Fundamentals of Futures and Options Markets John C. Hull Eighth Edition

Transcription

1 Fundamentals of Futures and Options Markets John C. Hull Eighth Edition

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3 Swaps basis points at the end of December 2011 as a result of concerns about European countries such as Greece. 7.7 VALUATION OF INTEREST RATE SWAPS We now move on to discuss the valuation of interest rate swaps. An interest rate swap is worth close to zero when it is first initiated. After it has been in existence for some time, its value may be positive or negative. Each exchange of payments in an interest rate swap is a forward rate agreement (FRA) where interest at a predetermined fixed rate is exchanged for interest at the LIBOR floating rate. Consider, for example, the swap between Microsoft and Intel in Figure 7.1. The swap is a three-year deal entered into on March 5, 2013, with semiannual payments. The first exchange of payments is known at the time the swap is negotiated. The other five exchanges can be regarded as FRAs. The exchange on March 5, 2014, is an FRA where interest at 5% is exchanged for interest at at the sixmonth LIBOR rate observed in the market on September 5, 2013; the exchange on September 5, 2014, is an FRA where interest at 5% is exchanged for interest at at the six-month LIBOR rate observed in the market on March 5, 2014; and so on. As shown at the end of Section 4.7, an FRA can be valued by assuming that forward rates are realized. Because it is nothing more than a portfolio of FRAs, an interest rate swap can also be valued by assuming that forward rates are realized. The procedure is as follows: 1. Calculate forward rates for each of the LIBOR rates that will determine swap cash flows. 2. Calculate the swap cash flows on the assumption that LIBOR rates will equal forward rates. 3. Discount the swap cash flows at the risk-free rate. Example 7.2 provides an illustration. The procedure requires us (a) to estimate a riskfree zero curve for discounting and (b) to estimate forward LIBOR rates for each of the payments underlying the swap. The next two sections deal with some technical issues concerned with the estimates and the difference between using LIBOR and OIS rates for discounting. These issues are of great concern to derivatives practitioners, but some readers, once they have understood Example 7.2, may wish to jump to Example 7.6 and Section ESTIMATING THE ZERO CURVE FOR DISCOUNTING Derivatives practitioners have traditionally used LIBOR and swap rates to define a risk-free zero curve and have used this curve to calculate discount rates when valuing derivatives. However, the credit crisis has caused this practice to be questioned. As explained in Sections 7.5 and 7.6, LIBOR rates incorporate a risk premium to allow for the possibility of a bank counterparty experiencing financial difficulties and defaulting on an interbank loan. Many derivatives dealers choose to use OIS rates when determining the discount rate for collateralized transactions and LIBOR/swap 171

4 172 CHAPTER 7 Example 7.2 Valuing an interest rate swap using FRAs Suppose that some time ago a financial institution entered into a swap where it agreed to make semiannual payments at a rate of 3% per annum and receive LIBOR on a notional principal of $100 million. The swap now has a remaining life of 1.25 years. Payments will therefore be made 0.25, 0.75, and 1.25 years from today. The risk-free rates with continuous compounding for maturities of 3 months, 9 months and 15 months are 2.8%, 3.2%, and 3.4%. We suppose that the forward LIBOR rates for the 3-month to 9-month and the 9-month to 15-month periods are 3.4% and 3.7%, respectively, with continuous compounding. Using equation (4.4), the 3-month to 9-month forward rate becomes 2 ðe 0:0340:5 1Þ, or 3.429%, with semiannual compounding. Similarly, the 9-month to 15-month forward rate becomes 3.734% with semiannual compounding. The LIBOR rate applicable to the exchange in 0.25 years was determined 0.25 years ago. We suppose that it is 2.9% with semiannual compounding. The calculation of swap cash flows on the assumption that LIBOR rates will equal forward rates and the discounting of the cash flows are shown in the following table. (All cash flows are in millions of dollars.) Time (years) Fixed cash flow Floating cash flow Net cash flow Discount factor Present value of net cash flow þ þ þ þ þ þ þ Total Consider, for example, the 0.75 year row. The fixed cash flow is 0:5 0:03 100, or $ million. The floating cash flow, assuming forward rates are realized, is 0:5 0: , or $ million. The net cash flow is therefore $ million. The discount factor is e 0:0320:75 ¼ 0:9763, so that the present value is 0:2145 0:9763 ¼ 0:2094. The value of the swap is obtained by summing the present values. It is $ million. (Note that these calculations do not take account of holiday calendars and day count conventions.) rates when determining the discount rate for noncollateralized transactions. 10 (See Section 2.5 for a discussion of the use of collateral in OTC markets.) In this section and the next, we explain how valuation is accomplished using both assumptions. We start by considering the construction of the risk-free zero curve. Determining Zero Rates for LIBOR Discounting When LIBOR rates and swap rates are used to define discount rates, the value of a newly issued floating-rate bond that pays LIBOR is always equal to its principal value 10 In J. Hull and A. White, LIBOR vs. OIS: The Derivatives Discounting Dilemma, Working Paper, University of Toronto, 2012, it is argued that OIS rates should be used for discounting both collateralized and noncollateralized transactions. 172

