Killing the Law of Large Numbers: Mortality Risk Premiums and the Sharpe Ratio

Abstract

We provide an overview of how the law of large numbers breaks down when pricing life-contingent claims under stochastic as opposed to deterministic mortality (probability, hazard) rates. In a stylized situation, we derive the limiting per-policy risk and show that it goes to a non-zero constant. This is in contrast to the classical situation when the underlying mortality decrements are known with certainty, per policy risk goes to zero. We decompose the standard deviation per policy into systematic and non-systematic components, akin to the analysis of individual stock (equity) risk in a Markowitz portfolio framework. Finally, we draw upon the financial analogy of the Sharpe Ratio to develop a premium pricing methodology under aggregate mortality risk. Copyright The Journal of Risk and Insurance, 2006.

Maximum Market Price of Longevity Risk Under Solvency Regimes: The Case Of Solvency II

“…Results differ from one approach to the other, but the standard formula and VaR approaches produce values that are quite similar to 0.25 usually suggested in the literature (see e.g. Milevsky et al [21] and Loeys et al [19]). …”

“…It distorts the distribution of the projected death probability to generate risk-adjusted death probabilities that can be discounted at risk-free interest rates (e.g., see Lin and Cox [18]). The last one applies the Sharpe ratio rule assuming that the risk premium required by investors to take on longevity risk is equal to the Sharpe ratio for other undiversifiable financial instruments (e.g., see Milevsky et al [21]). In this paper we adopt the risk-neutral approach.…”

A note on utility based pricing and asymptotic risk diversification

“…This solution has been originally proposed by Brennan and Schwartz [5,6], and then applied several times, in particular in the literature on variables annuities, see e.g. [1], [4], [14] or [15]. However, it seems to ignore the fact that playing with the law of large numbers on the diversifiable part of the risk requires selling a large number of contracts, and therefore may lead to huge positions on the financial market.…”

Valuation of mortality risk via the instantaneous Sharpe ratio: Applications to life annuities

“…Few, if any, have focused on the actual equilibrium compensation for this risk. Milevsky, Promislow, and Young (2006) provided some simple discrete-time examples, while Denuit and Dhaene (2007) used comonotonic methods to analyze this risk. A practitioneroriented paper by Smith, Moran, and Walczak (2003) used financial techniques to justify the valuing of mortality risk, although their approach is quite different from ours.…”