Chapter 7
Expected Return and Risk
Learning Objectives
• Explain how expected return and risk for
securities are determined.
• Explain how expected return and risk for
portfolios are determined.
• Describe the Markowitz diversification model
for calculating portfolio risk.
• Simplify Markowitz’s calculations by using the
single-index model.
Investment Decisions
• Involve uncertainty

Investors are trading a known present value for
some expected future value that is not known with
certainty
• Focus on expected returns

To estimate the returns from various securities,
investors must estimate the cash flows these
securities are likely to provide
• Goal is to reduce risk without affecting returns


Accomplished by building a portfolio
Diversification is key to effective risk management
Dealing with Uncertainty
• Risk that the expected return will not be realized
• Investors must think about return distributions, not
just a single return
• Use probability distributions

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

A probability should be assigned to each possible
outcome to create a distribution
A probability represents the likelihood of various
outcomes and is expressed as a decimal or fraction
Sum of the probabilities of all possible outcomes
must be 1.0
Can be discrete or continuous (eg., normal
distribution) (Fig.7.1 pg 188)
Calculating Expected Return
• Expected value (Expected rate of return)



Calculate the expected value in order to describe
the single most likely outcome from a particular
probability distribution
The weighted average of all possible return
outcomes, where each is weighted by its respective
probability of occurrence
Referred to as an ex ante or expected return
m
E(R )   Ripri
i1
Calculating Expected Return
E(R) = the expected return on a security
R_i = the ith possible return
pr_i = the probability of the ith return
m = the number of possible returns
m
E(R )   Ripri
i1
Calculating Risk
• Variance and standard deviation are used to
quantify and measure risk




Measure the spread or dispersion in the
probability distribution
The larger the dispersion the larger the variance
and standard deviation
Variance of returns: 2 = (Ri - E(R))2pri
Standard deviation of returns:
 =(2)1/2
Calculating Risk




Calculating a standard deviation using
probability distribution involves making
subjective estimates of the probabilities and the
likely returns
We cannot avoid making estimates because
future returns are uncertain
The relevant  in this situation is the ex ante
standard deviation and not the ex post standard
deviation based on realized returns
In this chapter we are interested in the variability
associated with future expected returns
Portfolio Expected Return
• Weighted average of the individual security
expected returns


Each portfolio asset has a weight, w, which
represents the percent of the total portfolio
value
The expected return on any portfolio can be
calculated as:
n
E(Rp )   w iE(Ri )
i1
Example: Portfolio Expected Return
• (Pg 191) Consider a three stock portfolio
consisting of stocks G, H, and I with expected
returns of 12%, 20%, and 17% respectively.
• Assume that 50% of investable funds is invested
in security G, 30% in H, and 20% in I.
• Calculate the expected return on the portfolio
Portfolio Risk
• Portfolio risk is not simply the sum of
individual security risks
• Emphasis is on the risk of the entire portfolio
and not on the risk of individual securities in
the portfolio
• Individual stocks are risky only if they add risk
to the total portfolio
Portfolio Risk
• Measured by the variance or standard
deviation of the portfolio’s return

Portfolio risk is not a weighted average of
the risk of the individual securities in the
portfolio
2
p
2
 i1 wi i
n
Portfolio Risk
• Although the expected return of a portfolio is a
weighted average of its expected returns, portfolio
risk is less than the weighted average of the risk
of the individual securities in a portfolio of risky
securities
 p   wi i
n
i 1
Risk Reduction in Portfolios
• Assume all risk sources for a portfolio of
securities are independent
• The larger the number of securities, the
smaller the exposure to any particular risk

“Insurance principle” the insurance company
reduces its risk by writing many policies
against many independent sources of risk
Risk Reduction in Portfolios
• Random (naïve) diversification




Diversifying without looking at relevant investment
characteristics such as expected return or industry
classification
An investor simply selects a relatively large number
of securities randomly
Marginal risk reduction gets smaller and smaller as
more securities are added (Fig. 7.2 pg 192)
Most finance textbooks contain similar diagrams,
with the number of stocks required to achieve
diversification varying depending upon the market
and the particular empirical study referred to in the
diagram (eg., 25 to 30 or 15 to 20)
Risk Reduction in Portfolios
How many securities are enough to diversify properly?


