Stochastic assessment of special-rate life annuities

Special-rate life annuities offer customized annuity rates, based on the lifestyle or health status of the individual. Their main purpose is to encourage the annuity demand, which is still underdeveloped in many markets; as better annuity rates are quoted for individuals showing a higher mortality profile, the number of individuals attracted by life annuities could increase. Providers should then gain larger pool sizes; however, this is possibly matched by a greater heterogeneity of the pool, due to several risk classes defined by the annuity design. Heterogeneity emerges not only in terms of different life expectancies, but also in respect of the dispersion of the lifetime distribution; indeed, situations resulting in a lower life expectancy also show greater variability of the lifetime. As it is well-known, pooling effects are reinforced by the pool size, while they are weakened by its heterogeneity, with a possibly unclear impact on the overall longevity risk to which the provider is exposed. In this paper we investigate the longevity risk profile of an annuity pool consisting of several risk classes. We consider both the idiosyncratic and aggregate components of the risk, by modelling the random number of deaths and assuming a stochastic mortality dynamics. The heterogeneity of risk classes is represented alternatively in a deterministic and stochastic setting. Our conclusions are in line with similar findings discussed in the literature, but obtained in a deterministic framework. Results suggest that the longevity risk profile of the provider is not significantly undermined by a greater pool heterogeneity, with a prevalence of the aggregate component whatever the pool composition.