Lecture 10
The Capital Asset Pricing Model
Preliminaries
Expectation, variance, standard error (deviation), covariance, and
correlation of returns may be based on
(i) fundamental analysis
(ii) historical data
Fundamental or Theoretical Analysis
Notation
S possible states
s probability of state s = 1,2,…,S
Rs likely return is state s
Example:
Suppose there are
4 business cycle states (boom, normal, recession, depression)
3 industry demand states
2 firm demand share states
3 firm cost states
Then, there are 4*3*2*3 = 72 possible states (or situations)
Expectation (mean)
S
  ER     sR s
s 1
Variance
S
  var R     s R s  ER 2
2
s 1
Standard error
  2 0
Covariance measures how two random variables are related
returns on stock A RAs s = 1,…,S
returns on stock B RBs s = 1,…,S
S
 AB  CovR A ,RB     s R As  ΕR A RBs  ΕRB 
s 1
Correlation is a normalized covariance
 AB
 AB
 corr R A ,RB  
sign  AB   sign AB 
 A B
Note !

2
AB
 
2
A
2
B

2
AB
2
 AB
 2 2  1  1   AB  1
 A B
Example:
Suppose we have a theoretical model that predicts
the following returns on stocks A and B in 3 states.
States
s
RA
RB
Boom
0.25
20%
5%
Normal
0.50
10% 10%
Recession 0.25
Expected
returns
0%
15%
 A  0.25 0.20   0.50 0.10   0.25 0.00   0.10
B  0.25 0.05   0.50 0.10   0.25 0.15   0.10
Variances
 A2  0.25 0.2  0.10 2  0.50 0.10  0.10 2  0.25 0.00  0.10 2  0.005
 B2  0.25 0.05  0.10 2  0.50 0.10  0.10 2  0.25 0.15  0.10 2  0.00125
Standard
errors
 A   A2  0.07071
 B   B2  0.03536
2
Covariance  AB  0.250.2  0.10.05  0.1  0.5 0.1  0.10.1  0.1
 0.250  0.10.15  0.1  0.0025
Correlation
 AB 
 AB
 0.0025

 1.0
 A B 0.070710.03536 
Returns on stocks A and B are perfectly negatively correlated.
Stocks A can be used as a hedge against the risk in holding
stock B
Historical Data Based Approach
From historical data, calculate the percentage returns R1, R2, …, RT
T
Sample average percentage return   R  R1  ...  R T  1
 Rt
T
Sample Variance

2

T
1

 Rt  R
T  1 t 1
T
t 1

2
Sample standard deviation (or standard deviation)
  2 0
Historical Data Based Approach (continued)
Sample covariance of returns on stocks A and B, calculated from
the historical samples of RA and RB
RA = (RA1, …, RAT) ; RB = (RB1, …, RBT)
 AB
RA


1 T

 R At  R A RBt  RB
T  1 t 1
1 T
  R At
T t 1
;
1 T
RB   RBt
T t 1
1 T
;  
R At  R A

T  1 t 1

 AB
1 T
2
2

 B   B ; B 
RBt  RB

 A B
T  1 t 1

Sample correlation of
RA and RB
 AB

A  
2
A
2
A


2
2
Expected Return and Variance of Returns
on Portfolios
A portfolio is an investment in N  2 stocks.
Let xn  [0,1] be the proportion invested in stock n.
N
Then
x
n 1
n
1
If the return on stock n is Rn, then the return on the portfolio is
N
Rp  x1R1  ...  x NRN   x nRn
n 1
and the expected return on the portfolio is
N
 N
 N
p  Ε  x nRn    x nΕRn   x n Rn
n1
 n1
 n1
Expected Return and Variance of Returns on
Portfolios (continued)
The variance of the returns on the portfolio is given by

2
p
 ΕR
 ΕR 
2
p
p


 Ε  x nRn   x n Rn 
n1
 n1

N


 Ε  x n Rn  ΕRn 
 n1

N
2
N
N
2
2
N n1
 Ε x n Rn  ΕRn   2Ε x nx m Rn  ΕRn Rm  ΕRm 
2
n1
n 2 m1
N n1
N
  x n  2 x nx m nm
2
n1
2
n
n 2 m1
Diversification
Consider a special case with
xn 
1
N
for each n  1,..., N .
Then
 p2
 N n1
    nm
N


