Stephen Richards

The survivor function from age \(x\) to age \(x+t\), denoted \({}_tp_x\) by actuaries, is a useful tool in mortality work. As mentioned in one of our earliest blogs, a basic feature is that the expected time lived is the area under the survival curve, i.e. the integral of \({}_tp_x\). This is easy to express in visual terms, but it often requires numerical integration if there is no closed-form expression for the integral of the survival curve.

In a previous blogs I have looked at seasonal fluctuations in mortality, usually with lower mortality in summer and higher mortality in winter.  The subject of excess winter deaths is back in the news, as the UK experienced heavy mortality in the winter of 2014/15, as demonstrated in Figure 1.

When making projections and forecasts, it can be instructive to compare them with what actually happened. In December 2002 the CMI published projections of mortality improvements that incorporated the so-called "cohort effect" (CMIB, 2002). These projections were in use by life offices and pension schemes in the United Kingdom from 2003 onwards.

In a recent posting I looked at the calculation of percentiles and quantiles, which underpin many calculations for ICA and Solvency II. Simply put, an \(\alpha\)-quantile is the value which is not expected to be exceeded \(\alpha\times 100\)% of the time. This value is denoted \(Q_{\alpha}\). Mathematically, for a continuous random variable, \(X\), and a given probability level \(\alpha\) we have:

$$\Pr(X\leq Q_\alpha)=\alpha$$