Fixed-coupon bonds
There are normally few zero-coupon bonds traded in the market, and they have a relatively short time to maturity, typically up to a year. The majority of bonds, especially those with longer times to maturity, are coupon bear-ing. The simplest kind are fixed-coupon bonds, providing predetermined payments (called coupons) at fixed dates.
The precise description of a fixed-coupon bond is:
• The holder of the bond receives deterministic coupon payments C1, . . . , CJ on the corresponding deterministic dates 0 < k1 < · · · <
kJ ≤ N.
• On the maturity date kJ the holder receives the deterministic face value F. The face value is received in addition to the final coupon due on that date.
Both zero-coupon bonds and fixed-coupon bonds are fixed income securities. We shall assume, for simplicity, that the interval between coupon dates is constant, kj+1− kj = c, j < J, with k1 ≤ c. The first coupon date indicates the day when the bond was issued, which is c− k1 units before today. So for instance with h= 1/12, k1= 2, k2 = 8, k3 = 14 (semi-annual coupon) we are dealing with a bond issued four months ago.
It is easy to compute the price of a fixed-coupon bond. The cash flow of fixed-coupon bond payments can be replicated by investing, at time 0,
• CjB(0, kj) in kj-bonds (to receive Cjat time kj), j = 1, . . . , J,
• FB(0, kJ) in kJ-bonds (to receive F at time kJ).
We know these dates and the amounts at time 0. We can use zero-coupon bonds with the corresponding maturities. Their prices are known, so we
can manufacture an investment replicating the coupon bond cash flow with certainty.
The total value of the replicating investment is the initial bond price:
P(0)=
J j=1
CjB(0, kj)+ FB(0, kJ). (6.4)
In practice unit bond prices are only known for short times to matu-rity (because zero-coupon bonds with longer maturities are rarely traded), whereas coupon bond prices are known for longer maturities. Therefore in practice in (6.4) P(0), Cj, and F will be known and B(0, kj) will have to be computed. To this end we need a sufficient number of equa-tions, in other words we need information about sufficiently many traded coupon bonds. The procedure of finding the structure of zero-coupon bonds with long maturities, known as bootstrapping, is illustrated by an example.
Example 6.9 Suppose that
• a one-year zero-coupon bond with face value £100 is trading at £91.80,
• a two-year bond with £10 annual coupons and face value £100 is trading at £103.95,
• a three-year bond with £10 annual coupons is trading at £105.15.
This leads to:
B(0, 1) = 0.9180, 10B(0, 1) + 110B(0, 2) = 103.95, 10B(0, 1) + 10B(0, 2) + 110B(0, 3) = 105.15.
Solving, we find
B(0, 2) 0.8616, B(0, 3) 0.7941.
In reality, the data may be scarce or irregular and the number of variables may be larger than the number of equations. The solution is to assume additional conditions based on linear interpolation.
Exercise 6.11 Assume that the unit is one month and find the zero-coupon bonds implied by a 4-month zero-zero-coupon bond trading at 98, and two coupon bonds with semi-annual coupons of 10, one matur-ing after 14 months tradmatur-ing at 101, and one maturmatur-ing after 12 months trading at 103.
The coupons are typically constant, Cj = C, and often determined as a percentage of the face value, called the coupon rate (not annualised)
C F.
Exercise 6.12 Let h = 1 and prove that the coupon rate equals the implied interest rate if and only if P(0)= F.
The price of the coupon bond at time n∈ [ki, ki+1) according to the cash flow replication as presented above will be
P(n)=
J j=i+1
CB(n, kj)+ FB(n, kJ) (6.5)
= C F
J j=i+1
B(n, kj)+ B(n, kJ)
F. (6.6)
This price is sometimes called the dirty price. Note that this price is not quoted in the markets.
Accrued interest is defined by
AI(n)= C n− ki
ki+1− ki,
which reflects the fact that between the coupon dates the holder should be compensated for the time the bond is held. For simplicity this is done in a linear way. The clean price is then defined as
CP(n)= P(n) − AI(n).
Variable-rate bonds
These are bonds for which the value of the coupons is not fixed in advance, but reset for every coupon period.
Definition 6.10
The coupon of a variable-rate bond is Cj=
kj− kj−1
hL(kj−1, kj)F.
The value of the coupon due at each time kj is determined one step earlier according to the rate L(kj−1, kj). The future rates are random so at the beginning we only know the first coupon.
Recall that with our conventions
L(kj−1, kj)= 1− B(kj−1, kj) (kj− kj−1)hB(kj−1, kj), so
Cj= F1− B(kj−1, kj) B(kj−1, kj) . Theorem 6.11
The initial price P(0) of a variable-rate bond is equal to F.
