Quantitative Methods for Financial Management Course … · 2019-10-14 · Quantitative Methods for Financial Management 4 University of London future values. The discipline that

Quantitative Methods for Financial Management

Course Introduction and Overview

Contents

1 Course Introduction and Objectives 3

2 The Course Authors 4

3 The Course Structure 5

4 Learning Outcomes 7

5 Study Materials 7

6 Study Advice 9

7 Assessment 9


2 University of London


Centre for Financial and Management Studies 3

1 Course Introduction and Objectives Welcome to the course on Quantitative Methods for Financial Management.

The aim of this course is to introduce the main concepts in the analysis of

financial securities, and to present and discuss the most important statistical

methods in applied economics and in financial management.

The use of mathematical and statistical models is rapidly becoming more

common in economic and financial analysis. The quantitative analysis of data

is often used as a guide in forecasting and in investment and portfolio deci-

sions. The literature on finance is increasingly relying on formal

mathematical models to explain the behaviour of security prices and rates of

return. It is therefore essential that you acquire a sound knowledge and

understanding of the most commonly used mathematical and statistical

methods, both in order to be able to read the recent literature on finance and

in order to develop further your professional ability in financial management.

This course starts by illustrating in Unit 1 the main types of financial securi-

ties: bonds and stocks (or shares). After defining each type of security, you

will see how we can decide among alternative investment strategies on the

basis of the expected returns that each one of them offers. The material

covered in this unit is the basis of all financial analysis, and it is crucial that

you make yourself perfectly familiar with all the concepts and methods of

this unit. The following two units introduce the main statistical ideas in

quantitative methods.

Unit 2 presents the central concepts of probability theory, which is that

branch of mathematics that deals with uncertainty. Since all financial deci-

sions are made in an uncertain environment, it is clear that the contents of

this unit are absolutely critical in all financial analysis.

Unit 3 explains what is meant by statistical inference: strictly speaking, this

deals with how to draw conclusions from a (small) random sample to a more

general (and possibly very large) population. Statistical inference is also

concerned with discovering regularities or general rules of behaviour, on the

basis of a sample of observations. Statistical inference can be applied, for

instance, for predicting future rates of return on securities, future private

investment spending, or other economic or financial variables. We can also

use statistical inference for testing whether certain hypotheses are statistically

confirmed by the observed data.

The methods for applying statistical inference to economics and finance are

studied in Units 4 to 7. The main model is regression analysis. This model

tries to explain how different economic or financial variables vary together.

For instance, the value of the stocks issued by a company could depend on

expectations about future interest rates, about the exchange rate, etc. It can

therefore be important to establish whether these variables are related to each

other, so that we can explain the value of a stock and possibly forecast its



future values. The discipline that applies regression analysis to economics

and finance is called econometrics.

Units 4 and 5 present the simplest example of econometric model, in which

we explain the behaviour of a variable we are interested in (aggregate in-

vestment spending, for instance) by using one explanatory variable (such as

the rate of interest). You will study the assumptions underpinning this model,

how to estimate the model, and how to use it for making statistical inferences

and for forecasting.

Unit 6 generalises the model examined in Units 4 and 5 to enable us to

explore the joint effect of several explanatory variables. For instance, private

investment spending could be a function both of interest rates (consistent

with a classical model of investment) and of expected changes in aggregate

demand (consistent with the accelerator model which is also referred to in the

course Macroeconomic Policy and Financial Markets). Unit 6 introduces the

multiple linear regression model, and explains how to carry out statistical

inference when more than one explanatory variable is present.

Unit 7 examines some more advanced topics in econometrics, and illustrates

how a number of issues in economics and finance can be analysed using

these advanced methods.

Finally, Unit 8 brings together all the main ideas and concepts of the course.

It explains the principles of investment under uncertainty and of portfolio

analysis. You will learn how to measure the risk of an investment project,

and study the principles of diversification. The unit also examines how the

econometric methods studied in the previous units can be applied to the

measurement and analysis of risk.

2 The Course Authors Dr Pasquale Scaramozzino is a Reader in Economics at the Centre for

Financial and Management Studies, SOAS, University of London, where he

is Academic Director for the PhD Programme. Dr Scaramozzino has taught

at the University of Bristol, at University College London and at Università

di Roma ‘Tor Vergata’. His research articles in finance and in economics

have been published in academic journals, including The Economic Journal,

Journal of Comparative Economics, Journal of Development Economics,

Journal of Environmental Economics and Management, Journal of Industrial

Economics, Journal of Population Economics, The Manchester School,

Metroeconomica, Oxford Bulletin of Economics and Statistics, Oxford

Economic Papers and Structural Change and Economic Dynamics. He has

also published extensively in medical statistics.

Dr Scaramozzino has taught Risk Management for the on-campus MSc in

Finance and Financial Law in London and has contributed to several off-

campus CeFiMS courses, including Mathematics and Statistics for Econo-



mists, Portfolio Analysis and Derivatives, Quantitative Methods for Finan-

cial Management and Managerial Economics.

Nir Vulkan is a University Lecturer in Business Economics at the Said

Business School, University of Oxford, and a Fellow of Worcester College.

He received his BSc (Maths and Computer Science) from Tel Aviv Univer-

sity and his PhD (in Economics) from University College, London. His

research interests are the economics of electronic commerce, and more

specifically, economic design, especially in the context of automated trading

and automated negotiations. He has published articles in major economics

journals and a number of AI journals. He co-operated with a number of

leading agent researchers from computer science and worked as a consultant

to Hewlett Packard for a number of years focusing on multi agent systems.

He is the author of The Economics of E-Commerce: A Strategic Guide to

Understanding and Designing the Online Marketplace, published by

Princeton University Press.

Nir has also co-authored the MSc Financial Management course on

Managerial Economics and tutored extensively for CeFiMs as well as

teaching in Mozambique and Singapore.

The work on adapting this course to the econometric software Eviews has

been done by Luca Deidda. Dr Deidda joined the Centre for Financial and

Management Studies at SOAS in 1999, as lecturer in financial studies. His

research focuses on financial and economic development, markets under

asymmetric information and welfare effects of financial development. He is

currently working at the Università di Sassari, Sardinia.

3 The Course Structure The course is divided into eight units of text and reading.

Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks

1.1 Introduction to Unit 1 1.2 Net Present Value 1.3 Annuities and Perpetuities 1.4 Valuing Bonds 1.5 Valuation of Common Stocks 1.6 Alternative Investment Criteria

Unit 2 Statistical Concepts and Probability Theory

2.1 Introduction 2.2 Moments of a Probability Distribution 2.3 Some Important Probability Distributions

Unit 3 Statistical Inference

3.1 Introduction 3.2 Estimation 3.3 Hypothesis Testing



Unit 4 The Classical Linear Regression Model

4.1 Introduction 4.2 The Meaning of Regression Analysis 4.3 The Regression Model and its Statistical Parameters 4.4 Actual and Fitted Values – the Regression Line and the Error Term 4.5 The Meaning of the Linearity Assumption 4.6 The Method of Ordinary Least Squares (OLS) 4.7 Some Examples

Unit 5 Statistical Inference in the Classical Linear Regression Model

5.1 Introduction 5.2 The Classical Linear Regression Model (CLRM) 5.3 The Variance and the Standard Error of the Parameter Estimators 5.4 Properties of the OLS estimators 5.5 Confidence Intervals and Hypothesis Testing 5.6 Goodness of Fit – the Correlation Coefficient r and the Coefficient of

Determination R 2 5.7 Forecasting

Unit 6 The Multiple Linear Regression Model

6.1 Introduction 6.2 The Multiple Linear Regression Model 6.3 OLS Estimation 6.4 The Multiple Coefficient of Determination 6.5 Hypothesis Testing in the Multiple Regression Model 6.6 An Exercise — The Demand for Money 6.7 Model Selection and the Adjusted Coefficient of Determination 6.8 Choice of the Functional Form

Unit 7 Topics in the Multiple Linear Regression Model

7.1 Introduction 7.2 Definition of Dummy Variables 7.3 Use of Dummy Variables to Compare Regressions 7.4 Autocorrelation of the Error Terms 7.5 Tests for Autocorrelation – the Durbin-Watson Test 7.6 Estimation of Models with Autocorrelated Disturbances 7.7 Dynamic Models and the Error Correction Mechanism 7.8 An Example 7.9 Conclusions

Unit 8 Risk Measurement and Investment Decisions

8.1 Introduction 8.2 Risk and Return 8.3 The Capital Asset Pricing Model 8.4 Arbitrage Pricing Theory (APT) 8.5 Estimation of the CAPM



4 Learning Outcomes When you have completed this course, you will be able to do the following:

• compute the Net Present Value of an investment project and apply the main investment evaluation criteria

• explain what is meant by probability and show how it can be applied in finance

• discuss the main concepts of statistical inference (estimation and hypothesis testing)

• explain and discuss how statistics can be applied to analyse relationships between financial variables

• apply statistical regression analysis to problems in finance

• measure the risk of a financial investment portfolio.

