9.4  Variables Remappings

In value-at-risk modeling, we frequently work with changes of variables. Every mapping is a change of variables. Consider a mapping that expresses a vector of discount factors ¹Q as a function φ of corresponding interest rates ¹R. We describe it schematically as

[9.48]

Here, mapping function φ reflects an exact relationship between ¹Q and ¹R. A change of variables can also be an approximation—and hence a remapping. Suppose futures prices are used to approximate forward prices ¹R. Based upon our convention that horizontal arrows denote exact relationships and vertical arrows denote approximations, we describe this relationship schematically as

[9.49]

A variables remapping is a remapping in which a key vector ¹R is approximated with some other risk vector. This can happen in two ways. Sometimes, a new key vector  directly approximates ¹R:

[9.50]

Other times, an approximating risk vector ¹Q is constructed as a function  of a new key vector :

[9.51]

In schematic [9.50], key vector  replaces key vector ¹R, but portfolio mapping function θ is unaffected. In schematic [9.51], both key vector ¹R and portfolio mapping function θ change.