Fat-tailed distribution

A fat-tailed distribution, also called a heavy-tailed distribution, gives extreme values a markedly higher probability than the Gaussian bell curve does: very large losses stay rare, but far less improbable than intuition suggests. Solvency II, Article 101: the SCR is calibrated to a 99.5% one-year VaR, the 1-in-200 loss. Under a normal distribution that quantile sits 2.58 standard deviations from the mean; under a fat-tailed distribution it sits much further out, and the capital number moves with it. Formally, the probability of exceeding a level x decays like a power of x, as x^−α, rather than exponentially as it does for the normal. The smaller the tail index α, the fatter the tail. Natural catastrophes and systemic cyber events follow this kind of distribution, as do market crashes. An underwriter who ignores it underprices and under-reserves. Mean and standard deviation turn misleading, pulled around by a handful of values; for a Pareto distribution with a tail index of 2 or less the variance is infinite, and at 1 or less so is the mean. Fat tails often get confused with high kurtosis. A leptokurtic distribution has more mass in the center and at the extremes than a normal one without necessarily decaying as a power law; the lognormal is heavy-tailed in the broad sense yet keeps all its moments finite. The telltale sign, in the fattest distributions, is that a single loss can weigh as much as all the others combined. So the work is done on the tail itself, with Tail Value at Risk rather than VaR alone, which sets a threshold and says nothing about what lies beyond it, and actuaries fit the tail with extreme value theory, a generalized Pareto distribution above a high threshold and a Hill estimator for the index. Nassim Taleb popularized the picture of Extremistan in The Black Swan (2007), where one event can dominate the whole series. Cyber most likely lives there, though the data does not yet let anyone fit the tail with confidence, which is why its insurability is still argued over.