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Unformatted text preview: FIN 367: Spring 2008
Test # 2: Portfolio Theory
March 5, 2008 SOLUTIONS
W NAME: SECTION: Time allowed: 1 hour, 15 minutes.
Maximum possible score: 100. NOTE: 0 If I cannot understand your calculations, I Will not be able to give you full credit even
if your ﬁnal answer is correct. You cannot say “these calculations are obvioUs.” o If a problem is missingsome key information that you think is necessary to solve
the problem, please ask me to clarify the question or state your confusions 'clearly.
Alternatively, please make appropriate assumptions, state them clearly and proceed.
No credit will be awarded if you fail to state your confusions or assumptions explicitly
even if the question is wrong. 0 Partial credits may be awarded if you Show your calculations or provide arguments
to support your answers. $10 I Certainty Equivalent (10 points, 10 minutes) Suppose the economy can be in one of the following two states: Boom or “good” state
and (ii) Recession or “bad” state. The good state occurs with a probability of 2/3 and the
bad state could occur with a probability of 1/3. An investor with an initial wealth of $70
is evaluating a gamble with the following net payoffs: $15 in the good state and ~$30
in the bad state. Assume that the utility function of an investor with wealth level W is = 2x/W. 1. ( 5 points) Compute the certainty equivalent (CE) and the insurance premium of the
gamble. Show your calculations clearly. 2. { 5 points) Draw the utility function of the investor who holds this gamble. Show the
following points on the utility graph very clearly: utility from the wealth in the
“bad” state, (ii) utility from the wealth in the “good” state, (iii) expected utility from
the uncertain wealth in the future, (iv) certainty equivalent, and (v) the insurance
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3 l% 'W II Asset Allocation and Risk Reduction (20 points, 15
minutes) Suppose the economy can be in one of the following three states: (i) Boom or “good” state,
(ii) Neutral state, and (iii) Recession or “bad” state. Each state can occur with an equal
probability. The following two risky securities are available at the beginning of a month: 0 Security 1: it is currently trading at $50 and at the end of the month, it is
expected to yield a $10 payoff in the good state, a $0 payoff in the neutral
state, and a —$10 payoff in the bad state. 0 Security 2: it is currently trading at $50 and at the end of the month, it is
expected to yield a $0 payoff in the good state, a $10 payoff in the neutral
state, and a $0 payoff in the bad state. In addition, you can purchase a call option on security 1 which expires at the end of the
month and has a strike price (or an exercise price) of $50. The option is currently trading
at $1. However, for simplicity, ignore the premium (i.e., the cost) you pay to purchase an
option. You are also allowed to short the two risky securities. 1. (3 points) Draw the payoff tree for the call option. Ignore the cost of the option. Please
label the tree clearly. 2. (5’ points) Draw the payoff tree if you short sell security 1 at the beginning of the
month and cover it (i.e., buy it back) at the end of the month. Please label the tree clearly.
3. (2 points) Will you ever short sell security 2? Please explain very brieﬂy (2—3 lines). 4. {12 points) Using one or more of the available securities, construct a portfolio that
will be riskless at the end of the month. Indicate clearly which assets you will hold in
the portfolio and the number of units of each asset in the portfolio. Please show your
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Rfsk‘ess III Basic Statistics (20 points, 15 minutes) Suppose the economy can be in one of the following two states: Boom or “good” state,
and (ii) Recession or_“bad” state. Each state can occur with an equal probability. At the
beginning of a month, you can purchase the following two risky securities in the market: 0 Security 1: It is currently trading at $50 and at the end of the month, it is
expected to yield a net $20 payoff in the good state and a —$10 net payoff
in the bad state. 0 Security 2: It is also currently trading at $50 but it’s payoff is contingent
upon the payoff of security 1. When security 1 yields a net payoff of $20,
security 2 yields a payoff of net —$10 with probability 9 / 10 and a net payoff
of $20 with probability 1/10. When security 1 yields a net payoff of —$10,
security 2 yields a net payoff of $20 with probability 9/10 and a net payoff
of ~$10 with probability 1 / 10. Compute the following:
1. (4 points) Compute the expected return (not expected payoff) of securities 1 and 2.
2. (4 points ) Compute the standard deviation of returns of securities 1 and 2.
3. (4 points) Compute the Sharpe Ratio of securities 1 and 2.
4. ( 8 points ) Compute the correlation between the returns of securities 1 and 2. Assume that the annual riskfree rate is 5%. Show all your calculations clearly. SeCuYH'j 1
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8 5 IV Capital Allocation with Different Borrowing and
Lending Rates (20 points, 15 minutes)  Consider the two securities listed below. Risky Security: a = E(R) = 15%, a = 20%.
Riskfree Security: Rf = 5%. You wish to form a portfolio combining the risky security and the riskfree security such that
you earn an expected return of 25%. 1. (2.5 points) What weights would you need to place in the risky and riskfree securities
to earn a 25% expected return? 2. (2.5 points) What is the standard deviation of this portfolio? 3. (2.5 points) Draw the capital allocation line (CAL). Label the points and the axes
clearly. 4. (2.5 points) What is the Sharpe Ratio of your portfolio? Now suppose that you cannot borrow at the riskfree rate. Instead, your broker charges
8% on borrowed funds. 5. (2.5 points) What weights would you need to place in the risky and riskfree securities
to earn a 25% expected return? 6. (2.5 points ) What is the standard deviation of this new portfolio?
7. (2.5 points) What is the Sharpe Ratio of this new portfolio? 8. (2.5 points ) Draw the new CAL alongwith the old CAL. Label the points and the axes
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minutes) Consider a simple economy with two risky securities and a riskfree security. The two risky
securities have the following characteristics: 0 Risky Security 1: Expected (or mean) annual return = M1 = 10%, Standard deviation
= 01 = o Risky Security 2: Expected (or mean) annual return 2 #2 = 10%, Standard deviation
= 0'2 2 The correlation between the two risky securities is p12 2 0.50. Assume that the annual
riskfree rate is 5%. 1. (10 points ) Draw the minimum variance frontier. Identify the minimum variance
portfolio (MVP) and (ii) the tangential portfolio (i.e., the optimal risky portfolio or the market portfolio You do not need to identify the locations of MVP and M
exactly. Just show their approximate locations. 2. { 5 points ) How will the shape of the minimum variance frontier change if the correlation
between the two risky securities p12 is —0.50 instead? Please show the new frontier
and identify the new locations of the MVP and M. Again, you do not have to identify
the locations of MVP and M exactly. Just show their approximate locations. 3. ( 5 points) Draw the mean variance frontier for two extreme cases: p12 = 1 and (ii)
P12 = *1. 4. (10 points) Consider the four cases now (p12 = —1, —0.5,0.5, 1). In which of these
cases (if any) do arbitrage opportunities exist? How will you take advantage of the
opportunity? Please explain brieﬂy. If you think arbitrage opportunities do not exist, please justify your answer. TRe like 30""in «Sn—ls :1. 4“ 2.
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