Pricing Life Insurance under Stochastic Mortality via the Instantaneous Sharpe Ratio: Theorems and Proofs

TL;DR

This paper introduces a new method for pricing life insurance policies under stochastic mortality risk using the instantaneous Sharpe ratio, resulting in a PDE-based valuation formula with practical numerical algorithms.

Contribution

It develops a novel pricing framework for life insurance in incomplete markets that incorporates the instantaneous Sharpe ratio and derives a PDE characterization of the contract price.

Findings

Price per contract converges to a solution of a linear PDE as contracts increase.

Risk-adjusted premiums exceed net premiums when hazard rates are stochastic.

Numerical algorithms illustrate the practical application of the theoretical results.

Abstract

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can interpret the limiting price as an expectation with respect to an equivalent martingale measure. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. We present a numerical example to illustrate our results, along with the corresponding algorithms.