Coherent Risk Measures

Before Artzner, Delbaen, Eber, and Heath's 1999 paper, "risk measure" meant whatever number a desk chose to report. Their contribution was to ask the foundational question: what properties must any sensible measure of risk satisfy? They wrote down four axioms, defined a measure obeying them as coherent, and then proved the industry's favorite, Value at Risk (VaR), is not coherent. This is one of the most influential theoretical results in risk management, and the reason regulators moved to Expected Shortfall (CVaR). Being able to state the four axioms and show precisely where VaR breaks is a rite of passage.

The setup

A risk measure $\rho$ maps a portfolio's future P&L (a random variable $X$, positive = gain) to a real number $\rho(X)$, the capital you must hold against it. Coherence is four axioms. Let $X, Y$ be P&Ls and take a risk-free rate normalized so cash grows trivially.

The four axioms

1. Monotonicity. If $X \le Y$ in every state (portfolio $Y$ always pays at least as much), then $\rho(Y) \le \rho(X)$. A dominated portfolio is at least as risky. Obvious, but it rules out measures like plain variance that ignore the direction of outcomes.

2. Translation invariance. For a certain cash amount $c$, $\rho(X + c) = \rho(X) - c$. Adding $1 of risk-free cash reduces required risk capital by exactly $1. This is what lets $\rho$ be interpreted as capital: $\rho(X)$ is the amount of cash that, added to the position, makes it "acceptable" ($\rho(X + \rho(X)) = 0$).

3. Positive homogeneity. For $\lambda \ge 0$, $\rho(\lambda X) = \lambda\,\rho(X)$. Doubling every position doubles the risk. (This ignores liquidity/size effects, a critique that motivates convex risk measures, which relax homogeneity + subadditivity to the single axiom of convexity.)

4. Subadditivity. $\rho(X + Y) \le \rho(X) + \rho(Y)$. Merging two portfolios cannot create risk, diversification can only help. This is the crucial, economically-loaded axiom. It is what guarantees that a firm's total risk is no more than the sum of its desks' risks, so that decentralized risk limits are conservative and capital can be sensibly allocated. A measure that violates it can tell you that splitting a book into two separately-capitalized entities reduces total required capital, an arbitrage against the risk system itself.

Together, homogeneity + subadditivity imply convexity: $\rho(\lambda X + (1-\lambda)Y) \le \lambda\rho(X) + (1-\lambda)\rho(Y)$, so a coherent measure is convex and can be minimized reliably over portfolios.

VaR fails subadditivity

VaR satisfies axioms 1–3 but can violate subadditivity. The cleanest counterexample uses two defaultable bonds. Let each bond, over the horizon, default independently with probability $4\%$, losing $100; otherwise it pays a small coupon, say gain $2. Consider the 95% VaR (loss not exceeded 95% of the time).

Single bond. The loss is $-2$ (a gain) with probability $96\%$ and $+100$ with probability $4\%$. Since $\Pr(\text{loss} \le -2) = 0.96 \ge 0.95$, the 95% quantile of loss is $-2$: \text{VaR}_{95\%}(\text{one bond}) = -\2$ (i.e. no capital, the default sits outside the 95% window).

Two independent bonds. Total loss is: both survive (prob $0.96^2 = 0.9216$, gain $4), exactly one defaults (prob $2\times0.96\times0.04 = 0.0768$, loss $98), both default (prob $0.0016$, loss $200). Now $\Pr(\text{loss} \le -4) = 0.9216 < 0.95$, so the 95% quantile lands in the one-default region: \text{VaR}_{95\%}(\text{portfolio}) = +\98$.

Compare: \text{VaR}(A+B) = \98$but$\text{VaR}(A) + \text{VaR}(B) = -$2 + (-$2) = -$4$. So

$\text{VaR}(A+B) = 98 \; > \; -4 = \text{VaR}(A) + \text{VaR}(B).$

Diversifying into two independent bonds increased measured VaR, a flat violation of subadditivity. The mechanism is generic: because VaR is a quantile, it ignores what happens beyond the threshold, and for skewed/discrete tail risks a single default can jump from outside the window to inside it upon aggregation. VaR is therefore not coherent, and using it to allocate capital across desks can reward hiding risk in independent tail events.

Expected shortfall is coherent

Expected Shortfall (CVaR), $\text{ES}_\alpha = \frac{1}{1-\alpha}\int_\alpha^1 \text{VaR}_u\,du$, satisfies all four axioms (Acerbi-Tasche 2002). Monotonicity, translation invariance, and homogeneity are inherited from the expectation. Subadditivity holds because ES is an average over the whole tail, and it admits the representation

$\text{ES}_\alpha(X) = \sup_{\mathbb{Q}\in\mathcal{Q}} \mathbb{E}_{\mathbb{Q}}[-X],$

a supremum of expectations over a set $\mathcal{Q}$ of probability measures ("generalized scenarios"). Every coherent risk measure has such a representation (the Artzner et al. structure theorem): coherence is equivalent to being a worst-case expected loss over a family of scenarios. Since a supremum of subadditive functionals (expectations are additive, hence subadditive) is subadditive, ES is subadditive by construction. Running the bond example through ES gives $\text{ES}(A+B) \le \text{ES}(A)+\text{ES}(B)$, diversification is properly rewarded.

Worked example (recap)

Redo the two-bond case under 95% ES rather than VaR. For the portfolio, the worst 5% of outcomes include the both-default ($200) and part of the one-default ($98) mass; averaging these tail losses gives an ES well below twice the single-bond ES, so $\text{ES}(A+B) < \text{ES}(A)+\text{ES}(B)$: the measure now records the diversification benefit that VaR perversely denied. Same portfolio, same data, a coherent measure and an incoherent one disagree on the sign of diversification.

Failure modes / caveats

• Positive homogeneity ignores liquidity. Doubling a huge position more than doubles its real risk (market impact); convex risk measures (Föllmer-Schied) drop homogeneity to capture this, keeping only convexity, see Portfolio Capacity.
• Coherence is necessary, not sufficient. A coherent measure can still be estimated badly; the axioms constrain the functional form, not the input model.
• Elicitability tension. ES is coherent but not elicitable, complicating backtesting; VaR is elicitable but not coherent. No standard measure is both, a genuine theoretical trade-off.

In interviews

Recite the four axioms crisply, monotonicity, translation invariance, positive homogeneity, subadditivity, and be able to say what each means economically, especially subadditivity ("diversification never increases risk," the one that makes decentralized risk limits sound). Then deliver the headline: VaR violates subadditivity, expected shortfall does not. Have the intuition ready even if not the full numbers, VaR is a quantile, so it ignores the tail beyond it and can jump upward when independent tail risks are combined. Bonus points for the representation theorem: coherent measures are exactly worst-case expectations over a scenario set, which is why ES $=\sup_\mathbb{Q}\mathbb{E}_\mathbb{Q}[-X]$ is subadditive. See Value at Risk (VaR) and Expected Shortfall (CVaR).