Interest Rate and Currency Swaps

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1 Interest Rate and Currency Swaps Eiteman et al., Chapter 14 Winter 2004 Bond Basics Consider the following: Zero-Coupon Zero-Coupon One-Year Implied Maturity Bond Yield Bond Price Forward Rate t r 0 (0,t) P (0,t) r 0 (t 1,t) 1 Year 6.00% % 2 Years 6.50% % 3 Years 7.00% % Note: the above forward rates are forward interest rates. 2

2 Bond Basics For each time to maturity t, bond prices are obtained as follows: P(0,t) = 1 ( 1 + r0 (0,t) ) t. 3 Bond Basics That is, P(0, 1) = 1 (1+r 0 (0,1)) 1 = = P(0, 2) = 1 (1+r 0 (0,2)) 2 = 1 (1.065) 2 = P(0, 3) = 1 (1+r 0 (0,3)) 3 = 1 (1.070) 3 =

3 Bond Basics For each time t 1 and t 2, the implied forward interest rate r(t 1,t 2 ) is such that (1 + r 0 (0,t 1 )) t 1 (1 + r 0 (t 1,t 2 )) t 2 t 1 = (1 + r 0 (0,t 2 )) t 2. This gives (1 + r 0 (t 1,t 2 )) t 2 t 1 = (1 + r 0(0,t 2 )) t 2 (1 + r 0 (0,t 1 )) t 1. 5 Bond Basics In the above table, we have r 0 (1,2) = (1 + r 0(0,2)) 2 (1 + r 0 (0,1)) 1 1 = (1.065)2 (1.060) 1 1 = % r 0 (2,3) = (1 + r 0(0,3)) 3 (1 + r 0 (0,2)) 2 1 = (1.070)3 (1.065) 2 1 = % 6

4 Bond Basics Note that (1 + r 0 (t 1,t 2 )) t 2 t 1 = (1 + r 0(0,t 2 )) t 2 (1 + r 0 (0,t 1 )) t 1 = P(0,t 1) P(0,t 2 ). 7 Bond Basics Combinations of actual zero-coupon bond yields also give us implied forward zero-coupon bond prices: P(t 1,t 2 ) = 1 (1 + r 0 (t 1,t 2 )) t = (1 + r 0(0,t 1 )) t1 2 t 1 (1 + r 0 (0,t 2 )) t 2 = P(0,t 2) P(0,t 1 ). 8

5 Bond Basics The implied forward zero-coupon bond prices in the present example are P(1,2) = P(0,2) P(0,1) = = P(2,3) = P(0,3) P(0,2) = = Forward Rate Agreements Consider the problem of a borrower who wishes to hedge against increases in the cost of borrowing. Suppose a firm expects to borrow $100m for 91 days, beginning 120 days from today, in June. The loan will be repaid in September. Suppose the effective quarterly interest rate at that time can either be 1.5% or 2%, implying a borrowing cost of $1.5m or $2m, a difference of $500,

6 Forward Rate Agreements To hedge against this uncertainty, the firm could enter into a forward rate agreement (FRA). A FRA is an over-the-counter contract that guarantees a borrowing or lending rate on a given notional amount. FRAs can be settled either at the initiation or maturity (in arrears) of the borrowing or lending transaction. 11 Forward Rate Agreements FRAs are forward contracts based on the interest rate and do not entail the actual lending of money. The borrower who enters a FRA is paid if a reference rate is above the FRA rate, and pays if the rate is below the FRA rate. 12

7 Forward Rate Agreements FRA Settlement in Arrears Let r FRA denote the FRA rate and let r q denote the prevailing quarterly rate at the time the loan was contracted. The payment to a borrower who would have previously entered into a FRA is then ( rq r FRA ) notional principal if the FRA is settled when the loan matures. 13 Forward Rate Agreements FRA Settlement in Arrears Suppose that, in the previous example, r FRA = 1.8%. Then the firm would receive ( rq ) $100m at the end of the loan period, which means ( ) $100m = $0.3m if r q = 1.5% ( ) $100m = + $0.2m if r q = 2.0%. 14

8 Forward Rate Agreements FRA Settlement at the Time of Borrowing In this case the payment made by one of the two parties to the other is simply the amount that would have been paid at the loan maturity discounted over the loan period. In the present example, the loan period is one quarter and thus the payment to the borrower would be r q r FRA 1 + r q notional principal. 15 Forward Rate Agreements FRA Settlement at the Time of Borrowing For the firm in our example, this gives $100m = $0.296m if r q = 1.5% $100m = + $0.196m if r q = 2.0%. 16

