The Term Structure of Interest Rates

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• The Term Structure of Interest Rates

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Overview of Term Structure of Interest Rates

• The relationship between yield to maturity and
  maturity.
• Information on expected future short term rates
  (short rate) might be implied from the yield
  curve. (short rate vs. spot (average) rate)
• The yield curve is a graph that displays the
  relationship between yield and maturity.
• Three major theories are proposed to explain the
  observed yield curve.

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Spot Rates on Yield Curves
Yields
Upward Sloping
Flat
Downward Sloping
Maturity
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Expected Interest Rates in Coming Years
Expected One-Year Short Rates in Coming
Years Year Interest Rate 0 (today) 8
1 10 2 11 3 11
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Pricing of Zero Coupon Bonds using Expected Rate
PVn Present Value of 1 in n periods r1
One-year short rate for period 1 r2 One-year
short rate for period 2 rn One-year short rate
for period n yn yield for a zero coupon security
with a maturity of n
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• Thus
• Short rates ? Spot rates
• But what about the other way around?

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Forward Rates from Observed Long-Term Rates
fn one-year forward rate for period n yn
yield for a security with a maturity of n
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Example of Forward Rates
4 yr 9.993 3yr 9.660 fn ? (1.0993)4
(1.0966)3 (1fn) (1.46373) / (1.31870)
(1fn) fn .10998 or 11 Note this is expected
rate that was used in the prior example.
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Downward Sloping Spot Yield Curve
Zero-Coupon Rates Bond Maturity 12 1 11.7
5 2 11.25 3 10.00 4 9.25 5
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Forward Rates for Downward Sloping Yield Curve

• 1yr Forward Rates
• 1yr (1.1175)2 / 1.12 - 1 0.115006
• 2yrs (1.1125)3 / (1.1175)2 - 1 0.102567
• 3yrs (1.1)4 / (1.1125)3 - 1 0.063336
• 4yrs (1.0925)5 / (1.1)4 - 1 0.063008

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Theories of Term Structure

• Expectations
• Liquidity Preference
• Upward bias over expectations
• Market Segmentation
• Preferred Habitat

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Expectations Theory

• Observed long-term rate is a function of todays
  short-term rate and expected future short-term
  rates.
• Long-term and short-term securities are perfect
  substitutes.
• Forward rates that are calculated from the yield
  on long-term securities are market consensus
  expected future short-term rates.

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Liquidity Premium Theory

• Long-term bonds are more risky.
• Investors will demand a premium for the risk
  associated with long-term bonds.
• The yield curve has an upward bias built into the
  long-term rates because of the risk premium.
• Forward rates contain a liquidity premium and are
  not equal to expected future short-term rates.

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Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Short Rates
Liquidity Premium
Maturity
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Liquidity Premiums and Yield Curves
Yields
Observed Yield Curve
Short Rates
Liquidity Premium
Maturity
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Market Segmentation and Preferred Habitat

• Short- and long-term bonds are traded in distinct
  markets.
• Trading in the distinct segments determines the
  various rates.
• Observed rates are not directly influenced by
  expectations.
• Preferred Habitat
• Modification of market segmentation
• Investors will switch out of preferred maturity
  segments if premiums are adequate.