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Unformatted text preview: FIN 397.1: SPRING 2007 Exam # 2: Portfolio Theory and Asset Pricing
March 22, 2007 (SOLUTIONS) NAME:
Time allowed: 1 hour, 15 minutes.
Minimum possible score: 100. NOTE: 0 HI 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 missing some key information that you think is necessary to solve
the probiem, please ask me to clarify the question or state your confusions clearly.
Aiternatively, 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. I Multiple Choice Questions (20 points, 15 minutes) 1. (2 points) Portfolio theory is most concerned with: (a) The elimination of systematic risk. .The effect of diversiﬁcation on portfolio risk. (c) The identiﬁcation of idiosyncratic (or unsystematic) risk. (d) Active portfolio management to enhance return. 2. (2 points) Which statement about portfolio diversiﬁcation is correct? (a) Proper diversiﬁcation can reduce or eliminate systematic risk. (‘0) Diversiﬁcation reduces the portfolio’s expected return because it reduces a port—
folio’s total risk. @As more securities are added to a portfolio, total risk typically would be expected
to fall at a decreasing rate. (d) The risk—reducing beneﬁts of diversiﬁcation do not occur meaningfully until at
least 30 individual securities are incinded in the portfolio. 3. (2 points) The term eﬁicient frontier refers to the set of portfolios that (a) yield the greatest return for a given level of risk.
(b) )involve the least risk for a given level of return. @both a and b above. (d) )None of the above answers are correct. 4. (2 points) The arbitrage pricing theory (APT) differs from the singlefactor capital
asset pricing model (CAPM) because the APT:
(a) Places more emphasis on market risk.
(b) Minimizes the importance of diversiﬁcation.
(c) Recognizes multiple unsystematic risk factors. .Recognizes multiple systematic risk factors. 5. (2 points) In contrast to the capitai asset pricing model (CAPM), arbitrage pricing
theory (APT):
(a) Requires that markets be in equilibrium.
(‘0) Uses risk premiums based on micro—economic variables. (0) Speciﬁes the number and identiﬁes speciﬁc factors that determine expected re
turns. .Does not require the restrictive assumptions concerning the market portfolio. 6. (3 points) Assume that we live in a world where the Capital Asset Pricing Model
(CAPM) holds. The expected return on the market is 15% and the expected return
on a stock with a beta of 1.2 is 18%. What is the risk—free rate? (a)2% l9:h_F+t.2(15’x§) 99%:0. (b) 4%
(c) 6%
(d) 8% r\
.none of the above 7. (3 points) Consider a simple economy with two risky securities with the following
characteristics:
0 Risky Security 1: Expected (or mean) return = 10%, Standard deviation = 15%.
o Risky Security 2: Expected (or mean) return = 15%, Standard deviation = 20%.
The correlation between the two risky securities is —O.50. What happens to the position
of the tangential portfolio when the riskfree interest rate rises?
(a) Its expected return decreases and its standard deviation increases. (b) Its expected return decreases and its standard deviation decreases. F (c) Its expected return increases and its standard deviation decreases. Its expected return increases and its standard deviation increases.
new A 01.9 8. (4 points) Consider an investor with a risk aversion level of A = 2.5 and a utility
function of the form: U = E(Rp) ~ §Ao§ E(Rp) is the expected return and Up is the
standard deviation of the investor’s portfolio. Both variables are in decimal units. This investor is choosing between a riskfree CD paying Rf = 3.85% and a risky stock
with E(R) = 9.9% and or 2: 22%. Which of these two securities should this investor
hold? (a) The investor should choose the riskfree CD. (b) The investor should choose the risky security.
@The investor should be indifferent between these two investments. (d) The answer cannot be determined from the information given.
UHIH‘J ‘FWM .‘mresi{Ka .14 CD : 55024;) —— o : 0.0385
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:_ oo‘ﬁ — Jix2$x (021) : 00 58'; ='> Ihdi'Fgfewf { II Certainty Equivalent (10 points, 5 minutes) Suppose the economy can be in one of the following two states: (i) Boom or “good” state,
and (ii) Recession or “bad” state. Each state can occur with an equal probability. An
investor with an initial wealth of $50 is evaluating a gamble with the following Bet payoffs:
$50 in the good state and —$50 in the bad state. Assume that the utility function of an
investor with wealth level W is u(W) 2 WV. 1. (8 points) Compute the certainty equivalent of the gamble. Show your calculations
clearly. 2. (2 points) What is the maximum premium the investor would be willing to pay to
avoid taking this gamble? v; $Ioo
N0: $50 35ij = Jixloo + lixo : $550 1/2.
