###### 1.8.6 Portfolio Remappings

A remapping is an approximation of one mapping by another. The result is always that some risk vector is approximated by another risk vector, and that will become our formal definition for “remapping”. As this book proceeds, we will find plenty of reasons to make such approximations. Let’s explain:

All the value-at-risk measures we have considered so far entail modest calculations. They apply to small portfolios that are easy to value. When we develop value-at-risk measures for real portfolios, this will change.

Every value-at-risk measure employs—either explicitly or implicitly—a primary mapping ^{1}*P* = θ(^{1}** R**). Primary mappings can be complicated. This occurs for two reasons:

**The mapping function θ may be complicated:**Mapping functions are formulas for marking-to-market a portfolio as of time 1. They are constructed using techniques of financial engineering. All the computational challenges that arise with financial engineering can arise with θ.**The key vector****may be complicated:**value-at-risk measures are sometimes implemented with 1,000 or more key factors . Also, the joint distribution of may be difficult to work with.

Such complexity can make it difficult to directly apply a transformation procedure. This is especially true if both a primary mapping and a transformation procedure employ the Monte Carlo method—resulting in nested Monte Carlo analyses.

Consider a portfolio holding 300 exotic derivatives, (, ) for *i* {1, 2, 3, … 300}. Each derivative can only be valued using the Monte Carlo method. The primary mapping has the form

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where key factors represent values for underliers, implied volatilities and discount factors. Valuing the mapping for a specific realization requires 300 Monte Carlo analyses, one for each derivative’s value:

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Consequently, valuing a realization of based upon one realization entails performing all 300 of those same Monte Carlo analyses.

Suppose we employ a Monte Carlo transformation procedure to calculate the portfolio’s value-at-risk. This will nest the 300 valuation Monte Carlo analyses within the Monte Carlo transformation. The Monte Carlo transformation might require that the portfolio be valued 10,000 times. This would require calculating realizations for *k* {1, 2, 3, … 10,000}, with each such valuation requiring 300 valuation Monte Carlo analyses. The entire analysis will entail 10,000(300) = 3,000,000 valuation Monte Carlo analyses. This is a staggering computational load.

To make a transformation less computationally expensive, we might replace a primary mapping ^{1}*P* = θ(^{1}** R**) with an approximation = (), which we call a remapping. In our (second) Australian equities example, we considered a simple remapping. The above example of nested Monte Carlo analyses illustrates an extreme case. Here, a remapping would be crucial.

Formally, a **remapping** is an approximation of a risk vector with some other risk vector . We describe remappings more generally in Chapter 9. For now, we are interested in remappings of . If we have a portfolio mapping ^{1}*P* = θ(^{1}** R**), such remappings may take three forms:

- Those that approximate
^{1}*P*= θ(^{1}) by replacing θ with an approximate mapping function , so = (*R*^{1}).*R* - Those that approximate
^{1}*P*= θ(^{1}) by replacing with alternative key vector , so = θ().*R* - Those that approximate
^{1}*P*= θ(^{1}) by replacing both θ and with alternatives and , so = ().*R*

The first and third forms are most common. Remappings of the first form often approximate a portfolio mapping function θ with a linear or quadratic polynomial to facilitate use of a linear or quadratic transformation. Those of the third form often replace a high-dimensional with a lower dimensional . Principal component analysis, which we discuss in Chapter 3, can be useful for this purpose. Remappings may be applied to primary mappings or to other remappings—approximating approximations.

###### Exercises

Below are informal descriptions of three portfolio mappings and three schematics of portfolio mappings. Match each description with the corresponding schematic. Recall that, in such schematics, horizontal arrows represent mappings while vertical arrows represent approximations (remappings).

- Portfolio value depends upon key factors representing exchange rates, implied volatilities, and interest rates in various currencies.
- A stock portfolio is modeled as a function of individual stocks’ single-period returns. For simplicity, all return pairs are assumed to have the same correlation.
- A portfolio holds options and futures on gold. Its market value is approximated as a quadratic polynomial of applicable risk factors.

*Schematic 1:*

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*Schematic 2:*

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*Schematic 3:*

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