Abstract
Life annuities are attractive mainly for healthy people. In order to expand their business, in recent years, some insurers have started offering higher annuity rates to those whose health conditions are critical. Life annuity portfolios are then supposed to become larger and more heterogeneous. With respect to the insurer’s risk profile, there is a trade-off between portfolio size and heterogeneity that we intend to investigate. In performing this, there is a second and possibly more important issue that we address. In actuarial practice, the different mortality levels of the several risk classes are obtained by applying adjustment coefficients to population mortality rates. Such a choice is not supported by a rigorous model. On the other hand, the heterogeneity of a population with respect to mortality can formally be described with a frailty model. We suggest adopting a frailty model for risk classification. We identify risk groups (or classes) within the population by assigning specific ranges of values to the frailty within each group. The different levels of mortality of the various groups are based on the conditional probability distributions of the frailty. Annuity rates for each class then can be easily justified, and a comprehensive investigation of insurer’s liabilities can be performed.
Highlights
The largely unanticipated dynamics of mortality experienced in the latest decades suggests developing new mortality models and to further explore those already known
A critical aspect that we focus on in this paper is understanding how the heterogeneity of the population with respect to mortality does impact on the liabilities of a life annuity provider
While models accounting for observable risk factors generally adopt pragmatic solutions to represent differential mortality, the modeling of heterogeneity due to unobservable risk factors has found an elegant and rigorous solution in the concept of frailty, described already by [1], but formally defined by [2]
Summary
The largely unanticipated dynamics of mortality experienced in the latest decades suggests developing new mortality models and to further explore those already known. Adopting models used to represent differential mortality arising from observable risk factors, in common actuarial practice, the higher or lower mortality level of a risk group is represented by applying adjustment coefficients to the population mortality rates. Such coefficients are chosen empirically, calibrated on the average ratio (possibly measured for age groups) between the annuitants’ and the population mortality; a model formally justifying such a difference is not adopted.
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