Expected Shortfall (CVaR)

Value at Risk (VaR) tells you the threshold of the tail; it says nothing about what happens once you cross it. Expected Shortfall, also called Conditional VaR (CVaR), Average VaR, or Expected Tail Loss, answers the follow-up question: "When things do go badly, how bad on average?" It is the average loss conditional on being in the worst $(1-\alpha)$ of outcomes. This one change fixes both of VaR's deepest flaws, its blindness to tail depth and its failure to reward diversification, which is why Basel's Fundamental Review of the Trading Book replaced VaR with ES for market-risk capital.

Definition

For loss $L$ at confidence level $\alpha$, expected shortfall is the average of the losses in the tail beyond the VaR quantile:

$\boxed{\;\text{ES}_\alpha = \mathbb{E}\big[L \,\big|\, L \ge \text{VaR}_\alpha\big] = \frac{1}{1-\alpha}\int_{\alpha}^{1} \text{VaR}_u\, du.\;}$

The second form makes the structure vivid: ES is the average of all VaRs from level $\alpha$ up to 100%, it integrates over the whole tail rather than reading a single point. By construction $\text{ES}_\alpha \ge \text{VaR}_\alpha$ always, since it averages losses that are all at least as large as VaR. (For continuous distributions the conditional-expectation and integral forms coincide; for distributions with atoms, ES uses the integral definition to stay coherent.)

Why it fixes VaR's tail-blindness

Consider two portfolios with the same 99% VaR of $10m. Portfolio A's losses beyond VaR are capped near $11m; Portfolio B occasionally loses $100m (it sold deep out-of-the-money options). VaR rates them identical. Expected shortfall does not: \text{ES}_A \approx \10.5\text{m}$while$\text{ES}_B might be \40m. Because ES depends on the entire shape of the tail, it cannot be gamed by pushing risk just past the quantile, the RiskMetrics-era loophole where traders lowered VaR while stockpiling catastrophic exposures. ES rewards you for the size of the tail, not merely its starting point.

Why it is coherent (and VaR is not)

The Coherent Risk Measures framework of Artzner et al. (1999) lists four axioms a sensible risk measure must satisfy: monotonicity, translation invariance, positive homogeneity, and subadditivity, $\rho(A+B) \le \rho(A)+\rho(B)$, i.e. diversification never increases risk. VaR can violate subadditivity: combining two portfolios can raise its measured risk, penalizing diversification and making it unsound for aggregating risk across desks. Expected shortfall satisfies all four axioms, it is a coherent risk measure. The reason is that ES is an average over the tail, and averaging is a linear operation that respects the diversification inequality, whereas a quantile (a single point) does not. This is the deep theoretical reason regulators switched.

The Rockafellar-Uryasev formulation

A practical bonus: ES can be minimized directly by convex optimization, without first computing VaR. Rockafellar and Uryasev showed that

$\text{ES}_\alpha = \min_{\zeta}\; \Big\{\, \zeta + \frac{1}{1-\alpha}\,\mathbb{E}\big[(L - \zeta)^+\big] \Big\},$

and the minimizing $\zeta$ is the VaR. With returns represented by scenarios (historical or Monte Carlo), this becomes a linear program: minimizing portfolio ES over weights $w$ is jointly convex in $(w,\zeta)$. That is a decisive practical advantage over VaR, whose quantile objective is non-convex and hard to optimize, you can build minimum-CVaR portfolios with off-the-shelf LP solvers.

Worked example

Suppose daily losses beyond the 99% level, from historical simulation of a $100m book, are the five worst returns in a 500-day window (the worst 1%): $-2.0\%, -2.4\%, -3.1\%, -4.5\%, -8.0\%$. The 99% VaR is the smallest of these tail losses, about $2.0m (the 5th-worst). The 99% ES is their average: $(2.0+2.4+3.1+4.5+8.0)/5 = 4.0\%$, i.e. $4.0m, twice the VaR. Notice the $8m disaster day: it is invisible to VaR (which just registers "exceeded") but pulls ES up sharply. For a Gaussian, the relationship is analytic: $\text{ES}_\alpha = \sigma\,\phi(z_\alpha)/(1-\alpha)$, so at 99% ES $= \sigma\,\phi(2.326)/0.01 = 2.665\sigma$ versus VaR $= 2.326\sigma$, ES is about 15% larger. Under fat tails the gap widens dramatically, which is exactly the tail information VaR discards.

Basel adoption

The 2008 crisis exposed VaR's tail-blindness in production. Under the FRTB (finalized 2016, implemented from 2023), the Basel Committee replaced 99% VaR with 97.5% Expected Shortfall as the market-risk capital standard, computed with stressed calibration and liquidity-horizon adjustments. The 97.5% ES was chosen to be roughly comparable in magnitude to the old 99% VaR under normal conditions while capturing tail severity. This is the clearest institutional endorsement that ES is the better measure.

Failure modes

• Harder to backtest. ES is not elicitable, there is no scoring function whose expected minimizer is ES, so direct backtesting is more delicate than VaR's simple exception count (though joint VaR-ES backtests and the Acerbi-Szekely tests exist). This was a genuine regulatory debate.
• Tail-estimation sensitivity. ES depends on the far tail, which is the least-sampled part of the distribution; a handful of extreme observations dominate the estimate, so it has higher sampling variance than VaR and is very sensitive to the model's tail assumptions.
• Still model-dependent. Gaussian ES understates fat tails just as Gaussian VaR does; the improvement is conceptual (what you measure), not a free pass on distributional risk.

In interviews

Define ES as the conditional expected loss beyond VaR, $\mathbb{E}[L\mid L\ge\text{VaR}_\alpha]$, and equivalently as the average of the tail quantiles. Give the two reasons it beats VaR: (1) it accounts for the size of tail losses, so it can't be gamed by pushing risk just past the quantile, and (2) it is subadditive/coherent while VaR is not, so it never penalizes diversification. Know the Gaussian ratio ($\text{ES}_{99\%}=2.67\sigma$ vs $\text{VaR}=2.33\sigma$) and that Basel FRTB uses 97.5% ES. If you want to impress, mention Rockafellar-Uryasev, ES minimization is a convex/LP problem, unlike VaR, and the honest caveat that ES is harder to backtest because it is not elicitable. See Value at Risk (VaR) and Coherent Risk Measures.