Finance

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Finance

By providing a solid theoretical basis, this book introduces modern finance to readers, including students in science and technology, who already have a good foundation in quantitative skills. It combines the classical, decision-oriented approach and the traditional organization of corpo-rate finance books with a quantitative approach that is particularly well suited to students with backgrounds in engineering and the natural sciences. This combination makes finance much more transparent and accessible than the definition-theorem-proof pattern that is common in mathemat-ics and financial econommathemat-ics. The book’s main emphasis is on investments in real assets and the real options attached to them, but it also includes extensive discussion of topics such as portfolio theory, market efficiency, capital structure and derivatives pricing. Finance: A Quantitative

Intro-ductionequips readers as future managers with the financial literacy necessary either to evaluate investment projects themselves or to engage critically with the analysis of financial managers.

A range of supplementary teaching and learning materials are available online at www. cambridge.org/wijst.

NICO VAN DER WIJSTis Professor of Finance at the Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology in Trondheim, where he has been teaching since 1997. He has published a book on financial structure in small business and a number of journal articles on different topics in finance.

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Finance

A Quantitative Introduction

NICO VAN DER WIJST

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Singapore, S˜ao Paulo, Delhi, Dubai, Mexico City Cambridge University Press

The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York www.cambridge.org

Information on this title: www.cambridge.org/9781107029224 c

Nico van der Wijst 2013

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press.

First published 2013

Printed and bound in the United Kingdom by the MPG Books Group

A catalog record for this publication is available from the British Library Library of Congress Cataloging in Publication data

Wijst, D. van der.

Finance : a quantitative introduction / Nico van der Wijst. pages cm

Includes bibliographical references and index. ISBN 978-1-107-02922-4

1. Finance–Mathematical models. 2. Options (Finance) 3. Corporations–Finance. 4. Investments. I. Title. HG106.W544 2013

332–dc23

2012038088 ISBN 978-1-107-02922-4 Hardback

Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

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Figures

1.1 The interlocking cycles of scientific and applied research 3

1.2 The angular spectrum of the fluctuations in the WMAP full-sky map 6

1.3 Risk–return relationship for Nasdaq-100 companies, October 2010 to

September 2011 7

2.1 The utility function U = 5W − 0.01W2 31

2.2 A two-dimensional utility function and U = 750 31

2.3 Indifference curves 32

2.4 Utility function U (W ) and uncertain values of (W ) 33

2.5 Consumption choices in a budget space 36

2.6 Investment opportunities and their continuous approximation 37

2.7 Investment opportunities and choices 37

2.8 Production and consumption choices with a financial market 38

2.9 Flows of funds through the financial system 40

2.10 Production and consumption choices 49

3.1 Nasdaq-100 index, 1-10-2010 to 30-9-2011 52

3.2 Daily returns Nasdaq-100 index, 4-10-2010 to 30-9-2011 53

3.3 Frequency of daily returns Nasdaq-100 index, 4-10-2010 to 30-9-2011 53

3.4 Diversification effect 57

3.5 Portfolios’ risk and return 61

3.6 Investment universe and choices along the efficient frontier 62

3.7 Efficient frontier 67

3.8 Portfolio composition versus risk 68

3.9 The capital market line 69

3.10 Portfolios of asset i and market portfolio M 71

3.11 Systematic and unsystematic risk 74

3.12 CML with different imperfections 79

4.1 Efficient and inefficient price adjustments 101

4.2 Weekly returns Microsoft, 29-10-2010 to 14-10-2011 102

4.3 Percentage return day t (x-axis) versus day t + 1 (y-axis) 104

4.4 Resistance and support line, Nasdaq-100 index 111

4.5 Moving averages, Nasdaq-100 index 112

4.6 Cumulative abnormal returns of Google 120

5.1 Modigliani–Miller proposition 2 145

5.2 Modigliani–Miller proposition 2 with taxes 148

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5.3 MM proposition 2 with limited liability, no taxes 149

5.4 Trade-off theory of capital structure 150

6.1 Returns and leverage 166

6.2 Timeline for rebalancing and discounting 169

6.3 Decision tree for calculation methods 174

7.1 Profit diagram for a call option 189

7.2 Profit diagrams for simple option positions 189

7.3 Profit diagrams for straddles 190

7.4 Profit diagrams for spreads 191

7.5 Payoff diagrams for the put–call parity 192

7.6 Arbitrage bounds on option prices 195

7.7 Geometric representation of market completeness 201

7.8 Binomial lattice and sample path 217

8.1 Geometric Brownian motion 227

8.2 Call option prices for σ = 0.5 (top), 0.4 and 0.2 (bottom) 236

8.3 Call option prices for T = 3 (top), 2 and 1 (bottom) 236

8.4 Option prices and option deltas 237

8.5 Implied volatility and volatility smile 241

9.1 Theoretical (ceteris paribus) effects of option interaction 277

10.1 Corporate securities as call option combinations 288

10.2 Option values of corporate claims on two projects 290

10.3 Default rates by rating category and year 293

12.1 The firm as nexus of contracts 332

12.2 Structure of an agency problem 334

12.3 Agency costs as a function of capital structure 341

12.4 Agency problems of cash and dividends 343

S2.1 Value as function of time 354

S2.2 Two utility functions 357

S4.1 caar for firms announcing dividend omissions 363

S7.1 Payoff diagram for an option position 371

S7.2 Profit diagrams for a butterfly spread 372

S7.3 Profit diagrams for spreads 373

S7.4 Payoff diagrams of spreads 373

S8.1 Lognormally distributed stock prices 379

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Tables

1.1 Milestones in the development of finance 2

2.1 Effective annual rates 12

2.2 Income statement ZX Co 19

2.3 Balance sheet ZX Co 20

2.4 Statement of cash flows ZX Co 23

2.5 Statement of retained earnings ZX Co 23

2.6 Accounting representation of the project 26

2.7 Financial representation of the project 27

2.8 Economic depreciation of the project 29

3.1 Models and modelling techniques 52

3.2 Asset returns in scenarios 54

3.3 Stock returns in scenarios 59

3.4 Portfolios of stock 1, and 2, 3 and 4 60

3.5 Uncle Bob’s portfolio October 2010 to October 2011 64

3.6 Uncle Bob’s return estimates 64

3.7 Portfolio optimization inputs 65

3.8 Uncle Bob’s optimal portfolio 66

3.9 Portfolios r and b 86

3.10 Arbitrage strategy 87

3.11 Portfolios r and b 95

4.1 Economic depreciation of a project 97

4.2 Autocorrelation coefficients 105

4.3 Coefficients of time series regression, five lags 105

4.4 Runs tests 106

4.5 Return horizons of momentum and contrarian strategies 109

4.6 Overview of fund performance studies 117

4.7 Returns around event day 120

4.8 Overview of event studies 124

5.1 Returns in three scenarios 144

5.2 (Un)levered cash flows 145

5.3 (Un)levered returns and values 146

6.1 Equity betas 177

6.2 Equity and asset betas 178

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7.1 Rights and obligations attached to options 186

