Robust portfolio techniques for mitigating the fragility of CVaR minimization and generalization to coherent risk measures

Abstract

The conditional value-at-risk (CVaR) has gained growing popularity in financial risk management due to the coherence property and tractability in its optimization. However, optimal solutions to the CVaR minimization are highly susceptible to estimation error of the risk measure because the estimate depends only on a small portion of sampled scenarios. The same is equally true of the other coherent measures. In this paper, by employing robust optimization modelling for minimizing coherent risk measures, we present a simple and practical way for making the solution robust over a certain range of estimation errors. More specifically, we show that a worst-case coherent risk minimization leads to a penalized minimization of the empirical risk estimate. The worst-case risk measure developed in this paper is different from the distributionally worst-case CVaR in Zhu and Fukushima's work of 2009, but these two worst-case risk measures can be simultaneously minimized. Additionally, inspired by Konno, Waki and Yuuki's work of 2002, we examine the use of factor models in coherent risk minimization. We see that, in general, factor model-based coherent risk minimization along the lines of that pursued by Goldfarb and Iyengar in 2003 becomes computationally intractable. Therefore, we apply a simplified version to the factor model-based CVaR minimization, and see that it improves on the performance, achieving better CVaR, turnover, standard deviation and Sharpe ratio than the empirical CVaR minimization and market benchmarks.

Keywords:

• Robust portfolio
• CVaR (conditional value-at-risk)
• Coherent risk measure
• Factor model
• Regularization

Acknowledgement

The research of the first author is partly supported by MEXT Grant-in-Aid for Young Scientists (B) 23710176.