### Chapter 3

#### Probability

# 3.1 Motivation

Value-at-risk measures are inherently probabilistic. A central question thatvalue-at-risk addresses is this: If a portfolio comprises holdings in various instruments, how is its market risk determined by theirs? In the parlance of probability, the question becomes: If a random variable is defined as a function of other random variables, how is its distribution determined by theirs?

As it relates to market risk, the question is addressed in Chapter 10, which discusses transformation procedures. That chapter draws on various techniques from this chapter, including:

- techniques for characterizing the distribution of a linear polynomial of a random vector;
- techniques for characterizing the distribution of a quadratic polynomial of a random vector;
- the central limit theorem;
- the inversion theorem.

Concepts from the present chapter underlie statistics and time series analysis, which are the topics of Chapter 4. They, in turn, are used in Chapter 7 to design inference procedures.

Finally, the present chapter describes principal component analysis, which is used in Chapter 9 with a category of portfolio remappings called, transparently, “principal component remappings.”