## Seasoned analysis

The importance of seasonal analysis was underscored by a recent letter form the UK insurance regulator. In a previous blog, I looked at quarterly seasonal variation in a portfolio of defined-benefit pensions, and in a more recent blog I looked at monthly seasonal variation in mortality in England & Wales. However, rather than split observation periods into ever-smaller units, why not analyse seasonal mortality continuously? And if we can model it continuously, can we detect seasonal variation in *any* portfolio?

At this point you could be forgiven for thinking that modelling seasonal mortality were irrelevant for underwriting pension-scheme buy-outs or longevity swaps. After all, any fluctuating effects will even out over the long-term for a portfolio of pensions in payment. However, there are some subtle potential benefits of allowing for seasonal variation:

**Enhancement**, i.e. the phenomenon whereby the inclusion of one statistically significant risk factor could improve the explanatory power of the other factors in the model.
**Flexibility**. Allowing for seasonal effects liberates the analyst from ensuring that an exposure period has equal numbers of each season. This is particularly useful when analysing portfolio experience for a bulk annuity or longevity swap, where the supplied data often cover a non-integral number of years and it is important to use all available data.

We can allow for seasonal effects in continuous time by adding a simple cyclic term to any existing model for mortality, \(\mu_{x,r,y}\), as follows:

\[\log\mu^*_{x,r,y} = \log\mu_{x,r,y} + e^\zeta\cos\left(2\pi(y-\tau)\right)\qquad (1)\]

where \(x\) denotes age, \(r\) denotes duration and \(y\) denotes calendar time. \(\tau\) represents the proportion of the year after 1st January when mortality peaks and where \(e^\zeta\) is the peak additional mortality at that point. There is an interactive online tool here to demonstrate this.

Since equation (1) is an addition to \(\log\mu_{x,r,y}\), the mortality hazard will be multiplied by a seasonal factor that fluctuates smoothly and continuously around 1. Table 1 shows the estimated seasonal mortality parameters for different portfolios in Northern Europe, where mortality tends to peak between late December and early February (seasonal effects are shifted by six months in the Southern Hemisphere):

Table 1. Seasonal peak mortality for selected international portfolios. Source: Richards (2019) plus own calculations.

Country | Portfolio nature | \(\hat\zeta\) | \(\hat\tau\) | Peak as % of average | Peak time of year |

Scotland |
Pension scheme |
-1.88 |
0.082 |
117% |
Jan 30th |

UK |
Insurer annuities |
-2.00 |
0.001 |
114% |
Jan 1st |

Canada |
Pension plan |
-2.04 |
0.109 |
114% |
Feb 9th |

England |
Pension scheme |
-2.02 |
0.071 |
114% |
Jan 26th |

Netherlands |
Pension scheme |
-2.25 |
0.052 |
111% |
Jan 20th |

France |
Insurer annuities |
-2.42 |
0.066 |
109% |
Jan 25th |

USA |
Pension plan |
-2.63 |
0.074 |
107% |
Jan 27th |

Table 1 shows that seasonal effects can be consistently detected in almost any portfolio — the Scottish pension scheme in the first row has fewer than 18,000 lives, while the UK annuity portfolio in the second row has just four years of experience. Seasonal effects are strong, hence they are relatively easy to detect using equation (1). Indeed, we recently went further in Richards et al (2020) and showed how seasonal variation increases with age, and is larger for low-income pensioners.

**References**

Richards, S. J. (2019) A Hermite-spline model of post-retirement mortality, *Scandinavian Actuarial Journal*, DOI: 10.1080/03461238.2019.1642239.

Richards, S. J., Ramonat, S. J., Vesper, G. and Kleinow, T. (2020) Modelling seasonal mortality with individual data, *Scandinavian Actuarial Journal*, DOI: 10.1080/03461238.2020.1777194.

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