## Division of labour

At this time of year insurers have commenced their annual valuation of liabilities, part of which involves setting a mortality basis.  When doing so it is common for actuaries to separate the basis into two components: (i) the current, or period, mortality rates and (ii) the projection of the future path of mortality rates (usually mortality improvements).  This sub-division is carried over into the regular Solvency II assessment of capital requirements, where there is always a minimum of two sub-risks for longevity:

1. Mis-estimation risk, i.e. the uncertainty over the current level of mortality.
2. Trend risk, i.e. the uncertainty over the future direction of improvements.

In practice a Solvency II assessment of longevity risk will contain more than this.  However, the above two components will be common to every Solvency II capital calculation for longevity risk — Richards, Currie & Ritchie (2012) list some further sub-risks that might be considered.

The separation of mis-estimation risk from trend risk is much more than an actuarial habit.  In fact, the nature of the investigations and calculations are different in almost every respect.  For example, a mis-estimation assessment is done using a portfolio's own experience data, where various portfolio-specific risk factors might be included.  Mis-estimation capital is therefore highly specific to a given portfolio — Richards (2014) gives two examples.

In contrast, few portfolios have a long enough time series of data to use in assessing trend risk, so population data are commonly used.  In theory this could make all industry participants in a country produce identical trend-risk assessments.  Table 1 gives a comparison of the features of mis-estimation risk and trend risk.

Table 1. Contrasting features of mis-estimation risk and trend risk.

FeatureMis-estimation riskTrend risk
Data source Portfolio's own experience. Population data.
Nature of data Individual lives. Grouped counts.
Procedure Estimation. Forecasting.
Model fit Critical relevance. Often poor, but this is not always relevant to forecast quality.
Fit assessment AIC, BIC, $$\chi^2$$ test.  Bootstrapping absolutely essential. AIC, BIC, $$\chi^2$$ test.  No amounts data, so no bootstrapping required.
Example models Makeham, Perks, Beard. Lee-Carter, Cairns-Blake-Dowd, Age-Period-Cohort.
Risk factors Age, gender and pension size as minimum, but often with many more portfolio-specific factors. Usually only age, gender and year of birth.
Reference Richards (2014). Richards, Currie & Ritchie (2012).

Although there are many differences between mis-estimation risk and trend risk, two particular aspects of trend risk stand out:

1. Basis risk, i.e. risk arising from using data other than that of the portfolio.  A mis-estimation assessment cannot be fully credible if it doesn't use the portfolio's own experience data.  In contrast, almost any assessment of trend risk includes basis risk because few portfolios have a long enough history of data of their own.
2. Model risk, i.e. risk arising from not knowing which (if any) model is the correct one to use.  With mis-estimation risk it is usually fairly straightforward to find a suitable mortality law to use, albeit work is required to find out which risk factors should be included.  In contrast, goodness-of-fit is not usually a useful decision-making criterion for selecting a projection model and the risk factors available are usually very few.  Cairns et al (2009) list some qualitative selection criteria for selecting a forecasting model for trend risk.

One consequence of the forced inclusion of basis risk and model risk is that a lot more actuarial judgement is required for setting capital for trend risk. As a result, trend risk and mis-estimation risk will probably always have to be handled separately.

References

Cairns, A. J. G., Blake, D., Dowd, K., Coughlan, G. D., Epstein, D., Ong, A. and Balevich, I. (2009) A quantitative comparison of stochastic mortality models using data from England and Wales and the United States, North American Actuarial Journal, 13(1), 1–35.

Richards, S. J., Currie, I. D. and Ritchie, G. P. (2012) A value-at-risk framework for longevity trend risk, British Actuarial Journal, 19(1), 116–167 (including discussion).

Richards, S. J. (2014) Mis-estimation risk: measurement and impact, British Actuarial Journal, 21(3), 429–475 (including discussion).

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Stephen Richards is the Managing Director of Longevitas