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Life insurance has been one of the many options for people with concerns about the uncertainty of their future. For a couple of parents, for example, they worry about what the future will bring for their children should they die. They need some financial protection from this kind of adversity to ensure the well-being of their children. Life insurance is the answer to that, because it is designed to protect against the serious financial impact that results from an individual’s death (Chen et al., 2017). An important variation of the single life insurance is the survivorship life insurance (a.k.a. last-to-die) which covers two or more lives. Under such contract, a death benefit is paid out only on the last death. Survivorship life insurance is useful for a few different reasons according to the needs of the policy holder, some of them are to; protect a family business, preserve an estate, give to a charity or leave an inheritance. It is a popular insurance product, particularly in the affluent market (Chen et al., 2017).

 

Insurance valuation is a very important tool for many insurance companies. It is through valuation that an insurance company is able to know its financial status at some point in time so that the company would be able to meet its future obligations. The valuation of an insurance policy comprises of 2 main components, they are: rates of return and mortality assumption. Traditionally, actuaries assume a constant rates of return and an independent mortality assumption in valuing joint-life insurance for the sake of simplicity. However, there has been considerable interest in the actuarial literature in studying the use of stochastic interest rate models for insurance valuations, such as Beekman and Fuelling (1990) and Frees (1990). Moreover, some empirical studies about joint life have shown the possibility of modelling dependent mortality between individuals in the same group of an insurance policy, especially in the two-life case, which comprises of a husband and a wife. Denuit et al. (2001) argue that a husband and wife are exposed to similar risks, since they share common lifestyles and may encounter common disasters. Jagger and Sutton (1991) show that there is an increased relative risk of mortality following spousal bereavement. All these findings show the possibility of modelling a dependent mortality, which may have a significant impact on risk management for joint-life insurance policies (Ji et al., 2011, p.357).

 

The objective of this study is to present the mathematical expressions based on basic actuarial concepts that are useful to do a valuation of a portfolio of survivorship life insurance with stochastic rates of return and dependent mortality. The valuation will be conducted by the means of calculation of the expected value and the variance of the loss random variable of the portfolio, by assuming an AR(1) process and Frank’s Copula to model the rates of return and the dependent mortality of the lifetimes of the policy holders, respectively. The loss random variable of the portfolio is defined to be the net difference between the present value of future benefits and the present value of future premiums of the portfolio. A significant result has not been reached yet, but the expected results of this study would be the calculation results of the expected value and standard deviations of the loss random variable of the portfolio.

 

A brief summary of the overall work will be explained. First of all, we present the mathematical expressions of the expected value and variance of the loss random variable, the AR(1) process, and the model of Frank’s Copula. And then, for illustration purposes, we use a hypothetical non-homogeneous portfolio to illustrate the methodology previously presented. In this illustration, the portfolio consists of several groups of homogeneous policies. Each group has its own characteristics with respect to ages of the lives insured, benefit amount and policy term. For simplicity purposes, we assume all policies are issued at the same time. We focus to present the valuation method rather than the estimation of the model parameters. Hence, we use the parameters estimated in Frees et al. (1996) to complete the necessary calculations.

 

Since it is still a new topic in its infancy, the idea of this study is to mainly show the possibility to value a survivorship life insurance with stochastic rates of return and dependent mortality assumption. The models used in this study are in no way the best models and the authors recommend further research regarding this subject.

dependent mortality; life insurance valuation; loss random variable; stochastic rates of return; survivorship life insurance.