Value-at-Risk (VaR), a widely used performance measure, answers the question:

what is the maximum loss with a specified confidence level? Although VaR is a very popular measure of risk, it has undesirable properties such as the lack of sub-additivity, i.e., VaR of a portfolio with two instruments may be greater than the sum of individual VaRs of these two instruments. Also, VaR is difficult to optimize when calculated using scenarios. In this case, VaR is non-convex, non-smooth as a function of positions, and it has multiple local extrema.

An alternative measure of loss, with more attractive properties, is Conditional Value-at-Risk (CVaR), see [6,7,8]. CVaR coincides in many special cases with Upper CVaR, which is the conditional expectation of losses exceeding VaR (also called Mean Excess Loss and Expected Shortfall), see [7]. However, Acerbi et al. [1,2] recently redefined Expected Shortfall in a manner consistent with the CVaR definition. Acerbi et al. [1,2] proved several important mathematical results on properties of CVaR, including asymptotic convergence of sample estimates to CVaR.

CVaR, is a coherent measure of risk [5,7] (sub-additive, convex, and other nice mathematical properties). CVaR can be represented as a weighted average of VaR and Upper CVaR. This seems surprising, in the face of neither VaR nor Upper CVaR being coherent. The weights arise from the particular way that CVaR "splits the atom" of probability at the VaR value, when one exists.

CVaR can be used in conjunction with VaR and is applicable to the estimation of risks with non-symmetric return-loss distributions. Although CVaR has not become a standard in the finance industry, it is likely to play a major role. CVaR is able to quantify dangers beyond value-at-risk, [1,6,7,8,10]. CVaR can be optimized using linear programming, which allows handling portfolios with very large numbers of instruments and scenarios. Numerical experiments indicate that for symmetric distributions the minimization of CVaR also leads to near optimal solutions in VaR terms because CVaR is always greater than or equal to VaR, [6]. Moreover, when the return-loss distribution is normal, these two measures are equivalent [6,7], i.e., they provide the same optimal portfolio. However, for skewed distributions, VaR optimal and CVaR optimal portfolios may be very different, [10]. Similar to the Markowitz mean-variance approach, CVaR can be used in return-risk analyses. For instance, we can calculate a portfolio with a specified return and minimal CVaR. Alternatively, we can constrain CVaR and find a portfolio with maximal return, see [4,7]. Also, we can specify several CVaR constraints simultaneously with various confidence levels (thereby shaping the loss distribution), which provides a flexible and powerful risk management tool.

Several case studies showed that risk optimization with the CVaR performance function and constraints can be done for large portfolios and a large number of scenarios with relatively small computational resources. For instance, a problem with 1,000 instruments and 20,000 scenarios can be optimized on a 700 MHz PC in less than one minute using the CPLEX LP solver. A case study on the hedging of a portfolio of options using CVaR is included in [6]. Also, the CVaR minimization approach was applied to the credit risk management of a portfolio of bonds, [3]. A case study on optimization of a portfolio of stocks with CVaR constraints is included in [4]. The numerical efficiency and stability of CVaR calculations are illustrated with an example of index tracking in [7]. Several related papers on probabilistic constrained optimization are included in [8].