8.5.2 Specific Portfolio
The current date is May 17, 2001. Based upon the calendar of Exhibit 8.3, a time line is plotted in Exhibit 8.15. The foreign exchange portfolio holds 17 contracts indicated in Exhibit 8.16. Based on our specifications in Section 8.5.1, for this example, contracts are not assets. Legs of contracts are. To calculate holdings ω, contracts must be broken into “legs”, as shown in Exhibit 8.17.
Based upon those calculations
where only active holdings are indicated. Define 1P = ω1S. Tomorrow, May 18, is a trading day, so τ(1) = 1. Our mapping 1S = φ(1Q) becomes
which is a quadratic polynomial. To express 1P in terms of 1R, we must specify matrix b and vector a that define mapping [8.56]. The linear interpolation is simple in principle. What complicates it is day counts. Let’s illustrate for the AUD discount factors . Key factors correspond to the specific loan periods for which Libor will be quoted at time t = 1. Loans commence 2 trading days from the date they are quoted. Since May 18 is a Friday, those 2 trading days are 4 actual days. Based upon BBA specifications, the loan periods are indicated in Exhibit 8.18.
1R has 15 key factors corresponding to AUD discount factors. In addition, because May 22 is our value date for time 1, the discount factor for 4 actual days maturity must be 1.0. Including this and our 15 key factors , we have 16 discount factors to interpolate between.
To illustrate the interpolation, consider discount factor for 136 actual days maturity as of time 1. By Exhibit 8.18, key factors and for 4- and 5-month maturities correspond to 129 and 157 actual days maturity. Linearly interpolating, we set
Accordingly, components b136,7 and b136,8 of the matrix b are .75 and .25. Remaining components b136,i are 0, as is component a136 of vector a. Now consider discount factor . For it, we know
so components b4,i are 0 for all i, but component a4 of vector a is 1.0. For maturities of less than 4 actual days, we extrapolate backward from the 4-day and 5-day discount factors. For the 3-day maturity, we set
so components b3,1 = –1 with remaining components b3,i all 0. Component a3 = 2. Continuing in this manner, we complete the matrix b and vector a. Our mapping [8.56] is
We often think of linear interpolation as an approximation, but [8.65] is not an approximation—it is not a remapping. Mathematically, [8.65] defines the random vector 1Q in terms of 1R. There is no independent mathematical definition of 1Q to approximate, so this is a mapping. The relationship is indicated in schematic [8.40].
Composing ω, φ, and , we obtain
Our portfolio mapping is quadratic. Note that it is more complicated than our earlier mapping [8.61]—it depends on 15 linear or quadratic expressions in terms of 1R while [8.61] depended on just 12 linear or quadratic expressions in terms of 1Q. This is due to our considering only a small portfolio. The dimensionality of 1R is 47 while that of 1Q is 1097. If we considered larger portfolios, our mapping [8.66] in terms of 1R would entail a maximum of 45 linear or quadratic expressions while the mapping [8.61] in terms of 1Q could entail as many as 1,095.