Journal of Risk

Optimization of conditional value-at-risk

R. Tyrrell Rockafellar and Stanislav Uryasev


A new approach to optimizing or hedging a portfolio of financial instruments to reduce risk is presented and tested on applications. It focuses on minimizing conditional value-at-risk (CVaR) rather than minimizing value-at-risk (VaR), but portfolios with low CVaR necessarily have low VaR as well. CVaR, also called mean excess loss, mean shortfall, or tail VaR, is in any case considered to be a more consistent measure of risk than VaR. Central to the new approach is a technique for portfolio optimization which calculates VaR and optimizes CVaR simultaneously. This technique is suitable for use by investment companies, brokerage firms, mutual funds, and any business that evaluates risk. It can be combined with analytical or scenario-based methods to optimize portfolios with large numbers of instruments, in which case the calculations often come down to linear programming or nonsmooth programming. The methodology can also be applied to the optimization of percentiles in contexts outside of finance.