The Risk and Term Structure of Interest Rates

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chapter 6
The Risk and Term Structure of Interest Rates
2
Risk Structure of Long Bonds in the United States
3
Bond Ratings
4
Increase in Default Risk on Corporate Bonds
5
Possibility of Increase in Default Risk in
Corporate Bonds

• Corporate Bond Market
• 1. Risk of corporate bonds ?, Dc ?, Dc shifts
  left

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Possibility of Increase in Default Risk in
Corporate Bonds

• Corporate Bond Market
• Risk of corporate bonds ?, Dc ?, Dc shifts left
• Higher expected interest rates in the future
  lower the expected return for existing long term
  bonds, decreases the demand, and shift the demand
  curve to the left.

7
Possibility of Increase in Default Risk in
Corporate Bonds

• Corporate Bond Market
• Risk of corporate bonds ?, Dc ?, Dc shifts left
• Higher expected interest rates in the future
  lower the expected return for existing long term
  bonds, decreases the demand, and shift the demand
  curve to the left. ?, Dc ?, Dc shifts left
• 3. Pc ?, ic ?

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Possibility of Increase in Default Risk in
Corporate Bonds

• Treasury Bond Market
• 4. Relative RETe on Treasury bonds ?, DT ?, DT
  shifts right

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Possibility of Increase in Default Risk in
Corporate Bonds

• Treasury Bond Market
• Relative risk of Treasury bonds ?, DT ?, DT
  shifts right
• Relative RETe on Treasury bonds ?, DT ?, DT
  shifts right

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Possibility of Increase in Default Risk in
Corporate Bonds

• Treasury Bond Market
• Relative risk of Treasury bonds ?, DT ?, DT
  shifts right.
• Relative RETe on Treasury bonds ?, DT ?, DT
  shifts right
• 6. PT ?, iT ?

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Corporate Bonds Become Less Liquid

• Outcome
• Risk premium, ic iT, rises
• Risk premium reflects not only corporate bonds
  default risk, but also lower liquidity

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Tax Advantages of Municipal Bonds
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Tax Advantages Municipal Bonds

• Municipal Bond Market
• 1. Tax exemption raises relative RETe on
  municipal bonds, Dm ?, Dm shifts right

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Tax Advantages Municipal Bonds

• Municipal Bond Market
• 1. Tax exemption raises relative RETe on
  municipal bonds, Dm ?, Dm shifts right
• 2. Pm ?, im ?

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Tax Advantages Municipal Bonds

• Treasury Bond Market
• 1. Relative RETe on Treasury bonds ?, DT ?, DT
  shifts left

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Tax Advantages Municipal Bonds

• Treasury Bond Market
• 1. Relative RETe on Treasury bonds ?, DT ?, DT
  shifts left
• 2. PT ?, iT ?

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Tax Advantages Municipal Bonds

• Outcome
• im lt iT

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Term Structure Facts to be Explained

• FACTS
• 1. Interest rates for different maturities move
  together

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Term Structure Facts to be Explained

• FACTS
• 1. Interest rates for different maturities move
  together
• 2. Yield curves tend to have steep slope when
  short rates are low and downward slope when short
  rates are high

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Term Structure Facts to be Explained

• FACTS
• 1. Interest rates for different maturities move
  together
• 2. Yield curves tend to have upward (positive)
  slope when short rates are low and downward slope
  when short rates are high
• 3. Yield curve is typically upward sloping

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Term Structure Facts to be Explained

• http//www.smartmoney.com/onebond/index.cfm?story
  yieldcurve

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Term Structure Facts to be Explained

• Three Theories of Term Structure
• Expectations Theory

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Term Structure Facts to be Explained

• Three Theories of Term Structure
• 1. Expectations Theory
• Segmented Markets Theory

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Term Structure Facts to be Explained

• Three Theories of Term Structure
• 1. Expectations Theory
• 2. Segmented Markets Theory
• 3. Liquidity Premium Theory

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Interest Rates on Different Maturity Bonds Move
Together
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Expectations Hypothesis

• Key Assumption
• Bonds of different maturities are perfect
  substitutes

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Expectations Hypothesis

• Key Assumption
• Bonds of different maturities are perfect
  substitutes
• Implication RETe on bonds of different
  maturities are equal

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Expectations Hypothesis

• Investment strategies for two-period horizon
• 1. Buy 1 of one-year bond and when it matures
  buy another one-year bond

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Expectations Hypothesis

• Investment strategies for two-period horizon
• 1. Buy 1 of one-year bond and when it matures
  buy another one-year bond.
• 2. Buy 1 of two-year bond and hold it.

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Expectations Hypothesis

• Expected return from strategy 2 (Buy 1 of
  two-year bond and hold it for two periods).
• A(1 i2t) Proceeds of the bond during the
  first year

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Expectations Hypothesis

• Expected return from strategy 2 (Buy 1 of
  two-year bond and hold it for two periods).
• That is (1 i2t)(1 i2t).

