بیمه عمر بیمه را با کمکهای خالص تضمین می کند

We extend the work of Milevsky et al., [Milevsky, M.A., Promislow, S.D., Young, V.R., 2005. Financial valuation of mortality risk via the instantaneous Sharpe ratio (preprint)] and Young, [Young, V.R., 2006. Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio (preprint)] by pricing life insurance and pure endowments together. We assume that the company issuing the life insurance and pure endowment contracts requires compensation for their mortality risk in the form of a pre-specified instantaneous Sharpe ratio. We show that the price Pm,nPm,n for mm life insurances and nn pure endowments is less than the sum of the price Pm,0Pm,0 for mm life insurances and the price P0,nP0,n for nn pure endowments. Thereby, pure endowment contracts serve as a hedge against the (stochastic) mortality risk inherent in life insurance, and vice versa.

Milevsky et al. (2005) present a framework for valuing (stochastic) mortality risk. They 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. They apply their method to price pure endowment contracts and show that the resulting price satisfies many desirable properties. Young (2006) applies the same method to price life insurance that pays at the moment of death of the insured. We extend their work by pricing life insurance and pure endowments together. We show that the price Pm,nPm,n for mm life insurances and nn pure endowments is less than the sum of the price Pm,0Pm,0 for mm life insurances and the price P0,nP0,n for nn pure endowments. Thereby, pure endowment contracts serve as a partial hedge against the (stochastic) mortality risk inherent in life insurance, and vice versa. In our framework the price of an additional life insurance contract can be defined as the marginal price, Pm+1,n−Pm,nPm+1,n−Pm,n, when the insurer already holds mm life insurances and nn pure endowments. Similarly, one can marginally price an additional pure endowment. Defining a price in this way is related to work of Stoikov (2005), in which he uses the principle of equivalent utility to find the indifference price of an additional financial contract. In this paper, we assume that the mortalities of the individuals buying the contracts are independent given the hazard rate, but that all the buyers share the same hazard rate. It is the positive correlation induced by the identical hazard rates that causes pure endowments to be a partial hedge against the mortality risk in life insurance. If, instead of assuming they are identical, we were to assume that the hazard rates of the buyers are positively correlated, then we would also obtain that pure endowment serves as a hedge to life insurance. The remainder of the paper is organized as follows. In Section 2, we present our modeling framework and then show how to price a portfolio of life insurance and pure endowment contracts via the instantaneous Sharpe ratio, the method proposed by Milevsky et al. (2005). In Section 3, we show that the price Pm,nPm,n for mm life insurances and nn pure endowments is monotone in mm and nn. We also show that Pm,n≤Pm1,n1+Pm2,n2Pm,n≤Pm1,n1+Pm2,n2, in which m1+m2=mm1+m2=m and n1+n2=nn1+n2=n (subadditivity). Thus, life insurance and pure endowments are hedges for each other. As a corollary of the subadditivity, we show that the marginal price of a life insurance and pure endowment contract satisfy are in fact reasonable prices. Section 4 concludes the paper.

In this paper, we developed a relative pricing methodology for pure endowments and life insurances, assuming that the insurer has already sold a portfolio of these instruments. In our model, we take the hazard rate and the interest rate to be stochastic. The market is incomplete since the insurer cannot fully hedge against the stochastic mortality risk. The insurer values a portfolio of pure endowment and insurance contracts by specifying the instantaneous Sharpe ratio of the optimal local hedging portfolio. We first show that the value of the portfolio with mm life insurances and nn pure endowments, Pm,nPm,n, satisfies subadditivity; that is, Pm,n≤Pm1,n1+Pm2,n2Pm,n≤Pm1,n1+Pm2,n2 for any m1,m2,n1,n2∈Nm1,m2,n1,n2∈N such that m1+m2=mm1+m2=m and n1+n2=nn1+n2=n. This property shows that for given market prices for insurance and pure endowments, the insurer is better off selling both the pure endowment and the life insurance instead of selling only one type of contract; that is, life insurance and pure endowment hedge each other partially. Pure endowments partially hedge life insurance when the mortalities of the two underlying groups of insureds are positively correlated. By assuming that all the individuals buying either a pure endowment or a life insurance policy share the same hazard rate, we are imposing a type of positive correlation. (The mortalities of the individuals are independent given the hazard rate, but the hazard rates are identical.) If, instead of assuming the hazard rates are identical, we were to assume that the hazard rates are positively correlated via the Brownian motions driving the diffusions, then we would also obtain that pure endowment serves as a hedge to life insurance. The greater the positive correlation, the greater the hedge. Proving this result would add complexity to our current model via the addition of a state variable for the second hazard rate. We chose to keep the model relatively simple to demonstrate the basic idea. We invite the interested reader to pursue this model. We also found that the relative price, Pm,n+1−Pm,nPm,n+1−Pm,n, for a pure endowment satisfies 0≤Pm,n+1−Pm,n≤hF.0≤Pm,n+1−Pm,n≤hF. Turn MathJax on Here FF is the price of a zero-coupon bond and h≤1h≤1 is a risk-adjusted probability of survival; see Eq. (3.13). Similarly, the relative price of a life insurance, Pm+1,n−Pm,nPm+1,n−Pm,n, satisfies 0≤Pm+1,n−Pm,n≤F.0≤Pm+1,n−Pm,n≤F. Turn MathJax on In future work, we plan to investigate the limits, View the MathML sourcelimn→∞limm→∞(Pm,n+1−Pm,n)andlimm→∞limn→∞(Pm+1,n−Pm,n), Turn MathJax on which are the relative prices of pure endowment and life insurance contracts, respectively, given that the insurer has sold arbitrarily many of these contracts (the saturation prices). We are also interested in determining the corresponding rates of convergence to see how well these limits might approximate the prices of contracts in real life.