Question

Exercise 4 (15 points) Suppose that each of two investments has a 1.5% chance of a loss of $5 million, a 4.5 % chance of a loss of 2 million, and a 94% chance ofa profit of $2 million. They are independent of each- other. a. What is the one-day VaR for one of the investments when the confidence level is 95%? 99%? (5 points) b. What is the 10-day VaR when the confidence level is 95 % ? (3 points) c. What is the one-day expected shortfall when the confidence level is 95%? 99 % ? (5 points) d. Suppose that the one-day VaR of a portfolio consisting of these two investments at a confidence level of 95% is $5 million. Is the subadditivity condition satisfied? (2 points)

Answer 1

1.5% chance loss of $ 5 millions

4.5% chance loss of $ 2 million

94% chance of profit of $2 million

a) The loss of $2 million extends from 94 percentile to 98.5 percentile

95% VaR is hence $ 2 million

the loss of $5 million extends from 98.5 percentile to 100 percentile

99% VaR is hence $ 5 million

b) 10 days VaR for 95% is (one day VaR * (10)^1/2 )

VaR(n days) = VaR(one day) * n^(1/2)

Hence 10 days VaR = $ 2 million * (10)^(1/2) = $ 6.32 million

c)  Expected shortfall is the loss when things turn south

95% confidence i.e. 5% in the tail

The problem is due to different graphs the two investments can have different expected shortfall due to the same VaR (since they are independent)

for 95% we can assume 80% profit of 2 million and 20 % loss of 2 million = $ 1.2 million

Sample Portfolio  

Again for 99% there is 20% chance the loss is 2 million and 80% of 5 million = 0.2* 2 + 0.8* 5 = $ 4.4 million

d) The subadditivity condition is not satisfied when two portfolio VaR is 5 million for 95% confidence