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
60 4,804 3,528.5 32 4,054 3,185.6 31
61 4,572 3,440.9 39 4,388 3,065.4 38
62 4,285 3,040.9 33 4,087 2,635.6 33
63 3,802 2,731.9 48 3,679 2,671.9 48
64 3,660 2,668.2 44 3,544 2,614.5 44
65 5,822 4,336.6 47 5,225 4,051.2 44

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.

Model types in Longevitas

Longevitas users can choose between seventeen types of survival model (μx) and seven types of GLM (qx). In addition there are a further seven extensions of the GLM models for qx to span multi-year data without violation of the independence assumption. Longevitas also offers non-parametric analysis, including Kaplan-Meier survival curves and traditional A/E comparisons against standard tables. 

Previous posts

Ahead of the curve

In an earlier post we looked at the implications for savers of the historically low interest rates in the UK. Low interest rates are a policy response to the unusual economic conditions in which the developed world currently finds itself.
Tags: Filter information matrix by tag: interest, Filter information matrix by tag: yield curve

Survival models for actuarial work

The CMI recently asked for an overview note on survival models.  Since this subject is of wider actuarial interest, we wanted to make this publically available.
Tags: Filter information matrix by tag: CMI, Filter information matrix by tag: survival models, Filter information matrix by tag: mortality

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