  We introduce performance-based regularization (PBR), a new approach to
addressing estimation risk in data-driven optimization, to mean-CVaR portfolio
optimization. We assume the available log-return data is iid, and detail the
approach for two cases: nonparametric and parametric (the log-return
distribution belongs in the elliptical family). The nonparametric PBR method
penalizes portfolios with large variability in mean and CVaR estimations. The
parametric PBR method solves the empirical Markowitz problem instead of the
empirical mean-CVaR problem, as the solutions of the Markowitz and mean-CVaR
problems are equivalent when the log-return distribution is elliptical. We
derive the asymptotic behavior of the nonparametric PBR solution, which leads
to insight into the effect of penalization, and justification of the parametric
PBR method. We also show via simulations that the PBR methods produce efficient
frontiers that are, on average, closer to the population efficient frontier
than the empirical approach to the mean-CVaR problem, with less variability.
