you have to take dependencies between the calculation elements, i.e. covariances, into account, for example between the acquisition payment and the residual value. The example starts at the level of the net present values (Co) that have already been determined. Two investment objects (object 1 and object 2) are considered. Using a computer simulation, a random generator determined a net present value for each object in 1000 calculation runs. This means that for each investment object 1000 calculated net present values are available. In this way, the variation of the target values is determined. Of course, in reality, in an investment that runs over several years with low performance, the management will take countermeasures. This aspect is not considered by the simulation. Output data and solutions are both visible in Fig. 6.30. The calculated net present values were divided into classes. The range of the classes was set at 200,000 euros each (class upper limit–class lower limit in Fig. 6.30). If the simulation determined a value within this range, the column “absolute frequency” was increased by one. The relative frequency and probability of occurrence in the following column is thus one-thousandth of the absolute frequency, since 1000 runs were performed. The relative frequency was multiplied by the middle of the class (middle of class in Fig. 6.30) (p
Co). The addition gives the expected value of the net present value EW (Co). X EWðC o Þ ¼ p j
Co j ð6:30Þ j

The standard deviation, therefore, is: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X
2 σ¼ 2 p j
Co j  EWðCo Þ j

ð6:31Þ

6.7 The Risk Analysis

285

class middle class class Object 1 lower limit upper limit (Co) -700000 -500000 -300000 -100000 100000 300000 500000 700000

-500000 -300000 -100000 100000 300000 500000 700000 900000

-600000 -400000 -200000 0 200000 400000 600000 800000 total

absolut relative frequency frequency p * Co 35 89 158 274 191 156 73 24 1000

0.035 0.089 0.158 0.274 0.191 0.156 0.073 0.024 1

-21000 -35600 -31600 0 38200 62400 43800 19200 75400

p * (Co - EW (Co))2 15965780600 20114459240 11983535280 1557733840 2965305560 16436964960 20089976680 12601083840 1.01715E+11 EW net present value 75400.00 standard deviation 318927.64 benefit(α = - 0,2) 11614.47

class middle class class absolut relative p * (Co - EW (Co))2 frequency frequency p * Co Object 2 lower limit upper limit (Co) -900000 -700000 -800000 4 0.004 -3200 3174098560 -700000 -500000 -600000 26 0.026 -15600 12407320640 -500000 -300000 -400000 90 0.09 -36000 21679617600 -300000 -100000 -200000 156 0.156 -31200 13192083840 -100000 100000 0 260 0.26 0 2143606400 100000 300000 200000 195 0.195 39000 2325304800 300000 500000 400000 160 0.16 64000 15296742400 500000 700000 600000 75 0.075 45000 19446348000 700000 900000 800000 26 0.026 20800 13077080640 900000 1100000 1000000 8 0.008 8000 6613157120 total 1000 1 90800 1.09355E+11 EW net present value 90800.00 standard deviation 330689.22 benefit(α = - 0,2) 24662.16

Fig. 6.30 Determination of EW (Co) and σ for object 1 and 2 (Source: Author)

α was set at 0.2 for the empirical analysis. A negative α reflects a risk-averse attitude. The absolute value provides information about the degree of risk attitude. The higher α, the more risk-averse the investor’s attitude is. The numerical solution can be seen in the last column of Fig. 6.30. For object 1, the values of the simulation fell into a total of eight classes. The expected value of the net present value is 75,400 euros. The standard deviation is 318,927.64 euros. At α ¼ 0.2, this results in a benefit value of 11,614 euros for this investment object 1. For object 2, values from the simulation fell into a total of ten classes. Here, the range of variation was higher. The expected value of the net present value is 90,800 euros. The standard deviation is 330,689.22 euros. At α ¼ 0.2, this results in a benefit value of 24,662 euros for investment object 2.

286

6 Investment Decisions in Uncertainty

The risk analysis, therefore, recommends the selection of object 2, as it has a higher benefit value of 24,662 euros than object 1 with 11,614 euros. Create your own excel sheet with the data used in Fig. 6.30 and reproduce the solution step by step. The risk analysis procedure has thus shown that it is a very well structured and formal procedure based on statistical theory. Theoretically, very qualified decision results are produced, which are at the same time very concrete and operational when IT is used. However, a problem is the amount of data to be obtained, the necessary knowledge of the expected probability distribution of the investment problem and the knowledge of the investor’s risk attitude. These are the difficulties for the use of theory in practical decision situations.

6.7.3

Section Results

In this section, you: • • • • • •

Got to know the importance of risk analysis. Reflected possible risk attitudes of an investor. Learned about the assumptions and prerequisites for risk analysis. Got to know the sequence of steps of the risk analysis. Applied the risk analysis to an example. Evaluated and questioned and the usefulness of the application for practical problems.

6.8

Portfolio Selection

The portfolio selection theory was developed by Harry M. Markowitz in 1952 (Markowitz 1952). In its original meaning, the theory examines the optimal mix of commercial papers in the portfolio. Basically, it is, therefore, an investment programme planning under uncertainty. The technique has been modified by various authors for different questions. The aim of portfolio selection theory is to optimise the composition of investment objects in an uncertain situation for a given budget in such a way that the expected value of the target values of the investment objects, i.e. the net present values or the returns, provides maximum benefit, taking into account the risk in the form of the standard deviation of the target values. The objective function of the risk-averse investor assumed in this model is the μ-σ decision rule known from Sect. 6.7. The theory is based on the possibility of improving the risk/return ratio of a portfolio compared to an investment in a single investment object by mixing investment alternatives in a portfolio.

6.8 Portfolio Selection

287

The diversification of investment alternatives is, therefore, the aim of this method. It is thus in contradiction to the current behaviour of many large companies, which are trying to concentrate on core competencies. The improvement of the risk/return ratio can only be achieved if the investment alternatives are not fully positively correlated with each other, i.e. if there are no completely parallel fluctuations in profit. An advantage of the model is the very systematic and thus intersubjective traceability of the model. However, a major problem of the practical implementation is the collection and processing of the necessary amount of data. For example, all net present values or returns of the considered investment objects and their probability distributions must be known, all variances of these returns and 0.5
(n2  n) covariances, n stands for the number of investment alternatives considered. Thus, with only five investment alternatives, five returns, five variances and ten covariances (0.5
(255)), i.e. 20 values, have to be calculated. Behind each return and thus also behind each variance and covariance is the formation of the return from the net present value formula and thus the presence of all calculation elements of the net present value formula, each of different uncertain situations. For ten investment alternatives, 65 values already have to be determined. Due to the power of IT, the actual calculation is less problematic, but each individual calculation element for the net present value formula must be estimated so that the uncertainty of the calculation results is again considerable. A further problem is that the formation of covariances is actually only possible for security papers, since capital market data are available for the calculation. In the case of real investments, an unfortunately very subjective estimation of the covariances is necessary, since for real investments there is practically never any time series data for net present values. Nevertheless, portfolio selection theory is a very valuable method for structuring a decision problem under uncertainty in investment programme planning and will now be presented in the next chapter.

6.8.1

Procedure for the Portfolio Selection Model According to Markowitz

At this point, the basic model of Markowitz is presented, all further developments from the literature on this theory that have been made over time are ignored, since most of the further developments require significantly more theory formation and significantly more calculation effort to obtain results. Markowitz has also assumed a one-period analysis and random divisibility of the security papers. The optimal portfolio is determined in two steps: • First, all efficient portfolios are determined. • From the efficient portfolios, the optimal portfolio is determined by means of the investor’s risk–benefit function.

288

6 Investment Decisions in Uncertainty

A portfolio is efficient if there is no portfolio with a higher expected return for a given risk level. Alternatively formulated, if a given expected return cannot be achieved at a lower risk level. As already mentioned, the indicator of a portfolio’s efficiency is the μ-σ decision rule. The possible portfolios, i.e. the composition of the given budget from several security papers, are determined as a measure of risk according to the expected value of the return on the total capital invested, which is calculated from the weighted expected values of the returns on the individual investments, and their dispersion. In this presentation, the example calculation is based on the assumption that the given capital is divided between only two assets.

6.8.2

Application Example for Portfolio Selection

We assume that we have 3000 euros available, which we want to divide completely between two different investments. Share 1 costs 200 euros, share 2 costs 150 euros. We consider the occurrence of four environmental states (U ) to be possible, the probabilities of occurrence ( p) of these environmental states are shown in Fig. 6.31. In the event of the occurrence of the environmental states, which we cannot influence, the returns (E(r)), indicated in Fig. 6.31, are obtained for each individual environmental state and weighted across all environmental states. Share 1 thus achieves an average return, i.e. the expected value of the return E(r), of 12.55%. Share 2 achieves the expected value of the return of 9.15%. The values are calculated using the formula for the expected value of the return: X E ðr i Þ ¼ r i ¼ p j r ij ð6:32Þ j

The variance σ 2 is the square of the standard deviation σ. The standard deviation is calculated by the formula sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X
2 ð6:33Þ σi ¼ p j r ij  r i j

In the presentation of the risk/return profiles of the two stock alternatives in Fig. 6.32, it quickly becomes clear that in a safe world, stock 1 would be chosen with a 12.55% return. expected return environmental σ2 U U2 U3 U4 E( r ) situation 1 probability p p1 = 0.3 p2 = 0.35 p3 = 0.15 p4 = 0.2 p = 1 probability p 0.3 0.35 0.15 0.2 1 Security share 1 share 2

Price

σ

Quantity 200 150

15 20

4 10

16 11

21 14

13 1

12.55 36.848 6.070 9.15 18.228 4.269

Fig. 6.31 Return and standard deviations of the sample stocks* (Source: Author)

6.8 Portfolio Selection

289

14 12.55

12

return

10

9.15

8 6 4 2 0

0

2

4 risk

6

8

Fig. 6.32 Risk–return representation of the sample shares (Source: Author)

14 12.55

12

9.83

return

10

9.15

8 6 4 2 0 0

1

2

3

4

5

6

7

risk

Fig. 6.33 Risk/return presentation of the example shares and combinations (Source: Author)

If we now allow arbitrary combinations of these two stocks, a surprising situation emerges, which is illustrated in Fig. 6.33. A combination of both stocks results in an overall lower risk level with a higher return than for the worse investment alternative. This is made possible by the fact that not both stocks react to market data in exactly the same direction. The exact data of various possible combinations are shown in Fig. 6.34. In Fig. 6.33 a combination of 20% (w1) shares 1 and 80% (w2) shares 2 has been assumed. This reduces the risk, as shown in Fig. 6.33. The dot is further to the left of the graph and, with a return of 9.83%, thus has a smaller portfolio risk, which is lower than the individual risks of the two individual stocks. The values are shown in Fig. 6.34. The expected value of the portfolio return E(rP) is calculated using the formula

290

6 Investment Decisions in Uncertainty Expected return on investment Environmental situation

σ2 E( r ) p = 1 p1 = 0,3 p2 = 0,35 p3 = 0,15 p4 = 0,2 0.3 0.35 0.15 0.2 1

Probability p Probability p security price Share 1 Share 2 mixture 20/80

U2

U1

U3

U4

σ

quantity 200 150

15 20

4 10 8.8

16 11 12

21 14 15.4

13 1 3.4

12.55 36.848 9.15 18.228 9.83 14.889

6.070 4.269 3.859

Fig. 6.34 Returns and standard deviations of the sample stocks and combinations* (Source: Author)

E ðr P Þ ¼ r P ¼

X

p j r Pj ¼

X
 p j w1 r 1j þ w2 r 2j ¼ w1 r 1 þ w2 r 2

j

ð6:34Þ

j

The standard deviation of the portfolio return (σ Portfolio, σ P) is obtained by the formula sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X
2 ð6:35Þ σP ¼ p j r Pj  r P j

