Investment Valuation and Appraisal: Exam Training with Exercises and Solutions (Springer Texts in Business and Economics) 3658330449, 9783658330446

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Springer Texts in Business and Economics

Kay Poggensee

Investment Valuation and Appraisal Exam Training with Exercises and Solutions

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

Investment Valuation and Appraisal Exam Training with Exercises and Solutions

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

ISSN 2192-4333 ISSN 2192-4341 (electronic) Springer Texts in Business and Economics ISBN 978-3-658-33044-6 ISBN 978-3-658-33045-3 (eBook) https://doi.org/10.1007/978-3-658-33045-3 # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 Translation from the German language edition: Klausurenkurs Invesitionsrechnung by Kay Poggensee. # Springer Fachmedien Wiesbaden 2016. All Rights Reserved. This work is subject to copyright. All rights are reserved 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 Fachmedien Wiesbaden GmbH part of Springer Nature. The registered company address is: Abraham-Lincoln-Str. 46, 65189 Wiesbaden, Germany

Preface

The investment valuation and appraisal module is part of the compulsory curriculum in most business administration courses around the world. Thus, this module is studied by a large number of students around the world and at each single university. Due to organisational reasons, there is hardly any other way for universities to assess competence by means of a written exam. In teaching, the investment valuation and appraisal module covers all relevant competences, i.e. professional competence in the forms of knowledge in breadth and depth and skills (instrumental and systemic skills) as well as personal competences in the forms of social competence and independence. Due to the limited time in which teachers and students meet each other directly in lectures, the focus there is on imparting specialist competence in the necessary breadth and depth of knowledge. The training of skills, i.e. instrumental and systematic problem-solving competence, in which the correct theory is independently applied to a specific data set in a practical case and a quantitative solution is determined and assessed, unfortunately moves into the background, since the training of these skills is particularly time-intensive and cannot be achieved for the breadth of knowledge of the entire field of teaching in the given time. In contrast to many other modules, in which the breadth and depth of knowledge is an exhaustive preparation for the examination, in investment valuation and appraisal, however, precisely these skills are necessary for successful professional activities in management and these are the subject of the competence assessment. This exam course manual closes the gap between the skills primarily conveyed in the lectures and the skills expected in exams and in practice. This investment valuation and appraisal exam course manual enables students to prepare for the written examination in an ideal way by self-study. It is also ideal for practical people as an object of illustration of specific quantitative business problems and their solutions. This investment valuation and appraisal exam course is based on the title Investment Valuation and Appraisal—Theory and Practice by the same author published by Springer. This exam course manual refers to the theory presented there and uses the same structure of the main chapters as in the textbook. Thus, for each chapter in the textbook, there is a chapter in this exam course manual with a mock exam based on the theory presented in the textbook. Sample solutions for the exercises, which v

vi

Preface

are all at the level of university examinations, are provided for each exercise, so that the reader can compare the solution with the correct sample solution. In some places, the solutions refer to the textbook. Formula numbers from the textbook are used identically in the solutions, so that the formulas and the theory necessary for understanding the solution in this exam course manual can be easily found in the textbook. The last main chapter of the investment valuation and appraisal exam course manual concludes with a module exam in investment theory, which is typical for many universities. Due to the large number of students at many universities, the examination in the investment valuation and appraisal module is offered as a hardcopy examination and is then worked on by the students using a pocket calculator as an assistant tool. The sample solutions are therefore provided both in the form of pocket calculator solutions and, where appropriate, digitally in the form of solutions using the Microsoft Office programme Excel. Calculation results for the many sample calculations and tasks created with Excel are available for download under the respective chapters on the homepage of Springer (https://link.springer.com, search term “Kay Poggensee”). In this way, all calculations and Excel formulas stored in the figures can be traced. This exam course manual is intended to help the reader to access the investment valuation and appraisal module more easily, to develop competence in this field more easily, to increase learning success and thus to be better prepared for the written module examination in investment valuation and appraisal and to be able to work on it more successfully. Since the textbook is used as literature in the investment valuation and appraisal module at some universities, the exercises of the exam course manual are certainly a valuable support for teachers and students. Only by working intensively with the exercises and the field of study, deficits in the exercises or areas that are not sufficiently considered in the exercises can be identified. I would be pleased if you would then send me your comments and suggestions for improvement ([email protected]). I would like to thank Mrs. Cyra-Helena Schmucker, BA, very much. 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. Finally, this investment valuation and appraisal exam course manual is a worthwhile investment, because investment theory is of high strategic and operational importance for companies, as this discipline usually means that relatively large amounts of capital are allocated for a relatively long time. The precise theoretical and practical knowledge of investment theory is therefore particularly important for students, because companies will especially seek, appreciate and promote employees with these skills, as they are particularly valuable to companies. Kremperheide, Germany January 2021

Kay Poggensee

Contents

1

Mock Exam: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Exercises Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Solutions Chapter Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2

2

Mock Exam: Static Investment Calculation Methods . . . . . . . . . . . . 2.1 Exercises Static Investment Calculation Methods . . . . . . . . . . . . . 2.2 Solutions Static Investment Calculation Methods . . . . . . . . . . . . .

9 9 11

3

Mock Exam: Dynamic Investment Calculation Methods . . . . . . . . . . 3.1 Exercises Dynamic Investment Calculation Methods . . . . . . . . . . . 3.2 Solutions Dynamic Investment Calculation Methods . . . . . . . . . . . 3.3 Excel-Based Solutions Dynamic Investment Calculation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Excel-Based Solutions Dynamic Investment Calculation Methods, Exercise a . . . . . . . . . . . . . . . . . . . . 3.3.2 Excel-Based Solutions Dynamic Investment Calculation Methods, Exercise b–f . . . . . . . . . . . . . . . . . .

21 21 24

32

Mock Exam: Selection of Alternatives and Investment Programme Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Exercises Part Selection of Alternatives . . . . . . . . . . . . . . . . . . . 4.2 Solutions Part Selection of Alternatives . . . . . . . . . . . . . . . . . . . 4.3 Excel-Based Solutions Part Selection of Alternatives . . . . . . . . . . 4.4 Exercises Part Investment Programme Planning . . . . . . . . . . . . . 4.5 Solutions Part Investment Programme Planning . . . . . . . . . . . . . 4.6 Excel-Based Solutions Part Investment Programme Planning . . . .

. . . . . . .

37 37 39 44 57 59 62

.

69

.

69

.

71

.

75

4

5

Mock Exam: Optimum Useful Life and Optimum Replacement Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Exercises Chapter Optimum Useful Life and Optimum Replacement Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Solutions Chapter Optimum Useful Life and Optimum Replacement Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Excel-Based Solutions Chapter Optimum Useful Life and Optimum Replacement Time . . . . . . . . . . . . . . . . . . . . . . . .

29 29

vii

viii

6

7

Contents

Mock Exam: Investment Decisions in Uncertainty . . . . . . . . . . . . . 6.1 Exercises Chapter Investment Decisions in Uncertainty . . . . . . . . 6.2 Solutions Chapter Investment Decisions in Uncertainty . . . . . . . . 6.3 Excel-Based Solutions Chapter Investment Decisions in Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Exercises Part Critical Value Calculation . . . . . . . . . . . . . . . . . . 6.5 Solutions Part Critical Value Calculation . . . . . . . . . . . . . . . . . . 6.6 Excel-Based Solutions Part Critical Value Calculation . . . . . . . . .

. . .

79 79 85

. 93 . 98 . 99 . 104

Mock Exam Investment Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Exercises Mock Exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Solutions Mock Exam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 Solutions Exercise 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Solutions Exercise 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Solutions Exercise 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Excel-Based Solutions Mock Exam . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Excel-Based Solutions Exercise 1 . . . . . . . . . . . . . . . . . . . 7.3.2 Excel-Based Solutions Exercise 2 . . . . . . . . . . . . . . . . . . . 7.3.3 Excel-Based Solutions Exercise 3 . . . . . . . . . . . . . . . . . . .

115 115 115 116 117 119 119 122 124 126 126 133 134

Tables of Financial Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

About the Author

Kay Poggensee is Director of the Institute of International Business Management at FH Kiel, University of Applied Sciences, Germany and has the chair of Investment Appraisal.

ix

1

Mock Exam: Introduction

Abstract

This chapter contains exercises and solutions of the topic “Introduction”. It is based on the textbook Investment Valuation and Appraisal—Theory and Practice published by Springer.

1.1

Exercises Chapter Introduction

Time: 60 Min, Credits: 30 (a) Formulate the meaning of the investment calculation from a strategic and operational point of view. (2 credits) (b) Formulate requirements for and effects of the implementation of an investment. (2 credits) (c) Formulate the aim and definition of the investment calculation. (2 credits) (d) Name relevant questions of the investment calculation. (3 credits) (e) Name different aspects of the individual business management disciplines of internal accounting. (4 credits) (f) Name and define the calculation elements of the investment calculation procedures. (2 credits) (g) Summarise investment calculation procedures into process groups and systematise them according to different aspects. (3 credits) (h) Structure investment calculation procedures based on the assumption of data certainty according to criteria in an overview graphic. (5 credits) (i) Name and describe phases of the process organisation of an investment calculation. (5 credits) (j) Identify and describe possible forecasting methods of data collection for the investment calculation. (2 credits)

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 K. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-33045-3_1

1

2

1.2

1

Mock Exam: Introduction

Solutions Chapter Introduction

(a) Formulate the meaning of the investment calculation from a strategic and operational point of view. (2 credits) Solution (Textbook Sect. 1.2) • Investment calculation is important from a strategic point of view, as it is the basis for long-term capital dispositions, usually on a larger scale. • Investment calculation is important from an operational point of view, as individual investment projects or investment programmes can be evaluated precisely and quantitatively. (b) Formulate requirements for and effects of the implementation of an investment. (2 credits) Solution (Textbook Sect. 1.2) • An investment should at least always create an improved situation compared to the initial situation. • In principle, investments carry technical and social progress into the areas in which they are invested. • The investment decision is made by the investor, not by the investment calculation. This is an academic model with assumptions that are detached from reality. The responsibility for the investment decision remains in the hands of the human investor. (c) Formulate the aim and definition of the investment calculation. (2 credits) Solution (Textbook Sect. 1.3) • The aim of investment calculation is to determine a generally quantitative result of an academic model, usually mathematical, on the basis of specific economic data from one or more investment objects, possibly also by means of expected value and known statistical distribution of defined data, using a defined procedure of a generally mathematical academic model, which serves as the basis for an investment decision. • The investment is a cash flow that begins with a payout. (d) Name relevant questions of the investment calculation. (3 credits) Solution (Textbook Sect. 1.3) • Determination of the absolute advantageousness of individual investments. The calculation techniques of statics and dynamics are suitable for this question. • Determination of the relative advantageousness of individual investments. Some calculation techniques, which take contradictory phenomena into account, are suitable for this question.

1.2 Solutions Chapter Introduction

3

• Determination of questions regarding the optimum useful life and the optimum replacement time. The calculation techniques of target value determination for each individual year of the useful life are suitable for this question. • Determination of the optimum investment and financing programme. The calculation techniques of account development planning and linear optimisation are suitable for this question. • Determination of investment decisions under data uncertainty. Calculation techniques that transform the data set or consider a risk–benefit function are suitable for this question. (e) Name different aspects of the individual business management disciplines of internal accounting. (4 credits) Solution (Textbook Sect. 1.4) (Fig. 1.1) 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, enterprise

Reference object

Single object, enterprise

Single object, enterprise

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

Enterprise

Fig. 1.1 (Textbook Fig. 1.7): Differences between the individual business management disciplines of internal accounting (Source: Author)

4

1

Mock Exam: Introduction

(f) Name and define the calculation elements of the investment calculation procedures. (2 credits) Solution (Textbook Sect. 1.4) (Fig. 1.2) Term Pay-out Disbursement In-payments Cash inflow Expenditure

Definition

Revenue

Monetary value of sales of goods and services per period

Costs Performance

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

Fig. 1.2 (Textbook Fig. 1.8): Calculation elements of the investment calculation methods (Source: Däumler, K.-D. and Grabe., J, 2007. Grundlagen der Investitions- und Wirtschaftlichkeitsrechnung. 12. A., Herne/Berlin, nwb Verlag Neue Wirtschafts-Briefe, p. 24)

(g) Summarise investment calculation procedures into process groups and systematise them according to different aspects. (3 credits) Solution (Textbook Sect. 1.5) Aspects of systematisation: • • • •

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

• • • •

Static investment calculation methods Dynamic investment calculation methods Simultaneous models of the capital budget Procedures for taking risk into account

(h) Structure investment calculation procedures based on the assumption of data certainty according to criteria in an overview graphic. (5 credits) Solution (Textbook Sect. 1.5) (Fig. 1.3)

Dynamic methods

Net present value method Horizon value method Annuity method Internal rate of return method Dynamic amorsaon calculaon

Stac methods

Cost comparison calculaon Profit comparison calculaon Profitability calculaon Stac amorsaon calculaon

Individual investment

Account development planning with regard to liquidity

Ulity value analysis

Aer removing the reinvestment premise and ficous investment: Net present value method Horizon value method Annuity method Internal rate of return method Dynamic amorsaon calculaon

Individual investment

Account development planning with regard to liquidity

Ulity value analysis

Linear opmizaon Dean Modell

Investment programme

Mul-dimensional target funcon

Account development planning regardless of liquidity

Investment programme

One-dimensional target funcon

Quantave target criteria

Assumpon of data security

Ulity value analysis

Individual investment

Ulity value analysis

Investment programme

One-dimensional target funcon

Qualitave target criteria

Ulity value analysis

Individual investment

Ulity value analysis

Investment programme

Mul-dimensional target funcon

1.2 Solutions Chapter Introduction 5

Fig. 1.3 (Textbook Fig. 1.9): Overview of investment calculation procedures on the assumption of data certainty (Source: Author)

6

1

Mock Exam: Introduction

(i) Name and describe phases of the process organisation of an investment calculation. (5 credits) Solution (Textbook Sect. 1.8)( Fig. 1.4)

Formulaon of objecves and alternaves Data collecon Investment calculaon Coordinaon with other business areas

Planning phase

Most favourable alternave

Decision phase

Fig. 1.4 (Textbook Fig. 1.12): Phases of the investment calculation (Source: Author)

Collecon of informaon

Excitaon phase

Phase of decision- making

Monitoring of project implementaon

Target-actual control Deviaon analysis Correcon decisions

Control phase

Phase of implementaon

Implementaon phase

Phase diagram of the investment calculaon

1.2 Solutions Chapter Introduction 7

8

1

Mock Exam: Introduction

(j) Identify and describe possible forecasting methods of data collection for the investment calculation. (2 credits) Solution (Textbook Sect. 1.9) (Fig. 1.5) Forecasting methods

Qualitative

Surveys

Quantitative

Creativity techniques

Univariate methods

Multivariate methods

Fig. 1.5 (Textbook Fig. 1.13): Overview of possible forecasting methods (Source: Kruschwitz, Lutz, p. 18)

2

Mock Exam: Static Investment Calculation Methods

Abstract

This chapter contains exercises and solutions of the topic “Static investment calculation methods”. It is based on the textbook Investment Valuation and Appraisal—Theory and Practice published by Springer. The formulas of the textbook are marked accordingly in this chapter.

2.1

Exercises Static Investment Calculation Methods

Time: 60 Min, Credits: 30 After your studies, you would like to supplement your management job with a parttime job to improve your income. You are planning to sell beach chairs, either with a pure online business with purchased chairs or with pure floor trading with beach chairs produced in the region. The online business has the project name “Beach Box Online”. Floor trading has the name “Beach Box Kiel”. Alternatively, you are considering continuing your already existing sideline activity of running diving courses in the Kiel Fjord. The diving courses have the project name “Baltic Diving”. Your time frame is 6 years; your calculation interest rate is 10%. The terms of the calculation elements correspond to the terms used in the textbook and are mainly shown in Fig. 2.1.

# Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 K. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-33045-3_2

9

10

2

Mock Exam: Static Investment Calculation Methods

Calculation element n (years) A (T€)

Beach Box Online Beach (BBO) (BBK) 6 6 140 120

Box

Kiel Baltic Diving (BD)

U (ek), k = 1 (T€) U (ek), k = 2 (T€) U (ek), k = 3 (T€) U (ek), k = 4 (T€) U (ek), k = 5 (T€) U (ek), k = 6 (T€)

80 90 100 100 110 120

90 90 90 90 90 90

65 65 65 65 65 65

B (ak), k = 1 (T€) B (ak), k = 2 (T€) B (ak), k = 3 (T€) B (ak), k = 4 (T€) B (ak), k = 5 (T€) B (ak), k = 6 (T€)

30 35 40 45 60 60

40 40 40 40 40 40

35 35 35 35 35 35

R (T€)

25

15

0

6 90

Fig. 2.1 Data set for the static exercise (Source: Author)

(a) Determine the costs of the two investment objects “Beach Box Online” and “Beach Box Kiel” according to the cost comparison calculation as a comparison of alternatives. Apply the procedure of primitive data collection. Assume the simple (linear) notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits) (b) Determine the costs of the two investment objects “Beach Box Online” and “Beach Box Kiel” according to the cost comparison calculation as a comparison of alternatives. Apply the procedure of improved data collection. Assume the simple (linear) notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits) (c) Determine the costs of the two projects “Beach Box Online” and “Beach Box Kiel” according to the cost comparison calculation as a comparison of alternatives. Apply the procedure of improved data collection. Assume the discrete notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits) (d) Comment on the suitability of the cost comparison calculation for the investment decision as a comparison of the alternatives “Beach Box Online” and “Beach Box Kiel”. (3 credits) (e) Determine the costs of the two investment objects “Beach Box Kiel” and “Baltic Diving” according to the cost comparison calculation in the replacement problem. The investment object “Baltic Diving” is the old object. Apply the

2.2 Solutions Static Investment Calculation Methods

(f)

(g)

(h)

(i) (j)

2.2

11

procedure of improved data collection. Assume the discrete notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits) Determine the profit of “Beach Box Online” according to the profit comparison calculation in case of a replacement problem. Assume that the old object “Baltic Diving” from Fig. 2.1 has been running for 2 years. Continue to assume a discrete notion of capital commitment. Apply the procedure of improved data collection. Make an investment decision based on the calculation results and document it. (3 credits) Determine two profitabilities according to the profitability calculation for the investment object “Beach Box Kiel”. Assume that • The simple notion of capital commitment is used to determine the first profitability. • The discrete notion of capital commitment applies for the determination of the second profitability. (3 credits) Determine two profitabilities according to the profitability calculation for the investment object “Baltic Diving”. Assume that • The discrete notion of capital commitment is used to determine the first profitability. • The constant notion of capital commitment applies for the determination of the second profitability. (3 credits) Determine the static amortisation period of the investment project “Beach Box Online” according to the accumulation calculation. (3 credits) Determine the static amortisation period of the investment project “Beach Box Kiel” according to the average calculation. (3 credits)

Solutions Static Investment Calculation Methods

After your studies, you would like to supplement your management job with a parttime job to improve your income. You are planning to sell beach chairs, either with a pure online business with purchased chairs or with pure floor trading with beach chairs produced in the region. The online business has the project name “Beach Box Online”. Floor trading has the name “Beach Box Kiel”. Alternatively, you are considering continuing your already existing sideline activity of running diving courses in the Kiel Fjord. The diving courses have the project name “Baltic Diving”. Your time frame is 6 years; your calculation interest rate is 10%. The terms of the calculation elements correspond to the terms used in the textbook and are mainly shown in Fig. 2.2.

12

2

Mock Exam: Static Investment Calculation Methods

Calculation element n (years) A (T€)

Beach Box Online Beach (BBO) (BBK) 6 6 140 120

Box

Kiel Baltic Diving (BD)

U (ek), k = 1 (T€) U (ek), k = 2 (T€) U (ek), k = 3 (T€) U (ek), k = 4 (T€) U (ek), k = 5 (T€) U (ek), k = 6 (T€)

80 90 100 100 110 120

90 90 90 90 90 90

65 65 65 65 65 65

B (ak), k = 1 (T€) B (ak), k = 2 (T€) B (ak), k = 3 (T€) B (ak), k = 4 (T€) B (ak), k = 5 (T€) B (ak), k = 6 (T€)

30 35 40 45 60 60

40 40 40 40 40 40

35 35 35 35 35 35

R (T€)

25

15

0

6 90

Fig. 2.2 Data set for the static exercise (Source: Author)

(a) Determine the costs of the two investment objects “Beach Box Online” and “Beach Box Kiel” according to the cost comparison calculation as a comparison of alternatives. Apply the procedure of primitive data collection. Assume the simple (linear) notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits)

Solution (Textbook Sect. 2.4.2) Due to the requirement to use the primitive data collection method, only data of the first year of the investment alternatives (U (ek), k ¼ 1 (T€), B (ak), k ¼ 1 (T€)) are considered for the determination of the annual revenue (U) and operating costs (B). Formula 2.6 from the textbook is the relevant formula and the data from the exercise must be entered accordingly. (2.1) = (2.6TB)1

1

TB here and in subsequent instances refer Textbook equation numbers.

2.2 Solutions Static Investment Calculation Methods

B1 þ

30, 000 þ þ

A1  R1 A1 þ R1 h A  R2 A2 þ R2  i B2 þ 2 i þ þ n1 2 n2 2 i

13

ð2:1Þ

h 140, 000  25, 000 140, 000 þ 25, 000 þ  0:1 40, 000 6 2 i

120, 000  15, 000 120, 000 þ 15, 000 þ  0:1 6 2

ð2:2Þ

In this situation, the costs for the BBO project would then amount to 57,416.67 euro and 64,250.00 euro for the BBK project. The Beach Box Online project would be preferable according to this approach. K BBO ð57, 416:67euroÞ < K BBK ð64, 250:00euroÞ

ð2:3Þ

(b) Determine the costs of the two investment objects “Beach Box Online” and “Beach Box Kiel” according to the cost comparison calculation as a comparison of alternatives. Apply the procedure of improved data collection. Assume the simple (linear) notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits) Solution (Textbook Sect. 2.4.2) Due to the requirement to use the improved data collection procedure, average values have to be calculated for the investment project “Beach Box Online”. This is not necessary for the investment project “Beach Box Kiel”, as the operating costs are the same every year. For the investment project “Beach Box Online”, this results in operating costs (B (ak)) of 45,000 euro. These values must now be used as the operating costs for the Beach Box Online project; otherwise, the solution is the same as for exercise a). (2.4) = (2.6TB) B1 þ

45, 000 þ þ

A1  R1 A1 þ R1 h A  R2 A2 þ R2  i B2 þ 2 i þ þ n1 2 n2 2 i

ð2:4Þ

h 140, 000  25, 000 140, 000 þ 25, 000 þ  0:1 40, 000 6 2 i

120, 000  15, 000 120, 000 þ 15, 000 þ  0:1 6 2

ð2:5Þ

In this situation, the costs for the BBO project amount to 72,416.67 euro and to 64,250.00 euro for the BBK project. The Beach Box Kiel project would be preferable according to this approach. K BBO ð72, 416:67euroÞ > K BBK ð64, 250:00euroÞ

ð2:6Þ

14

2

Mock Exam: Static Investment Calculation Methods

(c) Determine the costs of the two projects “Beach Box Online” and “Beach Box Kiel” according to the cost comparison calculation as a comparison of alternatives. Apply the procedure of improved data collection. Assume the discrete notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits) Solution (Textbook Sect. 2.4.2) First, the relevant formula for the solution must be identified. There are two possibilities which come to the same result, formulas 2.7 and 2.8 from the textbook: (2.7) = (2.7TB) (2.8) = (2.8TB) ! 1 A1  R1 þ A1nR A1  R1 1 B1 þ þ R1 þ n1 2 ! 2 A2  R2 þ A2nR h A2  R2 2 þ R2  i  i B2 þ þ ð2:7Þ i n2 2   h A1  R1 A1  R1 n1 þ 1 A  R2  þ þ R1  i B2 þ 2 n1 n1 2 n2 i   A2  R2 n2 þ 1 þ þ R2  i  n2 2

B1 þ

For the solution of the exercise, the data are entered into formula 2.7.   140  25 þ 14025 h 140  25 120  15 6 45 þ þ þ 25  0:1 40 þ 6 6 2 i   12015 120  15 þ 6 þ þ 15  0:1 2

ð2:8Þ

ð2:9Þ

For the BBO project, this calculation results in costs of 73,375.00 euro, and for the BBK project, the costs are 65,125.00 euro. The Beach Box Kiel project would be preferable according to this approach. K BBO ð73, 375:00euroÞ > K BBK ð65, 125:00euroÞ

ð2:10Þ

(d) Comment on the suitability of the cost comparison calculation for the investment decision as a comparison of the alternatives “Beach Box Online” and “Beach Box Kiel”. (3 credits) Solution (Textbook Sects. 2.2 and 2.4.1) First of all, there is general criticism of the static procedures:

2.2 Solutions Static Investment Calculation Methods

15

• 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 observed. • 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. • Profits and costs are assumed to remain constant over the years of the useful life. • Data certainty 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 payments of the alternatives are comparable. • The cost comparison calculation only makes sense if the useful lives of the alternatives are comparable. In general, as in this case in particular, comparative cost accounting is therefore less suitable for a qualified investment decision. (e) Determine the costs of the two investment objects “Beach Box Kiel” and “Baltic Diving” according to the cost comparison calculation in the replacement problem. The investment object “Baltic Diving” is the old object. Apply the procedure of improved data collection. Assume the discrete notion of capital commitment. Make an investment decision based on the calculation results and document it. (3 credits) Solution (Textbook Sect. 2.4.2) First, the relevant formula for the solution must be identified. There are two possibilities which come to the same result, formulas 2.15 and 2.16 from the textbook: (2.11) = (2.15TB) (2.12) = (2.16TB) ! new Anew  Rnew þ AnewnR Anew  Rnew h new Bnew þ þ Rnew  i Bold þ ð2:11Þ i nnew 2   h Anew  Rnew Anew  Rnew nnew þ 1  þ þ Rnew  i Bold Bnew þ nnew nnew 2 i

ð2:12Þ

For the solution of the exercise, the data are entered into formula 2.15 of the textbook.

16

2

40 þ

Mock Exam: Static Investment Calculation Methods

  120  15 þ 12015 120  15 < 6 þ þ 15  0:1 35 6 > 2

ð2:13Þ

For the BBK project, this calculation results in costs of 65,125.00 euro, and for the BD project, the costs are 35,000.00 euro. The Baltic Diving project would be preferable according to this approach. K BBK ð65, 125:00euroÞ > K BD ð35, 000:00euroÞ

ð2:14Þ

(f) Determine the profit of “Beach Box Online” according to the profit comparison calculation in case of a replacement problem. Assume that the old object “Baltic Diving” from Fig. 2.2 has been running for 2 years. Continue to assume a discrete notion of capital commitment. Apply the procedure of improved data collection. Make an investment decision based on the calculation results and document it. (3 credits) Solution (Textbook Sect. 2.5.2) First, the relevant formula must be identified. It becomes clear that the information in the exercise that the old object has already been running for 2 years is irrelevant for the calculation. There are two possibilities that come to the same result, formulas 2.54 and 2.55 from the textbook: (2.15) = (2.54TB) (2.16) = (2.55TB) ! new Anew  Rnew þ AnewnR Anew  Rnew h new U new  Bnew  þ Rnew  i U old  Bold  i nnew 2 ð2:15Þ   h Anew  Rnew Anew  Rnew nnew þ 1 U new  Bnew   þ Rnew  i U old  Bold  nnew nnew 2 i ð2:16Þ For the solution of the exercise, the data are entered into formula 2.54.   140  25 þ 14025 h 140  25 6  100  45  þ 25  0:1 65  35 6 2 i

ð2:17Þ

For the BBO project, this calculation results in an annual profit of 26,625.00 euro, and for the BD project, the annual profit is 30,000.00 euro. The Baltic Diving project would be preferable according to this approach. GBBO ð26, 625:00euroÞ < GBD ð30, 000:00euroÞ

ð2:18Þ

2.2 Solutions Static Investment Calculation Methods

17

(g) Determine two profitabilities according to the profitability calculation for the investment object “Beach Box Kiel”. Assume that • The simple notion of capital commitment is used to determine the first profitability. • The discrete notion of capital commitment applies for the determination of the second profitability. (3 credits) Solution (Textbook Sect. 2.6.2) First, the relevant formula for the solution must be identified. This is formula 2.82 for the profitability of the first bullet point. There are two possibilities for the profitability of the second bullet point, formulas (2.83) and (2.84) of the textbook: (2.19) = (2.82TB) (2.20) = (2.83TB) (2.21) = (2.84TB) Re nt ¼

UB AþR 2

UB Re nt ¼ ARþAR n þR 2 Re nt ¼ AR 2

UB  nþ1 n þR

ð2:19Þ

ð2:20Þ

ð2:21Þ

For the solution of the exercise, the data are entered into formulas 2.82 and 2.83. Re nt ¼

90  40 120þ15 2

¼ 74:07%

90  40 Re nt ¼ 12015þ12015 ¼ 65:57% 6 þ 15 2

ð2:22Þ

ð2:23Þ

For the BBK project, this calculation results in a static profitability of 74.07% in the first case and 65.57% in the second. (h) Determine two profitabilities according to the profitability calculation for the investment object “Baltic Diving”. Assume that • The discrete notion of capital commitment is used to determine the first profitability. • The constant notion of capital commitment applies for the determination of the second profitability. (3 credits)

18

2

Mock Exam: Static Investment Calculation Methods

Solution (Textbook Sect. 2.6.2) First, the relevant formula for the solution must be identified. For the profitability of the first bullet point, there are two possibilities which come to the same result: formulas 2.87 and 2.88 from the textbook. Formula 2.85 is the relevant formula for the profitability of the second bullet point. (2.24) = (2.87TB) (2.25) = (2.88TB) (2.26) = (2.85TB) Re nt ¼

UB AþAn 2

ð2:24Þ

UB Re nt ¼ A nþ1 2 n

ð2:25Þ

UB A

ð2:26Þ

Re nt ¼

For the solution of the exercise, the data are entered into formulas 2.87 and 2.85. Re nt ¼

Re nt ¼

65  35

¼ 57:14%

ð2:27Þ

65  35 ¼ 33:33% 90

ð2:28Þ

90þ90 6 2

For the BD project, this calculation results in a static profitability of 57.14% in the first case and 33.33% in the second. (i) Determine the static amortisation period of the investment project “Beach Box Online” according to the accumulation calculation. (3 credits) Solution (Textbook Sect. 2.7.2) First, the relevant formula for the solution must be identified. For the accumulation calculation to determine the static amortisation time, this is formula 2.112 from the textbook. (2.29) = (2.112TB) t m ðyearsÞ at A ¼

m X

ð e t  at Þ

ð2:29Þ

t

For the solution of the exercise, the data are entered into formula 2.112. The solution is shown in Fig. 2.3.

