Chapter 13 Fixed Income Portfolio Management Duration, Convexity

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Chapter 13Fixed Income Portfolio Management
Duration, Convexity

• Business 3059

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Important Terms

• Barbell strategy
• Bond swap
• Bullet strategy
• Confidence index
• convexity

• Flight to quality
• Laddered strategy
• Macaulay duration
• Modified duration
• Yield curve inversion

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Definitions Confidence Index

• Confidence Index is the ratio of the yield on AAA
  bonds to the yield on BBB bonds.
• It has an upper boundary of 1.0 because the yield
  on safe bonds should never exceed the yield on
  risky bonds.
• It is a measure of yield spreadas spreads widen,
  this indicates smart money is becoming
  increasingly pessimistic about the future. The
  confidence index will fall.
• Some equity analysts use this index to forecast
  trends in the equity markets based on the
  assumption that it takes longer for equity
  markets to respond to new expectations. They
  base this upon the assumption that equity markets
  have larger numbers of novice/inexperienced
  investorsbond markets are dominated by
  institutional money and portfolio managers.

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Yield Curves
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Definitions - Duration

• Is the first derivative of the bond-pricing
  equation with respect to yield.

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Types of Duration

• Macaulay duration a measure of time flow of
  cash from a bond.
• Modified duration a slight modification of
  Macaulays to account for semi-annual coupon
  payments
• Effective duration a direct measure of interest
  rate sensitivity of a bond price
• Empirical duration measures directly the
  percentage price change of a bond for an actual
  change in interest rates.

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Definitions Modified Duration

• Is Macaulays duration adjusted for semi-annual
  coupon payments

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Duration

• An alternative measure of bond price sensitivity
  is the bonds duration.
• Duration measures the life of the bond on a
  present value basis.
• Duration can also be thought of as the average
  time to receipt of the bonds cashflows.
• The longer the bonds duration, the greater is
  its sensitivity to interest rate changes.

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Duration and Coupon Rates

• A bonds duration is affected by the size of the
  coupon rate offered by the bond.
• The duration of a zero coupon bond is equal to
  the bonds term to maturity. Therefore, the
  longest durations are found in stripped bonds or
  zero coupon bonds. These are bonds with the
  greatest interest rate elasticity.
• The higher the coupon rate, the shorter the
  bonds duration. Hence the greater the coupon
  rate, the shorter the duration, and the lower the
  interest rate elasticity of the bond price.

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Duration

• The numerator of the duration formula represents
  the present value of future payments, weighted by
  the time interval until the payments occur. The
  longer the intervals until payments are made, the
  larger will be the numerator, and the larger will
  be the duration. The denominator represents the
  discounted future cash flows resulting from the
  bond, which is the bonds present value.

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Duration Example

• As an example, the duration of a bond with 1,000
  par value and a 7 percent coupon rate, three
  years remaining to maturity, and a 9 percent
  yield to maturity is

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Duration Example

• As an example, the duration of a bond with 1,000
  par value and a 7 percent coupon rate, three
  years remaining to maturity, and a 9 percent
  yield to maturity is

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Duration Example ...

• As an example, the duration of a zero-coupon bond
  with 1,000 par value and three years remaining
  to maturity, and a 9 percent yield to maturity is

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Duration

• is a handy tool because it can encapsule interest
  rate exposure in a single number.
• rather than focus on the formula...think of the
  duration calculation as a process...
• semi-annual duration calculations simply call for
  halving the annual coupon payments and
  discounting every 6 months.

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Duration Rules-of-Thumb

• duration of zero-coupon bond (strip bond) the
  term left until maturity.
• duration of a consol bond (ie. a perpetual bond)
  1 (1/R)
• where R required yield to maturity
• duration of an FRN (floating rate note) 1/2
  year

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Other Duration Rules-of-Thumb

• Duration and Maturity
• duration increases with maturity of a
  fixed-income asset, but at a decreasing rate.
• Duration and Yield
• duration decreases as yield increases.
• Duration and Coupon Interest
• the higher the coupon or promised interest
  payment on the security, the lower its duration.