This formula is empirically easier to determine using a different spelling. For example, it can be derived from Kruschwitz (2005): qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi σ P ¼ w21 σ 21 þ w22 σ 22 þ 2w1 w2 σ 12 ð6:36Þ σ 12 is the covariance of the return of share 1 with the return of share 2, i.e. a degree of the dependency of two random variables on each other. The covariance (σ 12) is calculated using the formula X

 σ 12 ¼ p j r 1j  r 1 r 2j  r 2 : ð6:37Þ j

Alternatively, the formula for the standard deviation of the portfolio return can also be written as: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð6:38Þ σ P ¼ w21 σ 21 þ w22 σ 22 þ 2w1 w2 ρ12 σ 1 σ 2 σ 12 in Eq. (6.38) is the correlation coefficient. The correlation coefficient, like the covariance, is a measure of the mutual dependence of two variables. However, the correlation coefficient is standardised to an interval between 1 and +1. If the coefficient is zero, both random variables are statistically independent of each other. If the coefficient has an absolute relatively high amount, then the distributions of both random variables are strongly dependent on each other. A relatively high positive correlation coefficient means that if, for example, stock 1 has a high return, this also applies to stock 2. It also means that if share 1 has a low

6.8 Portfolio Selection

291

Part of share 1 Part of share 2 Portfolio yield Portfolio risk 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

9.15 9.49 9.83 10.17 10.51 10.85 11.19 11.53 11.87 12.21 12.55

4.269 4.015 3.859 3.814 3.884 4.062 4.337 4.690 5.105 5.569 6.070

Fig. 6.35 Returns and standard deviations of the sample stocks in different portfolio mixes (Source: Author)

return, this also applies to share 2. Both distributions, therefore, fluctuate in the same direction. A relatively high negative correlation coefficient means an opposite fluctuation of the distribution. So, if stock 1 has a high return, stock 2 will tend to have a low return. If stock 1 has a low return, stock 2 will tend to have a high return. Markowitz uses this effect of statistical dependence. In Fig. 6.35, the two stocks defined by the data in Fig. 6.31 are now weighted in different proportions, each in 10% steps. From line to line in Fig. 6.35, the share of stock 1 increases by 10 percentage points and in return the share of stock 2 decreases by 10 percentage points of the total portfolio. In addition, the portfolio return and the portfolio risk are calculated according to the calculation method in Fig. 6.34. The calculation result is shown in Fig. 6.35. Figure 6.36 now shows the curve of all risk–return profiles of these two stocks. As shown in Fig. 6.35, the proportion of the individual stocks has been varied by 10 percentage points. The point with the lowest expected return would result from investing the entire capital in share 2, the point with the highest expected return would result from investing the entire capital in share 1. Before the actual interpretation of this Fig. 6.36, the presentation in Fig. 6.37 should be varied to discuss the significance of the correlation coefficient for the risk position. In Fig. 6.37, the continuous black curve that applies to the example is replaced by a linear hypothetical curve for other correlation coefficients. With a correlation coefficient ρ12 of +1, no risk reduction through diversification can be achieved. Both risk/return positions of the individual security papers share 1 and share 2 are linked linearly. A mix in the portfolio does not result in a risk reduction due to the mix. With a correlation coefficient ρ12 of 1, however, the expected value of the risk can be completely eliminated by diversification; the expected value of the risk can be

292

6 Investment Decisions in Uncertainty 14 12

return

10 8 6 4 2 0

0

1

2

3

4

5

6

7

risk

Fig. 6.36 Curve of risk/return positions for different portfolio mixes (Source: Author)

14

12

ρ12 = - 1 ρ12 = + 1

10

ρ12 = - 1

return

8

6

4

2

0

0

1

2

3

4

5

6

7

risk

Fig. 6.37 Curve of risk/return positions for different portfolio mixes and extreme correlation coefficients (Source: Author)

set to zero by a certain combination of the two security papers in the portfolio. In the example in Fig. 6.37, if the expected value of the return were between 10% and 11%, the expected value of the return would result in a risk-free situation according to the expected value.

6.8 Portfolio Selection

293

14

12

10

return

8

6

4

2

0

0

1

2

3

4

5

6

7

risk

Fig. 6.38 Efficient portfolios with different portfolio mixes (Source: Author)

Finally, the question must be asked as to which portfolio now provides the investor with the greatest benefit. First of all, it should be noted that not all portfolios are efficient. In Fig. 6.38 the perpendicular indicates that the curve on the lower branch forms positions that have the same risk level as positions on the upper branch. Positions on the lower branch are, therefore, not efficient, because at the same level of risk there are portfolios on the upper branch with a higher expected return. The efficient portfolios, therefore, only start at the point where the portfolio is at minimum risk and lie along the upper branch of the curve. In the two-security-paper-portfolio, the minimum risk point can be determined by means of the differential calculus by deriving the portfolio variance according to the quantity share of the individual share in the portfolio, zeroing and solving for this quantity share. The following sequence of steps must be applied; the procedure can be derived from Kruschwitz (2005). First, the formula for portfolio variance must be established. σ 2P ¼ w21 σ 21 þ w22 σ 22 þ 2w1 w2 σ 12

ð6:39Þ

It is to be derived according to the quantity of a share in the portfolio. The first derivation must be created. dσ 2P ¼ 2w1 σ 21  2σ 22 þ 2w1 σ 22 þ 2σ 12  4w1 σ 12 dw1 This derivative must be zeroed and solved for w1. Then you get:

ð6:40Þ

294

6 Investment Decisions in Uncertainty 20 18 Ф3

16 Ф2

14 Ф1

return

12 10 8 6 4 2 0

0

1

2

3

risk

4

5

6

7

Fig. 6.39 Optimal portfolio among efficient portfolios (Source: Author)

w1 ¼

σ 21

σ 22  σ 12 þ σ 22  2σ 12

ð6:41Þ

For our example, the data in Fig. 6.34, after the application of Eq. (6.37) to determine the covariance in Eq. (6.42), the following results: w1 ¼

18:2275  5:4675 ¼ 0:2891 36:8475 þ 18:2275  2
5:4675

ð6:42Þ

Share 1 thus accounts for 28.91% of the minimum-risk portfolio, the share of share 2 in the minimum-risk portfolio is, therefore, 71.09%. The return for this composition is 10.13%, the minimum risk measured in standard deviation is 3.8130. Which point on the line of efficient portfolios represents the optimal portfolio for the investor depends on his risk attitude. By no means does this have to be the portfolio with the minimum risk. The optimal portfolio is determined using the investor’s indifference curve, the concept of which we have already seen in Sect. 6.7 in Fig. 6.27. Here again, the problems already mentioned with the determination of the investor’s indifference curve arise. Figure 6.39 shows the representation with indifference curves. Figure 6.39 shows the dashed indifference curves of an investor assumed to be given. The higher the indifference curves are, the higher the level of benefit for the investor. The point at which the highest indifference curve touches the line of efficient portfolios forms the optimum. The risk level given there and the return determined there form the expected values of the planner and decide on the

6.8 Portfolio Selection

295

Part of share 1

Part of share 2

Portfolio return

Portfolio risk

μ-σP, α=0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

9.15 9.49 9.83 10.17 10.51 10.85 11.19 11.53 11.87 12.21 12.55

4.269 4.015 3.859 3.814 3.884 4.062 4.337 4.690 5.105 5.569 6.070

5.735 6.278 6.743 7.119 7.403 7.600 7.721 7.778 7.786 7.755 7.694

Fig. 6.40 Results of applying the μ-σ decision rule to the example portfolio* (Source: Author)

composition of the optimal portfolio. In Fig. 6.39 it was assumed that the indifference curve Ф2 just touches the line of the efficient portfolios. Where the black lines lead to the axes, optimal expected values of return and risk can then be read. If the investor’s indifference curves are not known, another objective function must be used to determine the composition of the optimal portfolio. Pragmatically, the investor can, therefore, simply select a point on the line of efficient portfolios as his optimal portfolio with the μ-σ decision rule, for example. The expected value of the return is used (μ) and weighted with the risk factor (α), the standard deviation of the portfolio (σ P) is subtracted. The highest calculated value indicates the composition of the optimal portfolio. The difference to the procedure used to determine the efficient portfolios is, therefore, that the standard deviation of the portfolio now receives a weighting factor α, which indicates the subjective risk attitude of the investor. Thus, the optimal portfolio among the efficient portfolios can be determined for the investor with this risk attitude. Theoretically, this procedure should be viewed very critically, as it does not necessarily have to fit the statistical distribution actually given in the practical problem. Pragmatically, however, a concrete solution can be found if the investor’s indifference curves are not known. For example, the risk factor (α) is set at 0.8. This results in the values in Fig. 6.40. Only portfolio variations in 10 percentage point steps were considered. According to the μ-σ decision rule, the portfolio with an 80% share of stock 1 and 20% share of stock 2 would be the optimal portfolio among the efficient ones. There the target value is 7.786, as shown in Fig. 6.40. If α is weighted differently in the decision rule, this ranking will naturally change. The calculation results can be seen in Fig. 6.40. The portfolio selection theory can, therefore, be used to determine combinations of assets for a given budget in a highly qualified manner. The model is theoretically sound and statistically very well supported for empirical applications. The riskminimum combination of assets can thus be found with confidence within the model. This is all very advantageous. However, the model also has some problems

296

6 Investment Decisions in Uncertainty

with practical application. It requires the collection of very extensive data, which must, therefore, be available first. Since the theory is supposed to make futureoriented decisions, the data must, therefore, be forecast. Here is the next problem of the model, the uncertainty. But even if all data were available and secure, the problem of determining the optimum remains. The investor must be able to specify his benefit function in order to select the optimal portfolio from the determined efficient portfolios in a theoretically sound manner. This requirement, too, cannot usually be met in practice, even though an operational solution can then be determined through pragmatic decision routines. In general, however, this solution is then again theoretically not correctly determined.

6.8.3

Section Results

In this section, you: • • • •

Got to know the theory of portfolio selection. Learned the assumptions and prerequisites of portfolio selection theory. Applied the portfolio selection theory to an example. Evaluated and questioned the usefulness of the application for practical problems.