5 Study Materials The materials provided for this course comprise the course guide, presented

in eight units of text covering the quantitative techniques most useful in

financial management, and two textbooks.

The Study Guide

As noted in the section on Course Structure, these are divided into eight units

of work. The units set out the main topics of study, guide your reading of the

textbooks and set exercises for you to complete. The course is designed so

that you should be able to complete one unit per week, but this does vary

according to how recently you have been involved in formal study. You may

well find that you get through the materials more quickly as you become

accustomed to studying them.

Textbooks

This course is based on two textbooks. The first one is

Richard Brealey, Stewart Myers and Franklin Allen (2008) Principles

of Corporate Finance, ninth edition, New York: McGraw-Hill

and the second one is

Damodar Gujarati and Dawn Porter (2010) Essentials of Econometrics, fourth edition, McGraw-Hill

Both textbooks are very well known and widely adopted for advanced

university study. They have been chosen for this course because they are both

extremely clear, and because each one of them contains a large number of

examples and exercises that complement the explanations and questions in

the units. The lecture notes in the units are closely related to the presentation

in the textbooks. We explain the main ideas and methods you must learn, and

point to where you can find an additional discussion in the textbooks. We



often try to offer a slightly different perspective, so that you can capture

additional features of the issues analysed.

Eviews

You have been provided with a copy of Eviews 6, Student Edition. This is the

econometrics software that you will use to do the exercises in the later units

of this course, and possibly also the data analysis part of your assignments.

The results presented in the units are from Eviews.

Instructions to install Eviews, and to register your copy of the software, are

included in the booklet that comes with the Eviews CD. (Your student edition

of Eviews will run for two years after installation, and you will be reminded

of this every time you open the program.)

There is excellent, comprehensive On-Line Help provided by Eviews. You

can access the Eviews Help information in a number of ways. Perhaps the

easiest is to go to Help on the top toolbar, then Eviews Help Topics...

This opens Internet Explorer and loads the Eviews Help files (these are

installed on your computer when you initially install Eviews). You can then

look through the Contents, use an A-Z Index, or use the Search facility.

Eviews Help Topics...links to the Users Guide I, Users Guide II, and the

Command Reference (more on Commands later). If you prefer, you can

access these pdf files directly, again via the Help button in Eviews. The pdf

file Users Guide I includes the contents pages for Users Guide I and Users

Guide II, and the entries in the contents pages link to the relevant pages in the

files. You can also search within the pdf files.

Important Note

You must register your copy of Eviews within 14 days of installing it on your computer. If you do not register your copy within 14 days, the software will stop working.

Eviews is very easy to use. Like any Windows program, you can operate it in

a number of ways:

• there are drop-down menus

• selecting an object and then right-clicking provides a menu of available operations

• double-clicking an object opens it

• keyboard shortcuts work.

There is also the option to work with Commands; these are short statements

that inform the program what you wish to do, and, once you have built up

your own vocabulary of useful Commands, this can be a very effective way

of working. You can also combine all of these ways of working with Eviews.

In Units 5 to 8 there are references to how Eviews helps with the exercises.



Although easy to use, Eviews is a very powerful program. There are ad-

vanced features that you will not use on this course, and you should not be

worried if you see these, either in the menus or the help files. The best advice

is to stay focused on the subject that is being studied in each unit, and to do

the exercises for the unit; this will reinforce your understanding and also

develop your confidence in using data and Eviews.

6 Study Advice The course units (or ‘Study Guide’) serve much as a lecture in a conventional

university setting, introducing you to the literature of the subject under study

and helping you to identify the core message of each reading you are as-

signed. As you work through the units, you should study the readings as

suggested and answer the questions set.

The objectives of the units are set out in the introductory section preceding

each unit, and it’s a good idea to review these when you have finished that

unit’s work to make sure that you can indeed complete each task suggested.

These are the sorts of issues you are likely to meet in examination questions

and your ability to write on them should prepare you well for success in the

course.

Throughout this course, it is essential that you do all the readings and solve

all the exercises you are asked to do. In quantitative methods, each idea

builds on the previous ones in a logical fashion, and it is important that each

idea is clear to you before you move on. You should therefore take special

care not to fall behind with your schedule of studies – if you follow your

schedule and keep up with the readings, exercises and assignments, by the

end of the course you will develop a good understanding of quantitative

methods.

Lastly, answers to the exercises are provided at the end of the unit, for you to

check that you have understood and done the exercises correctly. If you do

the exercises yourself, you will develop a good understanding of the course

materials, and the models and methods described in the units; you will also

become more confident using these methods and using Eviews.

7 Assessment Your performance on each course is assessed through two written

assignments and one examination. The assignments are written after

week four and eight of the course session and the examination is written

at a local examination centre in October.

The assignment questions contain fairly detailed guidance about what is

required. All assignment answers are limited to 2,500 words and are marked

using marking guidelines. When you receive your grade it is accompanied by

comments on your paper, including advice about how you might improve,



and any clarifications about matters you may not have understood. These

comments are designed to help you master the subject and to improve your

skills as you progress through your programme.

The written examinations are ‘unseen’ (you will only see the paper in the

exam centre) and written by hand, over a three hour period. We advise that

you practice writing exams in these conditions as part of you examination

preparation, as it is not something you would normally do.

You are not allowed to take in books or notes to the exam room. This means

that you need to revise thoroughly in preparation for each exam. This is

especially important if you have completed the course in the early part of the

year, or in a previous year.

Preparing for Assignments and Exams

There is good advice on preparing for assignments and exams and writing

them in Sections 8.2 and 8.3 of Studying at a Distance by Talbot. We rec-

ommend that you follow this advice.

The examinations you will sit are designed to evaluate your knowledge and

skills in the subjects you have studied: they are not designed to trick you. If

you have studied the course thoroughly, you will pass the exam.

Understanding assessment questions

Examination and assignment questions are set to test different knowledge and

skills. Sometimes a question will contain more than one part, each part

testing a different aspect of your skills and knowledge. You need to spot the

key words to know what is being asked of you. Here we categorise the types

of things that are asked for in assignments and exams, and the words used.

All the examples are from CeFiMS examination papers and assignment

questions.

Definitions

Some questions mainly require you to show that you have learned some concepts, by setting out their precise meaning. Such questions are likely to be preliminary and be supplemented by more analytical questions. Generally ‘Pass marks’ are awarded if the answer only contains definitions. They will contain words such as:

Describe Define Examine Distinguish between Compare Contrast Write notes on Outline What is meant by List



Reasoning

Other questions are designed to test your reasoning, by explaining cause and effect. Convincing explanations generally carry additional marks to basic definitions. They will include words such as:

Interpret Explain What conditions influence What are the consequences of What are the implications of

Judgment

Others ask you to make a judgment, perhaps of a policy or of a course of action. They will include words like:

Evaluate Critically examine Assess Do you agree that To what extent does

Calculation

Sometimes, you are asked to make a calculation, using a specified technique, where the question begins:

Use indifference curve analysis to Using any economic model you know Calculate the standard deviation Test whether

It is most likely that questions that ask you to make a calculation will also ask for an application of the result, or an interpretation.

Advice

Other questions ask you to provide advice in a particular situation. This applies to law questions and to policy papers where advice is asked in relation to a policy problem. Your advice should be based on relevant law, principles, evidence of what actions are likely to be effective.

Advise Provide advice on Explain how you would advise

Critique

In many cases the question will include the word ‘critically’. This means that you are expected to look at the question from at least two points of view, offering a critique of each view and your judgment. You are expected to be critical of what you have read.

The questions may begin

Critically analyse Critically consider Critically assess Critically discuss the argument that



Examine by argument

Questions that begin with ‘discuss’ are similar – they ask you to examine by argument, to debate and give reasons for and against a variety of options, for example

Discuss the advantages and disadvantages of Discuss this statement Discuss the view that Discuss the arguments and debates concerning

The grading scheme

Details of the general definitions of what is expected in order to obtain a

particular grade are shown below. Remember: examiners will take account of

the fact that examination conditions are less conducive to polished work than

the conditions in which you write your assignments. These criteria

are used in grading all assignments and examinations. Note that as the criteria

of each grade rises, it accumulates the elements of the grade below. As-

signments awarded better marks will therefore have become comprehensive

in both their depth of core skills and advanced skills.