5 Swaps 173 Example 7.3 Determining zero rates from swaps Suppose that the 6-month, 12-month, and 18-month LIBOR/swap zero rates have been determined as 4%, 4.5%, and 4.8% with continuous compounding and that the 2-year swap rate (for a swap where payments are made semiannually) is 5%. This 5% swap rate means that a bond with a principal of $100 and a semiannual coupon of 5% per annum sells for par. It follows that, if R is the 2-year zero rate, then 2:5e 0:040:5 þ 2:5e 0:0451:0 þ 2:5e 0:0481:5 þ 102:5e 2R ¼ 100 Solving this, we obtain R ¼ 4:953%. (Note that this calculation is simplified in that it does not take the swap s day count conventions and holiday calendars into account. See Section 7.2.) (or par value). The reason is that the bond provides a rate of interest of LIBOR and LIBOR is the discount rate so that the interest on the bond exactly matches the discount rate. As a result, the bond is fairly priced at par. As shown in Table 7.2, a swap can be regarded as the exchange of a fixed-rate bond for a floating-rate bond. In equation (7.1) we showed that, for a newly issued swap, the value of the fixed-rate bond equals the value of the floating-rate bond. We have just argued that the value of the floating-rate bond equals the notional principal. It follows that the value of the fixed-rate bond also equals the swap s notional principal. Swap rates therefore define a set of par yield bonds. For example, from the swap rates in Table 7.3 we can deduce that the two-year LIBOR/swap par yield is 6.045%, the threeyear LIBOR/swap par yield is 6.225%, and so on. 11 Section 4.5 shows how the bootstrap method can be used to determine the Treasury zero curve from Treasury bill and Treasury bond quotes. It can be used in a similar way to obtain the LIBOR/swap zero curve from the par yield bonds that are defined by swap rates. Example 7.3 illustrates this. Forward rates calculated from Eurodollar futures are sometimes used to help bootstrap the LIBOR/swap zero curve. A forward rate for time T 1 to time T 2 can be used to calculate the T 2 zero rate from the T 1 zero rate. Determining Zero Rates for OIS Discounting When OIS rates are used to define risk-free rates for discounting, the procedure for constructing the OIS zero curve is the similar to that just given for LIBOR/swap rates. The one-month OIS rate defines the one-month zero rate, the three-month OIS rate defines the three-month zero rate, and so on. When there are periodic settlements in the OIS contract, the OIS rate defines a par yield bond. Suppose, for example, that the fiveyear OIS rate is 3.5% with quarterly settlement. (This means that, at the end of each quarter, 3.5% is exchanged for the geometric average of the overnight rates during the quarter.) A bond paying a quarterly coupon at a rate of 3.5% per annum would be assumed to sell for par. LIBOR interest rate swaps are traded for longer maturities than OIS swaps. If the OIS zero curve is required for long maturities, a natural approach is to assume that 11 Analysts frequently interpolate between swap rates before calculating the zero curve. For example, for the data in Table 7.3, the 2.5-year swap rate could be assumed to be 6.135%, the 7.5-year swap rate could be assumed to be 6.696%, and so on. 173