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Expectations Hypothesis

• Expected return from strategy 2 (Buy 1 of
  two-year bond and hold it for two periods). That
  is
• (1 i2t)(1 i2t).
• If we subtract the 1 initial investment, we will
  have
• (1 i2t)(1 i2t) 1 1 2(i2t) (i2t)2
  1 2(i2t)
• Since (i2t)2 is extremely small, expected return
  is approximately 2(i2t)

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Expectations Hypothesis strategy 1

• Expected return from strategy 1 (Buy 1 of
  one-year bond and when it matures buy
  another one-year bond).
• A(1 it) Proceeds of the bond during the first
  year

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Expectations Hypothesis strategy 1

• Expected return from strategy 1 (Buy 1 of
  one-year bond and when it matures buy
  another one-year bond).
• A(1 it) Proceeds of the bond during the first
  year. Reinvest A for another year. At the end of
  the second year it will grow to
• A (1 iet1).
• That is (1 it)(1 iet1).

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Expected return from strategy 1

• Subtracting the initial investment, we have
• (1 it)(1 iet1)  1 1 it iet1
  it(iet1) 1
• Since it(iet1) is also extremely small, expected
  return is approximately
• it iet1

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Expected return from strategy 1

• From implication above expected returns of two
  strategies are equal Therefore
• 2(i2t) it iet1
• Solving for i2t
• it iet1
• i2t
• 2

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Expected return from strategy 1

• More generally for n-period bond
• it iet1 iet2 ... iet(n1)
• int
• n
• This implies Interest rate on long bond is equal
  to average short rates expected to occur over
  life of long bond

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Expected return from strategy 1

• Numerical example
• Assuming that one-year interest rates over the
  next five years are expected to be 5, 6, 7,
  8 and 9,
• Interest rate on two-year bond
• (5 6)/2 5.5
• Interest rate for five-year bond
• (5 6 7 8 9)/5 7

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Expectations Hypothesis and Term Structure Facts

• Explains why yield curve has different slopes
• 1. When short rates expected to rise in future,
  average of future short rates which is equal to
  int is above todays short rate therefore yield
  curve is upward sloping

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Expectations Hypothesis and Term Structure Facts

• Explains why yield curve has different slopes
• 1. When short rates expected to rise in future,
  average of future short rates int is above
  todays short rate therefore yield curve is
  upward sloping
• 2. When short rates expected to stay same in
  future, average of future short rates are same as
  todays, and yield curve is flat

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Expectations Hypothesis and Term Structure Facts

• Explains why yield curve has different slopes
• 1. When short rates expected to rise in future,
  average of future short rates int is above
  todays short rate therefore yield curve is
  upward sloping
• 2. When short rates expected to stay same in
  future, average of future short rates are same as
  todays, and yield curve is flat
• 3. Only when short rates expected to fall will
  yield curve be downward sloping

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Expectations Hypothesis and Term Structure Facts

• Expectations Hypothesis explains Fact 1 that
  short and long rates move together
• 1. Short rate rises are persistent. If they
  increase today, they tend to be high tomorrow,

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Expectations Hypothesis and Term Structure Facts

• Expectations Hypothesis explains Fact 1 that
  short and long rates move together
• 1. Short rate rises are persistent. If they
  increase today, they tend to be high tomorrow,
• If it ? today, expected future interest rates
  will rise
• (iet1, iet2 etc.) ? ? average of future rates ?
  ? int ?

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Expectations Hypothesis and Term Structure Facts

• Expectations Hypothesis explains Fact 1 that
  short and long rates move together
• 1. Short rate rises are persistent. If they
  increase today, they tend to be high tomorrow,
• If it ? today, expected future interest rates
  will rise
• (iet1, iet2 etc.) ? ? average of future rates ?
  ? int ?
• 3. That is, if it ? ? int ? Short and long rates
  move together

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Expectations Hypothesis Explains Fact 2 that
yield curves tend to have steep slope when short
rates are low and downward slope when short rates
are high

• 1. When short rates are low, they are expected to
  rise to normal level, and long rate (average of
  expected future short rates) will be well above
  todays short rate yield curve will have steep
  upward slope

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Expectations Hypothesis Explains Fact 2 that
yield curves tend to have steep slope when short
rates are low and downward slope when short rates
are high

• 1. When short rates are low, they are expected to
  rise to normal level, and long rates (average of
  future short rates) will be well above todays
  short rate yield curve will have steep upward
  slope
• 2. When short rates are high, they will be
  expected to fall in future, and long rate
  (average of future short rates) will be below
  current short rate yield curve will have
  downward slope

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Expectations Hypothesis Explains Fact 2 that
yield curves tend to have steep slope when short
rates are low and downward slope when short rates
are high

• Expectations Hypothesis Doesnt explain Fact 3
  that yield curve usually has upward slope
• The typical upward slope of the yield curve
  implies that the future short rates are going to
  be higher than the current short rates. However
  short rates are as likely to fall in future as to
  rise, so according to the expectations theory
  must be the yield curve must be flat and not
  upward slopping.

