7.7.2The pure expectations theory of interest rate structure

This theory assumes that present long-term interest rates depend entirely on future

short-term rates. Lenders are taken to be equally happy to hold short-term or long-

term securities. Their choice between them will depend only on relative interest

rates. It follows that, for instance, a series of ve one-year bonds is a perfect sub-

stitute for a ve-year bond. If this were so, the proceeds from investing say £1,000

for one year and then reinvesting the returns for another year and so on for ve

years must exactly equal the proceeds from buying a £1,000 ve-year bond at the

beginning.

Consider what would happen if this were not so. Suppose the proceeds from a

long-term bond were greater than from a series of short-term bonds. People would

buy long-term bonds, pushing up their price and pushing down the rate of interest

on them. This would continue until there was no advantage to be had from holding

the long-term bonds. Then people would be indifferent between the two types of

bond. Thus, the long-term interest rate would depend entirely on the expected

future short-term rates.

The simplest form of this theory assumes that lenders have perfect information

and know what is going to happen to short-term interest rates in the future. In this

case, the long-term interest rate will be an average of the known future short-term

rates. This relationship between long-term and short-term rates can be expressed in

the formula

(1 + i*)n(1 i)(1 i)(1 i)...(1 i)	(7.5)
123n

where i* is the interest rate payable each year on a long-term bond and nis the

number of years to maturity of the bond; iis the rate of interest payable now on a

1

one-year bond; iis the rate of interest which will be payable on a one-year bond in

2

a year’s time; iis the short-term rate two years into the future, and so on.

3

222

....

FINM_C07.qxd 1/18/07 11:33 AM Page 223

7.7 The structure of interest rates

It follows that if short-term rates are expected to rise, long-term rates will be

higher than the current short-term rates and the yield curve will slope upwards.

Box 7.3 provides a numerical illustration of this and Exercise 7.1 gives you some

practice with it.

Box 7.3

The pure expectations theory of interest rate structure – a numerical example

Assuming that lenders have perfect information, long-term interest rates will be an average

of the known future short-term rates. We assume that lenders know that short-term rates

over the next ve years will be:

year 18 per cent

year 210 per cent

year 311 per cent

year 412 per cent

year 59 per cent

Then, £1,000 invested in a one-year bond, with the proceeds being invested in a

further one-year bond in the subsequent year, will produce the following results:

	Principal	Interest rate	Interest	Capital interest
year 1	£1,000	8 per cent	£80	£1,080
year 2	£1,080	10 per cent	£108	£1,188

We can calculate that for a two-year bond taken out at the beginning of year one

to produce the same results it would need to pay an interest rate of 9 per cent – the

average of the two short-term rates. What does this mean for the yield curve?

We can see that because it is known that short-term interest rates will rise over

the following year (from 8 per cent to 10 per cent), the interest payable on the two-year

(long-term) bond must be greater than that payable on the one-year (short-term) bond.

That is, the yield curve will be sloping upwards.

Let us continue our gures, assuming that our investor continues to re-invest in

one-year bonds for each of the following three years. This will give us:

year 3£1,18811 per cent£131£1,319

year 4£1,31912 per cent£158£1,477

year 5£1,4779 per cent£133£1,610

It can be shown that, at the beginning of year one, the interest rate payable on three-

year bonds must have been 9.66 per cent (the average of 8, 10 and 11) and on four-year

bonds 10.25 per cent. In other words, as long as it is known that short-term interest rates

are going to rise, the yield on long-term bonds for the equivalent period must lie above

the short-term rate at the beginning of year one and must be rising. The yield curve will

be sloping up. However, what about the interest rate at the beginning of year one on a

ve-year bond? Because it is known that short-term interest rates will begin to fall in year

ve, so too will the interest rate on a ve-year bond. To produce a sum of £1,610 at the

end of ve years, the interest rate on a ve-year bond will need to be only 10 per cent

and the yield curve will begin to turn down.

223

....

FINM_C07.qxd 1/18/07 11:33 AM Page 224

Chapter 7 • Interest rates

Exercise 7.1In Box 7.3, assume that it is known that short-term interest rates in years 6, 7 and 8 will be:

year 64 per cent

year 79 per cent

year 813 per cent

What will be the interest rate at the beginning of year oneon six-year, seven-year and

eight-year bonds respectively? Draw the yield curve at the beginning of year one.