6.9

Case Study

You would like to found a new company together with a business partner. He has already carried out the business planning. The business plan is based on the data in Fig. 6.41. The acquisition payment is 100,000 euros. All values are given in euro. Residual values do not exist. You yourself estimate the market conditions to be more difficult and would like to use the correction procedure to determine what effects data changes have on the net present value, which is your decision criterion for this application case. Exercises: Exercise a) Calculate the net present value with a calculation interest rate of 10%. k

ek (euro)

ak (euro)

1

84,000

44,000

2

82,000

42,000

3 4

83,000 94,000

43,000 44,000

Fig. 6.41 Data for business planning (Source: Author)

6.9 Case Study

ek ak NEk (TEuro) (TEuro) (TEuro) BW

k 0 1 2 3 4 A= i1 = i2 = i3 = i4 = i5 = i6 = i7 =

297

84 82 83 94 100 0.1 0.12 0.02 0.07 0.05 0.09 0.11

44 42 43 44

40 40 40 50

Co, i=0.1

36.36 -63.64 33.06 -30.58 30.05 -0.53 34.15 33.62

Co, i=0.12 -64.29 -32.40 -3.93 27.85

Co, iek =0.12, Co, i=0.1;0.02 iak =0.05 -64.35 -32.58 -4.26 27.29

-66.90 -39.63 -17.70 5.84

Co, i rising -61.90 -26.97 3.92 36.86

ek (TEuro) ak (TEuro) Co,e k Co, a k - 10% + 10% - 10% + 10% 75.6 73.8 74.7 84.6

48.4 -71.27 46.2 -44.99 47.3 -21.18 48.4 6.56

-67.64 -38.05 -11.23 19.92

Co,ek ,ak -/+ 10% -75.27 -52.46 -31.88 -7.15

Fig. 6.42 Solutions to exercises a) to i) of the case study* (Source: Author)

Exercise b) Calculate the net present value with a calculation interest rate of 12%, increased by the lump-sum procedure. Exercise c) Calculate the net present value with double discounting using interest rates of 10% and 2%. Exercise d) Calculate the net present value by discounting all in-payments with 12% and all payouts with 5%. Exercise e) Calculate the net present value by discounting the net earnings of year 1 by 5% and the net earnings of all subsequent years by an interest rate increased by 2 percentage points. Exercise f) Calculate the net present value by expecting in-payments (ek) to reduce by 10%. The calculation interest rate is 10%. Exercise g) Calculate the net present value by expecting payouts (ak) to increase by 10%. The calculation interest rate is 10%. Exercise h) Calculate the net present value by calculating with 10% less inpayments (ek) and 10% more payouts (ak). The calculation interest rate is 10%. Exercise i) Calculate the critical acquisition payment. To do so, use the data from exercise a) as a basis. Solutions We present the solution in Fig. 6.42 in the spreadsheet. Since it is a matter of discounting and summing up the calculation elements, there are various correct procedures for formulating it in the spreadsheet. So far, the solution shown is only a suggestion from which you can deviate as long as you come to the same result. Of course, a solution with a pocket calculator is also possible. You learned the corresponding procedures in Chap. 3. Since in Fig. 6.42 the representation is limited to a maximum of two decimal places, the values in the figure are not as precise as in the following list. Exercise a) The net present value is: 33,624.75 euros Exercise b) The net present value is: 27,849.15 euros Exercise c) The net present value is: 27,293.97 euros Exercise d) The net present value is: 5842.43 euros Exercise e) The net present value is: 36,856.68 euros

298

6 Investment Decisions in Uncertainty

Calculation element Optimistic

Pessimistic

ek

+ 15

– 15

ak

– 10

+ 20

Fig. 6.43 Percentages of change of payment values for triple calculation (Source: Author)

ek ak NEk (TEuro) (TEuro) (TEuro) BW

k 0 1 2 3 4 A= i=

84 82 83 94 100 0.1

44 42 43 44

40 40 40 50

Co, i=0.1

NEkopt NEkpess Co, (TEuro) (TEuro) opt

36.36 -63.64 33.06 -30.58 30.05 -0.53 34.15 33.62

57 56.5 56.75 68.5

Co, pess

18.6 -48.18 -83.09 19.3 -1.49 -67.14 18.95 41.15 -52.90 27.1 87.94 -34.39

Fig. 6.44 Solution for the triple calculation of the case study* (Source: Author)

Exercise f) The net present value is: 6555.29 euros Exercise g) The net present value is: 19,917.77 euros Exercise h) The net present value is: 7151.70 euros Exercise i) The critical acquisition payment is 133,624.75 euros, since Eq. (6.4) can be applied (6.43 = 6.4) Akr ¼ C o þ A

ð6:43Þ

Furthermore, you now want to check the output data with a triple calculation. To do this, you start from the variations of in-payments and payouts shown in Fig. 6.43. Exercise j) Determine the probable, optimistic and pessimistic value according to the triple calculus. Solution Exercise j) We present the solution in the spreadsheet. Since it is a matter of discounting and summing up the calculation elements, there are various correct procedures for formulating the solution in the spreadsheet. So far, the solution shown is only a suggestion from which you can deviate as long as you come to the same result. You learned the corresponding procedures in Chap. 3. In this example, the optimistic situation results in a positive net present value of 87,935.93 euros, the probable situation in 33,624.75 euros and the pessimistic situation in a negative net present value of 34,393.42 euros. You can see the calculation steps of the triple calculation in Fig. 6.44, the results are also shown in Fig. 6.45. For this example, no clear decision can be made using this technique.

6.9 Case Study

299

Net present value

Euro

Co likely

33,624.75

Co optimistic

87,935.93

Co pessimistic

-34,393.42

Fig. 6.45 Net present values of the triple calculation (Source: Author)

Calculation element

Initial value (euro, units, %, years)

p

1000

q

50

l

10

m

5

r

8

x

6

af

5000

i

0.1

n

6

R

10,000

A

100,000

Fig. 6.46 Data record for the target value change calculation (Source: Author)

Because of the threat of loss in this unfavourable situation, you discard the project and plan your own new enterprise using the data from Fig. 6.46. Exercise k) Use the target value change calculation to determine the change in net present value in the event that the calculation elements turn out 20% less favourable than planned. For the useful life, the change is 1 year. In order to record the target value change contribution of the individual input variables without great calculation effort, it was assumed for this example that all input variables remain constant over the term. Of course, a calculation with annually different sizes is also possible. Present the net present value as a percentage deviation from the initial value. The labelling of the variables corresponds to the labelling in Sect. 6.4. Solution Exercise k) We present the solution of the target value change calculation in Fig. 6.47 in the spreadsheet. Of course, a solution with the pocket calculator using the techniques from Chap. 3 is also possible. The net present value in the initial situation is 80,290.69 euros. In the worst case, it drops to 36,738.09 euros. Relative changes are between 54.2% and 1.4%.

300

6 Investment Decisions in Uncertainty i= n=

0.1 6

change (%) output p q l m r x af i n R A

0 -20 -20 20 20 20 20 20 20 -1 -20 20

p q l m r x q af DSF R Abf A Co 1000 50 10 5 8 6 50 5000 4.355261 10000 0.564474 100000 80290.69 800 50 10 5 8 6 50 5000 4.355261 10000 0.564474 100000 36738.09 1000 40 10 5 8 6 40 5000 4.355261 10000 0.564474 100000 41006.24 1000 50 12 5 8 6 50 5000 4.355261 10000 0.564474 100000 78113.06 1000 50 10 6 8 6 50 5000 4.355261 10000 0.564474 100000 78113.06 1000 50 10 5 9.6 6 50 5000 4.355261 10000 0.564474 100000 78200.17 1000 50 10 5 8 7.2 50 5000 4.355261 10000 0.564474 100000 78200.17 1000 50 10 5 8 6 50 6000 4.355261 10000 0.564474 100000 75935.43 1000 50 10 5 8 6 50 5000 4.111407 10000 0.506631 100000 69933.74 1000 50 10 5 8 6 50 5000 3.790787 10000 0.620921 100000 58219.76 1000 50 10 5 8 6 50 5000 4.355261 8000 0.564474 100000 79161.75 1000 50 10 5 8 6 50 5000 4.355261 10000 0.564474 120000 60290.69

rel Co change (%) 0.00 -54.24 -48.93 -2.71 -2.71 -2.60 -2.60 -5.42 -12.90 -27.49 -1.41 -24.91

Fig. 6.47 Solution for the calculation of changes in target values* (Source: Author)

E1 HA 1

HA 2

U1,2 M-growth to newly founded

purchase

E2, E3

U3-U6 M-growth t1

high (U1)

HA 11

growth

high (U1)

HA 12

status quo

high (U3lU1) high (U4lU1)

zero (U2)

HA 11

growth

zero (U5lU2)

zero (U2)

HA 12

status quo

zero (U6lU2)

high (U1)

HA 21

status quo

high (U3lU1)

hoch (U1)

HA 22

sale

high (U4lU1)

zero (U2)

HA 21

status quo

zero (U5lU2)

zero (U2)

HA 22

sale

zero (U6lU2)

Fig. 6.48 Planning data for the decision tree method (Source: Author)

The investment of founding a new company, hereinafter referred to as HA 1, is still of interest to you. You want to use sequential planning to compare this investment with another investment where you can acquire an existing business, this option is called HA 2. To do this, you will use the flexible planning technique. The success of both activities depends primarily on market developments. It is, therefore, the considered environmental condition, which with a given probability occurs randomly, the market situation (M growth to). On favourable conditions, the market grows (high, U1), otherwise it stagnates (zero, U2). In period 2, you can either leave the company unchanged (HA 12, HA 21) or change it, that is, continue to grow if you started with the new company in to (HA 11) or sell part of the purchased company again (HA 22) if you started with the purchase in to. Again, economic success depends on the environmental situation market growth as a conditional probability. Therefore, you are now aware of the planning situation. The decision-making and environmental situations are systematised in Fig. 6.48. The probabilities of occurrence p (U ) and the economic consequences (Y) are shown in Figs. 6.49 and 6.50. The economic consequences are result values, i.e. necessary payments are already balanced. The values are discounted, i.e. related to the time zero.

6.9 Case Study

301

M-growth to

p (U)

high

p (U1) = 0.7 high

M-growth t1

p (U3lU1) = 0.6

p (U)

high

p (U1) = 0.7 zero

p (U4lU1) = 0.4

zero

p (U2) = 0.3 high

p (U5lU2) = 0.6

zero

p (U2) = 0.3 zero

p (U6lU2) = 0.4

Fig. 6.49 Probabilities of occurrence of environmental situations (Source: Author)

Fig. 6.50 Economic consequences (Y ) of the action alternatives under the environmental situations (Source: Author)

Y Y1 Y2 Y3 Y4 Y5 Y6 Y7 Y8

TEuro 110 40 60 55 90 20 55 54

Y Y9 Y10 Y11 Y12 Y13 Y14 Y15 Y16

TEuro 110 30 55 50 10 8 60 110

Exercise l) Determine the economic consequences of the action alternatives HA 1 and HA 2 with flexible planning and select the optimal alternative. Solution Exercise l) In flexible planning, where adjustments can still be made at time 1, the action alternative HA 2, the company acquisition, is selected. Compared to the alternative HA 1, founding a new company, which is expected to have a net present value of 76,000 euros, it provides a maximum economic consequence in the form of the expected net present value of 78,600 euros. The calculation can be seen in Fig. 6.51. This result now confuses you; in exercise k) you had determined with the target change calculation that the start-up is positive, but the sequential planning in exercise l) preferred the purchase of the company to the start-up. Since you are uncertain, you decide not to pursue both projects and instead use your capital to set up a venture capital fund that wants to give private equity capital to young entrepreneurs. This has already become known to interested parties who have applied to you for equity financing with their projects (HA). The young entrepreneurs have also already planned by means of a scenario technique which corporate profit (WK) will occur in which environmental situation (U ). Unfortunately, none of the young entrepreneurs can give a probability of occurrence ( p) of the environmental situation. You consider the planned environmental situations as complete and also believe that the expected values of the economic consequences (WK), which are given as net present values in million euros, are realistic. You summarised the individual projects (HA) of the young entrepreneurs in a table with the possible environmental situations (U ) and the economic consequences that are likely to occur. The data are shown in Fig. 6.52.

