Expected Shortfall vs VaR: Why Coherence Wins

In 2016, the Basel Committee’s Fundamental Review of the Trading Book (FRTB) replaced VaR with Expected Shortfall as the basis for market risk capital (BCBS, Minimum Capital Requirements for Market Risk, 2016). A regulator swapping out its risk measure is not a cosmetic change: every bank’s risk systems, capital reports, and internal-model approvals had to follow. The reason for the switch compresses into one sentence — VaR tells you where the threshold is, and nothing about what happens past it.

That blindness beyond the threshold is not an aesthetic complaint. It has a precise mathematical signature: VaR fails sub-additivity, meaning the reported risk of two portfolios combined can exceed the sum of their standalone risks. A measure that cannot even guarantee “diversification does not increase risk” will give bad advice at exactly the operations that matter most — merging books, setting concentration limits, allocating capital. Worse, the defect is exploitable in reverse: park the losses beyond the quantile and VaR simply cannot see them.

This article walks that line end to end: the definition of ES and its closed form under normality, a complete counterexample showing VaR violating sub-additivity, the proof that ES satisfies all the coherence axioms, and finally why VaR nonetheless keeps one piece of territory in practice — a piece of territory that, as it turns out, is also decided by what lies beyond the threshold.

The Definition of Expected Shortfall

Let $L$ be the portfolio loss over a holding period, with CDF $F_L$ and quantile function $F_L^{-1}$. The Value-at-Risk at confidence level $\alpha$ is the $\alpha$-quantile of the loss distribution:

The Expected Shortfall at the same level is the expected loss conditional on the loss already exceeding VaR:

In other words: VaR answers “where does the bad region start,” ES answers “how bad is the bad region on average.” For a continuous distribution the conditional expectation rearranges into an average of the quantile function over the tail (McNeil, Frey & Embrechts, 2015, §2.4):

This integral form will do repeated work below — both the sub-additivity proof and the estimator variance come out of it. To read it: for $\alpha = 0.975$, ES averages over the worst 2.5% of outcomes; for $\alpha = 0.99$, the worst 1%.

There is also an equivalent form that ties the two measures together directly:

The second term is the average excess beyond the threshold, and it is never negative — so ES is never smaller than VaR, with equality only when the loss distribution is bounded at the quantile. That term is precisely the information VaR throws away and ES recovers.

To get a feel for the gap, the normal distribution gives a ready-made worked example. For $L \sim \mathcal{N}(\mu, \sigma^2)$, the VaR is

and the Expected Shortfall is

where $\phi$ is the standard normal density. Writing $z_\alpha = \Phi^{-1}(\alpha)$ and $\alpha_c = 1 - \alpha$, this is

At $\alpha = 0.975$, $z_\alpha \approx 1.96$ and $\phi(1.96) \approx 0.0584$, so $\phi(z_\alpha)/\alpha_c \approx 0.0584/0.025 \approx 2.34$. For a zero-mean loss with $\sigma = 1\%$ per day, the 97.5% VaR is 1.96% and the 97.5% ES is 2.34% — about 19% larger. This is part of why FRTB chose 97.5% ES to replace 99% VaR: under normality the two are close in magnitude, so the switch does not jolt capital requirements — but under fat tails, ES at the same level pulls further away from VaR, which is exactly the behavior you want.

The Axioms of Coherent Risk Measures

To say precisely what is “wrong” with VaR, you need a standard to judge against. Artzner, Delbaen, Eber, and Heath (1999) proposed one: a set of axioms, and a risk measure $R$ satisfying all four is called coherent:

1. Translation invariance: holding a deterministic cash buffer $c$ should reduce measured risk by exactly $c$: $R(L - c) = R(L) - c$. Equivalently, shifting the loss by a constant shifts the risk by the same constant, in the same direction.
2. Sub-additivity: $R(L_1 + L_2) \le R(L_1) + R(L_2)$. Portfolio risk is at most the sum of standalone risks — diversification must never increase measured risk.
3. Positive homogeneity: $R(\lambda L) = \lambda R(L)$ for $\lambda > 0$. Double the position, double the risk.
4. Monotonicity: if $L_1 \le L_2$ pointwise, then $R(L_1) \le R(L_2)$. A loss distribution that is worse everywhere carries more risk.