7.2 Moneyness of options 187

7.3 Lower arbitrage bound on call options 194

7.4 Lower arbitrage bound on European put options 194

7.5 Upper arbitrage bound on European put options 195

7.6 Arbitrage bounds on and relations between option prices 197

7.7 Calculation of the pricing kernel 207

8.1 Transformed probabilities for a die 223

8.2 Option price determinants and their Greeks 234

8.3 Area in the left tail of the standard normal density function 254

9.1 Stock–real option analogy 258

9.2 Some common real options 258

9.3 Stock versus real option input parameters 260

9.4 Prisoners’ dilemma 280

9.5 Strategic investment timing 281

10.1 Two projects 289

10.2 Bankruptcy rates (%) in Norwegian industries 294

10.3 Logit estimates and example companies 296

10.4 Cash flow distribution 299

10.5 Joint cash flow distribution 300

10.6 Merger candidates’ data 301

10.7 Merger candidates’ values 302

10.8 Merger candidates’ values and benefits 303

11.1 LME official prices, US$ per tonne for 16 December 2009 310

11.2 Farmer’s closing position 311

11.3 Baker’s closing position 312

11.4 ZX Co. forward 1 316

11.5 ZX Co. forward 2 316

11.6 Cash-and-carry portfolio 317

11.7 Arbitrage portfolio 1 318

11.8 Arbitrage portfolio 2 319

11.9 Farmer’s closing position with perfect cross hedging 323

11.10 Farmer’s closing position with imperfect cross hedging 323

11.11 Currency rates 324

12.1 Agency problems and costs of debt 336

12.2 Agency problems and costs of equity 339

12.3 Corporate governance systems 346

12.4 Ownership effect of performance, literature overview 350

12.5 Two representations of a project 353

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Acronyms

APT Arbitrage Pricing Theory APV adjusted present value

BIS Bank for International Settlements caar cumulative average abnormal return CAPM Capital Asset Pricing Model CEO chief executive officer CFO chief financial officer CML capital market line DCF discounted cash flow EMH Efficient Market Hypothesis

FV future value

IPO initial public offering IRR internal rate of return

Nasdaq National Association of Securities Dealers Automated Quotations NPV net present value

NYSE New York Stock Exchange OCC opportunity cost of capital

OECD Organisation for Economic Cooperation and Development OTC over the counter

PV present value

S&P Standard & Poor’s

SDE stochastic differential equation SEC Securities and Exchange Commission SML security market line

VaR value at risk

WACC weighted average cost of capital

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Finance has undergone spectacular changes in the last four decades, both as a profession and as a scientific discipline. Before 1973 there were no option exchanges and there was no generally accepted model to price options. Today, the worldwide trade in derivative securities represents a much larger money amount than the global production of goods and services. The famous Black and Scholes option-pricing formula and its descendants are used in financial markets all over the world where an enormous number of derivative securities are traded every day. Professionals in sectors like engineering, telecommunica-tions and manufacturing regularly find that their projects are evaluated with techniques such as real options analysis. Understanding the basic concepts of finance is increasingly becoming a prerequisite for the modern work place.

Many scientific developments in finance are fuelled by the use of quantitative methods; finance draws heavily on mathematics and statistics. This gives students and professionals who are familiar with quantitative techniques an advantage in mastering the principles of finance. As the title suggests, this book gives an introduction to finance in a manner and ‘language’ that are attuned to an audience with quantitative skills. It uses mathematical notations and derivations where appropriate and useful. But the book’s main orientation is conceptual rather than mathematical; it explains core financial concepts without for-mally proving them. Avoiding the definition-theorem-proof pattern that is common in mathematical finance allows the book to use the more natural order of first presenting an insight from financial economics, then demonstrating its empirical relevance and prac-tical applicability, and concluding with a discussion of the necessary assumptions. This ‘reversed order’ reduces the scientific rigour but it greatly enhances the readability for novice students of finance. It also allows the more demanding parts to be skipped or made non-mandatory without loss of coherence.

The need for a book like this arose during the many years that I have been teaching finance to science and technology students. Their introductory years give these students a good working knowledge of quantitative techniques, so they are particularly well placed to study modern finance. However, almost all introductory textbooks in finance are writ-ten for MBA students, who have a much less quantitative background. In my experience, teaching finance to numerate students using an MBA textbook is an unfortunate combi-nation. It forces the teacher to supply much additional material to allow students to use their analytical skills and to highlight the quantitative aspects that are severely understated in MBA textbooks. Of course, there are many textbooks in finance that are analytically more advanced, but these are usually written for a second or third course. They assume familiarity with the terminology and basic concepts of finance, which first-time readers

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xv Preface

do not possess. This is also the case for introductory textbooks in financial economics, or the ‘theory of finance’. In addition, many of these books are written in the definition-theorem-proof pattern, which makes them, in my opinion, less suitable for introductory courses. Students’ first meeting with finance should be an appetizer that arouses their interest in finance as a science, shows them alternative uses for the quantitative techniques they have acquired, and welcomes them to the wonderful world of financial modelling. Formal proofs are not instrumental in that.

Readership

This book is primarily written for science and technology students who include a course in finance or project valuation in their study programmes. Most study programmes in mathematics, engineering, computer science and the natural sciences offer the opportu-nity to include such elective subjects; their typical place is late in the bachelor programme or early in the master programme. The book can be used as the only text for a course in finance or as one of several if other management aspects are included, such as project planning and organization. Given the limited room for these courses in most study pro-grammes the book has to be concise, but it takes students from discounting to the Black and Scholes formula and its applications. To limit its size, the main emphasis is on invest-ments in real assets and the real options attached to them. This is the area of finance that prospective natural scientists and engineers are most likely to meet later in their careers. Of course, a thorough analysis of such investments requires a theoretical basis in finance that includes portfolio theory and the pricing models based on it, market efficiency, capital structure, and derivatives pricing. Topics with a less direct connection with real assets are omitted, such as bond pricing, interest rate models, market microstructure, exotic options, cash and receivables management, etc.

I have also used the material in this book for intermediate courses in finance for busi-ness school students. The purpose of these courses is to deepen students’ theoretical understanding of finance and to prepare them for more specialized subjects in, for exam-ple, continuous-time finance and derivatives pricing. The step from an introductory MBA book to a specialized text is often too large, and this book can fruitfully be used to bridge the gap. It introduces students to techniques that they will meet in later courses, but in a much more accessible and less formal way than is usual in the specialized literature. Greater accessibility is increasingly required because of the growing diversity in business school students’ backgrounds. In my experience, students find the material in the book both interesting and demanding, but most students rise to the challenge and successfully complete the course.

