###### 1.8.3 Portfolio Mappings

In mathematics, a *mapping* is a *function*. The words are synonyms. In the context of value-at-risk, we reserve the word “mapping” for functions relating specific risk vectors to one another. If and are risk vectors, a mapping is a functional relationship:

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We call φ the **mapping function**.

A **portfolio mapping** is a mapping that defines a portfolio’s value as a function of some risk vector :

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Portfolio mappings play a simple but inevitable role in value-at-risk measures. Let’s focus on two of our earlier examples: Leavens’ PMMR and our Australian equities value-at-risk measure. To quantify a portfolio’s market risk, we must calculate the value of some function—value-at-risk metric or other PMMR—of and the conditional distribution of . We interpret as the portfolio’s market value at time 1, but this is not a definition. Mathematically, there are two ways we may define the random variable :

- we can directly specify a conditional distribution for ;
- we can define as a function of some random vector.

The first approach is hardly feasible. Portfolios and financial markets tend to be complicated, so it is difficult to directly specify a conditional distribution for . Inevitably, we define using the second approach—which leads to portfolio mappings. Both the Leavens and Australian equities value-at-risk measures define as a function of some asset vector :

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We interpret as a vector of accumulated values, but this is not a definition. To complete our definition of , we must mathematically define . As with , there are two ways to define :

- we can directly specify a conditional distribution for ;
- we can define as a function of some other random vector.

At this point, Leavens uses the first approach. He specifies a conditional distribution for and uses this to infer a binomial distribution for . We schematically represent Leavens’ portfolio mapping as

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The Australian equities value-at-risk measures don’t stop there. Rather than directly specify a joint distribution for , they define as a mapping of another random vector . We schematically represent the resulting portfolio mapping as

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No matter how many mappings are composed, ultimately must be defined as a function of some random vector for which we directly characterize a joint distribution. That random vector is the key vector . We denote the mapping function that relates to its key vector with θ. Accordingly, the notation

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recurs frequently in this text. An exception is if asset values are used as key factors. In this case, the relationship is

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and plays the dual role of asset vector and key vector.

Here we have described not only portfolio mappings, but also a general procedure for constructing them. Portfolio mappings constructed in this manner—starting with asset vector and holdings **ω**, and perhaps mapping to some key vector —are called **primary mappings**. The name distinguishes them from portfolio mappings constructed as remappings. All portfolio mappings stem from primary mappings. They either are left in that form, or are approximated using one or more remappings. We discuss primary mappings in Chapter 8.