2.2 Solutions Static Investment Calculation Methods

t (years) 0 1 2 3 4 5 6

A (T€) 140 140 140 140 140 140 140

U - B (T€) 0 80 – 30 = 50 90 – 35 = 55 100 – 40 = 60 100 – 45 = 55 110 – 60 = 50 120 – 60 = 60

Σ U - B (T€) 0 50 105 165 220 270 330

19

t amortisation (years)

t amortisation = 3 years

Fig. 2.3 Accumulation calculation to determine the static amortisation period (Source: Author)

(j) Determine the static amortisation period of the investment project “Beach Box Kiel” according to the average calculation. (3 credits) Solution (Textbook Sect. 2.7.2) First, the relevant formula for the solution must be identified. For the average calculation to determine the static amortisation time, this is formula 2.114 from the textbook. (2.30) = (2.114TB) t m ðyearsÞ ¼

A ð e  aÞ

ð2:30Þ

For the solution of the exercise, the data are entered into formula 2.114. 120 ¼ 3years ð90  40Þ

ð2:31Þ

Due to the assumption that calculation elements are incurred in arrears, we round up the result to the next full period. So, three years is the result.

3

Mock Exam: Dynamic Investment Calculation Methods

Abstract

This chapter contains exercises and solutions of the topic “Dynamic investment calculation methods”. It is based on the textbook Investment Valuation and Appraisal—Theory and Practice published by Springer. The calculation formulas of the textbook are marked accordingly in this chapter.

3.1

Exercises Dynamic Investment Calculation Methods

Time: 60 Min (Without Exercise a), Credits: 30 After your studies, you would like to supplement your management job with parttime work to improve your disposable income. You want to analyse four different investment projects for this purpose. To do so, you use the technique of dynamic investment calculation methods. The assumptions of the model of dynamic investment calculation methods apply. The useful life of all investment projects is 5 years; your calculation interest rate is 10 percent. The names of the calculation elements correspond to the names in the textbook. The calculation elements of 3 investment projects are shown in Fig. 3.1. The fourth is a company that trades with purchased beach chairs and is called “Beach Box Online”. You still have to calculate the necessary calculation elements from the data in the second table in Fig. 3.2. The 3 investment projects shown in Fig. 3.1 are named “Beach Box Kiel”, which is floor trading for beach chairs produced in the region, “Baltic Diving”, which focuses on the paid implementation of diving courses in the Kiel Fjord, and “Kiel Share”, an investment project which participates in a start-up from the region and has a planned exit from the project after 5 years at a price already agreed upon at the beginning of

Supplementary Information The online version of this chapter (https://doi.org/10.1007/978-3658-33045-3_3) contains supplementary material, which is available to authorized users. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 K. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-33045-3_3

21

22

3

Mock Exam: Dynamic Investment Calculation Methods

the participation. The annual payments received from the “Kiel Share” project in the form of dividends are in line with the annual payments for the support and administration of the investment, so that this project has no net payments during the term of the investment. Calculation element n (years) A (T€) i (%)

Beach Box Kiel (BBK) 5 160 0.1

Baltic Diving (BD) 5 140 0.1

ek, k = 1 (T€) ek, k = 2 (T€) ek, k = 3 (T€) ek, k = 4 (T€) ek, k = 5 (T€)

110 110 110 110 110

90 90 90 90 90

ak, k = 1 (T€) ak, k = 2 (T€) ak, k = 3 (T€) ak, k = 4 (T€) ak, k = 5 (T€)

50 50 50 50 50

40 40 40 40 40

160

0

R (T€)

Kiel Share (KS) 5 90 0.1

165

Fig. 3.1 Data set for the dynamics exercise (Source: Author)

The calculation elements of the investment project “Beach Box Online” are to be determined for the 5-year useful life. There is no residual value for this project. The acquisition payment for the purchase of the shares in this company amounts to 600,000.00 euro. When calculating, please note that the number of chairs sold in each year of use (i.e. from k ¼ 2 for the first time and then from year to year) increases by 5 chairs per quarter. When calculating, please note that the basic price of the chairs sold increases by 50 euro per quarter in each year of use (i.e. from k ¼ 2 for the first time and then from year to year). The price for the additional equipment remains unchanged in each quarter. 10 percent of the sales per quarter and over the term of use are made with the additional equipment “Sieger”; another 10 percent of the sales per quarter and over the term of use are made with the additional equipment “Landgraf”. Please accept the fact that these figures are not whole numbers, as this is a planning calculation. In addition, there is a special promotion “Student chair”, beach chairs for students, which are sold for 650.00 euro per piece for the entire useful life. These chairs are sold without special equipment. There, sales figures are expected to remain constant at 100 pieces per year. The payments for procurement and staff are constant in all years and quarters, a constant percentage of the payments per quarter.

3.1 Exercises Dynamic Investment Calculation Methods

Calculation bases Number of chairs sold (pieces, k = 1) Basic price per chair (€, k = 1) Additional equipment “Sieger” (€) Additional equipment “Landgraf” (€) Price special offer student chair (€) Number of special offer student chair (pieces) Payouts procurement beach chair as a share of in-payments Payouts staff as a proportion of in-payments Fixed payouts (€)

Winter 50 800 150 100 650 100

23

Spring 110 1200 250 200

Summer 90 1100 200 160

Autumn 40 700 125 100

2000

2000

1000

40% 10% 1000

Fig. 3.2 Planning data for the investment object Beach Box Online (Source: Author)

(a) Display the complete cash flow for the investment project “Beach Box Online” on a timeline. Document the calculation path of the individual calculation elements. (5 credits) (b) Calculate the net present value (Co) for each of the four investment projects. Interpret the result of the net present value method for all four investment projects. (5 credits) (c) Determine the horizon value (Cn) for each of the four investment projects. Use the independent calculation method of the horizon value method for the solution. Compare the result with the results from exercise (b) by adding interest on the net present values determined there or by discounting the horizon values. Interpret the result of the horizon value method for all four investment projects. (5 credits) (d) Determine the average annual surplus (DJÜ) for each of the four investment projects. Use the independent calculation method of the annuity method. Compare the result with the results from exercises (b) and (c) by distributing the net present values and horizon values determined there with the appropriate financial mathematical distribution factors. Interpret the result of the annuity method for all four investment projects. (5 credits) (e) Determine the returns on the four investment projects. Use a different approach for each investment project, taking into account the special cases of the internal rate of return method. Approximate solutions are permissible. For the “Beach Box Online” project, please use 25 percent as a second test interest rate. Interpret the result of the internal rate of return method for all four investment projects. (5 credits) (f) Determine the dynamic amortisation periods of the investment projects “Baltic Divinig” and “Kiel Share”. The maximum permissible payback period is 5 years. If you need an additional test useful life, please use 3 years. Interpret the result of the calculation for both investment projects. (5 credits)

24

3.2

3

Mock Exam: Dynamic Investment Calculation Methods

Solutions Dynamic Investment Calculation Methods

After your studies, you would like to supplement your management job with parttime work to improve your disposable income. You want to analyse four different investment projects for this purpose. To do so, you use the technique of dynamic investment calculation methods. The assumptions of the model of dynamic investment calculation methods apply. The useful life of all investment projects is 5 years; your calculation interest rate is 10 percent. The names of the calculation elements correspond to the names in the textbook. The calculation elements of 3 investment projects are shown in Fig. 3.3. The fourth is a company that trades with purchased beach chairs and is called “Beach Box Online”. You still have to calculate the necessary calculation elements from the data in the second table from Fig. 3.4. The 3 investment projects shown in Fig. 3.3 are named “Beach Box Kiel”, which is floor trading for beach chairs produced in the region, “Baltic Diving”, which focuses on the paid implementation of diving courses in the Kiel Fjord, and “Kiel Share”, an investment project which participates in a start-up from the region and has a planned exit from the project after 5 years at a price already agreed upon at the beginning of the participation. The annual payments received from the “Kiel Share” project in the form of dividends are in line with the annual payments for the support and administration of the investment, so that this project has no net payments during the term of the investment. Calculation element n (years) A (T€) i (%)

Beach Box Kiel (BBK) 5 160 0.1

Baltic Diving (BD) 5 140 0.1

ek, k = 1 (T€) ek, k = 2 (T€) ek, k = 3 (T€) ek, k = 4 (T€) ek, k = 5 (T€)

110 110 110 110 110

90 90 90 90 90

ak, k = 1 (T€) ak, k = 2 (T€) ak, k = 3 (T€) ak, k = 4 (T€) ak, k = 5 (T€)

50 50 50 50 50

40 40 40 40 40

160

0

R (T€)

Fig. 3.3 Data set for the dynamics exercise (Source: Author)

Kiel Share (KS) 5 90 0.1

165

3.2 Solutions Dynamic Investment Calculation Methods

25

The calculation elements of the investment project “Beach Box Online” are to be determined for the 5-year useful life from the table in Fig. 3.4. There is no residual value for this project. The acquisition payment for the purchase of the shares in this company amounts to 600,000.00 euro. When calculating, please note that the number of chairs sold in each year of use (i.e. from k ¼ 2 for the first time and then from year to year) increases by 5 chairs per quarter. When calculating, please note that the basic price of the chairs sold increases by 50 euro per quarter in each year of use (i.e. from k ¼ 2 for the first time and then from year to year). The price for the additional equipment remains unchanged in each quarter. 10 percent of the sales per quarter and over the term of use are made with the additional equipment “Sieger”; another 10 percent of the sales per quarter and over the term of use are made with the additional equipment “Landgraf”. Please accept the fact that these figures are not whole numbers, as this is a planning calculation. In addition, there is a special promotion “Student chair”, beach chairs for students, which are sold for 650.00 euro per piece for the entire useful life. These chairs are sold without special equipment. There, sales figures are expected to remain constant at 100 pieces per year. The payments for procurement and staff are constant in all years and quarters, a constant percentage of the payments per quarter. Calculation bases Number of chairs sold (pieces, k = 1) Basic price per chair (€, k = 1) Additional equipment “Sieger” (€) Additional equipment “Landgraf” (€) Price special offer student chair (€) Number of special offer student chair (pieces) Payouts procurement beach chair as a share of in-payments Payouts staff as a proportion of in-payments Fixed payouts (€)

Winter 50 800 150 100 650 100

Spring 110 1200 250 200

Summer 90 1100 200 160

Autumn 40 700 125 100

2000

2000

1000

40% 10% 1000

Fig. 3.4 Planning data for the investment object Beach Box Online (Source: Author)

(a) Display the complete cash flow for the investment project “Beach Box Online” on a timeline. Document the calculation path of the individual calculation elements. (5 credits) Solution You can find the solution in the Excel file or in Sect. 3.3.1, Figs. 3.5 and 3.6. (b) Calculate the net present value (Co) for each of the four investment projects. Interpret the result of the net present value method for all four investment projects. (5 credits)

26

3

Mock Exam: Dynamic Investment Calculation Methods

Solution (Textbook Sect. 3.4.1 and 3.4.2) First, the relevant formulas must be identified. These are partly different for the 4 investment projects. Relevant are formulas 3.8 and 3.10. (3.1) = (3.8TB) (3.2) = (3.10TB) Co ¼

n X

ðek  ak Þ  ð1 þ iÞk þ R  ð1 þ iÞn  A

ð3:1Þ

k¼1

Co ¼ ðe  aÞ  DSF ni þ R  ð1 þ iÞn  A

ð3:2Þ

You can find the solution in the Excel file or in Sect. 3.3.2, Figs. 3.7 and 3.8. Interpretation All 4 investment objects are worthwhile according to the net present value method. This means that in each case • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the calculation interest rate i • A net present value surplus in the amount of Co can be achieved. The most advantageous investment cannot be determined because a relative consideration is not possible using the net present value method. (c) Determine the horizon value (Cn) for each of the four investment projects. Use the independent calculation method of the horizon value method for the solution. Compare the result with the results from exercise (b) by adding interest on the net present values determined there or by discounting the horizon values. Interpret the result of the horizon value method for all four investment projects. (5 credits) Solution (Textbook Sect. 3.5) First, the relevant formulas must be identified. These are partly different for the 4 investment projects. Relevant are formulas 3.21 and 3.22. (3.3) = (3.21TB) (3.4) = (3.22TB) Cn ¼

n X

ðek  ak Þ  ð1 þ iÞnk þ R  A  ð1 þ iÞn

ð3:3Þ

k¼1

Cn ¼ ðe  aÞ  EWF ni þ R  A  ð1 þ iÞn

ð3:4Þ

3.2 Solutions Dynamic Investment Calculation Methods

27

You can find the solution in the Excel file or in Sect. 3.3.2, Figs. 3.7 and 3.8. Interpretation All 4 investment objects are worthwhile according to the horizon value method. This means that in each case • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the calculation interest rate i • A final surplus in the amount of the determined horizon value can be achieved. The most advantageous investment cannot be determined because a relative consideration is not possible using the horizon value method. (d) Determine the average annual surplus (DJÜ) for each of the four investment projects. Use the independent calculation method of the annuity method. Compare the result with the results from exercises (b) and (c) by distributing the net present values and horizon values determined there with the appropriate financial mathematical distribution factors. Interpret the result of the annuity method for all four investment projects. (5 credits) Solution (Textbook Sect. 3.6) First, the relevant formulas must be identified. These are partly different for the 4 investment projects. Relevant are formulas 3.30 to 3.38, which can be used for the solution in different ways. You can find the solution in the Excel file or in Sect. 3.3.2, Figs. 3.7 and 3.8. Interpretation All 4 investment objects are worthwhile according to the annuity method. This means that in each case • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the calculation interest rate i • A periodic surplus in the amount of the DJÜ determined can be achieved. The most advantageous investment cannot be determined because a relative consideration is not possible using the annuity method.

28

3

Mock Exam: Dynamic Investment Calculation Methods

(e) Determine the returns on the four investment projects. Use a different approach for each investment project, taking into account the special cases of the internal rate of return method. Approximate solutions are permissible. For the “Beach Box Online” project, please use 25 percent as a second test interest rate. Interpret the result of the internal rate of return method for all four investment projects. (5 credits) Solution (Textbook Sect. 3.7.1 and 3.7.2) First, the relevant formula must be identified. Here, the general possible solution is formula 3.52, which can always be used and must also be used for the “Beach Box Online” project. For the other three investment projects, the special case solutions of residual value equal to acquisition payment (formula 3.65), residual value equal to zero (formula 3.68) and two-payment case (formula 3.60) are also possible. (3.5) = (3.52TB) (3.6) = (3.65TB) (3.7) = (3.68TB) (3.8) = (3.60TB) r ¼ i1  Co1 

r¼

i2  i1 Co2  Co1

ð e  aÞ A

KWF nr ¼

ð e  aÞ A

rffiffiffi n R r¼ 1 A

ð3:5Þ

ð3:6Þ ð3:7Þ

ð3:8Þ

You can find the solution in the Excel file or in Sect. 3.3.2, Figs. 3.8, 3.9 and 3.10. Interpretation All 4 investment objects are worthwhile according to the internal rate of return method, as the calculated returns are all above the calculation interest rate. This means that in each case • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the return r can be achieved. The most advantageous investment cannot be determined because a relative consideration is not possible using the technique of the internal rate of return method.

3.3 Excel-Based Solutions Dynamic Investment Calculation Methods

29

(f) Determine the dynamic amortisation periods of the investment projects “Baltic Diving” and “Kiel Share”. The maximum permissible payback period is 5 years. If you need an additional test useful life, please use 3 years. Interpret the result of the calculation for both investment projects. (5 credits) Solution (Textbook Sect. 3.8) First, the relevant formula must be identified. Here is formula 3.77 as a general solution possibility, which can always be applied. For the two investment projects, special case solutions are also possible. (3.9) = (3.77TB) ndyn ¼ n1  Co1 

¼ KWF ndyn i

n2  n1 Co2  Co1

ð e  aÞ A

ð3:9Þ

ð3:10Þ

You can find the solution in the Excel file or in Sect. 3.3.2, Fig. 3.11. Both investment objects are worthwhile according to the method dynamic amortisation calculation. This means that • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the calculation interest rate i can be achieved within the specified payback period. The most advantageous investment cannot be determined because a relative consideration is not possible with the technique of calculating the dynamic payback period.

