Bubble Value at Risk: A Countercyclical Risk Management Approach, Revised Edition

Chapter 2

Essential Mathematics

This chapter provides the statistical concepts essential for the understanding of risk management. There are many good textbooks on the topic, see Carol Alexander (2008). Here, we have chosen to adopt a selective approach. Our goal is to provide adequate math background to understand the rest of the book. It is fair to say that if you do not find it here, it is not needed later. As mentioned in the preface, this book tells a story. In fact, the math here is part of the plot. Therefore, we will include philosophy or principles of statistical thinking and other pertinent topics that will contribute to the development of the story. And we will not sidetrack the reader with unneeded theorems and lemmas.

2.1 FREQUENTIST STATISTICS

Two schools of thought have emerged from the history of statistics—frequentist and Bayesian schools of thought. Bayesians and frequentists hold very different philosophical views on what defines probability. From a frequentist perspective, probability is objective and can be inferred from the frequency of observation in a large number of trials. All parameters and unknowns that characterize an assumed distribution or regression relationship can be backed out from the sample data. Frequentists will base their interpretations on a limited sample; as we shall see, there is a limit to how much data they can collect without running into other practical difficulties. Frequentists will assume the true value of their estimate lies within ...