INTRODUCTION TO
CORPORATE FINANCE
SECOND EDITION
Lawrence Booth & W. Sean Cleary
Prepared by Ken Hartviksen & Jared Laneus
Chapter 8
Risk, Return, and Portfolio Theory
8.1 Measuring Returns
8.2 Measuring Risk
8.3 Expected Return and Risk for Portfolios
8.4 The Efficient Frontier
8.5 Diversification
Booth/Cleary Introduction to Corporate Finance, Second Edition
2
Learning Objectives
8.1 Distinguish between “ex post” and “ex ante” returns and explain
how they are estimated.
8.2 Distinguish between arithmetic and geometric means.
8.3 Explain how common risk measures are calculated and what
they mean.
8.4 Describe what happens to risk and return when securities are
combined in a portfolio.
8.5 Explain what is meant by the “efficient frontier.”
8.6 Define diversification and explain why it is important to
investors.
8.7 Construct two-security portfolio risk-efficient frontiers.
Booth/Cleary Introduction to Corporate Finance, Second Edition
3
Measuring Returns
• Ex post returns are past or historical returns, while ex ante returns are
future or expected returns
• The income yield is the return earned by investors as a periodic cash
flow
• The capital gain (or loss) is the appreciation (depreciation) in the price
of an asset from some starting price, usually the purchase price or the
price at the start of a period
• The total return is the capital gain yield plus the income yield
• Equations 8-1, 8-2 and 8-3 can be summarized as:
total return  capital gain (loss) return  income yield
total return 
P1  P0 CF1 CF1  P1  P0


P0
P0
P0
Booth/Cleary Introduction to Corporate Finance, Second Edition
4
Measuring Returns
• Figure 8-1 shows the earnings yield on the S&P/TSX Composite
Index and the YTM on the long Canada bond since 1986
• Notice that usually yields on bonds exceed stocks, but this
relationship actually reversed during the recent financial crisis
Booth/Cleary Introduction to Corporate Finance, Second Edition
5
Measuring Returns
• Table 8-1 indicates that the
yield gap between common
shares and long Canada bonds
varies over time depending on
bond yields and the perceived
riskiness of equity investments
• The yield on the long Canada
bond is a fixed return earned
by buying and holding the bond
until maturity
• Investing in common shares,
however, should yield capital
gains over the long-term
Booth/Cleary Introduction to Corporate Finance, Second Edition
6
Measuring Returns
• Owning assets includes an attachment effect causing some investors to
refuse to accept capital losses in the total return calculation until they
are realized; paper losses, unlike realized losses, are perceived as not
being completely real
• Day traders and investors who must mark to market (i.e., carry
securities at the current market value regardless of whether they will be
sold at that price), on the other hand, do not suffer from attachment
effect
• Securities cannot be sold for the purchase price, but they can be sold
for the current market value and the funds reinvested elsewhere.
• This is the basic opportunity cost argument (that is, what is the best
alternative use for the funds tied up in the investment?)
• Marking securities to market reflects the economic value of past
investment decisions
Booth/Cleary Introduction to Corporate Finance, Second Edition
7
Measuring Average Returns
• The arithmetic mean is commonly used in statistics and is appropriate if
the investment horizon is one year, but the geometric mean is useful for
measuring the compound growth rate over multiple periods
• Standard deviation, the square root of variance, is a measure of risk
(see next slide)
Booth/Cleary Introduction to Corporate Finance, Second Edition
8
Measuring Average Returns
• Equation 8-4 gives the arithmetic mean and equation 8-5 the geometric
mean:
n
ri

Arithmetic mean (AM)  i 1
n
Geometric mean (GM)  (1  r1 )(1  r2 )(1  r3 )...(1  rn )
1
n
1
• Equation 8-6 can be used to estimate expected returns as a probabilityweighted average return:
n
ER   (ri  Prob i )
i 1
• For short-term forecasts a scenario-based approach (equation 8-6)
makes more sense, but for longer-run forecasts, the historical approach
(Equation 8-5) tends to be better because it reflects what actually
happens even if it was not expected
Booth/Cleary Introduction to Corporate Finance, Second Edition
9
Measuring Risk
• Risk is the probability of incurring harm, and for our purposes
harm means that the actual return from an investment is less
than its expected return
• Figure 8-2 shows the annual returns on common asset classes.
Notice that the asset classes with more variable returns are
considered more risky
Booth/Cleary Introduction to Corporate Finance, Second Edition
10
Measuring Risk
• Equation 8-7 gives the standard deviation (square root of the
variance) for a series of historical or ex post returns:
1 n
2
Ex post  
(
r

