Frailty and Risk Classification for a Life Annuity Portfolio

Pools of voluntary life annuities typically report lower mortality rates than the general population, as when the product is quoted at standard rates, it is attractive mainly for healthy people. In order to expand their portfolio, in recent years some insurers have started offering higher annuity rates to those whose health conditions are critical. The portfolio can then become larger, but also more heterogeneous. While a larger size implies an improved pooling effect, a higher risk profile follows from the increased heterogeneity. We intend to investigate the result of this trade-off.In performing this analysis, there is a second, and possibly, more important issue that we address. In actuarial practice, mortality rates for life annuities are usually obtained from a population (projected) life table. The different mortality level arising from the (self-)selection process is represented by applying adjustment coefficients to the population mortality rates. According to common actuarial practice, such coefficients are chosen empirically, calibrated on the average ratio (possibly measured for age-groups) between the annuitants' and the population mortality rates; conversely, a model formally justifying such a difference is not adopted.The heterogeneity of a population in respect of mortality can formally be described with a frailty model. The main findings of the model for a population as a whole are well-known. We suggest to adopt the 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. Conditional probability distributions for the frailty are obtained for each risk class, which allow us to describe the different levels of mortality of the various groups. Different values for the annuity rate derive from the different assumptions about the frailty level of the specific risk class.Finally, the value of insurer's liabilities is investigated, in terms of expected value and dispersion of the probability distribution of the present value of future benefits. Portfolios without and with risk classification are compared, so to measure the trade-off between portfolio size and heterogeneity