## 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:

- Mis-estimation risk, i.e. the uncertainty over the current level of mortality.
- 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.

Feature | Mis-estimation risk | Trend 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:

- 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.
- 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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