The Capital Asset Pricing
Model (CAPM)
Chapter 10
Individual Securities

The characteristics of individual securities
that are of interest are the:



Expected Return
Variance and Standard Deviation
Covariance and Correlation
Expected Return, Variance, and
Covariance
Rate of Return
Scenario Probability Stock fund Bond fund
Recession
33.3%
-7%
17%
Normal
33.3%
12%
7%
Boom
33.3%
28%
-3%
Consider the following two risky asset
world. There is a 1/3 chance of each state of
the economy and the only assets are a
stock fund and a bond fund.
Expected Return, Variance, and
Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
Expected Return, Variance, and
Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
E (rS )  1  (7%)  1  (12%)  1  (28%)
3
3
3
E (rS )  11%
Expected Return, Variance, and
Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
E (rB )  1  (17%)  1  (7%)  1  (3%)
3
3
3
E (rB )  7%
Expected Return, Variance, and
Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(7%  11%)  3.24%
2
Expected Return, Variance, and
Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
(12%  11%) 2  .01%
10.2 Expected Return, Variance, and
Covariance
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
1
2.05%  (3.24%  0.01%  2.89%)
3
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
14.3%  0.0205
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Stock fund
Rate of
Squared
Return Deviation
-7%
3.24%
12%
0.01%
28%
2.89%
11.00%
0.0205
14.3%
Bond Fund
Rate of
Squared
Return Deviation
17%
1.00%
7%
0.00%
-3%
1.00%
7.00%
0.0067
8.2%
Note that stocks have a higher expected return than bonds
and higher risk. Let us turn now to the risk-return tradeoff
of a portfolio that is 50% invested in bonds and 50%
invested in stocks.
The Return and Risk for Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The rate of return on the portfolio is a weighted average of
the returns on the stocks and bonds in the portfolio:
rP  wB rB  wS rS
5%  50%  (7%)  50%  (17%)
The Return and Risk for Portfolios
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The expected rate of return on the portfolio is a weighted
average of the expected returns on the securities in the
portfolio.
E (r )  w E (r )  w E (r )
P
B
B
S
S
9%  50%  (11%)  50%  (7%)
The Return and Risk for Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
The variance of the rate of return on the two risky assets
portfolio is
σ P2  (wB σ B )2  (wS σ S )2  2(wB σ B )(wS σ S )ρ BS
where BS is the correlation coefficient between the returns
on the stock and bond funds.
10.3 The Return and Risk for
Portfolios
Scenario
Recession
Normal
Boom
Expected return
Variance
Standard Deviation
Rate of Return
Stock fund Bond fund Portfolio
-7%
17%
5.0%
12%
7%
9.5%
28%
-3%
12.5%
11.00%
0.0205
14.31%
7.00%
0.0067
8.16%
squared deviation
0.160%
0.003%
0.123%
9.0%
0.0010
3.08%
Observe the decrease in risk that diversification offers.
An equally weighted portfolio (50% in stocks and 50%
in bonds) has less risk than stocks or bonds held in
isolation.
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.08%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.00%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
10.4 The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
100%
stocks
10.0%
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
We can consider other
portfolio weights besides
50% in stocks and 50% in
bonds …
% in stocks
Risk
Return
0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
8.2%
7.0%
5.9%
4.8%
3.7%
2.6%
1.4%
0.4%
0.9%
2.0%
3.1%
4.2%
5.3%
6.4%
7.6%
8.7%
9.8%
10.9%
12.1%
13.2%
14.3%
7.0%
7.2%
7.4%
7.6%
7.8%
8.0%
8.2%
8.4%
8.6%
8.8%
9.0%
9.2%
9.4%
9.6%
9.8%
10.0%
10.2%
10.4%
10.6%
10.8%
11.0%
Portfolio Return
10.4 The Efficient Set for Two Assets
Portfolo Risk and Return Combinations
12.0%
11.0%
10.0%
100%
stocks
9.0%
8.0%
7.0%
6.0%
100%
bonds
5.0%
0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0%
Portfolio Risk (standard deviation)
Note that some portfolios are
“better” than others. They have
higher returns for the same level
of risk or less.
return
Two-Security Portfolios with Various Correlations
100%
stocks
 = -1.0
100%
bonds
 = 1.0
 = 0.2

Portfolio Risk/Return Two Securities:
Correlation Effects




Relationship depends on correlation
coefficient
-1.0 <  < +1.0
The smaller the correlation, the greater the
risk reduction potential
If  = +1.0, no risk reduction is possible
return
The Efficient Set for Many Securities
Individual Assets
P
Consider a world with many risky assets; we
can still identify the opportunity set of riskreturn combinations of various portfolios.
return
The Efficient Set for Many Securities
minimum
variance
portfolio
Individual Assets
P
Given the opportunity set we can identify the
minimum variance portfolio.
return
The Efficient Set for Many Securities
minimum
variance
portfolio
Individual Assets
P
The section of the opportunity set above the minimum
variance portfolio is the efficient frontier.
return
Optimal Risky Portfolio with a Risk-Free Asset
100%
stocks
rf
100%
bonds

