## Analysis of VaR-iance

In recent years we have published a number of papers on stochastic mortality models. A particular focus has been on the application of such models to longevity trend risk in a one-year, value-at-risk (VaR) framework for Solvency II. However, while a small group of models has been common to each paper, there have been changes in the calculation basis, most obviously where updated data have been used. Sometimes these changes stemmed from more data being available, but, as Richard Willets covered in his blog, the ONS also restated the population estimates following the 2011 census. This makes it tricky to compare results between papers. We therefore thought it would be instructive to do a step-by-step analysis of the impact of these data changes over the years.

For the purposes of illustration we will use the projection model from Lee & Carter (1992):

\[\log\mu_{x,y} = \alpha_x + \beta_x\kappa_y\]

where \(\mu_{x,y}\) is the force of mortality at age \(x+\frac{1}{2}\) in the middle of year \(y\), and \(\alpha_x\), \(\beta_x\) and \(\kappa_y\) are parameters to be estimated by the method of maximum likelihood. We will smooth both \(\alpha_x\) and \(\beta_x\) to reduce the risk of forecast rates crossing over at adjacent ages. As Iain showed in his recent blog, the choice of identifiability constraints does not impact the projections. Table 1 shows the impact of the changing data on the best-estimate annuity reserve at age 70, discounting at 2% p.a in each case.

Table 1. Best-estimate annuity reserve at age 70 in 2011. Source: Own calculations using mortality-experience data for males in England & Wales and United Kingdom.

Description | Country | Period | Males | Females |

Before 2012 restatement of population estimates. |
England & Wales |
19612010 |
13.01 |
14.74 |

England & Wales |
1971-2010 |
13.14 |
14.66 |

After 2012 restatement of population estimates. |
United Kingdom |
1971-2010 |
12.94 |
14.56 |

Comparing the second and third rows of Table 1 shows the expected impact of the restated population estimates: with lower population estimates at older ages, improvements were lower and so the forecast annuity factors are also lower. Part of the drop between the second and third rows also comes from including the higher mortality of Scotland and Northern Ireland, but England & Wales account for around 88% of deaths and thus dominate the data.

What about the impact on the value-at-risk (VaR) capital requirements? Table 2 shows these, expressed as a percentage of the best-estimate annuity factor at age 70. For consistency the random-number seed was reset at the start of each run to ensure that the same sequence of pseudo-random numbers was used.

Table 2. 99.5% one-year, value-at-risk (VaR) capital requirements as percentage of best-estimate annuity reserve at age 70. Source: Own calculations using mortality-experience data for males in England & Wales and United Kingdom. 5,000 simulations according to the procedure outlined in Richards, Currie & Ritchie (2014).

Description | Country | Period | Males | Females |

Before 2012 restatement of population estimates. |
England & Wales |
19612010 |
3.293.56% |
2.042.23% |

England & Wales |
1971-2010 |
2.312.58% |
2.742.82% |

After 2012 restatement of population estimates. |
United Kingdom |
1971-2010 |
2.172.45% |
2.412.83% |

United Kingdom |
19712015 |
2.332.43% |
2.212.37% |

Comparing the first and second rows of Table 2 shows that the biggest impact of all the data changes comes from dropping the period 19611970 (this period is often discarded due to inconsistencies in the population estimates). However, the change has gone opposite ways for males and females: dropping the data from the 1960s reduces the male VaR capital, but increases it for females.

Comparing the second and third rows of Table 2 shows substantial overlap for the 95% confidence intervals for the VaR capital requirement. This suggests that the restatement of population estimates has not had much impact on the VaR capital, at least for the Lee-Carter model. If this seems surprising, the answer lies in the nature of the VaR calculation: over a single year the dominant driver of variation is random error, rather than parameter risk. This point is covered in detail in Kleinow & Richards (2016), who consider the respective roles of volatility and parameter uncertainty in VaR capital requirements. Finally, comparing the third and fourth rows of Table 2 shows a clear decrease in VaR capital for females arising from the lower mortality improvements since 2010, with a less-clear picture for males; the confidence intervals for both genders have narrowed as well.

While this analysis shows a broad tendency for VaR capital requirements for longevity trend risk to reduce over time in the UK, it also shows sensitivity to the choice of date range. The one-year, value-at-risk methodology is prescribed by the likes of Solvency II, but VaR capital requirements for longevity trend risk do not change in an easily predictable way.

**References**

Kleinow, T. and Richards, S. J. (2017) Parameter risk in time-series mortality forecasts, *Scandinavian Actuarial Journal*, 2017(9), pages 804828.

Lee, R. D. and Carter, L. (1992), Modeling and forecasting US mortality, *Journal of the American Statistical Association*, **87**, 659671.

Richards, S. J., Currie, I. D. and Ritchie, G. P. (2014) A value-at-risk framework for longevity trend risk, *British Actuarial Journal*, **Vol. 19, Part 1**, pages 116167 (with discussion).

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