1 1
N N  1  n 2 m 1

    n2   2
 NN  1
N  N n 1
N2









 0   2  2  1   2  
1. Variances are diversified away
2. Average covariance converges to covariance from economywide shocks affecting all stocks
- In a diversified portfolio, only systematic risk affects returns.
- Diversifiable or unsystematic (idiosyncratic) risk is irrelevant to
returns.
Diversification (continued)
Recall
A  B  0.10 ;  A  0.07071, B  0.03536
Suppose you invest $100 in stock A and $200 in stock B.
Returns on investment in assets A and B
States
Boom
Normal
Recession
s
0.25
0.50
0.25
RA
120 (20%)
110 (10%)
100 ( 0%)
RB
210 (5%)
220 (10%)
230 (15%)
The mean return on the portfolio is 10%.
ΕRp   Εx AR A  x BRB   x AΕR A   xBΕRB 

1
2
 10%   10%  10%
3
3
Total return
330 (10%)
330 (10%)
330 (10%)
Diversification (continued)
The mean return on the portfolio is a weighted average of
ER A and ERB
The standard deviation of the return on the portfolio is zero.
No risk!
Recall that the correlation
 AB between the returns on A and B
is -1. This implies that the variation in returns on either asset
can be completely offset by holding the right proportion of the
other asset.
Deriving an appropriate discount rate for risky
cash flows
1. The opportunity set for two assets
2. The opportunity set and efficient set with many securities
3. The efficient set with a riskless asset
4. The CAPM (capital asset pricing model) equation
5. A risk-return separation theorem
The opportunity set for two assets
Suppose there are two assets A and B in proportions
Then,
xB  1 x A since x A  xB  1 .
x AR A  1 x A RB 
 x A R A  1 x A RB
p  Rp 

 RB  x A R A  RB
 p2 


R
 Rp 

2
p
x R
x R
A
A
A
A


R 
 x BRB  x A R A  x B RB
 
 x B RB  x A RB  x B
 x 2A A2  2x A x B AB  x B2 B2
2
2
B
x A and xB .
The opportunity set for two assets (continued)

From p  RB  x A R A  RB
 , we have
xA 
 p  RB
R A  RB
Then we have
 p2
2 
  2  2


2
AB
B
 A


p
2


R A  RB






 2 RB A2  R A  RB  AB  R A B2 p
2
2
 RB A2  2R A RB AB  R A B2
Using the above equation, we can trace a feasible (or
opportunity) set of attainable  p and  p for given
R A , R B ,  A ,  B ,  AB ,  AB   AB A B 
.
Example
We are given the following parameter values,
R A  17.5%
; RB  5.5%
 A  25.86% ;  B  11.50% ;  AB  0.1639
For these values, the above equation becomes approximately
 p2  6.2394 p2  0.9880 p  0.0497
which looks like the following in
 ,   space.
p
p
Example (continued)
Opportunity set for assets A and B
Portfolio MV (minimum variance) has the lowest risk
obtainable with assets A and B.
Between B and MV, replacement of B by A increases  p and
reduces  p . This always happens if  AB  0 and may
happen for  AB  0 .
When  AB  1 , a riskless portfolio can be obtained by holding
A and B in right proportions.
The opportunity set and efficient set with many
securities
Suppose we add asset C, to the previous example, with the
parameter values
RC  10.5% ;  C  15.0% ; CA  0.20 ; CB  0.05
Each pair of securities ((A,B),(A,C),(B,C)) gives an opportunity set
Linear combination of portfolios in any of these opportunity set will
lead to additional curve in  ,   s space.
It can be shown that the opportunity set for N  3 assets is an
area bounded by a rectangular hyperbola.
Except for portfolios close to MV, the efficient set is very close to a
straight line. Also as the variance of the MV portfolio
decreases, the efficient set gets closer to a straight line.
The efficient set with a riskless asset
If one asset is riskless, the variance of returns on that asset, and
the covariance with returns on all other assets will be zero.
In the two security case discussed earlier, suppose B is riskless,
I.e.,  B  0 . Then from the above equation, we have
  p  RB 
 A   A p  RB A
p  
 R A  RB 
R A  RB
R A  RB