70% and above: Distinction As for the (60-69%) below plus:

• shows clear evidence of wide and relevant reading and an engagement with the conceptual issues

• develops a sophisticated and intelligent argument

• shows a rigorous use and a sophisticated understanding of relevant source materials, balancing appropriately between factual detail and key theoretical issues. Materials are evaluated directly and their assumptions and arguments challenged and/or appraised

• shows original thinking and a willingness to take risks

60-69%: Merit As for the (50-59%) below plus:

• shows strong evidence of critical insight and critical thinking

• shows a detailed understanding of the major factual and/or theoretical issues and directly engages with the relevant literature on the topic

• develops a focussed and clear argument and articulates clearly and convincingly a sustained train of logical thought

• shows clear evidence of planning and appropriate choice of sources and methodology

50-59%: Pass below Merit (50% = pass mark)

• shows a reasonable understanding of the major factual and/or theoretical issues involved

• shows evidence of planning and selection from appropriate sources,

• demonstrates some knowledge of the literature

• the text shows, in places, examples of a clear train of thought or argument

• the text is introduced and concludes appropriately



45-49%: Marginal Failure

• shows some awareness and understanding of the factual or theoretical issues, but with little development

• misunderstandings are evident

• shows some evidence of planning, although irrelevant/unrelated material or arguments are included

0-44%: Clear Failure

• fails to answer the question or to develop an argument that relates to the question set

• does not engage with the relevant literature or demonstrate a knowledge of the key issues

• contains clear conceptual or factual errors or misunderstandings

[approved by Faculty Learning and Teaching Committee November 2006]

Specimen exam papers

Your final examination will be very similar to the Specimen Exam Paper that

you received in your course materials. It will have the same structure and

style and the range of question will be comparable.

CeFiMS does not provide past papers or model answers to papers. Our

courses are continuously updated and past papers will not be a reliable guide

to current and future examinations. The specimen exam paper is designed to

be relevant to reflect the exam that will be set on the current edition of the

course.

Further information

The OSC will have documentation and information on each year’s

examination registration and administration process. If you still have ques-

tions, both academics and administrators are available to answer queries.

The Regulations are also available at ,

setting out the rules by which exams are governed.





UNIVERSITY OF LONDON Centre for Financial and Management Studies

MSc Examination Postgraduate Diploma Examination

for External Students

91DFM C219

91DFM C319

FINANCE

FINANCIAL MANAGEMENT


Specimen Examination

This is a specimen examination paper designed to show you the type of

examination you will have at the end of the Quantitative Methods for

Financial Management course. The number of questions required and the

structure of the examination will be the same, but the wording and

requirements of each question will be different. Good luck with your final

examination.

The examination must be completed in THREE hours. Answer FOUR

questions, comprising TWO questions from EACH section. Answer ALL

parts of multi-part questions.

The examiners give equal weight to each question; therefore, you are advised

to distribute your time approximately equally over four questions. The

examiners wish to see evidence of your ability to use technical models and of

your ability to critically discuss their mechanisms and application.

Statistical tables are provided as an enclosure.

Candidates may use their own electronic calculators in this examination

provided they cannot store text; the make and type of calculator MUST

BE STATED CLEARLY on the front of the answer book.

Do not remove this Paper from the Examination Room. It must be attached to your answer book at the end of the

examination.

© University of London, 2007 PLEASE TURN OVER



Section A (Answer TWO questions from this section)

1. Answer all parts of the question.

a. What should be the interest rate so that you prefer an annuity which pays £500 for 15 years, over another annuity which pays £800 for 10

years?

b. Calculate the price of a perpetuity with par value of £1000, a 13% coupon and current yield of 10%. How would your answer change if

the bond matured after 10 years?

c. ‘NPV is by the far the most robust evaluation criterion available to the financial manager’. Critically discuss this statement.


a. Two fair dice are thrown:

i. what is the probability of getting the same outcome in both?

ii. what is the probability of getting 5:5?

iii. what is the probability of getting 5:4?

iv. what is the probability of getting a sum (of both outcomes) which

is between 4 and 6 (inclusive of both)?

v. What is the probability of not getting a 6?

b. Suppose that the number of matches in a box are approximately normally distributed with mean 114, and standard deviation of 7. Find the probability that a matchbox choosen at random will contain a

number:

i. greater than 121?

ii. less then 97?

iii. between 110 and 123?

iv. the factory operates a quality control policy, where 15% of the

match-boxes containing the smallest number of matches are being re-packaged. How many matches in a box will ensure it does not have to be re-packaged?

c. Explain, using examples, the relationship between the t-distribution and the normal distribution.


a. To estimate the mean value of purchases of card holders in a month, a

credit card company takes a random sample of twelve monthly statements and obtains the following amounts (in dollars):

$91.21 $98.26 $143.62 $65.93 $95.08 $159.11 $34.27 $127.26 $211.87 $53.91 $139.53 $87.80

Assuming that the population distribution is normal, find a 90% confidence interval for the mean monthly value of purchases of all card

holders.



b. A manufacturer of detergent claims that the contents of boxes sold weigh on average at least 16 ounces. The distribution of weights is known to be normal, with standard deviation 0.4 ounces. A random

sample of 16 boxes yields a sample mean weight of 15.84 ounces.

Test the null hypothesis that the population mean weight is at least 16 ounces.

c. Explain the relationship between point and interval estimates. When

would you prefer to use one to the other? Explain your answer.

4. Can we diversify away all risk, and create a riskless portfolio? Answer this question, explaining first what is meant by the terms ‘risk’ and

‘diversification’ in the context of portfolio selection.

Section B (Answer TWO questions from this section)


Consider the following data on the rate of inflation (X) and on private

investment spending (Y) for the period 1988-1997.

Year Y X 1988 45.5 5.4 1989 44.8 6.2 1990 46.9 6.3 1991 48.2 4.2 1992 46.2 4.8 1993 45.2 5.7 1994 44.2 6.1 1995 46.3 7.5 1996 47.2 6.8 1997 48.6 4.2

a. Compute the OLS estimators for the linear regression model Y = B1 + B2 X + u. Show your computations in detail.

b. Tabulate the fitted values and the regression residuals.

c. Using a 5% significance level, test the null hypothesis that B1 = 1.

d. Carefully interpret your results.


The following regression equation on consumption expenditures (Y) and disposable income (X) has been estimated for the period 1968-1997

(millions of dollars).

Yt = 23.07 + 0.83 Xt R2 = 0.63

SE (4.09) (0.12)

a. Interpret the above equation;

b. Compute 95% confidence intervals for the regression coefficients;

c. Using a 1% significance level, test the null hypothesis that the

slope coefficient is equal to 1;

d. Find the F ratio and test the significance of the regression

coefficient. Compare your results with those obtained in (c);

e. Interpret your results.



7. Explain carefully what is meant by autocorrelation. What are its

consequences for econometric estimation? How can it be detected? What remedial measures can be taken for estimation if the regression residuals are

autocorrelated?

8. What is the multiple coefficient of determination? How can it be used for

model selection? Explain your answer in detail.

[END OF EXAMINATION]































Contents

1.1 Introduction to Unit 1 3

1.2 Net Present Value 3

1.3 Annuities and Perpetuities 6

1.4 Valuing Bonds 8

1.5 Valuation of Common Stocks 9

1.6 Alternative Investment Criteria 13

1.7 Summary 18

References 20

Answers to Unit Exercises 21



Unit Content

Unit 1 introduces the course and the general principles of financial manage-

ment. It starts by examining the implications of the fact that future cash flows

are worth less than an equivalent amount today. This allows us to set up the

fundamental formula for the rest of this course, the net present value of a

given project. We apply this method to the most common types of financial

instruments, stocks and bonds, and show how their current value can be

calculated from this general principle. Since the net present value depends on

future cash flows, this unit also touches on how to estimate these using a

simple growth formula. Finally, the unit discusses alternative investment

criteria, and their merits.

Learning Outcomes

When you have completed your study of this unit and its readings, you will

be able to

• explain the Net Present Value (NPV) of a given project and how it is computed

• compute the NPV under different capitalisation schemes

• define and discuss annuities and perpetuities

• value bonds and stocks

• explain and use some alternative investment evaluation criteria.

Readings for Unit 1

Richard Brealey, Stewart Myers and Franklin Allen (2008) Principles

of Corporate Finance, extracts from Chapters 2, 3, 4, 5 and all of Chapter 6.



1.1 Introduction to Unit 1

In this first unit we discuss the important decision constantly faced by the

financial manager of whether or not to invest in a given project. We intro-

duce the basic principle of finance – namely, that a dollar today is worth

more than a dollar tomorrow – and examine how it can be used to evaluate

the present value of the project in question. We then apply this rule to some

common financial securities: annuities, perpetuities, bonds and stocks. Since

we know something about the behaviour of these securities, we are able to

use the present value formula in more specific (and therefore more accurate)

ways. Finally, we introduce several other investment criteria and discuss their

advantages and disadvantages compared to the present value rule.

An important feature of this unit is that everything is discussed under the

simplifying assumption of perfect information – that is, we assume that we

know, when the decision is being made, the values of all parameters. We

come back to the same investment criteria in Unit 8, after you have learnt the

basic principles of modelling uncertainties, where we again discuss invest-

ment criteria, but within the context of uncertain outcomes. However, it is

important that you first learn how to use these rules within this simplified

framework.

1.2 Net Present Value

By far the most important investment criterion in finance, the Net Present

Value (NPV) rule, allows us to evaluate a stream of future cash flows (finite

or infinite) in today’s terms. This is important, because in most situations we

are concerned with the choice today of projects that may only pay back their

initial investment at some future date. But in order to do that, we first need to

establish what is the present value of a (single) future payment. The most

fundamental principle in finance states that a dollar today is worth more than

a dollar tomorrow. This is because it can be invested to start earning interest

immediately; waiting until tomorrow will lose the corresponding interest

income. In simple mathematical terms, the present value of a cash payment,

C, a year from now is given by:

Present Value (PV) = Discount Factor C

where, Discount Factor = 1/(1 + r) (1.1)

and r is the rate of interest you could have earned on the money had it been

invested between now and the date of the (single) payment; this is also

known as the opportunity cost of capital. Since the above holds true for any

payment, it can be summed over a stream of future payments. In other words,

the present value of an investment is given by the sum of its appropriately

discounted cash flows.



In reality, there is no reason why the discount factor should not change over

time. However, to make things simple, in this unit we will deal with the case

where the interest or discount rate is assumed to remain constant. This may

sound like a very restrictive assumption, but as you will learn in Unit 2, by

thinking of r as the expected interest rate, this assumption can be justified.

The assumption that r stays fixed over time allows us to find the discount

factors of cash flows at any time in the future, in similar terms. To see why,

consider the present value of one dollar in two years’ time. The dollar can be

invested and will be worth 1(1 + r) at the end of the first year. This sum can

be re-invested for a further year, and at the end of the second year it will be

worth 1(1 + r)(1 + r) = (1 + r)2. Solving backward, the present value of a cash flow C2 in two years time is C2/(1 + r)2. It is now easy to see how to

work out the present value of a cash flow in three, four or any given number

of years ahead.

Using these calculations, we can now look at the present value of an invest-

ment with a finite number, n, of annual cash flows:

PV =

C1

(1+ r)+

C2

(1+ r)2+

C3

(1+ r)3 ... +

Cn

(1+ r)n=

Ct

(1+ r)tt=1

n

(1.2)

Or, for that matter, an infinite number of cash flows (such as, for example,

the rents from an office building):1

PV =

C1

(1+ r)+

C2

(1+ r)2+

C3

(1+ r)3 ... =

Ct

(1+ r)tt=1

(1.2')

Finally, the initial cost of the investment needs to be added to the above equation. For this it is conventional to use C0 (where C0 is normally a nega-

tive number, corresponding to the initial cost). Together, this gives rise to the

concept of the Net Present Value (NPV) of an investment:

NPV = PV – required investment (1.3)

Exercises

1 Calculate the NPV of each of the following investments. The opportunity cost of capital is 20% for all four investments (or r = 0.20):

Investment Initial Cash Flow C 0 Cash Flow in Year 1 C1

1 –10,000 +20,000

2 –5,000 +12,000

3 –5,000 +5,500

4 –2,000 +5,000

Which investment is most valuable?

1 The sum of an infinite series of positive numbers may seem unbounded, but if these numbers

become smaller and smaller, as in our formula, the infinite sum may very well converge to a finite

number. For example, the infinite sum 0.5 + 0.52 + 0.53 +...+ 0.5n +... converges to 1.



2 An investment produces the following cash flows: $432 in the first year, $137 in the second, and $797 in the third. Assuming r = 0.15, what is the present value of this project?

Answers to exercise questions are provided at the end of the unit.

Reading

Please turn now to your textbook by Brealey, Myers and Allen, and read from the subsection ‘A Fundamental Result’ on p. 22 to the end of the chapter, p. 30; be sure that you can answer the following questions after you have finished it:

why can we assume that the discount rate is the same for all investors, regardless of their personal tastes, if we have a well-functioning capital market?

does the evidence support the assumption that managers act in such a way as to maximise the net present value?

do managers look after their own interests, or those of the company they manage?

1.2.1 Capitalisation Schemes

Since r is taken to be the annual opportunity cost of capital, it fits nicely with

present value calculations of investments that pay interest once a year. But

what if interest is paid more frequently than once per year? As you saw in the

previous section, the discount rate is based on what you could have earned on

your wealth, had it been invested from today. This figure is also known as the

Forward Rate. In this section, we shall discuss different capitalisation

schemes within the context of forward rates.

Consider an annual rate of 10%. This means that one dollar today will be

worth $1.10 at the end of one year from now. But what if the interest is paid

twice a year – that is, if 5% interest were paid after six months, and another

5% at the end of the year? Intuitively, we would expect to get a little bit more

than in the case of a one-off payment because the interest we received after

the first six months is being saved – and therefore receiving interest – during

the next six months. In mathematical terms, one dollar today will be worth

1(1 + 0.05)(1 + 0.05) = $1.1025 at the end of the year, which confirms our

intuition since 1.1025 > 1.10. This idea can easily be formulated in the

following way: the forward rate of $1, with r percent annual nominal interest

paid m times a year, over t years is:

F = (1 + r/m)mt (1.4)

The following table shows how a 10% annual discount rate gives rise to

different effective annual rates, based on the frequency of the interest pay-

ments. Notice that, for a given annual nominal interest rate, the effective

annual rate increases with the frequency of the payments.

Richard Brealey,

Stewart Myers and

Franklin Allen (2008) Principles of Corporate

Finance, the final

sections of Chapter 2, ‘Present Values, the

Objectives of the Firm

and Corporate Governance’.



Table 1.1 Effective Annual Interest Rate for Different Compounding Intervals (nominal interest rate r = 10%)

Compounding Interval

Interest Rate Factor

Effective Annual Rate

Interest Rate Factor

Annual (1 + r ) 10.00% 1.1000

Semi-Annual (1 + r /2)2 10.25% 1.1025

Quarterly (1 + r /4)4 10.38% 1.1038

Monthly (1 + r /12)12 10.47% 1.1047

Daily (1 + r /365)365 10.52% 1.1052

From Table 1.1 it is straightforward to see how the effective annual rate can

be computed: we simply calculate the value of one dollar after one year.

Formally,

(1 + reffective annual) = (1 + rquoted/m)m

Solving for the effective annual rate:

Effective annual rate = (1 + rquoted/m)m – 1 (1.5)

Of course, once the effective annual rate is known, we can go back to our

previous discussion of present value and substitute it for the discount rate.

For example, a single payoff of C at the end of t years which is being dis-

counted m times per year, is worth now:

PV =C

1+ (r /m)[ ]mt (1.6)

And the rest of the formulae can be modified in the same way.

As we have seen, the more frequently interest is paid, the higher will the

effective annual rate be (for a fixed nominal interest rate). Suppose, now, that

we take this idea to its limit – that is, suppose that interest is being paid

continuously (i.e. every fraction of a second). What would be the forward

rate in such a case? Formally, we need to take the mathematical limit of

equation (1.4) when m . The outcome of this is:

F = ert

which can be substituted in (1.6) as the interest factor, in order to get

PV = C . e–rt

This method is known as continuous time discounting, and is often used in

evaluating investments that pay interest very frequently.

1.3 Annuities and Perpetuities

In the previous section we introduced the NPV rule. Although formulae (1.2)

and (1.2') are straightforward to use, it turns out that, for many types of

common financial instruments, these can be simplified even further (perhaps

that is why Brealey, Myers and Allen use the title ‘looking for shortcuts’ for

their corresponding section, which you will read soon). The first type of



financial instrument we introduce is an annuity. An annuity is an asset that

pays a fixed sum at each equal period of time (year, quarter, etc.) during a

pre-specified and finite number of years. A fixed-payment mortgage loan is

an example of an annuity. When the sum, c, is paid annually over y years, the

present value of the annuity is:2

PV annuity =c

(1+ r)+

c

(1+ r)2+ +

c

(1+ r)t= c

1

r

1

r(1+ r)t (1.7)

The same formula can be used for annuities which pay quarterly or in any

other scheme, by simply replacing r in (1.7), which is the effective annual

rate corresponding to the scheme (as illustrated by the following exercise).

Exercise

Kangaroo Autos is offering free credit on a new $10,000 car. You pay $1,000 down and then $300 per month for the next 30 months. Turtle Motors next door does not offer free credit but will give you $1,000 off the list price.

If the rate of interest is 10% a year, which company is offering the better deal?

1.3.1 Perpetuities

A perpetuity is a special type of annuity, which is common enough to justify

a special subsection. It is an annuity whose payments continue to infinity.

Perpetuities are often issued by countries as a form of bond (and can, there-

fore, be seen as a way of financing debts). To find the present value of a

perpetuity which pays C forever, where the (effective) annual rate is r, all we

need to do is to take the limit of equation (1.7) when t . The second

component in the square brackets will tend to zero and the present value will

equal:

PV perpetuity = C

r (1.8)

Exercise

Find the present value of a perpetuity paying $50 every month under an interest rate of 12% per annum.

Reading

Please read now sections 3.1–3.4 in Brealey, Myers and Allen, pages 35–52, for more examples of the topics covered so far.

2 Here we are using the formula for the sum of a finite geometric series:

a(1 + x + x2 +...+xt) = a(1 – xt+1)/(1 – x), where a = c/(1 + r) and x = 1/(1 + r).

Richard Brealey, Stewart Myers and

Franklin Allen (2008)

Principles of Corporate Finance, Chapter 3

‘How to Calculate

Present Values’.



1.4 Valuing Bonds

Bonds are issued by companies or governments as a way to finance debts.

Each bond is issued with a coupon rate and a maturity date. The process

works as follows: the buyer pays a fixed sum of money, which we call the

principal (also known as the face value of the bond), and then receives

regular payments based on the coupon rates. This continues until the maturity

date, when he or she receives the last coupon payment plus the principal. For

example, a five-year US treasury bond with a coupon rate of 5% and a

principal of $2000, will pay the buyer 5% $2000 = $100 every year until

the last (fifth) year, when the buyer will receive $100 + $2000.

This description clearly spells out the cash flows from buying a bond. But, as

you saw in the previous sections, once the stream of cash flows has been

specified, formula (1.2) can be used to evaluate the present value of the bond.

Denoting by C the coupon payments, and by M the principal, and assuming a

constant opportunity cost of capital during the payment period, we have:

PV =C

1+ r( )+

C

1+ r( )2+ ...+

C + M( )

1+ r( )n (1.9)

By breaking the last payment into two parts, the coupon payment and the

principal, we can slightly modify equation (1.9) to obtain:

PV =

t =1

n C

1+ r( )t+

M

1+ r( )n (1.9')

This turns out to be useful: by examining the first part of (1.9') you can see

that it is identical to the PV of an annuity. But for annuities we have the much

simplified equation (1.7).3 Substituting into (1.9'), we get:

PV (bond) = 1

r

1

r(1+ r)n+

M

(1+ r)n (1.10)

Exercise

Suppose that a firm issues a $1,000 bond, and sets its coupon rate at 15%, which is identical to the market discount rate. Moreover, the market rate is expected to remain constant up to the bond’s maturity date, which is 15 years. Estimate the value of the bond both at the present time and at the beginning of its second year, in case you decide to sell it then.

The exercise above considers the case of bonds that pay interest on an annual

basis. Most bonds, however, pay interest on a semi-annual basis. So, to

compute the present value of such bonds, we need to adjust the present value

formula (1.10) to allow for intra-year compounding:

3 If you are not sure why equation (1.7) is so useful, try to calculate the present value of any of the

annuities described in the previous section by using the original discounted payments formula.



P =C /2

1+ r /2( )t

t=1

2n

+M

1+ r /2( )2n (1.9'')

Or, by using the same shortcut as before:

P =C

2

1

r / 2

1

r / 2 1+ r / 2( )2n +

M

1+ r / 2( )2n (1.10')

Study Note

It is important that you note, once again, that the annual coupon rate of a bond with semi-annual payments is not the effective annual rate the investor receives. The semi-annual interest rate considered above does not take the intra-year compounding into account. With intra-year payments, as we showed earlier in this unit, the effective annual rate will be higher than the coupon annual rate. In the specific case of a semi- annual compounding, the effective annual rate earned by the bondholder is equal to (1 + r/2)2 – 1. So, if the coupon rate is 8%, then the effective annual interest rate the bondholder receives is 8.16%. As a matter of convention, however, bond dealers always refer to the annual coupon rate as the interest rate paid by the bond, no matter whether it is paid on an annual or on a semi-annual interval. But you should bear in mind that whenever the bond pays semi-annual interest, the effective annual yield rate will be higher than the bond’s coupon rate.

Reading

Please now read the Chapter 3 Summary in Brealey, Myers and Allen, pages 53–54, and Section 4.1 of Chapter 4, pp. 59–63, for a review of bond valuation, and a summary of the techniques discussed in this section of the unit.

1.5 Valuation of Common Stocks

Stocks (or, shares, as they are better known in the UK and several other

countries) are issued by firms as a means of raising capital. Owners of these

shares are entitled to a proportion of the firm’s profits. The purpose of this

section is to give you the basic tools for the valuation of this very important

type of security. Although the present value principle is applied in a manner

similar to that used in the previous sections, the valuation of stocks requires

special attention.

Reading

Before we examine this, please read the introduction to Chapter 5, pp. 85–86, and Section 5.1, pp. 86–87, in Brealey, Myers and Allen for a fuller description of what stocks are and how they are traded in the US.

By owning shares, the investor is entitled to two types of income benefits:

i. dividend payments – typically based on the issuing firm’s earnings

Richard Brealey,

Stewart Myers and Franklin Allen (2008)

Principles of Corporate Finance, Chapter 3 summary and the first

section of Chapter 4

‘Valuing Bonds’

Richard Brealey,


Principles of Corporate Finance, Chapter 5 Introduction and

Section 5.1 ‘How

Common Stocks are Traded’.



ii. capital gains, which could be realised by re-selling the stock at a price higher than was initially paid for it.4

Therefore, an investor can compute the rate of return that s/he expects to

receive by the end of the next year (also known as the market capitalisation

rate)5 in the following way:

Market Capitalisation Rate = r =

DIV1 + P1 P0

P0

(1.11)

where DIV1 is the expected dividend to be paid over the current year and P1

is the expected price of the security at end of the year. Solving for the current value of the security, P0, we get:

P0 =

DIV1 + P1

1+ r (1.12)

where, as before, r is the market discount rate for securities of the same risk class. Implicitly, this assumes that, for given r and Pl, P0 is the equilibrium

value or ‘fair’ price for the stock. Further, we assume that the market ‘cor-

rects’ itself as shown by the following example.

Suppose that the stock dealer sets the price of the stock below P0. Profes-

sional investors will then buy large quantities of this security hoping to realise capital gains. The price of the stock will then be driven up to P0. This

is known as the No Arbitrage Principle, which we will come back to in Unit

8. The No Arbitrage Principle states that two investments that always deliver

the same returns, irrespective of the state of the world, must always have the same price. Similarly, if the stock price is set above P0, investors will sell

large quantities (or ‘go short’), thus driving the price back down to P0.

Of course, the problem of ‘fair’ pricing still remains when we have to deter-mine P1. In particular, different investors may have different expectations of

P1 and, as a direct result, will have different opinions as to how much they

would be willing to pay for the stock at time 0. Fortunately, there is a way

out. To see how, first notice that equation (1.12) can be generalised to

determine the price of the stock at time T–1, as a function of its dividend and

price at time T. Formally,

PT 1 =

DIVT + PT1+ r

(1.12')

Now, we can substitute the above expression into the (similar) expression for PT–2, and substitute that into the expression for PT–3 and so on, until we arrive

at the original price at time 0. What we get is that the price at time 0 depends

only on the cash flows provided by the dividend payments and the price at

time T. Formally,

4 Or at a lower price – that is, the capital gain may be negative. In this case the investor would incur a capital loss.

5 Note that this differs from what in the UK is known as market capitalisation of a company – that is, the share price and number of shares.



P0 =

DIVt

(1+ r)tt=1

T

+PT

(1+ r)T (1.13)

At this stage, you may be asking yourself, what did we gain by this exercise?

The price still depends on some unknown future price. Do not give up! Here

is the trick that will help us out of this problem: if the firm is expected to

survive into the far future, then T , and so PT/(1 + r)T 0 (this depends,

of course, on the rate of growth being smaller than r, which is a reasonable

assumption). That is, the significance of the price of the stock in the far

future to its current value becomes negligible. As a result, equation (1.13) can

now be written as:

P0 =

DIVt

(1+ r)tt=1

(1.14)

Equation (1.14) is fundamental in finance. What it is saying is that the current

value of the stock is determined by the present value of the expected divi-

dends to be paid by the firm.

Study Note

This pricing formula seems harmless at first sight but, at a closer look, rests on a very important assumption – the efficiency of financial markets. In particular, efficiency implies that all the available information that may have an effect on the price of the stock is immediately reflected in its price. In other words, individual investors cannot have beliefs that are inconsistent with the available information. If they do, then they must believe that the stock is either underpriced or overpriced.

In either case, these investors can be taken advantage of by professional investors who correctly interpret the information. In other words, investors holding beliefs not consistent with the available information will be wiped out from the market. We will come back to this in Unit 8, but what we can already see is that the ‘efficient markets’ assumption implies homogeneous beliefs, and fully justifies equation (1.14).

1.5.1 The Constant Growth Formula

Equation (1.14) eliminates some of the uncertainties in the valuation of

stocks. Still, it requires information about the flow of dividend payments.

If we believe that the dividends of a certain stock will increase along a stable

path, equation (1.14) can be simplified even further. In particular, denote by g the (constant) growth rate of the stock in question. That is, DIV2 =

DIV1(1 + g) and in general, DIVT = DIV1(1 + g) T–1. Substituting into (1.14),

and using the formula for the sum of a geometric series, we get6

6 Of course, the infinite sum of a geometric series is only defined when the ratio of the series are smaller than one. Applied to our case, this means that equation (1.15) holds only when (1 + g)/(1 +

r) < 1, which implies that g, the anticipated rate of growth of the stock dividends, is smaller than r, the discount rate.



P0 =DIV1(1+ g)

i 1

(1+ r)ii=1

=

DIV1

1+ r( )

11+ g( )

1+ r( )

=DIV1

(r g) (1.15)

A nice feature of equation (1.15) is that it holds even if the growth rate is not

constant – but ‘almost’ constant. By almost constant, we have in mind that

the growth rate may differ from one year to another, but that these different

growth rates are centred around a fixed growth pattern.

This is illustrated nicely in Figure 1.1.

Note that financial analysts use econometric methods (some of which are

illustrated in Units 4 to 7 of this course) to separate permanent from tempo-

rary dividend components. In the long run, the temporary component will

only have a negligible effect, and therefore equation (1.15) can be used by

substituting the permanent component into g.

Figure 1.1 Dividend Growth Rate Pattern

The growth pattern of dividends depends also on the investment decisions of

the firm in question. To make the case clear, one can think of two extremes:

on the one hand, the firm may distribute all of its profits to its shareholders,

making investors better-off in the short run, but making the company worse-

off in the long run. On the other hand, the firm might decide to re-invest all

of its profits in a way that maximises its long-run growth opportunities. It

should be easy to see the tension between these two extremes. In reality,

firms adopt investment behaviours which are somewhere in the middle.

However, the inverse relationship between generous dividends and long-term

growth always holds. A useful way of summarising this is:

P0 =

EPS1

r+ PVGO (1.16)

where the so-called Earning Per Share, EPS1 is the value of earnings per

share that the company could generate under the ‘generous dividend’ scheme

described above, and PVGO (Present Value of Growth Opportunities)

represents the proportion of profits re-invested in growth.



Reading

For a summary of the valuation of stocks and its relation to growth, please read now Sections 5.2–5.4 in Brealey, Myers and Allen, pp. 88–102, and the extract printed below.

What Do Price–Earnings Ratios Mean?

The price–earnings ratio is part of the everyday vocabulary of investors

in the stock market. People casually refer to a stock as ‘selling at a high

P/E’. You can look up P/Es in stock quotations given in the newspaper. (However, the newspaper gives the ratio of current price to the most

recent earnings. Investors are more concerned with price relative to

future earnings.) Unfortunately, some financial analysts are confused about what price–earnings ratios really signify and often use the ratios

in odd ways.

Should the financial manager celebrate if the firm’s stock sells at a high

P/E? The answer is usually yes. The high P/E shows that investors think

that the firm has good growth opportunities (high PVGO), that its earnings are relatively safe and deserve a low capitalization rate (low r),

or both. However, firms can have high price–earnings ratios not because

price is high but because earnings are low. A firm which earns nothing

(EPS = 0) in a particular period will have an infinite P/E as long as its shares retain any value at all.

Are relative P/Es helpful in evaluating stocks? Sometimes. Suppose you

own stock in a family corporation whose shares are not actively traded.

What are those shares worth? A decent estimate is possible if you can

find traded firms that have roughly the same profitability, risks, and growth opportunities as your firm. Multiply your firm’s earnings per

share by the P/E of the counterpart firms.

Does a high P/E indicate a low market capitalization rate? No. There is

no reliable association between a stock’s price–earnings ratio and the

capitalization rate r. The ratio of EPS to P0 measures r only if PVGO = 0 and only if reported EPS is the average future earnings the firm could

generate under a no-growth policy. Another reason P/Es are hard to

interpret is that the figure for earnings depends on the accounting procedures for calculating revenues and costs.

Brealey and Myers (2003) page 75.

1.6 Alternative Investment Criteria

The Net Present Value we have been using so far suggests that an investment

project is worthwhile if and only if the sum of discounted future profits

exceeds the initial investment cost. In other words, the manager should

choose to invest only in projects that have a positive NPV. This rule is not

only simple, it is also the best one we have. However, it is not the only rule.

In this section, we briefly describe three alternative investment criteria (these

are not the only possible criteria – Brealey, Myers and Allen describe four,

and further criteria exist as well). We will keep this discussion short and ask

you to read Chapter 6 in your textbook afterwards for more details.

Richard Brealey,


Principles of Corporate Finance, from Chapter 5 ‘The Value of

Common Stocks’.



1.6.1 The Payback Rule

The payback rule considers the period of time it takes for a project to pay

back its initial investment. The rule is then to prefer those projects with the

shortest payback period. Consider the following example, where three

projects – A, B, and C – each costing £2000, and with three annual payments,

are listed below:

Project

C0

Cl

C2

C3

Payback Period

NPV at r = 10%

A –2,000 +2,100 0 0 1 Year –91

B –2,000 +1,000 +1,000 +5,000 2 Years 3,492

C –2,000 +500 +1,000 +8,000 3 Years 5,291

As the table shows, project A will return the initial investment after one year,

B after 2 and C will only become profitable after 3 years. However, project C

has the highest NPV, with B trailing behind, and project A having a negative

NPV. What this example demonstrates, is that

i the payback rule tends to favour the short-lived projects

ii cash flows that come after the project has paid back the initial investment do not even enter the calculations.

However, if firms do not, for some reason, have access to long-term loans,

the payback period may be an important consideration, and the payback rule

may provide useful information. Of course, such considerations should come

second to the NPV rule – a project that pays back quickly, but which pro-

duces a negative NPV (like project A in our example above), should never be

chosen.

1.6.2 Internal Rate of Return

Consider once again equation (1.2) for the present value of cash flows.

Suppose now that the price of the investment and the cash flows are known.

We can now use the same formula to ask what kind of rates would equate the

discounted cash flows with a Net Present Value of zero:

NPV = C0 +

C1

(1+ IRR)+

C2

(1+ IRR)2+ ...+

Cn

(1+ IRR)n= 0 (1.17)

The Internal Rate of Return (IRR) rule then suggests that if this rate is higher

than that for assets of the same risk class, then the investment should be

undertaken. Since the IRR rule uses the same equation as the NPV rule, one

would expect that the final result of both rules should be the same. This is

true in the context of a single investment project with one payoff period but,

as we explain below, it is not necessarily true in the context of mutually

exclusive investment projects.

It is easy to see that equation (1.17) when solved for the IRR will generate an

n-order polynomial equation in IRR and so has n roots. This polynomial will

have a unique solution either when we consider a one-payoff cash flow



(because we have a linear function), or in the case of no inversions of signs in

the cash flows. That case is illustrated in Figure 1.2.

Figure 1.2 Cash Flow without Inversions

Unfortunately, this is not the case in most applications. To see why, consider

the following example of a two-year project with the following cash flows:

C0 = –4,000

C1 = 25,000

C2 = –25,000

Applying equation (1.17) to the above cash flows we get the following

equation:

4,000 +

25,000

(1+ IRR)

25,000

(1+ IRR)2= 0

which has two solutions: IRR = 25% and IRR = 400%.

What if 25% < r < 400% – say,

r = 30%?

The IRR does not provide us with a clear-cut solution to this problem.

However, no such difficulties exist when applying the NPV rule: for any

given r, the NPV rule returns a clear-cut answer (in our example, for r = 30%

the NPV is negative and the project should be rejected).

The second problem with the IRR rule occurs when equation (1.17) does not

have any solution. Consider, for example, a project with the following cash

flows:

C0 = +1,000

C1 = –3,000

C2 = +2,500

Substituting into equation (1.17) we get:

1,0003,000

(1+ IRR)+

2,500

(1+ IRR)2= 0



which has no real solution (try to solve it and see why). But for a given

discount rate, say 10%, the NPV rule has suggested that the project is worth-

while, since the NPV is equal to 339.

A third problem with the IRR rule is that we are unable to distinguish be-

tween projects which have the ‘opposite’ cash flows – known as ‘lending’

and ‘borrowing’ projects. For example, consider a project which costs £2000,

and which pays £1500 in the first year and £1000 in the second – and the

same project from the point of view of the borrower, who gets a positive cash

flow of £2000 initially, but then two negative payments of £1500 and £1000

over the next two years. Since the cash flows generated by these projects will

be identical, except for their signs, the corresponding polynomials will have

the same solutions (since the negative of the same cash flows will be equal to

zero if and only if the positive cash flows are equal to zero).

In the example above, IRR = 17.5% is the solution for both projects (try it

yourself, by substituting the above cash flows into equation 1.17). But, of

course, any r that is different from 17.5% will be good for one project and

bad for the other! To see why, consider the case when r = 10%. The NPV for

the first project (–2000, +1500, +1000) is now £326.45, whereas the NPV for

the opposite investment will be (not surprisingly) equal to –£326.45. Natu-

rally, rates greater than 17.5 will be preferred by the second project.

Finally, the NPV rule is superior to the IRR rule when it comes to making a

decision between two (or more) projects. Applying the IRR rule, we can only

find out whether each of the projects is profitable. However, if both are

profitable, the choice is not clear (it’s not true, in general, that the project

with the highest IRR is better, for a given discount rate). However, the NPV

rule lends itself to such comparisons – simply choose the project with the

highest NPV!

