# Pension size as a factor

In a previous blog I showed that there was often a statistically significant link between pension size and mortality.  It is clearly necessary to account for such a link in an actuarial mortality model, not least because people with larger pensions account for a disproportionate share of portfolio risk.

Pension size is essentially a continuous covariate, and a simple approach is to discretise, i.e. create non-overlapping ranges.  We sort the pensions within a portfolio, then define break-points such that equal numbers of lives fall into each band (or as close to equal as we can achieve).  Each life can then be assigned a size-band as a risk factor in the same way that they can have a gender; the only difference is that size-band is an ordinal factor, whereas gender is a categorical factor.  Table 1 shows the model fits for a pension scheme using various numbers of size-bands, where the fit is measured by Akaike's Information Criterion (AIC) .

Table 1. AIC using selected numbers of size-bands in a mortality model. Source: Richards (2020), ENG portfolio for a local-authority pension scheme in England.

Number of
size-bands
Lives per
size-band
AIC
2 42,193 150,179
3 28,128 150,129
4 21,094 150,085
5 16,876 150,042
10 8,437 149,973
20 4,208 149,972

Table 1 shows that some material improvements in fit are possible, but that there is little to be gained from using more than ten size-bands.  One problem with Table 1 is that many of the size-bands have similar mortality levels (not shown), so a natural question is whether we can improve the fit by using bands with unequal numbers of lives?

An alternative approach to size-bands with equal numbers of lives would be to optimise the breakpoints by starting with 20 bands (say) and merging adjacent bands with similar mortality. We do this by searching for merges that produce the lowest AIC for a given target number of levels (targeting the BIC would produce the same result, as the number of parameters is held constant). The process is iterative and the time taken to consider all possible merges increases with both the target number of levels and the initial number of size-bands. For this reason, we perform an exhaustive search of breakpoints when dealing with 2 or 3 target factor levels, but avoid a combinatorial explosion by adopting a more limited time-saving search algorithm for targeting four or more factor levels. In our implementation we have further reduced run-times by using parallel processing to spread calculations over 63 threads (Butenhof, 1997). The results of this optimisation process are shown in Table 2.

Table 2. AIC from optimising the breakpoints between size-bands in a mortality model. Source: Richards (2020), ENG portfolio for a local-authority pension scheme in England.

Number of
size-bands
Lives in
highest-
income
size-band
AIC
2 12,648 150,019
3 12,648 149,994
4 12,648 149,997
5 12,648 149,986
6 4,208 149,962
7 4,208 149,962
8 4,208 149,962
9 4,208 149,963

Table 2 shows that, for a given number of size-bands, the model fit is materially improved compared with the same number of levels in the equally-sized bands of Table 1.  With lives distributed unequally across six size-bands in Table 2 we have a better fit than an equal distribution of lives over ten bands in Table 1.  A further refinement might be to optimise from (say) 100 initial discretised size-bands instead of 20, although this would substantially increase the run-time in searching for the optimal breakpoints.

Turning a real-valued variate like pension size into a discrete factor is a neat approach, but it has drawbacks:

1. Information loss.  Creating such a discrete factor throws away information, which is seldom a good thing.  Specifically, it introduces discretisation error — mortality might not necessarily be homogeneous within a range, and there will also be jumps in mortality when crossing range boundaries.  For example, the boundary between the 18th and 19th size-bands for the portfolio used here is £11,115 p.a.; this means that a hypothetical Pensioner A receiving £11,100 p.a. might be treated as having quite different pension-related mortality than Pensioner B with £11,200 p.a., despite the fact that they receive near-identical incomes.
2. Computation time.  Note in Table 2 the anomalous jump in AIC moving from three to four size-bands even though lower AICs are subsequently found for five and then six bands. It is possible to conclude you have found an optimal number of bands and breakpoints, when a more exhaustive (but far more time-consuming) search might have found a further improvement, as shown here.
3. Extrapolation.  It is not obvious how to extrapolate mortality effects to pension sizes beyond those observed in the data set, say when creating an actuarial pricing basis.

While discretisation is a handy simplification, actuaries would therefore ideally like to treat pension-related mortality on a continuous basis (Richards, 2020).

References:

Butenhof, D. R. (1997) Programming with POSIX threads, Addison-Wesley, Boston, ISBN 978-0-201-63392-4.

Richards, S. J. (2020) Modelling mortality by continuous benefit amount, Longevitas working paper.

Written by: Stephen Richards
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## Size-bands for amounts

Mortality and other risks can be modelled by discretised amount by selecting the SizeBand factor (or ReserveSizeBand if reserves have been uploaded).  The number of size-bands is controlled by the Benefit/Reserve Bands? setting under Advanced Options