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 assets 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.