Posts Tagged ‘Risk’
The House of Modern Finance – The House of Cards
A Book Review of The (Mis)behaviour of Markets by Benoit Mandelbrot, Basic Books, 2004, 328 pages.
In the early 20th century the French mathematician Louis Jean-Baptiste Alphonse Bachelier (March 11, 1870 – April 28, 1946) came up with an original idea of how one could portray asset prices in mathematical formulas. Like flipping a coin, his idea was that the evolution of asset prices could be conceived of as purely random events, with a 50-50 chance of ticking up or ticking down as the time passes.
Today, the same fundamental building block constitutes the foundation of modern financial theory, although in a slightly different flavour, now called Brownian motion. The idea is simple. Imagine a drunk, walking from a point A to some unknown point B in the future. With every step he takes his memory goes blank. He doesn’t know where he’s going or remember where he came from. If you trace his path as he walks, and add the criterion that the size of his steps is given by the Gaussian distribution, you’ll end up with something similar to a Brownian motion.
Today, the efficient market hypothesis, CAPM, the Black Schole’s option formula and pretty much every other tool in the financial engineering toolbox, all relies on this particular foundation.
Orthogonal to this view is the polish born mathematician, Benoit Mandelbrot. The (Mis)behaviour of markets is the chronicle of his long lasting pursuit of understanding the financial markets. Instead of Brownian motion and Gaussian distribution, Mandelbrot base his views on fractal geometry, a mathematical branch he himself originated. Fractal comes from the Latin word “fractus” meaning broken. The idea is that a shape is broken down into smaller shapes, each echoing the large. Fractals can be found in many places in nature, like the British coast line, branches on a tree, or parts of a rock. The analogy to asset prices is the similarity between different frequencies of stock market data. We know that daily observations look very similar to lower frequency data, such as monthly observations. Hence the self similarity property of the parts to the whole.
Picture of the Mandelbrot Set.
By observing actual asset price dynamics Mandelbrot sought to create a realistic model of real life price movements. Instead of a Gaussian distribution, his research shows that asset returns exhibit so called “fat tails”. Furthermore he argues convincingly that asset prices experience both “long term dependence”, and “volatility clusters”, all assumed to be nonexistent by the now orthodox house of modern finance.
The fat tail distribution can perhaps best be explained by a gedanken experiment taken from the book The Black Swan written by Nicholas Nassim Taleb, an intellectual descendent of Benoit Mandelbrot. Imagine a population of 1000 people. When it comes to their length and weight its hard to imagine one additional person that could affect the average, other than by an incremental amount. We just don’t se people 20 meters tall or 5000 kilograms heavy. The occurrence of extreme events – that is far from the average – becomes increasingly rare the further from the mean we move. To borrow Mandelbrot’s phrasing, the physical attributes of height and weight don’t scale.
Now instead, imagine that the average of personal income is computed – instead of height or weight – over the same population. What happens to the average income if, lets say Bill Gates is added to the group of people. Clearly, he will not only affect the total income, but also to a large extent the average of the population.
While the Gaussian distribution appropriately describes the former, evidently income is of a completely different nature. Mandelbrot chose to call this characteristic by the name, wild randomness, in contrast to mild randomness. Inspired by Vilfredo Pareto, by using so called power laws, Mandelbrot manage to find the equivalent of the Pareto 80-20 rule for asset prices. According to Vilfred Pareto and the principle of factor sparsity, for many event 80 percent of the effects come from 20 percent of the causes. For example, Pareto showed the empirical tendency that eighty percent of Italians income went to twenty percent of the population.
By looking at historical data for cotton prices, Mandelbrot in an early essay showed that the price of cotton also displayed a pattern, much similar to the one Pareto found for income, and could be characterised by wild randomness. Instead of diminishing as observations went further away from the average, he argues that asset returns scale. According to the Gaussian distribution an event like the stock market crash 1987 could only occur once every billion billion years! Governed by the mathematics of power laws, the scaling property can be described as follows:
In market terms, a power law distribution implies that the likely hood of a daily or weekly drop exceeding 20 % can be predicted from the frequency of drops exceeding 10 %, and that the same ratio applies to a 10 % vs a 5 % drop. Thus the fat tails, and hence, the chance for the 1987 crash was certainly not one in a billion billion. [1]
Another of his contribution is the occurrence of long term dependence and volatility clusters in asset prices. According to the orthodox school, none of these phenomenons are neither supposed to happen (none the less be a major determinant of future events). For example, studies by Eugene Fama, Mandelbrot’s student, estimated the importance of long term dependence. He found that about 10 percent of returns during an eight year period could be attributed to the previous eight year period.
Mandelbrot’s theory of volatility cluster is one of his later achievements. He calls it multifractal time. The idea is as follows: Some days the market are “slow” while other days the market moves “fast” When the time is “fast” lots of large deviations occur in a short period of time, hence volatility clusters. Mandelbrot’s theory combines a fractal process with “clock time” in order to produce a process with what he “calls trading” time. In his words,
Price is a function of trading time, in turn trading time is a function of clock time.[2]
To sum up, why bother with such things as fat tails, long term dependence fractal time etc… The recent financial market crises currently in full swing should act as a fresh reminder. The (Mis)behaviour of market is a book about market risk, not Gaussian risk, but real risk. The type of risk that tells you the potential loss of your portfolio; not Betas, Alphas, CVAR, Sigmas etc… but real life probabilities, that is, at least if you want to believe Benoit Mandelbrot.
The (Mis)behaviour of markets is an important book. The current financial turmoil has exposed some of the troubles and internal weaknesess in present day models. The onset of the sub prime crises in the fall of 2007, is a schoolbook example of how events can turn bad, fast. Modern financial theory has shown unable to accomplish what it claims. While the modern financial house of cards is crumbling, practitioners are looking towards new alternatives. Time will tell if the pendulum once again swings in the direction of fractal geometry and Benoit Mandelbrot.
Sources:
[1] How The Financial Gurus Get Risk All Wrong: Benoit Mandelbrot and Nicholas Nassim Taleb, Fortune 2005
[2] The (Mis)behaviour of Markets: Benoit Mandelbrot, Basic Books 2004
