Life Insurance Mathematics with Random Life Tables

Abstract

When the insurer sells life annuities, projected life tables incorporating a forecast of future longevity must be used for pricing and reserving. To fix the ideas, the framework of Lee and Carter is adopted in this paper. The Lee-Carter model for mortality forecasting assumes that the death rate at age x in calendar year t is of the form exp(αx + (βxKt), where the time-varying parameter Kt reflects the general level of mortality and follows an ARIMA model. The future lifetimes are all influenced by the same time index Kt in this framework. Because the future path of this index is unknown and modeled as a stochastic process, the policyholders' lifetimes become dependent on each other. Consequently the risk does not disappear as the size of the portfolio increases: there always remains some systematic risk that cannot be diversified, whatever the number of policies. This paper aims to investigate some aspects of actuarial mathematics in the context of random life tables. First, the type of dependence existing between the insured life lengths is carefully examined. The way positive dependence influences the need for economic capital is assessed compared to mutual independence, as well as the effect of the timing of deaths through Bayesian credibility mechanisms. Then the distribution of the present value of payments under a closed group of life annuity policies is studied. Failing to account for the positive dependence between insured lifetimes is a dangerous strategy, even if the randomness in the future survival probabilities is incorporated in the actuarial computations. Numerical illustrations are performed on the basis of Belgian mortality statistics. The impact on the distribution of the present value of the additional variability that results from the Lee-Carter model is compared with the traditional method of mortality projection. Also, the impact of ignoring the dependence hat arises from the model is quantified.