Anything But Normal: An Underrated and Practically Ignored Economic Discovery
Fractals, Models, and Fun Times
Statistics, at first glance, seems like a relatively simple subject. Sure, you may need some mathematical chops, but in most professional settings, quantitatively versed individuals will have no problems applying principles of statistics to reality. Friends, this is a lie. In this blog post, I will profile some of the work of mathematician Benoit Mandelbrot in distributions of asset prices. I will then talk through some of the implications of this work, how you may be able to apply these principles in your work.
Benoit Mandelbrot: An Introduction
Benoit Mandelbrot was born on November 20th, 1924. He was born to a Lithuanian-Jewish family in Poland. He studied at the Ecole Polytechnique in France, then received a Master’s degree from the prestigious California Institute of Technology, before eventually earning a PhD from the University of Paris in Mathematical Sciences. His work would span many different fields, including information theory, linguistics, geometry, and economics.
The Variation of Certain Speculative Prices (1963)
In 1963, Mandelbrot published a landmark economics paper that caused a great deal of controversy at the time. This paper refuted the commonly held belief that changes of asset prices followed a random walk of a Gaussian distribution. He refuted this notion by demonstrating empirical evidence from Cotton prices in the 20th century that show changes too wide to be found in a normal distribution. He then identifies the distribution of this time series and talks through some of the implications.
Mandelbrot begins this paper by discussing the model contributed by Louis Bachelier (1870-1946) to analyze changes in asset prices. This model argues stock price changes follow the log transformation of a normal stochastic process. A stochastic process can be thought of as a sequence of random variables, in this case, normal random variables. Below is the equation demonstrating Bachelier’s model for price changes in the stock market.
In this equation, Z(t) is the price of the asset at a given time and L(t, T) follows a normal distribution.
Here, I grabbed stock price data from apple to demonstrate this model. According to Bachelier, the graph of log price changes should look like the graph on the left. In reality, it looks like the one on the right.
At first glance, the actual data does not look to terrible concerning. It actually looks more predictable than the simulated data. Looks, however, can be deceiving. Kurtosis is a statistical term used to measure the spread of a distribution, colloquially how thin or fat tailed the distribution (or dataset) is. The sample kurtosis for a normally distributed sample converges to 3 asymptotically. Anything more than 3 is considered “fat tailed.” The sample kurtosis of the dataset from Apple’s stock price is 7.839195. This means using a normal distribution would drastically underestimate the number of extreme values from either end of the distribution. Clearly, a normal distribution is not suitable to modelling the log-price changes in stock market prices.
The Stable Pareto Distribution
Enter, the stable pareto distribution family. A distribution can be said to be “stable” if a linear combination of i.i.d (independently and identically distributed) these distributions results in that same distribution. Below are examples of stable distribution
Essentially, Mandelbrot realized the distributions of these market changes were not normally distributed, and therefore could not be modeled using normal distributions. So, stochastic simulations of market events are possible, just not using the normal distribution. The tails of these stock price changes, however, can be modeled using a stable pareto distribution according to Mandelbrot.
The Problem That Remains
The issue that remains is that while this information is seemingly well known in academic circles, and relatively easy to verify, financial convention has not yet integrated these findings into risk management. The idea of using a Cauchy distribution to model stock price changes (which is what is recommended in the book The Physics of Wall Street) is still foreign to much of mainstream finance. We have a sort “good enough” idea of truth in relation to our models where we sacrifice truth on the altar of convenience and power of interpretation. Consider the fact that VaR (value at risk) models (which often use a normal distribution) are still in use despite their absolute failure in measuring financial risk over the past 20 years.
Please, do not sacrifice truth at the altar of scientific pretension.
For more on this topic, you may comment if you would like the (very basic) R code. Also check out this article on Towards Data Science about the topic.
Are Stock Returns Normally Distributed? | by Tony Yiu | Towards Data Science