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
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Quantitative Methods for Financial Management
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Course Introduction and Overview
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 decisions. 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 securities: 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 decisions 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
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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 investment 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 offcampus CeFiMS courses, including Mathematics and Statistics for Econo-
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Course Introduction and Overview
mists, Portfolio Analysis and Derivatives, Quantitative Methods for Financial 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 University 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.
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The Course Structure
The course is divided into eight units of text and reading.
Unit 1
1.1
1.2
1.3
1.4
1.5
1.6
Unit 2
2.1
2.2
2.3
Unit 3
3.1
3.2
3.3
Financial Arithmetic and Valuation of Bonds and Stocks
Introduction to Unit 1
Net Present Value
Annuities and Perpetuities
Valuing Bonds
Valuation of Common Stocks
Alternative Investment Criteria
Statistical Concepts and Probability Theory
Introduction
Moments of a Probability Distribution
Some Important Probability Distributions
Statistical Inference
Introduction
Estimation
Hypothesis Testing
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Unit 4
4.1
4.2
4.3
4.4
4.5
4.6
4.7
Unit 5
5.1
5.2
5.3
5.4
5.5
5.6
5.7
Unit 6
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
Unit 7
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
Unit 8
8.1
8.2
8.3
8.4
8.5
6
The Classical Linear Regression Model
Introduction
The Meaning of Regression Analysis
The Regression Model and its Statistical Parameters
Actual and Fitted Values – the Regression Line and the Error Term
The Meaning of the Linearity Assumption
The Method of Ordinary Least Squares (OLS)
Some Examples
Statistical Inference in the Classical Linear Regression Model
Introduction
The Classical Linear Regression Model (CLRM)
The Variance and the Standard Error of the Parameter Estimators
Properties of the OLS estimators
Confidence Intervals and Hypothesis Testing
Goodness of Fit – the Correlation Coefficient r and the Coefficient of
Determination R 2
Forecasting
The Multiple Linear Regression Model
Introduction
The Multiple Linear Regression Model
OLS Estimation
The Multiple Coefficient of Determination
Hypothesis Testing in the Multiple Regression Model
An Exercise — The Demand for Money
Model Selection and the Adjusted Coefficient of Determination
Choice of the Functional Form
Topics in the Multiple Linear Regression Model
Introduction
Definition of Dummy Variables
Use of Dummy Variables to Compare Regressions
Autocorrelation of the Error Terms
Tests for Autocorrelation – the Durbin-Watson Test
Estimation of Models with Autocorrelated Disturbances
Dynamic Models and the Error Correction Mechanism
An Example
Conclusions
Risk Measurement and Investment Decisions
Introduction
Risk and Return
The Capital Asset Pricing Model
Arbitrage Pricing Theory (APT)
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:
•
•
•
•
•
•
5
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.
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
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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 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.
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Although easy to use, Eviews is a very powerful program. There are advanced 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.
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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 assigned. 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.
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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,
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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 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 recommend 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:
10
Describe
Define
Examine
Distinguish between
Compare
Contrast
Write notes on
Outline
What is meant by
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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 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
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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. Assignments 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
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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 questions, 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
91DFM C219
91DFM C319
FINANCE
FINANCIAL MANAGEMENT
Quantitative Methods for 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
Centre for Financial and Management Studies
PLEASE TURN OVER
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Quantitative Methods for Financial Management
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.
2.
‘NPV is by the far the most robust evaluation criterion available to the
financial manager’. Critically discuss this statement.
Answer all parts of the question.
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.
3.
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Explain, using examples, the relationship between the t-distribution and
the normal distribution.
Answer all parts of the question.
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.
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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. 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)
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
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
Y
45.5
44.8
46.9
48.2
46.2
45.2
44.2
46.3
47.2
48.6
X
5.4
6.2
6.3
4.2
4.8
5.7
6.1
7.5
6.8
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.
6.
Answer all parts of the question.
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.
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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 for
model selection? Explain your answer in detail.
