Plotter for Hermite continuous amount effect

Benefit amounts are first transformed onto the [0,1) interval; see Richards (2020). Modelling for amounts-related mortality then uses three of the four Hermite basis functions to smoothly transition from the smallest pension to the largest. As is usual for mortality work, we operate on a logarithmic scale, so the values displayed below are additions to log(mortality) at age x0. The effect of benefit amount will automatically decrease with increasing age, as is standard for the Hermite model family.

Transformed amount (u) Amount effect added to log(mortality) © www.longevitas.co.uk

At its most basic the Hermite approach to continuous amounts mortality requires only two parameters and only one of the Hermite functions, h01, to describe a smooth curve for the adjustment to log(mortality) for transformed pension size. Leaving aside the transform parameter, there is just one mandatory parameter:

Mandatory parameters:
AmountUltimate Value for log(mortality) at age x0 for an infinite benefit amount.

The Hermite spline h10 controls the initial direction of the adjustment to log(mortality) as it leaves the smallest (zero) pension amount, while h11 controls the shape of the adjustment to log(mortality) as it approaches the ultimate effect, AmountUltimate. This gives rise to two optional parameters for the continuous amounts adjustment for log(mortality):

Optional parameters:
AmountGradientInitial Initial gradient of amounts effect with zero pension.
AmountGradientUltimate Gradient of amounts effect approaching infinite pension.
In addition to amounts-based mortality, one can also add an age-related time trend, selection effects and seasonal variation.

References

Richards, S. J. (2020) Modelling mortality by benefit amount, Longevitas working paper.