A final use that I have made of the book’s material is for a permanent education course aimed at professionals in science and technology and technical project leaders. After some years of work experience, many professionals feel the need for more knowledge about the way financial managers decide about projects, particularly how they value the flexibility in projects with real options analysis. The scope and depth of the book are sufficient to make such professionals competent discussion partners of financial managers in matters of project valuation, including the aspects of strategic value.

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Acknowledgements

I would not have enjoyed writing this book as much as I did without the support of many more people than can be mentioned here. I am grateful to my present and former PhD students, especially John Marius Ørke and Tom E. S. Farmen, Ph.D., Senior Adviser and Senior Portfolio Manager at Norway’s Central Bank. They were first in line to be asked to read and re-read the collection of lecture notes, exercises and manuscripts that grew into this book. I also want to thank Thomas Hartman and my other colleagues at the School of Business, Stockholm University. Teaching at the School of Business whetted my interest in the pedagogical features of the material in this book. I am indebted to Jaap Spronk at RSM/Rotterdam School of Management and to my other former colleagues at Erasmus University Rotterdam; this book owes much to discussions with them. A final word of thanks is due to my students who, over the years, have contributed in many ways to this book.

Nico van der Wijst

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Introduction

1

This opening chapter introduces finance as a scientific discipline and outlines its main research tools. We also take a brief look at the book’s central theme, the calculation of value, and the main ways to account for risk in these calculations. To illustrate the differences between finance and the natural sciences we compare results of a Nobel prize-winning financial model with measurements of a NASA space probe.

1.1 _...

Finance as a science

1.1.1 What is finance?

Finance studies how people choose between uncertain future values. Finance is part of economics, the social science that investigates how people allocate scarce resources, that have alternative uses, among competing goals. Both scarcity, i.e. insufficient resources to achieve all goals, and possible alternative uses are necessary ingredients of economic problems. Finance studies such problems for alternatives that involve money, risk and time. Financial problems can refer to businesses, in which case we speak of corporate finance, but also to individuals (personal finance), to governments (public finance) and other organizations. Financial choices can be made directly or through agents, such as business managers acting on behalf of stockholders or funds managers acting on behalf of investors. For the most part, we shall study choices made by businesses in financial markets, but the results have a wider validity. As we shall see, financial markets facil-itate, simplify and increase the possibilities to choose. Some typical problems we will look at are:

• Should company X invest in project A or not?

• How should we combine stocks and risk-free borrowing or lending in our investment portfolio?

• What is the best way to finance project C?

• How can we price or eliminate (hedge) certain risks? • What is the value of flexibility in investment projects?

Finance as a scientific discipline (also called the ‘theory of finance’ or ‘financial eco-nomics’) seeks to answer such questions in a way that generates knowledge of general validity. It evolved from the descriptive science it was about 100 years ago into the ana-lytic science it is now. Modern finance draws heavily on mathematics, statistics and other disciplines, and many scientists working in finance today started their careers in the nat-ural sciences. Table 1.1 lists some milestones in the development of finance over the past century as well as some of the people whose work we shall meet. The importance of their

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Table 1.1 Milestones in the development of finance

Name Period Nobel prize Topics

J. Fisher 1930s – Optimal investment/consumption

K. Arrow 1950s 1972 State-preference theory

G. Debreu 1950s 1983 State-preference theory

J. Nash 1950s 1994 Game theory

H. Markowitz 1950s 1990 Portfolio theory

F. Modigliani 1950–60s 1985 Capital structure, cost of capital M. Miller 1950–60s 1990 Capital structure, cost of capital

P. Samuelson 1960s 1970 Market efficiency

W. Sharpe 1960s 1990 Capital Asset Pricing Model

R. Merton 1970s 1997 Option pricing

M. Scholes 1970s 1997 Option pricing

F. Black 1970s – Option pricing

contributions is reflected in the Nobel prizes awarded to them: we will be standing on the shoulders of these giants.

Finance is also a tool box for solving decision problems in practice; this part is usually referred to as managerial finance. There is not always a direct relation between prac-tical decision making and scientific results. For a number of pracprac-tical problems there is no scientifically satisfactory solution. Conversely, some scientific results are still far from practical applications. But generally the insights from the theory of finance are also applied in practice and usually well beyond the strictly defined context they were derived in. Modern portfolio theory, Black and Scholes’ option-pricing formula, risk-adjusted discount rates and many more results all have found their way into practice and are now applied on a daily basis.

1.1.2 How does finance work?

As in many other sciences, the main tools in finance are the mathematical formulation (i.e. modelling) of theories and their empirical testing. What makes finance special among social sciences is that financial markets lend themselves very well to modelling and test-ing, as well as application of the results. The list of Nobel prizes in Table 1.1 is testimony to successful applications of these tools in finance.

Scientific research in finance usually has an actual problem as its starting point. The problem is first made manageable by making simplifying assumptions with regard to, for example, investor behaviour and the financial environment investors operate in. The stylized problem is then translated into mathematical terms (modelled) and the analytical power of mathematics is used to formulate predictions in terms of prices or hypotheses. The predictions are tested by confronting them with real-life data, such as prices in finan-cial markets, or accounting and other data. If the tests do not reject the theories we can apply their results to practical decisions, such as buying or selling in a financial market, accepting or rejecting an investment proposal, or choosing a capital structure for a project or a company. Alternatively, we can use the test results to adapt the theory. This gives a

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3 1.1 Finance as a science

full cycle of scientific research, from formal theories to tests and practical applications. Figure 1.1 illustrates the interlocking cycles of scientific and applied research.

Theoretical Empirical Practical

Assumptions Model Predictions Results Tests Prices/ Hypotheses Applications Conclusions/ Questions

Figure 1.1 The interlocking cycles of scientific and applied research

Option pricing is a good example to illustrate the workings of finance. For many years, finding a good model to price options was an actual and very relevant problem in finance. Black and Scholes cracked this puzzle by making the simplifying assumptions of greedy1 investors, a constant interest rate and stock price volatility and frictionless markets (we shall look at all these concepts later on). They then translated the problem in mathematical terms by formulating stock price changes as a stochastic differential equation and the option’s payoff at maturity as a boundary condition. The analytical power of mathematics was used to solve this ‘boundary value problem’ and the result is the famous Black and Scholes option-pricing formula. Empirical tests have shown that this formula gives good predictions of actual market prices. So we can use the model to calculate the price of a new option that we want to create and sell (‘write’ an option) or to hedge (i.e. neutralize) the obligations from another contract, e.g. if we have to deliver a stock in three months’ time. In fact, thousands of traders and investors use this formula every day to value stock options in markets throughout the world.

