Conditional value at risk

Conditional value at risk (CVaR), also called expected shortfall (ES), average value at risk (AVaR), or expected tail loss, is a Risk measure that quantifies the expected loss of an uncertain outcome conditional on that loss falling in the worst tail of its distribution. For a chosen confidence level, CVaR is the average of the losses that exceed the corresponding Quantile, and it therefore describes the typical severity of adverse outcomes rather than merely their threshold. CVaR was introduced as a tractable and theoretically well-behaved alternative to Value at risk (VaR), the quantile of the loss distribution, which reports a threshold that a loss will not exceed with a given probability but says nothing about how large the loss can be once that threshold is breached.

CVaR is a coherent risk measure in the sense of Artzner, Delbaen, Eber and Heath, meaning it satisfies the axioms of monotonicity, translation invariance, positive homogeneity and subadditivity; VaR, by contrast, is generally not subadditive and can penalize diversification. A defining practical property, established by Rockafellar and Uryasev, is that CVaR can be minimized through convex and, for empirical or scenario-based data, linear programming formulations, which makes it directly usable inside large-scale mathematical optimization and portfolio optimization models.

Definition

Let $L$ be a random loss with cumulative distribution function $F_L$, and let $\alpha \in (0,1)$ be a confidence level (commonly 0.95 or 0.99). The value at risk at level $\alpha$ is the $\alpha$-quantile of the loss,

${\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(L)=\inf \{x\in \mathbb{ℝ}:F_L(x)\ge \alpha \}.$

For a loss distribution with no probability atom at the quantile, the conditional value at risk is the conditional expectation of the loss given that it is at least as large as the VaR,

${\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(L)=\mathbb{𝔼}[L\mid L\ge {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(L)],$

equivalently the average of the worst $(1-\alpha )$ fraction of outcomes. Because averaging over a tail can be ambiguous when the loss distribution is discrete or has a jump at the quantile, the general definition of Rockafellar and Uryasev averages the quantile function over the tail,

${\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(L)=\frac{1}{1-\alpha }\underset{\alpha }{\overset{1}{\int }}{\mathrm{V}\mathrm{a}\mathrm{R}}_u(L)\mathrm{d}u,$

which coincides with the conditional-expectation form for continuous distributions and remains coherent for arbitrary distributions. Under this general definition CVaR equals expected shortfall as defined by Acerbi and Tasche, who showed that this tail-average form is coherent whereas several superficially similar "tail conditional expectation" definitions are not.

By construction ${\mathrm{C}\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(L)\ge {\mathrm{V}\mathrm{a}\mathrm{R}}_{\alpha }(L)$ for every distribution, so CVaR is always at least as conservative as VaR at the same confidence level.

Coherence and comparison with value at risk

Artzner and coauthors proposed four axioms that any sensible risk measure $\rho$ on losses should satisfy: monotonicity (larger losses carry larger risk), translation invariance (adding a certain loss $c$ raises risk by $c$), positive homogeneity (scaling a position scales its risk), and subadditivity, $\rho (L_1+L_2)\le \rho (L_1)+\rho (L_2)$, which formalizes the principle that a merged portfolio should be no riskier than its parts held separately. A measure satisfying all four is called coherent. Value at risk fails subadditivity in general, so aggregating positions can appear to increase measured risk and a firm can reduce reported VaR by splitting a book, an artefact with no economic basis. CVaR restores subadditivity and the other axioms, giving it a firmer theoretical footing for capital allocation and diversification analysis.

A further limitation of VaR is that it is insensitive to the shape and magnitude of losses beyond the quantile: two portfolios can share the same VaR while one carries far heavier tail losses. Because CVaR averages the entire tail, it reflects the severity of extreme events and discourages strategies that hide risk just past the VaR threshold.

Optimization formulation

The property that made CVaR widely adopted in practice is that it can be optimized directly. Rockafellar and Uryasev introduced the auxiliary function

$F_{\alpha }(x,t)=t+\frac{1}{1-\alpha }\mathbb{𝔼}[(L(x)-t{)}^{+}],$

where $x$ denotes decision variables (for example portfolio weights or staffing levels), $L(x)$ is the resulting loss, $t$ is an auxiliary scalar, and $(\cdot {)}^{+}=\max (\cdot ,0)$. They proved that minimizing $F_{\alpha }$ jointly over $x$ and $t$ yields the minimum CVaR, and that a minimizing $t$ equals the corresponding VaR, so the VaR is obtained "for free" as a by-product. The function $F_{\alpha }$ is convex in $(x,t)$ whenever the loss is convex in $x$, and jointly optimizing it avoids the need to compute VaR beforehand.

When the underlying uncertainty is represented by a finite set of scenarios or samples $L(x,{\omega }_1),\dots ,L(x,{\omega }_m)$, the expectation becomes a finite average and the positive-part terms are linearized with auxiliary variables $z_j\ge 0$, giving a linear program

$\underset{x,t,z}{\min }t+\frac{1}{(1-\alpha )m}\sum _{j=1}^m{z}_j\text{s.t.}{z}_j\ge L(x,{\omega }_j)-t,z_j\ge 0,$

for loss functions that are linear in the decisions. This scenario formulation scales to large problems and is the standard route for embedding tail-risk control into decision models.

Applications

CVaR is used across quantitative finance for portfolio construction, hedging and capital allocation, where its convex optimization formulation lets analysts minimize tail loss subject to return and budget constraints. Its regulatory appeal follows from coherence: because subadditivity rewards diversification and the tail average reflects the magnitude of extreme losses, expected shortfall was advanced as a more prudent basis for capital measurement than the non-coherent, tail-insensitive VaR. More broadly, minimizing CVaR is a form of stochastic optimization and is closely related to techniques for decision-making under uncertainty, including scenario-based and sample-average approaches.

Workforce management

In workforce management and contact-center staffing, service outcomes such as the service level or Average speed of answer vary from day to day because of stochastic call arrivals, agent absenteeism and handle-time variability. Planning to the mean day can leave the operation badly understaffed on adverse days, when demand spikes and shrinkage coincide. Casting the shortfall in service level (or the excess in wait time) as a loss $L$ lets a planner target CVaR rather than the average: minimizing, or constraining, the 95% CVaR of the daily service-level miss protects against the worst-day tail rather than the expectation, and the scenario linear program above maps naturally onto a Monte Carlo simulation of interval-level demand and staffing. This tail-risk framing generalizes ordinary service-level targeting and connects staffing decisions to the same convex machinery used in portfolio optimization; the specific formulation depends on how loss and scenarios are defined for a given operation.

See also

• Portfolio optimization
• Stochastic optimization
• Monte Carlo method
• Mathematical optimization