[END OF EXAMINATION]
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Management
Unit 1 Financial Arithmetic and
Valuation of Bonds and
Stocks
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
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Unit Content
Unit 1 introduces the course and the general principles of financial management. 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.
2
University of London
Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
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 introduce 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 investment 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.
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Quantitative Methods for Financial Management
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 investment with a finite number, n, of annual cash flows:
PV =
n
C1
C2
C3
Cn
Ct
+
+
...
+
=
2
3
n
t
(1+ r) (1+ r) (1+ r)
(1+ r)
t=1 (1+ r)
(1.2)
Or, for that matter, an infinite number of cash flows (such as, for example,
the rents from an office building):1
C1
C2
C3
Ct
PV =
+
+
... = 2
3
(1+ r) (1+ r) (1+ r)
(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 negative 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
1
2
3
4
Initial Cash Flow C 0
–10,000
–5,000
–5,000
–2,000
Cash Flow in Year 1 C1
+20,000
+12,000
+5,500
+5,000
Which investment is most valuable?
1
4
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.
University of London
Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
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?
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’.
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 payments. Notice that, for a given annual nominal interest rate, the effective
annual rate increases with the frequency of the payments.
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Table 1.1
Effective Annual Interest Rate for Different Compounding Intervals
(nominal interest rate r = 10%)
Compounding
Interval
Annual
Semi-Annual
Quarterly
Monthly
Daily
Interest
Rate Factor
(1 + r )
(1 + r /2)2
(1 + r /4)4
(1 + r /12)12
(1 + r /365)365
Effective
Annual Rate
10.00%
10.25%
10.38%
10.47%
10.52%
Interest
Rate Factor
1.1000
1.1025
1.1038
1.1047
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 discounted m times per year, is worth now:
PV =
C
mt
[1+ (r /m)]
(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
6
University of London
Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
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
c
1 c
1
+
+ +
= c 2
t
t (1 + r)
(1 + r) (1 + r)
r r(1 + r) (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, therefore, 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
Richard Brealey,
Stewart Myers and
Franklin Allen (2008)
Principles of Corporate
Finance, Chapter 3
‘How to Calculate
Present Values’.
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).
Centre for Financial and Management Studies
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Quantitative Methods for Financial Management
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
C
(C + M )
+
2 + ...+
n
(1+ r) (1+ r)
(1+ r)
(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:
n
M
C
+
PV = t
(1+ r) n
t = 1 (1+ r)
(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:
M
1
1
+
PV (bond) = n n
r r(1 + r) (1 + r)
(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
8
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.
University of London
Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
2n
M
C /2
t +
2n
(1+ r /2)
t=1 (1+ r /2)
P =
(1.9'')
Or, by using the same shortcut as before:
P=
M
C 1
1
2n +
2 r / 2 r / 2 (1 + r / 2 ) (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 semiannual 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 semiannual 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.
Richard Brealey,
Stewart Myers and
Franklin Allen (2008)
Principles of Corporate
Finance, Chapter 3
summary and the first
section of Chapter 4
‘Valuing Bonds’
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.
Richard Brealey,
Stewart Myers and
Franklin Allen (2008)
Principles of Corporate
Finance, Chapter 5
Introduction and
Section 5.1 ‘How
Common Stocks are
Traded’.
By owning shares, the investor is entitled to two types of income benefits:
i. dividend payments – typically based on the issuing firm’s earnings
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Quantitative Methods for Financial Management
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 ‘corrects’ itself as shown by the following example.
Suppose that the stock dealer sets the price of the stock below P0. Professional 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 determine 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 + PT
1+ 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
5
10
Or at a lower price – that is, the capital gain may be negative. In this case the investor would incur a
capital loss.
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.
University of London
Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
T
DIVt
PT
+
t
(1+ r)T
t=1 (1+ r)
P0 = (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:
DIVt
t
t=1 (1+ r)
P0 = (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 dividends 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.
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DIV1 (1+ g)
(1+ r) i
i=1
P0 = i1
DIV1
(1+ r) = DIV1
=
(1+ g) (r g)
1
(1+ r)
(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 temporary 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 worseoff 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
+ PVGO
r
(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.