In addition to stocks and bonds, consider a world that
also has risk-free securities like T-bills
return
Riskless Borrowing and Lending
100%
stocks
Balanced
fund
rf
100%
bonds

Now investors can allocate their money across
the T-bills and a balanced mutual fund
The Capital Market Line

Assumptions:

Rational Investors:




More return is preferred to less.
Less risk is preferred to more.
Homogeneous expectations
Riskless borrowing and lending.
σ P2  (wF σ F )2  (wAσ A )2  2(wF σ F )(wAσ A )ρFA   P  wA A
return
Riskless Borrowing and Lending
rf
P
With a risk-free asset available and the efficient
frontier identified, we choose the capital
allocation line with the steepest slope
return
Market Equilibrium
M
rf
P
With the capital allocation line identified, all investors choose a point
along the line—some combination of the risk-free asset and the
market portfolio M. In a world with homogeneous expectations, M is
the same for all investors.
return
The Separation Property
M
rf
P
The Separation Property states that the market
portfolio, M, is the same for all investors—they can
separate their risk aversion from their choice of the
market portfolio.
return
The Separation Property
M
rf
P
Investor risk aversion is revealed in their
choice of where to stay along the capital
allocation line—not in their choice of the line.
return
Market Equilibrium
100%
stocks
Balanced
fund
rf
100%
bonds

Just where the investor chooses along the Capital Market Line
depends on his risk tolerance. The big point though is that all
investors have the same CML.
return
Market Equilibrium
100%
stocks
Optimal
Risky
Porfolio
rf
100%
bonds

All investors have the same CML because they all have
the same optimal risky portfolio given the risk-free rate.
return
The Separation Property
100%
stocks
Optimal
Risky
Porfolio
rf
100%
bonds

The separation property implies that portfolio choice can
be separated into two tasks: (1) determine the optimal
risky portfolio, and (2) selecting a point on the CML.
return
Optimal Risky Portfolio with a Risk-Free
Asset
1
f
0
f
r
r
100%
stocks
First
Optimal
Risky
Portfolio
Second Optimal
Risky Portfolio
100%
bonds

The optimal risky portfolio depends on the
risk-free rate as well as the risky assets.
Expected versus Unexpected Returns


Realized returns are generally not equal to
expected returns
There is the expected component and the
unexpected component


At any point in time, the unexpected return can
be either positive or negative
Over time, the average of the unexpected
component is zero
Returns




Total Return = expected return + unexpected
return
Unexpected return = systematic portion +
unsystematic portion
Therefore, total return can be expressed as
follows:
Total Return = expected return + systematic
portion + unsystematic portion
Total Risk




Total risk = systematic risk + unsystematic
risk
The standard deviation of returns is a
measure of total risk
For well diversified portfolios, unsystematic
risk is very small
Consequently, the total risk for a diversified
portfolio is essentially equivalent to the
systematic risk
Portfolio Risk as a Function of the Number of
Stocks in the Portfolio

In a large portfolio the variance terms are effectively
diversified away, but the covariance terms are not.
Diversifiable Risk;
Nonsystematic Risk;
Firm Specific Risk;
Unique Risk
Portfolio risk
Nondiversifiable risk;
Systematic Risk;
Market Risk
n
Thus diversification can eliminate some, but not all of the
risk of individual securities.
Definition of Risk When Investors Hold the
Market Portfolio


The best measure of the risk of a security in a
large portfolio is the beta (b)of the security.
Beta measures the responsiveness of a
security to movements in the market portfolio.
bi 
Cov( Ri , RM )
 ( RM )
2
Total versus Systematic Risk

Consider the following information:





Standard Deviation
Security C
20%
Security K
30%
Beta
1.25
0.95
Which security has more total risk?
Which security has more systematic risk?
Which security should have the higher
expected return?
Security Returns
Estimating b with regression
Slope = bi
Return on
market %
Ri = a i + biRm + ei
Beta


Reuters
Yahoo
The Formula for Beta
bi 
Cov( Ri , RM )
 ( RM )
2
Your estimate of beta will depend upon your choice of a
proxy for the market portfolio.
Beta of a Portfolio
Stock
IBM
GM
Walmart
Portfolio


Amount
Invested
$6,000
$4,000
$2,000
$12,000
Portfolio
weights
50%
33%
17%
100%
Beta
0.90
1.10
1.30
0.450
0.367
0.217
1.03
The beta of a portfolio is a weighted average of the
beta’s of the stocks in the portfolio.
Mutual Fund Betas
Relationship of Risk to Reward

The fundamental conclusion is that the ratio of
the risk premium to beta is the same for every
asset.