302

6 Investment Decisions in Uncertainty to

Period 1

Period 2 0.6 82 HA 11

Y10

82

Y20

62

110

Z3

0.4

40

Y10/E2

0.6 58 HA 12

Z4

0.4

60 60 55

Z1

0.6 0.7

0.3

62 HA 11

76 HA 01

Z3

0.4

90 90

20

Y20/E2

0.6 54.6 HA 12

Z4

0.4

55 55 54

E1

0.6 78 HA 21

110 110

Z3

0.4

30

Y30/E3

HA 02 78.6

0.6

0.3 0.7

53 HA 22

Z4

0.4

55 50

Z2

0.6 9.2 HA 21 Y30

Z3

78

0.4

10 10 8

Y40/E3

Y40

80

0.6 80 HA 22

Z4

0.4

Fig. 6.51 Decision tree with flexible planning for the case study* (Source: Author)

60 60 110

6.9 Case Study

303

U2

U1

Alternatives for action (HA)

Environmental conditions (U) U3 U4 U5 U6

HA1

15

18

28

29

31

23

HA2

13

14

27

13

20

21

HA3

16

21

29

17

22

19

HA4

32

24

12

11

11

25

HA5

20

33

21

15

17

14

HA6

21

23

22

13

14

26

HA7

13

11

12

15

14

13

Fig. 6.52 Projects of young entrepreneurs with environmental situations and economic consequences (Source: Author)

Since you too are unable to estimate the probability of the environmental situation occurring, you decide to apply the techniques of decision-making under uncertainty. Exercise m) Check the projects in Fig. 6.52 for absolute dominance and for state dominance and then apply the following decision rules to the remaining projects one after the other and document the result: • • • • •

Maximax rule Minimax rule Hurwicz rule Laplace rule Savage–Niehans rule (minimum regret rule)

Solution Exercise m) Absolute dominance exists, since action alternative 7 is completely dominated by action alternative 3. Action alternative 7, therefore, does not need to be considered further. Action alternative 2 is subject to a state dominance of action alternative 1, so action alternative 2 needs not be considered further. There is no further state dominance. The Maximax rule prefers action alternative 5. Its maximum expected net present value in an environmental situation 2 is 33 million euros and, therefore, is greater than any other expected net present value in Fig. 6.52. The Minimax rule prefers alternative 3. Its maximum minimum expected net present value in environmental situations 1 is 16 million euros and, therefore, is greater than any other expected minimum net present value of the alternatives in Fig. 6.52. For the Hurwicz rule we have set a pessimism parameter of 0.6. The preferred action alternative is then HA 5, as can be seen in Fig. 6.53. The Laplace rule prefers action alternative 1, as can be seen in Fig. 6.54.

304

6 Investment Decisions in Uncertainty

Alternatives for action (HA)

Environmental conditions (U) Hurwicz U3 U4 U5 U6

U2

U1 HA1

15

HA3

16

HA4

32

31 29

21.2 11

19.4

33

HA5

21.4

13

HA6

14

21.6

26

18.2

Fig. 6.53 Results according to Hurwicz on the case study* (Source: Author)

U1 Alternatives for action (HA)

Environmental conditions (U) Laplace U3 U4 U5 U6

U2

HA1

15

18

28

29

31

23

24

HA3

16

21

29

17

22

19

20.67

HA4

32

24

12

11

11

25

19.17

HA5

20

33

21

15

17

14

20

HA6

21

23

22

13

14

26

19.83

Fig. 6.54 Laplace’s findings on the case study* (Source: Author)

U1 Alternatives for action (HA)

SavageEnvironmental conditions (U) Niehans U3 U4 U5 U6

U2

HA1

-17

-15

-1

0

HA3

-16

HA4

0

HA5

-12

HA6

-11

0

-3

-36

-12

0

-9

-17

-12

-9

-7

-56

-18

-20

-1

-65

0 -10

-8

-14

-14

-12

-60

-7

-16

-17

0

-61

Fig. 6.55 Savage–Niehans results for the case study* (Source: Author)

The Savage–Niehans rule prefers action alternative 1, as shown in Fig. 6.55, because it has the least deviation from the optimum. Since these decision routines do not provide clear results either, you also reject the idea of a venture capital fund. Somewhat demotivated, you decide to end the office day earlier than usual, when a glance at your diary reveals that a skating tournament of your skat club starts today at 8 pm in a nearby restaurant. There you could take part since months again and you could just manage to make it. Since you

6.9 Case Study

305

only find a parking space in an unlit side street, you take your notebook with the company data with you to the restaurant. At your table, you play with two business colleagues who are friends. Unfortunately, none of you have any points above the expected value, so you all do not qualify for the main round and cannot take part in the crunch time. Since you do not want to end the evening so early, you stay together in the conversation. You tell your colleagues about the failed attempt to invest your capital. Both are interested in a partnership with you, also have their notebooks with them and offer you the planning data. Since, in contrast to the previous problem, they are experienced entrepreneurs, they can also indicate probabilities of occurrence for their activities. Both colleagues have carried out planning calculations for the project for the offered partnership and offer you the simulation results, which you will transfer to your notebook. Both colleagues have planned using the net present value with a simulation model, both have performed exactly 1000 iterations. You summarise the information of both colleagues and store it in Fig. 6.56. Since the probability Fig. 6.56 Data for risk analysis (Source: Author)

class class class middle Object 1 lower limit upper limit (Co) -1700000 -1300000 -900000 -500000 -100000 300000 700000 1100000

-1300000 -1500000 -900000 -1100000 -500000 -700000 -100000 -300000 300000 100000 700000 500000 1100000 900000 1500000 1300000 total

class class Object 2 lower limit upper limit -1900000 -1500000 -1500000 -1100000 -1100000 -700000 -700000 -300000 -300000 100000 100000 500000 500000 900000 900000 1300000 1300000 1700000 1700000 2100000

absolut relative frequency frequency 7 63 131 236 144 205 175 39 1000

0.007 0.063 0.131 0.236 0.144 0.205 0.175 0.039 1

class middle absolut relative (Co) frequency frequency -1700000 14 0.014 -1300000 25 0.025 -900000 85 0.085 -500000 154 0.154 -100000 251 0.251 300000 188 0.188 700000 164 0.164 1100000 79 0.079 1500000 26 0.026 1900000 14 0.014 total 1000 1

306

6 Investment Decisions in Uncertainty

distributions are supposedly known, you want to apply the μ-σ rule and have a personal risk weighting of α ¼ 0.15. Exercise n) For the planning data from Fig. 6.56, determine the benefit values of object 1 and object 2 using the risk analysis. Solution Exercise n) We present the solution in the spreadsheet. Since these are linked arithmetical operations, there are several correct ways of formulating them in the spreadsheet. So far, the shown solution in Fig. 6.57 is only a suggestion from which you can deviate as long as you come to the same result. Theoretically, a solution with a pocket calculator is also possible. However, this is very calculation-intensive.

class class class middle Object 1 lower limit upper limit (Co) -1700000 -1300000 -900000 -500000 -100000 300000 700000 1100000

-1300000 -1500000 -900000 -1100000 -500000 -700000 -100000 -300000 300000 100000 700000 500000 1100000 900000 1500000 1300000 total

absolut relative frequency frequency p * Co 7 63 131 236 144 205 175 39 1000

p * (Co - EW (Co))2

0.007 -10500 0.063 -69300 0.131 -91700 0.236 -70800 0.144 14400 0.205 102500 0.175 157500 0.039 50700 1 82800

17536790880 88137997920 80273635040 34582458240 42600960 35681447200 1.16868E+11 57781457760 4.30904E+11 EW net present value 82800 standard deviation 656432.91 benefit (α = - 0,15) -15664.94

class class Object 2 lower limit upper limit -1900000 -1500000 -1500000 -1100000 -1100000 -700000 -700000 -300000 -300000 100000 100000 500000 500000 900000 900000 1300000 1300000 1700000 1700000 2100000

class middle absolut relative (Co) frequency frequency p * Co p * (Co - EW (Co))2 -1700000 14 0.014 -23800 44797276160 -1300000 25 0.025 -32500 48219136000 -900000 85 0.085 -76500 83106662400 -500000 154 0.154 -77000 53389557760 -100000 251 0.251 -25100 8947005440 300000 188 0.188 56400 8385822720 700000 164 0.164 114800 61264732160 1100000 79 0.079 86900 80779509760 1500000 26 0.026 39000 51778621440 1900000 14 0.014 26600 45926236160 total 1000 1 88800 4.86595E+11 EW net present value standard deviation benefit

Fig. 6.57 Risk analysis solution for the case study* (Source: Author)

88800 697563.3 -15834.5

6.9 Case Study

307 expected return

security share 1 share 2

price quantity 100 4000 80 5000

environmental situation

U1

probability p probability p

p1 = 0,2 p2 = 0,3 p3 = 0,35 p4 = 0,15 0.2 0.3 0.35 0.15

U2

6 25

U3

14 21

U4

18 4

16 12

Fig. 6.58 Expected return on both shares* (Source: Author)

Unfortunately, this skat evening did not put you on the right track on your way to investing the available capital. Both objects have a positive net present value, object 1 in the amount of 82,800 euros, object 2 in the amount of 88,800 euros, but both utility values are negative. Object 1 in the amount of 15,664.94 euros, object 2 in the amount of 15,834.50 euros. This results from the risk weighting α ¼ 0.15 for the standard deviation. You also have doubts that your colleagues have generated the data properly and that it is subject to normal distribution, which is the basis of this decision criterion. Therefore, you now decide to refrain from investing in companies and to invest the capital profitably on the capital market. Only two blue-chip shares are eligible for investment, companies you are convinced of because you have worked for them yourself in the past. The amount of capital to be invested is fixed for you, only the combination of both shares is still open for you. To do this, you want to apply the portfolio selection theory. You want to invest 400,000 euros, share 1 costs 100 euros, so you could buy 4000 shares if you only bought this share. Share 2 costs 80 euros, so you could buy 5000 shares if you only bought this one. You have spoken to some old colleagues in both companies who are senior managers there. Both have given you four scenarios with probabilities of occurrence and then expected returns. You share these evaluations. You have compiled the data in table form, which can be seen in Fig. 6.58. Exercise o) For the data in Fig. 6.58, determine the risk-minimum combination of stocks 1 and 2 using the portfolio selection theory. The proportion of the individual stocks only has to be varied by 10 percentage points. You do not want to make an exact percentage determination by differentiation. Solution Exercise o) First of all, we will present the solution in the spreadsheet. Since these are linked arithmetic operations, there are several correct procedures for formulating them in the spreadsheet. So far, the shown solution is only a suggestion from which you can deviate, as long as you come to the same result. Theoretically, a solution with a pocket calculator is also possible. However, this is very calculation-intensive.

308

6 Investment Decisions in Uncertainty

part of part of portfolio portfolio share 1 share 2 return risk 0 1 14.5 8.62 0.1 0.9 14.46 7.38 0.2 0.8 14.42 6.16 0.3 0.7 14.38 4.95 0.4 0.6 14.34 3.78 0.5 0.5 14.3 2.68 0.6 0.4 14.26 1.80 0.7 0.3 14.22 1.56 0.8 0.2 14.18 2.19 0.9 0.1 14.14 3.21 1 0 14.1 4.36 Fig. 6.59 Solution to the case study according to the portfolio selection theory* (Source: Author)

14.55

14.5 14.45

return

14.4 14.35 14.3 14.25 14.2 14.15 14.1

14.05

0

1

2

3

4

5

6

7

8

9

10

risk

Fig. 6.60 Curve of risk/return positions for different portfolio mixes for the case study (Source: Author)

You now find that for the simple objective of the minimum risk composition of the portfolio at the level of 10% steps participation of every share, share 1 must be represented in the portfolio at 70%, share 2 at 30%. Then the expected value of the return is 14.22%, the portfolio risk measured in the standard deviation is at this point minimal in the amount of 1.56. The values are shown in Fig. 6.59 and we illustrate them in a graph in Fig. 6.60.

6.10

Summary

309

This analysis has now convinced you. You can achieve a return of 14.22% and this at a relatively low risk level. You are satisfied with your risk analyses and have now found an attractive and relatively safe investment, at least according to the given plan values, which are only forecasts. You decide to invest the money in the capital market instead of starting or participating in risky ventures.