r
)
 i
n  1 i 1
Asset Class
Standard Deviation of Annual
Investment Returns (1938 to 2008)
Common shares
16.75%
Bonds
9.13%
• But this risk was not constant over the 1938 to 2008 period; Figure
8-3 (next slide) shows that the relative risk of equities versus bonds
changed over the 1938 to 2008 period: the returns on stocks are
less variable in recent years relative to bonds than they were in
earlier years
Booth/Cleary Introduction to Corporate Finance, Second Edition
11
Measuring Risk
• Equation 8-8 gives the standard deviation for a series of forecast
or ex ante returns:
Ex ante  
n
2
(Prob
)
(
r

ER
)

i
i
i
i 1
Booth/Cleary Introduction to Corporate Finance, Second Edition
12
Expected Return and Risk for Portfolios
• A portfolio is a collection of securities, such as stocks and bonds, that
are combined and considered a single investible asset
• The expected return on a portfolio is the weighted average of the
expected returns on the individual securities in the portfolio, as in
Equation 8-9:
n
ERP   ( wi  ERi )
i 1
where:
• ERP = expected return on the portfolio
• ERi = expected return on the security i
• wi = portfolio weight of security I
• Note that portfolio weights must sum to 1.
• For a two-security portfolio, equation 8-9 becomes equation 8-10:
n
ERP   ( wi  ERi )  ERB  w( ER A  ERB )
i 1
Booth/Cleary Introduction to Corporate Finance, Second Edition
13
Expected Return and Risk for Portfolios
Example: Security A earns 10% and security B earns 8%. Figure 8-4
shows the portfolio return varying the weights in each security
between 0% and 100%
Booth/Cleary Introduction to Corporate Finance, Second Edition
14
Expected Return and Risk for Portfolios
• The standard deviation of a two-security portfolio can be estimated
using Equation 8-11:
 P  w A2  A2  wB2  B2  2w A wB COV AB
where:
• σP = portfolio standard deviation
• COVAB = covariance of the returns of securities A and B
• The covariance is calculated using Equation 8-12:
n
COV AB   Prob(rA,i  rA )( rB ,i  rB )
i 1
Booth/Cleary Introduction to Corporate Finance, Second Edition
15
Expected Return and Risk for Portfolios
Example: Security A earns 10% and security B earns 8%; Figure 85 shows the portfolio standard deviation varying the weights in
each security between 0% and 100%
Booth/Cleary Introduction to Corporate Finance, Second Edition
16
The Correlation Coefficient
• The correlation coefficient is a statistical measure that identifies
how security returns move in relation to one another and is given
by Equations 8-13 or 8-14:
COV AB   AB A B   AB 
COV AB
 A B
• And, substituting into equation 8-11 gives equation 8-15:
 P  w A2  A2  wB2  B2  2w A wB  AB A B
Booth/Cleary Introduction to Corporate Finance, Second Edition
17
The Correlation Coefficient
• The correlation coefficient will range between -1 and +1
• Securities that have -1 correlation are perfectly negatively
correlated (as one goes up, the other goes down)
• Securities that have +1 correlation are perfectly positively
correlated (both go up together)
• Securities that have zero correlation have no relationship
• The closer the absolute value of the correlation is to 1, the
stronger the relationship between the securities
Booth/Cleary Introduction to Corporate Finance, Second Edition
18
The Correlation Coefficient
• Figure 8-6a shows positive correlation between Canadian and U.S.
stock returns:
Booth/Cleary Introduction to Corporate Finance, Second Edition
19
The Correlation Coefficient
• Figure 8-6b shows no correlation between T-Bill returns and
Canadian stock returns:
Booth/Cleary Introduction to Corporate Finance, Second Edition
20
The Correlation Coefficient
• Figure 8-7 shows a non-linear relationship between correlation
coefficient and standard deviation; notice that when perfect
negative correlation exists, there is a set of weights for which risk
is eliminated
Booth/Cleary Introduction to Corporate Finance, Second Edition
21
The Correlation Coefficient
• Figure 8-8 shows how variability changes with portfolio composition for
three special cases: perfect negative, perfect positive and no correlation
• Notice risk changes linearly when correlation is perfectly positive, but
non-linearly when risk is between -1 and +1
Booth/Cleary Introduction to Corporate Finance, Second Edition
22
Perfect Negative Correlation
Special Case
• Equation 8-16 gives the standard deviation of a portfolio where the
correlation between securities A and B is perfectly negative:
 P  w A  (1  w) B
• The portfolio weight of security A for this portfolio is then given by
equation 8-17
B
w
 A B
Example: In Figure 8-8 we see the portfolio weights that give zero risk if
