As we discussed in Section 1.8.6, remappings are approximations. They take many forms. Before exploring some of these, let’s first address the question: why remap? There are many reasons that motivate us to replace one portfolio mapping function θ with an alternative function and/or replace one key vector 1R with an alternative vector . Doing so may support one of three purposes:
- to facilitate a transformation procedure;
- to facilitate an inference procedure; or
- to facilitate another remapping.
9.1.1 Facilitating Transformations
Transformation procedures fall loosely into two categories:
- fast-running linear or quadratic transformations, which are applicable if a portfolio mapping function θ is a linear or quadratic polynomial; and
- slow-running numerical transformations—primarily Monte Carlo transformations—which entail repeated valuations of a portfolio mapping function θ.
If a primary mapping function θ can be approximated with a linear or quadratic polynomial , this will facilitate a fast-running linear or quadratic transformation. If a numerical transformation must be used, replacing a primary mapping function θ with another function that is easier to value may reduce the processing time required to perform the many valuations that numerical transformations require.
Quadratic remappings can also facilitate variance reduction in Monte Carlo transformations. Standard methods of variance reduction described in Chapter 10 are control variates and stratified sampling. Both apply directly to a primary portfolio mapping 1P = θ(1R), but they employ a quadratic remapping = (1R) to facilitate the variance reduction.
9.1.2 Facilitating Inference
Inference procedures generally require historical data. If data is unavailable for the key vector 1R of a primary mapping, a remapping may introduce an alternative key vector for which data is available. Also, certain time series models can only be applied in low dimensions. For this reason, we may replace a high-dimensional key vector 1R with a low-dimensional key vector .
9.1.3 Facilitating Another Remapping
Applying certain remappings can be computationally expensive. Computations to apply one remapping may be streamlined by first implementing another remapping. For example, applying certain function remappings requires multiple portfolio valuations. The computations for each portfolio valuation can be streamlined by first applying a holdings remapping. Function and holdings remappings are discussed later in this chapter.
9.1.4 Forms Of Remappings
In Section 1.8.6, we described remappings as taking three forms:
- Those that approximate 1P = θ(1R) by replacing θ with an approximate mapping function , so = (1R).
- Those that approximate 1P = θ(1R) by replacing with alternative key vector , so = θ().
- Those that approximate 1P = θ(1R) by replacing both θ and with alternatives and , so = ().
In this chapter, we shall consider several practical examples of remappings. These will encompass all three of the above forms.
Because it is an approximation, a remapping alters the output of a value-at-risk measure. In most cases, we can design remappings to have a modest or even trivial impact. However, any decision as to what is reasonable must entail human judgment as well as careful consideration of the intended application for the value-at-risk measure.