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A Scenario-Based Intuition for Expected Shortfall Subadditivity

Article Quant Q&A · Author: honeybadger
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Summary

The note gives an intuitive finite-scenario argument for why expected shortfall is subadditive, a property that supports recognizing diversification in portfolio risk. It describes expected shortfall as the average loss in the worst tail of outcomes, using a 99% threshold and 500 scenarios as an illustration. For that example, the tail average is approximated using the five worst losses.

For a portfolio split into two sub-portfolios, the scenarios that rank worst for the combined portfolio need not be the worst scenarios for either component on its own. Averaging each component’s losses over the combined portfolio’s worst scenarios therefore cannot exceed that component’s own average over its worst scenarios. Adding the component comparisons yields the subadditivity inequality. This is an explanatory discrete-scenario argument, not a full proof for every generic distribution. The answer also uses a particular loss sign convention and a tail-count approximation, so details require care when adapting it to other definitions or finite-sample estimators.

Key ideas

  • Expected shortfall measures average loss in the tail beyond a chosen quantile.
  • The worst scenarios for a combined portfolio need not be the worst scenarios for each sub-portfolio.
  • Each sub-portfolio’s expected shortfall is at least as large as its average loss over the combined portfolio’s worst scenarios.
  • Adding those component bounds gives the subadditivity result in the scenario approximation.

Tags

Risk managementStatisticsPortfolio construction
Full text
# Subadditivity of Expected Shortfall


# Subadditivity of Expected Shortfall












I am able to see why Expected Shortfall will be subadditive for normal distribution or a uniform distribution. I am trying to prove the result for any generic distribution. I came across many proofs available on the internet, but the math involved is too complex in all of them. Is there any simple explanation why ES is subadditive for any generic distributions?

## Answer by Gordon (score 4, accepted)

https://quant.stackexchange.com/a/35609

The expected shortfall is defined by \begin{align*} ES_{\alpha} = \frac{1}{1-\alpha}\int_{\alpha}^1 VaR_{p}(L) dp, \end{align*} where $L$ is the loss function. For the case with 500 scenarios, the $\alpha=99\,\%$ percentile VaR is approximately the $5^{\rm th}$ worst loss scenario. The expected shortfall can then be approximated by the average of the 5 worst losses, times $-1$ (we take $ES_{\alpha}$ to be positive). That is, \begin{align*} ES_{\alpha}(L) = -\frac{1}{5}\sum_{i=1}^5 L(i), \end{align*} where $L(i)$ is the $i^{\rm th}$ worst loss scenario. Assuming that the loss $L$ can be decomposed into the losses $L_1$ and $L_2$ from two sub-portfolios. That is $$L=L_1+L_2.$$ Then $$L(i) = L_1(i)+L_2(i).$$ However, it is easy to see that, though $L(1), \ldots, L(5)$ are the 5 worst loss scenarios of $L$, $L_j(1), \ldots, L_j(5)$, for $j=1, 2$, are not necessarily the 5 worst loss scenarios for $L_j$. In other words, for $j=1, 2$, \begin{align*} ES_{\alpha}(L_j) \ge -\frac{1}{5}\sum_{i=1}^5 L_j(i). \end{align*} Then \begin{align*} ES_{\alpha}(L) &= -\frac{1}{5}\sum_{i=1}^5 L(i)\\ &=-\frac{1}{5}\sum_{i=1}^5 L_1(i)-\frac{1}{5}\sum_{i=1}^5 L_2(i)\\ &\le ES_{\alpha}(L_1) + ES_{\alpha}(L_2). \end{align*}

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