securities A and B are perfectly negatively correlated are 27.76% in A and
72.24% in B.
Booth/Cleary Introduction to Corporate Finance, Second Edition
23
The Efficient Frontier
• For n securities, the expected portfolio return is still a weighted
average of individual security expected returns, while the
portfolio standard deviation must take the correlation of each
pair of securities into consideration to measure total risk
• Equation 8-18 calculates the risk of a three-security portfolio:
 P  wA2 A2  wB2 B2  wC2  C2  2wA wB  AB A B  2wA wC  AC A C  2wB wC  BC B C
• The more securities in a portfolio, the greater the relative impact
of the security co-movements on the overall portfolio’s risk and
the lower the relative impact of the individual risks
Booth/Cleary Introduction to Corporate Finance, Second Edition
24
Modern Portfolio Theory
• Harry Markowitz was awarded the Nobel Prize in Economics in
1990 for work on portfolio theory in the 1950s
• Markowitz assumed:
1. Investors are rational decision makers
2. Investors are risk averse, and so must be compensated for
assuming additional risk
3. Investor preferences are based on portfolio expected return and
risk, as measured by variance or standard deviation
• Based on these assumptions, efficient portfolios can be
constructed from a set of available securities
• Efficient portfolios offer the highest expected return for a given
level of risk or offer the lowest risk for a given expected return
Booth/Cleary Introduction to Corporate Finance, Second Edition
25
Modern Portfolio Theory
• Attainable portfolios can be constructed by combining the underlying
securities; unattainable ones cannot
• In Figure 8-10, portfolio A is unattainable, portfolios B, D and E lie along
the minimum variance frontier, and portfolio C is attainable only if
some portion of investible wealth remains uninvested (i.e., C is
inefficient)
• The blue line represents the minimum variance frontier, the risk-return
combinations available to investors from a given set of securities by
allowing portfolio weights to vary
Booth/Cleary Introduction to Corporate Finance, Second Edition
26
Modern Portfolio Theory
The minimum variance frontier can be divided into three sections:
• Portfolio E is the minimum variance portfolio (MVP), because it has the
minimum amount of risk available for any possible combination of
securities
• The segment of the frontier above E is the efficient frontier, while the
segment of the frontier below E is the dominated frontier
• The efficient frontier is the set of portfolios that offer the highest expected
return for their given level of risk; these are the only portfolios that
rational, risk-averse investors would hold
Booth/Cleary Introduction to Corporate Finance, Second Edition
27
Diversification
• Reduce a portfolio’s risk by spreading investment funds across several
assets, with the key being to choose assets whose returns are less than
perfectly positively correlated
• Even with random selection or naïve diversification, the risk of a
portfolio can be reduced
• Figure 8-11 shows that as the number of securities increases, the
diversification benefit decreases; eventually, additional securities are
superfluous. Also see Table 8-3 in the text
Booth/Cleary Introduction to Corporate Finance, Second Edition
28
Diversification
Equation 8-19 states:
Total risk = Market (systematic) risk + Unique (non-systematic) risk
• As Figure 8-11 shows, it is not possible to eliminate total risk
through diversification because market risk remains after
diversification
• Unique or non-systematic risk can, however, be eliminated with
sufficient amounts of diversification
• Diversification therefore adds value to a portfolio by reducing risk
without sacrificing return
• Most of the benefits of diversification can be achieved by
investing in 40 to 50 different securities
Booth/Cleary Introduction to Corporate Finance, Second Edition
29
Diversification
• Theoretically, international diversification could reduce more risk than
investing only in domestic capital markets because international
economies may not be well correlated
• But, evidence suggests the benefits of international diversification are
declining as global securities markets become more integrated
Booth/Cleary Introduction to Corporate Finance, Second Edition
30
Two-Security Portfolio Frontiers
• Equation 8A-1 shows the portfolio weight as we change our expected
return:
ERP  ERB
w
ERA  ERB
• Substituting equation 8A-1 into equation 8-15 gives equation 8A-2:
2
2
 ERP  ERB  2  ERP  ERB  2
 ERP  ERB   ERP  ERB 
P  
  A  1 
  B  2
 1 
  AB A B
ER

