|Abnormal Distributions and Substandard Deviations.
The Quant Problem.
Quantitative tools have clearly helped investment managers of all types to manage risk. Not so obviously, they have also hurt. And yet this should be obvious. And it should be one of the things we guard against as a matter of routine.
Quantitative investment analysis is one of many indirect outgrowths of the Logical Positivist School of philosophy. Another more direct outgrowth of Logical Positivism is the psychological school of Operationalism. B. F. Skinner stated this approach to research more clearly than I have ever heard an investment quant state it. In the 1945 Psychological Review, Skinner wrote, “Operationalism may be defined as talking about (1) one’s observations, (2) the manipulative and calculational procedures involved in making them, (3) the logical and mathematical steps that intervene between earlier and later statements, and (4) nothing else.”
The Logical Positivists wanted to eliminate nonsense from language by restricting what could be done with it. As such, Logical Positivism is at the core of some of the most powerful and successful ideas in human history. Which is not to say that Logical Positivism and Operationalism do not have problems. It is simply not possible to do research of any kind without violating Skinner’s directive, without talking about concepts and abstractions that cannot be defined within Skinner’s limits. In the social sciences, this is widely understood. Social scientists, therefore, typically make a distinction between operational and non-operational definitions and then spend a lot of time trying to reconcile the two. I have seen much less awareness of this distinction among investment quants. This lack of awareness results in massive wastes of time and money when developing and researching trading methods.
For example, there is a large literature on trend following methods with many of these methods selling for hundreds or even thousands of dollars. But if the basic concept is true, why are these trading methods, which are nothing more than operational definitions, so expensive? Why is it so important to get the operational definition exactly right?
Certainly, nothing in the basic concept (“a trend is more likely to continue than reverse.”) demands such precision. Much more likely than not, there is something wrong with the basic concept, with the non-operational definition.
Where risk management is concerned, our industry’s lack of distinction between operational and non-operational definitions is positively dangerous. Part of the hedge fund manager’s job is managing not risk but uncertainty; that is, managing everything of which he is unaware. This is a relatively difficult task. In part, perhaps, because it is logically impossible.
Uncertainty is usually managed, to the extent it is managed at all, by reducing it to risk, by assuming that future price changes will be randomly distributed in some manner.
On the one hand, reducing uncertainty to a probability distribution is incredibly dangerous--unless you are certain that you can fairly represent everything you do not know by some distribution or other. On the other hand, in most cases, there is no other practical assumption. This approach allows us to use the incredibly powerful tools of probability theory to manage our portfolios. But at the very least, however, we need to pick our distribution carefully.
Most risk management systems assume that price changes are normally or lognormally distributed. A normal distribution follows the familiar bell-shaped curve. A lognormal distribution is just a distribution of the logarithm of price changes. Roughly speaking, this just insures that the percentage changes in price are normally distributed.
Assuming that prices are normally (or lognormally) distributed is not an unreasonable idea. Normal distributions are relatively simple and have been well understood for a very long time. Many natural phenomena are normally distributed. And best of all, price distributions certainly look normal.
Unfortunately, they are not. In 1963, Mandelbrot noticed that the tails of the distributions of commodity returns and interest rates were fatter than they would be if the distributions were normal. Later, economists noticed that the distributions were skewed and unstable, that is, the distributions seemed to change over time.
It is important to understand that the differences between normal distributions and the distributions we have are subtle. Using tools that assume a normal distribution is like sitting in an old comfortable chair. Most of the time, everything is fine, just fine. Except that every once in a while, Angelina Jolie or Brad Pitt (depending on your taste in such matters) calls you and begs you for a date. And every once in a while, someone steps out from the closet behind you and slams a sledge hammer into your head.
Right now, there is no consensus among financial economists about how investment prices are really distributed. If I read the literature correctly, most researchers believe the proper distribution is a member of the stable Paretian family. But there is no guarantee of that. Mandelbrot, for example, believes the proper distribution is fractal. Worse, what assurance do we have that we even have a name for the proper distribution? On this single most important issue in risk management, despite decades of research, we hardly have a clue.
Svetlozar T. Rachev, Christian Menn and Frank J. Fabozzi’s recent book, Fat-Tailed and Skewed Asset Return Distributions is well worth reading. Be forewarned: this book is not an easy read. The writing is clear, if graceless, but the mathematical demands may be uncomfortably high for many people. Still, it makes a mathematically difficult subject as accessible as it can be made.
Even if we do not know how returns are distributed, we know that the stable Paretian family describes the returns better than the normal distribution. Even better, we have more powerful tools for managing risk and pricing options than we did twenty years ago. This book covers many of those tools. (And by the way, what the book does not cover is oddly revealing.) The hedge fund manager and fund of funds manager ignores these tools at his peril. Or, what is worse, at his investor’s peril.