Mildenhall’s Actuarial Geometry

Stephen J Mildenhall Actuarial Geometry Proc RTS 2006

This recommends the use of a Levy-process based model for insurance risk rather than simply adding a contingency and catastrophe premium. The key difference is that a Levy process based model can reflect ‘drift’ in risk, such as changes in flood risk due to climate change, and does not assume that on large enough scales risk tends proportionately to zero. The paper models NAIC (the US National Association of Insurance Commissioners)) data. It finds that  the time-base for the Levy process is not the usual calendar time, but needs to allow for ‘systematic time-varying contagion effects, such as weather patterns, inflation and level of economic activity, affecting all insureds’. In this sense the abstract is a bit misleading: the model is not simply Brownian or Poisson, but has a time-warp. Nor is it adequate to just model a ‘mixing variable’. A key finding is that, contrary to what has been widely supposed, risks do not tend to reduce proportionately with either volume of business or time.” Volume-related parameter risk, adjusted for company and pricing cycle effects, is shown to have a Laplace distribution—a surprising result.”


As in economics, there has been a lot of assumptions that have been claimed to be ‘mathematically based’, and those who doubt them have sometimes been criticised for being mathematically illiterate. In consequence those with experience of real systems (e.g. of economies or insurance) have tended to dismiss mathematical modelling as misleading and wrong. Here we are shown mathematical modelling being used in a positive sense, to construct models that support different conclusions and which fit the data better. I haven’t checked the details, but one needs to be aware of the dangers (risk?) of mis-interpreting the paper. The ‘old’ mathematical models have perhaps not been taken too seriously, and have been supplemented by other types of reasoning. It would be a mistake to take this new model too seriously, even if it passes more extensive theoretical validation. It is only a mathematical model, albeit a much better one, and needs to be treated with appropriate caution. It should also be noted that the time-warping provides a substrate which affects the risk. This seems reasonable for insurance but less so for economics, for example. But we can still apply the model to draw limited conclusions about economies.

The main conclusion that I think we can draw is not that ‘Life is a Levy process’ but that we should not simply assume a ‘washing out’ of risk with volume. With this insight we should look to the drivers of the ‘subordinator’ so that changes are not a random surprise, but something that we can engage with.

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