Question

Suppose that there are two identical and independent projects, each with a probability of 0.03 of a loss of $8m and a probability of 0.97 of a loss of $2m. Calculate the 96% VaR and expected shortfall for each project considered separately and the two projects combined. Comment on the quality of subadditivity for VaR and expected shortfall based on your results

Answer 1

Since P[X ? 2] = 0.97 and P[X ? 8] = 1, so the one-year 96% VaR for each project is $2 million.

When the projects are put in the same portfolio, there is a 0.03 × 0.03 = 0.0009 probability of a loss of $16 million, a 2 × 0.03 × 0.97 = 0.0582 probability of a loss of $10 million, and a 0.97 × 0.97 = 0.9409 probability of a loss of $4 million.

Let Y denote the loss random variable of the two projects. Since P[Y ? 4] = 0.9409, P[Y ? 10] = 0.9991 and P[Y ? 16] = 1, so the one-year 96% VaR for the portfolio is $10 million.

The total of the VaRs of the projects considered separately is $4 million. The VaR of the portfolio is therefore greater than the sum of the VaRs of the projects by $6 million. This violates the subadditivity condition.

Expected Shortfall:

Conditional that we are in the 4% tail of the loss distribution, there is therefore an 75% probability of a loss of $8 million and a 25% probability of a loss of $2 million. The expected loss is 0.75×8+0.25×2 or $6.5 million.

When the two projects are combined, of the 4% tail of the loss distribution, 0.09% corresponds to a loss of $16 million and 3.91% corresponds to a loss of $10 million. Conditional that we are in the 4% tail of the loss distribution, the expected loss is therefore (0.09/4)×16+(3.91/4)×10, or $10.135 million. Since 6.5+6.5 > 10.135, the expected shortfall measure does satisfy the subadditivity condition.