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Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
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.
Richard Brealey,
Stewart Myers and
Franklin Allen (2008)
Principles of Corporate
Finance, from Chapter
5 ‘The Value of
Common Stocks’.
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.
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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
A
B
C
–2,000
–2,000
–2,000
+2,100
+1,000
+500
0
+1,000
+1,000
0
+5,000
+8,000
Payback
Period
NPV at
r = 10%
1 Year
2 Years
3 Years
–91
3,492
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 produces 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
C2
Cn
+
+ ...+
=0
2
(1+ IRR) (1+ IRR)
(1+ IRR) n
(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
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Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
(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:
25,000
(1+ IRR)
4,000 +
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)
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2,500
(1+ IRR)2
= 0
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Quantitative Methods for Financial Management
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 worthwhile, since the NPV is equal to 339.
A third problem with the IRR rule is that we are unable to distinguish between 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. Naturally, 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
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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
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 recommend 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.
Richard Brealey,
Stewart Myers and
Franklin Allen (2008)
Principles of Corporate
Finance, Chapter 6
‘Making Investment
Decisions with the Net
Present Value Rule’.
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
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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 worthwhile 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.
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Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
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. Although 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
2
3
4
Brealey, Myers and Allen, question 14, p.55
Brealey, Myers and Allen, question 28, p.57
Brealey, Myers and Allen, question 27, p.57
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
c
Suppose that, two years after the initial offering, the going interest rate
had risen to 12%. At what price would the bonds sell?
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
d
At what price would the Beranek bonds sell if the yield on these bonds
were 10%?
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
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Quantitative Methods for Financial Management
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.)
8 Brealey, Myers and Allen, question 26, p.111
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.
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Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
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):
1
1
= $8, 973
PVkangaroo = $1, 000 + 300 30 0.008 0.008(1.008) • 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 manipulate 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
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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:
1
1,000
1
P = 150 15 +
15 = $1,000
0.15 0.15 (1 + 0.15 ) (1 + 0.15 )
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:
1
1,000
1
p = 150 14 +
14 = $1, 000
0.15 0.15 (1 + 0.15 ) (1 + 0.15 )
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 semiannual compounding is
rquoted = 11.7%, semi-annual compounding:
reffective =
0.117 2
1
1+
2 = 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.
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Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
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
1
1
p = 100 10 r r (1 + r ) 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
1
1
=
100 10 r r(1 + r) 100 1
(1 + r)10 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
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Quantitative Methods for Financial Management
16
PVB = $50
+
$1,000
(1.03) t (1.03)
PVB = 50 (12.5611) + 1,000(0.6232) = $1,251.22
t=1
16
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
DIV2
DIV3
1
DIV4
+
+
+
2
3
3
1+ r (1+ r) (1+ r) (1+ r) ( 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
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Unit 1 Financial Arithmetic and Valuation of Bonds and Stocks
50 =
3(1.06)
1
2
3
1
+
2 +
3 +
3 (1+ r) (1+ r) (1+ r) (1+ r) (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
and 5, and
years 6, 7, … The price of the stock is therefore:
5
for years 1, 2, 3, 4
for
DIVt
DIVt
DIVt
=
+
t
t
(1+ r)
(1+ r) t=6 (1+ r) t
t=1
t=1
P0 = =
DIV1 DIV1 (1+ g) DIV1 (1+ g)2 DIV1 (1+ g)3 DIV1 (1+ g) 4
+
+
+
+
(1+ r)2
(1+ r)
(1+ r)3
(1+ r) 4
(1+ r)5
DIV1 (1+ g) 4 (1+ g) DIV1 (1+ g) 4 (1+ g)2
+
+ ...
+
(1+ r)6
(1+ r) 7
DIV1 (1+ g) t DIV1 (1+ g) 4 (1+ g) t
DIV1
=
+
(1+ r)t
(1+ r) t=1 (1+ r) t
(1+ r)5
t=1
4
= $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,
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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 75
= 14.58
1.2 1.22
That is, this is an attractive project.
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