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Economic Meaning of Duration

• duration is a direct measure of the interest
  rate sensitivity or elasticity of an asset or
  liability. (ie. what impact will a change in YTM
  have on the price of the particular fixed-income
  security?)
• interest rate sensitivity is equal to
• dP - D dR/(1R)
• P
• Where P Price of bond
• C Coupon (annual)
• R YTM
• N Number of periods
• F Face value of bond

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Problems with Duration

• It assumes a straight line relationship between
  the changes in bond price given the change in
  yield to maturityhowever, the actual
  relationship is curvilineartherefore, the
  greater the change in YTM, the greater the error
  in predicted bond price using durationas can be
  seen

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The Problem with Duration
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Uses of Duration

• Immunization strategies
• If you equate the duration of an asset (bond)
  with the duration of a liability, you will
  (subject to some limitations) immunize your
  investment portfolio from interest rate risk.
• Used in predicting bond prices given a change in
  interest rates (yields)

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Predicting a Bond Price using Duration

• Price movements of bonds will vary proportionally
  with modified duration for small changes in
  yields.
• An estimate of the percentage change in bond
  price equals the change in yield times the
  modified duration.

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Predicting a Bond Price Using Duration
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Actual Bond Price
As you can see from the previous slide, using
modified duration and predicting an increase of
0.5 in yield, we forecast the bond price to be
936.64 where as it will actually be 937.27.
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Definitions - Convexity

• Bond convexity is the difference between the
  actual price change of a bond and that predicted
  by the duration statistic.
• It is the second derivative of the bond-pricing
  equation with respect to yield.
• The importance of convexity increases as the
  magnitude of rate changes increases.
• Other rules
• The higher the yield to maturity, the lower the
  convexity, everything else being equal.
• The lower the coupon, the greater the convexity,
  everything else being equal.
• The greatest convexity would be observed for
  stripped bonds at low yields.

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Second Derivative of the Bond Pricing Equation
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Convexity
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Computation of Convexity

• 3-year bond, 12 coupon, 9 YTM

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Bond Portfolio Strategy

• Selection of the most appropriate strategy
  involves picking one that is consistent with the
  objectives and policy guidelines of the client or
  institution.
• There are two basic types of strategies
• Active
• Laddered
• Barbell
• Bullet
• Swaps (substitution, inter-market or yield
  spread, bond-rating, rate anticipation)
• Passive
• buy and hold, and
• indexing

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Active Bond Portfolio Strategies

• Other authors categorize bond strategies as
  follows (see Frank K. Reilly and Keith C. Brown,
  Investment Analysis and Portfolio Management.)
• Passive Portfolio Strategies
• Buy and hold
• Indexing
• Active Management Strategies
• Interest rate anticipation
• Valuation analysis
• Credit analysis
• Yield spread analysis
• Bond swaps
• Matched-funding strategies
• Dedicated portfolio exact cash match
• Dedicated portfolio optimal cash match and
  reinvestment
• Classical (pure) immunization
• Horizon matching
• Contingent procedures (structured active
  management)
• Contingent immunization
• Other contingent procedures

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Buy-and-Hold Strategy

• Involves
• Finding issues with desired quality, coupon
  levels, term to maturity, and important indenture
  provisions, such as call features
• Looking for vehicles whose maturities (or
  duration) approximate their stipulated investment
  horizon to reduce price and reinvestment rate
  risk.
• A modified buy and hold strategy involves
• Investing with the intention of holding until
  maturity, however, they still actively look for
  opportunities to trade into more desirable
  positions.

