Chapter 11
RETURN AND RISK
THE CAPITAL ASSET PRICING MODEL (CAPM)
SLIDES
11.1 Key Concepts and Skills
11.2 Chapter Outline
11.3 Individual Securities
11.4 Expected Return, Variance, and Covariance
11.5 Expected Return
11.6 Expected Return
11.7 Variance
11.8 Variance
11.9 Standard Deviation
11.10 Covariance
11.11 Correlation
11.12 The Return and Risk for Portfolios
11.13 Portfolios
11.14 Portfolios
11.15 Portfolios
11.16 Portfolios
11.17 The Efficient Set for Two Assets
11.18 The Efficient Set for Two Assets
11.19 Portfolios with Various Correlations
11.20 The Efficient Set for Many Securities
11.21 The Efficient Set for Many Securities
11.22 Announcements, Surprises, and Expected Returns
11.23 Announcements, Surprises, and Expected Returns
11.24 Diversification and Portfolio Risk
11.25 Portfolio Risk and Number of Stocks
11.26 Risk: Systematic and Unsystematic
11.27 Total Risk
11.28 Optimal Portfolio with a Risk-Free Asset
11.29 Riskless Borrowing and Lending
11.30 Riskless Borrowing and Lending
11.31 Market Equilibrium
11.32 Market Equilibrium
11.33 Risk When Holding the Market Portfolio
11.34 Estimating β with Regression
11.35 The Formula for Beta
11.36 Relationship between Risk and Expected Return (CAPM)
11.37 Expected Return on a Security
11.38 Relationship Between Risk & Return
11.39 Relationship Between Risk & Return
11.40 Quick Quiz
CHAPTER ORGANIZATION
11.1 Individual Securities
11.2 Expected Return, Variance, and Covariance
Expected Return and Variance
Covariance and Correlation
11.3 The Return and Risk for Portfolios
The Expected Return on a Portfolio
Variance and Standard Deviation of a Portfolio
11.4 The Efficient Set for Two Assets
11.5 The Efficient Set for Many Securities
Variance and Standard Deviation in a Portfolio of Many Assets
11.6 Diversification
The Anticipated and Unanticipated Components of News
Risk: Systematic and Unsystematic
The Essence of Diversification
11.7 Riskless Borrowing and Lending
The Optimal Portfolio
11.8 Market Equilibrium
Definitions of the Market Equilibrium Portfolio
Definition of Risk When Investors Hold the Market Portfolio
The Formula for Beta
A Test
11.9 Relationship between Risk and Expected Return (CAPM)
Expected Return on Market
Expected Return on Individual Security
ANNOTATED CHAPTER OUTLINE
Slide 11.0 Chapter 11 Title Slide
Slide 11.1 Key Concepts and Skills
Slide 11.2 Chapter Outline
Lecture Tip: You may find it useful to emphasize the economic
foundations of the material in this chapter. Specifically, we assume:
-Investor rationality: Investors are assumed to prefer more money to
less and less risk to more, all else equal. The result of this assumption
is that the ex ante risk-return trade-off will be upward sloping.
-As risk-averse return-seekers, investors will take actions consistent
with the rationality assumptions. They will require higher returns to
invest in riskier assets and are willing to accept lower returns on less
risky assets.
-Similarly, they will seek to reduce risk while attaining the desired
level of return, or increase return without exceeding the maximum
acceptable level of risk.
Given the underlying assumptions above, this may be a good point at
which to discuss the increasingly popular field of behavioral finance. I.e.,
are investors rational?
1. Individual Securities
Characteristics include: expected return, variance, standard deviation,
covariance, and correlation.
Slide 11.3 Individual Securities
2. Expected Return, Variance, and Covariance
Slide 11.4 Expected Return, Variance, and Covariance
A. Expected Return and Variance
Let n denote the total number of states of the economy, Ri the return in
state i, and pi the probability of state i. Then the expected return,
R
−
, is
given by:
R
¿=∑
i=1
n
piRi
Example:
State of economy Probabilit
y
Return
(%)
Product
+1% change in
GDP
.25 -5 -1.25
+2% change in
GDP
.50 15 7.50
+3% change in
GDP
.25 35 8.75
Sums 1.00 E(R) =
15%
Slide 11.5 –
Slide 11.6 Expected Return
Variance measures the dispersion of points around the mean of a
distribution. In this context, we are attempting to characterize the
variability of possible future security returns around the expected return.
In other words, we are trying to quantify risk and return. Variance
measures the total risk of the possible returns.
State of Economy Probability Return (%) Squared
Deviation
Product
(Dev*Prob)
+1% change in GDP .25 -5 400 100
+2% change in GDP .50 15 0 0
+3% change in GDP .25 35 400 100
Total 1.00 E(R) = 15 2 = 200
Standard deviation = square root of variance = 14.14%
Calculating the Variance
Var (R)=σ2=∑
i=1
n
pi(Ri−R
¿)2
Slide 11.7 –
Slide 11.8 Variance
Slide 11.9 Standard Deviation
Lecture Tip: Some students experience confusion in understanding the
mathematics of the variance calculation. They may have the feeling that
they should divide the variance of an expected return by (n-1). Point out
that the probabilities account for this division. We divide by n-1 in the
historical variance because we are looking at a sample. If we looked at
the entire population (which is what we are doing with expected values),
then we would divide by n (or multiply by 1/n) to get our historical
variance. This is the same as saying that the “probability” of occurrence
is the same for all observations and is equal to 1/n.
