Remappings 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 ? with an alternative function 
or to replace one key vector 1 R 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.
Transformations fall loosely into two categories:
linear and quadratic transformations, which are applicable if a portfolio mapping function is a linear or quadratic polynomial; and
numerical transformations primarily Monte Carlo transformations which entail repeated valuations of a portfolio mapping function.
If a primary mapping function ? can be approximated with a linear or quadratic polynomial 
, this will facilitate a linear or quadratic transformation. If a numerical transformation is to be used, replacing a primary mapping function ? with another function 
that is easier to value may reduce the processing time required to perform the many valuations numerical transformations require.
Quadratic remappings can also facilitate variance reduction in Monte Carlo transformations. Standard methods of variance reduction described in Chapter 10 employ control variates and stratified sampling. Both apply directly to a primary portfolio mapping 1 P = ?( 1 R), but they employ a quadratic remapping 1 P = 
( 1 R) to facilitate the variance reduction.
Inference procedures generally require historical data. If data...
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