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This makes Sense bcaqse +k¢ mvrslo‘r («31% 4. lIVlCﬁY‘ I411 43 Fuhcl.‘0n is "n‘sk neu'thQ‘
NW III 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 = 22%.
Riskfree Security: Rf = 7%. You wish to form a portfolio combining the risky security and the riskfree security such that
you earn an expected return of 20%. 1. (2.5 points) What weights would you need to place in the risky and riskfree securities
to earn a 20% 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
9% on borrowed funds. 5. (2.5 points) What weights would you need to place in the risky and riskfree securities
to earn a 20% 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
clearly. Lei“ M : weight in +16 ‘h‘skj secuﬁia. => (100) : H u " 'm‘skfmc “
lbw + 70“”) = 20 g
__ [3 90 := __....
% 3L0 ::: l5 ' w " ’9'“ U D 8 Additional Space H i5!» + claws) : 20 _. ‘3 ‘? 600: ll w: ___ (1mg) z .— a, 96
The wa‘akf on +ke m‘skj assét’ “gases.
5? = we“ : _Ll_x22 : 40.337. é
SR? 2 32:3 : 0273. As expedcd, ~H~¢ SR
40 ‘ 33 deCTt’ASes. IV MVP and the Tangential Portfolio (10 points, 5 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) return = [14, Standard deviation = 01.
o Risky Security 2: Expected (or mean) return = r52, Standard deviation = 02.
The correlation between the two risky securities is p. 1. ( 5 points) Are there condition(s) under which the minimum variance portfolio would be
the tangential portfolio (i.e., the optimal risky portfolio)? Please explain very brieﬂy
(3—5 lines). You are encouraged to use a graph to support your argument. 2. { 5 points) Are there condition(s) under which the minimum variance portfolio could
not be the tangential portfolio (1.3., the optimal risky portfolio)? Please explain very
brieﬂy (35 lines). You are encouraged to use a graph to support your argument. (0 when +ke (.o‘ﬂdﬁ'lsbn is +1 or _'L. (1) All o+ker 015.25. 1 < 6 4 +1. V CAPM (20 points, 15 minutes) Suppose the economy can be in one of the following two states: state 1 and state 2. Each
state can occur with an equal probability. The annual return on the market and a certain .
security X in the two states are as follows: i a Market: at the end of the year, the market is expected to yield a return of i
30% in state 1 and a return of —$10% in state 2. ' 0 Security X: at the end of the year, the security is expected to yield a return
of —60% in state 1 and a return of $20% in state 2. Furthermore, assume that the annual riskfree rate is 5%. 1. ( 8 points) Calculate the beta of security X relative to the market. 2. {4 points) Draw the security market line (SML). Please label the axes and all points
(including the market portfolio, the riskfree security, and security X) in the graph
clearly. 3. (4 points) Calculate the alpha (i.e., the Jensen’s alpha) of security X. If possible,
identify J ensen’s alpha in the SML graph. 4. (4 points) Does an arbitrage opportunity exist? Please explain brieﬂy. . . , J. _ = ’.
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W An afbi‘hnje oFFo'rhnH'j en‘s’rs because dx #:0 11 VI Arbitrage (20 points, 20 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 A: it is currently trading at $3 and at the end of the month, it is
expected to yield a $4.50 net payoff in the good state, a $0 net payoff in the
neutral state, and a $0 net payoff in the bad state. a Security B: it is currently trading at $5 and at the end of the month, it is
expected to yield a $15 net payoff in the good state, a $6 net payoff in the
neutral state, and a $0 net payoff in the bad state. In addition, you can purchase a put option (Security 0) on security B which expires at the
end of the month and has a strike price (or an exercise price) of $11. The option is currently
trading at $2. You are also allowed to short the two risky securities. 1. ( 5 points) Which of the three securities (A, B, and C) are fairly priced? Which securities
are underpriced? Which securities are overpriced? 2. (9 points) Construct a purely riskless arbitrage portfolio using these securities. Mention
clearly which securities (and their quantities) you would long and/or short. 3. {3 points ) Calculate the net payoffs of the riskless arbitrage portfolio in the three states. 4. (3 points) How should the prices of securities A, B, and C change so that the existing
arbitrage opportunity disappears? Please explain briefly (23 lines) and give one ex
ample of prices for securities A, B, and C that will eliminate the existing arbitrage
opportunity. Hint: A purely riskless portfolio would have the same payoff in each state. I _
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