Abstract. Since the financial crisis of [2007][2008][2009], there has been a renewed interest in quantifying more appropriately the risks involved in financial positions. Popular risk measures such as variance and value-at-risk have been found inadequate because we now give more importance to properties such as monotonicity, convexity, translation invariance, positive homogeneity, and law invariance. Unfortunately, the challenge remains that it is unclear how to choose a risk measure that faithfully represents a decision maker's true risk attitude. In this work, we show that one can account precisely for (neither more nor less than) what we know of the risk preferences of an investor/policy maker when comparing and optimizing financial positions. We assume that the decision maker can commit to a subset of the above properties (the use of a law invariant convex risk measure for example) and that he can provide a series of assessments comparing pairs of potential risky payoffs. Given this information, we propose to seek financial positions that perform best with respect to the most pessimistic estimation of the level of risk potentially perceived by the decision maker. We present how this preference robust risk minimization problem can be solved numerically by formulating convex optimization problems of reasonable size. Numerical experiments on a portfolio selection problem, where the problem reduces to a linear program, will illustrate the advantages of accounting for the fact that the choice of a risk measure is ambiguous.

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Worst-case risk measures refer to the calculation of the largest value for risk measures when only partial information of the underlying distribution is available. For the popular risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), it is now known that their worst-case counterparts can be evaluated in closed form when only the first two moments are known for the underlying distribution. These results are remarkable since they not only simplify the use of worst-case risk measures but also provide great insight into the connection between the worst-case risk measures and existing risk measures. We show in this paper that somewhat surprisingly similar closed-form solutions also exist for the general class of law invariant coherent risk measures, which consists of spectral risk measures as special cases that are arguably the most important extensions of CVaR. We shed light on the one-to-one correspondence between a worst-case law invariant risk measure and a worst-case CVaR (and a worst-case VaR), which enables one to carry over the development of worst-case VaR in the context of portfolio optimization to the worst-case law invariant risk measures immediately.

Worst-case risk measures refer to the calculation of the largest value for risk measures when only partial information of the underlying distribution is available. For the popular risk measures such as Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR), it is now known that their worst-case counterparts can be evaluated in closed form when only the first two moments are known for the underlying distribution. These results are remarkable since they not only simplify the use of worst-case risk measures but also provide great insight into the connection between the worst-case risk measures and existing risk measures. We show in this paper that somewhat surprisingly similar closed-form solutions also exist for the general class of law invariant coherent risk measures, which consists of spectral risk measures as special cases that are arguably the most important extensions of CVaR. We shed light on the one-to-one correspondence between a worst-case law invariant risk measure and a worst-case CVaR (and a worst-case VaR), which enables one to carry over the development of worst-case VaR in the context of portfolio optimization to the worst-case law invariant risk measures immediately.

Data-driven distributionally robust optimization is a recently emerging paradigm aimed at finding a solution that is driven by sample data but is protected against sampling errors. An increasingly popular approach, known as Wasserstein distributionally robust optimization (DRO), achieves this by applying the Wasserstein metric to construct a ball centred at the empirical distribution and finding a solution that performs well against the most adversarial distribution from the ball. In this paper, we present a general framework for studying different choices of a Wasserstein metric and point out the limitation of the existing choices. In particular, while choosing a Wasserstein metric of a higher order is desirable from a data-driven perspective, given its less conservative nature, such a choice comes with a high price from a robustness perspective -it is no longer applicable to many heavy-tailed distributions of practical concern. We show that this seemingly inevitable trade-off can be resolved by our framework, where a new class of Wasserstein metrics, called coherent Wasserstein metrics, is introduced. Like Wasserstein DRO, distributionally robust optimization using the coherent Wasserstein metrics, termed generalized Wasserstein distributionally robust optimization (GW-DRO), has all the desirable performance guarantees: finite-sample guarantee, asymptotic consistency, and computational tractability. The worst-case expectation problem in GW-DRO is in general a nonconvex optimization problem, yet we provide new analysis to prove its tractability without relying on the common duality scheme. Our framework, as shown in this paper, offers a fruitful opportunity to 1