Calculating Expected Shortfall for Discrete Loss Distributions

Summary

The document addresses how to calculate expected shortfall when portfolio outcomes have a discrete distribution. It clarifies that the quantile-integral definition remains valid for discrete and continuous outcomes because integration is over the probability level, rather than over the outcome density. For a discrete distribution, value at risk is piecewise constant across intervals of probability levels.

When replacing the integral with a sum, each value-at-risk term must be weighted by the width of its probability interval, the discrete counterpart of the differential probability increment. Simply summing the values above a threshold, or dividing by the count of selected quantiles, does not generally reproduce the integral. The explanation gives the conceptual correction but no numerical example, and it does not spell out a specific finite-sample convention for probability mass at the threshold.

Key ideas

• The quantile-integral definition of expected shortfall applies to discrete distributions.
• Value at risk is piecewise constant as the probability level varies for discrete outcomes.
• A discrete sum must weight each value by its probability-interval width.
• Dividing by the number of selected quantiles is not generally the right correction.

Tags

# Expected shortfall in discrete cases


# Expected shortfall in discrete cases












Our professor gives us a definition in continuous cases: $$ \operatorname{ES}(p)=\frac{1}{1-p}\int_p^1 \operatorname{VaR}(z)\mathrm{d}z $$ where our value at risk (VaR) is defined as $$ \operatorname{VaR}(p)=-\inf \{x\mid F(x)=P(X\leqslant x)\geqslant 1-p\} $$ But I don't think this works in discrete cases: If we switch the integral $\int$ to discrete sum $\sum$, then the ES becomes $$ \operatorname{ES}(p)=\frac{1}{1-p}\sum_{p_i\geqslant p}\operatorname{VaR}(p_i) $$ This is not the average but the sum. Should I times a coefficient $\dfrac{1}{\text{number of }i \text{ such that }p_i\geqslant p}$? Can anyone help?

## Answer by Rylan (score 2)

https://quant.stackexchange.com/a/80721

https://en.wikipedia.org/wiki/Expected_shortfall

The form you have works whether $X$ (the portfolio gain or loss) is continuous or discrete. Note that the integral we are doing is not with respect to a pdf/pmf, but with a probability.

$\text{VaR}(z)$ will be piecewise constant if $X$ is discrete, so it can be written as a discrete sum as you note, but that sum will have to have a term along the lines of ($z_i - z_{i-1}$), which is the discrete analogue of $dz$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.