Capital Asset Pricing Model
Applied covariance:
Project Part 1
Review variance, covariance
 Variance:
square the deviations and
take expectation.
 Covariance: multiply the deviations and
take expectation.
Application
 Asset
B is the market portfolio
 Call it asset M.
 Everyone prefers to hold M, in theory
 Asset A is any asset.
 Think of adding a little A to the market
portfolio.
Question
 does
adding a little of asset A to the
market portfolio increase the risk?
 Yes if
2
 AM   M
No if
 AM   M
2
Derivation
 P  X A  A  (1  X A )  M  2 X A (1  X A ) AM
2
2
2
2
2
d
2
2
2
 P  2 X A  A  2(1  X A ) M
dX A
 2(1  X A ) AM  2 X A AM
at X A  0,
d
2
2
 P  2 M  2 AM
dX A
Beta measures risk
 How
much risk is added depends on
the relation of sigma AM and sigma
squared M
 Define beta
 AM
A 
2
M
Sum of squared errors
t T
F (b)   ( Dev A,t bDev M ,t )
2
t 1
Minimize it
F ' (b)  0
t T
F ' (b)   2( Dev A,t bDevM ,t ) DevM ,t  0
t 1
t T
 Dev
t 1
t T
A,t
DevM ,t  bDev
t 1
2
M ,t
0
Divide by T-1
t T
1
Dev A,t DevM ,t

T  1 t 1
b
t T
1
2
DevM ,t

T  1 t 1
The estimate of 
 Is
the ratio of sample covariance over
variance of the market.
 It’s beta, except for using sample
statistics instead of population values.
The story of CAPM
 Investors
prefer higher expected return
and dislike risk.
 All have the same information.
 Two (mutual) funds are sufficient to
satisfy all such investors:
The two funds:
 1)
The "risk-free" asset, i.e., Treasury
Bills
 2) The market portfolio consisting of all
risky assets held in proportion to their
market value.
The market portfolio
 Its
expected return is 8.5% over the TBill rate
 It bears the market risk
 Its beta is unity by definition.
Capital asset pricing model
E ( R j )  R f  E[ RM  R f ]   j
T-bill rate is known.
Market premium is known,
approximately 8.5%.
Estimate beta as in the project
Security market line
 It’s
straight.
 Risk-return relation is a straight line.
Why is it a straight line?
 Beta
is the measure of risk that matters.
 Given beta construct a portfolio with the
same beta by a mix of T-Bills (beta = 0)
and the market portfolio (beta = 1)
 Expected return on the portfolio is on
the SML.
 So any asset with the same beta must
also be on the SML.
Rate of return
expected by the market
E[RM]
Rf
beta
1
Review item
 Return
on asset A has a std dev of .05
 Return on asset B has a std dev of .07
 Correlation of return on asset A with
return on asset B is 1.
 What is the covariance of the returns?
Answer:
 Covar
= corr*stdevA*stdevB=.0035