Stochastic approximations in CBD mortality projection models
(2016) Journal of Computational and Applied Mathematics — Vol. 296, p. 102-115 (2016)
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- Abstract
- Actuarial calculations for life annuity portfolios need to account for the stochastic nature of decrements, including the systematic longevity risk coming from the unknown underlying life table and the theoretically diversifiable risk of random fluctuations around this life table. Deriving the exact distribution for the present value of life annuity payments, for instance, requires extensive simulations or numerical evaluations. Accurate approximations avoiding such time-consuming procedures are thus useful in insurance practice and this is precisely the topic of the present paper. In the mortality projection model proposed by Lee and Carter [1], conditional survival probabilities, given the time index future trajectory, are complicated functions of the single time index. As there is no analytical expression available for their distribution function, Denuit and Dhaene [2] used comonotonicity to approximate the sums of strongly correlated LogNormal random variables playing a central role in this framework. Expanding on this approach, Denuit [3] derived analytic approximations for the quantiles of the life annuity payments conditional expected present value given the time index. This is made by supplementing the comonotonic approximations for the conditional survival probabilities worked out in [2] with a second approximation of the same type for the life annuity conditional expected present value, given the time index. Denuit, Haberman and Renshaw [4] further studied the quality of these approximations, allowing for general AutoRegressive Integrated Moving Average (ARIMA) models instead of the simple random walk with drift adopted in the majority of papers using Lee–Carter methodology. These works allow the actuary to accurately quantify the systematic longevity risk in large portfolios. To make the approach also applicable to small or medium-size portfolios, Gbari and Denuit [5] developed accurate approximations for the numbers of survivors up to given ages and for the present value of the payments made in favor of a group of annuitants.
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Gbari, K. Y. A. S., & Denuit, M. (2016). Stochastic approximations in CBD mortality projection models. Journal of Computational and Applied Mathematics, 296, 102-115. https://doi.org/10.1016/j.cam.2015.09.020 (Original work published 2016)