Two independent bonds each default with probability 4%, losing 100 and paying nothing otherwise. Use them to show that 95% value at risk fails subadditivity, and give the expected shortfall for comparison.

Two independent bonds each default with probability 4%, losing 100 and paying nothing otherwise. Use them to show that 95% value at risk fails subadditivity, and give the expected shortfall for comparison.

Approach: Compute the 95% quantile of each single position, then the quantile of the sum, and compare. Then repeat with the tail average.

100. Each bond defaults with probability 0.04, which is below the 5% tail, so the 95% value at risk of each position is 0 and the two individual figures sum to 0. Held together the portfolio survives only if both do, with probability 0.96^2 = 0.9216, so a loss occurs with probability 7.84% and the 95% quantile of the combined loss is 100. The measure of the sum exceeds the sum of the measures, so diversification appears to create risk and subadditivity fails. Expected shortfall at the same level gives 0.04*100/0.05 = 80 for each bond and (0.0016*200 + 0.0484*100)/0.05 = 103.2 for the pair, comfortably under the sum of 160, since it is a coherent risk measure. The failure needs a jump in the loss distribution near the quantile, which credit, digital options and insurance books all have.

Follow-up: What loss distribution makes 99% value at risk subadditive while the 95% figure still fails?

Key concepts: value at risk, subadditivity, expected shortfall, coherent risk measure.