3.3

Excel-Based Solutions Dynamic Investment Calculation Methods

3.3.1

Excel-Based Solutions Dynamic Investment Calculation Methods, Exercise a

• Annual payments of the investment project Beach Box Online: Fig. 3.5. • Timeline for the investment project Beach Box Online: Fig. 3.6.

30

3

Mock Exam: Dynamic Investment Calculation Methods winter

K=1 Quanty Price Turnover Quanty "Sieger" Quanty "Landgraf" Price "Sieger" Price "Landgraf" Addional deposits "Sieger" Addional deposits "Landgraf" Σ deposits without student beach chair Payouts for procurement Payouts for staff Fixed payouts Nek without student beach chair Deposits student beach chair Payouts for procurement of student beach chair Payouts for staff of student beach chair Nek student beach chair ek total ak total Nek total

50 800 40000 5 5 150 100 750 500 41250 -16500 -4125 -1000 19625

winter K=2 Quanty Price Turnover Quanty "Sieger" Quanty "Landgraf" Price "Sieger" Price "Landgraf" Addional deposits "Sieger" Addional deposits "Landgraf" Σ deposits without student beach chair Payouts for procurement Payouts for staff Fixed payouts Nek without student beach chair Deposits student beach chair Payouts for procurement of student beach chair Payouts for staff of student beach chair Nek student beach chair ek total ak total Nek total

55 850 46750 5.5 5.5 150 100 825 550 48125 -19250 -4812.5 -1000 23062.5

spring 110 1200 132000 11 11 250 200 2750 2200 136950 -54780 -13695 -2000 66475

spring 115 1250 143750 11.5 11.5 250 200 2875 2300 148925 -59570 -14892.5 -2000 72462.5

summer 90 1100 99000 9 9 200 160 1800 1440 102240 -40896 -10224 -2000 49120

summer 95 1150 109250 9.5 9.5 200 160 1900 1520 112670 -45068 -11267 -2000 54335

autumn

year

40 700 28000 4 4 125 100 500 400 28900 -11560 -2890 -1000 13450

autumn 45 750 33750 4.5 4.5 125 100 562.5 450 34762.5 -13905 -3476.25 -1000 16381.25

299000

5800 4540 309340 -123736 -30934 -6000 148670 65000 -26000 -6500 32500 374340 -193170 181170

year

333500

6162.5 4820 344482.5 -137793 -34448.25 -6000 166241.25 65000 -26000 -6500 32500 409482.5 -210741.25 198741.25

Fig. 3.5 Annual payments for the investment project Beach Box Online (Source: Author)

3.3 Excel-Based Solutions Dynamic Investment Calculation Methods winter K=3 Quanty Price Turnover Quanty "Sieger" Quanty "Landgraf" Price "Sieger" Price "Landgraf" Addional deposits "Sieger" Addional deposits "Landgraf" Σ deposits without student beach chair Payouts for procurement Payouts for staff Fixed payouts Nek without student beach chair Deposits student beach chair Payouts for procurement of student beach chair Payouts for staff of student beach chair Nek student beach chair ek total ak total Nek total

60 900 54000 6 6 150 100 900 600 55500 -22200 -5550 -1000 26750

winter K=4 Quanty Price Turnover Quanty "Sieger" Quanty "Landgraf" Price "Sieger" Price "Landgraf" Addional deposits "Sieger" Addional deposits "Landgraf" Σ deposits without student beach chair Payouts for procurement Payouts for staff Fixed payouts Nek without student beach chair Deposits student beach chair Payouts for procurement of student beach chair Payouts for staff of student beach chair Nek student beach chair ek total ak total Nek total

Fig. 3.5 (continued)

65 950 61750 6.5 6.5 150 100 975 650 63375 -25350 -6337.5 -1000 30687.5

spring 120 1300 156000 12 12 250 200 3000 2400 161400 -64560 -16140 -2000 78700

spring 125 1350 168750 12.5 12.5 250 200 3125 2500 174375 -69750 -17437.5 -2000 85187.5

summer 100 1200 120000 10 10 200 160 2000 1600 123600 -49440 -12360 -2000 59800

summer 105 1250 131250 10.5 10.5 200 160 2100 1680 135030 -54012 -13503 -2000 65515

31 autumn 50 800 40000 5 5 125 100 625 500 41125 -16450 -4112.5 -1000 19562.5

autumn 55 850 46750 5.5 5.5 125 100 687.5 550 47987.5 -19195 -4798.75 -1000 22993.75

year

370000

6525 5100 381625 -152650 -38162.5 -6000 184812.5 65000 -26000 -6500 32500 446625 -229312.5 217312.5

year

408500

6887.5 5380 420767.5 -168307 -42076.75 -6000 204383.75 65000 -26000 -6500 32500 485767.5 -248883.75 236883.75

32

3

Mock Exam: Dynamic Investment Calculation Methods winter

K=5 Quanty Price Turnover Quanty "Sieger" Quanty "Landgraf" Price "Sieger" Price "Landgraf" Addional deposits "Sieger" Addional deposits "Landgraf" Σ deposits without student beach chair Payouts for procurement Payouts for staff Fixed payouts Nek without student beach chair Deposits student beach chair Payouts for procurement of student beach chair Payouts for staff of student beach chair Nek student beach chair ek total ak total Nek total

spring

70 1000 70000 7 7 150 100 1050 700 71750 -28700 -7175 -1000 34875

summer

130 1400 182000 13 13 250 200 3250 2600 187850 -75140 -18785 -2000 91925

110 1300 143000 11 11 200 160 2200 1760 146960 -58784 -14696 -2000 71480

autumn

year

60 900 54000 6 6 125 100 750 600 55350 -22140 -5535 -1000 26675

449000

7250 5660 461910 -184764 -46191 -6000 224955 65000 -26000 -6500 32500 526910 -269455 257455

Fig. 3.5 (continued)

timeline R ek ak Nek A k

374340 409482.5 -193170 -210741.25 181170 198741.25 -600000 k=0

K=1

K=2

446625 485767.5 -229312.5 -248883.75 217312.5 236883.75 K=3

0 526910 -269455 257455

K=4

K=5

Fig. 3.6 Timeline for the investment project Beach Box Online (Source: Author)

3.3.2

Excel-Based Solutions Dynamic Investment Calculation Methods, Exercise b–f

• Excel solution for net present value, horizon value and annuity: Fig. 3.7. • Pocket calculator solution for net present value, horizon value, annuity and return: Fig. 3.8. • Excel solution for calculating the return of Beach Box Online: Fig. 3.9. • Pocket calculator solution for determining the net present value of Beach Box Online to determine the return: Fig. 3.10. • Solution for calculating the dynamic amortisation period: Fig. 3.11.

3.3 Excel-Based Solutions Dynamic Investment Calculation Methods Exercise b - d)

0.10 NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 R k=5 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

0.10 NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 R k=5 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

0.10 NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 R k=5 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

0.10 NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 R k=5 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

33

Excel Solution

k=0 164700.00 164248.97 163270.10 161794.79 159859.30 0.00 813873.15 -600000.00 213873.15 0.263797 56419.20

k=1 181170.00

k=0 54545.46 49586.78 45078.89 40980.81 37255.28 99347.41 326794.62 -160000.00 166794.62 0.263797 44000.00

k=1 60000.00

k=0 45454.55 41322.31 37565.74 34150.67 31046.07 0.00 189539.34 -140000.00 49539.34 0.263797 13068.35

k=1 50000.00

k=0 0.00 0.00 0.00 0.00 0.00 102452.02 102452.02 -90000.00 12452.02 0.263797 3284.81

Beach Box Online k=3 k=4

k=2

k=5

198741.25 217312.50 236883.75 257455.00 0.00

Beach Box Kiel k=3

k=2

k=4

k=5

60000.00 60000.00 60000.00 60000.00 160000.00

Baltic Diving k=3

k=2

k=4

k=5

50000.00 0.00

k=5

k=5 (EW)

50000.00 50000.00

KielShare k=3

k=2

k=4

k=5 (EW) 87846.00 79860.00 72600.00 66000.00 60000.00 160000.00 526306.00 -257681.60 268624.40 0.163797 44000.00

k=5 (EW) 73205.00 66550.00 60500.00 55000.00 50000.00 0.00 305255.00 -225471.40 79783.60 0.163797 13068.35

50000.00

k=1

k=5 (EW) 265251.00 264524.60 262948.13 260572.13 257455.00 0.00 1310750.85 -966306.00 344444.85 0.163797 56419.20

0.00 0.00 0.00 0.00 0.00 165000.00

0.00 0.00 0.00 0.00 0.00 165000.00 165000.00 -144945.90 20054.10 0.163797 3284.81

Fig. 3.7 Excel solution for net present value, horizon value and annuity (Source: Author)

34

3

Mock Exam: Dynamic Investment Calculation Methods

Pocket Calculator Alternative: Σ(ek -ak ) x (1 + i)-k + R x Abf - A = Co 181,170.00 * 0.909091 + 198,741.25 * 0.826446 + 217,312.50 * 0.751315 + 236,883.75 * 0.683013 + 257,455.00 * 0.620921 + 0 * 0.620921 - 600,000 = 213,872.97 Σ(ek -ak ) x (1 + i)n-k + R - A * Auf = Cn 181,170.00 * 1.4641 + 198,741.25 * 1.331 + 217,312.50 * 1.21 + 236,883.75 * 1.1 + 257,455.00 * 1 + 0 * 1 - 600,000 * 1.61051 = 344,444.85 (Σek x (1 + i)-k + R * Abf) x KWF - (Σak x (1 + i)-k + A)* KWF = DJÜ, further formulas are possible, one line below it is calculated with (ek-ak) (181,170.00 * 0.909091 + 198,741.25 * 0.826446 + 217,312.50 * 0.751315 + 236,883.75 * 0.683013 + 257,455.00 * 0.620921 - 600,000) * 0.263797 = 56,419.05 Note: Differences between the spreadsheet solutions in figure 3.7 and the calculator solution in figure 3.8 are due to rounding differences Return=

22,88% see worksheet exercise e)

Alternative: (e-a) x DSF + R x Abf - A = Co 60,000 x 3.790787 + 160,000 x 0.620921 - 160,000 = 166,794.62 (e-a) x EWF + R - A x Auf = Cn 60,000 x 6.105100 + 160,000 - 160,000 x 1.610510 = 268,624.40 (e x DSF + R x Abf) x KWF - (a x DSF + A) x KWF = DJÜ (e + R x RVF) - (a + A x KWF) = DJÜ, as e and a are already constant, i.e. the same every year (110,000 + 160,000 x 0.163797) - (50,000 + 160,000 x 0.263797) = 44,000.00 Return=

0,375 (Formula 3.65)

r = 37.5%

Alternative: (e-a) x DSF + R x Abf - A = Co 50,000 x 3.790787 + 0 x 0.620921 - 140,000 = 49,539.35 (e-a) x EWF + R - A x Auf = Cn 50,000 x 6.105100 + 0 - 140,000 x 1.610510 = 79,783.60 (e x DSF + R x Abf) x KWF - (a x DSF + A) x KWF = DJÜ e - (a + A x KWF) = DJÜ, as e and a are already constant, i.e. the same every year 90,000 - (40,000 + 140,000 x 0.263797) = 13,068.42 KWFn=5 =

0,35714286 (Formula 3.68)

23% < r < 23.5%

Alternative: R x Abf - A = Co 165,000 x 0.620921 - 90,000 = 12,451.96 R - A x Auf = Cn 165,000 - 90,000 x 1.610510 = 20,054.10 (e x DSF + R x Abf) x KWF - (a x DSF + A) x KWF = DJÜ R x RVF - A x KWF = DJÜ, because e and a equal to 0 165,000 x 0.163797 - 90,000 x 0.263797 = 3,284.78 Return=

0,12888132 (Formula 3.60)

r = 12.89%

Fig. 3.8 Pocket calculator solution for net present value, horizon value, annuity and yield (Source: Author)

3.3 Excel-Based Solutions Dynamic Investment Calculation Methods

35

Exercise e) Rate of return Beach Box Online Beach Box Online variable value i1 10.00% Co1 213,873.153 i2 25.00% Co2 -35,215.162

Calculating r using linear interpolation: 0.1

Result:

0.25 NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 R k=5 SUM (BW / EW) minus A Co / Cn

-

213873.153

0.1500 -249,088.315

x

=

22.88%

k=4

k=5

The rate of return for the investment object is 22.88%.

k=0 144936.00 127194.40 111264.00 97027.58 84362.85 0.00 564784.84 -600000.00 -35215.16

k=1 181170.00

k=2

Beach Box Online k=3

198741.25 217312.50 236883.75

k=5 (EW) 442309.57 388166.50 339550.78 296104.69 257455.00 257455.00 0.00 0.00 1723586.54 -1831054.69 -107468.15

Fig. 3.9 Excel solution for calculating the return of Beach Box Online (Source: Author)

Alternative: Σ(e k -a k) x (1 + i)-k + R x Abf - A = Co 181,170.00 * 0.8 + 198,741.25 * 0.64 + 217,312.50 * 0.512 + 236,883.75 * 0.4096 + 257,455.00 * 0.32768 + 0 * 0.32768 - 600,000 = -35,215.16

Fig. 3.10 Pocket calculator solution for determining the net present value of Beach Box Online to determine the return (Source: Author)

36

3

Mock Exam: Dynamic Investment Calculation Methods

Exercise f) Baltic Diving variable value n1 5 Co1 49,539.338 n2 3 Co2 -15,657.401

Calculating n dyn using linear interpolation: 5

Result:

0.10 NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 R k=5 SUM (BW / EW) minus A Co / Cn

-

49539.338

x

-2.0000 -65,196.739

=

3.48

The dynamic amortisation period for the Baltic Diving investment object is 3.48 periods, rounded up to 4 periods based on the assumption of payment in arrears.

k=0 45454.55 41322.31 37565.74 0.00 0.00 0.00 124342.60 -140000.00 -15657.40

k=1 50000.00

k=2

k=3

k=4

k=5

50000.00 50000.00 0.00 0.00 0.00

The solution for KielShare must be 5 years, as there is no repayment before the residual value.

Alternative: KWF ndyn, i =

years

0.357143

3 years < n dyn < 4 years, therefore 4 years

Fig. 3.11 Solution for calculating the dynamic amortisation period (Source: Author)

k=5 (EW) 60500.00 55000.00 50000.00 0.00 0.00 0.00 165500.00 -186340.00 -20840.00

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

Abstract

This chapter contains exercises and solutions of the topic “Selection of alternatives and investment programme planning”. It is based on the textbook Investment Valuation and Appraisal—Theory and Practice published by Springer. The calculation formulas of the textbook are marked accordingly in this chapter.

4.1

Exercises Part Selection of Alternatives

Time: 60 Min, Credits: 30 After your studies, you would like to supplement your management job with parttime work to improve your income. To do so, you want to analyse four different investment projects. Your calculation interest rate is 10 percent. The names of the calculation elements correspond to the names in the textbook. The investment projects are called “Beach Box Online”, which is an online business with beach chairs purchased from other companies, “Beach Box Kiel”, which is floor trading with beach chairs produced in the region, “Baltic Diving”, which deals with the paid implementation of diving courses in the Kiel Fjord, and “Kiel Share” which is an investment project which participates in a start-up from the region. The calculation elements of the projects can be read in Fig. 4.1.

