Coherent Risk Measures: VaR and Expected Shortfall

The coherence axioms

Take two independent bonds. Each loses 100 units with probability 0.6% and otherwise loses nothing. At 99% confidence, each bond has Value at Risk (VaR) of zero. Put the bonds together and the portfolio's 99% VaR is 100.

That small example is the problem Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath formalised in their 1999 paper Coherent Measures of Risk. A capital measure should not reward a firm merely for placing two risks in separate boxes. Their answer was a set of four axioms: monotonicity, cash additivity (translation invariance in their terminology), positive homogeneity and subadditivity.

Expected shortfall (ES) obeys those rules. Quantile VaR does not always do so. The distinction matters, but "coherent" is not a synonym for accurate, conservative or crisis-proof. It is a consistency test.

We use losses throughout: positive numbers are bad and all monetary quantities use the same currency and horizon. This is the opposite sign from Artzner and colleagues' original net-worth convention. Our Value at Risk guide covers estimation methods; the expected shortfall guide goes further into implementation.

Let $L$ be a random loss and let $\rho(L)$ be the capital assigned to it. In the loss convention, coherence means:

$$\begin{aligned} L_1 \leq L_2\ \text{a.s.} &\Rightarrow \rho(L_1) \leq \rho(L_2) && \text{(monotonicity)}, \\ \rho(L+m) &= \rho(L)+m && \text{(cash additivity)}, \\ \rho(\lambda L) &= \lambda\rho(L),\quad \lambda \geq 0 && \text{(positive homogeneity)}, \\ \rho(L_1+L_2) &\leq \rho(L_1)+\rho(L_2) && \text{(subadditivity)}. \end{aligned}$$

Cash additivity is easy to misread. Here $m$ is a certain extra loss, so it raises required capital by $m$. Adding $m$ units of cash changes net loss to $L-m$ and reduces the measure by $m$. It also tells us the unit of $\rho$: if losses are in pounds, the result is in pounds.

Positive homogeneity assumes scale does not alter the economics. Doubling a liquid position may approximately double risk; doubling an illiquid position can more than double liquidation cost. This is why convex risk measures relax positive homogeneity while retaining diversification through convexity.

Subadditivity says a combined book needs no more capital than the sum of stand-alone books. It does not say every merger lowers risk. If two positions load on the same factor, equality or little diversification is entirely plausible.

Standard deviation and variance are not coherent loss measures. Both ignore certain shifts, so they fail cash additivity. Standard deviation also fails monotonicity, while variance scales with $\lambda^2$, not $\lambda$. These dispersion statistics can still be useful; they simply answer a different question.

VaR and the discrete counterexample

For confidence $q \in (0,1)$, define loss VaR as the lower $q$-quantile:

$$\operatorname{VaR}_q(L)=\inf\{\ell:\Pr(L\leq \ell)\geq q\}.$$

VaR is monotone, cash-additive and positively homogeneous. It is also subadditive for useful special cases, including jointly elliptical losses under standard conditions. It is not subadditive for arbitrary distributions.

Return to the two bonds. For each $L_i$,

$$\Pr(L_i=100)=0.006,\qquad \Pr(L_i=0)=0.994.$$

Since $\Pr(L_i\leq0)=99.4\%$, each stand-alone $\operatorname{VaR}_{0.99}$ is zero. Independence gives:

$$\Pr(L_1+L_2=0)=0.994^2=0.988036.$$

The probability of at least one default is $1-0.988036=1.1964\%$. The cumulative probability at loss zero is therefore below 99%, while the cumulative probability at loss 100 is $99.9964\%$. Hence

$$\operatorname{VaR}_{0.99}(L_1+L_2)=100>0+0.$$

Nothing about physical diversification has become harmful. The quantile has jumped across an atom in a discrete loss distribution. Credit and insurance portfolios often contain exactly this kind of rare-event structure, so it is more than a mathematical curiosity.

Expected shortfall, including ties

Expected shortfall at confidence $q$ is the average of the upper loss quantiles:

$$\operatorname{ES}_q(L)=\frac{1}{1-q}\int_q^1\operatorname{VaR}_u(L)\,du.$$

For a continuous distribution this equals the conditional mean beyond VaR. With discrete losses, the familiar expression $\mathbb{E}[L\mid L>\operatorname{VaR}_q(L)]$ can be wrong because it discards the fraction of the probability mass at the threshold needed to fill the tail. Carlo Acerbi and Dirk Tasche explain this distinction in On the Coherence of Expected Shortfall (2002).

