A) An investment has a 94% chance of making a profit of $3 million, a 2% chance of

losing $2 million, and a 4% chance of losing $10 million. What is the VaR and

expected shortfall for this investment when confidence level is 95%, 99%?

B) Suppose there is another independent investment with the following profile: a

93% change of making $7 million, and a 7% chance of losing $13 million. What

are the VaR and expected shortfall for this investment when the confidence level

is 95%? What are the VaR and expected shortfall for a portfolio consisting of this

investment and the one in A) when the confidence level is 95%? Is the

subadditivity condition satisfied for VaR and the expected shortfall? (Hint: in a

spreadsheet by first calculating all the joint probabilities and gains/losses and then

sort from gain to loss.)

C) There are two independent projects with cash flows uniformly distributed:

[-$4 million, $22 million] and [-$12 million, $55 million]. Please find the 95%

VaR and expected shortfall for each project and the two projects combined. Is the

subadditivity condition satisfied for VaR and the expected shortfall? (Hint: please

use the density function appended to this assignment and think about the precise

definition of VaR and expected shortfall.