Coherent Risk Measures

Motivation: why this matters in quant finance

Every risk manager needs a single number that answers "how bad can this get?" — one scalar that compresses a random future P&L distribution into a tractable figure. But not all ways of compressing are equal: some violate basic common-sense properties, and the ones that do have been implicated in real market blow-ups.

Value at Risk (VaR) is the industry workhorse, but it is not coherent — it can say that combining two portfolios produces more risk than the sum of their individual risks, which contradicts the intuition that diversification should not hurt. This failure isn't academic: it matters whenever a bank aggregates desk-level risks into a bank-level figure, or allocates capital across subsidiaries.

Artzner, Delbaen, Eber & Heath (1999) gave a clean axiomatic answer: write down the four properties any reasonable risk measure ought to satisfy, then ask which measures satisfy them. The answer — coherent risk measures — opens the door to Expected Shortfall (ES), now the Basel regulatory standard for market risk, and to a whole family of risk measures derived from acceptance sets and scenario analyses.

This note states the four axioms, proves VaR fails one of them, and derives the key dual representation that makes ES tractable.

The informal idea

Think of a risk measure

\rho(X)

as the minimum amount of cash the firm must hold so that the portfolio's position $X$ (a random net worth at horizon $T$ ) is acceptable. Four things ought to be true:

If portfolio $X$ is always at least as good as $Y$ , then $X$ should require no more capital than $Y$ .
Adding cash $c$ to the portfolio should reduce required capital by exactly $c$ .
Scaling the portfolio by $\lambda > 0$ should scale required capital by $\lambda$ .
Combining two portfolios into one should not require more capital than holding them separately — "a merger is at least as good as two separate firms."

The fourth is the diversification principle, and it is the one VaR breaks.

Formal definitions

Let

X, Y, \ldots

be random variables on a probability space

(\Omega, \mathcal{F}, \mathbb{P})

representing portfolio values at a fixed horizon

T

. A risk measure is any function

\rho : \mathcal{X} \to \mathbb{R}

on a suitable space of random variables

\mathcal{X}

\rho

is coherent if it satisfies:

(A1) Monotonicity. If

X \le Y

almost surely, then

\rho(X) \ge \rho(Y)

(A2) Translation invariance (cash invariance). For any constant

c \in \mathbb{R}

\rho(X + c) = \rho(X) - c

(A3) Positive homogeneity. For any

\lambda \ge 0

\rho(\lambda X) = \lambda\,\rho(X)

(A4) Subadditivity.

\rho(X + Y) \le \rho(X) + \rho(Y)

A risk measure satisfying only (A1) and (A2) is called a monetary risk measure. Adding (A4) gives a convex risk measure if we weaken (A3) to convexity

\rho(\lambda X + (1-\lambda)Y) \le \lambda\rho(X) + (1-\lambda)\rho(Y)

for

\lambda \in [0,1]

. Coherent means all four.

Value at Risk (VaR) — definition

At confidence level $\alpha \in (0, 1)$ (typically $\alpha = 0.05$ or $0.01$ ),

\text{VaR}_\alpha(X) := -\inf\{x \in \mathbb{R} : \mathbb{P}(X \le x) > \alpha\} = -F_X^{-1}(\alpha),

where $F_X$ is the CDF of $X$ . In words: $\text{VaR}_\alpha$ is the negative of the $\alpha$ -quantile of the P&L — the loss exceeded only with probability $\alpha$ .

Expected Shortfall (ES) — definition

\text{ES}_\alpha(X) := -\frac{1}{\alpha}\int_0^\alpha F_X^{-1}(u)\,du = -\mathbb{E}[X \mid X \le -\text{VaR}_\alpha(X)]\quad\text{(continuous case)}.

ES is the average loss in the worst $\alpha$ -fraction of cases — not the threshold of that worst fraction, but the expected loss once you're inside it.

Key properties

VaR is not subadditive

Claim. There exist

X, Y

with

\text{VaR}_\alpha(X + Y) > \text{VaR}_\alpha(X) + \text{VaR}_\alpha(Y)

Proof sketch (canonical counter-example). Let

X

and

Y

be the P&Ls of two independent defaultable bonds, each with face value

100

. Each bond defaults with probability

0.02

(and the bondholder loses

100

); otherwise the bondholder gains

5

in coupon. So individually

\mathbb{P}(X = 5) = 0.98

and

\mathbb{P}(X = -100) = 0.02

\alpha = 0.05

\text{VaR}_{0.05}(X) = -5

(negative of the

5\%

quantile, which is still

5

). The loss

-100

occurs with probability only

2\% < 5\%

, so is not captured.

Now consider $X + Y$ with independent defaults. The probability that at least one defaults is $1 - 0.98^2 \approx 0.0396 < 0.05$ . So the $5\%$ worst case is "exactly one defaults," producing $-100 + 5 = -95$ . Thus $\text{VaR}_{0.05}(X+Y) = 95$ .

Compare: $\text{VaR}_{0.05}(X) + \text{VaR}_{0.05}(Y) = -5 + -5 = -10$ .

95 > -10

— diversifying into two bonds increases VaR by

105

. This violates (A4).

