Plotter for Hermite age-related trend

This interactive page shows how Hermite interpolation can allow for age-related time trends in log(mortality). Time trends usually only have a weak signal in the short exposure periods typically found in insured portfolios. It is therefore important that the basic model for age-related time trends is simple and robust, i.e. that it can reliably capture the essence of the nature of age-related changes in mortality without being too easily distorted. The most basic formulation used here has four parameters:

  1. x2, the minimum age below which mortality is constant in time,
  2. x3, the maximum age above which mortality is also constant in time,
  3. TrendPeakAge, the age at which mortality change peaks, and
  4. TrendPeak, the peak value of mortality change at that age.
Age log(mortality) © www.longevitas.co.uk

We assume that mortality improvements will start at zero below age x2, peak at TrendPeakAge, then decline to zero at age x3. We scale the age, x, depending on which side of TrendPeakAge it falls:

v = (TrendPeakAge - x) / (TrendPeakAge - x2), if x < TrendPeakAge, or

v = (x - TrendPeakAge) / (x3 - TrendPeakAge), if x ≥ TrendPeakAge.

The age-related adjustment to log(mortality) at calendar time y in respect of trend is then:

h01(v)×TrendPeak×(y-2000), if x < TrendPeakAge, or

h00(v)×TrendPeak×(y-2000), if x ≥ TrendPeakAge.

This four-parameter minimal formulation assumes that mortality improvements are zero above the advanced age x3, which is reasonable. Similarly, mortality improvements at or below x2 are also assumed to be zero; while this is strictly speaking unlikely, it is a robust approach to there usually being little data at the youngest ages to reliably detect a time trend. For data sets with plenty of data at age x2, though, an optional fifth parameter allows a non-zero trend at the lowest ages.

Mandatory parameters:
x2 Lower bound of age range, often set by the analyst as the same minimum age for generating rate tables, i.e. x2=x0.
x3 Age above which trend becomes constant at zero. x3 will be set in configuration at a value less than x1, probably at around age 105-110.
TrendPeakAge Age at which trend peaks. Unlike all other x-values, this will be estimated from the data.
TrendPeak Peak trend value at TrendPeakAge.

For very large data sets, or for longer exposure periods, there is an optional parameter for controlling the rate of change at the youngest ages:

Optional parameters:
TrendYoungest Trend at age x2 (and lower ages). Default to zero when there is too little data to establish any kind of trend.

Finally, for experimental purposes only there are three further optional parameters for controlling the level and shape of change at the oldest ages:

Experimental parameters not usually used:
TrendGradientYoungest Initial gradient of trend at age x2. Best left at zero in most cases.
TrendOldest Trend value at age x3 and above. Usually assumed to be zero due to very limited data.
TrendGradientOldest Gradient of trend as x3 is approached. Best left close to zero.
This age-related time trend is added to the basic age pattern for log(mortality). One can also add selection effects or cyclic seasonal effects. For full details of the model and its implementation, see Richards (2019).