8.5.2 Specific Portfolio

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.

Exhibit 8.15: Time line for the example. Time is measured in trading days. Valuation is 2^nd-day.

Exhibit 8.16: Foreign exchange forward portfolio.

Exhibit 8.17: Calculating portfolio holdings for the foreign exchange forward portfolio. Only assets ¹S_i in which the portfolio has active holdings are indicated. Maturities are actual days as of time 0.

Based upon those calculations

[8.57]

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

[8.58]

[8.59]

[8.60]

We obtain

[8.61]

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.

Exhibit 8.18: Libor indicative interest rates, as compiled by the BBA, correspond to loans commencing and maturing on specific dates. This table indicates actual days from time t = 1 until commencement and maturity for AUD Libor quotes made at time 1, which is May 18, 2001. For AUD Libor, all loans commence spot. Because May 18 is a Friday, this is 4 actual days.

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

[8.62]

Accordingly, components b_136,7 and b_136,8 of the matrix b are .75 and .25. Remaining components b_136,i are 0, as is component a₁₃₆ of vector a. Now consider discount factor . For it, we know

[8.63]

so components b_4,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

[8.64]

so components b_3,1 = –1 with remaining components b_3,i all 0. Component a₃ = 2. Continuing in this manner, we complete the matrix b and vector a. Our mapping [8.56] is

[8.65]

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

[8.66]

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.