A recent study by Campbell, Lettau, Malkiel, and Xu
showed that, between 1962 and 1997 the market’s
overall volatility did not change whereas the volatility of
individual stocks increased sharply. As a result, investors
need more stocks today to adequately diversity (40
rather than 20)
Another recent study by Vladimir de Vassal examined
the period from 1993 to 1999 and found that with a
portfolio of 15 stocks, the probability of underperforming
the market benchmark by 100% or more was 13.5%, a
substantial risk. With a portfolio of 40 stocks the
probability declines to only 2.4%
Risk Reduction in Portfolios
• International diversification



Ignoring the hazards of foreign investing, such as
currency risk, we can conclude that if domestic
diversification is good, international diversification is
better (Fig 7.3 pg 195)
Traditional thinking focused on diversifying across
countries, but the current trend is to diversify across
industries and across countries simultaneously (Fig.
7.4 pg 196)
Since recent research suggests that industry factors
play as big a role (if not bigger) in obtaining
diversification benefits
Portfolio Risk and Diversification
p %
Total Portfolio Risk
35
20
Market Risk
0
10
20
30
40
......
Number of securities in portfolio
100+
International Diversification
p %
Domestic Stocks only
35
Domestic + International
Stocks
20
0
10
20
30
40
......
Number of securities in portfolio
100+
Markowitz Diversification
• Non-random diversification



Active measurement and management of
portfolio risk
Investigate relationships between portfolio
securities before making a decision to invest
Takes advantage of expected return and risk
for individual securities and how security
returns move together
Measuring Co-Movements in
Security Returns
• We need to consider two factors in order to
calculate risk of a portfolio as measured by the
variance or standard deviation:

Weighted individual security risks
•
•

Calculated by a weighted variance using the
proportion of funds in each security
For security i: (wi  i)2
Weighted co-movements between returns as
measured by the covariance between returns
•
•
Return covariances are weighted using the proportion
of funds in each security
For securities i, j: 2wiwj  ij
Correlation Coefficient
• Statistical measure of relative co-movements
between security returns (it measures how
security returns move in relation to one another)
• Limited to values between -1 and +1
mn = correlation coefficient between securities m
and n
 mn = +1.0 = perfect positive correlation
 mn = -1.0 = perfect negative (inverse) correlation
 mn = 0.0 = zero correlation
Correlation Coefficient
• With perfect positive correlation, the returns have a
perfect direct linear relationship. Knowing what the
return on one security will do allows an investor to
forecast perfectly what the other will do (Fig. 7.5 pg198)
• With perfect negative correlation, the securities’ returns
have an inverse linear relationship to each other.
Therefore, knowing the return on one security provides
full knowledge about the return on the other (Fig. 7.6 pg
198)
• With zero correlation, there is no relationship between
the returns on the two securities. Knowledge of the
return on one security is of no value in predicting the
return of the second security
Correlation Coefficient
• When does diversification pay?

Combining securities with perfect positive correlation
provides no reduction in risk
•

Risk of the resulting portfolio is simply a weighted
average of the individual risks of securities (Fig. 7.5 pg
198)
Combining securities with zero correlation reduces
the risk of the portfolio
•
•
If more securities with uncorrelated returns are added
to the portfolio, significant risk reduction can be
achieved
The portfolio risk cannot be eliminated in this case
Correlation Coefficient

Combining securities with negative correlation can
eliminate risk altogether
•

If the correct portfolio weights are chosen
In the real world, securities typically have some
positive correlation with each other since all security
prices tend to move with changes in the overall
economy (Fig. 7.7 pg 199,  = +0.55)
•
•
As a result, risk can be reduced it cannot be
eliminated
Any reduction in risk that does not adversely affect
return has to considered beneficial
Example: Correlation Coefficient
• (Pg 200) Over the 1998-2003 period the average
monthly return on Abitibi Consolidated (A) common
stock was 0.05%, and the standard deviation of
monthly returns was 10.69%
• During the same period, the average monthly return for
Air Canada’s (AC) common stock was -0.29%, and the
standard deviation of monthly returns was 22.42%
• The correlation between the returns on A and AC was
0.12 over this period.
• Calculate the return and standard deviation for an
equally weighted portfolio of these two securities.
Covariance
• Absolute measure of (the degree of association) the
extent to which two random variables, such as the
return on two securities, tend to covary, or move
together over time



