Lecture Lecture 10 10 The Capital Asset Pricing The Capital Asset Pricing Model Model
Dec 30, 2015
Lecture 10Lecture 10
The Capital Asset Pricing ModelThe Capital Asset Pricing Model
Expectation, variance, standard error (deviation), covariance, and correlation of returns may be based on
(i) fundamental analysis
(ii) historical data
Preliminaries
Fundamental or Theoretical Analysis
S possible states
s probability of state s = 1,2,…,S
Rs likely return is state s
Notation
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)
Example: Suppose there are
S
1sssRRE Expectation (mean)
S
1s
2ss ERRRvar 2Variance
02 Standard error
BBs
S
1sAAssBAAB RRRRR,RCov ΕΕ
returns on stock A RAs s = 1,…,S
returns on stock B RBs s = 1,…,S
Covariance measures how two random variables are related
ABAB signsign
11122
22222 AB
BA
ABABBAAB
BA
ABBAAB R,Rcorr
Correlation is a normalized covariance
Note !
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 0% 15%
Expected returns
0.100.150.250.100.500.050.25
0.100.000.250.100.500.200.25
B
A
Variances
0.001250.100.150.250.100.100.500.100.050.25
0.0050.100.000.250.100.100.500.100.20.252222
B
2222A
Standard errors
0.03536
0.07071
B
A
2
2
B
A
Covariance 0.00250.10.150.100.25
0.10.10.10.10.50.10.050.10.20.252AB
Correlation 1.0
0.035360.07071
0.0025
BA
ABAB
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
02 Sample standard deviation (or standard deviation)
Sample average percentage return
T
1tt
T1 RT
1
T
R... RR
Sample Variance
T
1t
2
t2 RR
1T
1
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)
T
1tBtB
T
1tAtA
BBt
T
1t
AAtAB
RT
1R ; R
T
1R
RRRR1T
1
Sample correlation of RA and RB
BA
ABAB
T
1t
2AAt RR
1T
122 ; AAA
T
1t
2BBt
2B RR
1T
1 ; 2BB
Expected Return and Variance of Returns on Portfolios
A portfolio is an investment in stocks.
Let be the proportion invested in stock n.
Then
2N
1xN
1nn
[0,1]xn
If the return on stock n is Rn, then the return on the portfolio is
N
1nnnNN11p RxRx...RxR
and the expected return on the portfolio is
N
1n
nn
N
1nnn
N
1nnnp RxΕRxRxΕ
N
2n
1n
1mnmmn
N
1n
2n
2
N
2nmm
1n
1mnnmn
2N
1nnn
2
2N
1nnnn
2N
1n
nn
N
1nnn
2pp
2p
xx2x
ΕRRΕRRxx2ΕΕRRxΕ
ΕRRxΕ
RxRxΕΕRRΕ
n
n
The variance of the returns on the portfolio is given by
Expected Return and Variance of Returns on Portfolios (continued)
Diversification
1. Variances are diversified away
2. Average covariance converges to covariance from economy-wide shocks affecting all stocks
Consider a special case with for each .
ThenN
1xn N1,...,n
2120
1NNN
1NN2
N
1
N
1
2
N
2n
1n
1mnm
2
N
1n
2n
2p
- In a diversified portfolio, only systematic risk affects returns.
- Diversifiable or unsystematic (idiosyncratic) risk is irrelevant to returns.
Diversification (continued)
Recall ;
Suppose you invest $100 in stock A and $200 in stock B.
0.10BA 0.035360.07071, BA
Returns on investment in assets A and B
States s RA RB Total return
Boom 0.25 120 (20%) 210 (5%) 330 (10%)
Normal 0.50 110 (10%) 220 (10%) 330 (10%)
Recession 0.25 100 ( 0%) 230 (15%) 330 (10%)
The mean return on the portfolio is 10%.
10%10%3
210%
3
1
RΕxRΕxRxRxΕRΕ BBAABBAAp
Diversification (continued)
The standard deviation of the return on the portfolio is zero.
No risk!
The mean return on the portfolio is a weighted average of
and AER BER
Recall that the correlation 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.
AB
Deriving an appropriate discount rate for risky cash flows
1. The opportunity set for two assets
3. The efficient set with a riskless asset
2. The opportunity set and efficient set with many securities
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 and .
Then, since .
Ax Bx
AB x1x 1xx BA
2B
2BABBA
2A
2A
2BBBABBAA
2BBAABBAA
2pp
2p
xx2xx
RxRxRxRx
RxRxRxRx
RR
BAAB
BAAA
BAAApp
RRxR
Rx1Rx
Rx1RxR
The opportunity set for two assets (continued)
From , we have . BAABp RRxR BA
BpA
RR
Rx
Then we have
2B
2AABBA
2A
2B
p2BAABBA
2AB
2p2
BA
2BAB
2A2
p
RRR2R
RRRR2
RR
2
Using the above equation, we can trace a feasible (or opportunity) set of attainable and for given BAABABABBABA RR ,,,,,
p p
Example
We are given the following parameter values,
0.1639 ; 11.50% ; 25.86%
5.5%R ; 17.5%R
ABBA
BA
For these values, the above equation becomes approximately
0.04970.98806.2394 p2p
2p
which looks like the following in space. pp ,
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 and
reduces . This always happens if and may
happen for .
p
p 0AB0AB
When , a riskless portfolio can be obtained by holding
A and B in right proportions.
1AB
The opportunity set and efficient set with many securities
Each pair of securities ((A,B),(A,C),(B,C)) gives an opportunity set
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.
