ترکیب محصول بهینه در شرکت های بیمه عمر با استفاده از ارزش مشروط در خطر

This paper proposes a Conditional Value-at-Risk Minimization (CVaRM) approach to optimize an insurer’s product mix. By incorporating the natural hedging strategy of Cox and Lin (2007) and the two-factor stochastic mortality model of Cairns et al. (2006b), we calculate an optimize product mix for insurance companies to hedge against the systematic mortality risk under parameter uncertainty. To reflect the importance of required profit, we further integrate the premium loading of systematic risk. We compare the hedging results to those using the duration match method of Wang et al. (forthcoming), and show that the proposed CVaRM approach has a narrower quantile of loss distribution after hedging—thereby effectively reducing systematic mortality risk for life insurance companies

Over the past decade, a longevity shock has spread across human society. Benjamin and Soliman (1993), McDonald et al. (1998), Grundl et al. (2006) and Stallard (2006) confirm that unprecedented improvements in population longevity have occurred worldwide. The decreasing trend in the mortality rate has created a great risk for insurance companies. The existing literature has proposed a number of solutions to mitigate the threat of longevity risk to life insurance companies. These solutions can be classified into three categories. The capital market solutions include mortality securitization (see, for example Dowd, 2003, Lin and Cox, 2005, Cairns et al., 2006a, Blake et al., 2006a, Blake et al., 2006b and Cox et al., 2006), survivor bonds (e.g. Blake and Burrows, 2001 and Denuit et al., 2007), and survivor swaps (e.g. Dowd and Blake, 2006 and Dowd et al., 2006). These studies suggest that insurance companies can transfer their exposures to the capital markets. Cowley and Cummins (2005) provide an excellent overview of the securitizations of life insurance assets and liabilities. The second set of solutions, the industry self-insurance solutions, include the natural hedging strategy of Cox and Lin (2007), the duration matching strategy of Wang et al. (forthcoming), and the reinsurance swap of Lin and Cox (2005). The advantages of these solutions are that the hedging does not require a liquid market and can be arranged at a lower transaction cost. Insurance companies can hedge longevity risk by themselves or with counterparties. The third kind of solution, known as mortality projection improvement, provides a more accurate estimation of mortality processes. As Blake et al. (2006b) propose, these studies fall into two areas: continuous-time frameworks (e.g. Milevsky and Promislow (2001), Dahl (2004), Biffis (2005), Schrager (2006), Dahl and Moller (2006)) and discrete-time frameworks, e.g., Brouhns et al. (2002), Renshaw and Haberman (2003) and Cairns et al. (2006b). Parameter uncertainty and model specification in relation to the mortality process have also attracted more attention in recent years. Among the industry self-insurance solutions, the natural hedging strategy suggests that life insurance can serve as a hedging vehicle against longevity risk for annuity products. Wang et al. (forthcoming) employ duration as a measure of the product sensitivity to mortality change, and propose a mortality duration matching (MDM) approach to calculate the optimal product mix. Their work, however, is based on several restrictive assumptions. First, they assume that future mortality changes involve parallel shifts in the mean, and do not measure the higher-order moments of the mortality risk distribution. Second, the MDM approach applies to only two products. Third, the MDM approach is a pure risk-reduction method because the profit loading is not considered during the hedging procedure. Fourth, Cairns (2000), Melnikov and Romaniuk (2006) and Koissi et al. (2006) suggest that parameter risk is crucial when dealing with longevity risk. The parameter uncertainty does not play a role in the MDM approach, since Wang et al. (forthcoming) consider the mortality shift only in terms of its mean. To overcome these problems, we employ the two-factor stochastic mortality model of Cairns et al. (2006b) and construct the Conditional Value-at-Risk Minimization (CVaRM) (Dowd and Blake, 2006) approach to control the possible loss. Managing products risk with parameter uncertainty is one feature of the CVaRM approach. The other feature is that we add the profit-loading constraint into the optimization. The premium-pricing principle suggested by Milevsky et al. (2006) is employed to estimate the required profit loadings, i.e., in order to compensate the stockholders bearing systematic mortality risk with the same Sharpe ratio as other asset classes in the economy.1 Furthermore, the CVaRM approach could be easily implemented using linear programming (Rockafellar and Uryasev, 2000), and insurance companies could adopt it as their own internal risk-management tool. The results of our simulation reveal that the proposed CVaRM approach yields a less dispersed product distribution after hedging and so effectively reduces systematic mortality risk for life insurance companies. The MDM approach, on the other hand, has a limited effect on the dispersion of the product distribution. In addition, the CVaRM approach considers not only risk reduction but also the required profit constraint. We found that the required loading substantially changes the optimal product mix and so cannot be ignored. The remainder of this article is organized as follows: Section 2 outlines the models, including the mortality model of Cairns et al. (2006b), the duration matching model of Wang et al. (forthcoming), the loading estimation of Milevsky et al. (2006) and the CVaRM approach. In Section 3 we introduce the mortality data, project future mortality and design the products. Section 4 presents the numerical examples in two scenarios: the two-product scenario without a required loading constraint and a multiple-product scenario with a required loading constraint. The hedging results of the MDM and CVaRM approaches are also compared in this section. Conclusions and implications are contained in Section 5.

This article proposes a new approach to optimize the insurer’s product mix under systematic mortality risk. By incorporating the natural hedging strategy of Cox and Lin (2007), the two-factor stochastic mortality model of Cairns et al. (2006b), and the Sharpe ratio-loading price of Milevsky et al. (2006), we construct a CVaRM approach to evaluate the product mix. We consider two numerical scenarios: the two-product case without a loading constraint and the multi-product case with a loading constraint. In the first scenario, the CVaRM approach exerts a better risk-reduction effect than the MDM approach. In the second scenario, the three-product example reveals a trade-off between the CVaR and the required loadings. The results show that the proposed CVaRM approach leads to an optimal product mix and effectively reduces the mortality risks associated with forecasting longevity patterns for life insurance companies. Some important issues for future research and practice clearly deserve further investigation. First, this paper deals with the parameter risk, but ignores the misspecification or modeling risk. For example, the real mortality process may not follow the CBD model. Second, this paper omits the basis risk of the mortality rate between life insurance and annuities because of the data limitations. Our numerical example assumes that the mortality processes for life insurance and annuities are the same. In fact, the mortality experiences may differ for these products. Third, in this study, the premium loadings for each product are decided individually by means of the Sharpe ratio. To maintain rigidity, they should be priced according to their contributions to the aggregated risk, in a way similar to the beta concept of the Capital Asset Pricing Model (CAPM). This work is beyond the scope of this paper, and so we leave it for future study. Finally, we illustrate the hedging strategy with a mortality term structure, but a flat interest-rate yield curve. An analysis of the combined effects of stochastic mortality and stochastic interest rate would offer more realistic results.