Optimal Risk Exchanges

TLDR

Optimal risk sharing and reinsurance treaty design aim to minimize variance or maximize utility, leading to linear quota‑sharing or stop‑loss arrangements, but these approaches can be non‑unique and may involve fixed side payments, prompting constraints to avoid reserve invasion. The authors incorporate long‑run fairness for each participant to produce a unique Pareto‑optimal risk pool with quota‑sharing‑by‑layers. They develop algorithms that implement this quota‑sharing‑by‑layers construction and provide numerical examples. When all utility functions are exponential, the model reduces to linear quota‑sharing‑by‑layers.

Abstract

The determination of optimal rules for sharing risks and constructing reinsurance treaties has important practical and theoretical interest. Medolaghi, de Finetti, and Ottaviani developed the first linear reciprocal reinsurance treaties based upon minimizing individual and aggregate variance of risk. Borch then used the economic concept of utility to justify choosing Pareto-optimal forms of risk exchange; in many cases, this leads to familiar linear quota-sharing of total pooled losses, or to stop-loss arrangements. However, this approach does not give a unique, risk-sharing agreement, and may lead to substantial fixed side payments. Gerber showed how to constrain a Pareto-optimal risk exchange to avoid invasion of reserves. To these ideas, the authors have added the actuarial concept of long-run fairness to each participant in the risk exchange; the result is a unique, Pareto-optimal risk pool, with “quota-sharing-by-layers” of the total losses. There are many interesting special cases, especially when all individual utility functions are of exponential form, giving linear quota-sharing-by-layers. Algorithms and numerical examples are given.