Expected Shortfall

VaR vs ES

ES captures tail severity that VaR can miss.

highlighted = computed this step

ES is at least VaR

Expected shortfall is always at least VaR for the same alpha. Here ES is $1,100.00, while VaR is $200.00.

ESα≥VaRα\text{ES}_{\alpha}\ge \text{VaR}_{\alpha}
VaR and expected shortfall summaryRisk measures are recomputed from the distribution.VaR and ES summaryMeasureAlphaExact lossDisplayVaR 95%19/2020000 cents$200.00VaR 99%99/100150000 cents$1,500.00ES 95%19/20110000 cents$1,100.00

A coherent tail measure

Expected shortfall is coherent and subadditive, so it handles tail aggregation better than VaR. Both measures still assume the stated distribution.

ES is subadditive; VaR is not always\text{ES is subadditive; VaR is not always}

Inputs, not forecasts

VaR and expected shortfall are model outputs from a stated loss distribution. Historical losses do not guarantee future losses, and estimated distributions carry uncertainty. This is descriptive, not investment advice.

risk measures are model outputs\text{risk measures are model outputs}