1. If the daily, 95% confidence level value at risk (VaR) of a portfolio is correctly estimated to be $10,000, one would expect that: a. In 1 out of 20 days, the portfolio value will decline by $10,000 or less. b. In 1 out of 95 days, the portfolio value will decline by $10,000 or less. c. In 1 out of 95 days, the portfolio value will decline by $10,000 or more. d. In 1 out of 20 days, the portfolio value will decline by $10,000 or more.

2. How many exceptions in backtesting a 90% VaR would be expected over a 250-day trading year?

a. 10

b. 15

c. 25

d. 50

3. Suppose that you have two investments, each of which has a 0.9% chance of a loss of $10 million and a 99.1% chance of a loss of $1 million. These two investments are independent of each other.

a. What is the VaR for one of the investments at the 99% confidence level?

b. What is the expected shortfall for one of the investments at the 99% confidence level?

c. What is the 99% VaR for a portfolio consisting of the two investments?

d. What is the 99% expected shortfall for a portfolio consisting of the two investments?

e. Verify that VaR does not satisfy the subadditivity condition in this example whereas expected shortfall does.

4. Suppose that the daily change in the value of a portfolio is normal with a mean of zero and a standard deviation of $2 million. Calculate the following VaRs: (1) one-day 97.5% VaR, (2) five-day 97.5% VaR, and (3) five-day 99% VaR. (Hint: The left-tailed z-value is 1.96 for the 97.5% confidence level.)