We investigate two methods for reducing estimation error in portfolio optimization with Conditional Value-at-Risk (CVaR ). The first method is nonparametric : penalize portfolios with large variances in mean and CVaR estimations. The penalized problem is solvable by a quadratically-constrained quadratic program, and can be interpreted as a chance-constrained program. We show the original and penalized solutions follow the Central Limit Theorem with computable covariance by extending M-estimation results from statistics. The second method is parametric : solve the empirical Markowitz problem instead if the log-return distribution is in the elliptical family (which includes Gaussian and t distributions), as then the population frontiers of the Markowitz and mean-CVaR problems are equivalent . Numerical simulations show both methods improve upon the empirical mean-CVaR solution under an elliptical model, with the Markowitz solution dominating. The penalized solution dominates under a non-elliptical model with heavy one-sided loss .
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 .
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investigate two methods for reducing estimation error in portfolio optimization with Conditional Value-at-Risk (CVaR
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
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first method is nonparametric : penalize
nonparametric PBR method penalizes
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variances
variability
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penalized problem is solvable by a quadratically-constrained quadratic program, and can be interpreted as a chance-constrained program. We show the original and penalized solutions follow the Central Limit Theorem with computable covariance by extending M-estimation results from statistics. The second method is parametric : solve
parametric PBR method solves
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if the log-return distribution is in the elliptical family (which includes Gaussian and t distributions), as then the population frontiers
of the empirical mean-CVaR problem, as the solutions
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. Numerical simulations show both methods improve upon the empirical
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
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solution under an elliptical model, with the Markowitz solution dominating. The penalized solution dominates under a non-elliptical model with heavy one-sided loss
problem, with less variability