Scientific research does not necessarily begin with a problem and assumptions, it can also start in other parts of the cycles in Figure 1.1. For instance, in the 1950s statisticians analyzed stock prices in the expectation of finding regular cycles in them, comparable to the pig cycles in certain commodities.2All they could find were random changes. These empirical results later gave rise to the Efficient Market Hypothesis, which was accurately

...

1 This is not a moral judgement but the simple assumption that investors prefer more to less, mathematically

expressed in the operator max[.].

2 Pig cycles are periodic fluctuations in price caused by delayed reactions in supply, named after cycles in pork prices

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and succinctly worded by Samuelson as ‘properly anticipated prices fluctuate randomly’. Similarly, Myers’ Pecking Order Theory of capital structure is based on the observation that managers prefer internal financing to external, and debt to equity.

1.2 A central issue

...

A central issue in finance is the valuation of assets such as investment projects, firms, stocks, options and other contracts. In finance, the value of an asset is not what you paid for it when you bought it, nor the amount the bookkeeper has written somewhere in the books. It is, generally, the present value of the cash flows the asset is expected to generate in the future or, in plain English, what the expected future cash flows are worth today. That value depends on how risky those cash flows are and how far in the future they will be generated. This means that value has a time and an uncertainty dimension; the pattern in time and the riskiness both determine the value of cash flows.

As we shall see in the next chapter, the time value of money is expressed in the risk-free interest rate. That rate is used to ‘move’ riskless cash flows in time: discount future cash flows to the present and compound present cash flows to the future. Since the rate is accumulated (compounded) over periods, the future value of a cash flow now increases with time. Similarly, cash flows further in the future are ‘discounted’ more and thus have a lower present value. We can express this a bit more formally in a general present-value formula (where t stands for time):

Value= t

ExpCash flows_t

(1 + discount ratet)t (1.1)

The numerator of the right-hand side of (1.1) contains the expected cash flow in each period. If the cash flow is riskless, the future amount is always the same, no matter what happens. Such cash flows can be discounted at the risk-free interest rate. If the cash flow is risky, the future amount can be higher or lower, depending on the state of the economy, for example, or on how well a business is doing. The size of a risky cash flow has to be expressed in a probabilistic manner, for example 100 or 200 with equal probabilities. The expectation then is the probability weighted average of the possible amounts:

ipiCFLi where pi is the probability and CFLi the cash flow. In the example, the expected cash flow is 0.5 × 100 + 0.5 × 200 = 150.

There are three different ways to account for risk in the valuation procedure. The first way is to adjust the discount rate to a risk-adjusted discount rate that reflects not only the time value of money but also the riskiness of the cash flows. For this adjustment we can use a beautiful theory of asset pricing, called the Capital Asset Pricing Model (CAPM) or, alternatively, the equally elegant and more general but less precise Arbitrage Pricing Theory (APT). The second way is to adjust the risky cash flows so that they become certain cash flows that have the same value as the risky ones. These certainty

equiva-lent cash flowscan be calculated with the CAPM or with derivative securities such as futures, and they are discounted to the present at the risk-free interest rate. The third way is to redefine the probabilities, that are incorporated in the expectations operator, in such a way that they contain pricing information. Risk is then ‘embedded’ in the probabili-ties and the expectation calculated with them can be discounted at the risk-free interest

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5 1.3 Difference with the natural sciences

rate. Changing probabilities is the essence of the Black–Scholes–Merton Option Pricing Theory, and it accounts for risk in a fundamentally different way than the CAPM or APT. We shall look at all three methods in detail and use them to analyze questions and topics such as these:

• What risks are there? Are all risks equally bad? Is risk always bad?

We will see that some risks don’t count and that risk can even be beneficial to some investments and people.

• Portfolio theory and valuation models

They demonstrate why investments should not be evaluated alone, but combined. • Market efficiency

that explains why you, and your pension fund, cannot quickly get rich if markets function properly.

• The variety of financial instruments

and how they helped to create the credit crunch. • Capital structure

or why some projects are easy to finance and others are not. • The wild beasts of finance: options and other derivatives

and why most projects and firms are options. • Real options analysis

and how flexibility can make unprofitable projects profitable. • Modern contracting and incentive theory

which explains why good projects can be turned down and bad projects can be accepted.

1.3 _...

Difference with the natural sciences

The natural sciences generally study phenomena that, at least in principle, can be very precisely measured. Moreover, the relations between different phenomena can often be accurately predicted from the laws of nature and/or established in experimental settings that control all conditions. As a result, observations in natural sciences such as physics and chemistry usually show little dispersion around their theoretically predicted values.

Finance, however, is a social science: it studies human behaviour. Controlled experi-ments are practically always impossible. Financial economists cannot keep firms in an isolated experiment, control all economic variables and then measure how firms react to changes in the interest rate that the experimenter introduces, for instance. They can only observe firms in some periods with low interest rates and other periods with high interest rates. But it is not only the interest rate that changes from period to period; everything else changes as well. Hence, financial data consist of noisy, real-life observations and not clean, experimental data. Furthermore, it is impossible to control for all other factors in the statistical analyses that are used to estimate financial relations. So these relations are necessarily incomplete. As a result, observations in finance usually are widely dis-persed around their theoretically predicted values. Science and technology students may need some time to acquaint themselves with the nature of financial relations. An extreme example from both sciences will illustrate the differences.

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Figure 1.2 plots data collected by NASA’s Wilkinson Microwave Anisotropy Probe (WMAP), a satellite that has mapped the cosmic microwave background radiation. That is the oldest light in the universe, released approximately 380,000 years after the birth of the universe 13.73 billion years ago. WMAP produced a fine-resolution full-sky map of the microwave radiation using differences in temperature measured from opposite directions. These differences are minute: one spot of the sky may have a temperature of 2.7251◦ Kelvin, another spot 2.7249◦ Kelvin. It took a probe of $150 million to measure them. Figure 1.2 shows the relative brightness (temperature) of the spots in the map versus the size of the spots (angle). The shape of the curve contains a wealth of information about the history of the universe (see NASA’s website at http://map.gsfc.nasa.gov/). The point here is that the observations, even of the oldest light in the universe, show very little dispersion.

Multipole moment T emper at u re fl u ct u ations [ µ K 2] 6000 10 100 90⬚ 2⬚ 0.5⬚ Angular size 0.2⬚ 500 1000 5000 4000 3000 2000 1000 0

Figure 1.2 The angular spectrum of the fluctuations in the WMAP full-sky map.