Supplementary Information The online version of this chapter (https://doi.org/10.1007/978-3658-33045-3_4) contains supplementary material, which is available to authorized users. # Springer Fachmedien Wiesbaden GmbH, part of Springer Nature 2021 K. Poggensee, Investment Valuation and Appraisal, Springer Texts in Business and Economics, https://doi.org/10.1007/978-3-658-33045-3_4

37

38

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

Calculation element

Beach Box Online (BBO)

Beach Box Kiel Baltic Diving (BBK) (BD)

n (years)

4

3

4

A (T€)

220

200

100

80

ek, k = 1 (T€)

120

110

90

75

ek, k = 2 (T€)

130

110

90

75

ek, k = 3 (T€)

140

110

90

75

ek, k = 4 (T€)

150

110

--

75

ek, k = 5 (T€)

--

110

--

--

ek, k = 6 (T€)

--

110

--

--

ak, k = 1 (T€)

50

50

40

25

ak, k = 2 (T€)

55

50

40

25

ak, k = 3 (T€)

60

50

40

25

ak, k = 4 (T€)

70

50

--

25

ak, k = 5 (T€)

--

50

--

--

ak, k = 6 (T€)

--

50

--

--

R (T€)

80

60

40

30

6

Kiel Share (KS)

Fig. 4.1 Data set for the exercise selection of alternatives (Source: Author)

(a) Calculate the net present value (Co) for each of the four investment projects considered. Interpret the result of the net present value method for all four investment projects. (4 credits) (b) Determine the horizon value (Cn) for each of the four investment projects considered. Use the independent calculation method of the horizon value method for the solution. Check the result with the results from exercise a) by adding interest on the net present values determined there or by discounting the horizon values. Interpret the result of the horizon value method for all four investment projects. (4 credits) (c) For each of the four investment projects considered, determine the net present value (Co*) after removing the reinvestment premise. Assume a reinvestment interest rate of 5 percent. Interpret the result of the net present value method for all four investment projects. (5 credits) (d) Determine the net present value (Co*) for the “Beach Box Online” investment project by removing the reinvestment premise. Assume a reinvestment interest rate of 5 percent in the first year. In the following years, the reinvestment interest rate falls by one percentage point per year. (2 credits)

4.2 Solutions Part Selection of Alternatives

39

(e) Compare the investment objects “Beach Box Kiel” and “Baltic Diving” in a complete comparison by means of fictitious investments. Compare on the basis of the net present values (Co). (3 credits) (f) Compare the investment objects “Beach Box Kiel” and “Kiel Share” in a complete comparison by means of fictitious investments. Compare on the basis of the horizon values, taking into account a reinvestment interest rate of 5 percent (Cn*). Free amounts can be invested at the reinvestment interest rate. (5 credits) (g) Compare the investment objects “Beach Box Kiel” and “Beach Box Online” in a complete comparison by means of fictitious investments. Compare on the basis of the net present values, taking into account a reinvestment interest rate of 5 percent (Co*). Free amounts can be invested at the reinvestment interest rate. What is your investment decision? (5 credits) (h) Check the results from exercise g) using the technique of the internal rate of return method. What is your investment decision now? (2 credits)

4.2

Solutions Part Selection of Alternatives

After your studies, you would like to supplement your management job with parttime work to improve your income. To do so, you want to analyse four different investment projects. Your calculation interest rate is 10 percent. The names of the calculation elements correspond to the names in the textbook. The investment projects are called “Beach Box Online”, which is an online business with beach chairs purchased from other companies, “Beach Box Kiel”, which is floor trading with beach chairs produced in the region, “Baltic Diving”, which deals with the paid implementation of diving courses in the Kiel Fjord, and “Kiel Share” which is an investment project which participates in a start-up from the region. The calculation elements of the projects can be read in Fig. 4.2.

40

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

Calculation element

Beach Box Online (BBO)

Beach Box Kiel Baltic Diving (BBK) (BD)

n (years)

4

3

4

A (T€)

220

200

100

80

ek, k = 1 (T€)

120

110

90

75

ek, k = 2 (T€)

130

110

90

75

ek, k = 3 (T€)

140

110

90

75

ek, k = 4 (T€)

150

110

--

75

ek, k = 5 (T€)

--

110

--

--

ek, k = 6 (T€)

--

110

--

--

ak, k = 1 (T€)

50

50

40

25

ak, k = 2 (T€)

55

50

40

25

ak, k = 3 (T€)

60

50

40

25

ak, k = 4 (T€)

70

50

--

25

ak, k = 5 (T€)

--

50

--

--

ak, k = 6 (T€)

--

50

--

--

R (T€)

80

60

40

30

6

Kiel Share (KS)

Fig. 4.2 Data set for the exercise selection of alternatives (Source: Author)

(a) Calculate the net present value (Co) for each of the four investment projects considered. Interpret the result of the net present value method for all four investment projects. (4 credits). Solution (Textbook Sects. 3.4, 3.4.1 and 3.4.2) First, the relevant formulas must be identified. These are partly different for the 4 investment projects. Relevant are formulas 3.8 and 3.10. (4.1) = (3.8TB) (4.2) = (3.10TB) Co ¼

n X

ðek  ak Þ  ð1 þ iÞk þ R  ð1 þ iÞn  A

ð4:1Þ

k¼1

Co ¼ ðe  aÞ  DSF ni þ R  ð1 þ iÞn  A

ð4:2Þ

You can find the solution in the Excel file or in Sect. 4.3, Figs. 4.3 and 4.4.

4.2 Solutions Part Selection of Alternatives

41

Interpretation All 4 investment objects are worthwhile according to the net present value method. This means that in each case • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the calculation interest rate i • A net present value surplus in the amount of Co can be achieved. The most advantageous investment cannot be determined because a relative consideration is not possible using the net present value method. (b) Determine the horizon value (Cn) for each of the four investment projects considered. Use the independent calculation method of the horizon value method for the solution. Check the result with the results from exercise a) by adding interest on the net present values determined there or by discounting the horizon values. Interpret the result of the horizon value method for all four investment projects. (4 credits). Solution (Textbook Sect. 3.5) First, the relevant formulas for the solution must be identified. These are partly different for the 4 investment projects. The relevant formulas are 3.21 and 3.22. (4.3) = (3.21TB) (4.4) = (3.22TB) Cn ¼

n X

ðek  ak Þ  ð1 þ iÞnk þ R  A  ð1 þ iÞn

ð4:3Þ

k¼1

Cn ¼ ðe  aÞ  EWF ni þ R  A  ð1 þ iÞn

ð4:4Þ

You can find the solution in the Excel file or in Sect. 4.3, Figs. 4.3 and 4.4. Interpretation All 4 investment objects are worthwhile according to the horizon value method. This means that in each case • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the calculation interest rate i • A final surplus in the amount of the determined horizon value can be achieved. The most advantageous investment cannot be determined because a relative consideration is not possible with the horizon value method.

42

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

(c) For each of the four investment projects considered, determine the net present value (Co*) after removing the reinvestment premise. Assume a reinvestment interest rate of 5 percent. Interpret the result of the net present value method for all four investment projects. (5 credits). Solution (Textbook Sect. 4.3–4.3.3) First, the relevant formulas must be identified. These are partly different for the 4 investment projects. The relevant formulas are 4.5 and 4.6, which can be used for the solution in different ways. (4.5) = (4.5TB) (4.6) = (4.6TB) " # n X nk Co ¼ ðek  ak Þ  ð1 þ iw Þ þ R  ð1 þ ik Þn  A ð4:5Þ k¼1

  Co ¼ NE  EWF niw þ R  ð1 þ ik Þn  A

ð4:6Þ

You can find the solution in the Excel file or in Sect. 4.3, Figs. 4.5 and 4.6. Interpretation All 4 investment objects are worthwhile according to the net present value method after removing the reinvestment premise. This means that in each case • A full recovery of the acquisition payment • Interest paid on all outstanding amounts at the reinvestment interest rate iw until the end of the useful life • The terminal value determined at the calculation interest rate ik • A net present value surplus in the amount of the calculated net present value Co* can be achieved. The most advantageous investment cannot be determined, as a relative consideration is only meaningful if the same capital budgets are compared over the same time. (d) Determine the net present value (Co*) for the “Beach Box Online” investment project by removing the reinvestment premise. Assume a reinvestment interest rate of 5 percent in the first year. In the following years, the reinvestment interest rate falls by one percentage point per year. (2 credits). Solution (Textbook Sect. 4.3.2) First, the relevant formula for the solution must be identified. The relevant formula is 4.5. (4.7) = (4.5TB)

4.2 Solutions Part Selection of Alternatives

" Co ¼

n X

43

# ð e k  ak Þ  ð 1 þ i w Þ

nk

þ R  ð1 þ ik Þn  A

ð4:7Þ

k¼1

You can find the solution in the Excel file or in Sect. 4.3, Figs. 4.7 and 4.8. (e) Compare the investment objects “Beach Box Kiel” and “Baltic Diving” in a complete comparison by means of fictitious investments. Compare on the basis of the net present values (Co). (3 credits) Solution (Textbook Sects. 4.4.1–4.4.3) First, the relevant approach must be identified. In principle, a complete comparison of advantages is possible in graphic form (Fig. 4.13 in the textbook) or in the accounted form (Fig. 4.14 in the textbook). The graphic form is much faster in processing. You can find the solution in the Excel file or in Sect. 4.3, Figs. 4.9, 4.10 and 4.11. (f) Compare the investment objects “Beach Box Kiel” and “Kiel Share” in a complete comparison by means of fictitious investments. Compare on the basis of the horizon values, taking into account a reinvestment interest rate of 5 percent (Cn*). Free amounts can be invested at the reinvestment interest rate. (5 credits) Solution (Textbook Sects. 4.4.1–4.4.3) First, the relevant approach must be identified. In principle, a complete comparison of advantages is possible in graphic form (Fig. 4.13 in the textbook) or in the accounted form (Fig. 4.14 in the textbook). The graphic form is much faster in processing. It corresponds to the procedure of the presentation in Fig. 4.18 in the textbook. You can find the solution in the Excel file or in Sect. 4.3, Figs. 4.12 and 4.13. (g) Compare the investment objects “Beach Box Kiel” and “Beach Box Online” in a complete comparison by means of fictitious investments. Compare on the basis of the net present values, taking into account a reinvestment interest rate of 5 percent (Co*). Free amounts can be invested at the reinvestment interest rate. What is your investment decision? (5 credits) Solution (Textbook Sect. 4.4.1–4.4.3) First, the relevant approach must be identified. In principle, a complete comparison of advantages is possible in graphic form (Fig. 4.13 in the textbook) or in the accounted form (Fig. 4.14 in the textbook). The graphic form is much faster in processing. It corresponds to the procedure shown in Fig. 4.18 in the textbook.

44

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

You can find the solution in the Excel file or in Sect. 4.3, Figs. 4.14 and 4.15. (h) Check the results from exercise g) using the technique of the internal rate of return method. What is your investment decision now? (2 credits) Solution (Textbook Sect. 4.3.3) First, the relevant approach must be identified. The relevant formula is 4.11: (4.8) = (4.11TB) rffiffiffiffiffiffiffiffiffiffi  n EW  r ¼ 1 ð4:8Þ A You can find the solution in the Excel file or in Sect. 4.3, Fig. 4.16.

4.3 • • • • • • • • • • • • • •

Excel-Based Solutions Part Selection of Alternatives

Excel solution for determining Co, Cn and DJÜ (solution a–b): Fig. 4.3. Pocket calculator solution to determine Co, Cn and DJÜ (solution a-b): Fig. 4.4. Excel solution to determine Co* (solution c-d): Fig. 4.5. Pocket calculator solution to determine Co* (solution c–d): Fig. 4.6. Excel solution to determine Co* with changing interest rates (solution c–d): Fig. 4.7. Pocket calculator solution to determine Co* with changing interest rates (solution c–d): Fig. 4.8. Accounted form of the fictitious investment with accumulated interest (solution e): Fig. 4.9. Accounted form of the fictitious investment with discounting (solution e): Fig. 4.10. Graphic form of the fictitious investment (solution e): Fig. 4.11. Accounted form of the fictitious investment (solution f–h): Fig. 4.12. Graphic form of the fictitious investment (solution f–h): Fig. 4.13. Accounted form of fictitious investment (solution f–h): Fig. 4.14. Graphic form of the fictitious investment (solution f–h): Fig. 4.15. Calculation of internal interest rates after fictitious investment (solution f–h): Fig. 4.16.

4.3 Excel-Based Solutions Part Selection of Alternatives Exercise a) - b)

Excel Solution

0.10 NE k=1 NE k=2 NE k=3 NE k=4 R k=4 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

k=0 63636.36 61983.47 60105.18 54641.08 54641.08 295007.17 -220000.00 75007.17 0.315471 23662.57

k=1 70000.00

NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 NE k=6 R k=6 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

k=0 54545.46 49586.78 45078.89 40980.81 37255.28 33868.44 33868.44 295184.08 -200000.00 95184.08 0.229607 21854.97

k=1 60000.00

0.10 NE k=1 NE k=2 NE k=3 R k=3 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

k=0 45454.55 41322.31 37565.74 30052.59 154395.19 -100000.00 54395.19 0.402115 21873.11

k=1 50000.00

0.10 NE k=1 NE k=2 NE k=3 NE k=4 R k=4 SUM (BW / EW) minus A Co / Cn KWF / RVF DJÜ

k=0 45454.55 41322.31 37565.74 34150.67 20490.40 178983.68 -80000.00 98983.68 0.315471 31226.46

k=1 50000.00

0.10

k=2

Beach Box Online k=3 k=4

45

k=5

K=6

k=4 (EW) 93170.00 90750.00 88000.00 80000.00 80000.00 431920.00 -322102.00 109818.00 0.215471 23662.57

k=5

k=6

k=6 (EW) 96630.60 87846.00 79860.00 72600.00 66000.00 60000.00 60000.00 522936.60 -354312.20 168624.40 0.129607 21854.97

75000.00 80000.00 80000.00 80000.00

k=2

Beach Box Kiel k=3 k=4

60000.00 60000.00 60000.00 60000.00 60000.00 60000.00

k=2

k=3

Baltic Diving k=4

k=5

50000.00 50000.00 40000.00

k=2

k=3

KielShare k=4

50000.00 50000.00 50000.00 30000.00

Fig. 4.3 Excel solution for determining Co, Cn and DJÜ (Source: Author)

k=3 (EW) 60500.00 55000.00 50000.00 40000.00 205500.00 -133100.00 72400.00 0.302115 21873.11

k=4 (EW) 66550.00 60500.00 55000.00 50000.00 30000.00 262050.00 -117128.00 144922.00 0.215471 31226.46

46

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

Pocket Calculator Solution Alternative: Σ(ek -ak ) x (1 + i) -k + R x Abf - A = Co 70,000.00 * 0.909091 + 75,000 * 0.826446 + 80,000.00 * 0.751315 + 80,000 * 0.683013 + 80,000 * 0.683013 - 220,000 = 75,007.17 Σ(ek -ak ) x (1 + i) n-k + R - A * Auf = Cn 70,000.00 * 1.331 + 75,000.00 * 1.21 + 80,000.00 * 1.1 + 80,000.00 * 1 + 80,000.00 * 1 - 220,000 * 1.4641 = 109,818.00

Note: Differences between the spreadsheet solutions and the calculator solution are due to rounding differences

Alternative: (e-a) x DSF + R x Abf - A = Co 60,000 x 4.355261 + 60,000 x 0.564474 - 200,000 = 95,184.10 (e-a) x EWF + R - A x Auf = Cn 60,000 x 7.715610 + 60,000 - 200,000 x 1.771561 = 168,624.40

Alternative: (e-a) x DSF + R x Abf - A = Co 50,000 x 2.486852 + 40,000 x 0.751315 - 100,000 = 54,395.20 (e-a) x EWF + R - A x Auf = Cn 50,000 x 3.31 + 40,000 - 100,000 x 1.331 = 72,400.00

Alternative: (e-a) x DSF + R x Abf - A = Co 50,000 x 3.169865 + 30,000 x 0.683013 - 80,000 = 98,983.64 (e-a) x EWF + R - A x Auf = Cn 50,000 x 4.641 + 30,000 - 80,000 x 1.4641 = 144,922.00