9 Forward Rate Agreements Synthetic FRAs Note that a future lending or borrowing rate can be locked in by trading zero-coupon bonds. Suppose for example that money will be borrowed at time t and the loan will be repaid at time t + s. The borrower wants to lock in r(t,t + s) in advance. How can this be done? 17 Forward Rate Agreements Synthetic FRAs Recall that (1 + r 0 (t,t + s)) s = P(0,t) P(0,t + s), where the subscript 0 is used to emphasize the fact that this rate is determined at time 0. 18

10 Forward Rate Agreements Synthetic FRAs Take s as the reference period. That is, s could be a quarter and thus r 0 (t,t + s) a quarterly rate. Then 1 + r 0 (t,t + s) = P(0,t) P(0,t + s). 19 Forward Rate Agreements Synthetic FRAs Consider a portfolio buying 1 zero-coupon bond maturing at time t and selling short 1 + r 0 (t,t + s) zero-coupon bonds maturing at time t + s. The payoff of this portfolio is (1 + r 0 (t,t + s))p(0,t + s) P(0,t) = 0 today, +1 at time t, (1 + r 0 (t,t + s)) at time t + s. 20

11 Forward Rate Agreements Synthetic FRAs The above payoff is the same as the payoff to a borrower entering a FRA to be settled in arrears with r FRA = r 0 (t,t + s). 21 Forward Rate Agreements Synthetic FRAs If the zero-coupon bond maturing at time t + s is repaid at time t, payoffs are (1 + r 0 (t,t + s))p(0,t + s) P(0,t) = 0 today, 1 (1+r 0(t,t+s)) (1+r t (t,t+s)) = r t(t,t+s) r 0 (t,t+s) 1+r t (t,t+s) at time t, which is the payoff to a borrower entering a FRA to be settled at the beginning of the loan period with r FRA = r 0 (t,t + s). 22

12 Forward Rate Agreements Synthetic FRAs In the previous slide, r t (t,t + s) denotes the interest from time t to time t + s as determined at time t. It is the time-t spot interest rate. 23 Forward Rate Agreements Synthetic FRAs Continuing the example of the firm willing to borrow $100m, suppose P(0, 211) = and P(0, 120) = The implied of forward rate for the 91-day period starting 120 days from now is then P(0, 120) P(0,211) 1 = = 1.8%. So the cost of times a zero-coupon bond maturing in 211 days is the same as a zero-coupon bond maturing in 120 days. 24

13 Eurodollars Futures The Eurodollar futures contract is one of the most widely used interest rate futures contract. Take the 3-month eurodollar futures as an example. The yield of a futures contract is calculated from the settlement price. If the settlement price of the 3-month eurodollar future maturing in March 2005 is 95.68, the annual yield over the 3-month period ending in March 2005 is expected to be = 4.32%, for a 3-month rate of 1.08%. 25 Eurodollars Futures Eurodollar futures can be used to hedge against interest risk as follows: Borrower: Sell Eurodollar futures. If interest rates go up, futures prices will decrease and the gains from the futures trades will compensate for the increased borrowing costs. Lender: Buy Eurodollar futures. 26

14 Interest Rate Swaps Suppose firm XYZ borrows at the London Interbank Offered Rate (LIBOR), which is a variable rate, but would prefer paying a fixed rate. The loan contract is for 3 periods and the actual and expected rates are as in the table used before. XYZ could enter into a swap agreement with a swap dealer wherein XYZ would pay ( LIBOR) notional principal to the swap dealer each period, % being the swap rate. 27 Interest Rate Swaps XYZ having to pay the LIBOR times the notional principal to whomever it borrowed the money each period, its net payoff is then % times the notional principal. Where does the rate % come from? 28

15 Interest Rate Swaps Let R denote the fixed rate of interest agreed upon in the swap agreement and let r t denote the variable LIBOR at time t. The payoff to the swap dealer per unit of the notional principal is then each period. R r t 29 Interest Rate Swaps The swap dealer could eliminate his own interest rate risk by entering into FRAs, in which case is net payoff each period would be R r t + r t r 0 (t 1,t) = R r 0 (t 1,t). 30

16 Interest Rate Swaps The loan being over three periods, the swap rate R must be such that R r 0 (0,1) 1+r 0 (0,1) + R r 0(1,2) + R r 0(2,3) (1+r 0 (0,2)) 2 (1+r 0 (0,3)) 3 which gives R = %. = R R (1.065) 2 + R (1.070) 3 = 0, 31 Interest Rate Swaps More generally, letting T denote the number periods covered by the swap agreement, R must be such that which gives T P(0,t)(R r 0 (t 1,t)) = 0, t=1 R = T t=1 P(0,t)r 0(t 1,t) T t=1 P(0,t). 32

17 Interest Rate Swaps Since r 0 (t 1,t) = P(0,t 1) P(0,t) 1, we can write R = T t=1 (P(0,t 1) P(0,t)) t=1 T P(0,t) = 1 P(0,T ) t=1 T P(0,t). 33 Interest Rate Swaps Swaps are contractual agreements to exchange a series of cash flows. If the agreement is for one party to swap its fixed interest rate payments for the floating interest rate payments of another, it is termed an interest rate swap. If the agreement is to swap currencies of debt service obligations, it is termed currency swap. A single swap may combine elements of both interest rate and currency swaps. 34

