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Value-at-Risk and Expected Shortfall

Value at Risk is a quantile of the loss distribution:

VaRα=inf⁡{ℓ:P(L≤ℓ)≥α}\text{VaR}_\alpha = \inf\{\ell : P(L \leq \ell) \geq \alpha\}

A 99% one-day VaR of $1m means: on 1% of days, expect to lose more than $1m.

Read that carefully. It is a threshold, not a maximum and not an expectation. It says nothing about how bad the 1% of days get.

Expected shortfall

ESα=E[L∣L>VaRα]\text{ES}_\alpha = E[L \mid L > \text{VaR}_\alpha]

The average loss given that you are in the tail. This is the question a risk manager actually wants answered: not "how bad before things get bad", but "how bad when they do".

Two portfolios can share a VaR while one loses $1.1m in its tail and the other $50m. Expected shortfall distinguishes them; VaR cannot.

The catastrophe is rarer than one day in a hundred, so it sits entirely beyond the 99% threshold and moves the VaR by nothing at all. Drag its probability above 1% and it crosses inside the quantile, at which point VaR finally notices it. That boundary is the whole weakness: the measure only sees what is common enough to reach.

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