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Finite-loss expected shortfall from tail mass

Expected shortfall divides the selected tail’s weighted loss by 1−c.

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Sort and group the outcomes

Sort distinct loss amounts from lowest to highest, adding probabilities for identical amounts. For confidence c, allocate exactly 1−c mass starting from the largest loss. Include a fractional boundary probability when the remaining mass is smaller than that outcome’s probability.

Use inclusion probabilities

ESc=∑iaiℓi1−c,0≤ai≤pi,∑iai=1−c\mathrm{ES}_c=\frac{\sum_i a_i\ell_i}{1-c},\qquad 0\leq a_i\leq p_i,\quad\sum_i a_i=1-c

The a values allocate the worst tail, rather than an arbitrary subset with the same mass.

Check the original 90% allocation

The selected mass is 0.10. Allocation is 0.05 to loss 50 and 0.05 to loss 10. The weighted loss sum is 3, yielding ES=30. Taking the whole 25% at loss 10 would include too much mass.

Connect to the quantile average

Equivalently, ES is the average of the loss quantile function over confidence levels from c to 1. This handles atoms without asserting that an unqualified conditional mean above VaR always equals the selected-tail average.

Retain the distribution assumption

The calculation summarises the supplied probabilities and losses. It neither estimates those inputs nor constructs a regulatory stressed/liquidity-adjusted measure.

Further references