Hedging Pure Endowments with Mortality Derivatives

TL;DR

This paper develops a pricing model for pure endowments using mortality forwards to hedge systematic mortality risk, incorporating correlation and market prices, with solutions via PDEs and martingale measures.

Contribution

It introduces a novel framework for pricing pure endowments with mortality forward hedging, accounting for correlation and unhedgeable risk via a PDE and martingale approach.

Findings

Hedging with mortality forwards can increase or decrease pure endowment prices.

The contract value satisfies a linear PDE as the number of contracts grows large.

Market prices and correlation influence hedging costs and pricing outcomes.

Abstract

In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or {\it hazard rates}. In this paper, we develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a pre-specified instantaneous Sharpe ratio. The major result of this paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation under an equivalent martingale measure. Another important result is that hedging with the mortality forward may raise or lower the price of this pure endowment comparing to its price without hedging, as determined in Bayraktar et al. [2009]. The market price of the reference mortality risk and the correlation between the two portfolios jointly determine the cost of hedging. We demonstrate our results using numerical examples.