Robustifying Convex Risk Measures for Linear Portfolios: A Nonparametric Approach

Abstract

This paper introduces a framework for robustifying convex, law invariant risk measures. The robustified risk measures are defined as the worst case portfolio risk over neighborhoods of a reference probability measure, which represent the investors' beliefs about the distribution of future asset losses. It is shown that under mild conditions, the infinite dimensional optimization problem of finding the worst-case risk can be solved analytically and closed-form expressions for the robust risk measures are obtained. Using these results, robust versions of several risk measures including the standard deviation, the Conditional Value-at-Risk, and the general class of distortion functionals are derived. The resulting robust risk measures are convex and can be easily incorporated into portfolio optimization problems, and a numerical study shows that in most cases they perform significantly better out-of-sample than their nonrobust variants in terms of risk, expected losses, and turnover.

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

“…Nevertheless, tractability results are available for special cases. Specifically, the worst case of a convex law-invariant risk measure with respect to a Wasserstein ambiguity set P reduces to the sum of the nominal risk and a regularization term whenever h(x, ξ) is affine in ξ and P does not include any support constraints [53]. Moreover, while this paper was under review we became aware of the PhD thesis [54], which reformulates a distributionally robust two-stage unit commitment problem over a Wasserstein ambiguity set as a semi-infinite linear program, which is subsequently solved using a Benders decomposition algorithm.…”

“…When (ξ ) is linear and the distribution of ξ ranges over a Wasserstein ambiguity set without support constraints, one can derive a concise closed-form expression for the worst-case risk of (ξ ) for various convex risk measures [53]. However, these analytical solutions come at the expense of a loss of generality.…”

Robust unit commitment with $$n-1$$ n - 1 security criteria

“…Compared to minimax robust optimization model, the distributionally robust formulation is obviously less conservative and hence more compelling in the circumstances where an optimal decision based on the former model may incur excessive economic and/or computational costs to prevent a rare event. Over the past few decades, DRO models have found many applications in operations research, finance and management sciences, see for instances Bertsimas and Popescu (2005), Delage and Ye (2010), Goh and Sim (2010), Mehrotra and Papp (2014), Wiesemann et al (2012), Wiesemann et al (2013) and Wozabal (2014) for various applications and numerical schemes. In particular, Bertsimas et al (2010) propose a distributionally robust formulation of two stage linear programming model with applications in finance and facility location planning.…”

Robust Portfolio Selection

“…Wozabal [62] introduces a framework for dealing with ambiguity of the distribution of random asset losses in the portfolio selection problem. He defines ''robustified'' risk measure versions of several portfolio risk measures, including the standard deviation, the CVaR, and the general class of distortion functionals that measure the worst-case portfolio risk over an ambiguity set of loss distributions, where an ambiguity set is defined as a neighborhood around a reference probability measure representing an investor's beliefs about the distribution of asset losses.…”