chapter 7 the risk and term structure of interest …jneri/econ330/files/lecture...term structure of...

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Chapter Seven

Chapter 6 – Part 2

Term Structure

of Interest Rates

Term Structure of Interest Rates • Why do bonds with the same default rate and

tax status but different maturity dates have different yields?

– Long-term bonds are like a composite of a series of short-term bonds.

– Their yield depends on what people expect to happen in the future.

• How do we think about future interest rates?

Term Structure of Interest Rates • The relationship among bonds with the same

risk characteristics but different maturities is called the term structure of interest rates.

• Comparing 3-month and 10-year Treasury yields we can see: 1. Interest rates of different maturities tend to move

together.

2. Yields on short-term bonds are more volatile than yields on long-term bonds.

3. Long-term yields tend to be higher than short-term yields – but NOT always

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Term Structure of Interest Rates

Yield Curve Yield Curve: A plot of the term structure, with the yield to

maturity on the vertical axis and the time to maturity on the

horizontal axis.

https://www.treasury.gov/resource-center/data-chart-center/Pages/index.aspx

http://www.stockcharts.com/freecharts/yieldcurve.php

Three Term Structure Theories 1. Pure Expectations Theory/Hypothesis - explains

the first two facts but not the third

2. Segmented Markets Theory - explains fact three

3. Liquidity Premium Theory combines the two theories to explain all three facts








http://www.stockcharts.com/freecharts/yieldcurve.php

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The Expectations Hypothesis • Assumes an investor will be indifferent between

holding a 2-year bond or a series of two 1-year bonds.

– bonds of different maturities are perfect substitutes for each other.

• The expectations hypothesis implies that the current 2-year interest rate should equal the average of current 1-year rate and the 1-year interest rate one year in the future.

The Expectations Hypothesis • If current interest rate is 5 percent and future

interest rate is expected to be 7 percent, then the current two-year interest rate will be (5+7)/2 = 6%.

• When interest rates are expected to rise, long-term interest rate will be higher than short-term interest rates.

– The yield curve will slope up.

• This also means: – If interest rates are expected to fall, the yield curve will

slope down.

– If expected to stay the same, the yield curve will be flat.

The Expectations Hypothesis

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The Expectations Hypothesis • If bonds of different maturities are perfect

substitutes for each other, then we can construct investment strategies that must have the same yields.

• Options:

1. Invest in a 2-year bond and hold it to maturity

• i2t is interest rate on a 2-year bond bought today, t.

• One dollar yields (1 + i2t)(1 + i2t) two years later.


2. Invest in two 1-year bonds, one today and one when the first matures.

– One-year bond today has interest i1t.

– One-year bond purchased in year 2 has interest ie1t+1, where e is expected.

– One dollar invested today returns

(1 + i1t)(1 + ie1t+1).


• The expectations hypothesis tells us investors will be indifferent between the two options.

• This means they must have the same return:

(1 + i2t)(1 + i2t) = (1 + i1t)(1 + ie1t+1)

• We can now write the two-year interest rate as the average of the current and future expected one-year interest rates:

i2t i1t i1t1

e

2

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• Returning to our example: • i1t = 5%

• ie1t+1 = 7%

• The interest rate on a 1-year bond is 5% and the interest rate on a 2-year bond is 6%.

%0.62

%7%5

2

1112

e

ttt

iii

A Note on Averages

• Geometric average of and =

• Arithmetic average =

• The arithmetic average is an approximation.

1ti 1 1ti

1/ 2

1 1 1((1 )(1 )) 1t ti i

1 1 1

2

t ti i

Quiz: 2 year investment horizon – 2 options

• Option/strategy 1:

• Invest $1,000 for 2-years at 8%:

• Ending Balance = $1,166.40

• Option/strategy 2:

• Invest $1,000 1-year at 6% and expect 9% one year later:

• Ending Balance = $1,155.40

• Which is the better strategy and why?

• What happens to S and D?

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A 3-year Bond:


• An N-year Bond:

• We can generalize this: a bond with n years to maturity is the average of n expected future one-year interest rates:

int i1t i1t1

e i1t2e ... i1tn1

e

n

Expectations Hypothesis - Arithmetic Average

In words: The interest rate on a bond with n years to maturity at time t is the average of the n expected future one-year rates.

