Conditional VaR (Expected Shortfall)

Conditional Value At Risk (CVaR) — also called expected shortfall, mean excess loss, or tail VaR — fixes VaR’s first and third defects by asking the follow-up question: given that we breach the VaR threshold, what is the average loss?

It is the mean of the tail rather than its boundary, so it is always at least as large as VaR, and it responds to how fat the tail is rather than merely where it starts.

A worked comparison. Take twenty monthly returns, sorted worst-first. Suppose the two worst are $-18.2\%$ and $-12.1\%$, and the rest are milder. At 90% confidence the worst two observations constitute the tail:

VaR reports the better of the two bad months. CVaR reports their average, and it is the number that would have told you what a bad quarter actually costs.

Why this is the theoretically correct choice, not merely the more conservative one. Artzner, Delbaen, Eber and Heath asked what properties any sensible risk measure $\rho$ must satisfy⁵¹ and arrived at four axioms defining a coherent risk measure:

Monotonicity

If portfolio $X$ always pays at least as much as $Y$, it cannot be riskier: $X\le Y\Rightarrow \rho (X)\ge \rho (Y)$.

Translation invariance

Adding cash reduces risk one-for-one: $\rho (X+c)=\rho (X)-c$.

Positive homogeneity

Doubling every position doubles the risk: $\rho (\mathit{\lambda X})=\mathit{\lambda \rho }(X)$ for $\lambda \ge 0$.

Subadditivity

Combining portfolios cannot create risk: $\rho (X+Y)\le \rho (X)+\rho (Y)$. This is diversification, stated as an axiom.

VaR satisfies the first three and fails the fourth.⁵² CVaR satisfies all four. That is not an aesthetic preference — it is the formal reason VaR misbehaves under aggregation and CVaR does not, and it is why the Basel framework moved bank market-risk capital from VaR to expected shortfall.

The practical objection to CVaR used to be tractability. Rockafellar and Uryasev demolished it:⁵³ minimizing CVaR reduces to a linear program, and their formulation computes VaR as a by-product of optimizing CVaR. It is now the easier of the two to optimize as well as the correct one.