Incorporating Asymmetric Distributional Information in Robust Value-at-Risk Optimization

Abstract

Value-at-Risk (VaR) is one of the most widely accepted risk measures in the financial and insurance industries, yet efficient optimization of VaR remains a very difficult problem. We propose a computationally tractable approximation method for minimizing the VaR of a portfolio based on robust optimization techniques. The method results in the optimization of a modified VaR measure, Asymmetry-Robust VaR (ARVaR), that takes into consideration asymmetries in the distributions of returns and is coherent, which makes it desirable from a financial theory perspective. We show that ARVaR approximates the Conditional VaR of the portfolio as well. Numerical experiments with simulated and real market data indicate that the proposed approach results in lower realized portfolio VaR, better efficient frontier, and lower maximum realized portfolio loss than alternative approaches for quantile-based portfolio risk minimization.

Distributionally Robust Optimization and Its Tractable Approximations

“…We consider upper bounds of directional deviations in order to characterize a family of distributions. We note that numerical values of these bounds can be estimated from empirical data, and we refer interested readers to Chen et al (2007), Natarajan et al (2008), or See and Sim (2009) for examples of how directional deviations can be estimated and used. These distributional properties characterize the family of distributions .…”

Robust portfolios: contributions from operations research and finance

“…This robust technique has obtained prodigious success since the late 1990s, especially in the field of optimization and control with uncertainty parameters Nemirovski 1998, 1999;El Ghaoui and Lebret 1997;Goldfarb and Iyengar 2003a). With respect to portfolio selection, the major contributions have come in the 21st century (see, for example, Rustem et al 2000;Costa and Paiva 2002;Ben-Tal et al 2002;Goldfarb and Iyengar 2003b;El Ghaoui et al 2003;Tütüncü and Koenig 2004;Pinar and Tütüncü 2005;Lutgens and Schotman 2006;Natarajan et al 2009;Garlappi et al 2007;Pinar 2007;Calafiore 2007;Huang et al 2008;Natarajan et al 2008a;Brown and Sim 2008;Natarajan et al 2008b;Shen and Zhang 2008;Elliott and Siu 2008;Zhu and Fukushima 2008). For a complete discussion of robust portfolio management and the associated solution methods, see Fabozzi et al (2007), Föllmer et al (2008), and the references therein.…”

A distributionally robust perspective on uncertainty quantification and chance constrained programming

“…Ambiguity sets of special interest include the Markov ambiguity set containing all distributions with known mean and support [48], the Chebyshev ambiguity set containing all distributions with known bounds on the first and second-order moments [12,14,22,31,39,46,49,51,52], the Gauss ambiguity set containing all unimodal distributions from within the Chebyshev ambiguity set [38,41], various generalized Chebyshev ambiguity sets that specify asymmetric moments [12,13,35], higher-order moments [7,30,45] or marginal moments [17,18], the median-absolute deviation ambiguity set containing all symmetric distributions with known median and mean absolute deviation [24], the Huber ambiguity set containing all distributions with known upper bound on the expected Huber loss function [15,48], the Wasserstein ambiguity set containing all distributions that are close to the empirical distribution with respect to the Wasserstein metric [19,34,40], the KullbackLeibler divergence ambiguity set and likelihood ratio ambiguity set [10,26,27,31,47] containing all distributions that are sufficiently likely to have generated a given data set, the Hoeffding ambiguity set containing all component-wise independent distributions with a box support [3,8,10], the Bernstein ambiguity set containing all distributions from within the Hoeffding ambiguity set subject to marginal moment bounds [36], several φ-divergence-based ambiguity sets [2,…”