### Chapter 8

#### Primary Portfolio Mappings

# 8.1 Motivation

Risk comprises both uncertainty and exposure. In Chapters 6 and 7, we have focused on the uncertainty component, compiling historical market data and designing inference procedures to characterize a conditional distribution for ^{1}** R**. We now turn to the exposure component, which is represented with a portfolio mapping. Portfolio mappings are specified with a mapping procedure.

When we specify a portfolio mapping, we may perceive this as somehow approximate. This perception is difficult to formalize. From an operational standpoint, a portfolio has no “true” portfolio mapping. How can a portfolio mapping be an approximation if there is no true mapping for it to approximate? In Section 1.8.3, we distinguished between primary portfolio mappings and remappings. This distinction allows us to formalize approximations while acknowledging that all portfolio mappings are models.

We construct a portfolio mapping by first specifying a primary mapping ^{1}*P*=θ(^{1}** R**). We choose the mapping function θ and key vector

^{1}

**to reflect as accurately as is reasonable our perception of how the market value of a portfolio will depend upon market variables. For certain value-at-risk measures, the primary mapping is all we need. These value-at-risk measures use the primary mapping as a practical tool, applying a transformation procedure directly to it. Other value-at-risk measures approximate the primary mapping with a remapping. For these value-at-risk measures, the primary mapping has more theoretical importance. It is a point of departure for defining the remapping. By defining precisely what the remapping is intended to approximate, the primary mapping formalizes the approximation, rendering it suitable for analysis. This allows us to analyze errors the remapping introduces or to objectively assess alternative approximations.**

*R*In this chapter, we describe how mapping procedures construct primary mappings. In the next chapter, we consider how they construct remappings.