###### 1.8.5 Transformation Procedures

A transformation procedure—or transformation—characterizes a conditional distribution for and uses this characterization to value a desired value-at-risk metric or other PMMR. According to Holton (2004), risk has two components:

- exposure, and
- uncertainty.

A portfolio mapping incorporates both. The characterization of a conditional distribution of reflects our uncertainty. The mapping function θ reflects our exposure. The challenge for a transformation procedure is to combine both components to characterize a conditional distribution for . To intuitively understand what this entails, consider some simple examples.

A portfolio’s value depends upon a single normally distributed key factor . The mapping function θ is a linear polynomial. The situation is depicted in Exhibit 1.8. The graph on the left depicts the mapping function θ. Evenly spaced realizations for have been mapped into corresponding realizations for . The resulting realizations of are also evenly spaced, indicating that θ imparts no distortions. Since is conditionally normal, will also be conditionally normal, as illustrated in the graph on the right.

^{1}

*R*

_{1}. This is illustrated intuitively by mapping evenly spaced realizations for

^{1}

*R*

_{1}through the mapping function. The output values for

^{1}

*P*are also evenly spaced, indicating that the mapping function causes no distortion. Since

^{1}

*R*

_{1}is conditionally normal, so is

^{1}

*P*.

For our second example, consider a portfolio comprising a single call option with a conditionally normal key factor as its underlier. To avoid a need for additional key factors, treat applicable interest rates and implied volatilities as constant. In Exhibit 1.9, the left graph depicts the familiar “hockey stick” mapping function of a call option. Evenly spaced realizations for do not map into evenly spaced realizations for , so the mapping function causes distortions. Since is conditionally normal, the resulting distribution of is conditionally non-normal, as illustrated on the right.

^{1}

*R*

_{1}. The result is a conditionally non-normal portfolio value

^{1}

*P*. This is illustrated intuitively by mapping evenly spaced realizations for

^{1}

*R*

_{1}through the mapping function. The corresponding values for

^{1}

*P*are not evenly spaced, reflecting how the mapping function distorts the distribution of

^{1}

*P*.

Our third example considers a long-short options position applied to a short position in the underlier. The mapping function θ, which is illustrated in the left graph of Exhibit 1.10, causes realizations of to cluster in two regions. If the underlier is conditionally normal, will have the dramatically non-normal conditional distribution shown on the right.

^{1}

*P*.

These are simple examples, especially since each entails a single key factor. Practical value-at-risk measures often entail 100 or more key factors. If a portfolio holds complex instruments such as exotic derivatives or mortgage-backed securities, a mapping function can be extremely complex. Such issues can make it difficult to design a practical transformation procedure.

In our examples of Section 1.6, we illustrated three types of transformations:

- linear transformations,
- quadratic transformations, and
- Monte Carlo transformations.

The first applies if a portfolio mapping function θ is a linear polynomial. The second applies if θ is a quadratic polynomial and is joint-normal. The third applies quite generally and is one example of a category of transformations called **numerical transformations**. We discuss transformation procedures in Chapter 10.

###### Exercises

Exhibit 1.11 illustrates three portfolio mapping functions θ for portfolios whose values depend upon a single key factor . As we did in Exhibits 1.8, 1.9, and 1.10, sketch what each conditional PDF of might look like assuming is conditionally normal with its mean at the mid-point of each graph.

Describe real-world portfolios whose mapping functions might appear like those of the previous exercise.