This chapter provides foundational material regarding statistical methods for the study of alternative investments in general and for the subsequent material in this book in particular. The use of statistics in performing hypothesis tests is addressed in detail in Chapter 8.

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.

**Ex post returns** are realized outcomes rather than anticipated outcomes. Future possible returns and their probabilities are referred to as expectational or **ex ante returns**. A crucial theme in understanding the analysis of alternative investments is to understand the differences and links between ex post and ex ante return data.

Often, predictions are formed partially or fully through analysis of ex post data. For example, the ex ante or future return distribution of a stock index such as the S&P 500 Index is often assumed to be well approximated by the ex post or historical return distribution. The direct use of past return behavior as a predictor of ...

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