Portfolio optimization with entropic value-at-risk

The entropic value-at-risk (EVaR) is a new coherent risk measure, which is an upper bound for both the value-at-risk (VaR) and conditional value-at-risk (CVaR). One of the important properties of the EVaR is that it is strongly monotone over its domain and strictly monotone over a broad sub-domain including all continuous distributions, whereas well-known monotone risk measures such as the VaR and CVaR lack this property. A key feature of a risk measure, besides its financial properties, is its applicability in large-scale sample-based portfolio optimization. If the negative return of an investment portfolio is a differentiable convex function for any sampling observation, the portfolio optimization with the EVaR results in a differentiable convex program whose number of variables and constraints is independent of the sample size, which is not the case for the VaR and CVaR even if the portfolio rate linearly depends on the decision variables. This enables us to design an efficient algorithm using differentiable convex optimization. Our extensive numerical study indicates the high efficiency of the algorithm in large scales, when compared with the existing convex optimization software packages. The computational efficiency of the EVaR and CVaR approaches are generally similar, but the EVaR approach outperforms the other as the sample size increases. Moreover, the comparison of the portfolios obtained for a real case by the EVaR and CVaR approaches shows that the EVaR-based portfolios can have better best, mean, and worst return rates as well as Sharpe ratios.