Technical Note—Closed-Form Solutions for Worst-Case Law Invariant Risk Measures with Application to Robust Portfolio Optimization

Abstract

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.

Frameworks and Results in Distributionally Robust Optimization

“…Moreover, they show that in the presence of additional information on the distribution, besides the first two moments, including constraints on the support and Kullback-Leibler divergence, an upper bound on the worst-case VaR can be obtained by solving a SDP. Motivated by the work in El Ghaoui et al [135], Li [242] showcases the results in the context of a risk-averse portfolio optimization problem. Unlike El Ghaoui et al [135] that considers polytopic and interval uncertainty sets for the mean and covariance, Lotfi and Zenios [254] assume that the unknown mean and covariance belong to an ellipsoidal uncertainty set.…”

“…They reformulate the chance constraints as binary second-order conic (SOC) constraints. Li [242] obtains a closed-form expression to the worst-case of the class of law invariant coherent risk measures, where the worst case is taken with respect to all distributions with the same mean and covariance matrix.…”

Model Aggregation for Risk Evaluation and Robust Optimization

“…For any F ∈ F p, w b ε (F w X ), as in Table 1. Using results of Li (2018), the WR portfolio optimization problem is min k and r 0 using the whole-period data. We can see that the robust value computed by the MA approach with 1 is always the largest one and that of SAA is always the smallest one; this is consistent with our intuition as MA with 1 is the most robust approach among them, and SAA is not conservative.…”

“…The two equalities in ( 26) can be safely replaced by inequalities E[F ] µ and var(F ) σ 2 in the problems we consider, and we omit the formulation with inequalities. The WR robust risk value for different risk measures based on this uncertainty set F µ,σ has been extensively studied in literature, see e.g., Ghaoui et al (2003), Zhu and Fukushima (2009), Natarajan et al (2010), Chen et al (2011), Li (2018) and the references therein.…”

A reverse Expected Shortfall optimization formula

“…We are interested in the worst-case value of a functional over L p (m, v). The special case of this problem when p = 2, i.e., the setting with mean and variance information, has been the most popular; see e.g., Ghaoui et al (2003), Li (2018) and Liu et al (2020) on various risk measures.…”