Born to be Wild(ly Random)
The (Mis)Behavior of Markets: A Fractal View of Risk, Ruin, and Reward
by Benoit B. Mandelbrot and Richard L. Hudson (Basic Books, 2004, $27.50)By Curtis Gary Dean
The title of this book provides an accurate description of its contents. Financial models often make the assumption that goods or security price changes are independent and normally distributed. The authors argue that price changes are much "wilder" than predicted by these models and are not truly independent. The authors refer to randomness that can be modeled by a normal distribution as "mild" randomness, but claim that markets actually demonstrate a "wild" randomness that cannot be modeled using the well-behaved normal. How can one model "wild" randomness? Use Mandelbrot's fractal geometry.
The name Mandelbrot should be familiar to many actuaries. He is credited with the invention of fractal geometry, a mathematical tool with applications in a variety of areas including image compression, computer animation, markets, and chaos theory, to name a few. Most readers have probably seen beautiful graphical representations of the Mandelbrot set. Benoit Mandelbrot is a mathematician with broad interests. If you look at his Web page at Yale, you will see that he is the Sterling Professor of Mathematical Sciences, Emeritus, Yale University, and IBM Fellow Emeritus, T.J. Watson Research Center, IBM.
What are fractals? Fractals are objects that display self-similarity at various scales. Magnifying a fractal reveals small-scale details similar to those of the larger-scale. As you drill down to finer and finer detail you still see patterns similar to those at higher levels. Check out Mandelbrot's Web site for an extensive primer on fractal geometry along with applications. Complex phenomena can be modeled by fractals using relatively few parameters.
Throughout the book the authors warn the reader that commonly used measures of risk such as standard deviation can be quite misleading when dealing with markets. One of Mandelbrot's students found that big stock price changes-changes more than five standard deviations from the average-occurred two thousand times more than expected under a normal distribution. Under the assumption of normality, the probability of a stock market fall as big as the one on Black Monday, October 19, 1987, is less than 1 in 1050.
VaR (Value at Risk) is another risk measure that is suspect since it typically assumes a normal distribution. Although it is widely discussed and used, VaR does not adequately measure the true exposure to large events in so many applications. This is a problem because VaR is supposed to help you measure exposure to large events! But, this should not be new information for many actuaries.
Mandelbrot's study of markets goes back more than four decades when he studied cotton prices. He found that there were too many big price jumps to conform to the standard normal model, not only in the cotton market, but in most other markets as well. For a normal distribution, the probability of large events drops exponentially with size. What model does Mandelbrot believe works better? A power law where probabilities decay as an inverse power of the size of events. Think Pareto! The book explains that the Pareto distribution got it name from Italian economist Vilfredo Pareto who used it to model the distribution of incomes in the upper-income levels of society.
The authors also challenge the idea that market events are independent through time. Although correlation coefficients for price changes may be near zero, a large price change in a market today may portend a large price change tomorrow. Clusters of volatility occur. Many markets also demonstrate medium-term and long-term memory, according to the authors.
I would highly recommend this book to anyone interested in analyzing or modeling risk in general and, in particular, risk in financial markets. I think that it would be great if many CAS members were familiar with the contents of this book. The second sentence in the CAS Centennial Goal is, "CAS members will be recognized as the leading experts in the evaluation of hazard risk and the integration of hazard risk with strategic, financial, and operational risk." The authors provide some useful insights to risk that you don't usually get from other sources.
The book is relatively short and written for lay readers, so our members will find the book readable. I did have trouble following some of the details at times, but the major ideas are clearly presented. There is an extensive bibliography at the end of the book that would be valuable to those wanting to do additional research. There are a number of typos in the book, but nothing serious.
