2.1. MOTIVATION

In this chapter, we describe a number of techniques of applied mathematics. Some may already be familiar to you. All will play a role in subsequent discussions of VaR. This opening section places them in that context.

Recall Section 1.8, which described a framework for modeling VaR. It includes a general schematic describing VaR measures, which we reproduce in Exhibit 2.1.

Exhibit 2.1: A reproduction of Exhibit 1.11, which is a general schematic for VaR measures. The techniques of applied mathematics described in this chapter are employed throughout the remainder of the book. They are especially important for discussions of mapping procedures in Chapter 9 and transformation procedures in Chapter 10.

A mapping procedure specifies a primary portfolio mapping ¹ P = ?( ¹ R). Sometimes, the mapping function ? or key vector ¹ R is computationally expensive to work with, making the subsequent application of a transformation procedure impractical. A solution is to replace the primary mapping with an approximation , which we call a portfolio remapping. There are many ways this might be done. Several are described in Chapter 9. In anticipation of that discussion, the present chapter covers a variety of techniques that are useful for constructing approximations. These include:

• gradient and gradient-Hessian approximations,

• ordinary interpolation,

• ordinary least squares.

Principal component analysis offers another technique of approximation. It is probabilistic, so...