Risk comprises both uncertainty and exposure. In Chapters 6 and 7, we 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, 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 = ?( 1 R). We choose the mapping function ? and key vector 1 R to reflect as accurately as is reasonable our perception of how the market value of a portfolio will depend upon market variables. For certain VaR measures, the primary mapping is all we need. These VaR measures use the primary mapping as a practical tool, applying a transformation procedure directly to it. Other VaR measures approximate the primary mapping with a remapping. For these VaR measures, the primary mapping has more theoretical importance. It is a point of departure for defining the remapping. By defining precisely...
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