Suppose we add asset C, to the previous example, with the parameter values
0.05 ; 0.20 ; 15.0% ; 10.5%R CBCACC
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 assets is an area bounded by a rectangular hyperbola.
3N
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 equilibrium, the riskless rate < return on MV. Hence, the opportunity set will be the tangent line from the riskless asset to the efficient set.
In the two security case discussed earlier, suppose B is riskless, I.e., . Then from the above equation, we have 0B
BA
AB
pBA
AA
BA
Bpp
RR
R
RRRR
R
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.
It is a market-valued weighted portfolio of all existing securities, I.e. market portfolio. A proxy commonly used is S&P 500.
What is the tangency portfolio?
Use of such a broad-based index as a proxy is justified since most investors hold diversified portfolios.
m
2mi
i R
R,RCov
Formula for beta
mi R,RCov
m2 R
covariance between the return on asset i and the return on market
variance of market portfolio
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.
Example
States Probability Economyshock
Firmshock
Marketreturn (%)
Firm return(%)
1234
0.250.250.250.25
RecessionRecession
BoomBoom
DownUp
DownUp
-5-51515
-15-51525
5%150.25150.2550.2550.25RR4
1smssm
5%25155150.25RR4
1sfssf
0.01RR4
1s
2mmss
2M
0.015RRRRR,RCov4
1s
mmsffssmf
Example (continued)
The beta coefficient for this firm is
1.5
0.01
0.015
R
R,RCov
m2
mf
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.
sloperR
is the contribution of this security to the portfolio risk.
If the tangency portfolio is the market portfolio with expected
return and standard deviation , then mR mR
m
m
R
rRslope
The CAPM equation (continued)
Equilibrium expected return on asset j :
jm
mj
R
rRrR
It can be shown that m
mjj R
R,RCov
Then we have
rRrRR
R,RCov
R
R,RCov
R
rRrR
mjm
m2
mj
m
mj
m
mj
rRrR mjj CAPM equation
The CAPM equation (continued)
(Expected return on a security)
= (current risk free interest rate)
+ (beta coefficient of the security)*(historical market risk premium)
rRrR mjj CAPM equation
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 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 to the extent that it contributes to the total market risk.
j
rRm
In practice, we usually estimate using linear regression using historical returns data on and
Regression
j
iR mR
terrorRR mtiiit
The CAPM equation (continued)
m
miT
t
mmt
T
t
mmtiit
i R
RRCov
RR
RRRR
2
1
2
1 ,ˆ
statistical (least squares) estimator for j
The Security Market Line (SML)
When
rR0
RR1
RR1
jj
mjj
mjj
The Security Market Line (SML) below graphs
expected return against beta, using the CAPM equation.
rRrR mjj
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 of A and of B, with beta coefficients and the expected return on the portfolio is
Ax Bx AB
BBAAp RxRx
Note that
implying
mBBmAA
mBBAAmp
R,RCovxR,RCovx
R,RxRxCovR,RCov
BBAAp xx
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.
State Prob. Returnon A
Returnon B
Returnon C
1234
0.100.400.400.10
0.250.200.150.10
0.250.150.200.10
0.100.150.200.25
There are 3 securities in the market with the following payoffs:
What are expected returns and standard deviations of the returns?
4
1sissi RR AR
BR
CR
0.1750.1750.175
2A2B2C
0.04030.04030.0403
4
1s
2iissi RR
Problem 10.13 from the text
What are covariances and correlations between the returns? For j = A,B,C and k = A,B,C
kks
4
1s
jjssjk RRRR
kj
jkjk
ABACBC
0.000625-0.001625-0.000625
ABACBC
0.385-1.000-0.385
Problem 10.13 from the text (continued)
Problem 10.13 from the text (continued)
What are expected returns and standard deviations of the portfolios?
ABP
ACP
BCP
5.0;5.0 BA xx
5.0;5.0 CA xx
5.0;5.0 CB xx
BABBAAP RRRxRxRAB
5.05.0
CAP RRRAC
5.05.0
CBP RRRBC
5.05.0
0.1750.1750.50.1750.5RxRxR BBAAPAB
0.175RRR ABBCAC PPP
2BAB
2A
B2BBABAA
2A
BBAA2P
0.250.2520.25
RVarxR,RCovx2xRVarx
RxRxVarAB
0.0335ABP 0
ACP 0.0224BCP
Problem 10.39 from the text
Suppose you have invested $30,000 in the following 4 stocks
S e c u r ity A m o u n tin v e s te d
B e ta ix rRrR mii
S to c k AS to c k BS to c k CS to c k D
5 ,0 0 01 0 ,0 0 08 ,0 0 07 ,0 0 0
0 .7 51 .1 01 .3 61 .8 8
5 /3 01 0 /3 0 8 /3 0 7 /3 0
0 .1 2 2 50 .1 6 1 00 .1 8 9 60 .2 4 6 8
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 denote the proportion invested in stock i (I=A,B,C,D)
and the beta coefficient of the stock i.i
ix
Problem 10.39 from the text (continued)
There are two ways to answer the question.
1. Calculate the beta coefficient for the portfolio, and get the expected return on the portfolio directly from CAPM equation.
P
1822.011.004.0 PmPP rRrR
293.1 DDCCBBAAP xxxx
2. Calculate the expected return individually for I = A,B,C,D and obtain the expected return on the portfolio as
1822.0 DDCCBBAAP RxRxRxRxR
iR
PR