calculating the value at risk (VaR) and expected shortfall for different investment scenarios.

Section I: Practice Problems (Show your work including hand-drawn CDF)
1. Suppose that each of the two investments has a 0.9% chance of a loss of $10 million, a
99.1% of a loss of $1 million, and 0% chance of a profit. The investments are
independent of each other.
a. What is the VaR for one of the investments when the confidence level is 99%?
b. What is the expected shortfall for one of the investments when the confidence
level is 99%?
c. What is the VaR for a portfolio consisting of the two investments when the
confidence level is 99%?
d. What is the expected shortfall for a portfolio consisting of the two investments
when the confidence level is 99%?
e. Show that in this example VaR does not satisfy the subadditivity condition
whereas expected shortfall does.
2. Suppose that the change in the value of a portfolio over a one-day time period is normal
and iid with a mean of zero and a standard deviation of $2 million. Compute the
following (In this case, you can still use the formula, $VaR=Z*Standard Deviation):
a. One-day $VaR with 97.5% confidence level
b. Five-day $VaR with 97.5% confidence level
c. Five-day $VaR with 99% confidence level
3. What difference does it make to your answer to Problem 2 if there is first-order daily
positive autocorrelation (persistence) with correlation parameter equal to 0.16? Compute
the following again with correlation parameter equal to 0.16:
a. One-day $VaR with 97.5% confidence level
b. Five-day $VaR with 97.5% confidence level
c. Five-day $VaR with 99% confidence level