Expected Shortfall

Expected Shortfall

Expected shortfall averages the worst tail of the stated distribution.

highlighted = computed this step

Worst-tail mass

Expected shortfall averages the worst 5% probability mass. Here that tail contains losses $500.00 and $1,500.00.

tail mass=1−α=1/20\text{tail mass}=1-\alpha=1/20
Loss distributionVaR and expected-shortfall rows are recomputed from the distribution.Loss distributionScenarioLossProbabilityCumulative1$-100.003/53/52$0.001/54/53$200.003/2019/204$500.001/5097/1005$1,500.003/1001

Conditional tail mean

The exact weighted tail loss is $55.00 divided by tail probability 1/20, giving expected shortfall $1,100.00.

ESα=11−α∑tailLp=$1,100.00\text{ES}_\alpha=\frac{1}{1-\alpha}\sum_{\text{tail}}Lp=\$1,100.00

Tail model dependence

Expected shortfall captures tail severity in this stated distribution. In real risk work, the tail distribution is estimated with uncertainty. This is descriptive, not investment advice.

ES depends on the stated tail\text{ES depends on the stated tail}