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Chapter 6

The Risk and Term Structure
of Interest Rates

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Preview

In this chapter, we examine the sources and causes of fluctuations in interest rates relative to one another, and look at a number of theories that explain these fluctuations.

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Learning Objectives

Identify and explain three factors explaining the risk structure of interest rates.
List and explain the three theories of why interest rates vary across maturities.

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Risk Structure of Interest Rates

Bonds with the same maturity have different interest rates due to:
Default risk
Liquidity
Tax considerations

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Figure 1 Long-Term Bond Yields, 1919–2014

Sources: Board of Governors of the Federal Reserve System, Banking and Monetary Statistics,

1941–1970; Federal Reserve Bank of St. Louis FRED database: http://research.stlouisfed.org/fred2

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Risk Structure of Interest Rates

Default risk: probability that the issuer of the bond is unable or unwilling to make interest payments or pay off the face value
U.S. Treasury bonds are considered default free (government can raise taxes).
Risk premium: the spread between the interest rates on bonds with default risk and the interest rates on (same maturity) Treasury bonds

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Figure 2 Response to an Increase in Default Risk on Corporate Bonds

Step 1. An increase in default risk shifts the demand curve for corporate bonds left . . .

Step 2. and shifts the demand curve for Treasury bonds to the right . . .

Step 3. which raises the price of Treasury bonds and lowers the price of corporate bonds, and therefore lowers the interest rate on Treasury bonds and raises the rate on corporate bonds, thereby increasing the spread between the interest rates on corporate versus Treasury bonds.

(a) Corporate bond market

(b) Default-free (U.S. Treasury) bond market

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Table 1 Bond Ratings by Moody’s, Standard and Poor’s, and Fitch

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Risk Structure of Interest Rates

Liquidity: the relative ease with which an asset can be converted into cash
Cost of selling a bond
Number of buyers/sellers in a bond market
Income tax considerations
Interest payments on municipal bonds are exempt from federal income taxes.

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Figure 3 Interest Rates on Municipal and Treasury Bonds

Step 1. Tax-free status shifts the demand for municipal bonds to the right . . .

Step 2. and shifts the demand for Treasury bonds to the left . . .

Step 3. with the result that municipal bonds end up with a higher price and a lower interest rate than on Treasury bonds.

(a) Market for municipal bonds

(b) Market for Treasury bonds

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Effects of the Obama Tax Increase on Bond Interest Rates

In 2013, Congress approved legislation favored by the Obama administration to increase the income tax rate on high-income taxpayers from 35% to 39%. Consistent with supply and demand analysis, the increase in income tax rates for wealthy people helped to lower the interest rates on municipal bonds relative to the interest rate on Treasury bonds.

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Term Structure of Interest Rates

Bonds with identical risk, liquidity, and tax characteristics may have different interest rates because the time remaining to maturity is different

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Term Structure of Interest Rates

Yield curve: a plot of the yield on bonds with differing terms to maturity but the same risk, liquidity and tax considerations
Upward-sloping: long-term rates are above
short-term rates
Flat: short- and long-term rates are the same
Inverted: long-term rates are below short-term rates

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Term Structure of Interest Rates

The theory of the term structure of interest rates must explain the following facts:

Interest rates on bonds of different maturities move together over time.

When short-term interest rates are low, yield curves are more likely to have an upward slope; when short-term rates are high, yield curves are more likely to slope downward and be inverted.

Yield curves almost always slope upward.

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Term Structure of Interest Rates

Three theories to explain the three facts:

Expectations theory explains the first two facts but not the third.

Segmented markets theory explains the third fact but not the first two.

Liquidity premium theory combines the two theories to explain all three facts.

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Figure 4 Movements over Time of Interest Rates on U.S. Government Bonds with Different Maturities

Sources: Federal Reserve Bank of St. Louis FRED database: http://research.stlouisfed.org/fred2/

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Expectations Theory

The interest rate on a long-term bond will equal an average of the short-term interest rates that people expect to occur over the life of the long-term bond.
Buyers of bonds do not prefer bonds of one maturity over another; they will not hold
any quantity of a bond if its expected return
is less than that of another bond with a different maturity.
Bond holders consider bonds with different maturities to be perfect substitutes.

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Expectations Theory

An example:

Let the current rate on one-year bond be 6%.

You expect the interest rate on a one-year bond to be 8% next year.

Then the expected return for buying two one-year bonds averages (6% + 8%)/2 = 7%.

The interest rate on a two-year bond must be 7% for you to be willing to purchase it.

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Expectations Theory

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Expectations Theory

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Expectations Theory

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Expectations Theory

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Expectations Theory

Expectations theory explains:
Why the term structure of interest rates changes at different times.
Why interest rates on bonds with different maturities move together over time (fact 1).
Why yield curves tend to slope up when short-term rates are low and slope down when short-term rates are high (fact 2).
Cannot explain why yield curves usually slope upward (fact 3)

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Segmented Markets Theory

Bonds of different maturities are not substitutes at all.
The interest rate for each bond with a different maturity is determined by the demand for and supply of that bond.
Investors have preferences for bonds of one maturity over another.
If investors generally prefer bonds with shorter maturities that have less interest-rate risk, then this explains why yield curves usually slope upward (fact 3).

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Liquidity Premium &
Preferred Habitat Theories

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 that responds to supply and demand conditions for that bond.
Bonds of different maturities are partial (not perfect) substitutes.

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Liquidity Premium Theory

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Preferred Habitat Theory

Investors have a preference for bonds of one maturity over another.
They will be willing to buy bonds of different maturities only if they earn a somewhat higher expected return.
Investors are likely to prefer short-term bonds over longer-term bonds.

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Figure 5 The Relationship Between the Liquidity Premium (Preferred Habitat) and Expectations Theory

Expectations Theory
Yield Curve

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Interest rates on different maturity bonds move together over time; explained by the first term in
the equation
Yield curves tend to slope upward when short-term rates are low and to be inverted when short-term rates are high; explained by the liquidity premium term in the first case and by a low expected average in the second case
Yield curves typically slope upward; explained
by a larger liquidity premium as the term to
maturity lengthens

Liquidity Premium &
Preferred Habitat Theories

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Figure 6 Yield Curves and the Market’s Expectations of Future Short-Term Interest Rates According to the Liquidity Premium (Preferred Habitat) Theory

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Figure 7 Yield Curves for U.S. Government Bonds

For an investment of $1

= today's interest rate on a one-period

bond

= interest rate on a one-period bond exp

ected for next period

= today's interest rate on the two-perio

d bond

Expected return over the two periods fro

m investing $1 in the

two-period bond and holding it for the t

wo periods

(1 + )(1 + )1

12()1

2()

Since () is very small

the expected re

=++-

turn for holding the two-period bond for

two periods is

If two one-period bonds are bought with

the $1 investment

(1)(1)1

1()1

()

() is extremely small

Simplifying we get

tttt

iiii

++-

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Both bonds will be held only if the expe

cted returns are equal

The two-period rate must equal the avera

ge of the two one-period rates

For bonds with longer maturities

ttt

iii

12(1)

...

The -period interest rate equals the ave

rage of the one-period

interest rates expected to occur over th

e -period life of the bond

eee

ttn

++-

+++

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