Distributionally robust goal-reaching optimization in the presence of background risk

TL;DR

This paper studies how background risk influences portfolio and reinsurance decisions when aiming to reach financial goals, using a novel quantile approach to derive explicit solutions despite non-concavity.

Contribution

It introduces a quantile formulation method to explicitly solve non-concave goal-reaching problems under dependence uncertainty, revealing how background risk affects solution parameters.

Findings

Background risk changes solution parameters but not the solution shape.

Explicit solutions are derived for complex non-concave problems.

Numerical examples confirm the robustness of the proposed solutions.

Abstract

In this paper, we examine the effect of background risk on portfolio selection and optimal reinsurance design under the criterion of maximizing the probability of reaching a goal. Following the literature, we adopt dependence uncertainty to model the dependence ambiguity between financial risk (or insurable risk) and background risk. Because the goal-reaching objective function is non-concave, these two problems bring highly unconventional and challenging issues for which classical optimization techniques often fail. Using quantile formulation method, we derive the optimal solutions explicitly. The results show that the presence of background risk does not alter the shape of the solution but instead changes the parameter value of the solution. Finally, numerical examples are given to illustrate the results and verify the robustness of our solutions.