Stephen Richards

Mortality at post-retirement ages has three apparent stages:

Happy New Year to all our readers!

Last week I presented at Longevity 14 in Amsterdam. A recurring topic at this conference series is index-based approaches to managing longevity risk. Indeed, this topic crops up so reliably, one could call it a hardy perennial.

Regular readers of this blog will know that we are strong advocates of the benefits of modelling mortality in continuous time via survival models. What is less widely appreciated is that a great many financial liabilities can be valued with just two curves, each entirely determined by the force of mortality, \(\mu_{x+t}\), and a discount function, \(v^t\).

Examining residuals is a key aspect of testing a model's fit. In two previous blogs I first introduced two competing definitions of a residual for a grouped count, while later I showed how deviance residuals were superior to the older-style Pearson residuals. If a model is correct, then the deviance residuals by age should look like random N(0,1) variables.

Assume we have a random variable, \(X\), with expected value \(\eta\) and variance \(\sigma^2\). Often we find ourselves wanting to know the expected value and variance of a function of that random variable, \(f(X)\). Fortunately there are some workable approximations involving only \(\eta\), \(\sigma^2\) and the derivatives of \(f\). In both cases we make use of a Taylor-series expansion of \(f(X)\) around \(\eta\):

\[f(X)=\sum_{n=0}^\infty \frac{f^{(n)}(\eta)}{n!}(X-\eta)^n\]