Chapter 12: Term Structure of Interest Rates

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Chapter 12 Term Structure of Interest Rates
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Chapter Summary

• Objective To explore the pattern of interest
  rates for different-term assets.
• Chapter 11 assumes all CF can be discounted at
  the same rate. This is unrealistic because Long
  Term CF are more risky than Short Term CF.

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Chaper Summary

• Term Structure
• Relationship between YTM and maturity
• The term structure under certainty
• Forward rates
• Theories of the term structure

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Yield Curves
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Overview of Term Structure of Interest Rates

• Information on expected future short term rates
  can be implied from yield curve
• 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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Yield Curves
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Recent yield curves April 2006
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CDN yield curve Feb 3, 2007
CDN yield curve Feb 3, 2007
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US yield curve Feb 3, 2007
US yield curve Feb 3, 2007
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Sources of term structure

• www.investinginbonds.com
• Published in Globe and Mail every Monday

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Term structure analysis under certainty

• Assumption Future one-period interest rates are
  known at time zero
• Zero-coupon bond prices found from these rates
• Forward rate expected future spot interest rate
• Under certainty, forward rates are equal to
  future short-term rates
• In reality, there is uncertainty so actual rates
  will be different from forward rates
• We will do exercises.

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Expected Interest Rates in Coming Years (Table
12.1)
Expected One-Year Rates in Coming
Years Year Interest Rate 0 (today) 8
1 10 2 11 3 11
a
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Long-Term Rates and Bond Prices using Expected
Rates
Time to Maturity Price of Zero YTM 1 925.9
3 8.00 2 841.75 8.995 3
758.33 9.660 4 683.18 9.993 1,000 Par
value zero
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Summary Reminder

• Objective To explore the pattern of interest
  rates for different-term assets.
• The term structure under certainty
• Forward rates
• Theories of the term structure
• Measuring the term structure

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Notations

• y current spot rate
• r f expected 1-yr spot rate in a year

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Two Alternative Scenarios
b
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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 using Table 12.2 Numbers
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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Uncertainty and forward rates

• Under certainty investors are indifferent
  between a short-term bond and a long-term bond
  sold before maturity, or between one long-term
  investment and a sequence of rolled-over
  short-term investments.
• However, there is uncertainty so we need a
  liquidity premium to compensate for the
  additional risk of investing long term.

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Two Alternative Scenarios
Riskier for short-term investors
Riskier for long-term investors
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Summary Reminder

• Objective To explore the pattern of interest
  rates for different-term assets.
• The term structure under certainty
• Forward rates
• Theories of the term structure

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Theories of Term Structure

• The Expectations Hypothesis
• Liquidity Preference
• Forward rates are higher than predicted by
  expectations hypothesis
• Market Segmentation
• Preferred Habitat

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Expectations Theory

• Forward rate is the markets expectation of
  future spot rates
• Long-term and short-term securities are perfect
  substitutes (ie. no liquidity premiums are
  required)

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Liquidity Premium Theory

• Assumes most investors are short-term investors
  (empirical evidence shows this true)
• Thus, long-term bonds are more risky
• Investors will demand a premium for the risk
  associated with long-term bonds (ie. liquidity
  premium)
• 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
  greater than expected future short-term rates

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Comparison of two theories
Liquidity Preference
Expectations Theory
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Market Segmentation and Preferred Habitat
Theories

• Market segmentation theory
• Short- and long-term bonds are traded in distinct
  markets, thus they arent substitutes
• Trading in the distinct segments determines the
  various rates
• Preferred Habitat Theory
• Extension of market segmentation theory
• Investors have a preferred maturity but will
  switch timeframes if premiums are adequate

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What does the empirical evidence say?

• Yield curves are mostly upward-sloping
• Liquidity premiums are hard to estimate and may
  not be constant
• Inverted yield curves generally point to
  declining interest rates
• Steeply rising yield curves are generally
  interpreted as signaling impending rate increases
  
• Do rising yield curves mean rates are increasing?
  Not necessarily it could be bec of liquidity
  premium

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Summary Reminder

• Objective To explore the pattern of interest
  rates for different-term assets.
• The term structure under certainty
• Forward rates
• Theories of the term structure
• Measuring the term structure

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Measuring the term structure- The bootstrapping
method

• Derive spot rates from bond yields of varying
  maturities
• Treat each coupon as a mini-zero coupon bond
• Use bonds of progressively longer maturities,
  starting from T-bills

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Example section 12.6, from Figure 11.1

• Observe prices and yields on August 17, 2001
  find the spot rate for December 1, 2002
• Observed yields 3.90, 4.04 for 6M and 12M,
  respectively
• Observed clean price for bond expiring on
  December 1, 2002 1002.29
• Dirty price clean price (time elapsed in
  semesters) x coupon

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Bootstrapping example (cont.)

• Solving, we find y34.16 annually

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Using Spot Rates to price Coupon Bonds

• A coupon bond can be viewed as a series of zero
  coupon bonds
• To find the value, each payment is discounted at
  the zero coupon rate
• Once the bond value is found, one can solve for
  the yield
• Its the reason for which similar maturity and
  default risk bonds sell at different yields to
  maturity

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Sample Bonds

• Assuming annual compounding

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Calculation of Price Using Spot Rates (Bond A)
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Calculation of Price Using Spot Rates (Bond B)
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Solving for the YTM

• Bond A
• Bond Price 978.54
• YTM 6.63
• Bond B
• Price 1,047.56
• YTM 6.61