Shared Compound Frailty Model With Application in Joint Life Annuity Insurance

Abstract

Observable risk factors (e.g., health condition) can explain heterogeneity in mortality among assureds; but modeling the risk profile of a heterogeneous life from unobservable risk factors, such as genetics is complex. This limits the application of only reported rating classes adopted for underwritten annuities. Insurance firms routinely disregard unreported risk factors perhaps because of difficulties in modeling. Although a number of research has been done in univariate frailty modeling to account for unobserved risks, the widely applied frailty mixture is the gamma. One major drawback of the gamma is that it is time-invariant. The scientific interest of the study is to account for time-varying heterogeneity using compound processes. For single-life insurance contracts, the non-central gamma compound process is suggested with the generalized exponential and generalized Weibull baselines to account for time-varying frailties and carry out valuations. On the other hand, grouping insureds in clusters such as joint annuities imposes statistical dependence between lifetimes. The dependence is a result of an unreported risk factor called the frailty that represents a weighted sum of shared lifestyles on mortality risk of group members. In fact, standard insurance valuation considers independence when pricing joint-life products. Different approaches to dependence modeling have been proposed in literature. However, these models consider separately either only the negative effects of dependence alone or positive effect of dependence. The study further proposes to apply the shared compound frailty approach in valuation of joint annuities to address time-varying heterogeneity effects positively and negatively associated with dependence. The positive stable distribution used entails the frailty distribution with the weighted exponential, generalized exponential and weighted Weibull as the base force of mortality distributions. In this study, Bayes inference based on Gibbs sampling is used to calibrate the base force of mortality distributions using a large Kenyan insurer term