Fig. 4.4 Pocket calculator solution to determine Co, Cn and DJÜ (Source: Author)

4.3 Excel-Based Solutions Part Selection of Alternatives Exercise c) 0.05 0.10 year (k) 1 2 3 4 4 EW* BW of EW* minus A Co*

Beach Box Online NEk EW* 70000.00 81033.75 75000.00 82687.50 80000.00 84000.00 80000.00 80000.00 80000.00 80000.00 407721.25 278479.10 -220000.00 58479.10

0.05 0.10 year (k) 1 2 3 4 5 6 6 EW* BW of EW* minus A Co*

Beach Box Kiel NEk EW* 60000.00 76576.89 60000.00 72930.38 60000.00 69457.50 60000.00 66150.00 60000.00 63000.00 60000.00 60000.00 60000.00 60000.00 468114.77 264238.58 -200000.00 64238.58

0.05 0.10 year (k) 1 2 3 3 EW* BW of EW* minus A Co*

Baltic Diving NEk EW* 50000.00 55125.00 50000.00 52500.00 50000.00 50000.00 40000.00 40000.00 197625.00 148478.59 -100000.00 48478.59

0.05 0.10 year (k) 1 2 3 4 4 EW* BW of EW* minus A Co* Cn*

KielShare NEk EW* 50000.00 57881.25 50000.00 55125.00 50000.00 52500.00 50000.00 50000.00 30000.00 30000.00 245506.25 167684.07 -80000.00 * (1 + ik)n 87684.07 = 128378.25

Fig. 4.5 Excel solution to determine Co* (Source: Author)

47

48

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

Pocket Calculator Solution Alternative: (Σ(ek -ak ) x (1 + iw) n-k + R) * (1 + ik)-n - A = Co* (70,000.00 * 1.157625 + 75,000.00 * 1.1025 + 80,000.00 * 1.05 + 80,000.00 * 1 + 80,000.00 * 1) * 0.683013 - 220,000 = 58,479.14

Note: Differences between the spreadsheet solutions and the calculator solution are due to rounding differences

Alternative: ((e-a) x EWF(n, iw) + R) * (1 + ik) -n - A = Co* (60,000 x 6.801913 + 60,000) * 0.564474 - 200,000 = 64,238.62

Alternative: ((e-a) x EWF(n, iw) + R) * (1 + ik) -n - A = Co* (50,000 x 3.1525 + 40,000) * 0.751315 - 100,000 = 48,478.63

Alternative: ((e-a) x EWF(n, iw) + R) * (1 + ik) -n - A = Co* (50,000 x 4.310125 + 30,000) * 0.683013 - 80,000 = 87,683.96

Fig. 4.6 Pocket calculator solution to determine Co* (Source: Author)

4.3 Excel-Based Solutions Part Selection of Alternatives

49

Exercise d) 0.05 0.10 year (k) 1 2 3 4 4 EW* BW of EW* minus A Co*

0.04 0.03 Beach Box Online NEk EW* 70000.00 81033.75 75000.00 81120.00 80000.00 82400.00 80000.00 80000.00 80000.00 80000.00 404553.75 276315.65 -220000.00 56315.65

0.02

Fig. 4.7 Excel solution to determine Co* with changing interest rates (Source: Author)

Pocket Calculator Solution Alternative: (Σ(ek -ak ) x (1 + iw) n-k + R) * (1 + ik)-n - A = Co* (70,000.00 * 1.157625 + 75,000.00 * 1.0816 + 80,000.00 * 1.03 + 80,000.00 * 1 + 80,000.00 * 1) * 0.683013 - 220,000 = 56,315.47

Fig. 4.8 Pocket calculator solution to determine Co* with changing interest rates (Source: Author)

ik =

k

0 1 2 3 3 0 1 2 3 3 3 4 5 6 6 3 4 5 6 6 6 0

190526.21

-100000.00

0.1 k=0 -100000.00

-100000.00

Version with accumulating k=3 -121000.00 -133100.00 55000.00 60500.00 50000.00 55000.00 90000.00 72400.00 -110000.00 -121000.00 -133100.00 50000.00 55000.00 60500.00 50000.00 55000.00 90000.00 72400.00 -100000.00 k=1 -110000.00 50000.00

-110000.00 50000.00

79640.00 -110000.00 50000.00

79640.00

k=4

-121000.00 55000.00 50000.00

87604.00 -121000.00 55000.00 50000.00

87604.00

k=5

96364.40 -133100.00 60500.00 55000.00 90000.00 72400.00 -133100.00 60500.00 55000.00 90000.00 72400.00

96364.40

k=6

22.98%

72400.00 337528.80

72400.00

96364.40

96364.40

Σk=6

4

Fig. 4.9 Accounted form of the fictitious investment with accumulated interest (Source: Author)

Baltic Diving Ak=0 NE k = 1 NE k = 2 NE + R k = 3 Cn k = 3 Ak=0 NE k = 1 NE k = 2 NE + R k = 3 Cn k = 3 Ak=3 NE k = 4 NE k = 5 NE + R k = 6 Cn k = 6 Ak=3 NE k = 4 NE k = 5 NE + R k = 6 Cn k = 6 Σ Cn Σ Co Return

50 Mock Exam: Selection of Alternatives and Investment Programme Planning

3 4 5 6

3 4 5 6

6 0 190,526.21

Ak=3 NE k = 4 NE k = 5 NE + R k = 6

Ak=3 NE k = 4 NE k = 5 NE + R k = 6

Σ Cn Σ Co Return

-82,644.63 37,565.74 34,150.67 55,882.92

90,000.00

90,000.00

-90,909.09 -100,000.00 41,322.31 45,454.55 37,565.74 41,322.31 61,471.21 67,618.33

-90,909.09 -100,000.00 41,322.31 45,454.55 37,565.74 41,322.31 61,471.21 67,618.33

50,000.00 81,818.18

50,000.00 81,818.18

Version with discounting k=3

Fig. 4.10 Accounted form of fictitious investment with discounting (Source: Author)

-75,131.48 34,150.67 31,046.07 50,802.65

-82,644.63 37,565.74 34,150.67 55,882.92

50,000.00 45,454.55 74,380.17

0 -100,000.00 1 45,454.55 2 41,322.31 3 67,618.33

Ak=0 NE k = 1 NE k = 2 NE + R k = 3

-75,131.48 34,150.67 31,046.07 50,802.65

50,000.00 45,454.55 74,380.17

k=1

0 1 2 3

k

0.1 k=0 -100,000.00 45,454.55 41,322.31 67,618.33

ik=

Baltic Diving Ak=0 NE k = 1 NE k = 2 NE + R k = 3

50,000.00 45,454.55 74,380.17

50,000.00 45,454.55 74,380.17

k=4

50,000.00 81,818.18

50,000.00 81,818.18

k=5

90,000.00

90,000.00

k=6

22.98%

337,528.80

Σk=6

4.3 Excel-Based Solutions Part Selection of Alternatives 51

52

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

A (euro)

200000

150000

Balc Diving 2

Balc Diving 4

Co = 54,395.19

Co = 54,395.19

100000

Balc Diving 3

Balc Diving 1

50000

Co = 54,395.19 0

Co = 54,395.19 1

2

3

4

5

6 years

Co total Baltic Diving after fictitious investment = 190,526.21 euro. Co Beach Box Kiel = 95,184.08 euro. With this assumption of interest rate Baltic Diving is preferable to Beach Box Kiel after fictitious investment calculation.

Fig. 4.11 Graphic form of the fictitious investment (Source: Author)

iw = ik =

k

0 1 2 3 4 4 1 2 3 4 4 0 6 0

135831.00

40000.00

k=0 -200000.00

0.05 0.1

44100.00

52500.00 50000.00

50000.00

42000.00

k=2 -242000.00 52500.00 50000.00

k=1 -220000.00 50000.00

Fig. 4.12 Accounted form of the fictitious investment (Source: Author)

KielShare A total k = 0 NE k = 1 NE k = 2 NE k = 3 NE + R k = 4 EW k = 4 NE k = 1 NE k = 2 NE k = 3 NE + R k = 4 EW k = 4 Investment AFI Σ Cn* Σ Co* Return*

Exercise f)

46305.00

55125.00 52500.00 50000.00

k=3 -266200.00 55125.00 52500.00 50000.00

k=4 -292820.00 57881.25 55125.00 52500.00 80000.00 245506.25 57881.25 55125.00 52500.00 80000.00 245506.25 48620.25

270670.64 53603.83

270670.64

257781.56

257781.56 51051.26

k=6 -354312.20

k=5 -322102.00

19.92%

270670.64 53603.83 240632.91

270670.64

Σk=6 -354312.20

4.3 Excel-Based Solutions Part Selection of Alternatives 53

54

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

A (euro)

* (1 + ik)6 -354312.2 200000

EW 40.000 (iw) = 53,603.83

53603.83

160000 150000

EW KS 2 = 245,506.25

* (1 + iw)2

EW KS 1 = 245,506.25

* (1 + iw)2

270670.64

100000 80000

50000

0

1

2

3

4

5

Fig. 4.13 Graphic form of the fictitious investment (Source: Author)

6 Σ Cn* =

270670.64

years 240632.91

iw = ik =

k

21000.00

20000.00

59367.56 33738.19

1 2 3 4 6 6 6 0 0

70000.00

k=1 -242000.00 60000.00

k=0 -220000.00

0 1 2 3 4 5 6 6 6

0.05 0.1

Fig. 4.14 Accounted form of fictitious investment (Source: Author)

Beach Box Kiel A total k = 0 NE k = 1 NE k = 2 NE k = 3 NE k = 4 NE k = 5 NE + R k = 6 EW BBK EW 20.000 BBK Beach Box Online NE k = 1 NE k = 2 NE k = 3 NE + R k = 4 EW BBO + FI BBO Σ EW* BBK incl FI Σ EW* BBO incl FI Co* BBK incl FI Co* BBO incl FI return* BBK return* BBO

Exercise g)

73500.00 75000.00

22050.00

k=2 -266200.00 63000.00 60000.00

77175.00 78750.00 80000.00

23152.50

k=3 -292820.00 66150.00 63000.00 60000.00

81033.75 82687.50 84000.00 160000.00 407721.25

24310.13

k=4 -322102.00 69457.50 66150.00 63000.00 60000.00

428107.31

25525.63

k=5 -354312.20 72930.38 69457.50 66150.00 63000.00 60000.00

449512.68

26801.91

k=6 -389743.42 76576.89 72930.38 69457.50 66150.00 63000.00 120000.00

14.47% 12.65%

449512.68 494916.68 449512.68

468114.77 26801.91

Σk=6 -389743.42

4.3 Excel-Based Solutions Part Selection of Alternatives 55

0

50000

80000

100000

160000 150000

200000

220000

BBK

FI BBK

1

Fig. 4.15 Graphic form of the fictitious investment (Source: Author)

279367.21 253738.19 -220000.00 -220000.00 59367.21 33738.19 BBK BBO

B B O

A (euro)

2

* (1 + ik)-n

3

4

EW BBO = 407,721.25

5

6 Σ EW* =

EW BBK = 468,114.77

* (1 + iw)2

EW 20.000 (i w) = 26,801.29

449512.68

0.1

years 494916.06 449512.68 BBK BBO

468114.77

26801.29

ik

FI BBO

iw 0.05

4

Σ BW* = `-A Σ Co* =

Exercise g )

56 Mock Exam: Selection of Alternatives and Investment Programme Planning

4.4 Exercises Part Investment Programme Planning Exercise g), h)

Co* BBK incl FI 59367.56

> >

Co* BBO incl FI 33738.19

return* BBK 14.47%

> >

return* BBO 12.65%

57

BBK is better than BBO, both are worthwhile

Fig. 4.16 Calculation of internal interest rates after fictitious investment (Source: Author)

4.4

Exercises Part Investment Programme Planning

Time: 60 Min, Credits: 30 In your newly founded consulting company “Kiel Consult”, you analyse an investment project which is initially planned with the following possible investment and financing options: • The company has equity of 50,000 €. • Investment 1 (I1) has an acquisition payment of 200,000 € in k ¼ 0 and annual net payments of 60,000 € for 5 years. The return on investment 1 is 15.25%. • Investment 2 (I2) has an acquisition payment of 100,000 € in k ¼ 0 and annual net payments of 40,000 € for 5 years. The return on investment 2 is 28.75%. • Investment 3 (I3) has an acquisition payment of 20,000 € at the end of k ¼ 3 and following to k ¼ 5 annual net payments of 12,000 € per year. The return on investment 3 is 13.10%. • Investment 4 (I4) has an acquisition payment of 120,000 € in k ¼ 0 and a residual value in k ¼ 5 of 160,000 €. Net payments do not exist. The return on investment 4 is 5.92%. • Free funds can be reinvested unlimitedly at 5% per annum (I50–I54). • Financing 1 (F1) is a loan of 200,000 € with an effective interest rate of 6% available in k ¼ 0. • Financing 2 (F2) is a loan of 170,000 € with an effective interest rate of 7% available in k ¼ 0. • Financing 3 (F3) is a fixed loan which is paid at the end of k ¼ 2 in the amount of 50,000 € and is repaid at the end of k ¼ 5 in the sum of 75,000 €. The return on financing 3 is 14.47%. • Financing 4 (F4) is a 1-year loan of 100,000 €, which must be repaid after the year, including 9% interest. It is available in every year. • Financing options F1 and F2 will bear interest annually in arrears and may be repaid annually at a variable rate, but interest must be paid in each period. • I1, I2, I3, F1 and F2, as far as they are taken into account in the following exercises, can only be performed once. Investment I4, as far as it is taken into account in the following exercises, is to be carried out at least once. Financing F3 must be carried out exactly once. Investment I5 and financing F4 can be divided as desired.

