Laying down the law

In actuarial terminology, a mortality "law" is simply a parametric formula used to describe the risk.  A major benefit of this is automatic smoothing and in-filling for areas where data is sparse.  A common example in modern annuity portfolios is that there is often plenty of data up to age 75 (say), but relatively little data above age 90.

For example, if we use a parametric formula like the Gompertz law:

log μx = α + βx

then we can use a procedure like the method of maximum likelihood to estimate α and β.  Once we have these values, we can generate mortality rates at any age we require, not just the ages at which we have data.

But which mortality law should one use?  In a recent paper (Richards, 2012) I outlined the structure of sixteen different survival models However, this is only a small subset of potential laws — what if you want to use a formula which isn't offered by your software package?  For example, you might want to try a quadratic extension to the Gompertz law:

log μx = α + β1x + β2x2

which is part of the Gompertz-Makeham family used by the CMI in its mortality graduations.  Fortunately, things are relatively straightforward when using the log-likelihood function directly.  Where the mortality law is too seldom used to be worth carrying out all the necessary mathematics, we can find the maximum-likelihood estimates with numerical differentiation.  This allows us to quickly fit and compare the Gompertz and "Quadratic Gompertz" models above, the results of which are shown in Table 1:

Table 1. Comparison of simple Age*Gender models for mortality in a large annuity portfolio.

ModelParametersAICImprovement
over Gompertz
Gompertz 4 385,322 -
Quadratic Gompertz 5 385,292 30
Perks 4 385,265 57

Table 1 shows that the Quadratic Gompertz model has led to an improvement when measuring the fit using Akaike's Information Criterion.  This means that the improvement in fit was large enough to justify the extra parameter, β2.

However, it is often the case that a simpler model can fit even better.  By way of illustration, we can also fit the same Age*Gender model using the logistic formula proposed by Perks (1932):

log [μx / (1-μx)] = α + βx

for 0 < μx < 1.  Perks' law has the same number of parameters as the Gompertz law, yet Table 1 shows that it fits better than either of the two Gompertz variants.

References

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

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Stephen Richards is the Managing Director of Longevitas
User-specified laws in Longevitas
Longevitas contains an expression evaluator for those who wish to use their own formulae for survival models or qx models.  A file of one or more formulae is uploaded and automatically validated before use.  The configuration option User-defined laws file allows you to choose which one of a number of alternative files to use.  To fit the model, simply select either "Survival(user law)" or "GLM(user law)" from the drop-down list.  Longevitas will use numerical differentiation to find the maximum-likelihood estimates for your parameters and also to estimate their standard errors.