###### 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.

^{nd}-day.

^{1}

*S*in which the portfolio has active holdings are indicated. Maturities are actual days as of time 0.

_{i}Based upon those calculations

[8.57]

where only active holdings are indicated. Define ^{1}*P* = **ω**^{1}** S**. Tomorrow, May 18, is a trading day, so τ(1) = 1. Our mapping

^{1}

**= φ(**

*S*^{1}

**) becomes**

*Q*[8.58]

[8.59]

[8.60]

We obtain

[8.61]

which is a quadratic polynomial. To express ^{1}*P* in terms of ^{1}** R**, we must specify matrix

**and vector**

*b***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**

*a**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.

*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.

^{1}** R** 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

*a*

_{136}of vector

**. Now consider discount factor . For it, we know**

*a*[8.63]

so components *b*_{4,i} are 0 for all *i*, but component *a*_{4} 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 *a*_{3 }= 2. Continuing in this manner, we complete the matrix ** b** and vector

**. Our mapping [8.56] is**

*a*[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 ^{1}** Q** in terms of

^{1}

**. There is no independent mathematical definition of**

*R*^{1}

**to approximate, so this is a mapping. The relationship is indicated in schematic [8.40].**

*Q*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 ^{1}** R** while [8.61] depended on just 12 linear or quadratic expressions in terms of

^{1}

**. This is due to our considering only a small portfolio. The dimensionality of**

*Q*^{1}

**is 47 while that of**

*R*^{1}

**is 1097. If we considered larger portfolios, our mapping [8.66] in terms of**

*Q*^{1}

**would entail a maximum of 45 linear or quadratic expressions while the mapping [8.61] in terms of**

*R*^{1}

**could entail as many as 1,095.**

*Q*