8.5 Example: Forwards
It is important to distinguish between the task of specifying a mapping procedure and that of specifying a mapping. When we design a value-at-risk measure, we do the former. We specify a mapping procedure for a class of portfolios to which we anticipate applying the value-at-risk measure. When the value-at-risk measure is implemented and in use, that mapping procedure then constructs a specific portfolio mapping for each portfolio to which the value-at-risk measure is applied. A mapping procedure specifies both a primary mapping as well as any remappings of that primary mapping. Remappings are discussed in the next chapter. For now, we are focused on primary mappings only.
In the example of the last section, we specified a procedure for specifying primary mappings for a class of international equity portfolios. You then used that procedure in Exercise 8.13 to specify a primary mapping for a specific portfolio. In the example of this section, we shall perform both tasks—both specifying a genreal procedure and then applying that procedure to a specific portfolio—this time for a class of foreign exchange portfolios.
8.5.1 Procedure
Assume a 1-day 95% AUD value-at-risk metric. An Australian foreign exchange trader holds forward positions in AUD, USD, and JPY. All contracts have maturities of less than 365 actual days. Because foreign exchange transactions typically settle in two trading days, adopt 2nd-day valuation. Count basis days as actual days. We shall construct a primary mapping 1P = θ(1R) of the form
[8.40]
Specify asset vector 1S as representing the accumulated value at time 1 of 1 million units of a particular currency to be received on a particular future date. Using component vectors, specify
[8.41]
where
[8.42]
and component vectors 1SUSD and 1S JPY are similar. Because currencies generally won’t be delivered on weekends or holidays, we have specified more assets than we need, but our choice of assets is notationally convenient.
Each forward contract comprises two “legs”—a long position in one currency and a short position in another. For example, a contract to deliver AUD 2MM in exchange for JPY 169MM on actual day 124 is represented as
[8.43]
In this manner, each contract is broken into individual holdings in specific assets represented by 1S. Summing such holdings across all contracts held by the portfolio, we obtain the portfolio holdings ω. We define 1P with the vector product
[8.44]
We define 1S as a mapping of a key vector 1R in two steps. First we define a mapping 1S = φ(1Q) where 1Q represents two spot exchange rates and discount factors for daily maturities out to 364 days. Next, a mapping 
employs linear interpolation to obtain the daily discount factors of 1Q from discount factors corresponding to quoted Libor rates. This yields a primary mapping
[8.45]
Specify 1Q with component vectors
[8.46]
where
[8.47]
[8.48]
and the two other discount curves 1QUSD and 1Q JPY are similar. In total, 1Q comprises 1,097 risk factors.
Foreign exchange professionals may wonder why we are modeling cash flows with 0 days to maturity, which would be physical cash. Since any cash position can be invested, at least overnight, shouldn’t the minimum maturity be 1? At time 0, the answer is yes, but we are modeling the portfolio as of time 1. If cash is invested overnight at time 0, it will be returned and have 0 days until maturity at time 1. In a similar vein, we don’t need to model discount factors at time 1 for maturities of 365 days because a cash flow with 365 days until maturity at time 0 will have 365 – τ(1) days until maturity at time 1.
Since we don’t know the present date—we are designing a mapping procedure that will apply for any date—we don’t know the value date for time 1. Discount factors 
, 
, and 
for maturity on that date will all equal 1.0. For convenience, we treat them as risk factors, but they are not random. Discount factors for maturities prior to the value date are accumulation factors. Representing accumulation of value to the value date, they are random but will exceed 1.0.
Define the mapping 1S = φ(1Q) with
[8.49]
[8.50]
[8.51]
where subscript terms τ(1) are used to account for the decline in the time-to-maturity of each cash flow over the value-at-risk horizon.
We now reduce our 1,097 risk factors to just 47 key factors with a mapping 1Q = (1R) that linearly interpolates between discount factors corresponding to quoted Libor rates. Specify 1R with component vectors:
[8.52]
where
[8.53]
[8.54]
and the two other discount curves 1RUSD and 1R JPY are similar. Define mapping 1Q = (1R) by setting corresponding exchange rates equal to each other
[8.55]
and linearly interpolating between the discount factors of 1RAUD, 1RUSD and 1R JPY to obtain the discount factors of 1QAUD, 1QUSD and 1Q JPY. The resulting mapping function is a linear polynomial
[8.56]
where, b is a 1,097 × 47 matrix, and a is a 1,097-dimensional vector. The actual matrix b and vector a depend upon the day counts for the times-to-maturity of quoted Libor rates at time 1. Let’s illustrate for an actual portfolio.