Annuity Uncertainty with Stochastic Mortality and Interest Rates

Abstract

Risk analysis in actuarial science has shifted its focus from diversifiable risk to systematic risk in the last 20 years or so. This article contributes further in this direction by proposing the concept of annuity rate to take account of systematic risk inherent in annuity products. The annuity rate is the conditional expectation of the annuity’s future payments, given the future paths of mortality and interest rates. We provide an empirical study to investigate the impact of the two systematic risk factors on the distribution of the annuity rate. In particular, we adopt the Lee-Carter and the Cairns-Blake-Dowd models for mortality risk, and the one-factor and two-factor CIR models for interest risk. Monte Carlo simulation is used to provide numerical illustrations of sensitivity analysis of the annuity rate and of risk assessment of a guaranteed annuity option.

Acknowledgments

This work was supported in part by the Natural Sciences and Engineering Research Council of Canada. The author thanks Bruce Jones for his valuable feedback during the writing of the article. The author also thanks anonymous reviewers for their helpful comments.

Notes

Our second case is different from the “no longevity risk” case discussed in Dowd et al. (2011)(see p. 242), where the mortality rates are kept unchanged in their simulation, excluding both mortality improvement and uncertainty in the improvement. As a result, the corresponding annuity values from this case are significantly lower than the values from the other two cases, leading to a different conclusion in their paper. However, we think allowing mortality improvement to be accounted is a more relevant scenario for our purpose.

FIGURE 1 Annuity Rate Fan Charts under LC Mortality Model and One-Factor CIR Interest Model. (color figure available online)