An investment has a 92% chance of making a profit of $10 million, a 1.5% chance of losing $4 million, a 3% of losing $6 million, and a 3.5% chance of losing $16 million.
A) What is the VaR for this investment when confidence level is 90%, 95%, or 99%?
B) What is the expected shortfall when the confidence level is 95%?
C) Suppose there is another independent investment with the following profile: a 94% change of making $3 million and a 6% change of losing $5 million. 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 the portfolio in C) when the confidence level is 95%?
E) Does VaR in C) satisfy the subadditivity condition? How about the expected shortfall in D)?
Hint: for A), please realize that we have only three possible values; go from the gain to the loss until the total probability is bigger than 95%. For C), enumerate all the possible combinations of losses and gains, and then identify the 95 percentile.