If the bond price is equal to the face value then we say that the bond trades at par. According to this theorem, a variable-coupon bond is trad-ing at par at time 0. Before we prove the theorem, we consider an example.
Example 6.12
Take time steps k1 = 1, k2 = 2 with constant rate L(0, 1) = L(1, 2) = R.
Then the coupons are given by C= RF and P(0)= C
1+ R+ C+ F
(1+ R)2 = RF
1+ R+ RF+ F (1+ R)2
= RF 1+ R+ F
1+ R = F.
This simple example illustrates the main idea of the theorem. Observe that the coupon rate must be computed according to simple interest, otherwise this argument would not work.
Proof of Theorem 6.11 If J= 1, then C1 = F1−B(0,kB(0,k1)1), so P(0)= B(0, k1)(C1+ F) = B(0, k1)F1− B(0, k1)
B(0, k1) + B(0, k1)F = F.
If J= 2, then C2= F1−B(kB(k1,k1,k2)2). The previous step yields P(k1)= B(k1, k2)(C2+ F) = F.
This is the key point. At time k1the bond price P(k1) represents the value of the cash flow at maturity, here time k2, namely the final coupon and face value. The holder of the bond will receive this amount with certainty in addition to the first coupon, so
P(0)= B(0, k1)(C1+ P(k1))= B(0, k1)(C1+ F) = F.
Induction gives the general result for any number of time steps.
Exercise 6.13 Having sold a variable-coupon bond with face value F design a hedging (replicating) strategy.
Interest rate swaps
Now we briefly consider interest rate swaps, which are contracts between two parties to exchange cash flows according to agreed criteria. For interest rate swaps this means the exchange of a floating-rate loan to a fixed-rate one, or conversely – this is usually called a plain vanilla swap. One party agrees to pay the other fixed-rate interest computed using the swap rate Rswapon a notional capital sum, while the second pays the first interest on the same sum and in the same currency, but at a floating rate determined by market conditions. The floating rate is usually based on the London Inter-bank Offer Rate (LIBOR), which acts a reference rate for Eurodollar bank deposits. Interest rate swaps can also be floating-for-floating, in which case two different reference rates are used to calculate the exchange pay-ments. In both instances the principal acts simply as a notional sum used to calculate interest payments. The condition which will enable us to find the swap rate is that the contract is initiated at no cost.
Thus a swap is a portfolio of forward contracts, with initial and final value zero. If there is no default, the swap reflects the difference between a fixed-rate and a floating-rate bond, and this fact can be used to price the swap. The fixed-rate buyer, in effect, has a long position in the floating-rate bond, and is short the fixed-rate bond underlying the swap. The value to him is the difference of the values of these two bonds. The position of the floating-rate payer is exactly the reverse. Swaps are used by banks to adjust the cash-flow characteristics of their assets or liabilities.
The party paying the fixed-rate stream (and receiving the floating-rate stream) is called the payer and the party receiving the fixed-rate stream (and paying the floating-rate stream) is called the receiver.
A swap amounts to exchanging at time 0 a fixed-coupon bond with constant coupon rate rswap (annual) for a floating-coupon bond, at no ini-tial cost. The two bonds have the same face value F and coupon dates 0< k1< · · · < kJ. If you swap a fixed rate for a floating rate in that way, at time step kjyou will receive the floating coupon FL(kj−1, kj)hcouponwhere hcoupon = (kj − kj−1)h is the time (in years) between the coupons, con-stant as we assumed above, while you will have to pay the fixed coupon Frswaphcoupon.
The valueΠ(0) of a swap to the receiver is the difference between the variable rate bond price
of a fixed-coupon bond with constant coupon rate rswap, as given by (6.3) with n= 0: allows us to compute the swap rate:
rswap= 1− B(0, kJ) hcouponJ
j=1B(0, kj).
It should be noted that there are other (similar) versions of the swap contract. The one described above is called a forward swap settled in arrears. The swap rate depends on kJso we introduce the self-explanatory notation rswap(0, kJ).
Exercise 6.14 Find the swap rates rswap(0, n) for the data from Ex-ample 6.1. Perturb one of the rates obtained by adding or subtracting x and reconstruct the bond prices. Analyse the impact of x on the prices obtained.
We have shown here that a swap can be decomposed into a portfolio of bonds, and so its value is completely determined by the bond price curve.
However, in practice, the swap market is so liquid, at so many maturities, that it is in fact swaps (through various swap rates) that drive bond prices!