Normal distributions vs. power laws: why the tail matters for risk

A normal distribution produces a bell curve: outcomes bunch around an average and extreme values fall off very quickly. A power law has no typical size. Small outcomes are most likely, but large ones remain far more probable than a bell curve would allow. For risk, the difference lies in the tails, because a model built on the wrong shape can badly understate how often severe losses occur. [1] [2]

According to [reactionwheel.net](https://reactionwheel.net/2015/06/power-laws-in-venture.html), normal distributions are common because of the Central Limit Theorem: many independent random inputs feeding one outcome tend to produce a normal curve. Height is the usual example. Finance theory leans heavily on normal or lognormal assumptions for prices and returns, and these underpin insurance and modern portfolio theory. [1]

Other processes behave differently. The same source notes that earthquakes, cities and wars have no typical size and are power-law distributed. The curve is written as p(x) = Cx^-alpha, where alpha sets the shape. Power laws have fat tails, meaning the curve falls off much more slowly than a normal one, and a smaller alpha means a heavier right tail. [1]

A paper on power laws in economics compares a normal distribution with a Student's t-distribution with three degrees of freedom, both set to a 3% daily standard deviation. Typical moves seen every ten or hundred days are similar. But the one-in-1,000 event is twice as large under the power law, and the one-in-10,000 event is three and a half times as large. The authors say a risk manager should take this seriously. [3]

The same paper applies this to the 21% S&P 500 fall of October 19, 1987. Under the normal hypothesis its probability is below 10^-16, while under the power law it is 0.08%, so it is to be expected. A risk-focused source, [riskhub.org](https://riskhub.org/blogs/tail-risk-explained), describes the 1987 crash as roughly a 20-sigma event under a normal model and says real markets produce 4-sigma and 5-sigma daily moves several times per decade. It also notes that Value at Risk says nothing about how bad the worst 1% of days are. [3] [4]

Choosing the wrong model has costs in both directions. Hubbard Research says using a lognormal when the data fit a power law could underestimate losses by potentially astronomical amounts, while using a power law for lognormal data could overprioritize small risks. [5]

The economics paper says it is not controversial that many economic data sets have fat tails. The debate is over how fat. Early work by Mandelbrot and Fama reported alpha below 2, which would mean the variance does not exist, but most later work reports alpha above 2. The authors say real differences between assets probably exist, though the tail exponent is hard to measure with reliable error bars and remains debated. [3]

The same paper says power-law behavior is clearer at short time scales and in large datasets. It argues the hypothesis can give more efficient estimates of extreme risk than non-parametric methods such as the bootstrap, which tend to underestimate risk outside the observed sample. [3]