6.10

Summary

In this chapter, we have dealt with the consideration of uncertainty in investment calculations and decisions. Only these techniques of investment calculation take real decision situations in the company into account, as they allow for uncertainty in the models, which are usually present in investment decisions in practice. The application of these techniques for corporate management will always mean that the complexity of the decision-making processes in the company increases. The higher the percentage of the investment capital in the company, the more important the techniques will get. The large amounts of empirical data required for the application of these techniques have now become more or less easily processable due to the improvement in IT performance. The procedures described in this chapter can be divided into different approaches. First, there are the correction methods, where calculation elements are usually changed in percentage for reasons of precaution. This technique is not very expensive, but theoretically not very complex and practically not very meaningful. Sensitivity analyses measure the effect of the change of one or more exogenous variables on an endogenous variable. These techniques show in a scenario-like manner what influence the change of individual calculation elements has on the target values. Three techniques are presented. This is the critical value calculation, which records the change in the target value when a calculation element is changed. The triple calculation calculates the investment problem using a probable, a pessimistic and an optimistic value. The target value change calculation determines how the target values of the dynamic investment calculation change when one or more calculation elements are varied by a certain percentage. These techniques thus determine the effects of data set changes. Causes or probabilities of data set changes are not analysed. When planning under uncertainty, defined decision rules are applied because of the unknown probability of environmental conditions occurring, so that an investment object can be selected in an uncertain situation. These decision rules are, however, theoretically very poorly substantiated and, as a rule, not everyone chooses the same investment alternative as the most favourable one. The consideration of risk by probability distributions is done in sequential planning, risk analysis and portfolio selection theory. These techniques require knowledge of the probability distribution of occurring calculation elements or target values. A decision routine in the form of a utility function or risk-relevant target criterion is used for the investment decision.

310

6 Investment Decisions in Uncertainty

Sequential planning is a decision tree procedure in which the investor determines the value of an alternative course of action from the backward addition of the possible economic consequences weighted with the probabilities of occurrence of the environmental conditions. In risk analysis and portfolio selection theory, probability distributions and stochastically based decision rules are used to select suitable investment objects. The problem here lies in the actual knowledge of the empirical probability distributions of the calculation elements in the future and in the measurement of the benefit of an investment decision by the investor. All techniques of considering uncertainty in investment decisions are useful methods for structuring the investment problem and help the investor to make a transparent and intersubjectively comprehensible investment decision. However, the analysis does not turn an uncertain future into a safe world, so the techniques certainly increase the probability of a meaningful investment decision, but there is no guarantee for a good investment decision by applying these techniques.

References Eisenführ, F., & Weber, M. (2002). Rationales Entscheiden (4. Aufl.). Berlin: Springer. Kruschwitz, L. (2005). Investitionsrechnung (10. Aufl.). München: Oldenbourg Verlag. Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance, 7, 77–92.

Tables of Financial Mathematics

This annex contains tables with financial mathematical factors for frequently used interest rates and useful lives. They are useful when the investment calculator is practicing the dynamic investment calculation methods with the pocket calculator, without the help of computer technology. All tables have the same structure. The relevant interest rate is specified in the table header. The six financial mathematical factors are in the columns, the years of useful life are in the rows. You can then read the empirical financial mathematical values in the corresponding cells. The fields of knowledge in the theory of financial mathematical factors and the corresponding applications of these factors to examples and an examination of the problems and requirements of the application of these factors are discussed in detail in Sect. 3.3 in this book. The table of financial mathematical factors as an Excel file is available on the homepage of Springer (https://link.springer.com, search term “Kay Poggensee”). By changing cell A2 in the Excel file, the values for the desired interest rate can be determined. It must be entered in decimal notation.

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. Poggensee, J. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-62440-8

311

312

Tables of Financial Mathematics

2% 0.02

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.980392 0.961169 0.942322 0.923845 0.905731 0.887971 0.870560 0.853490 0.836755 0.820348 0.804263 0.788493 0.773033 0.757875 0.743015 0.728446 0.714163 0.700159 0.686431 0.672971 0.659776 0.646839 0.634156 0.621721 0.609531 0.597579 0.585862 0.574375 0.563112 0.552071 0.500028 0.452890 0.410197 0.371528

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.020000 1.040400 1.061208 1.082432 1.104081 1.126162 1.148686 1.171659 1.195093 1.218994 1.243374 1.268242 1.293607 1.319479 1.345868 1.372786 1.400241 1.428246 1.456811 1.485947 1.515666 1.545980 1.576899 1.608437 1.640606 1.673418 1.706886 1.741024 1.775845 1.811362 1.999890 2.208040 2.437854 2.691588

n

1.020000 0.515050 0.346755 0.262624 0.212158 0.178526 0.154512 0.136510 0.122515 0.111327 0.102178 0.094560 0.088118 0.082602 0.077825 0.073650 0.069970 0.066702 0.063782 0.061157 0.058785 0.056631 0.054668 0.052871 0.051220 0.049699 0.048293 0.046990 0.045778 0.044650 0.040002 0.036556 0.033910 0.031823

n

0.980392 1.941561 2.883883 3.807729 4.713460 5.601431 6.471991 7.325481 8.162237 8.982585 9.786848 10.575341 11.348374 12.106249 12.849264 13.577709 14.291872 14.992031 15.678462 16.351433 17.011209 17.658048 18.292204 18.913926 19.523456 20.121036 20.706898 21.281272 21.844385 22.396456 24.998619 27.355479 29.490160 31.423606

1.000000 0.495050 0.326755 0.242624 0.192158 0.158526 0.134512 0.116510 0.102515 0.091327 0.082178 0.074560 0.068118 0.062602 0.057825 0.053650 0.049970 0.046702 0.043782 0.041157 0.038785 0.036631 0.034668 0.032871 0.031220 0.029699 0.028293 0.026990 0.025778 0.024650 0.020002 0.016556 0.013910 0.011823

1.000000 2.020000 3.060400 4.121608 5.204040 6.308121 7.434283 8.582969 9.754628 10.949721 12.168715 13.412090 14.680332 15.973938 17.293417 18.639285 20.012071 21.412312 22.840559 24.297370 25.783317 27.298984 28.844963 30.421862 32.030300 33.670906 35.344324 37.051210 38.792235 40.568079 49.994478 60.401983 71.892710 84.579401

Tables of Financial Mathematics

313

3% 0.03

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.970874 0.942596 0.915142 0.888487 0.862609 0.837484 0.813092 0.789409 0.766417 0.744094 0.722421 0.701380 0.680951 0.661118 0.641862 0.623167 0.605016 0.587395 0.570286 0.553676 0.537549 0.521893 0.506692 0.491934 0.477606 0.463695 0.450189 0.437077 0.424346 0.411987 0.355383 0.306557 0.264439 0.228107

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.030000 1.060900 1.092727 1.125509 1.159274 1.194052 1.229874 1.266770 1.304773 1.343916 1.384234 1.425761 1.468534 1.512590 1.557967 1.604706 1.652848 1.702433 1.753506 1.806111 1.860295 1.916103 1.973587 2.032794 2.093778 2.156591 2.221289 2.287928 2.356566 2.427262 2.813862 3.262038 3.781596 4.383906

n

1.030000 0.522611 0.353530 0.269027 0.218355 0.184598 0.160506 0.142456 0.128434 0.117231 0.108077 0.100462 0.094030 0.088526 0.083767 0.079611 0.075953 0.072709 0.069814 0.067216 0.064872 0.062747 0.060814 0.059047 0.057428 0.055938 0.054564 0.053293 0.052115 0.051019 0.046539 0.043262 0.040785 0.038865

n

0.970874 1.913470 2.828611 3.717098 4.579707 5.417191 6.230283 7.019692 7.786109 8.530203 9.252624 9.954004 10.634955 11.296073 11.937935 12.561102 13.166118 13.753513 14.323799 14.877475 15.415024 15.936917 16.443608 16.935542 17.413148 17.876842 18.327031 18.764108 19.188455 19.600441 21.487220 23.114772 24.518713 25.729764

1.000000 0.492611 0.323530 0.239027 0.188355 0.154598 0.130506 0.112456 0.098434 0.087231 0.078077 0.070462 0.064030 0.058526 0.053767 0.049611 0.045953 0.042709 0.039814 0.037216 0.034872 0.032747 0.030814 0.029047 0.027428 0.025938 0.024564 0.023293 0.022115 0.021019 0.016539 0.013262 0.010785 0.008865

1.000000 2.030000 3.090900 4.183627 5.309136 6.468410 7.662462 8.892336 10.159106 11.463879 12.807796 14.192030 15.617790 17.086324 18.598914 20.156881 21.761588 23.414435 25.116868 26.870374 28.676486 30.536780 32.452884 34.426470 36.459264 38.553042 40.709634 42.930923 45.218850 47.575416 60.462082 75.401260 92.719861 112.796867

314

Tables of Financial Mathematics

5% 0.05

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.952381 0.907029 0.863838 0.822702 0.783526 0.746215 0.710681 0.676839 0.644609 0.613913 0.584679 0.556837 0.530321 0.505068 0.481017 0.458112 0.436297 0.415521 0.395734 0.376889 0.358942 0.341850 0.325571 0.310068 0.295303 0.281241 0.267848 0.255094 0.242946 0.231377 0.181290 0.142046 0.111297 0.087204

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.050000 1.102500 1.157625 1.215506 1.276282 1.340096 1.407100 1.477455 1.551328 1.628895 1.710339 1.795856 1.885649 1.979932 2.078928 2.182875 2.292018 2.406619 2.526950 2.653298 2.785963 2.925261 3.071524 3.225100 3.386355 3.555673 3.733456 3.920129 4.116136 4.321942 5.516015 7.039989 8.985008 11.467400

n

1.050000 0.537805 0.367209 0.282012 0.230975 0.197017 0.172820 0.154722 0.140690 0.129505 0.120389 0.112825 0.106456 0.101024 0.096342 0.092270 0.088699 0.085546 0.082745 0.080243 0.077996 0.075971 0.074137 0.072471 0.070952 0.069564 0.068292 0.067123 0.066046 0.065051 0.061072 0.058278 0.056262 0.054777

n

0.952381 1.859410 2.723248 3.545951 4.329477 5.075692 5.786373 6.463213 7.107822 7.721735 8.306414 8.863252 9.393573 9.898641 10.379658 10.837770 11.274066 11.689587 12.085321 12.462210 12.821153 13.163003 13.488574 13.798642 14.093945 14.375185 14.643034 14.898127 15.141074 15.372451 16.374194 17.159086 17.774070 18.255925

1.000000 0.487805 0.317209 0.232012 0.180975 0.147017 0.122820 0.104722 0.090690 0.079505 0.070389 0.062825 0.056456 0.051024 0.046342 0.042270 0.038699 0.035546 0.032745 0.030243 0.027996 0.025971 0.024137 0.022471 0.020952 0.019564 0.018292 0.017123 0.016046 0.015051 0.011072 0.008278 0.006262 0.004777

1.000000 2.050000 3.152500 4.310125 5.525631 6.801913 8.142008 9.549109 11.026564 12.577893 14.206787 15.917127 17.712983 19.598632 21.578564 23.657492 25.840366 28.132385 30.539004 33.065954 35.719252 38.505214 41.430475 44.501999 47.727099 51.113454 54.669126 58.402583 62.322712 66.438848 90.320307 120.799774 159.700156 209.347996

Tables of Financial Mathematics

315

6% 0.06

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.943396 0.889996 0.839619 0.792094 0.747258 0.704961 0.665057 0.627412 0.591898 0.558395 0.526788 0.496969 0.468839 0.442301 0.417265 0.393646 0.371364 0.350344 0.330513 0.311805 0.294155 0.277505 0.261797 0.246979 0.232999 0.219810 0.207368 0.195630 0.184557 0.174110 0.130105 0.097222 0.072650 0.054288

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.060000 1.123600 1.191016 1.262477 1.338226 1.418519 1.503630 1.593848 1.689479 1.790848 1.898299 2.012196 2.132928 2.260904 2.396558 2.540352 2.692773 2.854339 3.025600 3.207135 3.399564 3.603537 3.819750 4.048935 4.291871 4.549383 4.822346 5.111687 5.418388 5.743491 7.686087 10.285718 13.764611 18.420154

n

1.060000 0.545437 0.374110 0.288591 0.237396 0.203363 0.179135 0.161036 0.147022 0.135868 0.126793 0.119277 0.112960 0.107585 0.102963 0.098952 0.095445 0.092357 0.089621 0.087185 0.085005 0.083046 0.081278 0.079679 0.078227 0.076904 0.075697 0.074593 0.073580 0.072649 0.068974 0.066462 0.064700 0.063444

n

0.943396 1.833393 2.673012 3.465106 4.212364 4.917324 5.582381 6.209794 6.801692 7.360087 7.886875 8.383844 8.852683 9.294984 9.712249 10.105895 10.477260 10.827603 11.158116 11.469921 11.764077 12.041582 12.303379 12.550358 12.783356 13.003166 13.210534 13.406164 13.590721 13.764831 14.498246 15.046297 15.455832 15.761861

