4.1 Motivation

In Section 1.2, we described risk as having two components: exposure and uncertainty. To quantify a portfolio’s market risk, a value-at-risk measure must describe both of these. Its mapping procedure describes exposure. Its inference procedure describes uncertainty.

Exhibit 4.1: A reproduction of Exhibit 1.12, which is a general schematic for value-at-risk measures. The statistical and time-series methods described in the present chapter underlie inference procedures. The discussion of statistics is also important for our discussion of the Monte Carlo method in Chapter 5.

In Chapter 2, we discussed various techniques of applied mathematics. Most of these will be employed in Chapter 9 to support mapping procedures. In the present chapter, we describe techniques from statistics and time series analysis. These will be used to support inference procedures, which we describe in Chapter 7.

The discussions of statistics—especially the notions of samples and estimators—will play an important role in our development of the Monte Carlo method in Chapter 5. Hypothesis testing and testing for autocorrelations will be used in our discussion of backtesting in Chapter 12.