## 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 in Table 1 below:

Table 1. Data available for *μ*_{x} and *q*_{x} modelling. Source: Small annuity portfolio of a UK life office, 2004–2006.

| Data available for *μ*_{x} | Data available for *q*_{x} |

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 *q*_{x} 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.

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