1.000000 0.485437 0.314110 0.228591 0.177396 0.143363 0.119135 0.101036 0.087022 0.075868 0.066793 0.059277 0.052960 0.047585 0.042963 0.038952 0.035445 0.032357 0.029621 0.027185 0.025005 0.023046 0.021278 0.019679 0.018227 0.016904 0.015697 0.014593 0.013580 0.012649 0.008974 0.006462 0.004700 0.003444

1.000000 2.060000 3.183600 4.374616 5.637093 6.975319 8.393838 9.897468 11.491316 13.180795 14.971643 16.869941 18.882138 21.015066 23.275970 25.672528 28.212880 30.905653 33.759992 36.785591 39.992727 43.392290 46.995828 50.815577 54.864512 59.156383 63.705766 68.528112 73.639798 79.058186 111.434780 154.761966 212.743514 290.335905

316

Tables of Financial Mathematics

8% 0.08

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.925926 0.857339 0.793832 0.735030 0.680583 0.630170 0.583490 0.540269 0.500249 0.463193 0.428883 0.397114 0.367698 0.340461 0.315242 0.291890 0.270269 0.250249 0.231712 0.214548 0.198656 0.183941 0.170315 0.157699 0.146018 0.135202 0.125187 0.115914 0.107328 0.099377 0.067635 0.046031 0.031328 0.021321

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.080000 1.166400 1.259712 1.360489 1.469328 1.586874 1.713824 1.850930 1.999005 2.158925 2.331639 2.518170 2.719624 2.937194 3.172169 3.425943 3.700018 3.996019 4.315701 4.660957 5.033834 5.436540 5.871464 6.341181 6.848475 7.396353 7.988061 8.627106 9.317275 10.062657 14.785344 21.724521 31.920449 46.901613

n

1.080000 0.560769 0.388034 0.301921 0.250456 0.216315 0.192072 0.174015 0.160080 0.149029 0.140076 0.132695 0.126522 0.121297 0.116830 0.112977 0.109629 0.106702 0.104128 0.101852 0.099832 0.098032 0.096422 0.094978 0.093679 0.092507 0.091448 0.090489 0.089619 0.088827 0.085803 0.083860 0.082587 0.081743

n

0.925926 1.783265 2.577097 3.312127 3.992710 4.622880 5.206370 5.746639 6.246888 6.710081 7.138964 7.536078 7.903776 8.244237 8.559479 8.851369 9.121638 9.371887 9.603599 9.818147 10.016803 10.200744 10.371059 10.528758 10.674776 10.809978 10.935165 11.051078 11.158406 11.257783 11.654568 11.924613 12.108402 12.233485

1.000000 0.480769 0.308034 0.221921 0.170456 0.136315 0.112072 0.094015 0.080080 0.069029 0.060076 0.052695 0.046522 0.041297 0.036830 0.032977 0.029629 0.026702 0.024128 0.021852 0.019832 0.018032 0.016422 0.014978 0.013679 0.012507 0.011448 0.010489 0.009619 0.008827 0.005803 0.003860 0.002587 0.001743

1.000000 2.080000 3.246400 4.506112 5.866601 7.335929 8.922803 10.636628 12.487558 14.486562 16.645487 18.977126 21.495297 24.214920 27.152114 30.324283 33.750226 37.450244 41.446263 45.761964 50.422921 55.456755 60.893296 66.764759 73.105940 79.954415 87.350768 95.338830 103.965936 113.283211 172.316804 259.056519 386.505617 573.770156

Tables of Financial Mathematics

317

10% 0.1

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.909091 0.826446 0.751315 0.683013 0.620921 0.564474 0.513158 0.466507 0.424098 0.385543 0.350494 0.318631 0.289664 0.263331 0.239392 0.217629 0.197845 0.179859 0.163508 0.148644 0.135131 0.122846 0.111678 0.101526 0.092296 0.083905 0.076278 0.069343 0.063039 0.057309 0.035584 0.022095 0.013719 0.008519

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.100000 1.210000 1.331000 1.464100 1.610510 1.771561 1.948717 2.143589 2.357948 2.593742 2.853117 3.138428 3.452271 3.797498 4.177248 4.594973 5.054470 5.559917 6.115909 6.727500 7.400250 8.140275 8.954302 9.849733 10.834706 11.918177 13.109994 14.420994 15.863093 17.449402 28.102437 45.259256 72.890484 117.390853

n

1.100000 0.576190 0.402115 0.315471 0.263797 0.229607 0.205405 0.187444 0.173641 0.162745 0.153963 0.146763 0.140779 0.135746 0.131474 0.127817 0.124664 0.121930 0.119547 0.117460 0.115624 0.114005 0.112572 0.111300 0.110168 0.109159 0.108258 0.107451 0.106728 0.106079 0.103690 0.102259 0.101391 0.100859

n

0.909091 1.735537 2.486852 3.169865 3.790787 4.355261 4.868419 5.334926 5.759024 6.144567 6.495061 6.813692 7.103356 7.366687 7.606080 7.823709 8.021553 8.201412 8.364920 8.513564 8.648694 8.771540 8.883218 8.984744 9.077040 9.160945 9.237223 9.306567 9.369606 9.426914 9.644159 9.779051 9.862808 9.914814

1.000000 0.476190 0.302115 0.215471 0.163797 0.129607 0.105405 0.087444 0.073641 0.062745 0.053963 0.046763 0.040779 0.035746 0.031474 0.027817 0.024664 0.021930 0.019547 0.017460 0.015624 0.014005 0.012572 0.011300 0.010168 0.009159 0.008258 0.007451 0.006728 0.006079 0.003690 0.002259 0.001391 0.000859

1.000000 2.100000 3.310000 4.641000 6.105100 7.715610 9.487171 11.435888 13.579477 15.937425 18.531167 21.384284 24.522712 27.974983 31.772482 35.949730 40.544703 45.599173 51.159090 57.274999 64.002499 71.402749 79.543024 88.497327 98.347059 109.181765 121.099942 134.209936 148.630930 164.494023 271.024368 442.592556 718.904837 1163.908529

318

Tables of Financial Mathematics

12% 0.12

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.892857 0.797194 0.711780 0.635518 0.567427 0.506631 0.452349 0.403883 0.360610 0.321973 0.287476 0.256675 0.229174 0.204620 0.182696 0.163122 0.145644 0.130040 0.116107 0.103667 0.092560 0.082643 0.073788 0.065882 0.058823 0.052521 0.046894 0.041869 0.037383 0.033378 0.018940 0.010747 0.006098 0.003460

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.120000 1.254400 1.404928 1.573519 1.762342 1.973823 2.210681 2.475963 2.773079 3.105848 3.478550 3.895976 4.363493 4.887112 5.473566 6.130394 6.866041 7.689966 8.612762 9.646293 10.803848 12.100310 13.552347 15.178629 17.000064 19.040072 21.324881 23.883866 26.749930 29.959922 52.799620 93.050970 163.987604 289.002190

n

1.120000 0.591698 0.416349 0.329234 0.277410 0.243226 0.219118 0.201303 0.187679 0.176984 0.168415 0.161437 0.155677 0.150871 0.146824 0.143390 0.140457 0.137937 0.135763 0.133879 0.132240 0.130811 0.129560 0.128463 0.127500 0.126652 0.125904 0.125244 0.124660 0.124144 0.122317 0.121304 0.120736 0.120417

n

0.892857 1.690051 2.401831 3.037349 3.604776 4.111407 4.563757 4.967640 5.328250 5.650223 5.937699 6.194374 6.423548 6.628168 6.810864 6.973986 7.119630 7.249670 7.365777 7.469444 7.562003 7.644646 7.718434 7.784316 7.843139 7.895660 7.942554 7.984423 8.021806 8.055184 8.175504 8.243777 8.282516 8.304498

1.000000 0.471698 0.296349 0.209234 0.157410 0.123226 0.099118 0.081303 0.067679 0.056984 0.048415 0.041437 0.035677 0.030871 0.026824 0.023390 0.020457 0.017937 0.015763 0.013879 0.012240 0.010811 0.009560 0.008463 0.007500 0.006652 0.005904 0.005244 0.004660 0.004144 0.002317 0.001304 0.000736 0.000417

1.000000 2.120000 3.374400 4.779328 6.352847 8.115189 10.089012 12.299693 14.775656 17.548735 20.654583 24.133133 28.029109 32.392602 37.279715 42.753280 48.883674 55.749715 63.439681 72.052442 81.698736 92.502584 104.602894 118.155241 133.333870 150.333934 169.374007 190.698887 214.582754 241.332684 431.663496 767.091420 1358.230032 2400.018249

Tables of Financial Mathematics

319

14% 0.14

Abf

Auf

KWF

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.877193 0.769468 0.674972 0.592080 0.519369 0.455587 0.399637 0.350559 0.307508 0.269744 0.236617 0.207559 0.182069 0.159710 0.140096 0.122892 0.107800 0.094561 0.082948 0.072762 0.063826 0.055988 0.049112 0.043081 0.037790 0.033149 0.029078 0.025507 0.022375 0.019627 0.010194 0.005294 0.002750 0.001428

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.140000 1.299600 1.481544 1.688960 1.925415 2.194973 2.502269 2.852586 3.251949 3.707221 4.226232 4.817905 5.492411 6.261349 7.137938 8.137249 9.276464 10.575169 12.055693 13.743490 15.667578 17.861039 20.361585 23.212207 26.461916 30.166584 34.389906 39.204493 44.693122 50.950159 98.100178 188.883514 363.679072 700.232988

n

1.140000 0.607290 0.430731 0.343205 0.291284 0.257157 0.233192 0.215570 0.202168 0.191714 0.183394 0.176669 0.171164 0.166609 0.162809 0.159615 0.156915 0.154621 0.152663 0.150986 0.149545 0.148303 0.147231 0.146303 0.145498 0.144800 0.144193 0.143664 0.143204 0.142803 0.141442 0.140745 0.140386 0.140200

n

0.877193 1.646661 2.321632 2.913712 3.433081 3.888668 4.288305 4.638864 4.946372 5.216116 5.452733 5.660292 5.842362 6.002072 6.142168 6.265060 6.372859 6.467420 6.550369 6.623131 6.686957 6.742944 6.792056 6.835137 6.872927 6.906077 6.935155 6.960662 6.983037 7.002664 7.070045 7.105041 7.123217 7.132656

1.000000 0.467290 0.290731 0.203205 0.151284 0.117157 0.093192 0.075570 0.062168 0.051714 0.043394 0.036669 0.031164 0.026609 0.022809 0.019615 0.016915 0.014621 0.012663 0.010986 0.009545 0.008303 0.007231 0.006303 0.005498 0.004800 0.004193 0.003664 0.003204 0.002803 0.001442 0.000745 0.000386 0.000200

1.000000 2.140000 3.439600 4.921144 6.610104 8.535519 10.730491 13.232760 16.085347 19.337295 23.044516 27.270749 32.088654 37.581065 43.842414 50.980352 59.117601 68.394066 78.969235 91.024928 104.768418 120.435996 138.297035 158.658620 181.870827 208.332743 238.499327 272.889233 312.093725 356.786847 693.572702 1342.025099 2590.564800 4994.521346