Lecture Tip: Each individual has their own level of risk tolerance. Some
people are just naturally more inclined to take risk, and they will not
require the same level of compensation as others for doing so. Our risk
preferences also change through time. We may be willing to take more risk
when we are young and without a spouse or kids. But, once we start a
family, our risk tolerance may drop.
B. Covariance and Correlation
Covariance is essentially a form of the variance calculation, but it
compares two assets rather than looking at a single security in isolation.
Take the deviation from the expected outcome for each security and
multiply them together. This replaces the squared deviation from the
variance calculation. From this point, the calculations are the same.
Slide 11.10 Covariance
The correlation standardizes the covariance by dividing by the product of
the standard deviations of the two assets. The result is a value between
positive 1 and negative 1.
Slide 11.11 Correlation
3. The Return and Risk for Portfolios
A portfolio is a collection of assets, such as stocks and bonds, held by an
investor.
Portfolios can be described by the percentage investment in each asset,
and these percentages are called portfolio weights.
Example: If two securities in a portfolio have a combined value of
$10,000 with $6000 invested in IBM and $4000 in GM, then the weight in
IBM = 6/10 = .6, and the weight in GM = 4/10 = .4.
Slide 11.12 The Return and Risk for Portfolios
A. The Expected Return on a Portfolio
The expected return on a portfolio is the sum of the product of the
expected returns on the individual securities and their portfolio weights.
Let wj be the portfolio weight for asset j and m be the total number of
assets in the portfolio; then
R
¿
p=∑
j=1
m
wjR
¿
j
This formula also works if you drop the expectations and just compute the
portfolio return in each state of the economy. This is useful for the
calculation of the portfolio variance in the next section.
Slide 11.13 –
Slide 11.14 Portfolios
B. Variance and Standard Deviation of a Portfolio
Portfolio Variance: Unlike expected return, the variance of a portfolio is
NOT the weighted sum of the individual security variances. Combining
securities into portfolios can reduce the total variability of returns.
Example: Consider a portfolio with equal amounts invested in three
stocks:
State of
Economy
Probabilit
y
Return on A
(%)
Return on B
(%)
Return on C
(%)
Return on
Portfolio (%)
+1% change in
GDP
.25 -5 0 20 5
+2% change in
GDP
.50 15 10 10 11.7
+3% change in
GDP
.25 35 20 0 18.3
Expected
Return
15 10 10 11.7
Variances and standard deviations:
Var(A) = .25(-5-15)2 + .5(15-15)2 + .25(35-15)2 = 200
Std. Dev.(A) = 14.14
Var(B) = .25(0 – 10)2 + .5(10 – 10)2 + .25(20 – 10)2 = 50
Std. Dev.(B) = 7.07107
Var(C) = .25(20 – 10)2 + .5(10-10)2 + .25(0-10)2 = 50
Std. Dev.(C) = 7.07107
Var(portfolio) = .25(5-11.7)2 + .5(11.7-11.7)2 + .25(18.3-11.7)2 = 22.1125
Std. Dev.(portfolio) = 4.7024
Notice that the portfolio variance is less than any of the individual
variances.
Lecture Tip: In most business programs, a course in elementary statistics
is a prerequisite for the introductory finance course. And, while students
are sometimes fuzzy on the details, they usually remember the general
concept of the correlation coefficient (and hopefully the covariance). They
almost always remember that the correlation coefficient is bounded by –1
and 1. You may find it useful to reintroduce them to the correlation
concept here to deepen their understanding of portfolio variance.
Specifically, for a two-asset portfolio, the portfolio variance is equal
to:
w1
2σ1
2+ w2
2σ2
2+ 2w1w2σ1σ2ρ1,2
or w1
2σ1
2+ w2
2σ2
2+ 2w1w2σ1,2
where 1,2 is the correlation coefficient and 1,2 is the covariance. When
you expand the equation to more assets, you will have a variance term for
each asset and a covariance term for each pair of assets. As you increase
the number of assets, it is easy to see that the correlation (covariance)
between assets is much more important in determining the portfolio
variance than the individual variances.
Reconsider the previous example.
The following covariances can be computed:
cov(A,B) = 100
cov(A,C) = -100
cov(B,C) = -50
Using the covariances and extending the formula above to three assets,
you can compute a portfolio variance and standard deviation:
var = (1/3)2(200) + (1/3)2(50) + (1/3)2(50) + 2(1/3)(1/3)(100) + 2(1/3)
(1/3)(-100) + 2(1/3)(1/3)(-50) = 22.22
standard deviation = 4.71%
This is just as we computed earlier, with a slight difference due to
rounding portfolio returns.