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Segmented Markets Theory

• Key Assumption Bonds of different maturities
  are not substitutes at all.

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Segmented Markets Theory

• Key Assumption Bonds of different maturities
  are not substitutes at all
• Implication Markets are completely segmented
  interest rate at each maturity determined
  separately, regardless of one another.

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Segmented Markets Theory

• Key Assumption Bonds of different maturities
  are not substitutes at all
• Explains Fact 3 that yield curve is usually
  upward sloping
• People typically prefer short holding periods and
  thus have higher demand for short-term bonds,
  which have higher price and lower interest rates
  than long bonds

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Segmented Markets Theory

• Key Assumption Bonds of different maturities
  are not substitutes at all
• Does not explain Fact 1 or Fact 2 because assumes
  long and short rates determined independently.

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Liquidity Premium Theory

• Key Assumption Bonds of different maturities
  are substitutes, but are not perfect substitutes

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Liquidity Premium Theory

• Key Assumption Bonds of different maturities
  are substitutes, but are not perfect substitutes
• Implication Modifies Expectations Theory with
  features of Segmented Markets Theory
• Investors prefer short rather than long bonds ?
  they must be paid positive liquidity (term)
  premium, lnt, to hold long-term bonds
• Results in following modification of Expectations
  Theory
• it iet1 iet2 ... iet(n1)
• int lnt
• n

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Relationship Between the Liquidity Premium and
Expectations Theories
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Numerical Example

• 1. One-year interest rate over the next five
  years 5, 6, 7, 8 and 9

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Numerical Example

• 1. Assume that one-year interest rate over the
  next five years 5, 6, 7, 8 and 9
• 2. Also assume that liquidity premiums for one to
  five-year bonds are
• 0, 0.25, 0.5, 0.75 and 1.0.

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Numerical Example

• 1. Assume that one-year interest rate over the
  next five years 5, 6, 7, 8 and 9
• 2. Also assume that liquidity premiums for one to
  five-year bonds are
• 0, 0.25, 0.5, 0.75 and 1.0.
• Interest rate on the two-year bond would be
• (5 6)/2 0.25 5.75

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Numerical Example

• 1. Assume that one-year interest rate over the
  next five years 5, 6, 7, 8 and 9
• 2. Also assume that liquidity premiums for one to
  five-year bonds are
• 0, 0.25, 0.5, 0.75 and 1.0.
• Interest rate on the five-year bond
• (5 6 7 8 9)/5 1.0 8

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Numerical Example

• 1. Assume that one-year interest rate over the
  next five years 5, 6, 7, 8 and 9
• 2. Also assume that liquidity premiums for one to
  five-year bonds are
• 0, 0.25, 0.5, 0.75 and 1.0.
• Interest rates on one to five-year bonds
• 5, 5.75, 6.5, 7.25 and 8.
• Comparing with those for the expectations theory,
  liquidity premium theory produces yield curves
  more steeply upward sloped

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Liquidity Premium Theory Term Structure Facts

• Explains all 3 Facts
• Explains Fact 3 of usual upward sloped yield
  curve by investors preferences for short-term
  bonds
• Explains Fact 1 and Fact 2 using same
  explanations as expectations hypothesis because
  it has average of future short rates as
  determinant of long rate

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Market Predictions of Future Short Rates panel
a future interest rates are expected to rise.
The liquidity premiums make it look steeper than
what the expectations theory suggests.
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Market Predictions of Future Short Rates

• Panel b This actually a flat curve. Investors
  are expecting the future rates to be the same as
  now. Its upward sloping look is due to the
  liquidity premiums.

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Market Predictions of Future Short Rates

• Panel c a flat look means investors expect
  interest rates to slightly fall. The liquidity
  premiums that they demand bring them back to the
  par with current rates.

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Market Predictions of Future Short Rates

• Panel d future interest rates are expected to
  fall.

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Term Structure Facts to be Explained

• Three Theories of Term Structure
• 1. Interest rates for different maturities move
  together
• 2. Yield curves tend to have steep slope when
  short rates are low and downward slope when short
  rates are high
• 3. Yield curve is typically upward sloping
• Expectations Theory explains facts 1 and 2, but
  not 3.
• Segmented Markets explains fact 3, but not 1 and
  2
• Liquidity Premium Theory Combines features of
  both Expectations Theory and Segmented Markets
  Theory to get Liquidity Premium Theory and
  explain all facts