Credit: NASA/WMAP Science Team

Figure 1.3 plots the return versus the risk of companies in the Nasdaq-100 index in the period 4 October 2010 to 30 September 2011. As the name suggests, this index includes 100 of the largest US and international non-financial securities listed on Nasdaq, the world’s first completely electronic stock market. Giants such as Apple, Adobe, Dell, Google, Intel and Microsoft are included in the data. Return is measured as the percent-age price return (changes in stock price over the year, adjusted for dividends). Risk is the company’s beta coefficient, which measures the contribution of the company’s stock to the variance of a well-diversified portfolio.3 The straight line is a theoretical model, the CAPM, for which Sharpe was awarded the Nobel prize in 1990. It gives the expected return of asset i, ri, as a function of its beta coefficient βi, the risk-free interest rate rf

...

3 The beta coefficients are calculated relative to the Nasdaq-100 index using daily returns (adjusted for events such

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7 1.3 Difference with the natural sciences

and the expected return on the market portfolio rm: E(ri)= rf + (E(rm)− rf)βi

In the period October 2010 to September 2011 the risk-free interest rate in the USA was close to zero, say 0.5 per cent, and the return of the Nasdaq-100 index was 9.5 per cent, so the relation is:

ri = 0.5 + (9.5 − 0.5)β

which is the formula of the plotted line. All these concepts are discussed in detail later on. The point here is that the observations, even of the largest US companies, show a very large dispersion, even around a Nobel prize-winning theoretical model.

0.5 1.0 1.5 2.0 –50 0 50 100 Beta Return

Figure 1.3 Risk–return relationship for Nasdaq-100 companies, October 2010 to

September 2011

Of course, this extreme example does not imply that financial economics cannot predict at all, nor that empirical relations cannot explain more than a few per cent of variations in, for example, stock prices. On the contrary, the Fama–French three-factor model, which we shall meet in Chapter 3, explained more than 90 per cent of the variance in stock returns when it was first estimated. But it illustrates that empirical relations in finance have a different character compared with those in the natural sciences.

There is an additional reason why financial relations are less precise. Financial eco-nomics studies how people choose between uncertain future values. These future values are to a very large extent unpredictable, not because financial economists are not good at their jobs but because properly functioning financial markets make them unpredictable. In such markets, accurate predictions ‘self-destruct’. For example, if news becomes avail-able from which investors can reliably predict that the value of a stock will double over the next month, they will immediately buy the stock and keep on buying it until the dou-bling is included in the price. So the ‘surprise’ is instantly incorporated into the price and

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the price changes over the rest of the month depend on new news, which is unpredictable by definition. Unlike most other sciences, financial economics has developed a coher-ent theory, the Efficicoher-ent Market Theory, that explains why some of its most important study-objects, such as stock price changes, should be unpredictable.

The same information effects that make future values unpredictable also make observed, historical data noisy, as Figure 1.3 shows. Returns (or stock price changes) can be regarded as the sum of an unobserved expected part, plus an unexpected part that is caused by the arrival of new information. There is a constant flow of economic news from all over the world and much of it is relevant for stock prices. As a result, the unex-pected part is large relative to the exunex-pected part. For example, daily stock price changes are typically in the range from −2 per cent to +2 per cent. If a stock is expected to have an annual return of 20 per cent and a year has 250 trading days, the daily expected return is 20/250 ≈ 0.08 per cent, very small compared with the observed values. Any model for expected stock returns will therefore have a high residual variance, i.e. explain only a small proportion of the variance of observed stock returns.

Finally, because efficient markets make price changes unpredictable, a high residual variance is a positive, not a negative, quality indicator of financial markets. The better financial markets function, the more unpredictable they are. To illustrate this, consider the results of studies which have shown that residual variance has increased over time (Campbell et al., 2001), that it increases with the sophistication of financial markets (Morck et al., 2000) and that it increases with the informativeness of stock prices (i.e. how much information stock prices contain about future earnings) (Durnev et al., 2003). Together, these elements give empirical relations in finance a distinct character compared with the natural sciences but also compared with other social sciences.

1.4 ...

In the eleven chapters that follow this introduction, the book discusses selected finan-cial topics in their approximate order of historical development. This is a natural way to acquaint students with the evolution of financial–economic thought and it ensures increasing sophistication of the analyses. The second chapter provides a basis for the later investigations by briefly recapitulating some fundamental concepts from financial calculus, accounting and micro economics. Most students will have met these concepts before, but the summary operationalizes their knowledge to the level necessary in later chapters. Chapter 2 also contains a short description of how financial markets work in practice. Modern portfolio theory is the subject of Chapter 3. It starts with Markowitz’s classic portfolio selection model and continues with the equilibrium pricing models that are its natural descendants, the Capital Asset Pricing Model and Arbitrage Pricing Theory. Chapter 4 covers market efficiency, a concept that is counter-intuitive to many stu-dents with a background in the natural sciences. A thorough discussion of the concept and the presentation of extensive empirical evidence aim to overcome their scepticism. The fifth chapter presents Modigliani and Miller’s ground-breaking analyses of the capi-tal structure and dividend decisions, along with Myers’ pecking order theory and recent

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9 1.4 Contents

empirical tests. The main insights from capital structure theory are applied, in Chapter 6, to the problem of valuing projects that are partly financed with debt.

From Chapter 7 onwards, option pricing pervades the analyses. The characteristics of options as securities are described in Chapter 7, which also lays the foundations for option pricing in state-preference theory. The binomial option pricing model completes the chapter’s analyses in discrete time. Chapter 8 surveys option pricing in continuous time. It introduces the technique of changing probability measure with an example from gambling and proceeds with an informal derivation of the celebrated Black and Scholes formula, followed by a discussion of some of its properties and applications. Option-pricing techniques are applied to a variety of real options in Chapter 9 and three other problems in corporate finance in Chapter 10.

Hedging financial risks is explored in Chapter 11, along with the pricing of the main derivative securities involved in the process. Hedging techniques are applied to cross hedging in commodity markets and foreign exchange rate risk. The final chapter exam-ines two more general problems in corporate finance, the agency relations that exist between the firm and its stakeholders, and corporate governance, the way in which firms are directed and controlled.

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Fundamental concepts and techniques

2

This chapter summarizes the basic concepts and techniques that are used throughout the other chapters. We first look at the time value of money and common interest rate calcula-tions. We then recapitulate how a firm’s accounting system records and reports financial data about the firm. An example illustrates how these techniques and data can be used for investment decisions. Subsequently, we introduce the economic concepts of utility and risk aversion, and their use in financial decision making. The chapter concludes with a brief look at the role of financial markets, both from a theoretical and practical perspective.

2.1 The time value of money

...