18 Interest Rate Swaps A borrower with floating-rate debt who believes that interest rates are about to increase may enter into a swap agreement to pay fixed/receive floating. Similarly, a borrower with fixed-rate debt who believes that interest rates are about to fall may enter a swap agreement to pay floating/receive fixed. 35 Currency Swaps All swaps being derived from the yield curve in each major currency, the fixed- to floating-rate interest rate swap in each currency allows to swap across currencies. The motivation for a currency swap is to replace cash flows scheduled in an undesired currency with flows in a desired currency. Look at Exhibit

19 Currency Swaps Swapping floating dollars into fixed-rate Swiss francs, say, would proceed as follows: 1. First determine the rate at which the floating dollar payments can be exchanged for fixed dollar payments; 2. Find the fixed rate in Swiss francs corresponding to the fixed rate in dollars. 37 Currency Swaps How are currency swap rates determined? Let P(0,t) Zero-coupon bond price maturing at time t, S 0 Spot rate at time 0 (dollars/desired currency), F 0 (t) Forward exchange rate at time t as of time 0 (dollars/desired currency), N Notional principal in dollars, R Fixed rate in desired currency, R Fixed rate in dollars. 38

20 Currency Swaps Without a swap agreement, the present value of the borrower s payments is PV = T P(0,t)RN + P(0,T )N. t=1 Note that if the bonds are sold at par, PV = N. 39 Currency Swaps In the desired currency, the notional principal is N/S 0 and the interest payment is R N/S 0 per period. The present value of the (hedged) desired currency payments is PV = T P(0,t)F 0 (t)r N/S 0 + P(0,T )F 0 (T )N/S 0. t=1 In equilibrium, we must have PV = PV. 40

21 Currency Swaps Example 1 Take, for example, a 3-year U.S. dollar bond with N = $100 and R = 6.95%. Let the spot and forward rates $/ be S 0 = , F 0 (1) = , F 0 (2) = and F 0 (3) = The annual yields are as before, i.e. P(0, 1) = , P(0, 2) = and P(0,3) = What rate would be paid if the debt payments were all made in euros? 41 Currency Swaps Example 1 First note that at the rate 6.95% the firm s bonds are sold at par and thus PV = N. So we need to find R such that PV = T P(0,t)F 0 (t)r N/S 0 + P(0,T )F 0 (T )N/S 0 = N. t=1 42

22 Currency Swaps Example 1 This gives R = 1 P(0,T )F 0(T )/S 0 T t=1 P(0,t)F 0(t)/S Currency Swaps Example 1 In the present example, we need R = = 6.43% for the swap agreement to have a zero net present value. 44

23 Currency Swaps Example 2 The problem is much simpler if we assume the exchange rate constant over the loan period (i.e. F 0 (t) = S 0 for all t) and the annual yield to be the same over any subperiod (r 0 (0,t) = r, say, for all t). Consider then the case of a US$ debt issue sold at par with coupon rate 5.56% and face value $10,000,000. What would be the equivalent Sfr rate? 45 Currency Swaps Example 2 If the current spot rate is Sfr1.5000/$, the rate R would be such that ,000,000 = Sfr15,000,000 would also be sold at par. If the annual yield in Switzerland is 2.01%, then R will be 2.01%. 46

24 Unwinding Swaps One of the partners to a swap may wish to terminate the agreement before it matures. If the present value of the contract is not zero at the time it is terminated, one partner will have to pay a termination fee to compensate the other. 47 Unwinding Swaps Take the example of a three-year pay Swiss francs/receive US$ currency swap on a notional principal of $10m at 5.56% arranged when the spot rate is Sfr1.5000/$. The equivalent Sfr loan is Sfr15,000,000 at 2.01%. 48

25 Unwinding Swaps If the exchange rate falls to Sfr after the first year, when the US two-year rate is 5.5% and the Sfr two-year rate is 2%, then the present value of the Sfr payments is 301, ,301, = Sfr15,002,912 = $10,240,896 and the present value of the US$ payments is 556, ,556, = $10,011, Unwinding Swaps If the borrowing firm wishes to terminate the swap agreement, it will have to pay to the swap dealer. 10,240,896 10,011,078 = $229,818 50

26 Interest Rate Caps and Floors Interest Rate Cap: Option limiting the maximum interest rate to be paid over a given period. Interest Rate Floor: Option limiting the minimum interest rate to be received over a given period. 51 Swaptions A swaption is an option to enter into a swap agreement on a pre-specified notional principal at a pre-specified strike rate. 52

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