Quantitative Methods for Financial Analyssis Sample-8

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Quantitative Methods for FinancialManagement

Course Introduction and Overview

Contents

1 Course Introduction and Objectives 32 The Course Authors 43 The Course Structure 54 Learning Outcomes 75 Study Materials 76 Study Advice 97 Assessment 9


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Quantitative Methods for Financial Management

2 University of London


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

This course starts by illustrating in Unit 1 the main types offinancial 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 ofprobability 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 privateinvestment 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 eachother, so that we can explain the value of a stock and possibly forecast its


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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 asthe 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 jointeffect 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

courseMacroeconomic Policy and Financial Markets). Unit 6 introduces the

multiple linear regression model, and explains how to carry out statisticalinference 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 themeasurement 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 Universit

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 taughtRisk Managementfor the on-campus MSc in

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

campus CeFiMS courses, includingMathematics and Statistics for Econo-


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

cial ManagementandManagerial 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 ofThe 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 softwareEviews 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 Universit di Sassari, Sardinia.

3 The Course StructureThe 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 Bonds1.5 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


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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 Coefficientrand the Coefficient ofDetermination 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 Money6.7 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 Disturbances7.7 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


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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 themain investment evaluation criteria

explain what is meant by probability and show how it can be appliedin finance

discuss the main concepts of statistical inference (estimation andhypothesis testing)

explain and discuss how statistics can be applied to analyserelationships 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 sothat 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-Hilland 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, andpoint to where you can find an additional discussion in the textbooks. We


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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 ofEviews 6, Student Edition. This is theeconometrics 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 fromEviews.

Instructions to installEviews, and to register your copy of the software, are

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

ofEviews 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 byEviews. You

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

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

This opens Internet Explorer and loads theEviews Help files (these are

installed on your computer when you initially installEviews). 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 inEviews. The pdf

file Users Guide Iincludes the contents pages for Users Guide Iand 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 NoteYou must register your copy ofEviews within 14 days of installing it on your computer. Ifyou 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 withEviews.

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


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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 precedingeach unit, and its a good idea to review these when you have finished that

units 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 specialcare 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 bycomments on your paper, including advice about how you might improve,


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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 thatyou 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 ofStudying 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, bysetting out their precise meaning. Such questions are likely to be preliminary and besupplemented by more analytical questions. Generally Pass marks are awarded if theanswer 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


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Reasoning

Other questions are designed to test your reasoning, by explaining cause and effect.Convincing explanations generally carry additional marks to basic definitions. They willinclude 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 willinclude 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 thequestion 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 anapplication of the result, or an interpretation.

Advice

Other questions ask you to provide advice in a particular situation. This applies to lawquestions and to policy papers where advice is asked in relation to a policy problem. Youradvice should be based on relevant law, principles, evidence of what actions are likely tobe 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 areexpected to look at the question from at least two points of view, offering a critique ofeach 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


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Examine by argument

Questions that begin with discuss are similar they ask you to examine by argument, todebate 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 criteriaof 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 engagementwith 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 andkey theoretical issues. Materials are evaluated directly and their

assumptions and arguments challenged and/or appraised

shows original thinking and a willingness to take risks60-69%: 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 andconvincingly a sustained train of logical thought

shows clear evidence of planning and appropriate choice of sources andmethodology

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

shows a reasonable understanding of the major factual and/ortheoretical 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


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45-49%: Marginal Failure

shows some awareness and understanding of the factual or theoreticalissues, but with little development

misunderstandings are evident

shows some evidence of planning, although irrelevant/unrelatedmaterial or arguments are included

0-44%: Clear Failure

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

does not engage with the relevant literature or demonstrate aknowledge 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 years

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.


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UNIVERSITY OF LONDON

Centre for Financial and Management Studies

MSc Examination

Postgraduate Diploma Examination

for External Students

91DFMC219

91DFMC319

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 questionsfrom 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 examinationprovided 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


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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 10years?

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 ifthe bond matured after 10 years?

c. NPV is by the far the most robust evaluation criterion available to thefinancial 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) whichis 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 approximatelynormally distributed with mean 114, and standard deviation of 7. Findthe 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 beingre-packaged. How many matches in a box will ensure it does nothave to be re-packaged?

c. Explain, using examples, the relationship between the t-distribution andthe 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 monthlystatements 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 cardholders.


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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. Whenwould 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)

5. Answer all parts of the question.Consider the following data on the rate of inflation (X) and on private

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

Year Y X1988 45.5 5.41989 44.8 6.21990 46.9 6.31991 48.2 4.21992 46.2 4.81993 45.2 5.71994 44.2 6.11995 46.3 7.5

1996 47.2 6.81997 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 thatB1 = 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 + 0.83Xt 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 Fratio and test the significance of the regression

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


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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 formodel selection? Explain your answer in detail.

[END OF EXAMINATION]


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Quantitative Methods for FinancialManagement

Unit 1 Financial Arithmetic andValuation of Bonds andStocks

Contents

1.1 Introduction to Unit 1 31.2 Net Present Value 31.3 Annuities and Perpetuities 61.4 Valuing Bonds 81.5 Valuation of Common Stocks 91.6 Alternative Investment Criteria 131.7 Summary 18References 20Answers to Unit Exercises 21


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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 iscomputed

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.


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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

thepresent 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 weknow, 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 offuture cash flows (finite

or infinite) in todays 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 ris 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.


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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 ofras the expectedinterest rate, this assumption can be justified.

The assumption that rstays 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 =C

1

(1+ r)+

C2

(1+ r)2+

C3

(1+ r)3

... +C

n

(1+ r)n=

Ct

(1+ r)t

t=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 =C

1

(1+ r)+

C2

(1+ r)2+

C3

(1+ r)3

... =C

t

(1+ r)t

t=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 theNet 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 costof capital is 20% for all four investments (or r= 0.20):

Investment Initial Cash Flow C0 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.


37/58



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

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

ReadingPlease turn now to your textbook by Brealey, Myers and Allen, and read from thesubsection A Fundamental Result on p. 22 to the end of the chapter, p. 30; be sure thatyou 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 capitalmarket?

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

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

1.2.1 Capitalisation SchemesSince ris 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 rpercent annual nominal interest

paid m times a year, over tyears 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 CorporateGovernance.


38/58



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

CompoundingInterval

InterestRate Factor

EffectiveAnnual Rate

InterestRate 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 ofCat the end oftyears 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 . ert

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 (perhapsthat 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


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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 overy 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 rin (1.7), which is the effective annual

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

ExerciseKangaroo 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 freecredit 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 PerpetuitiesAperpetuity 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 Cforever, 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:

PVperpetuity =C

r(1.8)

ExerciseFind the present value of a perpetuity paying $50 every month under an interest rate of12% per annum.

ReadingPlease read now sections 3.13.4 in Brealey, Myers and Allen, pages 3552, for moreexamples 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) andx = 1/(1 + r).

Richard Brealey,

Stewart Myers and

Franklin Allen (2008)

Principles of Corporate

Finance, Chapter 3

How to Calculate

Present Values.


40/58



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 theface value of the bond), and then receivesregular 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 Cthe coupon payments, and byMthe principal, and assuming aconstant 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 PVof 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 remainconstant up to the bonds maturity date, which is 15 years. Estimate the value of thebond both at the present time and at the beginning of its second year, in case you decideto 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 theannuities described in the previous section by using the original discounted payments formula.