Numerical example:

One-year interest rate over the next five years 5%, 6%, 7%, 8% and 9%:

Interest rate on a two-year bond:

(5% + 6%)/2 = 5.5%

Interest rate for a five-year bond:

(5% + 6% + 7% + 8% + 9%)/5 = 7%

Interest rate for one, two, three, four and five-year bonds are:

5%, 5.5%, 6%, 6.5% and 7%.

n

iiiii

e

nt

e

t

e

tt

nt

1121111 ....

This is the only interest rate that is

known at time t

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Expectations Hypothesis

Another example: One-year interest rate over the next five years 7%, 6%, 5%, 4% and 3%:

Interest rate on a two-year bond:

(7% + 6%)/2 = 6.5%

Interest rate for a five-year bond:

(7% + 6% + 5% + 4% + 3%)/5 = 5%

Interest rate for one, two, three, four and five-year bonds:

7%, 6.5%, 6%, 5.5% and 5%.

n

iiiii

e

nt

e

t

e

ttnt

1121111 ....

Side Note

n

iiiii

e

nt

e

t

e

ttnt

1121111 ....

n

rrrri

e

nt

e

nt

e

t

e

t

e

t

e

t

e

tt

nt

)(....)()()(11112121111111

Recall the Fisher Equation: i = r + πe

nn

rrrri

e

nt

e

t

e

t

e

t

e

nt

e

t

e

ttnt

)........ 11211111121111

This says long term interest rates equal the average

real interest rate and the average rate of inflation

expected over the life of the bond.

From the Fisher Equation: i = r + πe

• Holding r constant:

• If inflation is expected to rise in the future, expected

one-year interest rates will rise and the yield curve will

slope upward.

• If inflation is expected to fall in the future, expected

one-year interest rates will fall and the yield curve will

slope downward.

• If inflation is expected to remain the same in the future,

expected one-year interest rates will remain the same

and the yield curve will be flat.

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Expectations Theory: i = r + πe

i

2

1

2

e

tt

t

iii

tt

e

t iii 21 2

3

21

3

e

t

e

tt

t

iiii

tt

e

t iii 232 23

tnnt

e

nt innii )1()1( )1(

In general:

)(3 132

e

ttt

e

t iiii

From the formula for the yield on a 2-year bond:

From the formula for the yield on a 3-year bond:

Using the Expectations Theory to Solve for Expected

1-year (forward) Interest Rates

i

3

21

3

e

t

e

tt

t

iiii

tt

e

t iii 232 23

tnnt

e

nt innii )1()1( )1(

In general:

)(3 132

e

ttt

e

t iiii

Example: Calculate iet+2

i3t = 5% and i2t = 4%; iet+2 = 3(5%) – 2(4%) = 7%

Using the Expectations Theory to Solve for Expected

1-year (forward) Interest Rates

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Does this hypothesis explain the three observations we started with?

1. Interest rates of different maturities will move together. – Yes. Long term interest rates are averages of

expected future short-term interest rates.

2. Yields on short-term bonds will be more volatile than yields on long-term bonds. – Yes. Long-term rates are averages of short-term

rates, so changing one short-term rate has little effect on the long term rate.


3. This hypothesis cannot explain why long-term yields are normally higher than short term yields.

– It implies that the yield curve slopes upward only when interest rates are expected to rise.

– This hypothesis would suggest that interest rates are normally expected to rise.

Segmented Market Theory

• Bonds of different maturities are not perfect

substitutes for each other.

• Key assumptions:

• Investors have specific preferences about the maturity

or term of a security.

• Investors do not stray from their preferred maturity.

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Segmented Markets Hypothesis

• The slope of the yield curve is explained by different demand and supply conditions for bonds of different maturities.

• If the yield curve slopes up, it does so because the demand for short term bonds is relatively greater than the demand for long term bonds.

• Short term bonds have a higher price and a lower yield as a result of the relatively greater demand. So the yield curve slopes upward.


Price Price

0 0

S S

P2s

P1s P1

l

P2l

D1s

D2s

D1l

D2l

Quantity of Short-term Bonds Quantity of Long-term Bonds

Upward Sloping Yield Curve


• The segmented markets hypothesis explains

why….

• Yield curves typically slope upward.

• On average, investors prefer bonds with shorter

maturities that have less interest rate risk.

• Therefore, the demand for short term bonds is

relatively greater than the demand for long-term

bonds

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• But, the segmented markets hypothesis does not

explain why…

• Interest rates on different maturities move

together.

• The segmented markets hypothesis assumes that

short and long markets are completely segmented.

