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Solutions to Final Exam
ECON 337901 - Financial Economics
Boston College, Department of Economics
Peter Ireland
Spring 2015
Thursday, April 30, 10:30 - 11:45am
1. Risk Aversion and Portfolio Allocation
An investor has initial wealth Y0 = 100 and allocates the amount a to stocks, which provide
return rG = 0.35 in a good state that occurs with probability 1/2 and return rB = −0.15 in
a bad state that occurs with probability 1/2. The investor allocates the remaining Y0 − a to
a risk-free bond which provides the return rf = 0.05 in both states. The investor has von
Neumann-Morgenstern expected utility, with Bernoulli utility function of the logarithmic
form
u(Y ) = ln(Y ).
a. In general, the investor’s portfolio allocation problem can be stated mathematically as
max E{u[(1 + rf )Y0 + a(˜
r − rf )]},
a
where r˜ is the random return on stocks, but under the particular assumptions about the
Bernoulli utility function and stock returns made above, the problem can be written
more specifically as
max(1/2) ln(105 + 0.30a) + (1/2) ln(105 − 0.20a).
a
b. The first-order condition for the investor’s optimal choice a∗ is
(1/2)(0.20)
(1/2)(0.30)
−
= 0.
105 + 0.30a∗ 105 − 0.20a∗
This first-order condition leads to the numerical solution for a∗ as
(1/2)(0.30)
(1/2)(0.20)
=
105 + 0.30a∗
105 − 0.20a∗
(1/2)(0.30)(105 − 0.20a∗ ) = (1/2)(0.20)(105 + 0.30a∗ )
(0.30)(105) − (0.30)(0.20)a∗ = (0.20)(105) + (0.20) ∗ (0.30)a∗
(0.10)(105) = 2(0.30)(0.20)a∗
a∗ =
(0.10)(105)
= 87.5.
2(0.30)(0.20)
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c. A second investor also has vN-M expected utility with Bernoulli utility function of the
logarithmic form u(Y ) = ln(Y ), but has initial wealth Y0 = 1000 that is ten times as
large as the investor considered in parts (a) and (b) above. Although it is possible
to re-solve the entire problem after replacing the first investor’s Y0 = 100 with the
second investor’s Y0 = 1000, we know from class that because the logarthimic utility
function implies that the coefficient of relative risk aversion is constant, both of these
investors will allocate the same fraction of their wealth to stocks. Since this fraction
equals 0.875 for the first investor, it will equal 0.875 for the second investor as well.
With Y0 = 1000, this second investor will therefore choose a∗ = 875.
2. Portfolio Allocation and the Gains from Diversification
Asset 1 has expected return equal to µ1 = 10 and standard deviation of its return equal to
σ1 = 4; asset 2 has expected return equal to µ2 = 7 and standard deviation of its return
equal to σ2 = 2. The fraction of wealth in the portfolio allocated to asset 1 is w1 and the
fraction of wealth allocated to asset 2 is w2 .
a. Assuming that the correlation between the two returns is ρ12 = 1, the portfolio that sets
w1 = 1/2 and w2 = 1/2 has expected return equal to
µp = w1 µ1 + w2 µ2 = (1/2)10 + (1/2)7 = 5 + 3.5 = 8.5
and standard deviation of its return equal to
σp = [w12 σ12 + w22 σ22 + 2w1 w2 σ1 σ2 ρ12 ]1/2
= [(1/2)2 16 + (1/2)2 4 + 2(1/2)(1/2)(4)(2)(1)]1/2
√
= (4 + 1 + 4)1/2 = 9 = 3.
b. If instead the correlation between the two returns is ρ12 = 0, the same portfolio will have
still have expected return µp = 8.5, but the standard deviation of its return will equal
σp = [w12 σ12 + w22 σ22 + 2w1 w2 σ1 σ2 ρ12 ]1/2
= [(1/2)2 16 + (1/2)2 4 + 2(1/2)(1/2)(4)(2)(0)]1/2
√
= (4 + 1)1/2 = 5 = 2.24.
c. A risk-free asset has return rf = 5. Still assuming, as in part (b), that the correlation
between the two random returns is ρ12 = 0, the expected return on the portfolio that
allocates the fraction w1 = 1/4 of wealth to asset 1, w2 = 1/4 of wealth to asset 2, and
the remaining fraction wr = 1/2 to the risk-free asset is
µp = w1 µ1 + w2 µ2 + wr rf = (1/4)10 + (1/4)7 + (1/2)5 = 2.5 + 1.75 + 2.5 = 6.75.
There are a number of different ways of calculating the standard deviation of the return
on this portfolio of assets, but perhaps the easiest is to note that since the two risky
asset returns are assumed to be uncorrelated, and since the correlations between the
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risk-free return and each of the risky returns are zero, all of the “cross-products” in
the formula for the portfolio’s standard deviation equal zero, so that
√
σp = (w12 σ12 + w22 σ22 )1/2 = [(1/4)2 16 + (1/4)2 4]1/2 = (1 + 1/4)1/2 = 5/2 = 1.12.
3. Modern Portfolio Theory
The graph below traces out the minimum variance frontier from Modern Portfolio Theory.
a. No investor with mean-variance utility would ever choose to hold portfolio C, since
portfolio A offers a higher expected return with the same standard deviation.
b. Investor 2, holding portfolio B, is more risk averse than investor 1, holding portfolio
A, since investor 2 is accepting lower expected return in order to reduce the standard
deviation of his or her portfolio’s random return.
c. No, portfolio A does not exhibit mean-variance dominance over portfolio B since, while
it does have a higher expected return, the standard deviation of its return is higher as
well.
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4. The Capital Asset Pricing Model
The random return r˜M on the market portfolio has expected value E(˜
rM ) = 0.07 and the
return on risk-free assets is rf = 0.02.
a. According to the capital asset pricing model, the expected return on an asset with
random return that has βj = 1 is
E(˜
rj ) = rf + βj [E(˜
rM ) − rf ] = 0.02 + 1(0.07 − 0.02) = 0.07.
b. The expected return on an asset with random return that has βj = 0 is
E(˜
rj ) = rf + βj [E(˜
rM ) − rf ] = 0.02 + 0(0.07 − 0.02) = 0.02.
c. The expected return on an asset with random return that has βj = −0.20 is
E(˜
rj ) = rf + βj [E(˜
rM ) − rf ] = 0.02 − (0.20)(0.07 − 0.02) = 0.01.
5. The Market Model and Arbitrage Pricing Theory
This version of the arbitrage pricing theory is built on the assumption that the random
return r˜i on each individual asset i is determined by the market model
r˜i = E(˜
ri ) + βi [˜
rM − E(˜
rM )] + εi .
a. The APT implies that a well-diversified portfolio with beta βw will have random return
r˜w1 = E(˜
rw1 ) + βw [˜
rM − E(˜
rM )].
b. The APT also implies that the well-diversified portfolio with beta βw will have expected
return
E(˜
rw1 ) = rf + βw [E(˜
rM ) − rf ].
c. If you find another well-diversified portfolio with the same beta βw that has an expected
return E(˜
rw2 ) that is lower than the expected return given in the answer to part (b),
you can take a long position worth x in the portfolio described in parts (a) and (b)
and a short position worth −x in this new portfolio. This strategy is self-financing,
and since both portfolios are well-diversified and have the same betas, the strategy is
free of risk as well. It is therefore profitable for sure, and the larger the value of x, the
larger the profit you will make.
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