## Tuesday, October 19, 2010

### Rachev et al. Probability and Statistics for Finance

I make it a policy never to review a book until I have finished it. In the case of Probability and Statistics for Finance by Svetlozar T. Rachev, Markus Höchstötter, Frank J. Fabozzi, and Sergio M. Focardi (Wiley, 2010) I feel compelled to make an exception. The fault lies not in the book but in the reader. Although I am progressing at a reasonable clip given some of the gaps in my math background, I decided that it would be unfair to the authors to wait until I had read the last part of the book, multivariate linear regression analysis. So I’ll confine my brief remarks (brief, lest I demonstrate to the world that I really am a quant dolt) to the first three parts: descriptive statistics, basic probability theory, and inductive statistics. That’s 518 of the 632 pages of text and appendixes, so although it’s not a representative sample it’s a pretty good chunk.

I never studied probability and statistics formally but instead read assorted texts and watched educational videos. The result was an unstructured intellectual hodgepodge, little of it directly linked to financial issues. This book solves my scattershot problem, one that I suspect I am not alone in experiencing. First, it proceeds clearly and deliberately and does not skimp on equations. Second, it explains the strengths and weaknesses of particular concepts when applied to finance. And third, its examples, and there are many, are exclusively financial. For instance, we read about the application of the hypergeometric distribution in a Federal Reserve study to assess whether U.S. exchange-rate intervention resulted in a desired depreciation of the dollar. (p. 209) And we learn how to decompose the daily S&P 500 returns to determine whether there is any difference in price changes depending on the day of the week. (p. 157)

The authors balance mathematically appealing concepts with those that are “sometimes rather complicated, using parameters that are not necessarily intuitive.” (p. 277) As one might suspect, the latter are necessary to deal with extreme events. A chapter is devoted to such continuous probability distributions as the generalized extreme value distribution, the generalized Pareto distribution, the normal inverse Gaussian distribution, and the α-stable distribution. Yet even these alternative distribution models, the authors point out, tend to come up short in extreme circumstances. For instance, what was the likelihood that AIG could lose 60% of its value in a single day? According to a normal distribution model the probability was near zero. When the parameters for an α-stable distribution are selected to fit the AIG returns, the probability is still nearly negligible, only 0.003%. “That brings to light the immense risk inherent in the return distributions when they are truly α-stable.” (p. 292)

No book on statistics is complete without considering the formulation and testing of hypotheses. This section is surprisingly dense mathematically, which is where I got bogged down.

Probability and Statistics for Finance is not for the casual reader who comes from the “physics for poets” school of thought. It can be tough sledding. But for anyone who is serious about gaining a solid foundation in quantitative analysis this book is a very good place to start.