A handbook of parametric survival models for actuarial use

Abstract

Traditional actuarial techniques for mortality analysis are being supplanted by statistical models. Chief amongst these are survival models, which model mortality continuously at the level of the individual. An assumption of a mathematical form for the hazard function or, equivalently, the assumption of a continuous distribution for an individual's lifetime, leads automatically to smooth fitted mortality rates. This note gives an overview of the survival models commonly found in statistical packages and compares their suitability for actuarial work with the mortality ‘laws’ proposed by actuaries over the past two centuries. We find that the actuarial laws provide substantially better fits at post-retirement ages. We also give a common structure of parameterisation which gives consistent behaviour and interpretation of risk factors across all 16 survival models listed here. Finally, we consider the benefits of working directly with the log-likelihood function, including making allowance for the left truncation which is common for the data with which actuaries work.

Keywords:

• Survival models
• Mortality laws
• Left truncation

Acknowledgements

The author thanks Dr. Iain Currie, Professor Angus Macdonald and Andrew Howe for helpful comments. Any errors or omissions remain the sole responsibility of the author. Data validation and preparation were done using Longevitas, which was also used to fit all the models listed. Graphs were done in R.