ER
ER

ER
ER

ER
ER

ER
B 
A
B 
B 
A
B 
 A

 A
Booth/Cleary Introduction to Corporate Finance, Second Edition
31
Two-Security Portfolio Frontiers
Figure 8A-1 shows that:
• The risk of the portfolio falls, unless securities A and B are perfectly
correlated, as we add risky security A to the portfolio
• The decline in risk is much more dramatic if the securities are negatively
correlated
• Eventually, the risk reduction from holding A falls and portfolio risk is
minimized
Booth/Cleary Introduction to Corporate Finance, Second Edition
32
Two-Security Portfolio Frontiers
Example: Returning to the earlier example, where the correlation
coefficient was 0.379, the efficient frontier is given in Figure 8A-2
• Note that axes are flipped because it is customary to have expected
return on the vertical axis and risk on the horizontal axis
• Also, security weights fall below zero (short) or above one (borrowed to
invest) to obtain the entire IOS graphed in Figure 8A-2
Booth/Cleary Introduction to Corporate Finance, Second Edition
33
Value at Risk (VaR)
• Value at Risk (VaR) is a risk-management technique that measures
potential loss (in money terms) that could be exceeded (minimum loss) at a
given level of probability
• Example: a $1 million daily VaR at 5% means that there is a 5% chance of
losing at least $1 million in one day or, equivalently, a 95% chance that
daily losses will be less than $1 million
• There are three ways to estimate VaR:
1. The analytical (variance-covariance) method
2. The historical method
3. The Monte Carlo method
• Advantages: quantifies potential loss in simple terms, is widely accepted
by regulators and is versatile
• Disadvantages: can be difficult to estimate, can promote a false sense of
security for portfolio managers and may underestimate the severity of
worst-case scenarios
Booth/Cleary Introduction to Corporate Finance, Second Edition
34
Value at Risk (VaR):
The Analytical (Variance-Covariance) Method
• The analytical method assumes that portfolio returns are normally
distributed, and requires estimates of portfolio expected returns and
standard deviations
• Daily VaR = position dollar value × portfolio return volatility
• Portfolio return volatility = portfolio standard deviation multiplied by the Zscore appropriate for the required probability
• The main advantage of the analytical method is its simplicity , but its
disadvantage is the assumption that returns are normally distributed and
that estimates of return standard deviations are representative of the
future
• Example: Estimate the 5% daily VaR of a $2 million position in the market
which has a 2% daily standard deviation in price changes
• Note: the Z-score for 5% VaR is 1.65
• Daily VaR = $2 million × 1.65 × 0.02 = $66,000
Booth/Cleary Introduction to Corporate Finance, Second Edition
35
Value at Risk (VaR)
The Historical and Monte Carlo Methods
Historical Method
• Use actual daily returns from a specified past period to estimate future results
• Implicit assumption: the future will be like the past
• The key advantage is that this method is non-parametric: no probability
distribution needs to be assumed for any of the variables
• The key disadvantage is that this method may suffer from a lack of historical
data
Monte Carlo Method
• Random outcomes are simulated using a computer to examine the effects of
particular sets of risks using probability distributions for each variable of
interest
• Historical variance and covariance estimate are often used as inputs
• Non-normal probability can be used
• For large portfolios, this method can be computationally intensive
Booth/Cleary Introduction to Corporate Finance, Second Edition
36
Copyright
Copyright © 2010 John Wiley & Sons Canada, Ltd. All rights
reserved. Reproduction or translation of this work beyond that
permitted by Access Copyright (the Canadian copyright licensing
agency) is unlawful. Requests for further information should be
addressed to the Permissions Department, John Wiley & Sons
Canada, Ltd. The purchaser may make back-up copies for his or
her own use only and not for distribution or resale. The author
and the publisher assume no responsibility for errors, omissions,
or damages caused by the use of these files or programs or from
the use of the information contained herein.
Booth/Cleary Introduction to Corporate Finance, Second Edition
37