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The Actuary The magazine of the Institute & Faculty of Actuaries

Infinite variance

How many actuaries are familiar, I wonder, with distributions of infinite variance and their properties? The prevalence of terms such as variance, volatility, beta, and correlation (all typically relying on finite second moments) suggests the answer might be not many.

However, market statistics often display features such as fat tails and skewness that persist over time in a manner that does not fit easily (or possibly at all) in the world of finite variance, the central limit theorem and the normal (Gaussian) distribution. Other distributions of infinite variance, such as Cauchy and the L-stable family, do naturally display such characteristics though many are much less tractable analytically than the cosy world of Gauss.

For instance:

  • They do not give rise to Ito processes, so standard Black-Scholes methodology for formulating and solving equations for derivative prices breaks down (as do binomial methods). Consequent theories of dynamic hedging of risks using derivatives may also fail.
  • Variability of total returns doesn’t scale as ‘root N’ as the timeframe lengthens, but as a higher power – possibly as high as ‘N’. Extreme events aren’t necessarily averaged out.
  • Sample standard deviations from historical data (which, of course, are always finite and so potentially wrong by a factor of infinity) may not be a good basis for estimating future variability and risks. Ignoring the tails of the distribution could be very misleading.

All this has implications for, inter alia, investment decisions, asset allocation, and analysis of risks. Do current actuarial models capture the likely size, frequency, or impact of relatively extreme events? If they don’t, what risks might they obscure?