Efficient evaluation of risk allocations

TL;DR

This paper introduces a generating function method for efficiently computing conditional expectations of marginal risks given total portfolio risk, especially useful in complex, dependent, or heavy-tailed risk scenarios.

Contribution

The paper presents a novel generating function approach and algorithms for calculating risk allocations in complex portfolios, improving computational efficiency and handling dependencies.

Findings

Significant computational gains in large-scale risk-sharing problems

Effective in cases with dependent and heavy-tailed risks

Applicable to peer-to-peer insurance and Euler-based risk allocation

Abstract

Expectations of marginals conditional on the total risk of a portfolio are crucial in risk-sharing and allocation. However, computing these conditional expectations may be challenging, especially in critical cases where the marginal risks have compound distributions or when the risks are dependent. We introduce a generating function method to compute these conditional expectations. We provide efficient algorithms to compute the conditional expectations of marginals given the total risk for a portfolio of risks with lattice-type support. We show that the ordinary generating function of unconditional expected allocations is a function of the multivariate probability generating function of the portfolio. The generating function method allows us to develop recursive and transform-based techniques to compute the unconditional expected allocations. We illustrate our method to large-scale risk-sharing and risk allocation problems, including cases where the marginal risks have compound distributions, where the portfolio is composed of dependent risks, and where the risks have heavy tails, leading in some cases to computational gains of several orders of magnitude. Our approach is useful for risk-sharing in peer-to-peer insurance and risk allocation based on Euler's rule.