# An Optimal Multi-layer Reinsurance Policy under Conditional Tail Expectation

## TL;DR

This paper develops a method to construct optimal multi-layer reinsurance policies under the conditional tail expectation risk measure, extending traditional stop-loss policies and demonstrating their practical application through simulations.

## Contribution

It introduces a systematic extension of optimal stop-loss reinsurance to multi-layer policies and provides estimation methods for unknown parameters.

## Key findings

- Multi-layer reinsurance policies are optimal under CTE risk measure.
- The proposed policies outperform traditional single-layer policies.
- Simulation studies confirm practical applicability and parameter estimation methods.

## Abstract

A usual reinsurance policy for insurance companies admits one or two layers of the payment deductions. Under optimal criterion of minimizing the conditional tail expectation (CTE) risk measure of the insurer's total risk, this article generalized an optimal stop-loss reinsurance policy to an optimal multi-layer reinsurance policy. To achieve such optimal multi-layer reinsurance policy, this article starts from a given optimal stop-loss reinsurance policy $f(\cdot).$ In the first step, it cuts down an interval $[0,\infty)$ into two intervals $[0,M_1)$ and $[M_1,\infty).$ By shifting the origin of Cartesian coordinate system to $(M_{1},f(M_{1})),$ and showing that under the $CTE$ criteria $f(x)I_{[0, M_1)}(x)+(f(M_1)+f(x-M_1))I_{[M_1,\infty)}(x)$ is, again, an optimal policy. This extension procedure can be repeated to obtain an optimal k-layer reinsurance policy. Finally, unknown parameters of the optimal multi-layer reinsurance policy are estimated using some additional appropriate criteria. Three simulation-based studies have been conducted to demonstrate: ({\bf 1}) The practical applications of our findings and ({\bf 2}) How one may employ other appropriate criteria to estimate unknown parameters of an optimal multi-layer contract. The multi-layer reinsurance policy, similar to the original stop-loss reinsurance policy is optimal, in a same sense. Moreover it has some other optimal criteria which the original policy does not have. Under optimal criterion of minimizing general translative and monotone risk measure $\rho(\cdot)$ of {\it either} the insurer's total risk {\it or} both the insurer's and the reinsurer's total risks, this article (in its discussion) also extends a given optimal reinsurance contract $f(\cdot)$ to a multi-layer and continuous reinsurance policy.