These are not a mathematician’s fussiness. Each axiom corresponds to an operation a portfolio manager performs daily: adding a cash cushion (translation invariance), merging books (sub-additivity), adjusting leverage (positive homogeneity), comparing two positions (monotonicity). Violate one, and the measure lies to you at the corresponding operation (Föllmer & Schied, Stochastic Finance, 2011, ch. 4). VaR violates the second — precisely the one you lean on when setting limits and computing diversification benefit.

Why VaR Is Not Coherent

Intuition first, then the proof. VaR looks only at the threshold of the tail and at nothing beyond it. Two portfolios can have identical VaR while their worst-case losses differ by orders of magnitude. When you combine them, the dependence structure can pile loss mass beyond the threshold — neither standalone VaR moves, yet the combined VaR gets pushed up. So an outcome that is impossible under any coherent measure — “the whole is riskier than the sum of its parts” — is constructible for VaR.

Here is a clean construction. Let $L_1$ and $L_2$ be the losses of two portfolios.

$L_1$ takes the value 0 with probability 0.99 and 100 with probability 0.01. Its 99th percentile is 0, so $\text{VaR}_{0.99}(L_1) = 0$. $L_2$ has the identical distribution, so $\text{VaR}_{0.99}(L_2) = 0$. Note what has already happened: each portfolio’s VaR has completely hidden the 1%-probability loss of 100. Whatever lies beyond the threshold, VaR does not book it.

The dependence structure is the key variable, so start with a harmless baseline. Suppose $L_1$ and $L_2$ are perfectly positively correlated — the 100-loss always lands on the same day. Then $L_1 + L_2 = 2L_1$: 0 with probability 0.99, 200 with probability 0.01, and the 99th percentile is still 0. Sub-additivity holds with equality: $0 + 0 = 0$. So far, no problem.

Now change only the dependence, leaving both marginals untouched. Make $L_1$ and $L_2$ conditionally independent given a common Bernoulli shock $Z$: let $Z \sim \text{Bernoulli}(0.5)$, and

• If $Z = 1$: $L_1$ is 0 with prob 0.98 and 100 with prob 0.02; $L_2$ has the same conditional distribution, independent of $L_1$ given $Z$.
• If $Z = 0$: $L_1 = L_2 = 0$ deterministically.

First verify the marginals did not move: $P(L_1 = 100) = 0.5 \times 0.02 = 0.01$, so each of $L_1$, $L_2$ is still 0 with prob 0.99 and 100 with prob 0.01, and $\text{VaR}_{0.99}(L_1) = \text{VaR}_{0.99}(L_2) = 0$ as before.

But the combined loss $L_1 + L_2$ has support $\{0, 100, 200\}$ with probabilities:

• $L_1 + L_2 = 0$: $0.5 + 0.5 \cdot 0.98^2 = 0.5 + 0.4802 = 0.9802$
• $L_1 + L_2 = 100$: $0.5 \cdot 2 \cdot 0.98 \cdot 0.02 = 0.0196$ (one leg fails, the other doesn’t)
• $L_1 + L_2 = 200$: $0.5 \cdot 0.02^2 = 0.0002$ (both legs fail)

The 99th percentile is the smallest $\ell$ with $F(\ell) \ge 0.99$. Since $F(0) = 0.9802 < 0.99$ and $F(100) = 0.9998 \ge 0.99$, we get $\text{VaR}_{0.99}(L_1 + L_2) = 100$. Check sub-additivity:

Violated. What happened? The common shock raised the probability that at least one leg fails from 1% per leg to $0.5 \times (1 - 0.98^2) = 1.98\%$ — just past the 1% threshold line. Loss mass migrated from “beyond the threshold” (where VaR is blind) to “inside the threshold” (where VaR suddenly sees it). The marginal tails never moved, each portfolio still reports zero risk on its own, and yet 100 units of risk materialize on combination.

