Modeling mortality and pricing life annuities with Lévy processes

We consider the pricing of annuity-due under stochastic force of mortality. Similarly to Renshaw et al. (1996) and Sithole et al. (2000), the force of mortality will be defined using an exponential function of Legendre polynomials. We extend the approach of Ballotta and Haberman (2006) by conditionally adding α-stable Lévy subordinators in the force of mortality. In particular, we focus on the Gamma and Variance-Gamma processes in order to show how Lévy subordinators can capture mortality shocks. Generalized Linear Models is used to estimate coefficients of the explanatory variables and the Lévy process. For this purpose, the coefficients of the process are obtained by maximizing the log-likelihood function. We use the mortality data of males in Japan from 1998–2011 and the U.S. from 1965–2010 in order to compare our results with the model proposed by Renshaw et al. (1996). Some preferences are indicated based on Akaike’s information criterion, Bayesian information criterion, likelihood ratio test and Akaike weights to support the proposed model. We then use a cubic smoothing spline method to fit the interest rate curve and illustrate some over (under) estimations in the prices of annuities under the structure suggested by Renshaw et al. (1996).