41/58



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 NoteIt is important that you note, once again, that the annual coupon rate of a bond withsemi-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 intoaccount. With intra-year payments, as we showed earlier in this unit, the effectiveannual 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 thebondholder receives is 8.16%. As a matter of convention, however, bond dealers alwaysrefer to the annual coupon rate as the interest rate paid by the bond, no matter whetherit is paid on an annual or on a semi-annual interval. But you should bear in mind thatwhenever the bond pays semi-annual interest, the effective annual yield rate will behigher than the bonds coupon rate.

ReadingPlease now read the Chapter 3 Summary in Brealey, Myers and Allen, pages 5354, andSection 4.1 of Chapter 4, pp. 5963, 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 firms 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.

ReadingBefore we examine this, please read the introduction to Chapter 5, pp. 8586, andSection 5.1, pp. 8687, in Brealey, Myers and Allen for a fuller description of what stocksare 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 firms earnings

Richard Brealey,

Stewart Myers and



Finance, Chapter 3

summary and the first

section of Chapter 4

Valuing Bonds

Richard Brealey,

Stewart Myers and



Finance, Chapter 5Introduction and

Section 5.1 How

Common Stocks are

Traded.


42/58



ii. capital gains, which could be realised by re-selling the stock at a pricehigher 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 =DIV

1+ P

1 P

0

P0

(1.11)

whereDIV1is the expected dividend to be paid over the current year and P1

is the expectedprice of the security at end of the year. Solving for the current

value of the security, P0, we get:

P0=DIV

1+ P

1

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 rand Pl, P0is 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 theNo 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 selllarge 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 T1, as a function of its dividend and

price at time T. Formally,

PT1 =DIV

T

+ PT

1+ r(1.12')

Now, we can substitute the above expression into the (similar) expression forPT2, and substitute that into the expression for PT3 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,

4Or 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.


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P0=

DIVt

(1+ r)t

t=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 tosurvive into the far future, then T, and so PT/(1 + r)T0(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)t

t=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 NoteThis pricing formula seems harmless at first sight but, at a closer look, rests on a veryimportant assumption the efficiency of financial markets. In particular, efficiencyimplies that all the available information that may have an effect on the price of the stockis immediately reflected in its price. In other words, individual investors cannot havebeliefs that are inconsistent with the available information. If they do, then they mustbelieve that the stock is either underpriced or overpriced.

In either case, these investors can be taken advantage of by professional investors whocorrectly interpret the information. In other words, investors holding beliefs notconsistent with the available information will be wiped out from the market. We willcome back to this in Unit 8, but what we can already see is that the efficient marketsassumption implies homogeneous beliefs, and fully justifies equation (1.14).

1.5.1 The Constant Growth FormulaEquation (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) T1. 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 aresmaller 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.


44/58



P0=

DIV1(1+ g)

i1

(1+ r)i

i=1

=

DIV1

1+ r( )

11+ g( )1+ r( )

=DIV

1

(r g)(1.15)

A nice feature of equation (1.15) is that it holds even if the growth rate is notconstant but almost constant. By almostconstant, 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. Itshould 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=EPS

1

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.


45/58



ReadingFor a summary of the valuation of stocks and its relation to growth, please read nowSections 5.25.4 in Brealey, Myers and Allen, pp. 88102, and the extract printed below.

What Do PriceEarnings Ratios Mean?

The priceearnings 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 priceearnings ratios really signify and often use the ratios

in odd ways.

Should the financial manager celebrate if the firms stock sells at a high

P/E? The answer is usually yes. The high P/E shows that investors thinkthat 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 priceearnings 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 firms 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 stocks priceearnings 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 CriteriaThe 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,

Stewart Myers and



Finance, from Chapter

5 The Value of

Common Stocks.


46/58



1.6.1 The Payback Rule

Thepayback 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 C3PaybackPeriod

NPV atr = 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 negativeNPV. 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 bechosen.

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


47/58



(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:

1,000 3,000(1+ IRR)

+ 2,500(1+ IRR)

2= 0


48/58



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 lendingand 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, ofcourse, any rthat 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 NPVfor

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 theIRRrule, we can only

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

profitable, the choice is not clear (its not true, in general, that the projectwith the highestIRRis 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 IRRrule 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 IRRcan

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 higherNPV but much longermaturity. Here, a consideration of both rules would be advised.

