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The Actuary The magazine of the Institute & Faculty of Actuaries

Distributions of infinite variance

I was pleased to see Mr Kalia’s response (letters, November) to my letter. Perhaps I might add some comments.

I (perhaps mistakenly) tend to assume that models are likely to be framed in terms of forces (of interest, returns, etc) – ie using ‘delta’, not ‘i’ in formulae – and that the distribution that matters is of the force (delta) of increment/decrement of the value being modelled. With such models, problems sometimes cited when ‘i <= -1’ (which I suspect is what Mr Kalia meant in his letter in referring to losses exceeding world GDP) disappear. This applies whatever distribution, normal or otherwise, is adopted.

Following on from the suggestion in my earlier letter, looking at various market data series, my tentative expectation is that fitted distributions will tend to be skew, with longer tails on the downside than the upside. A consequence is that the probability of extreme gains is lower (possibly much lower) than the probability of correspondingly extreme losses. This goes a long way to mitigate concerns over extreme upside events. While I accept that there is some point at which extreme gains may instinctively feel unrealistic, I find it difficult to see quite how one might determine an appropriate level to set a bound on a model. Perhaps it is the instinctive feel that boundaries exist (upwards or downwards) that should be challenged and not reinforced by use of normal distributions – ‘six sigma’ thinking is for financial ostriches only!

Finally, I suspect that models are to be preferred that go further to model and capture downside risks than those based on normal distributions and that the use of such models should be in considering behaviour in adverse conditions rather than planning how to react to unanticipated windfalls.