In equilibrium, the riskless rate < return on MV. Hence, the
opportunity set will be the tangent line from the riskless asset
to the efficient set.
The efficient set with a riskless asset (continued)
Homogeneous expectations assumption
All investors have the same estimates on expectations, variances
and covariances.
Under homogeneous expectations, all investors would hold the
portfolio of risky assets represented by the tangency portfolio.
What is the tangency portfolio?
It is a market-valued weighted portfolio of all existing securities,
I.e. market portfolio. A proxy commonly used is S&P 500.
Use of such a broad-based index as a proxy is justified since
most investors hold diversified portfolios.
The efficient set with a riskless asset (continued)
The best measure of the risk of a security in a large portfolio is
the beta of the security, which measures the responsiveness
of the security to the movements in the market portfolio.
Formula for beta
i 
CovRi ,Rm 
 2 Rm 
CovRi ,Rm  covariance between the return on asset i
and the return on market
 2 Rm  variance of market portfolio
Example
States Probability Economy Firm
shock
shock
1
0.25
Recession Down
2
0.25
Recession
Up
3
0.25
Boom
Down
4
0.25
Boom
Up
Market
Firm return
return (%) (%)
-5
-15
-5
-5
15
15
15
25
4
Rm    sRms  0.25  5   0.25  5   0.25 15   0.25 15   5%
s 1
4
R f    sR f s  0.25  15  5  15  25   5%
s 1
4

    s Rms  Rm
2
M

2
 0.01
s 1
4



CovR f ,Rm     s R f s  R f Rms  Rm  0.015
s 1
Example (continued)
The beta coefficient for this firm is

CovR f ,Rm  0.015

 1.5
2
 Rm 
0.01
Returns on this firm’s stock magnify market returns.
The CAPM equation
Relationship between risk and expected return
If there is a riskless asset with return r, there is a straight line
trade off between risk and expected return for a security.
R  r  slope   

is the contribution of this security to the portfolio risk.
If the tangency portfolio is the market portfolio with expected
return Rm and standard deviation  Rm  , then
slope   Rm  r
 Rm 
The CAPM equation (continued)
Equilibrium expected return on asset j :
 Rm  r 
  j
R j  r  

  Rm  
It can be shown that
Then we have
j 
CovR j ,Rm 
 Rm 
Rm  r CovR j ,Rm 
Rj  r 

 Rm 
 Rm 

CovR j ,Rm 
 Rm 
2

CAPM equation R j  r   j Rm  r



 Rm  r   j  Rm  r


The CAPM equation (continued)

CAPM equation R j  r   j Rm  r

(Expected return on a security)
= (current risk free interest rate)
+ (beta coefficient of the security)*(historical market risk premium)
Finally, we established a way of determining appropriate
discount rate for risky cash flows. We first measure its risk
by its beta coefficient, and then obtain the required return
from the CAPM equation.
The CAPM equation (continued)
Interpretation
Recall that the variance of return on a diversified portfolio is
basically the “average covariance”. The beta coefficient for
asset j  j  can be considered as the share of overall market
risk contributed by asset j. Then CAPM equation says that an
asset shares the market excess return R m  r to the extent
that it contributes to the total market risk.


Regression
In practice, we usually estimate  j using linear regression
using historical returns data on Ri and Rm
Rit  i  iRmt  errort
The CAPM equation (continued)
statistical (least squares) estimator for  j
 R
T
ˆi 
t 1
it

 R i Rmt  R m
 R
T
t 1
mt
 Rm

2

CovRi , Rm 

 2 Rm 
The Security Market Line (SML)
When
 j  1  R j  Rm
 j  1  R j  Rm
j  0  Rj  r
The Security Market Line (SML) below graphs
expected return against beta, using the CAPM equation.