In the bond example, each 1% tail contains 0.6% default outcomes at 100 and 0.4% outcomes at zero. Its ES is therefore 60. For the combined portfolio, both bonds default with probability $0.006^2=0.0036\%$. The worst 1% contains that mass at 200 plus 0.9964% at 100:

$$\operatorname{ES}_{0.99}(L_1+L_2) =\frac{0.000036(200)+0.009964(100)}{0.01} =100.36.$$

The stand-alone sum is $60+60=120$, so subadditivity holds.

There is also a practical optimisation identity:

$$\operatorname{ES}_q(L) =\min_z\left[z+\frac{1}{1-q}\mathbb{E}(L-z)^+\right].$$

This representation makes scenario-based ES optimisation convex whenever portfolio loss is convex in the decision variables. It is the core of Rockafellar and Uryasev's 2000 CVaR optimisation paper. VaR optimisation is generally non-convex.

The following estimator handles a fractional observation at the tail boundary instead of selecting only losses strictly above VaR:

import numpy as np
from scipy.stats import norm

def empirical_var(losses, confidence=0.99):
    x = np.asarray(losses, dtype=float)
    return np.quantile(x, confidence, method="inverted_cdf")

def empirical_es(losses, confidence=0.99):
    x = np.sort(np.asarray(losses, dtype=float))[::-1]
    tail_mass = (1.0 - confidence) * x.size
    whole = int(np.floor(tail_mass))
    fraction = tail_mass - whole
    total = x[:whole].sum()
    if fraction:
        total += fraction * x[whole]
    return total / tail_mass

def normal_es(mean_loss, loss_vol, confidence=0.99):
    z = norm.ppf(confidence)
    return mean_loss + loss_vol * norm.pdf(z) / (1.0 - confidence)

Inputs must already be losses over a common horizon. Passing returns without negating them estimates the profitable tail.

Regulation and day-to-day use

The Basel Committee's minimum capital requirements for market risk use 97.5% ES in the internal-models approach, with liquidity-horizon adjustments and stress-period calibration. This is not simply a 2.5% tail average over the latest 250 daily P&Ls. Basel retained VaR-based backtesting as part of model approval and added further machinery for non-modellable risk factors.

The move from 99% VaR to 97.5% ES preserves a broadly similar level under normal losses while making tail severity matter. Coherence is one reason for preferring ES, not the whole regulatory design.

Do not generalise that choice to every regime. Solvency II Article 101 calibrates the insurance Solvency Capital Requirement to 99.5% VaR over one year, not ES. Internal models may use another measure only if they deliver equivalent policyholder protection.

ES also supports portfolio risk attribution, but there is a condition hidden in the usual sales pitch. If a positively homogeneous ES function is differentiable in position sizes, Euler's theorem gives contributions that add to total ES. At kinks or with discrete scenarios, marginal contributions may not be unique and need an allocation convention.

Backtesting is less direct than counting VaR breaches. ES depends on the sizes of tail losses, and no finite sample contains much information about a 1% tail. Joint VaR-and-ES scoring and regression-based tests are available, but desks should still inspect exceptions, scenario quality and factor coverage rather than rely on one p-value.

What coherence does not fix

A coherent measure can be estimated from a bad distribution. If a scenario engine omits a volatility regime, liquidity gap or basis break, ES will average the wrong tail with impeccable mathematics. Our regime-switching model guide shows one way distributions can change through time, while the quantitative risk management guide puts the measure beside limits, stress tests and governance.

ES is also statistically hungry. At 99% confidence, 1,000 equally weighted observations supply only 10 observations to the empirical tail. Serial dependence and overlapping horizons reduce the effective information further. Parametric smoothing lowers sampling noise by adding model risk.

Positive homogeneity can be the wrong economics for concentrated or illiquid books. Convex risk measures, stress losses and explicit liquidation-cost models can be better tools there. And a single cash number never replaces sensitivities, concentrations and named stress scenarios.

Coherence keeps aggregation honest. It cannot make the underlying view of the world honest for us.

Frequently Asked Questions

Which sign convention should a risk system use?

Either profits or losses can work, but every formula must use one convention consistently. This article uses positive losses, so a certain extra loss raises risk and cash held against the position lowers it.

How should expected shortfall handle losses equal to VaR?

Use the integrated-quantile definition. In a discrete sample, include just enough probability mass at the VaR threshold to fill the required tail; do not blindly average observations satisfying $L>\operatorname{VaR}$.

Is VaR always non-subadditive?

No. VaR is subadditive for important distribution classes, including many elliptical models. The coherence objection is that it can fail for other valid loss distributions, particularly discrete or strongly non-elliptical ones.

Does Basel FRTB eliminate VaR?

No. The internal-models capital measure is based on 97.5% ES, but VaR backtesting remains part of the framework. FRTB also applies liquidity horizons, stress calibration and separate treatment of non-modellable risk factors.