Defining the Par Term Structure
Step 1: Select an appropriate term structure
Based on the interest rate swap being priced an appropriate term structure or structures will be chosen. This is an important process because both the zero curve and the forward curves are derived from it which are in turn used to discount the cash flows and calculate the future coupon rates for the floating legs of the transaction.
Let us assume that we select the interbank offer rates as our par term structure. The interbank offer rate structure that we have chosen has quoted rates for terms 3 months, 6 months, 1 year, 2 years and 3 years as follows:
Interbank Offer Rates | Bid | Offer | Average |
3m | 11.87% | 12.12% | 12.00% |
6m | 11.99% | 12.24% | 12.12% |
1yr | 12.15% | 12.65% | 12.40% |
2yr | 12.27% | 12.77% | 12.52% |
3yr | 12.38% | 12.88% | 12.63% |
Step 2: Extending the term structure
If the term of the IRS is greater than the maximum tenor of the selected par term structure than rates from another available term structure having longer tenors may be used as a proxy to the selected par term structure to supplement it. For example a Treasury term structure that extends to longer tenors such as 20 years and 30 years may be used.
Tenor | Treasury Rate |
1yr | 12.21% |
2yr | 12.32% |
3yr | 12.36% |
4yr | 12.41% |
5yr | 12.45% |
6yr | 12.51% |
7yr | 12.56% |
8yr | 12.61% |
9yr | 12.60% |
10yr | 12.61% |
15yr | 12.85% |
20yr | 13.07% |
30yr | 13.21% |
Step 3: Creating a default term structure
Adding a spread
The rates quoted for respective term structures are based on the underlying instruments. Treasury term structures are based on treasury instruments that are usually compounded on semiannual basis. Interbank offer rates on the other hand tend to be based on corporates that are compounded on a quarterly basis. These rates, presented on an annualized basis, are therefore usually nominal rates compounded based on the payment frequency of the underlying instruments.
Depending on the IRS to be priced spreads (positive or negative) may need to be added to the rates to account for the differentials between quoting basis and the payment frequency of the IRS. The resulting rates would then be considered as been quoted on the appropriate basis.
For illustration purposes we assume that no spreads are added to any of the rates in the term structure. The default term structure in our example consists of 1 year, 2 years, 3 years and 4 years tenors. As mentioned above our primary term structure is the interbank term structure with rates for tenors exceeding the maximum tenor being taken from the Treasury term structure. The resulting default term structure is as follows:
Tenor (in years) | Default Term Structure |
1 | 12.15% |
2 | 12.27% |
3 | 12.38% |
4 | 12.41% |
b) Interpolating par rates for intermediate tenors
Depending on the payment frequency of the IRS, par rates for intermediate tenors may need to be calculated. For example if the payment frequency was semiannual then a rate for tenor 1.5 years would be required. However this is not readily available in the default term structure. The rate would be interpolated from the rates of the adjacent tenors, i.e. the 1 year and 2 year rates as follows:
T= intermediate tenor =1.5 years
T_{1}= 1 year
T_{2}=2 years
If the payment frequency was quarterly then the rates for tenors 1.25 years, 1.5 years and 1.75 years would also be required for the period 1 – 2 years. The rate for 1.25 years would be calculated from the available 1 year and 2 year rates using interpolation as follows:
T= intermediate tenor =1.25 years
T_{1}= 1 year
T_{2}=2 years
In our example however, we assume that the underlying payment frequency for the IRS is annual. Therefore our default par term structure is as given above, i.e.:
Tenor (in years) | Default Term Structure |
1 | 12.15% |
2 | 12.27% |
3 | 12.38% |