[Paper Review] Financial Valuation of Mortality Risk via the Instantaneous Sharpe Ratio: Applications to Pricing Pure Endowments

Moshe A. Milevsky, S. David Promislow+1 more|ArXiv.org|May 9, 2007

Insurance, Mortality, Demography, Risk Management17 references21 citations

TL;DR

This paper proposes a risk-based pricing framework for mortality-contingent claims, such as pure endowments, using the instantaneous Sharpe ratio to price non-diversifiable stochastic mortality risk in incomplete markets. It shows that even with infinitely many contracts, a positive risk charge persists if the hazard rate is stochastic, reflecting undiversifiable systematic risk.

ABSTRACT

We develop a theory for pricing non-diversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortality-contingent claim requires compensation for this risk in the form of a pre-specified instantaneous Sharpe ratio. We prove that our ensuing valuation formula satisfies a number of desirable properties. For example, we show that it is subadditive in the number of contracts sold. A key result is that if the hazard rate is stochastic, then the risk-adjusted survival probability is greater than the physical survival probability, even as the number of contracts approaches infinity.

Motivation & Objective

• Address the limitation of traditional actuarial methods that assume perfect diversification via the law of large numbers.
• Model the pricing of pure endowments in an incomplete market where mortality risk cannot be fully hedged.
• Account for stochastic mortality risk—where hazard rates follow a random process—as a non-diversifiable risk component.
• Develop a valuation framework that incorporates investor risk aversion through the instantaneous Sharpe ratio.
• Decompose the total risk charge into finite portfolio risk and stochastic mortality risk components.

Proposed method

• Assume the insurer requires a pre-specified instantaneous Sharpe ratio as compensation for bearing mortality risk.
• Model the hazard rate as a mean-reverting diffusion process with stochastic volatility: $ d ilde{\lambda}_t = a dt + b(\tilde{\lambda}_t - \underline{\lambda}) dW^\lambda_t $.
• Use the Feynman-Kac theorem to derive a nonlinear PDE for the risk-adjusted survival probability and solve it via a transformation to a linear PDE.
• Define the risk charge as the difference between the actual price and the risk-neutral price, decomposed into finite portfolio and stochastic mortality components.
• Apply the theory to pure endowments, where the payoff depends on survival to maturity T.
• Use the representation $ P^{(n)} = F \varphi^{(n)} $ and $ P^{\alpha 0} = F \varphi^{\alpha 0} $, with $ \varphi^{\alpha 0} $ solving a PDE involving the Sharpe ratio.

Experimental results

Research questions

• RQ1 How can mortality risk be priced in an incomplete market where hedging is not frictionless?
• RQ2 What happens to the risk charge per contract as the number of policies approaches infinity when the hazard rate is stochastic?
• RQ3 Can the instantaneous Sharpe ratio be used to derive a consistent, economically meaningful pricing formula for pure endowments?
• RQ4 How does the risk charge decompose into finite portfolio risk and stochastic mortality risk?
• RQ5 Does the risk charge vanish in the limit of large portfolios when the hazard rate is stochastic?

Key findings

• The risk charge per contract is subadditive in the number of contracts, satisfying a key property of coherent risk measures.
• When the hazard rate is deterministic, the risk charge per contract vanishes as the number of contracts approaches infinity, consistent with diversification.
• When the hazard rate is stochastic, the risk charge per contract remains positive even in the limit of infinite contracts, indicating undiversifiable risk.
• The stochastic mortality risk charge is strictly positive when the volatility parameter $ b > 0 $, and zero only when $ b = 0 $, confirming the existence of systematic risk.
• The risk-adjusted survival probability exceeds the physical survival probability when the hazard rate is stochastic, even in large portfolios.
• The total risk charge decomposes cleanly into two components: one due to finite portfolio size and another due to stochastic mortality, with the latter persisting asymptotically.