A State-Space Estimation of the Lee-Carter Mortality Model and Implications for Annuity Pricing

Abstract

A common feature of retirement income products is that their payouts depend on the lifetime of policyholders. A typical example is a life annuity policy which promises to provide benefits regularly as long as the retiree is alive. Consequently, insurers have to rely on best estimate life tables, which consist of age-specific mortality rates, in order to price these kind of products properly. Recently there is a growing concern about the accuracy of the estimation of mortality rates since it has been historically observed that life expectancy is often underestimated in the past (so-called longevity risk), thus resulting in longer benefit payments than insurers have originally anticipated. To take into account the stochastic nature of the evolution of mortality rates, Lee and Carter (1992) proposed a stochastic mortality model which primarily aims to forecast age-specific mortality rates more accurately. The original approach to estimating the Lee-Carter model is via a singular value decomposition, which falls into the least squares framework. Researchers then point out that the Lee-Carter model can be treated as a state-space model. As a result several well-established state-space modeling techniques can be applied to not just perform estimation of the model, but to also perform forecasting as well as smoothing. Research in this area is still not yet fully explored in the actuarial literature, however. Existing relevant literature focuses mainly on mortality forecasting or pricing of longevity derivatives, while the full implications and methods of using the state-space representation of the Lee-Carter model in pricing retirement income products is yet to be examined. The main contribution of this article is twofold. First, we provide a rigorous and detailed derivation of the posterior distributions of the parameters and the latent process of the Lee-Carter model via Gibbs sampling. Our assumption for priors is slightly more general than the current literature in this area. Moreover, we suggest a new form of identification constraint not yet utilised in the actuarial literature that proves to be a more convenient approach for estimating the model under the state-space framework. Second, by exploiting the posterior distribution of the latent process and parameters, we examine the pricing range of annuities, taking into account the stochastic nature of the dynamics of the mortality rates. In this way we aim to capture the impact of longevity risk on the pricing of annuities. The outcome of our study demonstrates that an annuity price can be more than4% under-valued when different assumptions are made on determining the survival curve constructed from the distribution of the forecasted mortality rates. Given that a typical annuity portfolio consists of a large number of policies with maturities which span decades, we conclude that the impact of longevity risk on the accurate pricing of annuities is a significant issue to be further researched. In addition, we find that mis-pricing is increasingly more pronounced for older ages as well as for annuity policies having a longer maturity.