1. finance
2. investing
3. portfolio

Final Exam
ECON 337901 - Financial Economics
Boston College, Department of Economics
Peter Ireland
Spring 2015
Thursday, April 30, 10:30 - 11:45am
This exam has five questions on four pages; before you begin, please check to make sure
that your copy has all five questions and all four pages. The five questions will be weighted
equally in determining your overall exam score.
Please circle your final answer to each part of each question after you write it down, so that
I can find it more easily. If you show the steps that led you to your results, however, I can
award partial credit for the correct approach even if your final answers are slightly off.
1. Risk Aversion and Portfolio Allocation
Consider the portfolio allocation problem faced by an investor who has initial wealth Y0 =
100. The investor allocates the amount a to stocks, which provide return rG = 0.35 (a 35
percent gain) in a good state that occurs with probability 1/2 and return rB = −0.15 (a
15 percent loss) 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 (a five percent gain)
in both states. The investor has von Neumann-Morgenstern expected utility, with Bernoulli
utility function of the logarithmic form
u(Y ) = ln(Y ).
a. Write down a mathematical statement of this portfolio allocation problem.
b. Write down the numerical value of the investor’s optimal choice a∗ .
c. Suppose that 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. Still assuming
that stocks 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 and that the
risk-free bond provides the return rf = 0.05 in both states, what is the numerical value
of a∗ measuring the amount that this second investor optimally allocates to stocks?
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2. Portfolio Allocation and the Gains from Diversification
Consider portfolios formed from two risky assets, the first with expected return equal to
µ1 = 10 and standard deviation of its return equal to σ1 = 4 and the second with expected
return equal to µ2 = 7 and standard deviation of its return equal to σ2 = 2. Let w1 denote
the fraction of wealth in the portfolio allocated to asset 1 and w2 the fraction of wealth
allocated to asset 2.
a. Calculate the expected return and the standard deviation of the return on the portfolio
that sets w1 = 1/2 and w2 = 1/2, assuming that the correlation between the two
returns is ρ12 = 1.
b. Calculate the expected return and the standard deviation of the return on the same
portfolio that sets w1 = 1/2 and w2 = 1/2, assuming instead that the correlation
between the two returns is ρ12 = 0.
c. Finally, suppose that in addition to the two risky assets described above, there is also a
risk-free asset with return rf = 5. Still assuming, as in part (b), that the correlation
between the two random returns is ρ12 = 0, calculate the expected return and the
standard deviation of the 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.
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3. Modern Portfolio Theory
The graph below traces out the minimum variance frontier from Modern Portfolio Theory.
Each of the three portfolios shown, A, B, and C, lies on the minimum variance frontier: each
one provides the minimized variance σP2 for a given mean or expected return µP . Portfolio A
has a higher expected return than portfolio C, however, even though both returns have the
same standard deviation. Thus, portfolios A and B lie on the efficient frontier, but portfolio
C does not.
In answering each part of this question, assume, as Harry Markowitz did when he invented
Modern Portfolio Theory, that all investors have mean-variance preferences, that is, utility
functions that are increasing in their portfolio’s expected return and decreasing the variance
or standard deviation of their portfolio’s random return. Assume, as well, that there is no
risk-free asset, so that all investors much choose portfolios on or inside the minimum variance
frontier.
a. Would any investor ever choose to hold portfolio C? Here, you can just say “yes” or
“no;” you don’t have to explain why.
b. Suppose you observed one investor – call him or her “investor 1” – holding portfolio
A and another investor – call him or her “investor 2” – holding portfolio B. Which
investor is more risk averse: investor 1 or investor 2? Again, you can just say “investor
1” or “investor 2;” you don’t have to explain why.
c. Does portfolio A exhibit mean-variance dominance over portfolio B? Here, once more,
you can just say “yes” or “no;” you don’t have to explain why.
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4. The Capital Asset Pricing Model
Suppose that 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, what is the expected return on an asset
with random return that has a “beta” equal to βj = 1?
b. According to the capital asset pricing model, what is the expected return on an asset
with random return that has a “beta” equal to βj = 0?
c. According to the capital asset pricing model, what is the expected return on an asset
with random return that has a “beta” equal to βj = −0.20?
5. The Market Model and Arbitrage Pricing Theory
Consider a version of the arbitrage pricing theory that 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
where, as we discussed in class, E(˜
ri ) is the expected return on asset i, r˜M is the return
on the market portfolio and E(˜
rM ) is the expected return on the market portfolio, βi is the
same beta for asset i as in the capital asset pricing model, and εi is an idiosyncratic, firmspecific component. Assume, as Stephen Ross did when developing the APT, that there are
enough assets for investors to form many well-diversified portfolios and that investors act to
eliminate all arbitrage opportunities that may arise across all well-diversified portfolios.
a. Write down the equation, implied by the APT, for the random return r˜w1 on a welldiversified portfolio with beta βw .
b. Write down the equation, implied by the APT, for the expected return E(˜
rw1 ) on this
well-diversified portfolio with beta βw .
c. Suppose that 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 E(˜
rw1 ) given in your
answer to part (b), above. Describe briefly (a sentence or two is all that it should take)
the trading opportunity provided by this discrepancy that is free of risk, self-financing,
but profitable for sure.
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