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 represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. 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. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.
We propose a pricing rule for life insurance when interest rates and mortality rates are stochastic by applying the method developed and expounded upon by Milevsky et al., 2005 and Milevsky et al., 2007. In the case addressed in this paper, their method amounts to targeting a pre-specified Sharpe ratio for a portfolio of bonds that optimally hedges the life insurance, albeit only partially.
Actuaries often assume that one can eliminate the uncertainty associated with mortality by selling a large number of insurance contracts. This assumption is valid if the force of mortality is deterministic. Indeed, if the insurer sells enough contracts, then the average deviation of actual results from what is expected goes to zero, so the risk is diversifiable. However, because the insurer can only sell a finite number of insurance policies, it is impossible to eliminate the risk that experience will differ from what is expected. The risk associated with selling a finite number of insurance contracts is what we call the finite portfolio risk.
On the other hand, if the force of mortality for a population is stochastic, then there is a systematic (that is, common) risk that cannot be eliminated by selling more policies. We call this risk the stochastic mortality risk, a special case of stochastic parameter risk. Even as the insurer sells an arbitrarily large number of contracts, the systematic stochastic mortality risk remains.
We argue that mortality is uncertain and that this uncertainty is correlated across individuals in a population–mostly due to medical breakthroughs or environmental factors that affect the entire population. For example, if there is a positive probability that medical science will find a cure for cancer during the next thirty years, this will influence aggregate mortality patterns. Biffis (2005), Schrager (2006), Dahl (2004), as well as Milevsky and Promislow (2001), use diffusion processes to model the force of mortality, as we do in this paper. One could model catastrophic events that affect mortality widely, such as epidemics, by allowing for random jumps in the force of mortality.
In related work, Blanchet-Scalliet et al. (2005) value assets that mature at a random time by using the principle of no arbitrage by focusing on equivalent martingale measures; the resulting pricing rules are, therefore, linear. Dahl and Møller (2006) take a similar approach in their work. However, for insurance markets, one cannot assert that no arbitrage holds, so we use a different method to value life insurance contracts and our resulting pricing rule is non-linear, except in the limit.
We value life insurance by assuming that the insurance company is compensated for its risk via the so-called instantaneous Sharpe ratio of a suitably-defined portfolio. Specifically, we assume that the insurance company picks a target ratio of expected excess return to standard deviation, denoted by αα, and then determines a price for a life insurance contract that yields the given αα for the corresponding portfolio. One might call αα the market price of mortality risk, but it appears in the pricing equation in a non-linear manner. However, as the number of life insurance policies increases to infinity, then this αα is the market price of risk in the “traditional” sense of pricing in financial markets in that it acts to modify the drift of the hazard rate process.
In this paper, we assume that the life insurance company does not sell annuities to hedge the stochastic mortality risk. In related work, Bayraktar and Young (2007a) allow the insurer to hedge its risk partially by selling pure endowments to individuals whose stochastic mortality is correlated with that of the buyers of life insurance.
We obtain a number of results from our methodology that one expects within the context of insurance. For example, we prove that if the hazard rate is deterministic, then as the number of contracts approaches infinity, the price of life insurance converges to the net premium under the physical probability for mortality and the risk neutral probability for interest rates. In other words, if the stochastic mortality risk is not present, then the price for a large number of life insurance policies reflects this and reduces to the usual expected value pricing rule in the limit. Alternatively, one can see that in the limit the average cash flow is certain, hence, the price becomes that of a zero-coupon bond with a rate of discount modified to account for the rate of dying.
An important theorem of this paper is that as the number of contracts approaches infinity, the limiting price per risk solves a linear partial differential equation and can be represented as an expectation with respect to an equivalent martingale measure as in Blanchet-Scalliet et al. (2005). Therefore, we obtain their results as a limiting case of ours. Moreover, if the hazard rate is stochastic, then the value of the life insurance contract is greater than the net premium, even as the number of contracts approaches infinity. Milevsky et al. (2005) obtain similar results when pricing pure endowments.
The remainder of this paper is organized as follows. In Section 2, we present our financial market, describe how to use the instantaneous Sharpe ratio to price life insurance payable at the moment of death, and derive the resulting partial differential equation (pde) that the price A=A(1)A=A(1) solves. We also present the pde for the price A(n)A(n) of nn conditionally independent and identically distributed life insurance risks. In Section 3, we study properties of A(n)A(n); our valuation operator is subadditive and satisfies a number of other appealing properties. In Section 4, we find the limiting value of View the MathML source1nA(n) and show that it solves a linear pde. We also decompose the risk charge for a portfolio of life insurance policies into a systematic component (due to uncertain aggregate mortality) and a non-systematic component (due to insuring a finite number of policies). In Section 5, we present a numerical example that illustrates our results, along with the corresponding algorithms that we use in the computation. Section 6 concludes the paper.
We developed a risk-adjusted pricing method for life insurance by assuming that the insurance company is compensated for its risk in the form of a given instantaneous Sharpe ratio of an appropriately-defined (partially) hedging portfolio. Because the market for insurance is incomplete, one cannot assert that there is a unique price. However, we believe that the price that our method produces is a valid one because of the many desirable properties that it satisfies. In particular, we studied properties of the price of nn conditionally independent and identically distributed life insurance contracts. In Theorem 3.3, we showed that the risk charge per person decreases as nn increases, and in Corollary 3.4, we showed that the price is subadditive with respect to nn.
Arguably our main results are those dealing with the limiting price per person. Our most important result is that the limiting price solves a linear pde (Theorem 4.2), and we provided a probabilistic representation of the limiting price as an expectation with respect to an equivalent martingale measure (4.4). From (4.6), one can interpret the instantaneous Sharpe ratio as the market price of mortality risk. Our work, therefore, extends that of Blanchet-Scalliet et al. (2005) and Dahl and Møller (2006).
We proved that if the hazard rate is deterministic, then the risk charge per person goes to zero as nn goes to infinity (Theorem 4.2 and Remark (1)). Moreover, we proved that if the hazard rate is stochastic, then the risk charge person is positive as nn goes to infinity, which reflects the fact that the mortality risk is not diversifiable in this case (Theorem 4.2 and Remark (2)). Additionally, in Eq. (4.15), we decomposed the per-risk risk charge into the finite portfolio and stochastic mortality risk charges.
Milevsky et al. (2005) proved similar properties for the risk-adjusted price of pure endowments, including the limiting property; see Milevsky et al. (2007) for an elementary discussion of this general topic. Because of these properties, we anticipate that our pricing methodology will prove useful in pricing risks for many insurance products. For example, Bayraktar and Young (2007a) apply the method in this paper to price both pure endowments and life insurance. They showed that the price of the two products combined is less than the sum of the individual prices, an intuitively pleasing result. An interesting extension would be to allow the hazard rate to exhibit jumps, as a model for catastrophes.
We also believe that our valuation method will be useful in pricing risks in other incomplete markets. For example, Bayraktar and Young (2007b) price options on non-traded assets for which there is a traded asset that is correlated to the non-traded asset. Then, they to price options in the presence of stochastic volatility, and the instantaneous Sharpe ratio in this case equals the market price of volatility risk, similar to interpreting the instantaneous Sharpe ratio in this paper as the market price of mortality risk in (4.6).