320

Tables of Financial Mathematics

15% 0.15

Abf

Auf

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.869565 0.756144 0.657516 0.571753 0.497177 0.432328 0.375937 0.326902 0.284262 0.247185 0.214943 0.186907 0.162528 0.141329 0.122894 0.106865 0.092926 0.080805 0.070265 0.061100 0.053131 0.046201 0.040174 0.034934 0.030378 0.026415 0.022970 0.019974 0.017369 0.015103 0.007509 0.003733 0.001856 0.000923

KWF

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.150000 1.322500 1.520875 1.749006 2.011357 2.313061 2.660020 3.059023 3.517876 4.045558 4.652391 5.350250 6.152788 7.075706 8.137062 9.357621 10.761264 12.375454 14.231772 16.366537 18.821518 21.644746 24.891458 28.625176 32.918953 37.856796 43.535315 50.065612 57.575454 66.211772 133.175523 267.863546 538.769269 1083.657442

n

1.150000 0.615116 0.437977 0.350265 0.298316 0.264237 0.240360 0.222850 0.209574 0.199252 0.191069 0.184481 0.179110 0.174688 0.171017 0.167948 0.165367 0.163186 0.161336 0.159761 0.158417 0.157266 0.156278 0.155430 0.154699 0.154070 0.153526 0.153057 0.152651 0.152300 0.151135 0.150562 0.150279 0.150139

n

0.869565 1.625709 2.283225 2.854978 3.352155 3.784483 4.160420 4.487322 4.771584 5.018769 5.233712 5.420619 5.583147 5.724476 5.847370 5.954235 6.047161 6.127966 6.198231 6.259331 6.312462 6.358663 6.398837 6.433771 6.464149 6.490564 6.513534 6.533508 6.550877 6.565980 6.616607 6.641778 6.654293 6.660515

1.000000 0.465116 0.287977 0.200265 0.148316 0.114237 0.090360 0.072850 0.059574 0.049252 0.041069 0.034481 0.029110 0.024688 0.021017 0.017948 0.015367 0.013186 0.011336 0.009761 0.008417 0.007266 0.006278 0.005430 0.004699 0.004070 0.003526 0.003057 0.002651 0.002300 0.001135 0.000562 0.000279 0.000139

1.000000 2.150000 3.472500 4.993375 6.742381 8.753738 11.066799 13.726819 16.785842 20.303718 24.349276 29.001667 34.351917 40.504705 47.580411 55.717472 65.075093 75.836357 88.211811 102.443583 118.810120 137.631638 159.276384 184.167841 212.793017 245.711970 283.568766 327.104080 377.169693 434.745146 881.170156 1779.090308 3585.128460 7217.716277

Tables of Financial Mathematics

321

16% 0.16

Abf

Auf

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.862069 0.743163 0.640658 0.552291 0.476113 0.410442 0.353830 0.305025 0.262953 0.226684 0.195417 0.168463 0.145227 0.125195 0.107927 0.093041 0.080207 0.069144 0.059607 0.051385 0.044298 0.038188 0.032920 0.028380 0.024465 0.021091 0.018182 0.015674 0.013512 0.011648 0.005546 0.002640 0.001257 0.000599

KWF

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.160000 1.345600 1.560896 1.810639 2.100342 2.436396 2.826220 3.278415 3.802961 4.411435 5.117265 5.936027 6.885791 7.987518 9.265521 10.748004 12.467685 14.462514 16.776517 19.460759 22.574481 26.186398 30.376222 35.236417 40.874244 47.414123 55.000382 63.800444 74.008515 85.849877 180.314073 378.721158 795.443826 1670.703804

n

1.160000 0.622963 0.445258 0.357375 0.305409 0.271390 0.247613 0.230224 0.217082 0.206901 0.198861 0.192415 0.187184 0.182898 0.179358 0.176414 0.173952 0.171885 0.170142 0.168667 0.167416 0.166353 0.165447 0.164673 0.164013 0.163447 0.162963 0.162548 0.162192 0.161886 0.160892 0.160424 0.160201 0.160096

n

0.862069 1.605232 2.245890 2.798181 3.274294 3.684736 4.038565 4.343591 4.606544 4.833227 5.028644 5.197107 5.342334 5.467529 5.575456 5.668497 5.748704 5.817848 5.877455 5.928841 5.973139 6.011326 6.044247 6.072627 6.097092 6.118183 6.136364 6.152038 6.165550 6.177198 6.215338 6.233497 6.242143 6.246259

1.000000 0.462963 0.285258 0.197375 0.145409 0.111390 0.087613 0.070224 0.057082 0.046901 0.038861 0.032415 0.027184 0.022898 0.019358 0.016414 0.013952 0.011885 0.010142 0.008667 0.007416 0.006353 0.005447 0.004673 0.004013 0.003447 0.002963 0.002548 0.002192 0.001886 0.000892 0.000424 0.000201 0.000096

1.000000 2.160000 3.505600 5.066496 6.877135 8.977477 11.413873 14.240093 17.518508 21.321469 25.732904 30.850169 36.786196 43.671987 51.659505 60.925026 71.673030 84.140715 98.603230 115.379747 134.840506 157.414987 183.601385 213.977607 249.214024 290.088267 337.502390 392.502773 456.303216 530.311731 1120.712955 2360.757241 4965.273911 10435.648773

322

Tables of Financial Mathematics

18% 0.18

Abf

Auf

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.847458 0.718184 0.608631 0.515789 0.437109 0.370432 0.313925 0.266038 0.225456 0.191064 0.161919 0.137220 0.116288 0.098549 0.083516 0.070776 0.059980 0.050830 0.043077 0.036506 0.030937 0.026218 0.022218 0.018829 0.015957 0.013523 0.011460 0.009712 0.008230 0.006975 0.003049 0.001333 0.000583 0.000255

KWF

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.180000 1.392400 1.643032 1.938778 2.287758 2.699554 3.185474 3.758859 4.435454 5.233836 6.175926 7.287593 8.599359 10.147244 11.973748 14.129023 16.672247 19.673251 23.214436 27.393035 32.323781 38.142061 45.007632 53.109006 62.668627 73.948980 87.259797 102.966560 121.500541 143.370638 327.997290 750.378345 1716.683879 3927.356860

n

1.180000 0.638716 0.459924 0.371739 0.319778 0.285910 0.262362 0.245244 0.232395 0.222515 0.214776 0.208628 0.203686 0.199678 0.196403 0.193710 0.191485 0.189639 0.188103 0.186820 0.185746 0.184846 0.184090 0.183454 0.182919 0.182467 0.182087 0.181765 0.181494 0.181264 0.180550 0.180240 0.180105 0.180046

n

0.847458 1.565642 2.174273 2.690062 3.127171 3.497603 3.811528 4.077566 4.303022 4.494086 4.656005 4.793225 4.909513 5.008062 5.091578 5.162354 5.222334 5.273164 5.316241 5.352746 5.383683 5.409901 5.432120 5.450949 5.466906 5.480429 5.491889 5.501601 5.509831 5.516806 5.538618 5.548152 5.552319 5.554141

1.000000 0.458716 0.279924 0.191739 0.139778 0.105910 0.082362 0.065244 0.052395 0.042515 0.034776 0.028628 0.023686 0.019678 0.016403 0.013710 0.011485 0.009639 0.008103 0.006820 0.005746 0.004846 0.004090 0.003454 0.002919 0.002467 0.002087 0.001765 0.001494 0.001264 0.000550 0.000240 0.000105 0.000046

1.000000 2.180000 3.572400 5.215432 7.154210 9.441968 12.141522 15.326996 19.085855 23.521309 28.755144 34.931070 42.218663 50.818022 60.965266 72.939014 87.068036 103.740283 123.413534 146.627970 174.021005 206.344785 244.486847 289.494479 342.603486 405.272113 479.221093 566.480890 669.447450 790.947991 1816.651612 4163.213027 9531.577105 21813.093666

Tables of Financial Mathematics

323

20% 0.2

Abf

Auf

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

0.833333 0.694444 0.578704 0.482253 0.401878 0.334898 0.279082 0.232568 0.193807 0.161506 0.134588 0.112157 0.093464 0.077887 0.064905 0.054088 0.045073 0.037561 0.031301 0.026084 0.021737 0.018114 0.015095 0.012579 0.010483 0.008735 0.007280 0.006066 0.005055 0.004213 0.001693 0.000680 0.000273 0.000110

KWF

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

1.200000 1.440000 1.728000 2.073600 2.488320 2.985984 3.583181 4.299817 5.159780 6.191736 7.430084 8.916100 10.699321 12.839185 15.407022 18.488426 22.186111 26.623333 31.948000 38.337600 46.005120 55.206144 66.247373 79.496847 95.396217 114.475460 137.370552 164.844662 197.813595 237.376314 590.668229 1469.771568 3657.261988 9100.438150

n

1.200000 0.654545 0.474725 0.386289 0.334380 0.300706 0.277424 0.260609 0.248079 0.238523 0.231104 0.225265 0.220620 0.216893 0.213882 0.211436 0.209440 0.207805 0.206462 0.205357 0.204444 0.203690 0.203065 0.202548 0.202119 0.201762 0.201467 0.201221 0.201016 0.200846 0.200339 0.200136 0.200055 0.200022

n

0.833333 1.527778 2.106481 2.588735 2.990612 3.325510 3.604592 3.837160 4.030967 4.192472 4.327060 4.439217 4.532681 4.610567 4.675473 4.729561 4.774634 4.812195 4.843496 4.869580 4.891316 4.909430 4.924525 4.937104 4.947587 4.956323 4.963602 4.969668 4.974724 4.978936 4.991535 4.996598 4.998633 4.999451

1.000000 0.454545 0.274725 0.186289 0.134380 0.100706 0.077424 0.060609 0.048079 0.038523 0.031104 0.025265 0.020620 0.016893 0.013882 0.011436 0.009440 0.007805 0.006462 0.005357 0.004444 0.003690 0.003065 0.002548 0.002119 0.001762 0.001467 0.001221 0.001016 0.000846 0.000339 0.000136 0.000055 0.000022

1.000000 2.200000 3.640000 5.368000 7.441600 9.929920 12.915904 16.499085 20.798902 25.958682 32.150419 39.580502 48.496603 59.195923 72.035108 87.442129 105.930555 128.116666 154.740000 186.688000 225.025600 271.030719 326.236863 392.484236 471.981083 567.377300 681.852760 819.223312 984.067974 1181.881569 2948.341146 7343.857840 18281.309940 45497.190750

324

Tables of Financial Mathematics

22% 0.22

Abf

Auf

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

KWF

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

0.819672 1.220000 0.671862 1.488400 0.550707 1.815848 0.451399 2.215335 0.369999 2.702708 0.303278 3.297304 0.248589 4.022711 0.203761 4.907707 0.167017 5.987403 0.136899 7.304631 0.112213 8.911650 0.091978 10.872213 0.075391 13.264100 0.061796 16.182202 0.050653 19.742287 0.041519 24.085590 0.034032 29.384420 0.027895 35.848992 0.022865 43.735771 0.018741 53.357640 0.015362 65.096321 0.012592 79.417512 0.010321 96.889364 0.008460 118.205024 0.006934 144.210130 0.005684 175.936358 0.004659 214.642357 0.003819 261.863675 0.003130 319.473684 0.002566 389.757894 0.000949 1053.401842 0.000351 2847.037759 0.000130 7694.712191 0.000048 20796.561453

n

1.220000 0.670450 0.489658 0.401020 0.349206 0.315764 0.292782 0.276299 0.264111 0.254895 0.247807 0.242285 0.237939 0.234491 0.231738 0.229530 0.227751 0.226313 0.225148 0.224202 0.223432 0.222805 0.222294 0.221877 0.221536 0.221258 0.221030 0.220843 0.220691 0.220566 0.220209 0.220077 0.220029 0.220011

n

0.819672 1.491535 2.042241 2.493641 2.863640 3.166918 3.415506 3.619268 3.786285 3.923184 4.035397 4.127375 4.202766 4.264562 4.315215 4.356734 4.390765 4.418660 4.441525 4.460266 4.475628 4.488220 4.498541 4.507001 4.513935 4.519619 4.524278 4.528096 4.531227 4.533792 4.541140 4.543858 4.544864 4.545236

