“Beauty lies in the eye of the beholder.” To claim that a mathematical model be elegant or even graceful may seem immodest. Please bear with me …
Mortality models based on Hermite splines can surely claim simplicity. At the same time, they can express the key features of mortality in remarkable richness. With only few parameters, they capture the age shape of the mortality curve over the full adult age range. Their flexibility allows Hermite spline models to distinguish between a variety of risk factors in a natural, and indeed graceful way.
In sport, the ability to transmit power in a seemingly effortless way is often described as graceful. Star athletes are admired for attaining a level of skill which even amateurs can recognize. In a similar way, the smooth mortality curve à la méthode Hermite conveys meaning without undue complexity; an advantage gained from the parsimony of the model, in which each parameter bears meaning and can be interpreted.
Figure 1: CPM2024 combined mortality rates for male and female retirees and suriving spouses
Referring to the chart in Figure 1, the differences between male and female mortality rates at the beginning and end of the age range are each governed by a single parameter. For different risk groups, the difference in mortality is typically greatest around age 50. The Hermite spline framework exploits this by characterizing the difference between retirees and surviving spouses with only three parameters, which can be interpreted as the level and slope of the force of mortality at age 50 and an adjustment centred around age 80. All three can differ for widows and widowers.
Figure 2: Applying splines to mortality
The parameters of the age component of log(\mu_x) are shown in Figure 2 and all have an intuitive interpretation:
- \alpha (“alpha”) is the log force of mortality at the beginning of the age range, here age 50.
- \omega (“omega”) is the log force of mortality at the end of the age range, here age 110.
- m_0 is the gradient of the mortality curve at age 50.
- \eta (“eta”) is a shape parameter with greatest effect mid-age range, here age 80.
The age range can be extended naturally to younger ages using only one or two additional parameters. Figure 3 shows four parameters describing younger age mortality, of which \omega(young) and m_1(young) coincide with the parameters \alpha and m_0 for the pensioner age range shown in Figure 2.
Figure 3: Extension to younger ages
Ageing gracefully, an objective which is becoming increasingly important to a growing demographic group, ultimately depends on how mortality rates evolve over time. The study of mortality trends, until now usually a separate field of research and modelling, can be included within the development of base tables by employing the frugal yet flexible Hermite spline model. In the recently published 2024 Canadian Pensioner Mortality Research Project it was possible to measure how the model parameters that govern the shape of the mortality curve changed over time when calibrated to the data pool collected by the Canadian Institute of Actuaries.
The Canadian pensioner mortality tables CPM2024 are the first industry tables to have been derived using the Hermite spline mortality model, which was first proposed by Richards (2020). While mathematically similar to the approach of the Continuous Mortality Investigation of the Institute and Faculty of Actuaries, the Hermite spline method gives the coefficients meaning and thus makes predictions more powerful.
Capturing mortality curves and mortality dynamics within the same model framework is a substantial step towards being able to measure and understand the ageing process, thanks to Charles Hermite and his spline basis. Find out more about Hermite spline mortality models from the CIA’s CPM2024 Research Report and check out how the COVID-19 pandemic was considered when measuring the trends.
