Financial Markets and Institutions (2-downloads)

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We can conduct the same steps for bonds with a longer maturity so that we can exam-

ine the whole term structure of interest rates. Doing so, we will find that the inter-

est rate of i

nt

on an n-period bond must be

(2)

Equation 2 states that the n-period interest rate equals the average of the one-

period interest rates expected to occur over the n-period life of the bond. This is a

restatement of the expectations theory in more precise terms.

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The analysis here has been conducted for discount bonds. Formulas for interest rates on coupon

bonds would differ slightly from those used here, but would convey the same principle.

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Chapter 5 How Do Risk and Term Structure Affect Interest Rates?

101

The expectations theory is an elegant theory that explains why the term struc-

ture of interest rates (as represented by yield curves) changes at different times.

When the yield curve is upward-sloping, the expectations theory suggests that short-

term interest rates are expected to rise in the future, as we have seen in our numer-

ical example. In this situation, in which the long-term rate is currently higher than

The one-year interest rates over the next five years are expected to be 5%, 6%, 7%, 8%,

and 9%. Given this information, what are the interest rates on a two-year bond and a

five-year bond? Explain what is happening to the yield curve.

Solution

The interest rate on the two-year bond would be 5.5%.

where

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year 1 interest rate 

= 5%

=

year 2 interest rate

= 6%

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=

number of years 

= 2

Thus,

The interest rate on the five-year bond would be 7%.

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nt

=

where

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t

=

year 1 interest rate 

= 5%

=

year 2 interest rate 

= 6%

=

year 3 interest rate 

= 7%

=

year 4 interest rate 

= 8%

=

year 5 interest rate 

= 9%

n

=

number of years 

= 5

Thus,

Using the same equation for the one-, three-, and four-year interest rates, you will be

able to verify the one-year to five-year rates as 5.0%, 5.5%, 6.0%, 6.5%, and 7.0% respec-

tively. The rising trend in short-term interest rates produces an upward-sloping yield curve

along which interest rates rise as maturity lengthens.

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E X A M P L E   5 . 3 Expectations Theory

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102

Part 2 Fundamentals of Financial Markets

the short-term rate, the average of future short-term rates is expected to be higher

than the current short-term rate, which can occur only if short-term interest rates

are expected to rise. This is what we see in our numerical example. When the yield

curve is inverted (slopes downward), the average of future short-term interest rates

is expected to be lower than the current short-term rate, implying that short-term

interest rates are expected to fall, on average, in the future. Only when the yield curve

is flat does the expectations theory suggest that short-term interest rates are not

expected to change, on average, in the future.

The expectations theory also explains fact 1, which states that interest rates

on bonds with different maturities move together over time. Historically, short-term

interest rates have had the characteristic that if they increase today, they will tend

to be higher in the future. Hence a rise in short-term rates will raise people’s expec-

tations of future short-term rates. Because long-term rates are the average of

expected future short-term rates, a rise in short-term rates will also raise long-term

rates, causing short- and long-term rates to move together.

The expectations theory also explains fact 2, which states that yield curves tend

to have an upward slope when short-term interest rates are low and are inverted

when short-term rates are high. When short-term rates are low, people generally

expect them to rise to some normal level in the future, and the average of future

expected short-term rates is high relative to the current short-term rate. Therefore,

long-term interest rates will be substantially higher than current short-term rates,

and the yield curve would then have an upward slope. Conversely, if short-term rates

are high, people usually expect them to come back down. Long-term rates would then

drop below short-term rates because the average of expected future short-term rates

would be lower than current short-term rates, and the yield curve would slope down-

ward and become inverted.

3

The expectations theory is an attractive theory because it provides a simple

explanation of the behavior of the term structure, but unfortunately it has a major

shortcoming: It cannot explain fact 3, which says that yield curves usually slope

upward. The typical upward slope of yield curves implies that short-term interest

rates are usually expected to rise in the future. In practice, short-term interest rates

are just as likely to fall as they are to rise, and so the expectations theory suggests

that the typical yield curve should be flat rather than upward-sloping.

Market Segmentation Theory

As the name suggests, the market segmentation theory of the term structure sees

markets for different-maturity bonds as completely separate and segmented. The

interest rate for each bond with a different maturity is then determined by the sup-

ply of and demand for that bond, with no effects from expected returns on other

bonds with other maturities.

