The Capital Asset Pricing Model
Lecture XXIV
.Literature
 Most
of today’s materials comes from
Eugene F. Fama and Merton H. Miller The
Theory of Finance (Hinsdale, Illinois:
Dryden Press, 1972) Chapter 7.
 The
primary literature is:
 Lintner,
John. “Security Prices, Risk, and
Maximal Gain from Diversification.” Journal
of Finance 20(Dec. 1965): 587-615.
 Lintner, John. “The Valuation of Risk Assets
and the Selection of Risky Investments in Stock
Portfolios and Capital Budgets.” Review of
Economics and Statistics 47(Feb. 1965): 13-37.
 Mossin,
Jan. “Equilibrium in a Capital Asset
Market.” Econometrica 34(Oct. 1966): 768-83.
 Sharpe, W. F. “Capital Asset Prices: A Theory
of Market Equilibrium under Conditions of
Risk.” Journal of Finance 19(Sept. 1964): 42542.
Setting Up the Market
 Perfect
Markets: We assume that all markets
are competitive for goods and investments.
 All
goods and investments are infinitely
divisible.
 Information is costless.
 There are no transaction costs.
 No individual is large enough to effect the
price.
 Firms:
 All
goods are produced by firms.
 These firms purchase the factors of production
in the first period, produce output, and market
their goods in the second period.
 In addition, these firms do not have any capital
of their own and must raise this capital by
issuing stock.
 Consumers:
Consumers begin with an
endowment w1. The consumer’s choice
problem (initially) is twofold.
 First,
the consumer must decide how much to
consume in this period, c1, and how much to
invest, h1. This investment will earn a rate of
return h1(1+Ri) which will be consumed in the
second period, c2.
 Second,
the consumer must decide how to
invest h1, that is how to divide it up between a
wide array of assets.
 In
general, this implies two decision
dimensions. The first is intertemporal
(across time). In this decision the consumer
has a time preference, the preference
between consuming now and consuming
later.
c2
1/(1+r)
U(c1,c2)
c1
 The
second is the risk or uncertainty on the
investment. Both of these questions can be
represented in the utility function.
 Market Equilibrium: Assuming that firms
supply investment and consumers demand
investment opportunities, we hypothesize
that there exists an equilibrium where the
supply of stocks equals the demand of
stocks.
.Risk Equilibrium From the
Consumer’s Point of View
 At
the outset, we assume that consumers are
risk averse and that risk can be
characterized using the normal distribution
function.
 Given
these assumptions, we can use the
Expected Value-Variance, or in this case the
Expected Value-Standard Deviation approach to
expected utility/risk efficiency.
N
~
m p   m i xi
i 1
where mp is the expected return from the
portfolio, mi is the expected return on a specific
asset, and xi is the level of asset i held in a
specific portfolio.
 Similarly,
the standard deviation of the
portfolio can be written as
N N

~
~
s p   s ij xi x j 
 j 1 i 1

1
2
where sp is the standard deviation of a
particular portfolio and sij is the covariance
between asset i and asset j.
 In
addition, we impose a portfolio balance
condition
N
~
 xi  1
i 1
 We
can reformulate the risk measure, standard
deviation of the portfolio, to analyze the
contribution of each asset to the overall risk of
the portfolio:
s p2
sp 
sp
N

N
~
~
s
x
 ij i x j
j 1 i 1
sp
 N

~
  s ij xi 
N

 ~
x j  i 1
 sp 
j 1




 The
risk of a particular asset xj is then
dependent on weighted covariances
between asset j and the returns in the rest of
the portfolio. Remember that the xis are
weights in the general portfolio. This raises
two points:
 First,
note that the risk of an individual asset
depends on the portfolio weights and the risk of
the portfolio.
 Second,
the risk of a particular asset depends
both on its own variance and the variance of the
remaining assets in the portfolio. Thus, as the
number of assets becomes large and the
portfolio becomes well diversified, the risk of a
particular asset is more dependent on the
covariance with other assets in the portfolio
than on its own risk.
 Following
our previous discussions of Expected
Value-Variance frontier, we assume that
consumers choose the portfolio that minimizes
risk for any given level of expected income.
However, in this case risk is parameterized by
the standard deviation instead of the variance.
 N N