2.1.1 Sources of time value

The time value of money can be summarized in the simple statement that e1 now has a higher value than e1 later. The time value of money springs from two sources: time preference and productive investment opportunities. Time preference, or ‘human impa-tience’ as the economist Fisher (1930) calls it, is the preference for present rather than future consumption. This is more than just impatience. Some consumption cannot be postponed for very long, for example the necessities of life. For other goods, the time pattern of people’s consumptive needs is almost inversely related to the time pattern of their incomes. People want to buy houses when they are young and starting families, but if they had to accumulate the necessary money by saving, only a few could afford to buy a house before retirement age. Moreover, postponing consumption involves risk. Even if the future money is certain, the beneficiary, or the consumptive opportunity, may no longer be around. As a result, people require a compensation for postponing consumption and are willing to pay a premium to advance it.

The alternative to consumption is using money for productive investments. Produc-tive means that the investment generates more than the original amount. This is found in its simplest form in agriculture where grain and livestock can be consumed directly or cultivated to give a larger harvest after some time. But the same principle applies to invest-ments in machinery, infrastructure or human capital: by giving up consumption today we can increase consumption later.

The time value of money is expressed in a positive risk-free interest rate.1 In free markets, this rate is set by supply and demand which, in turn, are determined by factors such as the amounts of money people and businesses hold, the availability of productive

...

1 Real-life interest rates often contain other elements as well, such as compensation for risk and inflation. We will

deal with risk-adjusted rates later and assume no inflation.

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11 2.1 The time value of money

investment opportunities and the aggregate time preference. Governments and central banks are big players on money markets.

A consequence of the time value is that money amounts on different points in time cannot be directly compared. We cannot say that e100 today is worth less or more than e110 next year. To make that comparison we have to ‘move’ amounts to the same point in time, adjusting for the time value. The process of moving money through time is called compounding or discounting, depending on whether we move forward or backward in time.

2.1.2 Compounding and discounting

Interest is compounded when it is added to the principal sum so that it starts earning interest (i.e. interest on interest). How much and when interest is compounded can be agreed upon in different ways. In its simplest form compounding takes place after the period for which the interest rate is set. For example, if you deposit e100 at a bank at a 10 per cent yearly interest rate compounded annually, then after one year 10 per cent or e10 is added to your account. Your new principal becomes e110 and in the second year you earn 10 per cent interest over e110, or e11, so that after two years the principal becomes e121, etc. In formula form, the future value after T years (t = 1, 2, . . . , T ), FVT,is the present value, PV, times the compounded interest rate r:

F VT = P V (1 + r)T

The same principle applies to discounting, i.e. moving money in the opposite direction. A future value of e100 at time T has a value of e100/1.1 = e90.90 at T − 1. This, in turn, has a value of e90.90/1.1 = e82.60 at T − 2, etc. In the formula we simply move the interest rate factor to the other side of the equation:

P V = F VT (1 + r)T

Of course, we can also re-write the formula to give an expression for the interest rate (or discretely compounded return):

r = T

F VT P V − 1

Note that this is the geometric average rate, which is lower than the arithmetic average if the interest rate fluctuates over time. The differences between the two averages are discussed in appendix 3A.

The period after which interest is compounded is not necessarily the same as the period for which the interest rate is set. For example, corporate bonds usually pay interest twice a year, even though the interest is set as an annual rate. An 8 per cent bond then pays 4 per cent every six months. It is easy to demonstrate what happens to the future value FVT if a variable compounding frequency, n, is introduced: F VT = P V (1 +r_n)T n. So semi-annual compounding of 10 per cent per year gives (1 + 0.1₂ )2= 1.1025 or an effective annual

rateof 10.25 per cent. Table 2.1 gives the effective annual rates for some compounding frequencies of 10 per cent per year.

In the limit, as n → ∞, the time spans over which compounding takes place become infinitesimal and compounding becomes continuous. Then, an expression for the effective

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Table 2.1 Effective annual rates

Compounding Effective annual

frequency n rate % (10% annual)

yearly 1 10.00 semi-annually 2 10.25 quarterly 4 10.38 monthly 12 10.47 weekly 52 10.51 continuously ∞ 10.52

annual rate is found by multiplying T n by r/r and splitting the term in n/r and rT :

F VT = P V 1 + r n n/rrT

Defining c = n/r this becomes:

F VT = P V

1 +1 c

crT

cincreases with n and as both approach infinity, (1 +1_c)capproaches the constant e, the base of natural logarithms:

lim c→∞ 1 +1 c c = e = 2.71828....

so that the formula for the future value becomes:

F VT = P V erT or P V = F VTe−rT

Re-writing this for the interest rate we get F VT/P V = erT. Taking natural logarithms produces:

lnF VT P V = ln e

rT _{= rT}

These logarithmic returns are easy to calculate from a series of daily or weekly closing prices on a stock exchange, say S0, S1, S2, etc.: lnSS1₀, ln

S2

S₁, etc. They are additive over time, the compound return S1

S₀ × S2 S₁ can be calculated as ln S1 S₀ + ln S2 S₁= ln er1_{+ ln e}r2_{= r}

1+ r2. To obtain a weekly return we can simply add up the five daily returns. This makes log returns convenient to use in a times series context. On the other hand, log returns are non-additive across investments. The logarithmic transformation is non-linear, so that the log of a sum is not equal to the sum of the logs.

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By contrast, the discretely compounded returnsS1−S0

S₀ , S2−S1

S₁ ,etc. are easily aggregated across investments because the ‘weighted returns’ (return × its weight) are additive. For two stocks A and B the weighted returns in period 1 are:

S₀A S₀A+ S₀B × S₁A− S₀A S₀A + S₀B S₀A+ S₀B × S₁B− S₀B S₀B = S₁A− S₀A S₀A+ S₀B + S₁B − S₀B SA₀ + S₀B

This property makes it attractive to use these returns in a portfolio context. But discrete compounding makes the returns non-additive over time: 5 per cent return over ten years gives 62.9 per cent (1.0510) and not 50 per cent.

2.1.3 Annuities and perpetuities

Payments and receipts often come in series and their regularity can be exploited to make them more easy to handle.

The present value of annuities

An annuity is a series of equal payments at regular time intervals. Annuities derive their name from annual payments but the valuation principles apply to other time intervals as well, provided the interest rate is properly adjusted. Suppose we have a series of n payments2 of amount A, starting at the end of the period. If the period interest rate is r the present value, PV, of this annuity can be written as:

P V = A 1 + r + A (1 + r)2 + · · · + A (1 + r)n

Of course, we can discount the individual terms and sum the results. However, it is easy to recognize a geometric series in these payments, with A/1+r as the first term, n as the number of terms and 1/1+r as the common ratio. The sum3of this series is:

P V = A

1 + r

1 −_1+r1 n

1 −_1+r1 (2.1)

Note that n in (2.1) refers to the number of terms in the series and not to the number of discounting periods, although the two coincide in this case. Also note that the formula calls for the first term in the series, A/(1 + r), which, in this case, does not coincide with

...