1.6.3 The Profitability Index Rule

Theprofitability index is defined as the ratio between the investments

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


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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

projects PVwith 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 flowsand 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 profitabilityindex is useful as an additional investment criterion.

ReadingFor 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 insection 6.4 and which completes the reading.

Some More Elaborate Capital Rationing Models

The simplicity of the profitability-index method may sometimesoutweigh its limitations. For example, it may not pay to worry aboutexpenditures in subsequent years if you have only a hazy notion of futurecapital availability or investment opportunities. But there are alsocircumstances in which the limitations of the profitability-index methodare intolerable. For such occasions we need a more general method forsolving the capital rationing problem. We begin by restating the problem

just described. Suppose that we were to accept proportionxA of project Ain our example. Then the net present value of our investment in theproject would be 21xA. Similarly, the net present value of our investmentin project B can be expressed as 16xB and so on. Our objective is to selectthe set of projects with the highest total net present value. In other words

we wish to find the values ofx that maximizeNPV = 21xA + 16xB + 12xC + 13xD

Our choice of projects is subject to several constraints. First, total cashoutflow 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 cannotpurchase more than one of each. Therefore we have

0 xA1, 0 x

B1,

Collecting all these conditions, we can summarize the problem as:

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

Richard Brealey,

Stewart Myers andFranklin Allen (2008)


Finance, Chapter 6

Making Investment

Decisions with the Net

Present Value Rule.


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Subject to

10xA

+ 5xB

+ 5xC

+ 0xD

10

30xA

5xB

5xC

+ 40xD

10

0 xA1, 0 xB1,

One way to tackle such a problem is to keep selecting different values forthe xs, noting which combination both satisfies the constraints and givesthe highest net present value. But its smarter to recognize that theequations above constitute a linear programming (LP) problem. It can behanded to a computer equipped to solve LPs.

The answer given by the LP method is somewhat different from the onewe obtained earlier. Instead of investing in one unit of project A and oneof project D, we are told to take half of project A, all of project B, andthree-quarters of D. The reason is simple. The computer is a dumb, butobedient, pet, and since we did not tell it that the xs had to be wholenumbers, it saw no reason to make them so. By accepting fractionalprojects, it is possible to increase NPV by $2.25 million. For many

purposes this is quite appropriate. If project A represents an investment in1,000 square feet of warehouse space or in 1,000 tons of steel plate, itmight be feasible to accept 500 square feet or 500 tons and quitereasonable to assume that cash flow would be reduced proportionately. If,however, project A is a single crane or oil well, such fractionalinvestments 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 xs to integers.

Brealey and Myers (2003) pp. 10708.

1.7 SummaryIn 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 bemodified 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.


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

addition and not insteadof the NPV rule.

Revision Exercises

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

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

2 Brealey, Myers and Allen, question 28, p.57

3 Brealey, Myers and Allen, question 27, p.574 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 bondssuch as these fell to 6%. At what price would the bonds sell?

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

c Suppose that the conditions in Part a) existed that is, interest rates fellto 6 per cent two years after the issue date. Suppose further that theinterest rate remained at 6% for the next eight years. What would happento 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 yield12%. What would be the price of the Beranek bonds?

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

d How would your answers to Parts a), b) and c) change if the bonds werenot 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 4per cent a year in perpetuity. If you require a return of 12 per cent on yourinvestment, how much should you be prepared to pay for the stock? [FromBrealey 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, mayrun 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 itsrecovery 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 ofreturn from buying the stock at this price? Assume dividends of $1, $2, and $3


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for years 1, 2, and 3. A little trial and error will be necessary to find r

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Quantitative Methods for Financial Analyssis Sample-8