# NUMERICAL EXAMPLE: BOOTSTRAPPING AN INTEREST RATE CURVE

In order to put into practice what we have seen throughout this chapter, we shall give an example of what it means to bootstrap a simple interest rate swap curve. We shall show what can be accomplished in a so-called closed

TABLE 2.4 Market inputs used to bootstrap the interest rate curve in our example.

 Maturity Instrument type Instrument code Market quote ID Deposit ID 0.389 % 1W Deposit 1W 0.444 % 1M Deposit 1M 0.341 % 2M Deposit 2M 0.351 % 3M Deposit 3M 0.380 % 6M Deposit 6M 0.560 % 9M FRA 6MX3M 0.650 % 12M FRA 9MX3M 0.670 % 2Y Swap 2Y 0.630 % 3Y Swap 3Y 0.810 % 4Y Swap 4Y 1.030 % 5Y Swap 5Y 1.240 % 7Y Swap 7Y 1.670 % 10Y Swap 10Y 1.850 % 15Y Swap 15Y 1.960 %

form, without using a solver routine, and then discuss what the further steps might be.

Let us imagine that we have the market inputs shown in Table 2.4 at our disposal. In particular we have deposit rates, Forward Rate Agreements (FRA), and interest rate swaps. Deposit rates from one day up to six months, forward rate agreements each with three-month lengths starting in six months' time and nine months' time, respectively, and finally interest rate swaps for maturities from two years onward. We assume the date of the data is taken to be January 3, 2012.

## The Short End of the Curve: Deposits and FRAs

Our initial objective is to find those discount factors Dt such that the market instruments used are priced correctly. As far as the deposits are concerned it is fairly simple. Using the fact that the deposit rate rj is related to the discount factor DT by

we can obtain the needed discount factors. (In the above we assume, as in Equation 2.2, linear compounding. We also assume, here and throughout this exercise as stated in the footnote in Section 2.1, that the correct day count convention has been applied when using the time variable T.) For example, using the overnight deposit we obtain, assuming an actual day count by which there are 366 days in the first year,

Similarly, using the weekly deposit, we obtain that

Since the date on which the data in Table 2.4 is assumed to have been collected is January 3, 2012, 31 days elapse in the first month and thus the discount factor corresponding to the one-month point will be given by

The remaining discount factors derived from deposits can be obtained in a similar fashion.

The information contained in the two FRAs can be used to find one discount factor each if we remind ourselves of the relation between floating rates and discount factors, namely that a forward rate contained in a FRA between and is given by

where is the time elapsed calculated according to the correct day count convention. If we take the first FRA we see that , moreover we know, having calculated it from deposits, that the six-month discount factor is equal to 0.99722. Combining these two pieces of information to find the discount factors corresponding to the nine months, we have

Similarly, using the discount factor obtained previously and the rate we have from the second FRA, we can obtain the one- year discount factor