Life insurance policies with cash flows subject to random interest rate changes

TL;DR

This paper develops a PDE framework for life insurance reserves with stochastic interest rates and interest-dependent benefits, providing explicit solutions under the Vasicek model and exploring new contract types based on investment performance.

Contribution

It introduces a PDE model for reserves with interest rate-dependent benefits and premiums, including explicit solutions and simulations under the Vasicek model, and proposes new interest-linked contract structures.

Findings

Explicit reserve solutions under Vasicek model

Numerical reserve surface computations

Example of a reinsurance treaty based on investment thresholds

Abstract

The main purpose of this work is to derive a partial differential equation for the reserves of life insurance liabilities subject to stochastic interest rates where the benefits and premiums depend directly on changes in the interest rate curve. In particular, we allow the payment streams to depend on the performance of an overnight technical interest rate, making them stochastic as well. This opens up for considering new types of contracts based on the performance of the insurer's returns on their own investments. We provide explicit solutions for the reserves when the premiums and benefits vary according to interest rate levels or averages under the Vasicek model and conduct some simulations computing reserve surfaces numerically. We also give an example of a reinsurance treaty taking over pension payments when the insurer's average returns fall under some specified threshold.