6 174 CHAPTER 7 the spread between the OIS rates and LIBOR swap rates is the same at the long end as it is for the longest OIS maturity for which there is reliable data. Suppose, for example, that there is no reliable data on OIS swaps for maturities longer than five years. If at the five-year point the LIBOR OIS spread is 20 basis points, this could be assumed to continue at the same level beyond five years. LIBOR OIS basis swaps (see Section 7.16), which have longer maturities than OIS swaps, can also sometimes be used. 7.9 FORWARD RATES The calculation of forward LIBOR rates depends on the rates used for discounting. This means that again we must consider two cases. Forward Rates When LIBOR Discounting Is Used When LIBOR/swap rates are used to determine risk-free discount rates, forward LIBOR rates can be calculated using equation (4.5). (Note that the data given in Example 7.2 is consistent with the assumption of LIBOR discounting. For example, the three-month and nine-month rates with continuous compounding were 2.8% and 3.2% and the forward rate given for the three to nine month period was given as 3.4% with continuous compounding. Because 0:032 0:75 0:028 0:25 0:5 ¼ 0:034 this is what equation (4.5) would give.) An alternative approach to calculating the forward rates, which can be used in more general situations, is to use ensure that swaps of all maturities are worth zero. This approach is illustrated in Example 7.4. The example shows that the forward rate calculated for the 18 to 24 month period using the approach agrees with the forward rate given by equation (4.5). Forward Rates When OIS Discounting Is Used When OIS rates are assumed to determine risk-free discount rates, it is necessary to follow an approach similar to that in Example 7.4 and bootstrap forward rates in such a way that swaps of different maturity have a value of zero. This is illustrated in Example 7.5. The OIS zero rates for 6, 12, 18, and 24 months, with continuous compounding, are assumed to be 3.8%, 4.3%, 4.6%, and 4.75% (about 20 basis points less than the corresponding LIBOR/swap zero rates in Example 7.4). The 6-month LIBOR rate, 6 to 12 month LIBOR forward rate, and the 12 to 18 month LIBOR forward rate are assumed to have been already calculated and to be the same as in Example 7.4. The calculations in Example 7.5 show that the 18 to 24 month forward rate is 5.483%, about 0.2 basis points less than when LIBOR is assumed to be risk-free. In the highly competitive interest rate swap market, the impact of this small difference when combined with the impact of lower discount rates has an effect of swap valuations which market participants cannot ignore. 174

7 Swaps 175 Example 7.4 Bootstrapping LIBOR forward rates with LIBOR discounting Consider again the data in Example 7.3. The 6-, 12-, 18-month, and (calculated) 24-month rates are, with continuous compounding, 4%, 4.5%, 4.8%, and 4.953%, respectively. The 6-month rate is 4.040% with semiannual compounding. From equation (4.5), the forward rate for the period between 6 and 12 months is 5% with continuous compounding or 5.063% with semiannual compounding. Similarly, the forward rate for the period between 12 and 18 months is 5.4% with continuous compounding or 5.474% with semiannual compounding. We suppose that we do not know the forward rate for the 18 to 24 month period. The two-year swap rate is 5%. This means that a swap where LIBOR is exchanged for 5% is worth zero. It follows that the sum of the values of (a) the known exchange at 6 months and (b) the forward contracts corresponding to the exchanges at 12, 18, and 24 months must be zero. This can be used to determine the forward rate for the 18 to 24 month period. The value of the first payment, assuming fixed is paid and floating is received on a principal of 100, is 0:5 ð0:4040 0:5000Þ100 e 0:040:5 ¼ 0:4704 The value of the second payment is 0:5 ð0: :000Þ100 e 0:0451 ¼ 0:0301 The value of the third payment is 0:5 ð0: :000Þ100 e 0:0481:5 ¼ 0:2203 The total value of the first three payments is 0:4704 þ 0:0301 þ 0:2203 ¼ 0:2199. Suppose that the (assumed unknown) forward rate for the final payment is F with semiannual compounding. For the swap to be worth zero, we must have 0:5 ðf 0:05Þ100 e 0: ¼ 0:2199 This gives F ¼ 0:05486, or 5.486%. This is 5.412% with continuous compounding. It is consistent with using equation (4.5) in conjunction with the 18-month zero rate of 4.8% and the 24-month zero rate of 4.953% because ð2:0 0: :5 0:048Þ=0:5 ¼ 0: VALUATION IN TERMS OF BONDS Principal payments are not exchanged in an interest rate swap. However, as illustrated in Table 7.2, we can assume that principal payments are both received and paid at the end of the swap without changing its value. By doing this, we find that a swap where fixed cash flows are received and floating cash flows are paid can be regarded as a long position in a fixed-rate bond and a short position in a floating-rate bond, so that V swap ¼ B fix B fl where V swap is the value of the swap, B fl is the value of the floating rate bond underlying 175