1.000000 0.450450 0.269658 0.181020 0.129206 0.095764 0.072782 0.056299 0.044111 0.034895 0.027807 0.022285 0.017939 0.014491 0.011738 0.009530 0.007751 0.006313 0.005148 0.004202 0.003432 0.002805 0.002294 0.001877 0.001536 0.001258 0.001030 0.000843 0.000691 0.000566 0.000209 0.000077 0.000029 0.000011

1.000000 2.220000 3.708400 5.524248 7.739583 10.442291 13.739595 17.762306 22.670013 28.657416 35.962047 44.873697 55.745911 69.010011 85.192213 104.934500 129.020090 158.404510 194.253503 237.989273 291.346913 356.443234 435.860746 532.750110 650.955134 795.165264 971.101622 1185.743978 1447.607654 1767.081337 4783.644738 12936.535267 34971.419051 94525.279331

Tables of Financial Mathematics

325

25% 0.25

Abf

Auf

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

KWF

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

0.800000 1.250000 0.640000 1.562500 0.512000 1.953125 0.409600 2.441406 0.327680 3.051758 0.262144 3.814697 0.209715 4.768372 0.167772 5.960464 0.134218 7.450581 0.107374 9.313226 0.085899 11.641532 0.068719 14.551915 0.054976 18.189894 0.043980 22.737368 0.035184 28.421709 0.028147 35.527137 0.022518 44.408921 0.018014 55.511151 0.014412 69.388939 0.011529 86.736174 0.009223 108.420217 0.007379 135.525272 0.005903 169.406589 0.004722 211.758237 0.003778 264.697796 0.003022 330.872245 0.002418 413.590306 0.001934 516.987883 0.001547 646.234854 0.001238 807.793567 0.000406 2465.190329 0.000133 7523.163845 0.000044 22958.874039 0.000014 70064.923216

n

1.250000 0.694444 0.512295 0.423442 0.371847 0.338819 0.316342 0.300399 0.288756 0.280073 0.273493 0.268448 0.264543 0.261501 0.259117 0.257241 0.255759 0.254586 0.253656 0.252916 0.252327 0.251858 0.251485 0.251186 0.250948 0.250758 0.250606 0.250485 0.250387 0.250310 0.250101 0.250033 0.250011 0.250004

n

0.800000 1.440000 1.952000 2.361600 2.689280 2.951424 3.161139 3.328911 3.463129 3.570503 3.656403 3.725122 3.780098 3.824078 3.859263 3.887410 3.909928 3.927942 3.942354 3.953883 3.963107 3.970485 3.976388 3.981111 3.984888 3.987911 3.990329 3.992263 3.993810 3.995048 3.998377 3.999468 3.999826 3.999943

1.000000 1.000000 0.444444 2.250000 0.262295 3.812500 0.173442 5.765625 0.121847 8.207031 0.088819 11.258789 0.066342 15.073486 0.050399 19.841858 0.038756 25.802322 0.030073 33.252903 0.023493 42.566129 0.018448 54.207661 0.014543 68.759576 0.011501 86.949470 0.009117 109.686838 0.007241 138.108547 0.005759 173.635684 0.004586 218.044605 0.003656 273.555756 0.002916 342.944695 0.002327 429.680869 0.001858 538.101086 0.001485 673.626358 0.001186 843.032947 0.000948 1054.791184 0.000758 1319.488980 0.000606 1650.361225 0.000485 2063.951531 0.000387 2580.939414 0.000310 3227.174268 0.000101 9856.761315 0.000033 30088.655381 0.000011 91831.496158 0.000004 280255.692865

326

Tables of Financial Mathematics

30% 0.3

Abf

Auf

(1 + i) − n (1 + i)

n 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50

KWF

DSF

RVF

EWF

i (1 + i ) n − 1 n i(1 + i) (1 + i) − 1 n (1 + i ) n − 1 i * (1 + i) n (1 + i ) − 1 i

0.769231 1.300000 0.591716 1.690000 0.455166 2.197000 0.350128 2.856100 0.269329 3.712930 0.207176 4.826809 0.159366 6.274852 0.122589 8.157307 0.094300 10.604499 0.072538 13.785849 0.055799 17.921604 0.042922 23.298085 0.033017 30.287511 0.025398 39.373764 0.019537 51.185893 0.015028 66.541661 0.011560 86.504159 0.008892 112.455407 0.006840 146.192029 0.005262 190.049638 0.004048 247.064529 0.003113 321.183888 0.002395 417.539054 0.001842 542.800770 0.001417 705.641001 0.001090 917.333302 0.000839 1192.533293 0.000645 1550.293280 0.000496 2015.381264 0.000382 2619.995644 0.000103 9727.860425 0.000028 36118.864808 0.000007 134106.816713 0.000002 497929.222979

n

1.300000 0.734783 0.550627 0.461629 0.410582 0.378394 0.356874 0.341915 0.331235 0.323463 0.317729 0.313454 0.310243 0.307818 0.305978 0.304577 0.303509 0.302692 0.302066 0.301587 0.301219 0.300937 0.300720 0.300554 0.300426 0.300327 0.300252 0.300194 0.300149 0.300115 0.300031 0.300008 0.300002 0.300001

n

0.769231 1.360947 1.816113 2.166241 2.435570 2.642746 2.802112 2.924702 3.019001 3.091539 3.147338 3.190260 3.223277 3.248675 3.268211 3.283239 3.294800 3.303692 3.310532 3.315794 3.319842 3.322955 3.325350 3.327192 3.328609 3.329700 3.330538 3.331183 3.331679 3.332061 3.332991 3.333241 3.333308 3.333327

1.000000 1.000000 0.434783 2.300000 0.250627 3.990000 0.161629 6.187000 0.110582 9.043100 0.078394 12.756030 0.056874 17.582839 0.041915 23.857691 0.031235 32.014998 0.023463 42.619497 0.017729 56.405346 0.013454 74.326950 0.010243 97.625036 0.007818 127.912546 0.005978 167.286310 0.004577 218.472203 0.003509 285.013864 0.002692 371.518023 0.002066 483.973430 0.001587 630.165459 0.001219 820.215097 0.000937 1067.279626 0.000720 1388.463514 0.000554 1806.002568 0.000426 2348.803338 0.000327 3054.444340 0.000252 3971.777642 0.000194 5164.310934 0.000149 6714.604214 0.000115 8729.985479 0.000031 32422.868084 0.000008 120392.882695 0.000002 447019.389044 0.000001 1659760.743264

Index

A Absolute advantageousness, 87 Absolute dominance, 274 Account development planning, 176–180 Accumulating interest, 93 Accumulation calculation, 72 Accumulation factor, 94 Acquisition payment, 92 Alternative selection, 148 Ambiguity of the internal rate of return, 166–172 Annual replacement, 228 Annuity, 115 Annuity criterion, 115 Annuity method, 115 Answer report, 189 Arrears, 95 Asset concept, 11 Average annual incoming payments, 118 Average annual outgoing payments, 118 Average annual surplus, 115 Average calculation, 73

C Calculation elements, 92 Calculation interest rate, 92 Capital commitment, 41 Capital recovery factor, 97 Cash-based criterion, 87 Cash flow-based concept, 9 Certainty equivalent, 241 Change of the algebraic (+/-) sign, 170 Comparison of alternatives, 37 Complete advantageousness comparison, 157 Concept of net present value equivalence, 207 Constant commitment of capital, 43 Correction procedures, 244 Cost comparison calculation, 45

Creativity techniques, 24 Critical interest rate, 250 Critical maximum values, 249 Critical minimum values, 250 Critical rationalism, 29 Critical useful life, 250 Critical value calculation, 249

D Dean Model, 180 Debt service, 36 Decision tree, 265 Discount factor, 94 Discounting, 93 Discount sum factor (DSF), 96 Discrete reduction of capital, 41 Disinvestment, 21 Distribution factors, 97 Dominance rule, 274 Double discounting, 245 Dual variables, 194 Dynamic amortisation calculation, 131 Dynamic amortisation time, 131

E Economic consequences, 264 Economic principle, 70 Endogenous return on capital, 194 Engineering formula, 46 Environmental conditions, 264 Equidistant, 95

F Fictitious investment, 157 Flexible planning, 264 Forcing (Zwang), 185

# The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 K. Poggensee, J. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-030-62440-8

327

328 Formulas for cost comparison calculation, 46 Formulas for the profit comparison calculation, 55 Formulas of the profitability calculation, 66, 67 Formulas of the static amortisation calculation, 73–75

H Horizon value, 110 Horizon value criterion, 111 Horizon value method, 110 Hurwicz-rule, 276

I Imponderables, 10 Improved procedure, 33 Information as an economic problem, 244 Initial matrix, 185 Integer numbers, 185 Interest component, 36 Interest rate assumption, 145 Internal interest rate, 122 Internal rate of return, 122 Internal rate of return criterion, 122 Internal rate of return method, 122 Interpolation line, 125 Investment, 6 Investment chain, 207 Investment ratio, 2 Investment without residual value, 129

J Joint activities (VB), 185

L Laplace-rule, 277 Linear optimization, 184 Linear reduction of capital, 41 Liquidity restriction, 187 Lump-sum procedure, 245

M Marginal analysis, 212 Marginal in-payments (GE), 228 Marginal loss values, 193 Marginal payouts (GA), 212 Marginal surplus, 228 Maximax rule, 275

Index Maximum sum restrictions (HSR), 187 Minimax rule, 276 Minimum sum (MSR) restrictions, 184 Multivariate methods, 24

N Net earnings NEk, 99 Net present value Co, 98 Net present value criterion, 100 Net present value method, 98, 99 Non-negativity condition, 185

O One-time factors, 94 Operating costs, 36 Opportunity cost principle, 28 Optimal replacement time, 225–234 Optimization models, 177 Optimum useful life, 208–225 Organisational structure, 18

P Perpetual annuity, 127 Portfolio selection theory, 286 Present value, 104 Primitive method, 33 Process organisation, 20–22 Profitability calculation, 62 Profit comparison calculation, 54 Programme planning, 142

Q Qualitative methods, 23 Quantitative methods, 24

R Recovery share, 36 Regula falsi, 124 Reinvestment interest rate, 151 Relative advantageousness, 87 Replacement option after expired useful life, 229 Replacement problem, 34 Residual value distribution factor (RVF), 97 Residual value equal to the acquisition payment, 128, 129 Revenue, 36 Rigid planning, 264

Index Risk, 241 analysis, 279 averse attitude, 241 neutrality, 241 seeking, 241 situations, 243 Roll-back procedure, 264

S Savage-Niehans-rule, 278 Sensitivity analyses, 249 Sensitivity report, 189 Sequential planning, 264 Simplex algorithm, 185 Single investment, 34 Singular criterion, 87 State dominance, 274 Static amortisation calculation, 70 Sum factors, 95 Surveys, 23

T Target criterion, 87 Target value change calculation, 260–264

329 Terminal value, 151 Terminal value factor (EWF), 96 Terminal value maximisation, 184 Test (or trial) interest rates, 125 Test useful lives, 134 Time preference, 9 Triple calculation, 258–260 Two-payment case, 127

U Uncertainty, 240, 272 Uncertainty situations, 243 Uniform, 95 Univariate methods, 24 Useful life, 203 Useful life optimization, 204 Utility value analysis, 172

W Withdrawal concept, 11

Y Yield concept, 11