The key assumption in market segmentation theory is that bonds of different

maturities are not substitutes at all, so the expected return from holding a bond of

3

The expectations theory explains another important fact about the relationship between short-term

and long-term interest rates. As you can see in Figure 5.4, short-term interest rates are more volatile

than long-term rates. If interest rates are mean-reverting—that is, if they tend to head back down

after they are at unusually high levels or go back up when they are at unusually low levels—then an

average of these short-term rates must necessarily have less volatility than the short-term rates them-

selves. Because the expectations theory suggests that the long-term rate will be an average of future

short-term rates, it implies that the long-term rate will have less volatility than short-term rates.

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Chapter 5 How Do Risk and Term Structure Affect Interest Rates?

103

one maturity has no effect on the demand for a bond of another maturity. This the-

ory of the term structure is at the opposite extreme to the expectations theory, which

assumes that bonds of different maturities are perfect substitutes.

The argument for why bonds of different maturities are not substitutes is that

investors have strong preferences for bonds of one maturity but not for another,

so they will be concerned with the expected returns only for bonds of the matu-

rity they prefer. This might occur because they have a particular holding period

in mind, and if they match the maturity of the bond to the desired holding period,

they can obtain a certain return with no risk at all.

4

(We have seen in Chapter 3 that

if the term to maturity equals the holding period, the return is known for certain

because it equals the yield exactly, and there is no interest-rate risk.) For example,

people who have a short holding period would prefer to hold short-term bonds.

Conversely, if you were putting funds away for your young child to go to college,

your desired holding period might be much longer, and you would want to hold

longer-term bonds.

In market segmentation theory, differing yield curve patterns are accounted

for by supply-and-demand differences associated with bonds of different matu-

rities. If, as seems sensible, investors have short desired holding periods and gen-

erally prefer bonds with shorter maturities that have less interest-rate risk,

market segmentation theory can explain fact 3, which states that yield curves typ-

ically slope upward. Because in the typical situation the demand for long-term

bonds is relatively lower than that for short-term bonds, long-term bonds will have

lower prices and higher interest rates, and hence the yield curve will typically

slope upward.

Although market segmentation theory can explain why yield curves usually tend

to slope upward, it has a major flaw in that it cannot explain facts 1 and 2. First,

because it views the market for bonds of different maturities as completely seg-

mented, there is no reason for a rise in interest rates on a bond of one maturity to

affect the interest rate on a bond of another maturity. Therefore, it cannot explain

why interest rates on bonds of different maturities tend to move together (fact 1).

Second, because it is not clear how demand and supply for short- versus long-term

bonds change with the level of short-term interest rates, the theory cannot explain

why yield curves tend to slope upward when short-term interest rates are low and

to be inverted when short-term interest rates are high (fact 2).

Because each of our two theories explains empirical facts that the other can-

not, a logical step is to combine the theories, which leads us to the liquidity pre-

mium theory.

Liquidity Premium Theory

The liquidity premium theory of the term structure states that the interest rate

on a long-term bond will equal an average of short-term interest rates expected to

occur over the life of the long-term bond plus a liquidity premium (also referred to

as a term premium) that responds to supply-and-demand conditions for that bond.

4

The statement that there is no uncertainty about the return if the term to maturity equals the hold-

ing period is literally true only for a discount bond. For a coupon bond with a long holding period,

there is some risk because coupon payments must be reinvested before the bond matures. Our analysis

here is thus being conducted for discount bonds. However, the gist of the analysis remains the same for

coupon bonds because the amount of this risk from reinvestment is small when coupon bonds have the

same term to maturity as the holding period.

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104

Part 2 Fundamentals of Financial Markets

The liquidity premium theory’s key assumption is that bonds of different matu-

rities are substitutes, which means that the expected return on one bond does influ-

ence the expected return on a bond of a different maturity, but it allows investors

to prefer one bond maturity over another. In other words, bonds of different matu-

rities are assumed to be substitutes but not perfect substitutes. Investors tend to pre-

fer shorter-term bonds because these bonds bear less interest-rate risk. For these

reasons, investors must be offered a positive liquidity premium to induce them to

hold longer-term bonds. Such an outcome would modify the expectations theory

by adding a positive liquidity premium to the equation that describes the relationship

between long- and short-term interest rates. The liquidity premium theory is thus

written as

(3)

where l

nt

is the liquidity (term) premium for the n-period bond at time t, which is

always positive and rises with the term to maturity of the bond, n.

The relationship between the expectations theory and the liquidity premium the-

ory is shown in Figure 5.5. There we see that because the liquidity premium is always

positive and typically grows as the term to maturity increases, the yield curve implied

by the liquidity premium theory is always above the yield curve implied by the expec-

tations theory and generally has a steeper slope. (Note that for simplicity we are

assuming that the expectations theory yield curve is flat.)

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Expectations Theory

Yield Curve

Liquidity

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