2 The symbols n and c are frequently re-used for counters and constants.

3 The sum S of a geometric series of n terms starting with s and with common ratio c (c = 1) is found by first writing

out the sum, then multiplying both sides by the common ratio and then subtracting the latter from the former so that all but two terms drop from the right hand side of the equation:

S= s + sc + sc2+ · · · + scn−1 −(Sc = sc + sc2+ · · · + scn−1+ scn) S− Sc = s − scn

Solving for S then gives: S = s1 − c n 1 − c

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the size of the annuity A.4This becomes clear when we define the annuity such that it starts today: P V = A + A 1 + r + A (1 + r)2 + · · · + A (1 + r)n−1

In this case the first term is A while the number of terms still is n, one more than the number of discounting periods. The sum of this series is:

P V = A

1 −_1+r1 n

1 −_1+r1 (2.2)

The following example illustrates the valuation of annuities. Suppose you just inherited e1 million from a distant relative. The accountancy firm that handles the estate suggests to make the money available over a longer period of time, so that you are not tempted to spend it all at once. They offer to pay out in fifteen amounts of e100, 000, one now and one after each of the following fourteen years. If the interest rate is 10 per cent, is this a fair offer? To answer that question we calculate the present value of the payments using equation (2.2) for the sum of a 15-year annuity of e100, 000 starting now:

P V = 100 000 1 − 1 1.1 15 1 −_1.11 = 836 670

Of course, we get the same result if we use (2.1) to calculate the sum of the fourteen future end-of-period payments and add e100, 000 for the payment now:

P V = 100 000 1.1

1 −_1.11 14

1 −_1.11 + 100 000 = 836 670

So the offer is not fair; the value of the proposed payments is much less than the inher-itance. The nominal value of the fifteen payments, e1.5 million, can be deceptive since it ignores the time value. For example, the last payment has a present value of only 100 000/1.114= 26 333.

How large should the 15 annuities be to make their value equal to the e1 million inheritance? To answer that question we rewrite equation (2.2) into an expression for the annuity given the present value. This is simply a matter of moving A and PV to other sides of the equation:

A= P V 1 −

1 1+r 1 −_1+r1

n

Inserting the numbers gives:

A= 100 000 1 − 1 1+0.1 1 −_1+0.11 15 = 119 520

...

4 The first term is somewhat obscured if (2.1) is simplified by multiplying the terms in the denominators (1 + r)(1 −

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For an end-of-period annuity we perform the same operation on (2.1):

A

1 + r = P V

1 −_1+r1 1 −_1+r1 n

Multiplying both sides by (1+r) we get:

A= P V r

1 −_1+r1 n

These calculations are often used in practice, when money is borrowed to buy business assets like machinery or private assets like cars. It is not unusual that the buyer borrows at least a part of the purchasing price and agrees to pay back the loan plus interest in a number of equal amounts, calculated as above. In such cases, end-of-period annuities are more logical, since it doesn’t make sense to borrow money and immediately pay back some of it. Annuities to pay back a loan with interest are also known as amortization

factors.

The future value of annuities

Annuities can also be saved so that they accumulate to some future value. The calculation of future annuity values rests on the same principle of geometric series summation, but since amounts are brought forward in time, the common ratio is the compounding factor. It is customary to calculate future values on the moment that the last payment is made, i.e. when the whole sum becomes available. This means that the last payment did not earn any interest, the last but one payment earned interest over one period, etc. The first payment earned interest over n-1 periods. The geometric series character is clearest when the payments are written out in the reverse order of time, the last payment first, etc. The future value, FV, of a series of n payments of amount A is then:

F V = A + A(1 + r) + A(1 + r)2+ ... + A(1 + r)n−1

This is a geometric series of n terms with first term A and common ratio (1+r). The sum of that series is:

F V = A1 − (1 + r) n 1 − (1 + r) = A

(1 + r)n− 1

r (2.3)

This formula can be used to calculate the future value of the fifteen payments of e119, 520 that have the same value as the e1 million inheritance today:

F V = 119 5201.1 15_{− 1}

0.1 = 3 797 447

It is easy to check this with the future value of the e1 million today: 1 000 000 × 1.114= 3 797 498. Note that the number of periods is fourteen, one less than the number of terms.

It is more common to calculate the annuity given the future value. This is done, for example, to calculate the yearly contributions to a so-called sinking fund, into which

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home-owners in an apartment block set aside money to pay for major work like replacing the roof. Rewriting (2.3) for the annuity given the future value is, again, simply a matter of moving A and PV to other sides of the equation:

A= F V r

(1 + r)n_{− 1}

This formula can be used to calculate how much the home-owners should set aside each year if the roof on their apartment building needs to be replaced in 10 years at a cost of e75, 000. If the interest rate is 10 per cent, the annual amount is:

75 000 0.1

(1 + 0.1)10_{− 1} = 4 706

Growing annuities

The valuation formulas for annuities can be extended to incorporate a constant growth factor. Growing annuities are usually defined as end-of-period payments, but they can also start immediately. Consider a series of n payments, starting today, of amount A that grows with g per cent each period. If the period interest rate is r, as before, the present value of this annuity can be written as:

P V = A + A(1 + g) (1 + r) + A(1 + g)2 (1 + r)2 + · · · + A(1 + g)n−1 (1 + r)n−1

Using the sum formula for a geometric series with first term A and common ratio (1+g)_(1+r) this equals: P V = A 1 −(1+g)_(1+r) n 1 −(1+g)_(1+r) (2.4)

When the annuity starts at the end of the period it is convenient to define the first term as A(1+g)

(1+r) so that the growth and interest factor accumulate over the same number of periods. The present value then is:

P V = A(1 + g) (1 + r) + A(1 + g)2 (1 + r)2 + · · · + A(1 + g)n (1 + r)n P V = A(1 + g) (1 + r) 1 − (1+g) (1+r) n 1 −(1+g)_(1+r)

Working out the terms in the denominators this becomes:

P V = A(1 + g)

1 −(1+g)_(1+r) n

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For example, if the interest rate is 10 per cent, the present value of a series of five payments that starts with e100 at the end of the period and grows with 5 per cent per period is:

P V = 100

1 −(1+0.05)_(1+0.1) 5

0.1 − 0.05 = 415.06

An expression for the future value of a growing annuity can be derived along the same lines as before, i.e. looking backwards on the moment that the last payment is made. That last payment is A(1 + g)n and has not earned any interest. The last but one payment is A(1 + g)n−1 and earned one period interest, which can be written as A(1 + g)n _1+g1+r. The last but two payment is A(1 + g)n

(1+r)2 (1+g)2

, etc. So we have a geometric series with first term A(1 + g)n and common ratio_1+g1+r.The sum of this series is the future value:

F V = A(1 + g)n 1 −1+r_1+g n 1 − 1+r 1+g

Multiplying the growth factor from the first term, (1 + g)n,with the numerator we get:

F V = A(1 + g)

n_{− (1 + r)}n 1 −_1+g1+r

Note that this formula calls for A, which is not the first term of the end-of-period annuity, that is A(1+g), but the first term divided by (1+g). With this formula we calculate the future value of the series of five payments above as:

F V = (100/1.05)(1 + 0.05) 5_{− (1 + 0.1)}5 1 − 1+0.1 1+0.05 = 668.46

Checking this with the compounded present value we see that the result is the same: 415.06 × (1 + 0.1)5= 668.46. Expressions for an annuity given the present or future value are a matter of simple algebra.

