Worst-Case Conditional Value-at-Risk with Application to Robust Portfolio Management

Abstract

This paper considers the worst-case Conditional Value-at-Risk (CVaR) in the situation where only partial information on the underlying probability distribution is available. The minimization of the worst-case CVaR under mixture distribution uncertainty, box uncertainty, and ellipsoidal uncertainty are investigated. The application of the worst-case CVaR to robust portfolio optimization is proposed, and the corresponding problems are cast as linear programs and second-order cone programs that can be solved efficiently. Market data simulation and Monte Carlo simulation examples are presented to illustrate the proposed approach.

Robust portfolios: contributions from operations research and finance

“…As a result, the "optimal" portfolios frequently have extreme or counterintuitive weights for some assets (Best and Grauer 1991;Broadie 1993;Chopra and Ziemba 1993). This obstacle also applies to strategies using risk measures other than variance such as VaR and CVaR (El Ghaoui et al 2003;Ceria and Stubbs 2006;Zhu and Fukushima 2008;Huang et al 2008).…”

“…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.…”

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

“…The worst case CVaR with respect to the Fréchet class of distributions for α ∈ (0, 1) is defined as (see Natarajan et al (2009a) and Zhu and Fukushima (2009)):…”

The Regularization Aspect of Optimal-Robust Conditional Value-at-Risk Portfolios

“…Portfolio optimization using CVaR is implemented by dualizing its representation via the scenario set [23]. When the probability distribution is unknown but can be restricted to a scenario set containing all the potential distributions, Robust Conditional Value-at-Risk (Robust CVaR) can be defined as the worst-case CVaR when the distribution varies in this set [11,25]. The scenario set is constructed by structuring randomness in two stages and concentrating uncertainty at the first stage.…”

“…The scenario set is constructed by structuring randomness in two stages and concentrating uncertainty at the first stage. Again, dualizing the representation enables portfolio optimization using Robust CVaR [11,25].…”