Conditional Value-at-Risk Robust Optimization

We propose using the well-known conditional value at risk (CVaR) risk measure as a new methodology for incorporating robustness into portfolio optimization. Robustness in portfolio optimization can address the poor out-of-sample performance of the classical mean-variance optimization problems. Using many estimates of the covariance matrix and mean return vector, we incorporate robustness by finding a portfolio that performs well, on average, for a worst-case subset of these estimates, rather than for a single estimate. This becomes a bilevel integer program that we reformulate into a tractable form under appropriate conditions. We present numerical results that compares this CVaR robust method to a distributionally robust optimization approach that uses the Wasserstein metric to measure robustness. Theoretically, we extend the existing work of stochastic programming by linking the CVaR robustness to the sample average approximation. Specifically, we show that the CVaR robustness problem provides an upper bound, in expectation, to stochastic programming problems. We derive various asymptotic convergence results