A risk manager keeping books in VaR alone would read the merger of these two portfolios as “zero diversification cost, zero concentration problem,” because zero plus zero is zero. The reverse reading is worse: since parking losses beyond the quantile drives VaR to zero, the measure institutionally rewards constructing portfolios that are quiet almost always and catastrophic when they break. Selling deep out-of-the-money options is the market’s off-the-shelf version of that structure.

In fairness, violations of this kind require deliberately engineered dependence. Danielsson, Jorgensen, Samorodnitsky, and Sarma (2005) argue that for most fat-tailed distributions encountered in practice, VaR is in fact sub-additive deep enough in the tail — the violations are special cases, not the norm. But regulatory capital exists precisely to guard against special cases: a hole in the measure that exists and is constructible will eventually be constructed.

Why ES Is Coherent

ES averages in everything beyond the threshold, so the trick above does not work on it — pushing losses past the quantile makes ES larger, not invisible. That intuition can be made rigorous: ES satisfies all four axioms (Föllmer & Schied, 2011, ch. 4).

Translation invariance and positive homogeneity read straight off the integral representation $\text{ES}_\alpha(L) = \frac{1}{\alpha_c} \int_\alpha^1 F_L^{-1}(u)\, du$: shift the loss by a constant and every quantile shifts by that constant; scale the loss by $\lambda$ and every quantile scales by $\lambda$. Monotonicity is the same move: if $L_1 \le L_2$ pointwise then $F_{L_1}^{-1}(u) \le F_{L_2}^{-1}(u)$ for all $u$, and the integrals compare pointwise.

The non-trivial axiom is sub-additivity, and the proof goes through the dual representation:

where $\mathcal{Q}$ is the set of probability measures absolutely continuous with respect to the physical measure, with Radon–Nikodym derivative bounded by $1/\alpha_c$. Read it as: ES equals the worst expected loss over all pessimistic reweightings that inflate tail probabilities by at most a factor of $1/\alpha_c$. With this representation, sub-additivity is one line — for any $Q$, $\mathbb{E}_Q[L_1 + L_2] = \mathbb{E}_Q[L_1] + \mathbb{E}_Q[L_2] \le \text{ES}_\alpha(L_1) + \text{ES}_\alpha(L_2)$, and taking the supremum on the left finishes it. A supremum of linear functionals is automatically sub-additive; VaR, being a single quantile, admits no such representation. As an aside, ES belongs to the family of distortion risk measures, whose mathematics is the same as Yaari’s (1987) dual theory of choice — sub-additivity there corresponds to concavity of the distortion function.

The Bill for Coherence: What ES Costs in Practice

If ES dominates mathematically, why did VaR hold a monopoly for twenty years? The answer is the same sentence again: information beyond the threshold is not free. The extra tail that ES looks at has to be paid for with data.

VaR can be read off by sorting 1000 historical losses. ES requires estimating a conditional expectation in the tail — at $\alpha = 0.99$ with 1000 observations, “the worst 1%” is 10 data points, and you are representing an entire tail by the average of 10 numbers. The asymptotic variance of the ES estimator is roughly (McNeil, Frey & Embrechts, 2015, estimation theory in ch. 2):

The $1/(1-\alpha)$ factor means: the higher the confidence level, the fewer effective observations, the worse the variance — ES at $\alpha = 0.99$ is two orders of magnitude noisier than at $\alpha = 0.5$. Getting a stable ES at high confidence means either stacking tens of thousands of observations or fitting a parametric tail model (the EVT article covers that route).

