Question

Suppose that you have two investments, each of which has a 0.9% chance of a loss of $10 million and a 99.1% chance of a loss of $1 million. These two investments are independent of each other.

(1) What is the VaR for one of the investments at the 99% confidence level?

(2) What is the expected shortfall for one of the investments at the 99% confidence level?

(3) What is the 99% VaR for a portfolio consisting of the two investments?

(4) What is the 99% expected shortfall for a portfolio consisting of the two investments?

(5) Verify that VaR does not satisfy the subadditivity condition in this example whereas expected shortfall does.

Answer 1

Solution

There are two independent investments and each investment has

0.9% chance of a loss of $10 million

99.1% chance of a loss of $1 million

0% chance of profit

Explanation of the given factors:

Diagram:

So, as per the distribution up to a probability of 0.991 there is a loss of $ 1 million and the remaining probability of 0.09 there is a loss of $ 10 million.

Answers:

A) When the confidence level is 99%, it is below 99.1% and also as mentioned in the above diagram the corresponding Loss is $ 1 million.

So, VaR for one of the investment is $ 1 million.

B) When the confidence level is 99%,

Out of the remaining 1%; 0.1% has a chance of a loss of $1 million 0.9% has a chance of a loss of $10 million.

This implied there is 10% chance of loss of $1 million and 90% chance of a loss of $10 million.

So, the expected shortfall for one of the investments = 0.1x$1million + 0.9x$10million = $9.1 million

C)When there is portfolio for two investments from the given statements it can be derived that,

There is 0.009x0.009 =0.000081 probability of loss of $(10+10) million =$20 million

There is 2x0.009x0.991 =0.017838 probability of loss of $(10+1) million =$11 million

There is 0.991x0.991 =0.982081 probability of loss of $(1+1) million =$2 million

As 0.982081+0.017838 = 0.999919 or 99.9919%

When the confidence level is 99%, the VaR is $11 million

D) When the confidence level is 99%, the remaining tail of the distribution is 1% or 0.01

0.000081/0.01 = 0.0081 probability of loss of $20 million

And, (1-0.0081) = 0.9919 probability of loss of $11 million

The expected shortfall for portfolio for two investments = 0.0081x$20 million + 0.9919x$11 million = $11.0729 million

E)Sum of VaR of the investments separately = $(1+1) million =$2 million [Derived from Answer A]

VaR of the portfolio of the two investments = $11 million [From Answer C]

As, VaR of the portfolio of the two investments are greater than Sum of VaR of the investments separately; this does not satisfy subadditivity condition.

Sum of Shortfalls of the investments separately = $(9.1+9.1) million =$18.2 million [Derived from Answer B]

Shortfalls of the portfolio of the two investments = $11.0729 million [From Answer D]

As, Shortfalls of the portfolio of the two investments are less than Sum of Shortfalls of the investments separately; this satisfies subadditivity condition.