7.7.3Term premiums

However, people do not have perfect information about the future course of short-

term interest rates. All they can have are estimates of these rates, which are subject

to the risk of error. The further into the future we try to look, the greater is the

chance that we shall be wrong. Suppose a lender acquires a long-term bond that

pays an interest rate related to expected short-term interest rates but then nds

that short-term rates rise above the expected level. As current interest rates rise, the

prices of existing bonds fall, and bond holders suffer a capital loss. As we have seen

in Chapter 6, the longer the time to maturity of the bond, the greater will be the fall

in bond price. It follows that the risk of capital loss associated with any given error

in forecasting future interest rates, the capital risk,is greater for long-term than for

short-term bonds.

People respond to risk in different ways. If they have the same attitude to both

the risk of loss and the prospect of gain, the two balance out and they are said to be

risk-neutral. In this case, the yield curve reects what investors expect to happen

to short-term rates of interest. In equilibrium, the outcome is exactly the same as

with perfect certainty. However, now it is possible that investors’ expectations will

be wrong and thus that the market will not be in equilibrium. This case is set out

formally in Box 7.4.

Box 7.4

The expectations view of the term structure of interest rates under uncertainty assuming risk neutrality

Assume that there are two types of bond available to investors. They are identical in all

ways but one – one bond is for one year only (short-term bond); the second bond is for

two years (long-term bond). A person who wishes to invest for two years has a choice of

two strategies.

Strategy A

Buy a one-year bond now and when it matures in one year’s time, use the funds to buy

a second one-year bond. The investor knows the current rate of interest on one year

bonds, i, but does not know what the rate of interest on one-year bonds will be in

s

one year’s time. However, he holds the same expectation about that rate (which we shall

call the expected future short-term rate, i) as everyone else in the market.

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7.7 The structure of interest rates

Strategy B

Buy a two-year bond now. The investor knows the current rate of interest on a two-year

bond (i*).

Calculations

The investor can thus calculate the certain return to Strategy B and the expected return

to Strategy A.

2

The certain return to strategy B we can write as (1 i*). All interest rates, remember, are

written as decimals. Thus, if i* were 5 per cent, a £100 bond would return in two years’

time: £(100) (1 0.05)²£100 1.1025 £110.25.

The expectedreturn to Strategy A, we can write as: (1 i)(1 i).

s

Now we can calculate the value that iwould need to be for the two strategies to pro-

duce the same return at the end of two years: that is, the value imust take for the investor

to be indifferent between the two strategies. We call this value f. By denition,

2

(1 i)(1 f) (1 i*)²

s2

2

and, (1 f) (1 i*)/(1 i).

2s

Let ibe 4 per cent and i* be 5 per cent. Then, (1 f) 1.1025/1.04 1.06 and

s2

f0.06 or 6 per cent.

2

In other words, if investors are to be indifferent between the two strategies, the

expected short-term rate of interest in one year’s time must be 6 per cent. Then no one

would have a reason for changing the balance of their present holdings of one-year and

two-year bonds: the market would be in equilibrium.

That is, our equilibrium condition for the market is that i(the expected future short-

term rate of interest) be equal to f(the rate of interest that causes the two strategies to

2

produce the same return at the end of two years).

It follows that if people expect the short-run rate in one year’s time to be greater than

f(i.e. if), they will shift from long to short bonds without limit. In the reverse case

22

(if) they will shift from short to long bonds without limit.

2

In the latter case, capital risk would increase, but there is both an upside risk (interest

rates fall and bond prices rise) and a downside risk (interest rates rise and bond prices

fall). Risk-neutral investors will see the upside and downside risk as offsetting each other.

In equilibrium, with if, the term structure indicates what the market expects to happen

2

to short rates, just as under conditions of certainty. Thus, if ii* and fi, then ii; people

s 2 ss

are expecting short-run rates to rise: the yield curve will be rising. Equally, if ii* but fi,

s2s

then iiand people are expecting short rates to fall: the yield curve will be falling.

s

If we next assume that people generally do not much like taking risks (they are

risk averse), we can argue that lenders will have to be paid a rather higher rate

of interest than the average of the expected but uncertain short-term rates to per-

suade them to buy longer-term bonds. This addition to the interest rate is sometimes

known as a risk premium, but is better referred to as a term premium to avoid con-

fusion with the risk premium paid to offset exchange rate risk or default risk. It

is also sometimes referred to as a liquidity premium but this, strictly speaking, is

incorrect since ‘liquidity’ refers to the speed and ease with which an asset can be

converted into money without risk. A long-term bond can be converted into cash as

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Chapter 7 • Interest rates

quickly and as easily as a short-term bond through the secondary market. The term

premium is justied by the extra risk of capital loss associated with a long-term

bond, not with any greater difculty involved in converting the bond into cash.

Term premium:The addition to the rate of interest needed to persuade capital

risk-averse savers to lend for longer periods.

If we apply this to the example in Box 7.3, the rate of interest on ve-year bonds

will need to be greater than 10 per cent because, although lenders are as likely to

overestimate as to underestimate future short-term rates, since we have assumed them

to be risk-averse, they will be more worried by the possibility of underestimating

them. (In the jargon we can say that they are more worried by the downside risk

than they are attracted by the upside risk.) We can say that interest rates on long-term

securities will include a term premium. This provides an extra reason for an upward-

sloping yield curve; but yield curves may still slope down, despite the inclusion of

a term premium in the long-term rates. For instance, if people expect short-term

rates to fall in the future, they have an incentive to buy long-term bonds now to

‘lock in’ to current rates. However, this increased demand for long bonds will force

up their price and force down long interest rates. This effect may be strong enough

to outweigh the term premium included in long rates.

Why are borrowers willing to pay this premium? Borrowers raise funds in order

to invest – to acquire assets that will produce a prot at a rate higher than they

are paying to borrow. However, they do not wish (and may indeed not be able) to

repay the loan until they have earned their prots. If their investment projects are

long-term ones (as, for example, are most purchases of capital equipment), they will

prefer to borrow long (matching long-term liabilities to long-term assets). Thus, we

can summarise much of the foregoing by saying that lenders would (other things

being equal) rather lend short term whereas borrowers often wish to borrow long.

Borrowers may thus be prepared to pay a higher rate of interest on long-term funds

than the average of expected short-term rates of interest in order to obtain funds in

the form they prefer.

So far, we have been assuming that all lenders have the same attitudes. Specically,

we have been assuming that all risk-averse lenders are worried about capital risk.

However, some savers may have no plans to sell their bonds before the maturity

date. Since they know that at maturity they will be paid the face value of the bond,

such savers have no reason to be worried about capital risk. Rather, their concern

might be with the size of interest payments that they receive every six months. For

them, long-term bonds provide greater certainty than short-term ones. People buy-

ing fteen-year bonds and intending to hold them until maturity know how much

income they will receive for the whole of that period. People buying a series of

fteen one-year bonds do not know this since the income they receive each year will

depend on what happens to short-term interest rates in the future. They face the risk

that short-term interest rates might fall. In other words, they face an income risk.

We have seen in Chapter 4 that some institutions, for example pension funds and

life insurance companies, have a good idea of liabilities well into the future and wish

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