## Why use survival models?

We and our clients much prefer to analyse mortality continuously, rather than in yearly intervals like actuaries used to do in previous centuries. Actuaries normally use μx to denote the continuous force of mortality at age x, and qx to denote the yearly rate of mortality. For any statisticians reading this, μx is the continuous-time hazard rate.

We are sometimes asked why we prefer using μx, to which the lazy answer would be that this is what the CMI Technical Standards Working Party recommends, and it is how the the CMI has graduated all its tables since the early 1990s. Using μx to model mortality has a number of advantages, but here we will illustrate the simplest one.

One immediate advantage of modelling μx is that it allows each and every piece of data to contribute to the model. In contrast, modelling qx involves throwing away data where the policyholder could not have completed a full year of exposure. To illustrate, consider the data in Table 1 below:

Table 1. Data available for μx and qx modelling.  Source: Small annuity portfolio of a UK life office, 2004–2006.

Data available for μx Data available for qx
Age Lives Time lived Deaths Lives Time lived Deaths
604,8043,528.5324,0543,185.631
614,5723,440.9394,3883,065.438
624,2853,040.9334,0872,635.633
633,8022,731.9483,6792,671.948
643,6602,668.2443,5442,614.544
655,8224,336.6475,2254,051.244

Table 1 shows that the requirement for a full year's exposure for the qx model reduces the data available for any mortality investigation.  The differences may appear modest — 16% fewer lives at age 60, for example — but one should always seek to use all available information if possible.

Assume we have a random variable, $$X$$, with expected value ... Read more