Perpetuities

Perpetuities are annuities with an infinite number of payments. In technical terms this means that n becomes infinite, as does the future value. To see the effect on annuity present value, look at the formula for a growing end-of-period annuity in (2.5). The term affected by the number of periods is the ratio ((1 + g)/(1 + r))n. For all r > g,

lim n→∞

1+g 1+r

n

= 0 so that the present value of the perpetuity becomes:

P V = A(1 + g)

r− g (2.6)

Recall that A(1 + g) is the first term. This formula is known as the Gordon growth model and is frequently used in practice. The simplification to an annuity without growth (g = 0) is straightforward:

P V = A

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Because their present values are so easy to calculate, perpetuities are often included in exercises and exam questions. But they are not just theoretical constructs. Shares are permanent investments and their valuation is commonly based on an infinite stream of dividend payments that are assumed to grow over time. There are also perpetual bonds, called consols. The issuers of such bonds are not obliged to redeem them (although they sometimes have a right to do so). Their value is entirely based on the perpetual stream of interest payments.

2.2 _...

The accounting representation of the firm

Finance is primarily concerned with market values, but accounting data (or book values) are often used in their place. This generally is the case when we look at non-traded parts of the firm, such as specific asset categories or bank loans, for which no market values are available. To produce these data, accounting uses its own set of rules (or accounting

prin-ciples), that are framed by law and professional organizations such as the International Accounting Standards Board.5These rules have to cover exceptional situations as well as large, diversified companies, so they are both extensive and complex. Since we only look at simple, stylized situations, we can disregard most accounting issues. But even in those situations accounting values and concepts may differ from the corresponding market val-ues and financial concepts, so it is important to know what the differences are and which accounting data to use.

2.2.1 Financial statements

In its very essence, a firm’s accounting system records two things: the flows of goods and money through the firm and the effects these flows have on the firm’s assets (its posses-sions) and the claims of various parties against these assets (liabilities and equity). The recorded data are reported in four financial statements that, together with the explanatory notes, make up the financial report. Firms have to publish a financial report at least yearly, but many stock exchanges also require a quarterly report from their listed firms. The four statements are the income statement, the balance sheet, the statement of cash flows and the statement of stockholders’ equity. The latter can also be published in an abbrevi-ated form, known as the statement of retained earnings. The purpose of these reports is to allow outsiders to evaluate the firm’s performance and to assess its financial position and prospects.6We shall have a brief look at all of them and introduce some accounting concepts along the way.

The income statement

The income statement reports the firm’s revenues, costs and profits over a particular period. It should give insight into the size and profitability of the firm’s operations.

...

5 www.iasplus.com/index.htm

6 Financial reports can also play a role in determining the amount of taxes a firm has to pay, but large firms usually

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19 2.2 The accounting representation of the firm

Table 2.2 Income statement ZX Co

year ended 31 December: 2011 2012

Sales 250 300

− Cost of goods sold 175 200

Gross profit 75 100

− Personal cost 15 20

− Depreciation 15 20

− other cost 5 10

Operating income 40 50

+ Financial revenue (interest received) 3 4 − Interest paid and other financial cost 13 14

Profit before taxes 30 40

− Income taxes 9 12

± Income/loss from discontinued operations – –

Net profit 21 28

A considerable part of the accounting rules concerns the allocation of revenues and costs to particular periods but, as we shall see, this is much less relevant in finance. A simpli-fied example of an income statement is presented in Table 2.2. The specification of costs and revenues can vary with the characteristics of the firm and the industry. For instance, the entry ‘cost of goods sold’ is common when the firm’s activities include an element of trade, where goods are bought and re-sold after relocation and/or processing. The pur-chasing price of those goods is ‘costs of goods sold’ which is subtracted from sales to find gross profit. In manufacturing these two entries have little meaning and are usually replaced by the single entry ‘cost of raw materials’. Similarly, costs can be broken down by their nature, as is done in Table 2.2, but also by their function (marketing, distribu-tion, occupancy, administrative costs, etc.). The distinction between revenues and costs from normal operations and from financial transactions and incidental events (discontin-ued operations) is made to give investors a clearer picture of the firm’s prospects. These are usually based on the firm’s normal operations rather than financial transactions or incidental events as the sale of assets from discontinued operations.

Depreciation is the accounting way of spreading the costs of long-lived assets over time. When the purchasing price of these assets is paid, the payment is not recorded as a cost on the income statement but as an investment in assets on the balance sheet. This book value is gradually reduced (depreciated) over time by recording a predetermined amount of depreciation as a cost on the income statement in each period. As a result, costs and profits are more evenly distributed over time. It follows that depreciation is not a cash outflow in itself, i.e. no payment is made to parties outside the firm. But depreciation does influence the firm’s cash flows because it is deductible from taxable income and, hence, reduces the amount of taxes the firm has to pay. This is illustrated in Section 2.3.

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Table 2.3 Balance sheet ZX Co

at 31 December: 2011 2012

Assets:

Property, plant and equipment 225 250

− accumulated depreciation − 90 − 110

Financial assets 35 40

Intangible assets 20 25

other non-current assets 10 10

Total non-current assets 200 215

Cash, bank and marketable securities 35 40

Accounts receivable 40 50

Inventories 20 25

other current assets 5 10

Total current assets 100 125

Total assets 300 340

Liabilities and equity:

Issued capital 50 50

Retained earnings 130 150

Total equity 180 200

Long-term bank loans 55 60

other non-current borrowings 10 15

Total non-current liabilities 65 75

Account payable 45 50

other current borrowings 10 15

Total current liabilities 55 65

Total equity and liabilities 300 340

The balance sheet

A typical balance sheet is presented in Table 2.3. It gives the firm’s assets (its possessions, or what the firm’s capital is invested in) and the claims against these assets (liabilities and equity, or from which sources the firm’s capital was raised). In economic terms, the balance sheet is meant to give insight into the resources the firm has at its disposal and the firm’s financial structure, i.e. the relative importance of the various sources of capital. Note that the balance sheet refers to a particular date, not a period. The combined value of