6.3 Bonds: supply, demand and price

Thus it follows that the value of the whole stream of payments is the sum of this

progression. If P is the present value or price of the bond, then

	n
	1
P	∑C	(6.3)
	t1i)t
	(1

In the case of an irredeemable bond, the payments go on for ever and t tends to

innity. This means that the series

	1
C		(6.4)
	i)t
	(1

is converging on zero and the present value Pof the sum of the series can be more

conveniently written as

PC/i

(6.5)

This can be conrmed by taking the coupon of any undated government bond

from the Financial Timesand dividing it by the current long-term rate of interest.

Exercise 6.1

(a)On 6 May 2006, long-term interest rates were about 4.6 per cent. Calculate a price 1 for the undated bonds ‘Treasury 2/2%’ on that day.

(b)	What would the price have been if long-term interest rates had been 1.6 per cent? Answers at end of chapter

However, most bonds in fact mature and so our formula has to include a valu-

ation of the payment received on maturity. In this case Pis found as follows:

		n
			1G1	J
P		∑C	M		(6.6)
			
		t1	i)tI	i)nL
			(1 (1 

where Mis the maturity value of the bond.

Or, more compactly,

C	M
P		(6.7)
∑t	n
(1 i)	(1 i)

Remember that in calculating Pwe have assumed that the next coupon payment

is one year away. This is a way of saying that the last coupon payment has just been

made. In practice, however, we shall often want to price bonds at dates which lie

between two coupon payments. We might want to price a bond, for example, where

three months have elapsed since the last coupon payment (and there are three months

to run to the next half-coupon payment, or nine months to the next full payment).

If we continue with our assumption of single, annual coupon payments, it is clear that

if we buy a bond three months after its last coupon payment, we have to wait only

nine months to get the next coupon, and waiting nine months for a given payment

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Chapter 6 • The capital markets

is better than waiting a whole year. (On the face of it, we would do best if we bought

bonds which had, say, only a month to go before the next coupon payment and then

sold them after one month in order to buy more bonds near their payment date.)

And of course we can do precisely that; and so can everyone else. The result is that

the price that investors actually have to pay for a bond reects not just the elements

in eqn 6.6 and eqn 6.7 but also the position of the transaction date relative to the

coupon payment. In effect, the behaviour of the bond’s price reects the amount

of accrued interest, that is to say, the share of the coupon that is represented by

the number of days since the last coupon payment divided by the coupon payment

period. If we are three months into an annual cycle, the price that we pay should

¹/4

include a premium equal to C. This ensures that one-quarter of the coupon

goes to the seller of the bond, who has, after all, held the bond for three months

of the coupon cycle.

‘Clean’ price:The price of a bond ignoring any interest which may have accrued since

the last coupon payment.

The need to allow for accrued interest introduces the distinction between ‘clean’

and ‘dirty’ bond prices. The clean price is the price without accrued interest. In

practice, the clean price will be the price we should pay if we bought the bond

immediately after a coupon payment, so that we would have to wait for the entire

payment period before receiving the next payment. Pin eqn 6.6 and eqn 6.7 is a

clean price, since we calculated it on the assumption that we had to wait a year for

the coupon. The dirty price is always the price we actually pay and it will normally

include accrued interest in addition to the clean price.

‘Dirty’ price:The price of a bond, including any accrued interest.

At rst sight, it may seem strange to use bond pricing formulae (like eqn 6.6 and

eqn 6.7) which give us a price which will only rarely be the price that people actually

pay, but there is a logic to it. First of all, it is quite easy to calculate accrued interest

and to add it to the clean price. There are no unknowns or uncertainties. The pre-

mium is simply the fraction of the payment period times the coupon payment. The

coupon payment and period are both xed and the fraction can be calculated from

a calendar. The interesting inuences on a bond’s price are the variables whose values

can change: in particular interest rates and (we shall see later) expectations about

changes in interest rates. Because it is the clean price which is subject to unpredict-

able changes, bond prices quoted in the nancial press are usually clean prices.

In Box 6.2 and Exercise 6.2, we have two 8% bonds. The bond in Box 6.2 matures

in three years, while the bond in the exercise matures in ve years. The box shows

the price of the three-year bond when interest rates are 6 per cent and again when

they are 9 per cent, while the exercise requires the price of the ve-year bond to be

calculated for each of these interest rates.

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