The data hunger feeds directly into the second problem: backtesting. VaR backtests have a clean null hypothesis — is the frequency of realized losses exceeding predicted VaR what it should be? Kupiec’s test just counts exceptions (details in the backtesting article). The equivalent ES test must ask whether the average loss in the worst $1-\alpha$ fraction matches the predicted ES, and the conditioning event itself depends on the predicted VaR — the object under test is tangled up with its own conditioning. To date there is no consensus regulatory backtest for ES. FRTB’s actual practice sidesteps the problem: it keeps counting VaR exceptions under a traffic-light framework, using VaR backtests to discipline the ES model indirectly.

Then there is interpretability, which gets underrated. VaR is a threshold: “our 99% VaR is USD 10M” lands with any committee member. “Our 99% ES is USD 14M” first requires explaining what a conditional tail mean is. When your audience is a non-quantitative board, that difference does daily work.

To see the actual gap, run it:

import numpy as np
from scipy import stats

# Compare VaR and ES for sample loss distribution
np.random.seed(7)
losses = -np.random.normal(0.001, 0.02, size=10_000)   # negative returns = losses

alpha = 0.99
var_99 = np.quantile(losses, alpha)
es_99 = losses[losses > var_99].mean()

print(f"99% VaR: {var_99:.5f}")
print(f"99% ES:  {es_99:.5f}")
print(f"Ratio ES/VaR: {es_99/var_99:.3f}")

For Gaussian losses the ratio ES/VaR converges to $\phi(z_\alpha)/[(1-\alpha) z_\alpha]$, roughly 1.15 at $\alpha = 0.99$. Under fat tails it is larger — the thicker the tail, the more VaR misses and the more ES recovers.

What ES Doesn’t Fix, and Where Each Measure Belongs

Coherence repairs the one defect — blindness beyond the threshold — but risk measures have other failure modes, and ES touches none of them. Both VaR and ES assume a fixed holding period and a portfolio that can be liquidated at marked prices; in a crisis neither assumption holds, and liquidity adjustment is a separate extension (the LA-VaR article). Both share model risk: misspecify the loss distribution and VaR and ES are wrong together — if the worst 1% of scenarios in your model are not the worst 1% in reality, coherence buys you nothing. Aggregation across time has no free answer either: the square-root-of-time rule $\text{ES}_{10} = \text{ES}_1 \sqrt{10}$ holds only for iid normal losses; in general the relationship depends on the joint distribution across days. And one point has to be repeated to every user of the number: ES is a tail average, not a bound. A 99% ES of USD 10M means the worst 1% of days lose USD 10M on average — some of those days will lose far more.

With those accounts settled, the division of labor between the two measures writes itself. Capital allocation, concentration limits, any decision that involves “what is the risk after combining” — use ES: it punishes concentration hidden in fat tails, its axioms line up with portfolio logic, and it is the current regulatory baseline. Daily reporting and audit stay with VaR — “did we breach yesterday” is a count, while the equivalent ES audit is statistically delicate. Most mature risk systems compute both and let each do its own job.

There is a cheap probe for whether a risk report deserves trust. When someone quotes a 1-day 99% VaR of USD 5M, ask what the 99% ES is and how the conditional tail mean was estimated. An answer means the tail has actually been looked at. No answer means only the location of the threshold was ever estimated — and the region beyond it, the part VaR cannot see, was never looked at by anyone.

References

Artzner, P., Delbaen, F., Eber, J.-M., & Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228.

Basel Committee on Banking Supervision. Minimum Capital Requirements for Market Risk. 2016 (revised 2019).

Danielsson, J., Jorgensen, B. N., Samorodnitsky, G., & Sarma, M. (2005). Subadditivity re-examined: the case for Value-at-Risk. FMG Discussion Papers.

Föllmer, H., & Schied, A. (2011). Stochastic Finance (3rd ed.). De Gruyter.

McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools (rev. ed.). Princeton University Press. Chs. 2, 4.

Yaari, M. E. (1987). The dual theory of choice under risk. Econometrica, 55(1), 95-115.