Tractable Robust Expected Utility and Risk Models for Portfolio Optimization

Abstract

Expected utility models in portfolio optimization is based on the assumption of complete knowledge of the distribution of random returns. In this paper, we relax this assumption to the knowledge of only the mean, covariance and support information. No additional assumption on the type of distribution such as normality is made. The investor's utility is modeled as a piecewise-linear concave function. We derive exact and approximate optimal trading strategies for a robust or maximin expected utility model, where the investor maximizes his worst case expected utility over a set of ambiguous distributions. The optimal portfolios are identified using a tractable conic programming approach. Using the optimized certainty equivalent (OCE) framework of Ben-Tal and Teboulle [6], we provide connections of our results with robust or ambiguous convex risk measures, in which the investor minimizes his worst case risk under distributional ambiguity. New closed form expressions for the OCE risk measures and optimal portfolios are provided for two and three piece utility functions. Computational experiments indicate that such robust approaches can provide good trading strategies in financial markets.

Data-driven distributionally robust optimization using the Wasserstein metric: performance guarantees and tractable reformulations

“…We first investigate the worst-case expectations of convex and concave piecewise affine loss functions, which arise, for example, in option pricing [8], risk management [34] and in generic two-stage stochastic programming [6]. Moreover, piecewise affine functions frequently serve as approximations of smooth convex or concave loss functions.…”

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.…”

Robustness to Dependency in Portfolio Optimization Using Overlapping Marginals

“…Some of the common classes of distributions employed in the financial literature are distributions with first and second moment information (see for example, El Ghaoui et al (2003), Natarajan et al (2009a), and Delage and Ye (2010)) and multivariate normal distributions with parameter uncertainty in the mean and covariance matrix (see Garlappi et al (2007) aspect of such an approach is to identify the cover structure E to balance over-fitting the data and getting overly conservative solutions due to lack of information. In order to construct the cover E, we use time-dependent correlation information of the asset returns.…”

“…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)):…”

“…Several models have been proposed to capture the uncertainty (or ambiguity) in distributions. This includes the class of distributions with information on the first and second moments (see El Ghaoui et al (2003), Delage and Ye (2010), Natarajan et al (2009a), Zymler et al (2013)), the Fréchet class of distributions with information on the univariate marginal distributions (see Meilijson and Nadas 3 (1979), Denuit et al (1999)) and the Fréchet class of distributions with information on the multivariate marginals of non-overlapping subsets of asset returns (see Doan and Natarajan (2012), Garlappi et al (2007), Rüschendorf (1991)). In this paper, we adopt a more general representation of distributional uncertainty where the multivariate marginals possibly overlap with each other.…”