Suppose that each of two investments has a 4% chance of a loss of $10 million, a 2% chance of a loss of $1 million, and a 94% chance of a profit of $1 million. They are independent of each other.

a. What is the VaR for one of the investments when the confidence level is 95%?

b. What is the expected shortfall when the confidence level is 95%?

c. What is the VaR for a portfolio consisting of the two investments when the confidence level is 95%? (When combining the two investments, note that each investment has 3 possible outcomes.)

d. What is the expected shortfall for a portfolio consisting of the two investments when the confidence level is 95%?

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