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 μ_x to denote the continuous force of mortality at age x, and q_x to denote the yearly rate of mortality. For any statisticians reading this, μ_x is the continuous-time hazard rate.

We are sometimes asked why we prefer using μ_x, 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 μ_x to model mortality has a number of advantages, but here we will illustrate the simplest one.

One immediate advantage of modelling μ_x is that it allows each and every piece of data to contribute to the model. In contrast, modelling q_x involves throwing away data where the policyholder could not have completed a full year of exposure. To illustrate, consider the data below from a life office which wanted to investigate anti-selection at retirement ages between 60 and 65 over the 2004-2006 period:

As you can see, the requirement for a full year's exposure for the q_x model has reduced the data available, especially at the key retirement ages of 60 and 65. At age 60 especially, the q_x model has less than half the exposure time to drawn upon and under two-thirds of the deaths. Since this office was interested in anti-selection at retirement, the ability to use more of the available data was a key reason to prefer modelling μ_x instead of q_x.

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