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%?

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

Value at Risk 203

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

AppendixLO1