Approximations for life annuity contracts in a stochastic financial environment

Abstract

In the traditional approach to life contingencies only decrements are assumed to be stochastic. In this contribution we consider the distribution of a life annuity (and a portfolio of life annuities) when also the stochastic nature of interest rates is taken into account. Although the literature concerning this topic is already quite rich, the authors usually restrict themselves to the computation of the first two or three moments. However, if one wants to determine, e.g. capital requirements using more sophisticated risk measures like Value-at-Risk or Tail Value-at-Risk, more detailed knowledge about underlying distributions is required. For this purpose, we propose to use the theory of comonotonic risks developed in Dhaene et al. [Dhaene, J., Denuit, M., Goovaerts, M., Kaas, R., Vyncke, D., 2002a. The concept of comonotonicity in actuarial science and finance: theory. Insur. Math. Econ. 31 (1), 3–33; Dhaene, J., Denuit, M., Goovaerts, M., Kaas, R., Vyncke, D., 2002b. The concept of comonotonicity in actuarial science and finance: applications. Insur. Math. Econ. 31(2), 133–161], which has to be slightly adjusted to the case of scalar products. This methodology allows to obtain reliable approximations of the underlying distribution functions, in particular very accurate estimates of upper quantiles and stop-loss premiums. Several numerical illustrations confirm the very high accuracy of the methodology.