In other words, the reward-to-risk ratio is constant and
equal to:
Re ward / Risk 
E ( Ri )  RF
bi
Market Equilibrium

In equilibrium, all assets and portfolios must
have the same reward-to-risk ratio and they
all must equal the reward-to-risk ratio for the
market
E ( RA )  R f
bA

E ( RM )  R f
bM
Relationship between Risk and Expected Return
(CAPM)

Expected Return on the Market:
R M  RF  Market Risk Premium
• Expected return on an individual security:
Ri  RF  βi  ( R M  RF )
Market Risk Premium
This applies to individual securities held within well-diversified
portfolios.
Expected Return on an Individual Security

This formula is called the Capital Asset
Pricing Model (CAPM)
Ri  RF  βi  ( R M  RF )
Expected
return on
a security
RiskBeta of the
=
+
×
free rate
security
Market risk
premium
• Assume bi = 0, then the expected return is RF.
• Assume bi = 1, then Ri  R M
Expected return
Relationship Between Risk & Expected
Return
Ri  RF  βi  ( R M  RF )
RM
RF
1.0

b
The slope of the security market line is equal to the market
risk premium; i.e., the reward for bearing an average amount
of systematic risk.
Expected
return
Relationship Between Risk & Expected
Return
13.5%
βi  1.5
RF  3%
R M  10%
3%
1.5
b
R i  3%  1.5  (10%  3%)  13.5%
Total versus Systematic Risk

Consider the following information:





Standard Deviation
Security C
20%
Security K
30%
Beta
1.25
0.95
Which security has more total risk?
Which security has more systematic risk?
Which security should have the higher
expected return?
Summary and Conclusions


This chapter sets forth the principles of modern portfolio
theory.
The expected return and variance on a portfolio of two
securities A and B are given by
E(rP )  wA E(rA )  wB E(rB )
σ P2  (wAσ A )2  (wB σ B )2  2(wB σ B )(wAσ A )ρAB
• By varying wA, one can trace out the efficient set of portfolios. We
graphed the efficient set for the two-asset case as a curve, pointing out
that the degree of curvature reflects the diversification effect: the lower
the correlation between the two securities, the greater the diversification.
• The same general shape holds in a world of many assets.
Summary and Conclusions
The efficient set of risky assets can be combined with
riskless borrowing and lending. In this case, a rational
investor will always choose to hold the portfolio of risky
securities represented by the market portfolio.
• Then with
borrowing or
lending, the
investor selects a
point along the
CML.
return

M
rf
P
Summary and Conclusions

The contribution of a security to the risk of a welldiversified portfolio is proportional to the covariance of
the security's return with the market’s return. This
contribution is called the beta.
bi 
Cov( Ri , RM )
 2 ( RM )
• The CAPM states that the expected return on a security is
positively related to the security’s beta:
Ri  RF  βi  ( R M  RF )
Expected (Ex-ante) Return, Variance and
Covariance
 Expected Return: E(R) = S (ps x Rs)
 Variance: 2 = S {ps x [Rs - E(R)]2}
 Standard Deviation = 
 Covariance: AB = S {ps x [Rs,A - E(RA)] x [Rs,B -
E(RB)]}
 Correlation Coefficient: AB = AB / (A B)
Risk and Return Example
State
Port.
Prob. T-Bills IBM HM XYZ Market
Recession
Below Avg.
Average
Above Avg.
Boom
0.05
0.20
0.50
0.20
0.05
E(R)=
 =
8.0% (22.0%) 28.0% 10.0% (13.0%)
8.0
(2.0)
14.7 (10.0)
1.0
8.0
20.0
0.0
7.0
15.0
8.0
35.0
(10.0) 45.0
29.0
8.0
50.0
(20.0) 30.0
43.0
Expected Return and Risk of IBM
E(RIBM)= 0.05*(-22)+0.20*(-2)
+0.50*(20)+0.20*(35)+0.05*(50) = 18%
IBM2 = 0.05*(-22-18)2+0.20*(-2-18)2
+0.50*(20-18)2+0.20*(35-18)2
+0.05*(5018)2 = 271
IBM =16.5%
Covariance and Correlation
COV IBM&XYZ = 0.05*(-22-18)(10-12.5)+
0.20*(-2-18)(-10-12.5)+0.50*(20-18)(7-12.5)+
0.20*(35-18)(45-12.5)+0.05*(50-18)(30-12.5)
=194
Correlation = 194/(16.5)(18.5)=.6355
Risk and Return for Portfolios (2 assets)
 Expected Return of a Portfolio:
E(Rp) = XAE(R)A + XB E(R)B
 Variance of a Portfolio:
p2 = XA2A2 + XB2B2 + 2 XA XB AB