~
~
min s p    s ij x j xi 
 j 1 i 1

N
s.t.
m p   m i ~xi  m *
i 1
N
~
 xi  1
i 1
1
2
N
N




~
~
L  s p  1  m p   m i xi   2 1   xi 
i 1


 i 1 
L s p
 ~  1m k  2  0
~
x
x
k
k
N
L
 m p   mi ~
xi  0
1
i 1
N
L
 1  ~
xi  0
2
i 1
s p
s p
 1m k  ~  1m l
~
xk
xl
1  s p s p 
m k  m l   ~  ~ 
1  xl
xk 
mp
sp
N
N



s

s
1
p
p
~
~
~




x
m

m

x

x



i
j
i
i
i
~
~




x

x
i 1
i

1
i

1
1 
j
i 
N
1  s p N ~ s p 
m j  m p   ~   xi ~ 
1  x j i 1 xi 
 N

~
  s ij xi 
N
s p
~
i 1


s p   xj
 ~ 
 sp 
xk
j 1




N
~
s
 ik xi
i 1
sP
N
N s ~
N N s x x

s
x
p
ij j
ij i j
~
~
 xi ~x   xi  s   s  s p
N
i 1
i
i 1
j 1
P
j 1 i 1
p

1  N s ij xi x j
m j  mp  
s p 

1  i 1 s p


1  s jp
m j  mp  
s p 

1  s p

The Role of the Riskless Asset
 The
equilibrium presented above does not
yield an estimable representation because
different investors may have different risk
preferences. One way around this
ambiguity is to introduce a riskless asset.
 The
riskless asset reduces the potential number
of efficient portfolios to a single portfolio.
mp
rp
rf
sp
sp
 Within
this equilibrium, there is only one
efficient portfolio of assets. Any degree of risk
aversion can construct a risk preferred position
by holding a combination of the single efficient
portfolio and borrowing or lending at the
riskless rate.
 Substituting xm for xp or letting the index
portfolio be the market efficient portfolio, we
have

1  s m
m j  m m   ~  s m 
1  x j

1
1

m

m
 rf 
sm
 m m  rf  s m
m j  rf  
 ~
 s m  x j
 m m  rf  s jm
 rf  

 sm  sm
 rf   j m m  rf 
s jm
j  2
sm
rit     i rmt   it
Supply of Stocks from the
Firm
 We
start by assuming that each firm sells
stocks at a price Pi. Investors are willing to
bid on these stocks based on the future
value of the firm at the end of the year, Vi.
The bid price and the value at the end of the
year determine the rate of return:
Vi  Pi
Ri 
Pi
Given that the future value of the firm
implies some risk, the rate of return is risky.
In addition, given the preceding proof we
know that investors value the investment
under the capital market equilibrium.
 Mathematically, the price and value of the
market portfolio becomes
N
N
i 1
i 1
Pm   Pi and Vm   Vi
Vm  Pm 1  Rm 
E V j   Pj
Pj
 E Rm   R f
 Rf  
sm

 s jm

 Pjs m
Incorporating Risk Using
CAPM
 Both
risk adjustments come directly from
the last equation. The first approach is
referred to as the risk adjusted discount rate
(RADR). Reducing the preceding
expression to the CAPM formula
R j  R f   j Rm  R f 
 This
risk adjusted discount rate can be used
in present value analysis
ECF I t 
N
PV  
i 1
1  R
f

  j Rm  R f  
i
 Reformulating
1
j
 This
R
j
this equality slightly
 R f   RM  R f
approach leads to transforming the
annual rates of return into “certainty
equivalents” based on the market portfolio:
 1 
E CF I t 

N 

j

PV  
i
1  Rm 
i 1