R j  r   j  Rm  r

Slope of the SML is the risk premium. For the S&P500 and
US treasury bills, the risk premium is about 8.5%. (The
book uses 9.2%, which is based on Ibbotson et. al study).
This estimate is often used as a forecast for the risk
premium on stocks in the future.
SML (continued)
The SML applies to portfolios as well as individual securities. For
a portfolio with x A of A and xB of B, with beta coefficients  A
and B the expected return on the portfolio is
p  x A R A  xB RB
Note that CovRp ,Rm   Covx AR A  x BRB ,Rm 
 x A CovR A ,Rm   x BCovRB ,Rm 
implying
 p  x A  A  xB B
Hence, the portfolio also will be on the SML.
The SML should not be confused with the efficient set.
A Risk-Return Separation Theorem
An investment will be worth taking only if it is at least as desirable
as what is already available in the financial markets.
A new investment will be worthwhile if and only if it is outside
(above) the efficient set (or the risk-return budget constraint).
No matter where individual would choose to be on the efficient
set, an investment can only make them better off if it is above
the efficient set.
If the two financial separation theorems did not hold, then the
firms would need to know the inter-temporal and risk-return
preferences of each owner to decide desirable investments.
Problem 10.13 from the text
There are 3 securities in the market with the following payoffs:
State
Prob.
1
2
3
4
Return
on A
0.25
0.20
0.15
0.10
0.10
0.40
0.40
0.10
Return
on B
0.25
0.15
0.20
0.10
Return
on C
0.10
0.15
0.20
0.25
What are expected returns and standard deviations of the
returns?
4
Ri    sRis
s 1
4

 i    s Ris  Ri
s 1

2
RA
RB
RC
0.175
0.175
0.175
 A2
 B2
 C2
0.0403
0.0403
0.0403
Problem 10.13 from the text (continued)
What are covariances and correlations between the returns?
For j = A,B,C and k = A,B,C

4

 jk    s R js  R j Rks  Rk

s 1
 jk
 AB
 AC
 BC
 jk

 j k
0.000625
-0.001625
-0.000625
 AB
 AC
 BC
0.385
-1.000
-0.385
Problem 10.13 from the text (continued)
What are expected returns and standard deviations of the
portfolios?
PAB
PAC
x A  0.5; xB  0.5
x A  0.5; xC  0.5
RPAC  0.5RA  0.5RC
PBC
xB  0.5; xC  0.5
RPBC  0.5RB  0.5RC
RPAB  x A RA  xB RB  0.5 RA  0.5 RB
RPAB  x A R A  x B RB  0.5 0.175   0.5 0.175   0.175
RPAC  RPBC  RPAB  0.175
 P2  Var x AR A  x BRB 
AB
 x 2A Var R A   2x A x BCovR A ,RB   x B2 Var RB 
 0.25 A2  20.25  AB  0.25 B2
P
AB
 0.0335
P
AC
0
 P  0.0224
BC
Problem 10.39 from the text
Suppose you have invested $30,000 in the following 4 stocks
Security
Stock A
Stock B
Stock C
Stock D
Amount
invested
5,000
10,000
8,000
7,000
Beta
0.75
1.10
1.36
1.88
xi
5/30
10/30
8/30
7/30
R
i
 r  
i
R m
 r
0.1225
0.1610
0.1896
0.2468
The risk free rate is 4% and the expected return on the market
portfolio is 15%. Based on the CAPM, what is the expected
return on the above portfolio?
Let x i denote the proportion invested in stock i (I=A,B,C,D)
and i the beta coefficient of the stock i.
Problem 10.39 from the text (continued)
There are two ways to answer the question.
1. Calculate the beta coefficient  P for the portfolio, and get
the expected return on the portfolio directly from CAPM
equation.
R P  r   P Rm  r   0.04  0.11 P  0.1822
 P  x A  A  xB  B  xC  C  xD  D  1.293
2. Calculate the expected return R i individually for I =
A,B,C,D and obtain the expected return R P on the
portfolio as
R P  x A R A  xB R B  xC RC  xD R D  0.1822