8.3  Primary Mappings

The construction of a primary mapping always begins—explicitly or implicitly—with a row vector of holdings ω and an asset vector ¹S. We define

[8.10]

We can let ¹S be our key vector, in which case [8.10] is our primary mapping. More often, we select some other key vector ¹R. Using asset valuation formulas φ_i for each asset ¹S_i, we specify a mapping ¹S = φ(¹R). The composition θ = ωφ defines a primary mapping:

[8.11]

Our discussion of primary mappings must address the four constructs: ¹S, ω, ¹R, and φ.

8.3.1 Specifying ¹S and ω

Asset values ¹S_i represent the accumulated value at time 1 of one unit of some asset held at time 0. Selecting what instruments to represent with ¹S entails a number of definitional issues.

Any institution that implements a value-at-risk measure will link the system (directly or manually) to some portfolio accounting system. That system will define a universe of instruments it can account for. As a practical matter, it makes sense to define ¹S to conform closely to this. However, the correspondence need not be identical. Assets do not need to conform to any legal or accounting notions of what might constitute an asset. In many cases, it makes sense to define assets creatively or break certain instruments into components, each of which is represented with a different asset. Doing so can minimize the dimensionality of ¹S—and possibly reduce the number of active holdings we need to model for a portfolio.

Asset values ¹S_i typically reflect mid-market valuations, but they can also represent bid or offer valuations. Since mapping [8.10] defines the portfolio ¹P in terms of ¹S, we must decide whether we wish ¹P to reflect mid-market, bid, or offer valuations. This depends upon our intended interpretation of our value-at-risk measure, but it also depends upon the historical market data available to us. If the data we have is indicative offer prices, then it is practical to let assets ¹S_i—and hence our portfolio ¹P—represent offer valuations.

Another issue is valuation method. Should asset values—and hence portfolio values—reflect cash or some n^th-day valuation? It may be convenient to decide this issue based upon applicable settlement conventions. In foreign exchange and money markets, spot transactions tend to have 2^nd-day settlement. Corporate bonds typically settle in 3 days. Futures settle (margin) daily, which facilitates cash valuation.

With a clear definition of what constitutes a unit of each asset ¹S_i, data from a portfolio accounting system can be accessed to construct holdings ω.

8.3.2 Specifying ¹R and θ

If our primary mapping is to employ some key vector ¹R other than ¹S, we must specify a mapping

[8.12]

The mapping function φ is a vector of component functions φ_i, each of which values some asset:

[8.13]

From an implementation standpoint, the mapping φ corresponds to a library of financial engineering models—a model library. If a value-at-risk measure is to be used in a fixed-income trading environment, the library will have models for valuing swaps, caps, floors, swaptions, bonds, etc. If it is to be used in an energy trading environment, the library will have models for valuing energy forwards, futures, options, etc.

The model library that defines φ requires various inputs in order to value assets, and these are the key factors ¹R_i. This does not mean that selecting key factors is as simple as selecting some model library and seeing what inputs it requires. Our choice of financial variables to represent with ¹R will be determined by a number of considerations. To some extent, these must drive our selection of a model library.

To avoid ¹R having a singular covariance matrix, it is desirable that key factors be linearly independent. If we model two prices as key factors, we should not model the spread between them as a third key factor. To avoid multicollinearity, it is desirable to avoid key factors that are highly correlated or are in some other sense “almost” linearly dependent. If we model interest rates as key factors, it does not make sense to also model the corresponding discount factors or forward rates. It is desirable that the dimensionality of ¹R not be too great. Specifying 20 or 100 key factors is reasonable. Specifying 10,000 is not.

None of the above issues is critical. A subsequent remapping can tidy up problems, but we should not rely too heavily on one to do so. Applying a remapping can be computationally intensive or unstable. Fitting a quadratic approximation in 10,000 dimensions is not a task to be taken lightly!

An important consideration in selecting a model library for a value-at-risk measure is the fact that its purpose is different from that of a model library that might support a front office. A front-office library—one that supports trading—is intended to calculate market values ⁰s_i for instruments as of time 0. A middle-office library—one that supports risk measurement—is intended to calculate accumulated values ¹s_i for instruments as of time 1. If an instrument generates cash flows during the interval (0,1], a middle-office library must accumulate these until time 1 and include them in its calculation of accumulated value for that instrument. If a value-at-risk metric depends upon ⁰p, the middle office model library will also need to calculate current asset values ⁰s_i.