Inflation%20and%20the%20Yield%20Curve

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Inflation and the Yield Curve

• Week 6 October 1, 2005

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Inflation Premium and Returns

• Fixed income securities lose purchasing power
  with inflation investors adjust expected real
  rate for inflation
• The Fisher Equation reflects this if ? is
  expected inflation and R is expected real
  ratewhere r is the nominal rate and
  approximately we say that r R ?, ignoring
  the cross-product term as small as of multiple of
  two fractions

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Inflation Premium

• Fisher hypothesis is that inflation premium
  equals expected inflation ?
• Harrod-Mundel theorize that inflation reduces
  wealth in real money balances, causing increased
  saving and decreasing real rate, thus R declines
  r is constant
• Darby-Feldstein note look at after-tax returns
  and argue that r increases more than ? to
  maintain same return after inflation

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Nominal and Real Returns

• We have looked at nominal T-Bill yields and ex
  post real returns using realized inflation
• Expected real returns include a premium for
  expected inflation
• Realized real returns differ from expected
  returns because
• Expected inflation changes
• Unexpected inflation occurs

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Definitions

• The term structure is defined as the relation
  between yield (to maturity) and term or time to
  maturity
• Term structure and yield curve are the same
• Term structure is defined as of a given date
  (e.g. today, or January 1, 1996)
• Term structure is given for fixed incomes of a
  given risk class (usually Treasuries)

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Types of Term Structures

• Upward sloping, sometimes called normal or rising
  term structure, can be steep or gentle
• Downward sloping, or falling term structure
• A flat or horizontal term structure
• A humped term structure
• Four years ago, we observed a unique U-shaped
  term structure, not seen since
• The Treasury yield curve is published daily in
  the Wall Street Journal

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Term Structure

• Is drawn as of a given date but changes over
  time.
• Our objective is to look at the pure term effect
  on yields
• Coupon issues have different payments, so tax and
  reinvestment rates blur pure term effect
• Best to use T-Bills or zero coupon bonds (called
  Strips with Treasuries)

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Short and Long Treasury Rates
Source FRB St. Louis
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Treasury Rates since 1970
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Term Structure Shifts in 1980s
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Term Structure in 1990s
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Term Structure in 1999
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U-Shaped Yield Curves 2000-01
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Term Structure Last 2002
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Current and 2003-5 Yield Curves
Source FRBoard Release H15
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Treasury Strips

• Treasury pays coupon interest and principal to
  holders of its many bonds frequently
• Coupons and principal can be stripped from bonds
  and sold separately
• E.g. the 5-3/4 Aug 10 will pay 28,750 for every
  1 million outstanding in August, 2010
• Aug 10 stripped interest ci sold for 7408 on
  Sept. 26, 2002, that is, .7425 times face value
  of coupon payments. What does it sell for now?

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The Forward Rate of Interest

• The spot rate on current (time t) observable rate
  on securities of maturity n, that is, the current
  market rate, is noted
• The forward rate is the rate now (t) on an
  one-period investment beginning tk periods in
  the future

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Definition of Forward Rates

• Forward rates are often unquoted or unavailable
• Forward rates are implicit in the term structure
  of spot rates since spot rates of different
  maturities define forward rates
• Note there is no one-period forward rate now but
  only for future or forward periods

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Solving for Forward Rates 1

• Two spot rates differing by maturity one year
  can be used to solve for forward rates
• Example using Treasury strips of one and two year
  maturities from Sept. 26, 2002

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Solving for Forward Rates 2

• Example using T-Bills of 3 and 6 months
• In this case, we have to annualize the
  three-month (.25 year) forward rateor, in
  other words, rates are not expected to change
  much on three-month bills 1.54

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Interpretation of Forward Rates

• Forward rates are merely defined as relations
  between observed spot rates
• Theories of the term structure can be discussed
  in terms of what forward rates mean
• Three basic theories
• Unbiased or pure expectations theory
• Market segmentation or preferred habitats
• Uncertainty, term or liquidity premiums

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Expectations Theory

• Forward rates represent expectations of future
  spot rates
• Economic reasoning is based on arbitrage if for
  any horizon, we haveor for three-period
  example

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Expectations Theory (continued)

• In the examples, we make more if we invest in the
  shorter maturity bond and reinvest at the
  expected future spot rate
• Note that the expected future spot rate is the
  same as the definition of the forward rate if
  equation holds as equality
• Under pure expectations, the forward rate is the
  unbiased expectation of expected future spot rates

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Expectations Theory (example)

• Using previous examples, if we haveand we
  believe the Treasury rate in one year will be
  higher than 2.94, we will buy the one-year
  Treasury and plan to reinvest at higher rates in
  one year
• Arbitrage drives returns into line with
  expectations so that forward expected rate

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Example of Short Sale
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Term Structure and Expectations

• Rearranging our forward rate definition,or the
  spot rate is the geometric average of future
  forward or expected future spot rates
• A downward sloping term structure implies falling
  rates, a rising term structure increasing rates,
  and a flat structure unchanging rates

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Expectations Theory

• Simple and based on basic principles
• Assumes investors are indifferent to risk of
  holding period returns (risk neutral)
• Arbitrage is a powerful economic force
• Widely used in analysis of market events
• Students should be able to do calculations and
  interpret term structure in terms of future
  expected spot rates
• We discuss the evidence on theory later

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Market Segmentation/Preferred Habitat Theory

• Investors have different asset maturities and
  want to match asset/liability interest rate risks
• Risk-matching is assumed to be much more
  important than possible profit opportunities
  (extreme risk aversion)
• Very little supporting evidence and little room
  for arbitrage or risk-taking

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Term or liquidity premiums

• Theory combines some aspects of other two
• Borrowers and lenders may have different concerns
  in interest rates -- lenders may prefer long-term
  fixed rates, investors fear gains and losses from
  longer investments
• Longer rates may have therefor have a term or
  liquidity premium
• Forward rates would be biased because of the
  premium

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Recent Term Structure Theories

• Cox-Ingersoll-Ross (CIR)
• Continuous time models using option-pricing
  techniques
• Arbitrage portfolios determine prices of risk
• Risk factors are state of technology (I.e. real
  return on capital)
• CIR show basis for term premiums
• Lattice models (e.g. Black-Derman-Toy) gaining
  widespread use to price derivatives

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Next (October 5)

• Read over Fama-Bliss and Black-Derman-Toy for
  Oct. 5
• Read text Chapter 8 and KMV reading posted on
  website for Oct. 12 class
• See me concerning the group project if you have
  remaining questions or need help
• Take-home examination will be distributed October
  13 to be returned by next class, so review old
  exams and raise any questions you have with me