\square

The pathology comes from VaR ignoring tail shape beyond the quantile. Anything heavier than the quantile contributes nothing to VaR; merging two such bets can push a previously hidden tail into the quantile range.

Expected Shortfall is coherent

Theorem (Acerbi-Tasche).

\text{ES}_\alpha

is coherent.

The proof is easy on (A1)-(A3); subadditivity uses that $\text{ES}_\alpha$ has a dual representation (see below) as a supremum over a convex set of probability measures.

Dual representation

Theorem (Artzner et al.). Every coherent risk measure

\rho

on a finite

\Omega

admits a representation

\rho(X) = \sup_{\mathbb{Q} \in \mathcal{Q}} \mathbb{E}^\mathbb{Q}[-X],

where

\mathcal{Q}

is a convex set of probability measures absolutely continuous with respect to

\mathbb{P}

(called the generating set or the set of scenarios).

Interpretation. A coherent risk measure is always a worst-case expected loss across a family of scenarios. VaR fails to have this representation because it is not a supremum over any convex scenario set — it's a quantile, which is discontinuous.

For Expected Shortfall, the generating set is explicit:

\mathcal{Q}_\alpha = \{\mathbb{Q} \ll \mathbb{P} : d\mathbb{Q}/d\mathbb{P} \le 1/\alpha\}.

This is why ES is sometimes called "the average of VaRs above the threshold" — it weights the tail probability uniformly on the worst $\alpha$ -fraction.

Worked example — portfolio of two assets

Let asset A return

+10

with probability

0.9

and

-50

with probability

0.1

. Let asset B return

+10

with probability

0.9

and

-50

with probability

0.1

, independently of A. Consider the equal-weight portfolio

P = \tfrac12(A + B)

\alpha = 0.1

Individual $\text{VaR}_{0.1}(A)$ . Quantile at $10\%$ : loss $50$ occurs with $\mathbb{P} = 0.1$ , so $\text{VaR}_{0.1}(A) = 50$ .
Individual $\text{VaR}_{0.1}(B)$ . Same: $50$ .
Portfolio distribution. Outcomes of $P$ : $+10$ (both up, prob $0.81$ ), $-20$ (one up one down, prob $0.18$ ), $-50$ (both down, prob $0.01$ ). So $\mathbb{P}(P \le -20) = 0.19 > 0.1$ but $\mathbb{P}(P \le -50) = 0.01 < 0.1$ , meaning the $10\%$ quantile is $-20$ and $\text{VaR}_{0.1}(P) = 20$ .
Subadditivity check for VaR. $\text{VaR}_{0.1}(A+B) = 2 \cdot 20 = 40 \le 50 + 50 = 100$ . VaR happens to satisfy subadditivity here because the distributions are symmetric and the extreme tail is small enough.
$\text{ES}_{0.1}(A)$ = average loss in worst 10% = $50$ (the entire worst decile is the loss event). Same for B: $50$ .
$\text{ES}_{0.1}(P)$ . Worst 10% of $P$ consists of probability 0.01 on $-50$ and probability 0.09 on $-20$ , so average $= (0.01 \cdot 50 + 0.09 \cdot 20)/0.1 = (0.5 + 1.8)/0.1 = 23$ .
Subadditivity check for ES. $\text{ES}_{0.1}(A+B) = 2 \cdot 23 = 46 \le 50 + 50 = 100$ . $\checkmark$

Both measures diversify correctly here because the tail is not pathological. Change the probabilities to make the individual tails just barely under $\alpha$ , and VaR's subadditivity can flip.

Common confusions and pitfalls

"VaR is the expected loss." No — VaR is a quantile, not an expectation. ES is the conditional expectation in the tail.
"ES dominates VaR." $\text{ES}_\alpha(X) \ge \text{VaR}_\alpha(X)$ always, because ES averages the tail that begins at the VaR threshold.
Coherent ≠ good. Coherent is a minimum requirement; many other properties (comonotonic additivity, law invariance, elicitability) are additionally desirable. VaR is elicitable; ES is not — a technical cost that matters for backtesting.
ES at $\alpha = 0.025$ ≈ VaR at $\alpha = 0.01$ . Under normality; the two are calibrated to roughly equal capital under Basel. Under fat tails the relationship breaks and ES is the conservative choice.
Empty scenarios. When fitting ES from a finite sample at small $\alpha$ , you might have only one or two sample losses in the tail. Careful estimation (e.g., extreme-value theory) is needed.

Where this goes next

Backtesting (Risk Models) — how to validate ES despite non-elicitability.
Risk Management — situates these measures in the regulatory and operational landscape.
Convex risk measures (weakening positive homogeneity) — the right framework when liquidity risk makes large positions non-linearly riskier than small ones.

Coherent Risk Measures

Motivation: why this matters in quant finance

The informal idea

Formal definitions

Value at Risk (VaR) — definition

Expected Shortfall (ES) — definition

Key properties

VaR is not subadditive

Expected Shortfall is coherent

Dual representation

Worked example — portfolio of two assets

Common confusions and pitfalls

Where this goes next

Exercises