Question 1 -
Consider the following data for securities A B, and C: R-A = 20%; R-B = 10%; R-C = 8%; σA =4%; σB = 2%; σC = 2%; ρAB = 0.4; ρAC = 0.2; ρBC = -1.0.
a. What is the expected return and standard deviation of a portfolio constructed by placing 60% of your money in A and 40% in B?
b. What is the expected return on the portfolio constructed from among the above three securities that has the smallest possible risk?
c. If an investor had to place 100% of his or her money in only one of the above three securities,
1. Which security would a risk-neutral investor pick?
2. What can you say about the preference ordering of the three securities for a risk-averse investor?
Question 2 -
You are evaluating two risky investments, A and B, which have the following distributions:
|
Probability
|
Return on A
|
Return on B
|
|
0.6
|
20%
|
30%
|
|
0.4
|
10%
|
10%
|
a. What are the expected returns and standard deviations for A and B?
b. Suppose that an investor must pick either A or B to hold in some combination with the riskless asset (RF = 8%). Which risky asset should the investor choose?
Question 3 -
Assume that the returns of assets A and B depend on the state of the market and that the return of asset C depends on the state of the weather. Assume that the state of the market is independent of the state of the weather. Given the following
|
State of Market
|
Probability
|
RA
|
RB
|
State of weather
|
Probability
|
RC
|
|
Good
|
0.2
|
18%
|
5%
|
Good
|
0.3
|
12%
|
|
Average
|
0.6
|
12%
|
10%
|
Average
|
0.6
|
15%
|
|
Poor
|
0.2
|
6%
|
5%
|
Poor
|
0.1
|
18%
|
a. What is the mean return of asset C?
b. What is the standard deviation of asset C?
c. What is the covariance between each pair of assets?
d. Which of the following two portfolios dominates the other portfolio 1 consisting of ½ invested in A and ½ invested in B, or portfolio 2 consisting of ½ invested in B and ½ invested in C?
Question 4 -
Consider the following data for securities A, B, and C: R-A = 20%; R-B = 10%; R-C = 8%; σA =4%; σB = 2%; σC = 2%; ρAB = 0.4; ρAC = 0.2; ρBC = -1.0.
a. What is the expected return and standard deviation of a portfolio constructed by placing 60% of your money in A and 40% in B?
b. What is the expected return on the portfolio constructed from among the above three securities that has the smallest possible risk?
C. If an investor had to place 100% of his or her money in only one of the above three securities,
1. Which security would a risk-neutral investor pick?
2. What can you say about the preference ordering of the three set-unties for a risk-averse investor?
Question 5 -
Consider a firm with a portfolio of traded assets worth $90 million which has been optimized to minimize the portfolio's risk, while maximizing its overall expected return. The firm's investment portfolio has two assets with normally distributed returns. The firm holds 550 Million in the first asset which has an expected annual return of25% and annual volatility of 35%. The firm has invested $40 Million in the second asset which has an expected annual return of 35% and annual volatility of 25%. The correlation coefficient between the returns of both assets is 20%.
a. Compute the 5% annual VaR for the firm's investment portfolio?
b. Consider a trade by the firm where it sells $20 Million of the second asset and buys a similar amount of the first asset. By how much would your VaR that was calculated in part (a) change?
c. Discuss (i) some of the shortcomings of VaR, (ii) Historical and Analytical VaR.
Question 6 -
Use the historical method and the following information for the last 120 days of returns to calculate an approximate VaR for a portfolio of $20 million using a probability of 0.05;
|
Less than -0%
|
5
|
|
-10% to -5%
|
18
|
|
-5% to 0%
|
42
|
|
0% to 5%
|
36
|
|
5% to 10%
|
15
|
|
Greater than 10%
|
4
|