Lecture 12: Fixed Income Securities

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Lecture 12 Fixed Income Securities

• The following topics will be covered
• Discount Bonds
• Coupon Bonds
• Interpreting the Term Structure of Interest Rates
• Basic of Term Structure Models

Materials from Chapter 10 and 11 (briefly) of CLM
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Zero-coupon Bonds basic notations

• For zero-coupon bonds, the yield to maturity is
  the discount rate which equates the present value
  of the bonds payments to its price.
• where Pnt is the time t price of a discount bond
  that makes a single payment of 1 at time tn,
  and Ynt is the bond yield to maturity. We have,
• Expressed in log form, we have

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Yield Curve of Zero-coupon Bonds

• Term structure of interest rates is the set of
  yields to maturity, at a given time, on bonds of
  different maturities. Yield spread SntYnt-Y1t,
  or in log term sntynt-y1t, measures the shape of
  the term structure.
• Yield curve plots Ynt or ynt against some
  particular date t.

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Return for Discount Bonds (1)

• Define Rn,t1 as the 1-period holding-period
  return on an n-period bond purchased at time t
  and sold at time t1
• Writing in the log form, we have
• Holding period return is determined by the
  beginning-o-period yield (positively) and the
  change in the yield over the holding period
  (negatively).

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Return for Discount Bonds (2)

• The log bond price today is the log price
  tomorrow minus the return today.
• We can solve this difference equation forward and
  get
• We can also get
• The log yield to maturity on a zero-coupon bond
  equals the average log return period if the bond
  is held to maturity

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Forward Rate

• The forward rate is defined to be the return on
  the time tn investment of Pn1,t/Pnt
• where, in the forward rate, n refers to the
  number of periods ahead that the 1-period
  investment is to be made, and t refers to the
  date at which the forward rate is set.

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Coupon Bonds

• Coupon bonds can be viewed as a package of
  discount bonds
• There is no analytical solution for yield to
  maturity of coupon bonds
• Unlike the yield to maturity on a discount bond,
  the yield to maturity on a coupon bond does not
  necessarily equal the per-period return if the
  bond is held to maturity.
• The yield to maturity equals the per-period
  return on the coupon bond held to maturity only
  if coupons are reinvested at a rate equal to the
  yield to maturity.
• Two cases
• Selling at par
• perpetuity

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Duration

• Macaulay duration
• See the example on page 402
• Duration is the negative of the elasticity of a
  coupon bonds price with respect to its gross
  yield (1Ycnt)
• Modified duration

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Immunization

• Implications firms with long-term zero-coupon
  liabilities, such as pension obligations, they
  may wish to match or immunize these liabilities
  with coupon-bearing Treasury bonds.
• Zero-coupon Treasury bonds are available, they
  may be unattractive because of tax clientele and
  liquidity effects, so the immunization remain
  relevant.
• If there is a parallel shift in the yield curve
  so that bond yields of all maturities move by the
  same amount, then a change in the zero-coupon
  yield is accompanied by an equal change in the
  coupon bond yield

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Limitations

• A parallel shift of the term structure
• Works for small change in interest rates
• Cash flows are fixed and dont change when
  interest rate changes.
• Callable securities

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Loglinear Model for Coupon Bonds

• Starting from the loglinear approximate return
  formula, we have

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Estimating Zero-coupon Term Structure

• If the prices of discount bonds P1Pn maturing at
  each coupon date is known, then the price of a
  coupon bond is
• If coupon bond prices are known, then we can get
  the implied zero-coupon term structure

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Spline Estimation

• When there are more than one price for each
  maturity, statistical methods should be used. One
  way is regression
• In practice the term structure of coupon bonds is
  usually incomplete. McCulloch (1971, 1975)
  suggest to write Pn as a function of maturity
  P(n)
• Assume P(n) to be a spline function. The term
  "spline" is used to refer to a wide class of
  functions that are used in applications requiring
  data interpolation and/or smoothing.

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Tax Effect

• US Treasury bond coupons are taxed as ordinary
  income while price appreciation on a coupon
  bearing bond purchased at a discount is taxed as
  capital
• Thus there is a tax effect
• Page 411, CLM

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Pure Expectation Hypothesis (PEH)

• PEH

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Alternatives to Pure Expectation Hypothesis

• Expectation hypothesis
• Considering term premia
• Preferred habitat
• Different lenders and borrowers may have
  different preferred habitats
• Time varying of term premia

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Term Structure Models -- Motivations

• Starting from the general asset pricing
  condition introduces
• 1Et(1Ri,t1)Mt1
• Fixed-income securities are particularly easy to
  price. When a fixed-income security has
  deterministic cash flows, it covaries with the
  stochastic discount factor only because there is
  time-variation in discount factors.
• PntEtPn-1,t1Mt1
• It can be solved forward to express the n-period
  bond price as
• PntEtPn-1,t1Mt1

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Affine-Yield Models

• Assume that the distribution of the stochastic
  discount factor Mt1 is conditionally lognormal
• Take logs of PntEtPn-1,t1Mt1, we have