Still, the IRR rule has some advantages. First, it can be useful if we believe it

is likely that the discount rate could rise (and therefore investments with a

shorter time horizon will turn more profitable). If this is the case, it would be

safer to choose the investment with the highest IRR. Second, the IRR can

prove useful if a firm is faced with financial constraints and has to decide

between a project which has a higher IRR and pays the initial outlay back

much quicker, and a project which has a higher NPV but much longer

maturity. Here, a consideration of both rules would be advised.

1.6.3 The Profitability Index Rule

The profitability index is defined as the ratio between the investment’s

present value and its initial cash outflow. Formally,

Profitability Index = PV/C0 (1.18)

This rule is simply to accept an investment whenever its profitability index is

greater than one. Of course, the index will be greater than one if and only if

the present value is greater than the cash outflow, which is exactly when the



project will be accepted under the NPV rule. In other words, for any given

project, the two rules will always return the same decision.

The difference lies in the fact that the profitability index compares the

project’s PV with the initial investment cost in the form of a ratio rather than

in the form of a difference between the present value of positive cash flows

and the present value of negative cash flows (or costs). This means that when

considering the choice between two projects, the two criteria may recom-

mend different choices. In particular, the NPV will be expressed in real-terms

while the profitability index is expressed in relative terms. That is, a limited

investment that requires £10 and has a present value of £20 will have a

fantastic profitability index of 2, but in real terms will only make £10 for the

firm! A project which costs £100,000 and has a present value of £110,000

has a profitability index of 1.1, but actually earns the firm a nice profit of

£10,000. Bearing this in mind, it is easy to see why Brealey, Myers and Allen

recommend that the NPV should be preferred. Of course, the profitability

index is useful as an additional investment criterion.

Reading

For a more detailed discussion of these rules, please now read Chapter 6 of Brealey, Myers and Allen, and then the extract reprinted below, which relates to the example in section 6.4 and which completes the reading.

Some More Elaborate Capital Rationing Models

The simplicity of the profitability-index method may sometimes outweigh its limitations. For example, it may not pay to worry about

expenditures in subsequent years if you have only a hazy notion of future

capital availability or investment opportunities. But there are also circumstances in which the limitations of the profitability-index method

are intolerable. For such occasions we need a more general method for

solving the capital rationing problem. We begin by restating the problem just described. Suppose that we were to accept proportion xA of project A

in our example. Then the net present value of our investment in the

project would be 21xA. Similarly, the net present value of our investment

in project B can be expressed as 16xB and so on. Our objective is to select the set of projects with the highest total net present value. In other words

we wish to find the values of x that maximize

NPV = 21xA + 16xB + 12xC + 13xD Our choice of projects is subject to several constraints. First, total cash

outflow in period 0 must not be greater than $10 million. In other words,

10xA + 5xB + 5xC + 0xD 10 Similarly, total outflow in period 1 must not be greater than $10 million:

–30xA – 5xB – 5xC + 40xD 10 Finally, we cannot invest a negative amount in a project, and we cannot

purchase more than one of each. Therefore we have

0 xA 1, 0 xB 1, … Collecting all these conditions, we can summarize the problem as:

Maximize 21xA + 16xB + 12xC + 13xD

Richard Brealey,


Principles of Corporate Finance, Chapter 6 ‘Making Investment

Decisions with the Net

Present Value Rule’.



Subject to

10xA + 5xB + 5xC + 0xD 10

–30xA – 5xB – 5xC + 40xD 10

0 xA 1, 0 xB 1, … One way to tackle such a problem is to keep selecting different values for

the x’s, noting which combination both satisfies the constraints and gives

the highest net present value. But it’s smarter to recognize that the

equations above constitute a linear programming (LP) problem. It can be handed to a computer equipped to solve LPs.

The answer given by the LP method is somewhat different from the one

we obtained earlier. Instead of investing in one unit of project A and one

of project D, we are told to take half of project A, all of project B, and

three-quarters of D. The reason is simple. The computer is a dumb, but obedient, pet, and since we did not tell it that the x’s had to be whole

numbers, it saw no reason to make them so. By accepting “fractional”

projects, it is possible to increase NPV by $2.25 million. For many purposes this is quite appropriate. If project A represents an investment in

1,000 square feet of warehouse space or in 1,000 tons of steel plate, it

might be feasible to accept 500 square feet or 500 tons and quite

reasonable to assume that cash flow would be reduced proportionately. If, however, project A is a single crane or oil well, such fractional

investments make little sense. When fractional projects are not feasible,

we can use a form of linear programming known as integer (or zero-one) programming, which limits all the x’s to integers.

Brealey and Myers (2003) pp. 107–08.

1.7 Summary

In this unit we have shown how a combination of simple mathematical

techniques (such as the sum of a geometric series) and basic economic

principles (like time discounting, and the no-arbitrage principle) can be used

to evaluate financial securities. In particular, we have defined the present

value of an investment – as equal to the sum of its discounted cash flows. We

then defined the net present value as the difference between the present value

and the initial outflow. We concluded that an investment project is worth-

while if and only if it has a positive NPV.

Although the NPV was defined for annual returns, we showed how it can be

modified to any other capitalisation scheme. Since interest from payment is

re-invested, we concluded that the effective annual rate increases with the

frequency of payments (for a fixed nominal annual rate).

We then applied the NPV formula for securities where we know the pattern

of cash flows: annuities, perpetuities and bonds. Applying the same rule to

common stocks, we showed that the ‘fair’ price for a stock is simply equal to

the sum of its discounted dividend payments. We then showed that by

understanding the relationship between growth and generous dividend

schemes, we can impose more structure on the formula for the price of a

given stock.



Finally, we introduced three more investment criteria: the payback rule,

which measures the time it takes for a project to pay back its initial outflow,

the Internal Rate of Return rule, which seeks the rate at which the project

becomes profitable, and the Profitability Index rule, which looks at the ratio

of the present value and the initial outflow, instead of the difference. Al-

though each of these three methods has some advantages, we showed that the

NPV rule is superior, and that it is advisable to use these other rules only in

addition and not instead of the NPV rule.

Revision Exercises

To gain confidence in the use of the methods introduced in this unit, you should try the following exercises.

1 Brealey, Myers and Allen, question 14, p.55



4 Suppose Ford Motor Company sold an issue of bonds with a 10-year maturity, a $1,000 par value, a 10% coupon rate and semi-annual interest payments.

a Two years after the bonds were issued, the going rate of interest on bonds such as these fell to 6%. At what price would the bonds sell?

b Suppose that, two years after the initial offering, the going interest rate had risen to 12%. At what price would the bonds sell?

c Suppose that the conditions in Part a) existed – that is, interest rates fell to 6 per cent two years after the issue date. Suppose further that the interest rate remained at 6% for the next eight years. What would happen to the price of the Ford Motor Company bonds over time?

5 The bonds of the Beranek Corporation are perpetuities with a 10% coupon. Bonds of this type currently yield 8%, and their par value is $1,000.

a What is the price of the Beranek bonds?

b Suppose interest rate levels rise to the point where such bonds now yield 12%. What would be the price of the Beranek bonds?

c At what price would the Beranek bonds sell if the yield on these bonds were 10%?

d How would your answers to Parts a), b) and c) change if the bonds were not perpetuities but had a maturity of twenty years?

6 You believe that next year the Superannuation Company will pay dividend of $2 on its common stock. Thereafter you expect dividends to grow at a rate of 4 per cent a year in perpetuity. If you require a return of 12 per cent on your investment, how much should you be prepared to pay for the stock? [From Brealey and Myers (2003) question 6, p.84.]

7 Vega Motor Corporation has pulled off a miraculous recovery. Four years ago, it was near bankruptcy. Now its charismatic leader, a coporate folk hero, may run for president.

Vega has announced a $1 per share dividend, the first since the crisis hit. Analysts expect an increase to a “normal” $3 as the company completes its recovery over the next three years. After that, dividend growth is expected to settle down to moderate long-term growth of 6 per cent.

Vega stock is selling at $50 per share. What is the expected long-run rate of return from buying the stock at this price? Assume dividends of $1, $2, and $3



for years 1, 2, and 3. A little trial and error will be necessary to find r. (From Brealey, Myers and Allen (2006), question 14, p.81.)


9 Brealey, Myers and Allen, question 5, p. 137. Solve parts a and c only.

References

Black, D (1990) Financial Market Analysis, London: McGraw-Hill.

Brealey, Richard A, Stewart C Myers and Franklin Allen (2008)

Principles of Corporate Finance, Ninth edition, New York: McGraw-Hill International.

Brealey, Richard A, Stewart C Myers and Franklin Allen (2006)

Principles of Corporate Finance, Eighth edition, New York: McGraw-Hill International.

Brealey, Richard A and Stewart C Myers (2003) Principles of Corporate

Finance, Seventh edition, New York: McGraw-Hill International.



