## Monday, April 23, 2012

### Miller, Mathematics and Statistics for Financial Risk Management

Mathematics and Statistics for Financial Risk Management by Michael B. Miller (Wiley, 2012) doesn’t take the reader from 0 to 60 in 300 pages; the acceleration is probably more in the range of 25 to 50. Miller assumes that “readers have a solid grasp of algebra and at least a basic understanding of calculus.”

Ten chapters cover the following topics: some basic math, probabilities, basic statistics, distributions, hypothesis testing and confidence intervals, matrix algebra, vector spaces, linear regression analysis, time series models, and decay factors.

In each chapter Miller explains the key concepts and frequently illustrates them with sample problems (answers included). For instance, in the section on correlation the sample problem is: “X is a random variable. X has an equal probability of being -1, 0, or +1. What is the correlation between X and Y if Y = X2?” (p. 56) At the end of each chapter is a set of problems; the answers are given at the end of the book. A companion website provides further Excel examples.

Although this book carefully and clearly explains the basic math and statistics at the heart of risk management, it does more. It also uses math and statistics to explain the shortcomings of some financial risk management tools. VaR is an obvious candidate. One criticism of VaR is that it is not a subadditive risk measure. “Subadditivity is basically a fancy way of saying that diversification is good, and a good risk measure should reflect that.” (p. 120) Miller offers a sample problem that demonstrates a violation of subadditivity. Here’s the question: “Imagine a portfolio with two bonds, each with a 4% probability of defaulting. Assume that default events are uncorrelated and that there is a recovery rate of 0%. The bonds are currently worth \$100 each. If a bond defaults, it is worth \$0; if it does not, it is still worth \$100. What is the 95% VaR of each bond separately? What is the 95% VaR of the bond portfolio?” It turns out that in this case VaR “seems to suggest that holding \$200 of either bond would be less risky than holding a portfolio with \$100 of each. Clearly this is not correct. For this portfolio, VaR is not subadditive.” (pp. 120-21)

At every turn this book shows the relevance of mathematical and statistical concepts to risk management. They are no longer the desiccated notions found in most textbooks but assume a sense of vibrancy. So, if you’re trying to hone your skills, this book is a great place to start.