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 r0(0, t) P (0, t) r0(t−1, t)
1 Year 6.00% 0.943396 6.00000%
2 Years 6.50% 0.881659 7.00236%
3 Years 7.00% 0.816298 8.00705%
For each time t1 and t2, the implied forward interest rate r(t1,t2) is such that (1 + r0(0,t1))t1× (1 + r0(t1,t2))t2−t1 = (1 + r0(0,t2))t2. This gives (1 + r0(t1,t2))t2−t1 = (1 + r0(0,t2))t2 (1 + r0(0,t1))t1 . 5
Bond Basics
In the above table, we have
Note that (1 + r0(t1,t2))t2−t1 = (1 + r0(0,t2))t2 (1 + r0(0,t1))t1 = P(0,t1) P(0,t2) . 7
Bond Basics
Combinations of actual zero-coupon bond yields also give us implied forward zero-coupon bond prices:
The implied forward zero-coupon bond prices in the present example are P(1, 2) = P(0, 2) P(0, 1) = 0.881659 0.943396 = 0.934559 P(2, 3) = P(0, 3) P(0, 2) = 0.816298 0.881659 = 0.925865 9
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.
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.
FRA Settlement in Arrears
Let rFRA denote the FRA rate and let rq 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 − rFRA
¢
× 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, rFRA = 1.8%. Then the
firm would receive
¡
rq − 0.018 ¢
× $100m
at the end of the loan period, which means
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
rq − rFRA
1 + rq
× notional principal.
15
Forward Rate Agreements
FRA Settlement at the Time of Borrowing
For the firm in our example, this gives 0.015 − 0.018
1.015 × $100m = − $0.296m if rq = 1.5% 0.020 − 0.018
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 + r0(t,t + s))s =
P(0,t) P(0,t + s),
Synthetic FRAs
Take s as the reference period. That is, s could be a quarter and thus r0(t,t + s) a quarterly rate. Then
1 + r0(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 + r0(t,t + s) zero-coupon bonds maturing at time t + s. The payoff of this portfolio is
(1 + r0(t,t + s))P(0,t + s) − P(0,t) = 0 today,
+1 at time t,
Synthetic FRAs
The above payoff is the same as the payoff to a borrower entering a FRA to be settled in arrears with rFRA = r0(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 + r0(t,t + s))P(0,t + s) − P(0,t) = 0 today, 1 − (1+r0(t,t+s)) (1+rt(t,t+s)) = rt(t,t+s)−r0(t,t+s) 1+rt(t,t+s) at time t,
Synthetic FRAs
In the previous slide, rt(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) = 0.95836 and P(0, 120) = 0.97561. The implied of forward rate for the 91-day period starting 120 days from now is then
P(0, 120)
P(0, 211) − 1 =
0.97561
0.95836 − 1 = 1.8%.
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 100 − 95.68 = 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.
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
(0.069548 − LIBOR) × notional principal
to the swap dealer each period, 6.9548% 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 6.9548% times the notional principal.
Let R denote the fixed rate of interest agreed upon in the swap agreement and let ˜rt denote the variable LIBOR at time t.
The payoff to the swap dealer per unit of the notional principal is then
R − ˜rt
each period.
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
The loan being over three periods, the swap rate R must be such that R−r0(0,1) 1+r0(0,1) + R−r0(1,2) (1+r0(0,2))2 + R−r0(2,3) (1+r0(0,3))3 = R−.060 1.060 + R−.070024 (1.065)2 + R−.080071 (1.070)3 = 0, which gives R = 6.9548%. 31
Interest Rate Swaps
More generally, letting T denote the number periods covered by the swap agreement, R must be such that
Since r0(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 t=1P(0,t) = 1 − P(0, T ) ∑T t=1P(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 aninterest rate swap.
• If the agreement is to swap currencies of debt service obligations, it is termed currency swap.
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.
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, S0 ≡ Spot rate at time 0 (dollars/desired currency),
F0(t) ≡ Forward exchange rate at time t as of time 0
(dollars/desired currency),
Without a swap agreement, the present value of the borrower’s payments is PV = T
∑
t=1 P(0,t)RN + P(0, T )N.Note that if the bonds are sold at par, PV = N.
39
Currency Swaps
In the desired currency, the notional principal is N/S0 and the interest payment is R∗N/S0 per period. The present value of the (hedged) desired currency payments is
PV0 =
T
∑
t=1
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 $/€ be S0 = 1.3000,
F0(1) = 1.3030, F0(2) = 1.3100 and F0(3) = 1.3200. The annual yields are as before, i.e. P(0, 1) = 0.9434, P(0, 2) = 0.8817 and
P(0, 3) = 0.8163
What rate would be paid if the debt payments were all made in euros?
41
Currency Swaps
Example 1
First note that at the rate 6.95% the firm’s bonds are sold at par and thus PV = N. So we need to find R∗ such that
PV0 =
T
∑
t=1
Example 1 This gives R∗ = 1 − P(0, T )F0(T )/S0 ∑T t=1P(0,t)F0(t)/S0 . 43
Currency Swaps
Example 1In the present example, we need
R∗ = 1 − .8289
.9456 + .8884 + .8289 = 6.43%
Example 2
The problem is much simpler if we assume the exchange rate constant over the loan period (i.e. F0(t) = S0 for all t) and the annual yield to be the same over any subperiod (r0(0,t) = r, say, for all t).
Consider then the case of a US$ 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
1.5000 × 10, 000, 000 = Sfr15, 000, 000
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 $10m at 5.56% arranged when the spot rate is Sfr1.5000/$.
If the exchange rate falls to Sfr1.4650 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, 500 1.020 +
15, 301, 500
1.0202 = Sfr15, 002, 912 = $10, 240, 896 and the present value of the US$ payments is
556, 000 1.055 + 10, 556, 000 1.0552 = $10, 011, 078. 49
Unwinding Swaps
If the borrowing firm wishes to terminate the swap agreement, it will have to pay
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
