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

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.