PRICING AN INTEREST RATE SWAP
Figure 8.1 portrays an interest rate swap in the customary box-and-arrow format. Party A and Party B agree to exchange an interest rate that varies from period to period, specifically 3-month LIBOR (hence, it's the “floating” rate), for a fixed rate of 3.40% on a quarterly basis for two years. Net settlement payments are in arrears, meaning 3-month LIBOR is determined at the beginning of the period and then a payment for the rate difference, times the notional principal, times the fraction of the year, is made at the end of the period. Importantly, there is no exchange of principal at initiation or at maturity. That's why the principal is merely “notional” – it's the scale factor for the transaction.
FIGURE 8.1 Two-Year, Quarterly Net Settlement, Interest Rate Swap 3.40% Fixed versus 3-Month LIBOR
Party A, the fixed-rate payer and floating-rate receiver, sometimes is said to be the “buyer” of the swap, or is “long” the swap. Party B, the fixed-rate receiver and floating-rate payer, then is the “seller” and is “short” the swap. In this context, the reference rate (here 3-month LIBOR) is the presumptive commodity, and the fixed rate is the price paid or received for it. In practice, Party A is often just called the payer and Party B is the receiver, referring to the fixed-rate leg of the transaction.
Settlement payments are easily calculated on interest rate swaps. They depend on the specific day-count convention, payment frequency, and the amount of notional principal. For simplicity, I assume quarterly settlements on a 30/360 basis for both rates, although actual/360 is common with actual transactions. Suppose the notional principal is $60 million and 3-month LIBOR is 2.25%. Party A owes its counterparty $172,500 at the end of the quarter.
If 3-month LIBOR is 3.90%, Party B owes $75,000.
Where does the fixed rate of 3.40% on this 2-year interest rate swap come from? The answer is that it is the “average” of the first two years of the forward curve, specifically the sequence of forward rates on 3-month LIBOR. Where those come from is the real challenging question that I get to later. For now, let's take as given the rates in Table 8.1. The notation is
TABLE 8.1 Initial 3-Month LIBOR Forward Curve and Calculated Implied Spot Rates and Discount Factors
Time Period |
Forward Rates |
Time Period |
Implied Spot Rates |
Discount Factors |
0×3 |
0.5000% |
0×3 |
0.5000% |
0.998752 |
3×6 |
1.5821% |
0×6 |
1.0407% |
0.994817 |
6×9 |
2.6694% |
0×9 |
1.5829% |
0.988222 |
9×12 |
3.7647% |
0×12 |
2.1272% |
0.979008 |
12×15 |
3.7468% |
0×15 |
2.4506% |
0.969922 |
15×18 |
4.4047% |
0×18 |
2.7756% |
0.959358 |
18×21 |
5.0696% |
0×21 |
3.1025% |
0.947352 |
21×24 |
5.7427% |
0×24 |
3.4316% |
0.933943 |
the same as in Chapter 5 for implied forwards. The 3x6 forward rate of 1.5821% means the 3-month rate on a transaction between months 3 and 6 – the first number is the starting month and the second the ending month. The 21 x 24 forward rate of 5.7427% means the 3-month rate between months 21 and 24. (Alternate notation for these forward rates is 3m3m and 21m3m, where the first number is the forward time period and the second is the tenor of the rate.) The rates are annualized as usual, all for a periodicity of 4 because of the assumed 30/360 day-count. This forward curve is quite steep. A true believer in the expectations theory is expecting a dramatic jump in rates over the next year. Also, notice the quirky 12 x 15 rate – the forward curve is not always smooth or monotonically increasing or decreasing.
These forward rates can be used to bootstrap the sequence of implied spot rates, which also are shown in Table 8.1. They are annual rates for quarterly compounding, same as the LIBOR forwards. Technically, they are computed as the geometric average of the forward rates. A few examples illustrate the process.
Notice the bootstrapping – the result for one time frame is used as an input in the next as we move out along the term structure of spot rates.
The “average” of the 3-month LIBOR forward curve is the solution for SFR (swap fixed rate) in the lengthy expression to follow. The idea is that each forward rate is “monetized” by multiplying by the notional principal (assumed to be 100) and by the fraction of the year (0.25). That amount is discounted using the corresponding implied spot rate. SFR, which is the same for every period, is “monetized” as well using the same notional principal, the same day-count factors, and the same spot rates.
For the 2-year swap on 3-month LIBOR, SFR = 3.40%.
The key idea in this swap pricing equation is that the forward curve indicates the sequence of “hedge-able” future 3-month spot rates. The noarbitrage condition is that the present values of each leg of the swap are equal. Therefore, the initial value of the interest rate swap is zero to both counterparties. Notice that if we included the costs and risks of hedging the future spot rates, our forward curve would become a forward cone or cylinder. Then we would not get a precise answer; instead we would have a range for the swap fixed rate. Another way of expressing this is that 3.40% is the mid-market rate around which a swap market maker will build the bid-ask spread to reflect the costs and risks of hedging.
This approach portrays a general model of swap pricing that clearly begs for spreadsheet analysis using the discount factors instead of the spot rates. Recall from Chapter 5 that spot rates and discount factors contain the same information. Spot rates are more visual in that it is easier to see the discounting process at work, whereas discount factors lend themselves to the precision produced on a spreadsheet. Using the discount factors in
Table 8.1, which are calculated using equation 5.9, the same swap fixed rate is obtained: SFR = 3.40%.
This is a 2-year plain vanilla swap in that the notional principal is constant and the transaction starts immediately. We could have the notional principal vary from period to period or even be zero for the first few periods. The same pricing model can be used to calculate the fixed rate for such “flavored” varieties as a varying notional principal swap or a forward-starting swap. The principle is that the swap fixed rate is the “average” of the relevant segment of the LIBOR forward curve, appropriately weighted. Notice that I'm neglecting transactions costs and counterparty credit risk in these calculations.
Where does the forward curve for the reference rate come from? It could be the sequence of implied forward rates, as calculated in Chapter 5, given a series of cash market bonds issued by banks. However, there is a Eurodollar futures contract on 3-month LIBOR, and the observed rates from that futures market serve as a foundation for the forward curve. In general, market makers set the bid-and-ask rates on over-the-counter (OTC) derivatives such as forwards and swaps based on the cost and risk of the most efficient way of hedging risk. When available, actively traded futures contracts offer the best hedge. When a commercial bank is the OTC market maker, it is in effect a financial risk intermediary between an end user and the futures exchange.