1. There is a risk-free asset with the rate of return 0.1. The market has only one risky

asset A, whose expected rate of return is 0.2 and standard deviation is 1. A risk-averse

investor constructs a portfolio with h fraction of her funds on the risk-free asset. She

chooses to short-sell the risky asset and buy more of the risk-free asset; that is, she

sets h > 1. Is this investment strategy optimal? Why? [Hint: if a random variable X

has variance V ar(X), then for any number a, V ar(aX) = a2V ar(X), no matter a is

positive or negative.]

2. There are three risky securities (A, B, and C) in the market. ?AB= 0.

E(r)

?

A

0.2

1

B

0.3

2

C

0.21

3

(a) If a risk-averse investor prefers security C to security A. Will the investor prefer

security B to security A?

(b) Are there risk-averse investors who prefer A to B? [Hint: draw a graph to answer

this question.]

(c) P = h ? A + (1 ? h) ? B is a portfolio. If an investor prefers A to B. Then does

the investor prefer P to C for all h ? (0,1)?

3. Consider a capital market with only two risky assets A and B. Their standard devia-

tions are 1 and 2, respectively. There is no risk-free asset.

(a) When the correlation coefficient ?AB= 0, construct a portfolio, whose variance

is strictly less than 1. [Hint: you may want to try the portfolio that puts more

weights on the security with the lower standard deviation.]

(b) Show that when the correlation coefficient ?AB = ?1, there exists a portfolio

consisting of A and B, whose standard deviation is 0. In particular, specify how

the portfolio is constructed.

[Hint: to solve this question, you may need the following formula to calculate the

variance of a portfolio P = hA? A + hB? B:

?2

P= h2

A?2

A+ 2hAhB?AB?A?B+ h2

B?2

B.]

MGMT 141 Homework 2 October 9, 2013 Chong Huang 1. There is a risk-free asset with the rate of return 0.1. The market has only one risky asset A, whose expected rate of return is 0.2 and standard deviation is 1. A risk-averse investor constructs a portfolio with h fraction of her funds on the risk-free asset. She chooses to short-sell the risky asset and buy more of the risk-free asset; that is, she sets h > 1. Is this investment strategy optimal? Why? [Hint: if a random variable X has variance V ar(X), then for any number a, V ar(aX) = a2 V ar(X), no matter a is positive or negative.] 2. There are three risky securities (A, B, and C) in the market. AB = 0. E(r) A 0.2 1 B 0.3 2 C 0.21 3 (a) If a risk-averse investor prefers security C to security A. Will the investor prefer security B to security A? (b) Are there risk-averse investors who prefer A to B? [Hint: draw a graph to answer this question.] (c) P = h A + (1 h) B is a portfolio. If an investor prefers A to B. Then does the investor prefer P to C for all h (0, 1)? 3. Consider a capital market with only two risky assets A and B. Their standard deviations are 1 and 2, respectively. There is no risk-free asset. (a) When the correlation coecient AB = 0, construct a portfolio, whose variance is strictly less than 1. [Hint: you may want to try the portfolio that puts more weights on the security with the lower standard deviation.] (b) Show that when the correlation coecient AB = 1, there exists a portfolio consisting of A and B, whose standard deviation is 0. In particular, specify how the portfolio is constructed. [Hint: to solve this question, you may need the following formula to calculate the variance of a portfolio P = hA A + hB B: 2 2 2 P = h2 A + 2hA hB AB A B + h2 B .] A B 1