###### 9.4.1 Example: Coffee Spreads

Consider our coffee wholesaler from Section 8.7. Based upon the primary mapping we constructed, the portfolio has exposures to a vector of coffee spreads:

[9.52]

Many of these coffees trade inactively, so historical data is unavailable for their spreads. We address the problem by identifying eight coffees that are actively traded, and consider a vector of spreads for these

[9.53]

We cannot directly approximate ^{1}* R^{ Spreads}* with . The former has 30 components, whereas the latter has only 8. offers among its components no reasonable proxy for many of the components of

^{1}

*. We solve the problem by approximating*

**R**^{ Spreads}^{1}

*with a 30-dimensional vector*

**R**^{ Spreads}^{1}

**comprising “blends” of the components of . Blends are specified by a knowledgeable coffee trader. The remapping has form**

*Q*[9.54]

The mapping function is a linear function that can be represented with a matrix:

[9.55]

For each row of the matrix, components sum to 1. This is a practical requirement, but it is not logically necessary. The remapping provides us with key vector for which historical data is available.