Question 1.
Consider a portfolio that has equal amounts of $10 invested in two assets. Suppose returns on the two assets are jointly normally distributed. The annual expected returns and variance or return on the first asset are given by
m1 = 0.10 s1^2 = 0.04
And those on the second asset are given by
m2 = 0.05 s2^2 = 0.03
Consider three cases:
The correlation between the return is r = 0
The correlation between the return is r = +0.50
The correlation between the return is r = -0.50
For each case, identify the 99% Value-at-Risk of the portfolio. Explain the pattern of dependence of VaR on the correlation.
Question 2.
You are given a portfolio of three assets whose returns are jointly normally distributed with the following mean vector and covariance matrix:
0.20 0.08 0.02 0.02
0.10 0.02 0.06 0.03
0.15 0.02 0.03 0.07
a. Compute the 95% VaR for the portfolio if we invest $1 in the first asset. $2 in the second asset, and $3 in the third asset.
b. How much does each asset’s holding contribute to the overall VaR risk?
Question 3.
You are given a portfolio of two assets whose returns are jointly distributed with the following vector and covariance matrix.
0.20 0.08 0.04
0.10 0.04 0.06
a. Compute the 95% VaR of the portfolio if $1 is invested in the first asset and $1 is invested in the second.
b. Compute the risk-contribution of each asset to the VaR.
c. Is the current portfolio weighting optimal? If not, suggest a better one.