58

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

(a) Draw up an account development plan. Consider investments I1 to I4 and, if necessary, I5. Assume that investments I1 to I4 are each carried out exactly once. To finance them, use financing F1, F2, F3 and your equity. Financings F1 and F2 can be repaid at variable rates; interest must be paid annually in arrears on the tied-up capital of the financing. The entitlements of the equity investor are paid in residual terms after the implementation of the programme. The interest claim for the equity is 4%. The relatively more expensive repayable loan is repaid first. (4 credits). (b) Draw up an account development plan. Consider investments I1 to I4 and, if necessary, I5. Assume that investments I1 to I4 are each carried out exactly once. To finance them, use financing F1, F2, F3 and your equity. Financings F1 and F2 can be repaid at variable rates; interest must be paid annually in arrears on the tied-up capital of the financing. List the value development of the equity over time explicitly with the annual interest. The interest claim for the equity is 4%. The relatively more expensive repayable loan is repaid first. (7 credits) (c) Calculate the terminal value, the horizon value and the achievable return on equity from the results of the account development plan. (3 credits) (d) Use the Dean Model to determine the optimum investment and financing programme. Use the investments I1–I5 and the financings F1–F4. Assume that investments I1–I4 and financings F1–F3 are each considered exactly once. Please take investment I5 and financing F4 in the amount of 100,000 euro at the time k ¼ 0 into account. For this exercise, assume that all investments and financings are divisible at will. (3 credits). After the account development plan has been drawn up, further discussions with the creditors of the financing show that loans F1 and F2 cannot be repaid variably but only at the end of the term as a fixed loan. Interest is still payable annually in arrears. In order to develop the new optimum investment and financing programme, carry out a linear optimisation with the planning data. (e) Set up the LP initial matrix. (5 credits) The scope line of the optimum solution determined contains the values that can be seen in Fig. 4.17:

I1

I2

I3

I4

I50

I51

I52

I53

I54

F1

1.00 1.00 1.00 1.00 0.00 0.00 0.32 1.02 2.07 1.00

F2

F3

F40

0.00 1.00 1.70

Fig. 4.17 Scope line of the optimum LP solution (Source: Author)

F41

F42

0.97 0.00

F43

F44

EK

0.00 0.00 1.00

4.5 Solutions Part Investment Programme Planning

59

In this solution, all activities are formulated on the basis of 1000 € units in the initial matrix. On the objective of maximising the terminal value, you have determined a target value of 201,948.02 euro for the entire investment programme. The dual variables for the liquidity restriction of the individual period (k) have the following values in Fig. 4.18 in the optimum solution: Period (k) Dual variable (d)

0 1.375

1 1.262

2 1.158

3 1.103

4 1.05

5 1

Fig. 4.18 Dual variables for the liquidity restriction of the individual periods (k) (Source: Author)

(f) Determine the return on equity. Compare this result with the return on equity from exercise c) and explain why the two results differ. (2 credits) (g) You are considering whether you would like to make an additional 10,000 € in year 3 available for the project from your private assets. What impact would this have on the terminal value? What would be the return on investment of this additional 10,000 € from year 3 onward? (2 credits) (h) Calculate the liquidity of the optimum investment programme of year 4. (2 credits) (i) What is the increase of the terminal value resulting from the additional implementation of a further investment I2? (2 credits)

4.5

Solutions Part Investment Programme Planning

In your newly founded consulting company “Kiel Consult”, you analyse an investment project which is initially planned with the following possible investment and financing options: • The company has equity of 50,000 €. • Investment 1 (I1) has an acquisition payment of 200,000 € in k ¼ 0 and annual net payments of 60,000 € for 5 years. The return on investment 1 is 15.25%. • Investment 2 (I2) has an acquisition payment of 100,000 € in k ¼ 0 and annual net payments of 40,000 € for 5 years. The return on investment 2 is 28.75%. • Investment 3 (I3) has an acquisition payment of 20,000 € at the end of k ¼ 3 and following to k ¼ 5 annual net payments of 12,000 € per year. The return on investment 3 is 13.10%. • Investment 4 (I4) has an acquisition payment of 120,000 € in k ¼ 0 and a residual value in k ¼ 5 of 160,000 €. Net payments do not exist. The return on investment 4 is 5.92%. • Free funds can be reinvested unlimitedly at 5% per annum (I50–I54).

60

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

• Financing 1 (F1) is a loan of 200,000 € with an effective interest rate of 6% available in k ¼ 0. • Financing 2 (F2) is a loan of 170,000 € with an effective interest rate of 7% available in k ¼ 0. • Financing 3 (F3) is a fixed loan which is paid at the end of k ¼ 2 in the amount of 50,000 € and is repaid at the end of k ¼ 5 in the sum of 75,000 €. The return on financing 3 is 14.47%. • Financing 4 (F4) is a 1-year loan of 100,000 €, which must be repaid after the year, including 9% interest. It is available in every year. • Financing options F1 and F2 will bear interest annually in arrears and may be repaid annually at a variable rate, but interest must be paid in each period. • I1, I2, I3, F1 and F2, as far as they are taken into account in the following exercises, can only be performed once. Investment I4, as far as it is taken into account in the following exercises, is to be carried out at least once. Financing F3 must be carried out exactly once. Investment I5 and financing F4 can be divided as desired. (a) Draw up an account development plan. Consider investments I1 to I4 and, if necessary, I5. Assume that investments I1 to I4 are each carried out exactly once. To finance them, use financing F1, F2, F3 and your equity. Financings F1 and F2 can be repaid at variable rates; interest must be paid annually in arrears on the tied-up capital of the financing. The entitlements of the equity investor are paid in residual terms after the implementation of the programme. The interest claim for the equity is 4%. The relatively more expensive repayable loan is repaid first. (4 credits). Solution (Textbook Sect. 4.7.2) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.21. (b) Draw up an account development plan. Consider investments I1 to I4 and, if necessary, I5. Assume that investments I1 to I4 are each carried out exactly once. To finance them, use financing F1, F2, F3 and your equity. Financings F1 and F2 can be repaid at variable rates; interest must be paid annually in arrears on the tied-up capital of the financing. List the value development of the equity over time explicitly with the annual interest. The interest claim for the equity is 4%. The relatively more expensive repayable loan is repaid first. (7 credits) Solution (Textbook Sect. 4.7.2) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.22. (c) Calculate the terminal value, the horizon value and the achievable return on equity from the results of the account development plan. (3 credits)

4.5 Solutions Part Investment Programme Planning

61

Solution (Textbook Sects. 4.3.3 and 4.7.2) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.23. (d) Use the Dean Model to determine the optimum investment and financing programme. Use the investments I1–I5 and the financings F1–F4. Assume that investments I1–I4 and financings F1–F3 are each considered exactly once. Please take investment I5 and financing F4 in the amount of 100,000 euro at the time k ¼ 0 into account. For this exercise, assume that all investments and financings are divisible at will. (3 credits) Solution (Textbook Sect. 4.8.1) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.24. After the account development plan has been drawn up, further discussions with the creditors of the financing show that loans F1 and F2 cannot be repaid variably but only at the end of the term as a fixed loan. Interest must still be paid annually in arrears. In order to develop the new optimum investment and financing programme, carry out a linear optimisation with the planning data. (e) Set up the LP initial matrix. (5 credits). Solution (Textbook Sect. 4.9.1) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.25. The scope line of the optimum solution determined contains the values that can be seen in Fig. 4.19:

I1

I2

I3

1.00 1.00 1.00

I4

I50

1.00 0.00

I51

I52

0.00 0.32

I53

I54

F1

1.02

2.07

1.00 0.00

F2

F3

F40

F41

F42

F43

1.00

1.70

0.97

0.00

0.00 0.00

F44

EK 1.00

Fig. 4.19 Scope line of the optimum LP solution (Source: Author)

In this solution, all activities are formulated on the basis of 1000 € units in the initial matrix. On the objective of maximising the terminal value, you have determined a target value of 201,948.02 euro for the entire investment programme. The dual variables for the liquidity restriction of the individual period (k) have the following values in Fig. 4.20 in the optimum solution: Period (k) Dual variable (d)

0 1.375

1 1.262

2 1.158

3 1.103

4 1.05

5 1

Fig. 4.20 Dual variables for the liquidity restriction of the individual periods (k) (Source: Author)

62

4

Mock Exam: Selection of Alternatives and Investment Programme Planning

(f) Determine the return on equity. Compare this result with the return on equity from exercise c) and explain why the two results differ. (2 credits) Solution (Textbook Sect. 4.9.2.3) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.30. (g) You are considering whether you would like to make an additional 10,000 € in year 3 available for the project from your private assets. What impact would this have on the terminal value? What would be the return on investment of this additional 10,000 € from year 3 onward? (2 credits) Solution (Textbook 4.9.2.3) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.31. (h) Calculate the liquidity of the optimum investment programme of year 4. (2 credits) Solution (Textbook Sect. 4.9.2.3) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.32. (i) What is the increase of the terminal value resulting from the additional implementation of a further investment I2? (2 credits) Solution (Textbook Sect. 4.9.2.3) You can find the solution in the Excel file or in Sect. 4.6, Fig. 4.33.

4.6 • • • • • • • • • • • • •

Excel-Based Solutions Part Investment Programme Planning

Account development plan with deduction of equity: Fig. 4.21. Account development plan with debt and equity account: Fig. 4.22. Dynamic results of the account development plan: Fig. 4.23. Dean Model of the investment programme: Fig. 4.24. LP initial matrix for the investment programme: Fig. 4.25. LP solution matrix for the investment programme: Fig. 4.26. Answer report of the LP solution matrix for the investment programme: Fig. 4.27. Limits report of the LP solution matrix for the investment programme: Fig. 4.28. Sensitivity report of the LP solution matrix for the investment programme: Fig. 4.29. Determination of the return on equity from the LP solution matrix: Fig. 4.30. Effect of a change in capital on the LP solution matrix: Fig. 4.31. Determining the liquidity from the LP solution matrix: Fig. 4.32. Effect of an additional investment execution on the LP solution matrix: Fig. 4.33.

4.6 Excel-Based Solutions Part Investment Programme Planning

63

Exercise a) simple form, T euro k

I1 0 1 2 3 4 5

I2 -200 60 60 60 60 60

I3

I4

-100 40 40 40 40 40

FK3

Ne

-120 50 -20 12 12

160

FK1 Interest

FK1 -420 100 150 80 112 272

200 200.00 162.47 92.22 0.00 0.00

FK1 Repayment FK2

12.00 12.00 9.75 5.53 0.00

0.00 37.53 70.25 92.22 0.00

170 93.90 0.00 0.00 0.00 0.00

FK2 Interest

FK2 Repayment FK3

11.90 6.57 0.00 0.00 0.00

FK3 Interest

0 0.00 50.00 50.00 50.00 0.00

76.10 93.90 0.00 0.00 0.00

FK3 Repayment I5

0.00 0.00 0.00 0.00 25.00

Interest I5 0 0.00 0.00 0.00 14.25 272.00

0.00 0.00 0.00 0.00 50.00

0.00 0.00 0.00 0.00 0.71

Balance -370 -293.90 -212.47 -142.22 -35.75 211.96

Fig. 4.21 Account development plan with deduction of equity (Source: Author)

Exercise b) Form with EK, T euro k

I1 0 1 2 3 4 5

I2 -200 60 60 60 60 60

I3 -100 40 40 40 40 40

I4

FK3

Ne

-120 50 -20 12 12

160

FK1 -420 100 150 80 112 272

FK1 FK1 Interest Repayment FK2

200 200.00 162.47 92.22 0.00 0.00

12.00 12.00 9.75 5.53 0.00

0.00 37.53 70.25 92.22 0.00

170 93.90 0.00 0.00 0.00 0.00

FK2 FK2 Interest Repayment FK3 11.90 6.57 0.00 0.00 0.00

76.10 93.90 0.00 0.00 0.00

0 0.00 50.00 50.00 50.00 50.00

FK3 FK3 Interest Repayment EK 0.00 0.00 0.00 0.00 25.00

0.00 0.00 0.00 0.00 0.00

EK Interest I5

50 52.00 54.08 56.24 58.49 60.83

2.00 2.08 2.16 2.25 2.34

0 0.00 0.00 0.00 14.25 272.00

Fig. 4.22 Account development plan with debt and equity account (Source: Author)

Exercise c)

EW Cn EK return

211.95760 151.12496 33.49%

Fig. 4.23 Dynamic results of the account development plan (Source: Author)

Interest I5 Balance -420 0.00 -345.90 0.00 -266.55 0.00 -198.46 0.00 -94.25 0.71 151.12

64

4

r, i eff, %

29

Mock Exam: Selection of Alternatives and Investment Programme Planning

I2, (28.75%, 100T€)

28 27 26 25 24 23 22 21 20 19 18 17 16

I1, (15.25%, 200T€) F3, (14.47%, 50T€)

15 14

I3, (13.10%, 20T€)

13 12 11 10

F4, (9% 100T€)

9 8

F2, (7%; 170T€)

7

F1, (6%, 200T€)

6

I4, (5.92%, 120T€) I5, (5%, 100T€)

5 4 3 2 1 50

100

150

200

250

300 3 20 350 370 400

440

470 500 520 540

Capital budget (T€)

cut off rate, 320 T€

According to the Dean model, investments I2, I1 and I3 are carried out and financed with the F1 and 120 T€ from F2. The optimum investment volume is 320 T€.

Fig. 4.24 Dean Model of the investment programme (Source: Author) Initial matrix Exercise e) k Restriction I1 I2 I3 200 100 0 LR0

I4

I50 120

1 LR1

-60

-40

2 LR2

-60

-40

3 LR3

-60

-40

20

4 LR4

-60

-40

-12

5 LR5 HSR I1 HSR I2 HSR I3 MSR I4 HSR F1 HSR F2 Constraint F3

-60 1

-40

-12

I51

I52

I53

I54

F1

100 -105

100 -105

100 -105

100 -105

-160

F2

F3

F40

F41

-200

-170

-100

12

11.9

109

12

11.9

12

11.9

100

12

11.9

-105

212

181.9

-50

F42

F43

F44

Z

RHS ≤

-100 109

-100 109

-100

50

≤

0

≤

0

≤

0

109 -100

≤

0

109

1≤ ≤ ≤ ≤ ≥ ≤ ≤ =

0 1 1 1 1 1 1 1

75

1 1 1 1 1 1

Fig. 4.25 LP initial matrix for the investment programme (Source: Author) Solution matrix I1 Scope Liquidity d 0 1 2 3 4 5

I2 1.00

1.375 1.262 1.158 1.103 1.050 1.000

I3 1.00

I4 1.00

I50 1.00

I1 I2 I3 I4 I50 200.00 100.00 0.00 120.00 -60.00 -40.00 0.00 0.00 -60.00 -40.00 0.00 0.00 -60.00 -40.00 20.00 0.00 -60.00 -40.00 -12.00 0.00 -60.00 -40.00 -12.00 -160.00

0.00

I51 I52 I53 I54 F1 F2 F3 F40 F41 F42 0.00 0.32 1.02 2.07 1.00 0.00 1.00 1.70 0.97

0.00 0.00 0.00 0.00 0.00 0.00

I51 I52 I53 I54 F1 F2 F3 F40 F41 F42 0.00 0.00 0.00 0.00 -200.00 0.00 0.00 -170.00 0.00 0.00 0.00 0.00 0.00 12.00 0.00 0.00 185.30 -97.30 0.00 31.94 0.00 0.00 12.00 0.00 -50.00 0.00 106.06 0.00 -33.54 101.54 0.00 12.00 0.00 0.00 0.00 0.00 0.00 0.00 -106.62 206.62 12.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -216.95 212.00 0.00 75.00 0.00 0.00

F43

F44 0.00 0.00 0.00

EK

F43

F44 0.00 0.00 0.00 0.00 0.00 0.00

EK

0.00

0.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 0.00

Fig. 4.26 LP solution matrix for the investment programme (Source: Author)

1.00 balance -50.00 0.00 0.00 0.00 0.00 0.00

0.00 0.00 0.00 0.00 0.00 -201.948

4.6 Excel-Based Solutions Part Investment Programme Planning

65

Microsoft Excel 15.0 Answer Report Target Cell (min.) Cell Name Original Value Final Value $W$28 balance -201.94802 -201.94802

Adjustable Cell Cell Name $C$20 Scope I1 $D$20 Scope I2 $E$20 Scope I3 $F$20 Scope I4 $G$20 Scope I50 $H$20 Scope I51 $I$20 Scope I52 $J$20 Scope I53 $K$20 Scope I54 $L$20 Scope F1 $M$20 Scope F2 $N$20 Scope F3 $O$20 Scope F40 $P$20 Scope F41 $Q$20 Scope F42 $R$20 Scope F43 $S$20 Scope F44 $T$20 Scope $V$20 EK

Constraints Cell Name $W$23 balance $W$24 balance $W$25 balance $W$26 balance $W$27 balance $W$28 balance $C$20 Scope I1 $C$20 Scope I1 $D$20 Scope I2 $E$20 Scope I3 $F$20 Scope I4 $G$20 Scope I50 $H$20 Scope I51 $I$20 Scope I52 $J$20 Scope I53 $K$20 Scope I54 $L$20 Scope F1 $M$20 Scope F2 $N$20 Scope F3 $O$20 Scope F40 $P$20 Scope F41 $Q$20 Scope F42 $R$20 Scope F43 $S$20 Scope F44 $T$20 Scope $D$20 Scope I2 $E$20 Scope I3 $F$20 Scope I4 $L$20 Scope F1 $M$20 Scope F2 $N$20 Scope F3 $V$20 EK $V$20 EK

Original Value Final Value 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 0.00 0.00 0.00 0.00 0.32 0.32 1.02 1.02 2.07 2.07 1.00 1.00 0.00 0.00 1.00 1.00 1.70 1.70 0.97 0.97 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 1.00 1.00

Cell Value

Formula 0.00 $W$23=0 1.02 $J$20>=0 2.07 $K$20>=0 1.00 $L$20>=0 0.00 $M$20>=0 1.00 $N$20>=0 1.70 $O$20>=0 0.97 $P$20>=0 0.00 $Q$20>=0 0.00 $R$20>=0 0.00 $S$20>=0 0.00 $T$20>=0 1.00 $D$20 130653.39

Cn BBK 110468.25

0

1

2

3

4

CnKSn=4 = 70,999.00

5

6

7

8

70999

years

Fig. 7.19 Graphic form of the fictitious investment (source: author) Exercise e) 5.00% 10.00% NE k=1 NE k=2 NE k=3 NE k=4 NE k=5 NE k=6 NE k=7 NE k=8 R k=8 SUM (BW* / EW*) minus A Co* / Cn*

k=0

k=1 25,000

k=2

k=3

Beach Box Kiel k=4 k=5

k=6

k=7

k=8

35,000 45,000 55,000 65,000 45,000 45,000 35,000 50,000 216,214.27 -200,000.00 16,214.27

k=8 (EW) 35177.51 46903.35 57432.67 66852.84 75245.63 49612.50 47250.00 35000.00 50000.00 463474.50 -428717.76 34756.74

Co* of Beach Box Kiel is 16,214.27 euro.