Not limited to values between -1 and +1
Sign interpreted the same as correlation (+, -, 0)
The formulas for calculating covariance and the
relationship between the covariance and the correlation
coefficient are:
m
 AB   [R A ,i  E(R A )][R B,i  E(R B )]pri
i 1
 AB   AB  A  B
Covariance
The covariance allows us to measure the amount of comovement and incorporate it into any measure of portfolio risk
(covariance formula is similar to the variance formula)
σ_AB = the covariance between securities A and B
R_A,i = one estimated possible return on security A
E(R_A) = expected return for security A
m = the number of likely outcomes for a security for the period
pr_i = the probability of attaining a given return R_A,i
m
 AB   [R A ,i  E(R A )][R B,i  E(R B )]pri
i 1
 AB   AB  A  B
Calculating Portfolio Risk
• Encompasses three factors



Variance (risk) of each security
Covariance between each pair of securities
(σ_AB = ρ_AB σ_A σ_B)
Portfolio weights for each security
• Goal: select weights to determine the
minimum variance combination for a given
level of expected return
Calculating Portfolio Risk
• Generalizations

The smaller the positive correlation between
securities, the better
•

The only case where there are no risk reduction
benefits obtained from two-security diversification
occurs when the correlation coefficient is +1
As the number of securities increases:
•
•
The importance of covariance relationships
increases
The importance of each individual security’s risk
(variance) decreases
Calculating Portfolio Risk
• The number of relevant covariances for an
n-security portfolio equals n(n-1)
• For example, the number of relevant
covariances in a 100-security portfolio
would equal 100(100-1) = 9,900. On the
other hand, the number of relevant
variances will be 100
Calculating Portfolio Risk
• Two-Security Case:
 p  (w   w   2wAwB AB )
2 2
A A
2 2
B B
1/ 2
• N-Security Case:
n
n
n
  ( w    wi w j ij ) (i  j )
P
i 1
2
i
2
i
1/ 2
i 1 j 1
Example: Portfolio Risk
• Prove that in the two-security case, the
portfolio standard deviation will be the
weighted average of the standard deviations
of the individual securities when the
correlation coefficient is equal to +1
Example: Portfolio Risk
• (Pg 202) We have an equally weighted
portfolio that is compromised of stock A (TR
= 26.3%, σ = 37.3%) and stock B (TR =
11.6%, σ = 23.3%)
• Calculate the standard deviation of the
portfolio if the correlation between A and B
is: +1, +0.5, +0.15, 0, -0.5, and -1)
Simplifying Markowitz Calculations
• Markowitz full-covariance model

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Allows us to determine the portfolio expected return and
risk
Can be used to determine the optimal portfolio
combinations (Ch. 8)
Main problem is its complexity
Requires a covariance between the returns of all
securities in order to calculate portfolio variance
Full-covariance model becomes burdensome as number
of securities in a portfolio grows
• n(n-1)/2 unique covariances for n securities
• Therefore, Markowitz suggests using an index to
simplify calculations
The Single-Index Model
• Developed by William Sharpe
• Relates returns on each security to the
returns on a common stock index, such
as the S&P/TSX Composite Index
• Expressed by the following equation:
Ri  i  iRM  ei
The Single-Index Model
Ri  i  iRM  ei
R_i = the total return on security i
α_i = the part of security i’s return independent of the
market performance (intercept coefficient)
β_i = a coefficient that measures the expected change in
the dependent variable R_i given a change in the
independent variable R_M
R_M = the total return on the market index
e_i = the random residual error
The Single-Index Model
• Divides return into two components

a unique part, α_i
•

Is a micro-event, affecting an individual company, but
not all companies in general (e.g., strike or
resignation of CEO)
a market-related part, β_iR_M
•
Is a macro-event, affecting all (or most) firms (e.g.,
inflation or oil prices)
Example: The Single-Index Model
• (Pg 206) Assume that the return for the market
index for period t is 12%, with α_i = 3%, and β_i
= 1.5
• Use the single index model to calculate the
return for stock i for time t
• Assuming that the actual return on stock i for
period t in the previous example is 19%,
calculate the error term
The Single-Index Model
 (slope coefficient) measures the sensitivity of a
stock to the market movements
The single-index model assumes that


Residuals for different securities are uncorrelated
Securities are only related in their common response
to the market
•
•
Securities covary together only because of their
common relationship to the market index
Security covariances depend only on market risk and
can be written as:
 ij  i  j M2