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Indexing Strategy

• The manager builds a portfolio that will match
  the performance of a selected bond-market index
  such as the Lehman Brothers Index, Scotia McLeod
  bond index, etc.
• In such a case, the bond manager is NOT judged on
  the basis of risk and return compared to an
  index, BUT on how closely the portfolio tracks
  the index.
• Tracking error equals the difference between the
  rate of return for the portfolio and the rate of
  return for the bond-market index.
• When a portfolio has a return of 8.2 percent and
  the index an 8.3 percent return, the tracking
  error would be 10 basis points.

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Active Portfolio Strategies

• There are three sources of return from holding a
  fixed-income portfolio
• coupon income
• any capital gain (or loss),
• reinvestment income
• in general, the following factors affect a
  portfolios return
• changes in the level of interest rates
• changes in the shape of the yield curve
• changes in the yield spreads among bond sectors
• changes in the yield spread (risk premium) for a
  particular bond (perhaps the default risk
  associated with a particular bond increases or
  decreases)

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Manager Expectations vs. Market Consensus

• A money manager who pursues an active strategy
  will position a portfolio to capitalize on
  expectations about future interest rates.
• But the potential outcome (as measured by total
  return) must be assessed before an active
  strategy is implemented.
• The primary reason for assessing the potential
  outcome is that the market (collectively) has
  certain expectations for future interest rates,
  and these expectations are embodied in the market
  price of bonds.

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Yield Curve Strategies

• Yield curve strategies involve positioning a
  portfolio to capitalize on expected changes in
  the shape of the yield curve.
• A shift in the yield curve refers to the relative
  change in the yield for each Treasury maturity.
• A parallel shift in the yield curve refers to a
  shift in which the change in the yield on all
  maturities is the same.
• A nonparallel shift in the yield curve means that
  the yield for each maturity does not change by
  the same number of basis points.
• Historically, two types of nonparallel yield
  curve shifts have been observed a twist in the
  slop of the yield curve and a change in the
  humpedness of the yield curve.

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Upward Parallel Shift
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Downward Parallel Shift
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Nonparallel Shifts

• A nonparallel shift in the yield curve means that
  the yield for each maturity does not change by
  the same number of basis points.
• Historically, two types of nonparallel yield
  curve shifts have been observed a twist in the
  slope of the yield curve and a change in the
  humpedness of the yield curve.

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Yield Curve Shifts

• A flattening of the yield curve means that the
  yield spread between the yield on a long-term and
  a short-term Treasury has decreased.
• A steepening of the yield curve means that the
  yield spread between a long-term and a short-term
  Treasury has increased.

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Flattening Twist
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Steepening Twist
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Non-parallel Yield Curve Shifts

• A change in the humpedness of the yield curve is
  referred to as a butterfly shift.
• This is also an example of a non-parallel shift.

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Positive Butterfly
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Negative Butterfly
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Yield Curve Shifts 1979-1990

• Frank Jones found that the three types of yield
  curve shifts are NOT independent. (parallel,
  twists and butterfly)
• The two most common shifts
• a downward shift combined with a steepening of
  the yield curve, and
• an upward shift combined with a flattening of the
  yield curve.

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Upward shift/flattening/positive butterfly
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Downward shift/steepening/negative butterfly
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Yield Curve Shifts and returns

• Jones found that parallel shifts and twists in
  the yield curve are responsible for 91.6 of
  Treasury returns, while 3.4 of the returns is
  attributable to butterfly shifts, and the
  balance, 5, to unexplained factor shifts.
• This indicates that yield curve strategies
  require a forecast of the direction of the shift
  and a forecast of the type of twist.

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Yield Curve Strategies

• In portfolio strategies that seek to capitalize
  on expectations based on short-term movements in
  yields, the dominant source of return is the
  change in the price of the securities of the
  portfolio.
• This means that the maturity of the securities in
  the portfolio will have an impact on the
  portfolios return
• a total return over a 1-year investment horizon
  for a portfolio consisting of securities all
  maturing in 1 year will not be sensitive to
  changes in how the yield curve shifts 1 year from
  now.
• In contrast, the total return over a 1-year
  investment horizon for a portfolio consisting of
  securities all maturing in 30 years will be
  sensitive to how the yield curve shifts because,
  1 year from now, the value of the portfolio will
  depend on the yield offered on 20-year securities.

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Yield Curve Strategies(bullet, barbell, and
ladder)
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Yield Curve Strategies

• Each of these strategies (bullet, barbell,
  ladder) will result in different performance when
  the yield curve shifts.
• The actual performance will depend on both the
  type of shift and the magnitude of the
  shift.thus, no general statements can be made
  about the optimal yield curve strategy.

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Duration and Yield Curve Shifts

• Duration is a measure of the sensitivity of the
  price of a bond or the value of a bond portfolio
  to changes in market yields.
• A portfolio with a duration of 4 means that if
  market yields increase by 100 basis points, the
  portfolio will change by approximately 4.
• If a portfolio of bonds is made up of 5-year,
  10-year and 20-year bonds, and the portfolios
  duration is 4the portfolios value will change
  by 4 if the yields on all bonds change by 100
  basis points. That is, it is assumed that there
  is a parallel yield curve shift.

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Analysis of Expected Yield Curve Strategies

• The proper way to analyze any portfolio strategy
  is to look at its potential total return.
• Example
• consider the following two yield curve
  strategies
• Bullet portfolio 100 bond C
• Barbell portfolio 50.2 bond A and 49.8 bond B

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Three Hypothetical Treasury Securities

• The bullet portfolio consists of only bond C, the
  10-year bond. All principal is received when
  bond C matures in 10 years.
• The barbell portfolio consists of almost equal
  amount of the short-term and long-term
  securities. The principal will be received at two
  ends of the maturity spectrum. (5 year and 20
  year dates).

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Barbell and Bullet Duration

• The dollar duration of the bullet portfolio per
  100-basis-point change in yield is 6.43409.
• Dollar duration is a measure of the dollar price
  sensitivity of a bond or a portfolio.
• The dollar duration for the barbell portfolio is
  just the weighted average of the dollar duration
  of the two bonds
• .502(4.005) 0.498(8.882) 6.434

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Dollar Convexity

• Duration is just a first approximation
• Convexity is the second derivative and simply
  gives a more accurate indication of sensitivity
  of bond price to a change in interest rates.
• Both duration and convexity assume a parallel
  shift in the yield curve!!
• Two general rules for convexity
• The higher the yield to maturity, the lower the
  convexity, everything else being equal
• The lower the coupon, the greater the convexity,
  everything else being equal.
• Managers should seek high convexity while meeting
  other constraints in their bond portfoliosby
  doing so, they minimize the adverse effects of
  interest rate volatility for a given portfolio
  duration.

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Swaps

• Swaps are used to do one of four things
• Increase current income
• Increase yield to maturity
• Improve the potential for price appreciation with
  a decline in interest rates
• Establish losses to offset capital gains or
  taxable income.

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Substitution Swap

• Purpose- to increase current yield
• Assumes market inefficiencythat results in
  equally risky bonds (default risk, same duration)
  to have different pricesthis is an arbitrage
  action
• In an efficient market, we expect few of these
  situations to arise.

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Intermarket or Yield Spread Swap

• Purpose- to take advantage of expected changes in
  the default risk premiums that may occur as a
  result of changes in market optimism or
  pessimism.
• A confidence index measures these changes.

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Bond-Rating Swap

• Purpose- to take advantage of expected changes in
  the default risk premiums that may occur as a
  result of changes in bond ratings.
• Fundamental analysis of the prospectus of the
  individual issuer and of their financial health
  is used to predict changes in bond ratings (but
  this must be done in conjunction with analysis of
  changes in the overall market returns (ie. yield
  curve changes).)

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Rate Anticipation Swap

• The purpose is to take advantage of expected
  changes in interest rates by positioning the bond
  portfolio with an appropriate duration, AND an
  appropriate default risk category.