Dual utilities on risk aggregation under dependence uncertainty

Abstract

Finding the worst-case value of a preference over a set of plausible models is a well-established approach to address the issue of model uncertainty or ambiguity.In this paper, we study the worst-case evaluation of Yaari's dual utility functionals of an aggregate risk under dependence uncertainty along with its decision-theoretic implications. To arrive at our main findings, we introduce a technical notion of conditional joint mixability. Lower and upper bounds on dual utilities with dependence uncertainty are established and, in the presence of conditional joint mixability, they are shown to be exact bounds. Moreover, conditional joint mixability is indeed necessary for attaining these exact bounds when the distortion functions are strictly inverse-S-shaped. A particular economic implication of our results is what we call the pessimism effect. We show that a (generally non-convex/non-concave) dual utility-based decision maker under dependence uncertainty behaves as if she had a more pessimistic risk-averse dual utility but free of dependence uncertainty.

Optimizing distortion riskmetrics with distributional uncertainty

“…This result implies Theorem 4.11 of Wang et al (2019) on the robust risk aggregation problem based on dual utilities with inverse-S-shaped distortion functions.…”

“…Indeed, any set closed under concentration and containing U[0, 1] has infinitely many elements. Another example that is closed under concentration within {(0, p)} is the set of all possible distributions of the sum of several Pareto risks; see Example 5.1 of Wang et al (2019).…”

“…Figure 1: An example of h (left) and ĥ (right) with J h = {t 1 , t 2 , t 3 , t 4 , t 5 }; the dashed lines represent h * and ( ĥ) * , respectively For all distortion functions h ∈ H, from Proposition 1, there exist (countably many) disjoint open intervals on which ĥ = h * . Using a similar notation to Wang et al (2019), we define the set…”

“…Another example is the difference between two distortion risk measures, which is clearly not increasing or convex in general. Optimizing non-convex distortion riskmetrics under distributional uncertainty is difficult and results are available only for special cases; see Li et al (2018), Cai et al (2018), Zhu and Shao (2018), Wang et al (2019), and Bernard et al (2020), all with an increasing distortion function.…”

Bilateral Risk Sharing With Heterogeneous Beliefs and Exposure Constraints

“…The construction of the convex envelope with an inverse-S shaped distortion is studied by Ghossoub (2019b) and Wang et al (2019). We make the following assumption.…”