Chapter 6 The Risk and Term Structure of Interest Rates

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Chapter 6 The Risk and Term Structure of
Interest Rates

• -Examine the relationship of the various interest
  rates with
• 1. the same term to maturity, but different risk
  structure three components
• 2. the same risk structure, but different term to
  maturity three theories

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How many interest rates do we have?-From Wall
Street Journal, BONDS, RATES CREDIT MARKETS
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Different interest rates for the same type of
bonds-From Federal Reserve, Fred Database
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What relationship do you observe?
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Risk Structure of Interest Rates

• Default
• Liquidity
• Income tax considerations
• Note Interest rate risk vs Risk structure of
  interest rates

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I. Default Risk

• Default risk occurs when the issuer of the bond
  is unable or unwilling to make interest payments
  or pay off the face value
• U.S. T-bonds are considered default free (Why?)
• All other bonds are considered to be subject to
  default risk
• Corporate bonds
• Mortgage-backed bonds
• Municipal bonds

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Credit rating is the simplest way to measure
default risk

• Rating Agencies Moodys and SP
• Investment-grade bonds Baa/BBB and above
• Speculative-grade bonds (or junk bonds) below
  Baa/BBB

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Risk Premium

• Risk premium is the spread between the interest
  rates on bonds with default risk and the interest
  rates on T-bonds
• If the probability that the issuer of the bond
  may default rises, the default risk of the bond
  increases, and the risk premium between the
  interest rates on this bond and default-free
  T-bonds with the same term to maturity increases
  too.

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Equilibrium analysis tells you why risk premium
exists
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II. Liquidity

• Liquidity is the ease with which an asset can be
  converted into cash
• The more liquid an asset is, the more desirable
  it is (ceteris paribus), and the lower the
  interest rate is. Bond issuers must pay higher
  interest to compensate investors for the lower
  liquidity
• T-bonds are the most liquid long-term bonds
  because their markets are deep corporate bonds
  are much less liquid
• The spread between the interest rates on
  corporate bonds and T-bonds reflect not only
  their different default risk, but also their
  different liquidity.
• We can use the same figure used in explaining
  risk premium to explain how the liquidity premium
  is derived from equilibrium analysis

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III. Income Tax Considerations

• Why the interest rates of municipal bonds (munis)
  are even lower than the interest rates of T-bonds
  though munis are not default-free and less
  liquid?
• Interest payments on munis are exempt from
  federal income taxes, sometimes state income
  taxes, which attracts rich people to invest in
  munis.

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Application Effects of Bush Tax Cut on Bond
Interest Rates

• 2001 tax cut reduces the top income tax bracket
  from 39 to 35
• Munis are less desirable to rich people with the
  income tax rate cut than without the tax cut
• Investors somewhat switch demand from munis to
  T-bills. The demand curve for munis shifts to the
  left, which lowers their prices and raises their
  interest rates.
• The demand curve for T-bills shifts to the right,
  which raises their prices and lowers their
  interest rates.

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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
• Yield curve is the main way to illustrate the
  term structure of interest rates. It is a plot of
  the yield on bonds with the same risk, liquidity
  and tax considerations against their different
  terms to maturity.

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Current Treasury Yield Curve-From Bloomberg
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Interest rates of your bank saving account.
Similar patter as the Treasury yield curve.
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We observed different shapes in history
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Different shapes of yield curves exist in history

• Upward-sloping long-term rates are above
  short-term rates
• Flat short- and long-term rates are the same
• Downward-sloping/Inverted long-term rates are
  below short-term rates
• Other irregular shapes like U-shape, inverted
  U-shape, or mixtures
• How to explain them?

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Do you observe any pattern?
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Facts Theory of the Term Structure of Interest
Rates Must Explain

• 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 slope upward in most cases

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Three Major Theories for Term Structure of
Interest Rates

• Expectations Theory explains the first two
  facts but not the third
• Segmented Markets Theory explains fact three but
  not the first two
• Liquidity Premium Theory combines the two
  theories to explain all three facts

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I. Expectations Theory

• Assumption Bonds with different terms to
  maturity can be perfect substitutes, or say,
  investors do not prefer bonds of one maturity
  over another
• Conclusion 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
• -Why upward-sloping yield curve? People expect
  the interest rate to go up.
• -Why downward-sloping yield curve? People expect
  the interest rate to go down.

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Example

• Let the current interest rate on one-year bond be
  6.
• You expect the interest rate on a one-year bond
  to be 8 next year.
• The interest rate on a two-year bond must be (6
  8)/2 7 for you to be willing to purchase
  it.

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Why?
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Why? (cont.)
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Pros and Cons

• Explains why the term structure of interest rates
  changes at different times
• Explains why interest rates on bonds with
  different maturities move together over time
  (fact 1)
• Explains 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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II. 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 and supply
  of that bond
• Investors have preferences for bonds of one
  maturity over another
• If investors have short desired holding periods
  and 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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III. Liquidity Premium Theory

• Assumptions Bonds of different maturities are
  substitutes but not perfect substitutes.
  Investors prefer short-term bonds to long-term
  bonds because short-term bonds are more liquid.
• Conclusion The interest rate on a long-term bond
  will the interest rate predicted by the
  expectation theory plus a liquidity premium.

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The Formula

• is the n-year interest rate at time t
• The first term on the right side of the equation
  is the interest rate predicted by the expectation
  theory w/o considering liquidity preference
• The second term, ,is the liquidity premium
  for the n-year bond at time t, which is always
  positive and rises with the term to maturity n.

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A Numerical Example

• Suppose we study the yield curve with two
  interest rates with 1-year and 2-year to
  maturity.
• 1-year interest rate 4
• We expect 1-year interest rate to
• -Case I increase to 5
• -Case II be unchanged 4
• -Case III decrease to 3
• The liquidity premium for the 2-year bond over
  the 1-year bond is 1

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Expectation vs Liquidity Premium

• Based on the expectation theory,
• -Case I two-year interest rate(45)/24.5
  the yield curve is upward-sloping
• -Case II two-year interest rate(44)/24
  the yield curve is flat
• -Case III two-year interest rate(43)/23.5
  the yield curve is downward-sloping

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Expectation vs Liquidity Premium

• Based on the liquidity premium theory,
• -Case I two-year interest rate(45)/215.5
  the yield curve is upward-sloping (even steeper)
• -Case II two-year interest rate(44)/215
  the yield curve is upward-sloping
• -Case III two-year interest rate(43)/214.5
  the yield curve is upward-sloping
• --Fact 3 Yield curves slope upward in most cases

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Explanation of the Facts using the Liquidity
Premium Theory

• Fact 1 that interest rates on different maturity
  bonds move together over time is explained by the
  first term in the equation
• Fact 2 that yield curves tend to slope upward
  when short-term rates are low and to be inverted
  when short-term rates are high is explained by
  both an expected high future interest rate and
  the liquidity premium term in the first case and
  by an expected strongly low future interest rate
  in the second case
• Fact 3 that yield curves typically slope upward
  is explained by the liquidity premium which
  dominates the effect of expected low future
  interest rate.