###### 1.8.7 Summary

Recall our definition of measure from Section 1.1:

A measure is an operationally defined procedure for assigning values.

A value-at-risk measure is an operation that assigns a value to a portfolio. That operation comprises various procedures, which we have defined. Exhibit 1.12 relates these to one another in a general schematic. Specific value-at-risk measures vary in certain respects, but most conform generally to the scheme of Exhibit 1.12. They accept both a portfolio’s holdings and historical market data as inputs. A **mapping procedure** specifies a portfolio mapping function θ, which may reflect a **primary portfolio mapping** or a **portfolio remapping**. An **inference procedure** characterizes a conditional distribution for . It generally employs techniques of time-series analysis.

The outputs of the mapping and inference procedures reflect the two components of risk. The mapping function θ reflects exposure. The characterization of the conditional distribution of reflects uncertainty. A **transformation procedure** combines these two components to somehow characterize the conditional distribution of . To do so, it may employ results from probability theory as well as methods of numerical integration, such as the Monte Carlo method. The characterization may take many forms—a probability density function (PDF), a characteristic function, certain parameters of the distribution of , a realization of a sample4 from the distribution of , etc.

Finally, the characterization of the conditional distribution for is used, with the value of ,5 to calculate the desired value-at-risk metric.

While the above summary is presented for value-at-risk measures, it is equally applicable to measures of any PMMR. The only difference occurs at the very end, where the characterization of the conditional distribution of is used, together with the value of , to calculate the desired PMMR.