Question

We have 2 independent investments. Each of them may have a 1% chance of a loss of $10m, a 2% chance of a loss of $5m, a 3% chance of a loss of $1m and a 94% chance of a profit of $1m. Question. The VaR 0.95 for each single investment is... The espected shortfall (ES)0.95 for each single investment is...

Answer 1

The VaR at significance level=0.95 (or probability p=0.05) estimates how much an investment is likely to lose(minimum) with the given probability in a particular amount of time. It can be defined as the maximum loss if we exclude all the worse possible outcomes whose combined probability is p.

In this case,we exclude the bottom 5% of worst likely outcomes(i.e. 1% chance of loss of 10 million,2% chance of loss of $5 million,an another 2% chance of loss of 1 million) and so the Maximum possible loss is 1 million (1% chance of loss of 1 million is still remaining)

The expected shortfall at 0.95 level means the Expected outcome of the worst 5% cases. It is different from VaR as it also considers the values of those worst 5%

The expected outcome of losses of the worst 5% is 1/5*10 million+2/5*$5 million+2/5*$1 million= $6 million

So the expected shortfall at this level is $6 million