# Don't fear the integral!

Actuaries denote with \({}_tp_x\) the probability that a life alive aged exactly \(x\) years will survive a further \(t\) years or more. The most basic result in survival analysis is the following relationship with the instantaneous mortality hazard, \(\mu_x\):

\[{}_tp_x = e^{-H_x(t)}\qquad(1)\]

where \(H_x(t)\) is the *integrated hazard*:

\[H_x(t) = \int_0^t\mu_{x+s}ds\qquad(2).\]

It is at this point that cortisol levels spike for some people, often because it is a long time since they last saw an integral. In the case of actuaries, this can lead them to seek safety in the comfort of standard tables based on \(q_x\). However, even if you have forgotten everything you ever knew about integration, equation (2) poses no hurdle for mortality work. The reason is that \(H_x(t)\) is just a function, and Richards (2008, Table 5) has a list of explicit formulae for the seven most common mortality models used by actuaries. For example, assume that you want to integrate the Gompertz mortality hazard \(\mu_x=\exp(\alpha+\beta x)\). Table 5 of Richards (2008) has this formula to use:

\[H_x(t) = \frac{(e^{\beta t}-1)}{\beta}e^{\alpha+\beta x}\qquad(3).\]

Thus, the integrated hazard between age 60 and 80 with \(\alpha=-12\) and \(\beta=0.12\) is 0.6874017. No integration is required because it has already been done. For the more adventurous, Richards (2012, Table 2) lists formulae for \(H_x(t)\) for a further ten models.

But what if you have an unusual mortality model that isn't listed? Then \(H_x(t)\) can be approximated numerically using R's \(\tt integrate\) function. The following snippet of R code illustrates this using the same Gompertz model as an example:

```
dAlpha = -12.0
dBeta = 0.12
GompertzHazard = function(x)
{
return(exp(dAlpha+dBeta*x))
}
integrate(GompertzHazard, 60, 80)
```

If you paste the above into an R console you will see output like this:

`0.6874017 with absolute error < 7.6e-15`

which is the same result using the closed-form expression in equation (3). Use \(\tt help(integrate)\) within your R console to find out how to fine-tune your control over this function. And if your IT department won't let you use R, then there are some surprisingly accurate approximations available for integration; see also Appendix D of Macdonald et al (2018).

So, with explicit formulae freely available for most mortality models - and accurate approximations for all others - you can use survival models without ever having to work out a single integral yourself. Unless you really want to, that is.

**References: **

Macdonald, A. S., Richards, S. J. and Currie, I. D. (2018) Modelling Mortality with Actuarial Applications, *Cambridge University Press*, ISBN 978-1-107-04541, doi 10.1017/9781107051386.

Richards, S. J. (2008) Applying survival models to pensioner mortality data, *British Actuarial Journal*, **14(2)**, pages 257-303, doi 10.1017/S1357321700001720.

Richards, S. J. (2012). A handbook of parametric survival models for actuarial use, *Scandinavian Actuarial Journal*, **2012(4)**, pages 233–257, doi 10.1080/03461238.2010.506688.

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