The Liquidity Premium Theory • In order to explain all 3 facts we need to extend

the expectations hypothesis to include risk.

• Bondholders face both inflation and interest-rate risk.

– The longer the term of the bond, the greater both types of risk.

The Liquidity Premium Theory Inflation Risk

• Real return is what matters and computing real return from nominal return requires a forecast of expected future inflation.

– The further into the future we look, the greater the uncertainty.

– A bond’s inflation risk increases with its time to maturity.

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The Liquidity Premium Theory

Interest-rate risk

• arises from the mismatch between the investor’s investment horizon and a bond’s time to maturity.

– If a bondholder plans to sell a bond prior to maturity, changes in the interest rate generate capital gains or losses.

– The longer the term of the bond, the greater the price change for a given change in interest rates and the larger the potential for capital losses.

The Liquidity Premium Theory

• Investors require compensation for the increase in risk they take for buying longer term bonds.

• We can think about bond yields as having two parts:

– One that is risk free - explained by the expectations hypothesis.

– One that is a risk premium - explained by inflation and interest-rate risk.

The Liquidity Premium Theory • Together this forms the liquidity premium theory

of the term structure of interest rates.

• We can add the risk premium (rpn) to our previous equation to get:

• The liquidity premium theory explains all three of our observations about the term structure of interest rates.

n

iiiirpi

e

nt

e

t

e

ttnnt

1121111 ....

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Numerical Example

Term in years (n)

1 2 3 4 5

One year interest rate

expectations 5% 6% 7% 8% 9%

Liquidity premium 0% 0.25% 0.5% 0.75% 1.0%

Pure expectations

predicted n-year bond

interest rates

5% 5.5% 6% 6.5% 7%

Actual n-year bond

interest rates,

accounting for liquidity

preference

5% 5.75% 6.5% 7.25% 8%

5% 6%

2

5% 6% 7%

3

5 6 7 8%

4

5 6 7 8 9%

5

Relationship Between the Liquidity Premium and

Expectations Theories

(if short term interest rates are

expected to remain constant)

Information Content of Interest Rates:

Term Structure

• When the yield curve slopes down,

it is called inverted

• An inverted yield curve

is a very valuable forecasting tool

• It signals an economic downturn

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Market

Predictions

of Future

Short

Rates

The Information Content of Interest Rates

• Risk spreads provide one type of information, the term structure another.

• We can apply what we have just learned to recent U.S. economic history to show how forecasters use these tools.

Information in the Risk Structure of Interest Rates

• The immediate impact of a pending recession is to raise the risk premium on privately issued bonds.

– Note that an economic slowdown or recession does not affect the risk of holding government bonds.

– The impact of a recession on companies with high bond ratings is also usually quite small.

• The lower the initial grade of the bond, the more the default-risk premium rises as general economic conditions deteriorate.

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Information in the Risk Structure of Interest Rates

• Panel A of Figure 7.8 shows the annual GDP growth over four decades superimposed on shading that shows the dates of recessions.

– During shaded periods growth is negative.

• Panel B of figure 7.8 shows GDP growth against the spread between yields on Baa-rated bonds and U.S. Treasury bonds.

– risk spread rises during recessions.

• During financial crises, people take cover.

• They sell risky investments & buy safe ones.

• An increase in the demand for government bonds coupled with a decrease in the demand for virtually everything else is called a flight to quality.

– This leads to an increase in the risk spread.

• The 1998 Russian default on its bonds led to a serious flight to quality causing the financial markets to cease to function properly.

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Information in the Term Structure of Interest Rates

• Information on the term structure, particularly the slope of the yield curve - helps to forecast general economic conditions. – The yield curve usually slopes upward.

– On rare occasions, short-term interest rates exceed long-term yields leading to an inverted yield curve.

• This is a valuable forecasting tool because it predicts a general economic slowdown. – Indicates policy is tight because policymakers are

attempting to slow economic growth and inflation.


• Figure 7.9 shows GDP growth and the slope of the yield curve, measured as the difference between the 10-year and 3 month yields: term spread.

• Panel A shows GDP growth together with the term spread at the same time.

• Panel B shows GDP growth in the current year against the slope of the yield curve one year earlier. – The two lines clearly move together.

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• When the term spread falls, GDP growth tends to fall one year later.

• This shows that the yield curve is a valuable forecasting tool.

chapter 7 the risk and term structure of interest …jneri/econ330/files/lecture...term structure of...

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