The Capital Asset Pricing Model
Ming Liu
Industrial Engineering and Management Sciences,
Northwestern University
Winter 2009
Returns to financial securities
P0: security price at time 0
P1: security price at time 1
DIV1: dividend at time 1
r = total return = dividend yield + capital gain
rate
r = DIV1/P0+(P1-P0)/P0
(random variable)
ri : the return on security i,
Decompose this return ri into that part
correlated with the market and that part
uncorrelated with the market
rm = the return on the market portfolio
εi = the specific return of firm i
Systemic and Idiosyncratic Risk
ri=αi+ βi rm + εi
systemic risk
undiversifiable risk
beta risk
market risk
idiosyncratic risk
diversifiable risk
non-systematic risk
"Beta" (β) an asset market risk parameter, represents straight-line inclination degree.
E is average "residual" yield, describing an average asset yield deviation from "fair" yield as
shown by the central line.
ri=αi+βi rm +εi
The larger is βi , the more subject to market risk
is this firm.
The larger is σ[i] the more important is firmspecific risk.
 [ri ]    [rm ]  ( [ i ])
2
i
2
2
Example
Decomposing the Total Risk of a Stock
Considering two stocks:
2
A: An automobile stock with βA=1.5,  [ A ]  .10
2
B: An oil exploration company with βB=0.5,  [ B ]  .18
The variance of the market return is  [rm ]2  .04
What is the total risk of each stock?
 [rA ]2  1.52  0.04  0.10  0.19
 [rB ]2  0.52  0.04  0.18  0.19
Which has a higher expected rate of
return?
Portfolio risk
1. Decompose each security return into
systematic and idiosyncratic risk:
ri=αi+βirm+εi
2. Form a portfolio of these securities, with
portfolio weights w1, w2, …, wn . (sum to one)
3. The portfolio rate of return is a weighted
average of the individual returns
rp = w1r1+w2r2+…+wnrn
rp = w1[α1+β1rm+ε1]
+ w2[α2+β2rm+ε2]
+ …
+ wn[αn+βnrm+εn]
Rearrange to get
zero
rp = α*+β*rm+ε*,
where
α*:= w1α1+w2 α 2+…+wn α n
β*:= w1 β1+w2 β2+…+wn β n
ε*:= w1 ε1+w2 ε2+…+wn εn
Conclusions
• β of portfolio is weighted-average β
• Well diversified -> risk only from βrm term
The standard deviation of a well diversified
portfolio:
 [rp ]   *  [rm ]
Construct the market portfolio
• The market portfolio includes every security in the
market
• The weight of each security in the portfolio is
proportional to its relative size in the economy
• A common proxy measure for the market portfolio
is the S&P 500 index.
http://www.indexarb.com/indexComponentWtsSP5
00.html
The Capital Asset Pricing Model
Market model
ri=αi+βirm+εi
with αi=(1-βi) rf
ri=(1-βi) rf+βirm+εi
ri=(1-βi) rf+βirm+εi
• Does this restricted case make sense?
What does it imply for the return on a risk-free
asset (βi=0)?
What does it imply about the return on an asset
that has the same market risk as the market
portfolio (βi=1)?
• The CAPM equation can be rewritten as
ri-rf=βi (rm –rf )+εi
The CAPM can also be written as a linear
relationship between the β of a security and
its expected rate of return,
E(ri )-rf=βi (E (rm )–rf )
E(ri ) : expected rate of return on the security
E (rm): expected rate of return on the market
portfolio
rf : the risk free rate
βi : the security’s beta
The Security Market Line
E(ri )
E(ri )=rf+βi (E (rm )–rf )
E(ri )=(1- βi)rf+βi E (rm )
E(rA )
E(rB )
rf
βB=0.5
βA=1.5
βi
Example
Using the Security Market Line (SML)
The β of Cisco Systems is about 1.37.
The risk free rate rf=0.07
Expected risk premium on market E (rm )–rf =0.06
The expected rate of return on CSCO:
E (ri )  0.07  1.37  0.06  0.1522
How to get β?
• If we know
 σ[ri] ----- standard deviation of ri
 σ[rm] ----- standard deviation of rm
 ρim----- correlation between ri and rm
cov( ri , rm )  im [ri ] [rm ]  im [ri ]
i 


2
2
 [rm ]
 [rm ]
 [rm ]
How to get β?
•
ˆ imˆ [ri ]
ˆ
Estimate beta:  i 
ˆ [rm ]
ri=αi+ βi rm + εi
•
http://finance.yahoo.com/
• CAPM serves as a benchmark
– Against which actual returns are compared
– Against which other asset pricing models are compared
• Advantages:
– Simplicity
– Works well on average
• Disadvantages:
– What is the true market portfolio and risk free rate?
– How do you estimate beta?
– Standard deviation not a good measure of risk.