8.2.6 Example: Random Discount Curve
Again, suppose today is May 10, 2001, but now we want to construct the 2nd-day valuation discount curve for May 11. That’s tomorrow! The discount curve we are about to construct is random. Tomorrow, just as we did today, we will observe USD Libor rates for overnight, 1-week and 2-week loans. Denote these rates with random variables 1R1, 1R2, and 1R3. From these we will be able to calculate daily 2nd-day valuation discount factors. Denote these random variables 1Qi, corresponding to discount factors for i basis days, i = 0, 1, … , 15. Note that these are maturities, so we can measure them relative to any date. In this case, we measure them from the time the Libor rates will be quoted. That is time 1, which is May 11.
Based upon the calendar of Exhibit 8.3, we determine the basis day counts for the loans associated with the three Libor rates. These are indicated in Exhibit 8.10:
We are using 2nd-day valuation. From May 11, 2 trading days will be 4 basis days, so our value date is May 15. The discount factor 1Q4 is not random; it will equal 1.0. Discount factors for maturities of less than 4 basis days will be accumulation factors. They are random, but we know they will exceed 1.0. As before, we assume the tom-next rate equals the overnight rate 1R1. We obtain discount factors by accumulating to the value date:
[8.5]
[8.6]
We obtain the discount factor 1Q11 from the 1-week Libor rate by discounting 7 days, from May 22 back to the value date May 15:
[8.7]
We are only interested in discount factors for maturities out to 15 actual days. However, for interpolating some of the later maturities, we need the 18-day discount factor:
[8.8]
Linearly interpolating between values, we obtain the remaining discount factors. With Exhibit 8.11 and the formulas it references, we have defined a mapping 1Q = φ(1R).
Exercises
Measure time t in trading days. The spot-next USD Libor rate for time 1 corresponds to a short-term loan commencing spot (in two trading days) and maturing one trading day after it commences. Assume basis days equal actual days. Use the day count function τ to represent:
- the number of basis days from time 0 until that loan commences and until it matures;
- the number of basis days from time 1 until that loan commences and until it matures.
Assume basis days equal actual days. Use a 1-day value-at-risk horizon and 2nd-day valuation. Represent discount factors as 1Qi, where i is the number of basis days from time 1 to the maturity of the cash flow being discounted. Suppose today is November 7, 2002. A USD 1MM cash flow will be received on November 15, 2002. You represent its value at the end of the value-at-risk horizon with the formula
[8.9]
In this formula, what should the subscript i be? (see calendar)
Solution