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face value of the bond isH, the payment at timeTi (i= 1,2, . . . , n−1) equalsHδlTTii−1, while the

final payment at timeTnequalsH(1 +δlTTii−1). If we defineT0=T1−δ, the datesT0, T1, . . . , Tn−1

are often referred to as the reset dates of the bond.

Let us look at the valuation of a floating rate bond. We will argue that immediately after each reset date, the value of the bond will equal its face value. To see this, first note that immediately after the last reset date Tn−1, the bond is equivalent to a zero-coupon bond with a coupon rate

equal to the market interest rate for the last coupon period. By definition of that market interest rate, the timeTn−1value of the bond will be exactly equal to the face valueH. In mathematical

terms, the market discount factor to apply for the discounting of timeTn payments back to time

Tn−1 is (1 +δlTTnn−1)

−1_{, so the time T}

n−1 value of a payment of H(1 +δlTTnn−1) at time Tn is

precisely H. Immediately after the next-to-last reset dateTn−2, we know that we will receive a

payment ofHδlTn−1

Tn−2 at timeTn−1and that the timeTn−1value of the following payment (received

at Tn) equals H. We therefore have to discount the sum HδlT_Tn_n−−₂1 +H = H(1 +δl_TTn_n−₋1₂) from

Tn−1 back to Tn−2. The discounted value is exactly H. Continuing this procedure, we get that

immediately after a reset of the coupon rate, the floating rate bond is valued at par.

We can also derive the value of the floating rate bond between two payment dates. Suppose we are interested in the value at some timetbetweenT0 andTn. Introduce the notation

i(t) = min_{i_{∈ {}1,2, . . . , n_}:Ti> t},

so thatTi(t)is the nearest following payment date after timet. We know that the following payment

at timeTi(t) equalsHδl Ti(t)

Ti(t)−1 and that the value at time Ti(t) of all the remaining payments will

equalH. The value of the bond at timetwill then be

(2.1) Bfl t =H(1 +δl Ti(t) Ti(t)−1)B Ti(t) t , T0≤t < Tn.

This expression also holds at payment dates t = Ti, where it results in H, which is the value excluding the payment at that date.

While few floating rate bonds are traded, the results above are also very useful for the analysis of interest rate swaps studied in Section 2.9.

2.3

Forwards on bonds

A forward contract is an agreement between two parties on a trade of a given asset or security at a given future point in time and at a price that is already fixed when the agreement is made. We will refer to this price as the delivery price. Usually, the delivery price is set so that the net present value of the future transaction (computed at the agreement date) is equal to zero. This value of the delivery price is called the forward price. Note that the forward price will depend on the time of delivery and the underlying asset to be transacted. In the following we will take a closer look at forwards on zero-coupon bonds and coupon bonds. For a general introduction to forwards, see Hull (2003).

Consider an agreement on a delivery at time T of a zero-coupon bond maturing at time S

(whereS > T) for a price ofK. We assume that the face value of zero-coupon bonds is 1 (dollar). The following theorem gives formulas for the timetvalueVtT,Sof a long position in such a forward contract and for the forward price of the zero-coupon bond. Here and throughout the text, BT t denotes the timetprice of a zero-coupon bond that pays 1 (dollar) at timeT.

2.3 Forwards on bonds 24

Theorem 2.1 The unique no-arbitrage value at time t of a forward with delivery at time T of a zero-coupon bond maturing at time S at the delivery priceK is given by

(2.2) VtT,S=BtS−KBtT.

The unique no-arbitrage forward price on the zero-coupon bond is

(2.3) FtT,S= BS t BT t . Proof: Consider two portfolios, namely

(A) a long position in the forward contract and Kzero-coupon bonds maturing at timeT, (B) a zero-coupon bond maturing at timeS (i.e. the underlying bond).

At timeT the forward yields a payoff ofBS

T −K, so that the total timeT value of portfolio A is

BS

T−K+K=BST, which is identical to the value of portfolio B. Since none of the portfolios yield payments before timeT, the no-arbitrage principle implies that they must have the same current price, i.e.

VtT,S+KBtT =BSt,

from which (2.2) follows. Since the forward price FtT,S is that value ofK that makes V T,S

t = 0,

equation (2.3) follows directly from (2.2). 2

At the delivery timeT the gain or loss from the forward position will be known. The gain from a long position in a forward written on a zero-coupon with face valueH and maturity atSis equal toH(BS

T−F

T,S

t ). If we write the spot bond priceBTS in terms of the spot LIBOR ratelTS and the forward bond priceFtT,S in terms of the forward LIBOR rateL

T,S

t , it follows from (1.8) and (1.9) that the gain is equal to

HHBTS−F T,S t =H 1 1 + (S₋T)lS T − 1 1 + (S₋T)LT,St ! = (S−T)(L T,S t −lST)H 1 + (S₋T)lS T 1 + (S−T)L T,S t . (2.4)

An investor with a long position in the forward will realize a gain if the spot bond price at delivery turns out to be above the forward price, i.e. if the spot interest rate at delivery turns out to be below the forward interest rate when the forward position was taken. We can think of a short position in a forward on a zero-coupon bond as a way to lock in the borrowing rate for the period between the delivery date of the forward and the maturity date of the bond.

Next, let us consider a forward on a coupon bond. As before, let T be the delivery date and

K be the delivery price. The underlying coupon bond is assumed to yield payments at the points in timeT1< T2<· · ·< Tn, whereT < Tn. The timeTi payment is denoted byYi,i= 1,2, . . . , n. The timetvalue of the bond is therefore given by

Bt=

X

Ti>t YiBtTi,

where the sum is over all future payment dates; cf. (1.2) on page 6. LetVtT,kup denote the timet value of this forward contract. We then have the following result: