Term Structure of Interest Rates

1
Term Structure of Interest Rates

• Earlier, we assumed a single interest rate. Now
  we relax this assumption and consider different
  interest rates by periods. The result will be
  more realistic models and analysis.

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• Factors affecting bond yield to maturity
• - quality (lower quality ? higher yield
  expected)
  
• - time to maturity long bonds tend to offer
  higher yields than short bonds of the same
  quality (more risk in long bonds)
  
• Inverted Yield Curve. Sometimes long bonds have
  lower yields than short bonds. Short-term rates
  may increase rapidly, but investors may believe
  the increase is temporary, so that long-term
  yields remain near their previous levels.

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• Important question to consider in studying a
  bond. How does its yield and maturity compare
  with others in its risk class? A bond which is
  far from the curve is usually there for a good
  reason bond-specific call features, or concerns
  about insolvency.

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• Term Structure
• Term structure provides a more basic theory than
  is available by just using a yield curve. It
  provides more insight into interest rates than
  the yield curve does.
• Basic Idea. The interest rate charged (or paid)
  for money depends on the length of time that the
  money is held.
  
• The spot rate st is the rate of interest,
  expressed in yearly terms, charged (or paid) for
  money held from the present until time t. Both
  interest and principal are paid at time t.

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• Examples
• 1) One-year spot rate, s1 5.
• You deposit 1,000 in a bank today. At the end
  of a year the bank pays you 1,000 ? 1.05
  1,050.
  
• 2) Two-year spot rate, s2 5.5
• You deposit 1,000 in a bank today. At the end
  of 2 years the bank pays you 1,000 ? 1.0552
  1,113.03.
  
• 3) t-year spot rate, st
• You deposit 1,000 in a bank today. At the end
  of t years the bank pays you 1,000 ? (1st)t

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• Discount Factors and Present Value
• For each spot rate st there is a discount factor
  dt. The discount factor is the reciprocal of the
  growth factor the spot rate defines.
  
• If, using the spot rate st, an amount M grows to
  
  
• M gt M in t years, then M dt ? M. The
  discount factor is the factor by which future
  cash flows must be multiplied to obtain an
  equivalent present value. That is, dt 1/gt .
• Given CFS (x0, x1, , xn)
• PV x0 d1 x1 d2 x2 dn xn.

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• Determining the Spot Rate
• If we know the prices of a series of zero-coupon
  bonds with various maturity dates, we can use
  their yields to define the spot rates. Such
  information may be sparse, or unavailable.
  However there are some other ways to determine
  spot rates.

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• Example. Using one spot rate to compute
  another.
  
• The known interest rate of a 1-year Treasury bill
  gives s1.
  
• Now consider a 2-year bond, with price P, coupon
  payments of C, and face value F. The price
  should satisfy
  
• P C/(1s1) (C F)/(1s2)2.
• We can solve this equation for s2, since we know
  all the other terms.
  
• Similarly, if we find a 3-year bond, by knowing
  s1 and s2 we can compute s3. Continuing in this
  way, we can find s4, then s5, etc.
  

Solve for s2.
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• Example 4.3 Construction of a zero (p. 77).
  Cancellation approach.
  
• Bond A is a 10-year bond with a 10 coupon, and
  PV of PVA. Bond B is a 10-year bond with an 8
  coupon, and PV of PVB. Bonds A and B have prices
  of 98.72 and 85.89 respectively.
• A 10 years, 10, PA 98.72, normalized face
  value 100
  
• B 10 years, 8, PB 85.89, normalized face
  value 100.

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• Buy uA units of bond A, uB units of bond B.
• Bond A coupon value CA 1 ? 100
• Bond B coupon value CB 0.8 ? 100
• key idea choose uA, uB to get uA ? CA uB ? CB
  , then uBB -uAA is a zero coupon bond because
  coupon payments of the new instrument equal zero
  
• With uA0.8 and uB1 we have uA ? CA uB ? CB

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• Price of new instrument equals
• P uBPB -uAPA 1 ? 85.89 - 0.8 ? 98.72 6.914
• Face value equals
• F uBFB -uAFA1 ? 100 - 0.8 ? 100 20
• Interest rate s10 can be found from the equation
• 6.914 20/(1s10)10 ? 6.914 20/(1s10)10
• Conclude s10 11.2065

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Forward Rates

• Example. A bank offers spot rates s1, s2.
• 1) We put 100 in a 2-year account. Its value at
  the end of 2 years is 100 ? (1s2)2.
  
• 2) Alternatively, we put 100 in a 1-year account
  today. We know its value in 1 year will be 100 ?
  (1s1). We also arrange today to loan the
  proceeds for 1 year, beginning in a year. The
  loan will accrue interest at a rate of f which we
  agree upon today. The rate f is called the
  forward rate. At the end of 2 years we receive
• 100 ? (1s1) ? (1f).

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• Forward rates are interest rates for money to be
  borrowed between two dates in the future, but
  under terms agreed upon today.
  
• Two alternatives
• - returns 100 ? (1s2)2
• - returns 100 ? (1s1) ? (1f )
• By the comparison principle, we should get the
  same return either way
  
• (1s2)2 (1s1) ? (1f ) ? f (1s2)2
  /(1s1) 1
  

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• Example 4.4. s1 7, s2 8.
• f (1s2)2 /(1s1) 1 (1.08)2 /(1.07)
  1 0.09009 ? 9.01.
  
• An arbitrage argument justifies the comparison
  principle.
  
• This argument assumes no transaction costs, and
  identical borrowing and lending rates. These are
  not bad assumptions for large transactions and
  highly competitive lending/borrowing markets.
• Even if we do not make the arbitrage assumption,
  the comparison principle is reasonable. If there
  were a difference in rates between the two
  alternatives, both investors and borrowers
  involved in 2-year loans would choose the best
  alternative. Market forces would tend to adjust
  the rates.

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• Forward rate definition. The forward rate
  between times t1 and t2 with t1 by f(t1,t2). It is the rate of interest, agreed
  upon today, charged for borrowing money at time
  t1 which is to be repaid (with interest) at time
  t2.
• Notation Comment. If t1 i, t2 j, we write
  fij or fi,j instead of f(t1,t2).
  
• Forward rates are usually expressed on an
  annualized basis, just like spot rates and other
  rates.

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• Remarks
• The forward rate for lending may differ from that
  for borrowing.
  
• Calculated forward rates, determined from a set
  of spot rates, are often called implied forward
  rates.
  
• A market forward rate is one charged in the
  market.
  

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• m periods per year
• (1sj/m)j (1si/m)i ? (1fi,j/m)j-I
• forward rate equals
• fi,j m (1sj/m)j/(1si/m)i1/(j-i) - m
• Continuous time
• est? t? est t ef(t,t?) (t?-t)
• forward rate equals
• f(t,t?) st? t? st t/(t? t)

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Term Structure Explanations

• Basic Question. Why are the yield curves, and
  spot rate curves, almost never flat?
  
• - If the curves were flat, they would be like
  common interest rates.
  
• - Why do they usually slope upwards?
• Three standard answers to the question
• - Expectations Theory
• - Liquidity Preference (liquidity risk, interest
  rate risk
  
• s)
• - Market Segmentation

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• Expectations Theory Explanation
• Spot rates are determined by expectations of
  what rates will be in the future.
  
• - If the curve slopes upwards, it is because the
  market believes that the 1-year rate will likely
  go up next year. (e.g., inflation.)
  
• - An expectation is only an average guess, not
  definite information.
  
• Expectations can be expressed in terms of forward
  rates more precisely.
  

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• Weaknesses of expectations theory
• The market expects rates to increase whenever the
  spot rate curve slopes upwards practically all
  the time.
  
• Rates do not go up as often as expectations would
  imply.

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• Liquidity Preference (liquidity and interest
  rate risk)
  
• In this context, liquidity means investors
  prefer short-term bonds to long-term bonds. They
  may need to sell their bonds soon, and know
  short-term bonds are less sensitive to interest
  rate changes than long-term bonds. To induce
  investors into long-term instruments, better
  rates must be offered.
• (Liquidity usually means that the investments can
  be easily bought and sold. A 1-year treasury
  note would be liquid real estate might not be.
  Long-term bonds are in fact liquid in the sense
  that they are easily bought or sold.)

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• Moderated Version of Market Segmentation
• Although the market is basically segmented,
  individual investors are willing to shift
  segments if the rates in an adjacent segment are
  substantially more attractive than those of the
  main target segment.
• This willingness to shift means rates in adjacent
  segments must bear some relationship to each
  other, and be a curve instead of a jumble of
  disjointed numbers.

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• Short Rates
• Short rates are the forward rates spanning a
  single time period.
  
• The set of short rates completely specifies a
  term structure. Thus, they can be considered
  just as fundamental as spot rates, which also
  completely specify a term structure.

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• We can compute spot rates from short rates.
  Interest earned from time 0 to time k is the same
  as interest that would be earned by rolling over
  an investment each year. That is,
• (1sk)k (1r0)(1r1) ... (1rk-1)
• If we knew the short rates, we could compute sk.
  
  
• Example
• (1s2) 2 (1r0)(1r1) ? (1.0645)2
  (1.06)(1.069019)
  
• See file FORWARDS.XLS with an example.

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• Invariance Theorem
• This theorem basically says the following. If
  interest rates evolve according to expectation
  dynamics, and you invest a sum of money in F-I
  securities in almost anyway whatsoever for n
  years, then you get the same amount at the end of
  year n. In other words, under expectation
  dynamics, how you invest in F-I securities does
  not much matter. Every investment strategy would
  yield the same result as investing in a single
  zero-coupon bond for n years.

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• Implications of Invariance Theorem.
• The result is useful for structuring an actual
  F-I portfolio. It implies that the only reason
  for selecting a mixture of bonds must be due to
  anticipated deviations from expectation dynamics
  deviations of the realized short rates from
  their originally implied values.
• In a sense, expectation dynamics is the simplest
  assumption about the future, because it implies
  your investment results do not depend upon your
  investment strategy.

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• Running Present Value
• Running PV is an alternative, recursive approach
  to computing PV.
  
• - is the preferred calculation approach in later
  chapters,
  
• - uses concepts of expectation dynamics,
• - does not require the assumption that rates
  follow expectations dynamics.

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• Example for insight
• CFS (x0,x1,x2,x3,x4) with PV PV(0)
• PV(0) ? x0 d1 x1 d2 x2 d3 x3 d4 x4
• x0 d1x1 d2/d1 x2 d3/d1 x3
  d4/d1 x4
  
• x0 d1 PV(1)
• recall dik dij djk, i

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• Continuing recursively we have
• PV(4) x4
• PV(3) x3 d34 PV(4)
• PV(2) x2 d23 PV(3)
• PV(1) x1 d12 PV(2)
• PV(0) x0 d01 PV(1)

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• Thus running PV always uses short rates to
  determine the discount factors.
  
• rk fk,k1 is the short rate at time k.
• dk,k1 1/(1fk,k1) 1/(1rk)
• Reminder. The above rates are all computed
  starting with the spot rates, sk
  
• rk fk,k1 (1sk1)k1/(1sk)k - 1, for
  all k,

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• Present value updating
• PV(k) xk dk,k1 ? PV(k1), k 0, 1., ...,
  n-1
  
• dk,k1 1/(1fk,k1) 1/(1rk)
• To carry out the computations in a recursive
  manner the process is initiated at the final
  time. One first calculates PV(n) xn and then
  continue the process backward.

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• Flat Spot Rates
• If every spot rate sk r, then every forward
  rate r. This means every dk,k1 1/(1r).
  This means, for the CFS (x0,x1, ..., xn), RPV
  gives
• PV(n) xn
• PV(k) xk dk,k1 ? PV(k1), k 0, 1., ...,
  n-1
  
• PV(k) xk 1/(1r) ? PV(k1), k 0, 1., ...,
  n-1
  

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Floating Rate Bonds

• A floating rate note or bond has
• a fixed face value,
• a fixed maturity,
• its coupon payments are tied to current (short)
  rates of interest.
  

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• Comments
• We do not know the exact value of future coupon
  payments until 6 months before they are due.
  
• This makes it look difficult to assess the value
  of such a bond.
  
• In fact, the value is easy to deduce.

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• Reminder. Yied of the par bond equals the
  interest rate. Value of par bond at coupon
  payment times equal to face value. A par bond
  with a 10,000 face (or par value) pays 10,000
  at maturity.
• Theorem 4.1 Floating Rate Value. The value of
  a floating rate bond is equal to par at any reset
  point.
  

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Portfolio Duration Revisited

• Earlier duration ideas were based on the idea of
  a yield that was invariant with time. Changes in
  yield,e.g., spot rate changes, can create risk.
  In the term structure framework, we consider a
  different, yet related measure of risk.
• A flat spot rate curve would represent yield.
  Changes in the yield would thus move the curve up
  or down, creating parallel shifts.

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• Shifting spot rate curve
• Initial spot rates
• s1, s2, ..., sn
• Shifted spot rates
• s1 ?, s2 ?, ..., sn ?

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• The term ? is the amount of the shift (? ? 0)
  

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• Fisher-Weil Duration (continuous compounding)
• CFS (xt0, xt1, xt2, ..., xtn)
• Spot Rate Curve st0, st1, st2, ..., stn
• Present Value of CFS
• PV xt0 e-st0 t0 xt1 e-st1 t1 xt2 e-st2 t2
  ... xtn e-stn tn
  
• Fisher-Weil Duration DFW
• (t0 xt0 e-st0 t0 t1 xt1 e-st1 t1 t2 xt2 e-st2
  t2 ...
  
• tn xtn e-stn tn) / PV

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• Fisher-Weil Duration uses
• xt0 e-st0 t0/PV as the weight of t0,
• xt1 e-st1 t1/PV as the weight of t1
• xt2 e-st2 t2/PV as the weight of t2
• ...
• xtn e-stn tn/PV as the weight of tn
• These weights total to one. DFW is between the
  minimum and maximum of the ti (when each weight
  is nonnegative).
  

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• Price/Present Value of CFS as function of ?
• P(?) xt0 e-(st0 ?) t0 xt1 e-(st1 ?) t1
  ... xtn e-(stn ?) tn
  
• When ? 0 we get the current PV. The rate of
  change of P(?) in the neighborhood of 0 is the
  rate of change of the price as ? changes from 0.
  Denote by PV?(0) the first derivative of P(?)
  evaluated at ? 0. Since the derivative of eax
  wrt x is aeax, we find
• P?(0) - (t0 xt0 e-st0 t0 t1 xt1 e-st1 t1
  t2 xt2 e-st2 t2 ...
  
• tn xtn e-stn tn) DFW ? PV(0)

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• Calculating of sensitivities
• Note it is essentially the same formula we found
  for yield sensitivity in Ch. 3. ( (1/P) dP/d?
  - DM ).
  

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• Discrete-Time Compounding
• For the CFS (x0, x1, x2, ..., xn) with
  compounding m times per year, and spot rate sk
  for each period k, P(0) is the present
  value/price of the CFS. When we add ? to each
  spot rate in the equation for P(0) we get P(?).

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• Discrete-Time Compounding (Contd)
• The derivative of P(?) evaluated at 0, P?(0), is
  the rate of change of the price with respect to ?
  when ? is currently 0
  

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• Quasi-modified duration
• Quasi-modified duration satisfies
• P?(0)/P(0) DQ.

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Immunization Revisited

• The earlier approach in Chapter 3, when there is
  only one interest rate, generalizes in a direct
  manner to protect against a spot rate parallel
  shift.

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• Example illustrating immunization approach
• We have a 1 M obligation payable in 5 years
  (Spot rate curve from Table 4.4, s5
  9.7542855).
  
• We wish to choose an immunized bond portfolio.
• We consider 2 bonds, each with face value 100
• Bond B1 12-year, 6 bond with price 65.95
• Bond B2 5-year, 10 bond with price 101.66.

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• Approach (Matching equations, Goal Programming)
• We want to find x1 and x2, the number of units of
  bonds 1 and 2 respectively, to purchase to
  immunize our portfolio.
  

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• Steps
• Compute P1 65.95, P2 101.66, PVs of bonds
  1 and 2.
  
• Compute D1, D2, the quasi-modified durations of
  bonds 1 and 2.
  
• Compute PV, the present value of the obligation,
  
  
• 1 M/(1s5)5 627,903.01.
• Compute D, the quasi-modifed duration of the
  obligation, 5/(1s5) 4.56.
  
• Solve the equations and get
• x1 2,208.17, x2 4,744.03

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• Table 4.5. Immunization Results. The overall
  portfolio is immunized against parallel shifts in
  the spot rate curve.
  

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• Steps
• dt 1/(1st)t is the discount factor for year t
• Multiply each cash flow by dt to get contribution
  to PV
  
• Total PV contributions to get PV for each bond
  (65.95, 101.66).
  
• -PV?1 gives the contributions to PV?1 multiply
  each cash flow by t and by (1st)-(t1) to get
  these contributions.
  
• Total the -PV?1 contributions to get -PV?1
  466.00
  
• Divide 466 by 65.95 to get the duration for bond
  1.
  
• Similarly for bond 2.

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• Summary
• CFS (x0, x1, x2, ..., xn)
• n
• P(?) ? xk 1 (sk ?)/m-k
• k0
• is the PV of the CFS with a shift of ? and
  compounding m times per year.
  

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• Summary (Contd)
• Note PV P(0) is the PV with no shift.
• n
• P?(0) - ? (k/m) xk 1 sk/m-(k1)
• k0
• is the rate of change of P(?) with respect to ?
  when ? 0.
  
• DQ -P(0)/P(0) is the quasi-modified duration