Answers to Unit Exercises

Section 1.2.1

1 Using formula (1.3), we have

Investment 1 NPV = –10,000 + 20,000/1.2 = $6,667

Investment 2 NPV = –5,000 + 12,000/1.2 = $5,000

Investment 3 NPV = –5,000 + 5,500/1.2 = –$417

Investment 4 NPV = –2,000 + 5,000/1.2 = $2,167

So, Investment 1 is the most valuable since it has the highest NPV.

2 Using formula (1.2), we get:

PV = 432/(1 + 0.15) + 137/(1 + 0.15)2 + 797/(1 + 0.15)3 = $1,003.28.

In other words, any price up to $1003.28, for this investment, would be a good price (compared with the cost). For any price above it, the investment is not worthwhile.

Section 1.3

You can use formula (1.7) to solve this problem provided that you find the

monthly rate equivalent to a 10% annual interest rate. That is, we want to

find a monthly interest rate which, when applied to a certain principal amount

P, produces the same final investment value F at the end of the period. This

can be done by manipulating equation (1.5) – in its original form – where

(1 + rannual)1 = (1 + rmonthly)12

Re-arranging this we obtain:

rmonthly = (1 + rannual)1/12 – 1 = 0. 008

The present value of the annuity to be paid to Kangaroo Autos can then be

computed with the help of formula (1.7):

PVkangaroo = $1,000 + 3001

0.008

1

0.008(1.008)30= $8,973

• Since a car from the other company – Turtle Motors – costs $9,000, the Kangaroo offer is definitely a better deal.

Section 1.3.1

Since C in this example is paid on a monthly basis, we have to first manipu-

late equation (1.5) to find the monthly compounded rate equivalent to a 12%

annual rate. Solving in the same way as in the previous exercise, we get:

rmonthly = [(1 + 0.12)]1/12 – 1 = 0.0095

So, the present value of the perpetuity, using equation (1.8), is:

PV = $50/0.0095 = $5,263



Section 1.4.1

The annuity paid by the bond is now C = 0.15 x 1,000 = $150. With a

discount rate of 15% we can then use equation (1.10) to get:

P = 150

1

0.15

1

0.15 1+ 0.15( )15 +

1,000

1+ 0.15( )15 =$1,000

Here we can see that the coupon rate is equal to the market rate of discount,

its present value equals its face value and, thus, the bond is realistically

priced at issue.

The present value at the beginning of the second year can be calculated in the

same way, noting that the time to maturity is now 14 years. So, the present

value of the bond at the beginning of year 2 is:

p = 1501

0.15

1

0.15 1+ 0.15( )14 +

1,000

1+ 0.15( )14 = $1,000

This is not a coincidence: provided the coupon rate is equal to the market, the

present value of the bond does not change over the successive payoff periods.

Section 1.7 Revision Exercises

1 You can calculate the present value of the l0-year stream of cash inflows either manually – using formula (1.2) with C 1 = C 2 = . . . = C

10 = 170 – or using the respective conversion factor provided in Brealey, Myers and Allen’s Appendix A Table 3. However you prefer to do it, you should get that the present value of this stream of cash inflows is 170,000 (5.216) = $886,720. Thus, the corresponding

NPV = –800,000 + 886,720 = $86,720.

At the end of five years, the factory’s value will be the present value of the five remaining $170,000 cash flows. Following the same procedure to compute present values, we have that

PV = $170,000 (3.433) = $583,610

2 To be able to compare these different interest-compounding schemes, you have to find the effective interest rate for each one of them. Using formula (1.5), we have that the effective annual rate for the semi-annual compounding is

rquoted = 11.7%, semi-annual compounding:

reffective = 1+0.117

2

2

1

= 12.04% per annum

For continuous compounding, we have seen that the value of the

principal at the end of the year is given by F = Petrquoted.



Pe trquoted = P(1 + reffective)t

erquoted – 1 = reffective

e0.115 – 1 = 12.187% p.a.

Since the continuous compounding scheme at a quoted rate of 11.5% yields the highest effective annual rate, this should be chosen.

The futures of these distinct investments after 20 years are:

r = 12%, annual compounding:

F = P (1 + 0.12)20 = 9.64P

r = 11.7%, semi-annual compounding:

F = P (1 + 0.117/2)2 20 = 9.72P

r = 11.5%, continuous compounding:

F = Pe 20 0.115 = 9.97P

where P is the initial investment.

3 You can approach this problem by solving for the present value of (1) $100 a year for 10 years and (2) $100 a year in perpetuity, with the first cash flow at year 11. If this is a fair deal, these present values must be equal, and thus we can solve for the interest rate, r.

The present value of $100 for 10 years is

p = 1001

r

1

r 1+ r( )10

while the present value at year 10 of $100 a year forever is:

P10 = 100/r

At t = 0, this present value is:

P = [1/(1 + r )10] [100/r].

Equating these two expressions for the present value yields

1001

r

1

r(1+ r)10 =

1

(1+ r)10100

r

To solve this equation manually, you have to use a method of trial and error. You will find that r = 7.18% is the solution. Note that the interest rate is such that your payment doubles in 10 years.

4a Two years after the bond was issued, the bond will have an 8-year (or 16-semester) maturity while the relevant discount rate has fallen to 6%. So the bond’s value is



PVB =t=1

16 $50

1.03( )t+

$1,000

1.03( )16

PVB = 50 12.5611( ) +1,000 0.6232( ) =$1,251.22

b Applying the same formula as before but now with r/2 = 0.06, you will find that PVB = $898.94.

c You should answer this question in an intuitive manner. With a fall in interest rates, the price of the bond will rise, as illustrated in item a).

However, as time passes (i.e. as t becomes smaller), you can see from the formula above that its second term becomes relatively more important, thus gradually offsetting the impact of the fall in the interest rate on the value of the bond. In the limit (i.e. when the bond approaches maturity), it is easy to see that the component of its value converges towards zero at year 10, when the value of the bond hits $1,000 plus accrued interest.

5a Using the perpetuity valuation formula, we have

PVB = C/r = $100/0.08 = $1,250.

b PVB = $100/0.12 = $833.33

c PVB = $100/0.01 = $1,000 (which, as expected, is equal to the par

value)

d Applying the familiar bond valuation formula at an interest of 8%, you should find

PVB = $1,196.36

For r = 12% : PVB = $ 850.61

For r = 10% : PVB = $1,000

The end result is that if the bonds are selling at a premium, the value of the 20-year bond will be less than the value of the perpetuity, while the perpetuity will have a lower value if the bonds are selling at a discount. The value of the shorter, 20-year bond fluctuates less than the longer, perpetual bond because the value of the perpetuity’s distant coupon payments fluctuates significantly as the cost of capital changes.

6 Using the constant growth formula (1.15), you find that

P0 = DIV1 / (r – g) = 2 / (0.12 – 0.04) = $25

7 As we have discussed in this unit, the value of a common stock is equal to the present value of its expected dividends. In the case of this example, expected dividends are variable up to the third year, then growing at a constant rate thereafter. So we have

P =DIV1

1+ r+

DIV2

1+ r( )2+

DIV3

1+ r( )3+

1

1+ r( )3

DIV4

r g( )

(where the coefficient 1/(1 + r)3 for DIV4 is simply converting the value

of stock from year 3 into year 0). Substituting the value of P, DIVs and g into the above expression gives you



50 =1

1+ r( )+

2

1+ r( )2+

3

1+ r( )3+

1

1+ r( )3

3 1.06( )

r 0.06( )

By trial and error (or using a financial calculator if you have one) you will find that r = 11% solves this equation and it thus provides the expected long-run rate of return offered by the stock.

8a The expected growth rate of dividends is:

The expected long-run rate of return from purchasing the stock can be computed as

In order to compute PVGO, we need to know EPS. This can be obtained from the definition of the plowback ratio:

Solving for EPS we obtain

Hence,

b The expected growth rate of dividends is

for years 1, 2, 3, 4

and 5, and for

years 6, 7, … The price of the stock is therefore:

P0 =DIVt

(1+ r)tt=1

=DIVt

(1+ r)tt=1

5

+DIVt

(1+ r)tt=6

=

DIV1

(1+ r)+

DIV1(1+ g )

(1+ r)2+

DIV1(1+ g )2

(1+ r)3+

DIV1(1+ g )3

(1+ r)4+

DIV1(1+ g )4

(1+ r)5

+

DIV1(1+ g )4(1+ g )

(1+ r)6+

DIV1(1+ g )4(1+ g )2

(1+ r)7+ ...

=

DIV1

(1+ r)

DIV1(1+ g )t

(1+ r)tt=1

4

+

DIV1(1+ g )4

(1+ r)5(1+ g )t

(1+ r)tt=1

= $114.81.

The price of the stock increases from $100 to $114.81.

9a From the number of changes in signs, we know that the maximum possible number of internal rates of return is two. Using trial and error,



or a financial calculator that solves IRR equations, you should find that there is only one positive IRR that is equal to 50%.

c

NPV = 100 +

200

1.2

75

1.22=14.58

That is, this is an attractive project.

Documents

Quantitative Methods for Financial Management Course … · 2019-10-14 · Quantitative Methods for Financial Management 4 University of London future values. The discipline that