Fig. 7.20 Excel solution to determine Co* of Beach Box Kiel (source: author)

5.00% 10.00% NE k=1 NE k=2 NE k=3 NE k=4 R k=4 S UM (BW * / EW *) minus A Co* / Cn*

k=0

k=1 55, 000

KielShare k=2 k=3 55,000 55,000

196, 063. 71 -160,000. 00 36, 063. 71

Co* of KielShare is 36,063.71 euro.

Fig. 7.21 Excel solution to determine Co* of Kiel Share (source: author)

k=4

k=4 (EW) 63, 669. 38 60, 637. 50 57, 750. 00 55,000 55, 000. 00 50,000 50, 000. 00 287, 056. 88 -234, 256.00 52, 800. 88

7.3 Excel-Based Solutions Mock Exam

7.3.2

133

Excel-Based Solutions Exercise 2

• Determination of the optimum useful life of a one-time investment (solution a): Fig. 7.22. • Determination of the optimum useful life for an infinitely repeated investment (solution b): Fig. 7.23. • Determination of the optimum useful life of the new object (solution c): Fig. 7.24. • Determination of the optimum replacement time of the old object, possibility of replacement after a multi-year period (solution c): Fig. 7.25. Exercise a) Maximising net present value (Co): 10.00% k 0 1 2 3 4 5 6 7 8

ek 30,000.00 30,000.00 25,000.00 25,000.00 20,000.00 20,000.00 15,000.00 15,000.00

Result:

ak 3,000.00 3,500.00 4,000.00 4,500.00 5,000.00 5,500.00 6,000.00 6,500.00

NEk BW of NEk ∑ BW of NEk 27,000.00 24,545.45 24,545.45 26,500.00 21,900.83 46,446.28 21,000.00 15,777.61 62,223.89 20,500.00 14,001.78 76,225.67 15,000.00 9,313.82 85,539.49 14,500.00 8,184.87 93,724.36 9,000.00 4,618.42 98,342.78 8,500.00 3,965.31 102,308.09

Rk 100,000.00 60,000.00 50,000.00 45,000.00 40,000.00 35,000.00 25,000.00 15,000.00 5,000.00

BW of Rk 54,545.45 41,322.31 33,809.17 27,320.54 21,732.25 14,111.85 7,697.37 2,332.54

BW Co 79,090.90 -20,909.10 87,768.59 -12,231.41 96,033.06 -3,966.94 103,546.21 3,546.21 107,271.74 7,271.74 107,836.21 7,836.21 => Co = max! 106,040.15 6,040.15 104,640.63 4,640.63

The optimum useful life of a one-time investment is 6 years, since the net present value of € 7,836.21 is the maximum.

Fig. 7.22 Determination of the optimum useful life of a one-time investment (source: author)

Exercise b) Maximising average annual surplus (DJÜ) 10.00% k 0 1 2 3 4 5 6 7 8

ek 30,000.00 30,000.00 25,000.00 25,000.00 20,000.00 20,000.00 15,000.00 15,000.00

ak 3,000.00 3,500.00 4,000.00 4,500.00 5,000.00 5,500.00 6,000.00 6,500.00

NEk BW of NEk ∑ BW of NEk 27,000.00 24,545.45 24,545.45 26,500.00 21,900.83 46,446.28 21,000.00 15,777.61 62,223.89 20,500.00 14,001.78 76,225.67 15,000.00 9,313.82 85,539.49 14,500.00 8,184.87 93,724.36 9,000.00 4,618.42 98,342.78 8,500.00 3,965.31 102,308.09

Rk 100,000.00 60,000.00 50,000.00 45,000.00 40,000.00 35,000.00 25,000.00 15,000.00 5,000.00

BW of R k 54,545.45 41,322.31 33,809.17 27,320.54 21,732.25 14,111.85 7,697.37 2,332.54

BW 79,090.90 87,768.59 96,033.06 103,546.21 107,271.74 107,836.21 106,040.15 104,640.63

Co -20,909.10 -12,231.41 -3,966.94 3,546.21 7,271.74 7,836.21 6,040.15 4,640.63

KWF 1.100000 0.576190 0.402115 0.315471 0.263797 0.229607 0.205405 0.187444

DJÜ -23,000.01 -7,047.62 -1,595.17 1,118.73 1,918.27 => DJÜ = max! 1,799.25 1,240.68 869.86

Result: The optimum useful life for an infinitely repeated investment is 5 years, as the annuity of € 1,918.27 is the maximum. Total net present value DJÜ/i =

19182.7 euro

Fig. 7.23 Determination of the optimum useful life for an infinitely repeated investment (source: author)

134

7

Exercise c) 10.00% k 0 1 2 3 4

Mock Exam Investment Calculation

Baltic Diving, new object ek 65,000.00 60,000.00 55,000.00 50,000.00

ak 10,000.00 15,000.00 20,000.00 25,000.00

NEk BW of NEk ∑ BW of NEk 55,000.00 50,000.00 50,000.00 45,000.00 37,190.08 87,190.08 35,000.00 26,296.02 113,486.10 25,000.00 17,075.34 130,561.44

Rk 120,000.00 70,000.00 60,000.00 50,000.00 45,000.00

BW of R k 63,636.36 49,586.78 37,565.74 30,735.61

BW 113,636.36 136,776.86 151,051.84 161,297.05

Co -6,363.64 16,776.86 31,051.84 41,297.05

KWF 1.100000 0.576190 0.402115 0.315471

DJÜ -7,000.00 9,666.67 12,486.40 13,028.01

=> DJÜ new object

Fig. 7.24 Determination of the optimum useful life of the new object (source: author)

10.00% k 0 1 2 3 4

Result:

Beach Box Online, old object ak NEk BW of NEk ∑ BW of NEk ek 20,000.00 5,000.00 15,000.00 13,636.36 13,636.36 20,000.00 5,500.00 14,500.00 11,983.47 25,619.83 15,000.00 6,000.00 9,000.00 6,761.83 32,381.66 15,000.00 6,500.00 8,500.00 5,805.61 38,187.27

Rk 40,000.00 35,000.00 25,000.00 15,000.00 5,000.00

BW of R k 31,818.18 20,661.16 11,269.72 3,415.07

KWF 1.100000 0.576190 0.402115 0.315471

RVF 1.000000 0.476190 0.302115 0.215471

GÜ 6,000.00 3,619.04 1,468.27 505.49

=> GÜ (= DJÜ) old

The new project "Baltic Diving" should be realised, the current project "Beach Box Online" should be terminated, as the formal condition GÜold < DJÜnew is fulfilled.

Fig. 7.25 Determination of the optimum replacement time of the old object, possibility of replacement after a multi-year period (source: author)

7.3.3

Excel-Based Solutions Exercise 3

• Account development plan with deduction of equity (solution a–b): Fig. 7.26. • Calculation of terminal value, horizon value and return on equity from the results of the account development plan (solution a–b): Fig. 7.27. • LP initial matrix of the investment programme (solution c–f): Fig. 7.28. • LP solution matrix of the investment programme (solution c–f): Fig. 7.29. • Response report of the LP solution matrix of the investment programme: Fig. 7.30. • Limits report of the LP solution matrix of the investment programme: Fig. 7.31. • Sensitivity report of the LP solution matrix of the investment programme: Fig. 7.32. • Determining the return on equity from the LP solution matrix (solution c–f): Fig. 7.33. • Effect of a change in capital on the LP solution matrix (solution c–f): Fig. 7.34. • Determining the liquidity from the LP solution matrix (solution c–f): Fig. 7.35.

7.3 Excel-Based Solutions Mock Exam Simple form, T euro Exercise a) k I1 I2 I3 I4 FK3 Ne 0 -400 -200 -200 1 120 70 2 120 70 -40 100 3 120 70 20 4 120 70 20 5 120 70 20 300

FK1 -800 190 250 210 210 510

135

FK1 Interest FK1 Repayment FK2

400 400. 00 339.85 150. 24 0. 00 0.00

24. 00 24.00 20. 39 9.01 0.00

0.00 60.15 189. 61 150. 24 0.00

FK2 Interest FK2 Repayment FK3

300 155.00 0.00 0.00 0.00 0.00

0 145. 00 0. 00 155.00 100.00 0.00 100. 00 0.00 100. 00 0.00 0.00

21.00 10.85 0.00 0.00 0.00

FK3 Interest FK3 Repayment Investment Investment interest Balance 0 -700 0. 00 0. 00 0. 00 0.00 -555. 00 0.00 0.00 0.00 0.00 -439.85 0. 00 0. 00 0. 00 0.00 -250. 24 0. 00 0. 00 50. 74 0.00 -100. 00 40.00 100.00 370.00 2.54 423.28

Fig. 7.26 Account development plan with deduction of equity (source: author)

EW Cn

423.28177 301.61648

EK return

33.45%

The return is lower than for exercise d) because more worthwhile investments are made there (I4), the choice of financing is optimised there and is not predetermined (F4 is taken into account) and the form of repayment of F1 and F2 is different.

Fig. 7.27 Calculation of terminal value, horizon value and return on equity from the results of the account development plan (source: author)

initial matrix k Restriction I1 I2 0 400 -120 1 LR1

I3

I4

200 -70

I50 200

2 LR2

-120

-70

40

3 LR3

-120

-70

-20

4 LR4

-120

-70

-20

5 LR5 -120 HSR I1 1 HSR I2 HSR I3 MSR I4 HSR F1 HSR F2 Constraint F3

-70

-20

I51 100 -105

I52

I53

I54

-105

F1

F2 -400 24

100 100

24

21

24

21

-105 100

24

21

-105

424

321

-105 100 -300

F3

F40

-300 21

F41 -100 109

-100

F42

F43

F44

Z ≤ ≤

-100 109

-100 109

-100

RHS 100 0

≤

0

≤

0

109 -100

≤

0

109

1≤ ≤ ≤ ≤ ≥ ≤ ≤ =

0 1 1 1 1 1 1 1

140

1 1 1 1 1 1

Fig. 7.28 LP initial matrix of the investment programme (source: author)

solution matrix I1 Scope Liquidity k d 0 1 2 3 4 5

I2 1.00

1.50 1.38 1.26 1.16 1.06 1.00

I3 1.00

I4 1.00

I50 3.75

I1 I2 I3 I4 I50 400.00 200.00 0.00 749.87 -120.00 -70.00 0.00 0.00 -120.00 -70.00 40.00 0.00 -120.00 -70.00 -20.00 0.00 -120.00 -70.00 -20.00 0.00 -120.00 -70.00 -20.00 -1124.81

0.00

I51 I52 I53 I54 F1 F2 F3 F40 F41 F42 0.00 0.00 0.00 0.00 1.00 1.00 1.00 5.50 4.54

0.00 0.00 0.00 0.00 0.00 0.00

I51 I52 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

I53 0.00 0.00 0.00 0.00 0.00 0.00

F43 2.90

F44 1.51 0.00 0.00

I54 F1 F2 F3 F40 F41 F42 F43 F44 0.00 -400.00 -300.00 0.00 -549.87 0.00 0.00 0.00 0.00 0.00 24.00 21.00 0.00 599.36 -454.36 0.00 0.00 0.00 0.00 24.00 21.00 -100.00 0.00 495.25 -290.25 0.00 0.00 0.00 24.00 21.00 0.00 0.00 0.00 316.38 -151.38 0.00 0.00 24.00 21.00 0.00 0.00 0.00 0.00 165.00 0.00 0.00 424.00 321.00 140.00 0.00 0.00 0.00 0.00 0.00

Fig. 7.29 LP solution matrix of the investment programme (source: author)

EK 1.00 EK Balance -100.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 -449.80853

136

7 Microsoft Excel 15.0 Answer Report Target Cell (min.) Cell Name Original Value

Adjustable Cell Cell Name $B$20 s c ope I1 $C$20 s c ope I2 $D$20 s c ope I3 $E$20 s c ope I4 $F$20 scope I50 $G$20 scope I51 $H$20 scope I52 $I$20 scope I53 $J$20 scope I54 $K$20 s c ope F1 $L$20 s c ope F2 $M$20 s c ope F3 $N$20 scope F40 $O$20 scope F41 $P$20 scope F42 $Q$20 scope F43 $R$20 scope F44 $S$20 s c ope $U$20 E K

Constraints Cell Name $V$23 balanc e $V$24 balanc e $V$25 balanc e $V$26 balanc e $V$27 balanc e $V$28 balance $B$20 s c ope I1 $B$20 s c ope I1 $C$20 s c ope I2 $D$20 s c ope I3 $E$20 s c ope I4 $F$20 s c ope I50 $G$20 s c ope I51 $H$20 s c ope I52 $I$20 s c ope I53 $J$20 s c ope I54 $K$20 s c ope F1 $L$20 s c ope F2 $M$20 s c ope F3 $N$20 scope F40 $O$20 scope F41 $P$20 scope F42 $Q$20 scope F43 $R$20 s c ope F44 $S$20 s c ope $C$20 s c ope I2 $D$20 s c ope I3 $E$20 s c ope I4 $K$20 s c ope F1 $L$20 s c ope F2 $M$20 s c ope F3 $U$20 E K $U$20 E K

Mock Exam Investment Calculation

Final Value

Original Value Final Value Integer 1.00 1.00 Contin 1.00 1.00 Contin 1.00 1.00 Contin 3.75 3.75 Contin 0.00 0.00 Contin 0.00 0.00 Contin 0.00 0.00 Contin 0.00 0.00 Contin 0.00 0.00 Contin 1.00 1.00 Contin 1.00 1.00 Contin 1.00 1.00 Contin 5.50 5.50 Contin 4.54 4.54 Contin 2.90 2.90 Contin 1.51 1.51 Contin 0.00 0.00 Contin 0.00 0.00 Contin 1.00 1.00 Contin

Cell Value 0.00 0.00 0.00 0.00 0.00 -449.81 1.00 1.00 1.00 1.00 3.75 0.00 0.00 0.00 0.00 0.00 1.00 1.00 1.00 5.50 4.54 2.90 1.51 0.00 0.00 1.00 1.00 3.75 1.00 1.00 1.00 1.00 1.00

Formula $V$23= 0 $I$20> =0 $J$20>=0 $K$20>= 0 $L$20> =0 $M$20>= 0 $N$20>=0 $O$20>=0 $P$20>=0 $Q$20>=0 $R$20>= 0 $S$20>= 0 $C$20