Risk comprises uncertainty and exposure. A VaR measure represents uncertainty with a characterization of the conditional distribution of 1 R. It represents exposure with a portfolio mapping 1 P = ?( 1 R), which may be a primary mapping or a remapping. To quantify market risk for a portfolio, we need to combine these two components. This is the purpose of a transformation procedure.
A transformation procedure or transformation represents risk with a characterization of the conditional distribution of 1 P. It then uses that characterization to value a VaR metric. The value for that metric is the output of the VaR measure.
That characterization of the conditional distribution of 1 P may be a standard deviation, PDF, characteristic function, or some other representation. If the characterization is sufficiently general to support any reasonable VaR metric, we say the transformation is complete. Otherwise, it is incomplete.
In Section 1.8, we described three categories of transformations:
linear transformations,
quadratic transformations, and
Monte Carlo transformations.
Linear and quadratic transformations apply to linear and quadratic portfolios, respectively. Monte Carlo transformations apply generally to all portfolios. In this chapter, we describe all three types of transformations.
Linear transformations are most widely used. In practice, many portfolios are linear or so nearly linear that they can be accurately approximated with a linear remapping. Linear transformations are easy to implement, run in real time and are exact. The only reason to not implement a linear transformation is if a portfolio...
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