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2.5.2 The Long End of the Curve: Interest Rate Swaps

We have used the information contained in all the deposits and the FRAs to build the short end of the curve. We are going to use interest rate swaps to build the remaining long end of the curve. We know that interest rate swaps are made of two legs, each made of a series of cash flows. In order to be able to solve one variable at a time, the goal of a closed-form methodology, we need to use an inductive process, one that uses knowledge recently built in order to solve for the variable at hand. This is also the definition of the term bootstrapping.

We know that a single currency fixed for floating swap with maturity T and paying a swap rate Cт is given by

where we have allowed for the two legs to have different payment frequencies. If we add a final payment of the entire principal on both sides, we can use the assumption that a floating leg with final principal payment is always equal to unity, and therefore

The essence of a bootstrapping process is to check up to where we have data points, go one step further, and solve for that extra data point. We then repeat the process extending it by one point and consider the previous unknown as the last known data point. This means that in the above we assume that we know everything except from Dt, which we are trying to solve for. Since DT appears twice, let us isolate it by writing the above in a slightly different form and a slightly different notation

(2.20)

where Tn indicates the time of the last cash flow, the one taking place at maturity, and Tn-1 the one immediately preceding it. Let us now use an important piece of information, the fact that the frequency of the fixed leg in the swap is annual (the one of the floating leg is of no concern to us) and the accrual method is Act/Act, meaning that the time between two points as a fraction of one year is calculated by taking the actual number of days between those two points and dividing it by the actual number of days in that year. From this information we can see that. Since we know the

discount factor for the first year, we need to use the information contained in the quote for the interest rate swap maturing in two years' time, that is,

which, using the actual values, becomes

Using the value for D2Y we can now proceed with the next data point and solve for D3Y using Equation 2.20. We need to calculate

which becomes

For the following years the principle is the same, only with longer summations. Once we have found the discount factor corresponding to year five, we notice that the next point corresponds to year seven. Since we need to find the discount factor D6Y, a reasonable approach is to assume that the market

TABLE 2.5 The output of the bootstrapping process: the discount factors, the zero rates (annually and continuously compounded), and the one-year forward rates.

Maturity

Discount factor

Zero rate (ann. comp.)

Zero rate (cont. comp.)

One-year forward rate

ID

0.99999

0.39 %

0.39 %

1W

0.99992

0.44 %

0.44 %

1M

0.99971

0.34 %

0.34 %

2M

0.99943

0.35 %

0.35 %

3M

0.99906

0.38 %

0.38 %

6M

0.99723

0.56 %

0.56 %

9M

0.99560

0.59 %

0.59 %

12M

0.99394

0.61 %

0.61 %

0.65 %

2Y

0.98753

0.63 %

0.63 %

1.18 %

3Y

0.97607

0.81 %

0.81 %

1.71 %

4Y

0.95968

1.03 %

1.03 %

2.12 %

5Y

0.93981

1.25 %

1.24 %

2.59 %

6Y

0.91604

1.47 %

1.46 %

3.07 %

7Y

0.88879

1.70 %

1.68 %

2.19 %

8Y

0.86975

1.76 %

1.74 %

2.32 %

9Y

0.85002

1.82 %

1.80 %

2.46 %

10Y

0.82964

1.88 %

1.87%

2.12 %

11Y

0.81241

1.91 %

1.89 %

2.17 %

12Y

0.79514

1.93 %

1.91 %

2.22 %

13Y

0.77786

1.95 %

1.93 %

2.27 %

14Y

0.76057

1.97%

1.95 %

2.33 %

15Y

0.74328

2.00 %

1.98 %

quote for year six, which we do not have, must be the linear interpolation between the quotes for year five and year seven, which we have. Thus

The same approach applies to the other points where we do not have an actual market quote. Table 2.5 shows the discount factors obtained using the process above and also, as a reference, the corresponding zero rate obtained with annual or continuous compounding, which are also plotted in Figure 2.9a.

The advantage of this approach is that it is quite trivial to implement. It could also be extended to approximate, and let us stress the term approximate, the discount factors obtained in the presence of a cross currency basis.

a) The zero rates of the bootstrapped curve; b) the one-year forward rates calculated from the bootstrapped discount factors.

FIGURE 2.9 a) The zero rates of the bootstrapped curve; b) the one-year forward rates calculated from the bootstrapped discount factors.

We have seen in Section 2.2.5.2 that in some foreign currencies, if we want to calculate the discount factors, we need to include the information contained in a cross currency basis swap, a swap where, as shown in Equation 2.8, a flat floating (usually USD) leg is exchanged for a floating foreign leg plus some basis. Using the bootstrap process shown above we can approximate

the foreign discount factors, and maintain the ease of calculation, by simply adding the basis onto the swap quote, that is, in Equation 2.20 have CT + bC instead of CT, where bC is the cross currency basis.

Using the previous bootstrap methodology for cross currency basis swaps is an approximation, which nonetheless is considered sufficiently valid to be used by many commercially successful trade entry and management softwares. It is an approximation because it assumes that to put the basis onto the fixed leg of a swap is equivalent to adding it on to the floating leg of a swap. Purely in terms of values, if the swap is at par and the term structure of expected forward rates is more or less flat (let us remind ourselves that one can very roughly consider the fixed rate of a swap as the average of the expected forward floating rates), the resulting discount factors will be similar to the ones obtained by solving simultaneously Equations 2.10 and 2.11. Using the above process for currency basis swaps will fail completely, but not when applied to more exotic situations, such as the one presented by cross currency basis swaps in Mexican or Chilean Pesos (MXN and CLP respectively). In the case of MXN, the cross currency basis swap is not, as one would expect, an exchange of USD LIBOR flat for MXN LIBOR plus basis, but MXN LIBOR flat versus USD LIBOR plus basis

In the above we can immediately see how we cannot assume the USD leg to be equal to par (the point of view of a USD-centered institution, which we have always taken), the usual convenient way of eliminating from the equation and concentrating on solving for the discount factors of the currency under consideration. In the above we need to solve simultaneously for and and this cannot be done in a closed form.

 
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