Statistical Foundations

This chapter provides foundational material regarding statistical methodologies for the study of alternative investments in general and for the subsequent material in this book in particular. The chapter begins with a discussion of probability distributions and computations of returns. The middle of the chapter discusses the normal distribution and the use of higher moments to describe and measure probability distributions. The chapter concludes with a discussion of related risk measures, including value at risk.


Risky assets experience unexpected value changes and therefore unexpected returns. If we assume that investors are rational, the more competitively traded an asset, the more these unexpected price changes may be random and unpredictable. Hence, asset prices and asset returns in competitively traded markets are typically modeled as random variables. Frequency and probability distributions therefore provide starting points for describing asset returns.

3.1.1 Frequency Distributions

The top panel of Exhibit 3.1 contains a table of monthly return data from the S&P 500 for 40 months. The middle of Exhibit 3.1 contains a frequency distribution, and the bottom of the exhibit contains a histogram of the monthly S&P 500 returns. A histogram is simply a graphical representation of a frequency distribution. Note that the returns illustrated in Exhibit 3.1 tend to cluster around the middle (i.e., the central ...

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