ECON337901: Problem Set 7
Problem Set 7
ECON337901 - Financial Economics
Boston College, Department of Economics
Peter Ireland
Spring 2015
Due Tuesday, April 21
1. Re-Deriving the CAPM, Part I
Consider an economy with a risk-free asset with return rf and two risky assets, one with
random return r˜1 with expected value E(˜
r1 ) and standard deviation σ1 and the second
with random return r˜2 with expected value E(˜
r2 ) and standard deviation σ2 . Assume for
simplicity that the returns on the two risky assets are uncorrelated, so that a portfolio with
share w1 allocated to risky asset 1, share w2 allocated to risky asset 2, and the remaining
share 1 − w1 − w2 allocated to the risk-free asset has return with expected value
µP = (1 − w1 − w2 )rf + w1 E(˜
r1 ) + w2 E(˜
r2 )
and variance
σP2 = w12 σ12 + w22 σ22 .
Suppose that all investors in this economy have identical levels of wealth and identical
preferences over the mean and variance of the returns on their portfolios described by the
utility function
A
2
σP2 ,
U (µP , σP ) = µP −
2
where a higher value of A again corresponds to a higher level of economy-wide risk aversion.
By substituting the expressions for µP and σP2 into the utility function, the problem solved
by the “representative investor” can be described as one of choosing w1 and w2 to maximize
A
(w12 σ12 + w22 σ22 ).
(1 − w1 − w2 )rf + w1 E(˜
r1 ) + w2 E(˜
r2 ) −
2
What are the first-order conditions for the investor’s optimal choices w1∗ and w2∗ ?
2. Re-Deriving the CAPM, Part II
In the same economy described in question 1, above, suppose that there are equal shares of
asset 1 and asset 2 in total, so that the random return on the market portfolio is
1
1
r˜1 +
r˜2 .
r˜M =
2
2
Thus, the expected return on the market portfolio equals
1
1
E(˜
rM ) =
E(˜
r1 ) +
E(˜
r2 )
2
2
1
and, again using the fact that the two asset returns are uncorrelated, the variance of the
return on the market portfolio equals
2
2
1
1
1
2
2
2
σM =
(σ12 + σ22 ).
σ1 +
σ2 =
2
2
4
Although the two individual asset returns are uncorrelated with each other, both are correlated with market return, since
σ1M = E{[˜
r − E(˜
r1 )][˜
rM − E(˜
r )]}
1
M
1
1
= E [˜
r1 − E(˜
r1 )]
[˜
r1 − E(˜
r1 )] +
[˜
r2 − E(˜
r2 )]
2
2
1
1
2
=
E{[˜
r1 − E(˜
r1 )] } +
E{[˜
r1 − E(˜
r1 )][˜
r2 − E(˜
r2 )]}
2
2
1
σ12
=
2
measures the covariance between the return on asset 1 and the return on the market portfolio
and, similarly,
1
σ22
σ2M =
2
measures the covariance between the return on asset 2 and the return on the market portfolio.
These results imply that the betas on assets 1 and 2 are
β1 =
1 2
σ
σ1M
2 1
=
1
2
2
σM
(σ1 + σ22 )
4
β2 =
1 2
σ
σ2M
2 2
.
=
1
2
2
σM
(σ1 + σ22 )
4
and
Since, by assumption, all investors in this economy are identical, it must be true that in
equilibrium the first-order conditions for w1 and w2 that you derived for the individual
investor when answering question 1 must be satisfied with w1∗ = 1/2 and w2∗ = 1/2. That is,
investors as a group must hold the two risky assets in the same proportion in which those
assets are supplied.
Can you use those first-order conditions, with w1∗ = 1/2 and w2∗ = 1/2 imposed, together
with the expressions for E(˜
rM ), β1 , and β2 shown above to re-derive the CAPM relations
E(˜
rj ) − rf = βj [E(˜
rM ) − rf ]
for each risky asset j = 1 and j = 2?
2
3. CAPM Betas and Expected Returns
Here are some stocks with high and low betas:
High Beta Stocks
Company
Beta
Caterpillar
General Electric
General Motors
US Steel
1.25
1.15
1.20
1.60
Low Beta Stocks
Company
Beta
AT&T
Coca-Cola
General Mills
Pfizer
0.75
0.70
0.65
0.80
Suppose that the risk-free rate is rf = 0.02 (2 percent) and that the expected return on the
market portfolio is E(˜
rM ) = 0.08 (8 percent), then use the CAPM to calculate the expected
returns on each of these eight stocks.
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