Bond Pricing Duration and Convexity

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Lecture 7

• Bond Pricing Duration and Convexity
• Managing Bond Portfolios

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Review of Bond Pricing

• The bond price is
• C and r should be consistent with the frequency
  of coupon payments.
• Is coupon being paid annually or semiannually?

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Yield-to-Maturity (YTM)

• It is the interest rate (IRR) at which the PV of
  the future payments is equal to the current
  price.
• If a bond maturing in n years from now makes two
  payments of C/2 each year, then the YTM is the
  value of r such that

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Current Yield

• It is the bonds annual coupon payment divided by
  the current bond price.
• It is similar to the concept of dividend yield
  for shares.
• The current yield is higher than the coupon rate
  if the bond is selling at a discount. The current
  yield is lower than the coupon rate if the bond
  is selling at a premium.

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Yield-to-Call

• Some bonds are callable, which given the right to
  the firm to retire the bond prior to the maturity
  date.
• Yield-to-call for callable bonds is just like the
  YTM except that the time until call replace time
  maturity, and the call price replaces the par
  value.

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Example 1

• What is the price of a semi-annual coupon with 8
  coupon rate and 30-year maturity? The
  yield-to-maturity is 10.
• Answer
• The YTM is greater than the coupon rate, so the
  bond is sold at a discount.

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Example 1 (continued)

• If the bond is callable in 10 years at a call
  price of 1,100, what is the yield-to-call?
• Suppose the current price is 870.
• Solve for r.
• Using Excel IRR function. IRR 5.37.
• YTC 25.37 10.73

8
Price-Yield Curve

• Bond prices and yields are inversely related.
• An increase in a bonds yield results in a
  smaller price decline than the price gain
  associated with a decrease of equal magnitude in
  yield.
• Prices of long-term bonds tend to be more
  sensitive to interest rate changes than prices of
  short term bonds.

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Price-Yield Curve (contd)

• As maturity increases, price sensitivity to yield
  changes increases at a decreasing rate.
• Interest rate risk is inversely related to the
  bonds coupon rate.
• Bond prices are more sensitive to changes in
  yields when the bond is selling at a lower
  initial YTM.
• If a bond is sold at par, then its YTM must be
  equal to its coupon rate.

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Zero-Coupon Bonds

• Zero coupon bonds do not pay coupons.
• The price of a zero is
• Origin The U.S. Treasury STRIP program.
• A coupon bond is a combination of a series of
  zeros. Short cut to calculate bond price.
• Australian zero coupon rates can be found in the
  Datastream.

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Floating-Rate Bonds

• Floating-rate bonds (floaters) link interest
  payments with some measure of current market
  rates, such as LIBORs.
• The coupon payments of inverse floaters are
  negatively related to a reference rate.
• A floater may have a cap and/or a floor.

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Price of a Floater

• The price of a floater depends on (1) the spread
  over the reference rate, and (2) any restrictions
  on the resetting of the coupon rate.
• The price of a floater will trade close to its
  par value as long as (1) the spread above the
  reference rate that the market requires is
  unchanged and (2) neither the cap nor the floor
  is reached.

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Practical Example 1 PBL-PARS

• Publishing and Broadcasting Limited (PBL) issued
  PARS in September 1999. It is listed on the
  Australian Stock Exchange (code PBLHA).
• PBL issued 3 million notes with a face value of
  100 to raise 300mil.
• Interest payments are four times a year.
• No maturity date perpetual unless redeemed by
  the issuer.
• It is rated BBB by standard and Poors.

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PBL-PARS
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Convertible Bonds

• Convertible bonds give the holders an option to
  exchange each bond for a specified number of
  shares of common stock of the firm.
• The conversion ratio gives the number of shares
  for which each bond may be exchanged.

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Convertible Bonds An Example

• Suppose a bond that is issued at par value of
  1000 is convertible into 40 shares of a firms
  stock. The current stock price is 20, so it is
  not profitable to convert the bond into stocks.
  However, when the stock price rises to 30, then
  it is profitable to do so.
• The value of a convertible bond is the sum of the
  comparable straight bond and the option value of
  conversion.

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Conversion Value and Premium

• Given the current stock price at 20, the
  conversion value is 800. If the bond were sold
  at 960, then the conversion premium would be
  160.
• Conversion is voluntary, but most convertibles
  are also callable at the discretion of the firm.

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Practical Example 2 DIWGA Convertible Notes

• Issuer Djerriwarrh Investments Limited
• Quoted date12 July 2004
• Issue type Unsecured Convertible Notes
• Number on issue40,000,000
• Face value3.90 per note
• Maturity date30 September 2009 (unless converted
  earlier)
• Interest rate The notes will bear interest at a
  rate of 6.5 per cent per annum on the face value
  of the New Note accruing from the allotment date
  and payable semi-annually on each interest
  payment date
• Interest payment dates First payment on 30
  September 2004 (for the period from the Allotment
  date to 30 September 2004) and then on 31 March
  and 30 September each year until 30 September
  2009.

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Practical Example 2 DIWGA Convertible Notes
(continued)

• Conversion details The new Notes may be
  converted into Ordinary shares on a one for one
  basis on 31 March or 31 September of each year
  from the Allotment date to Maturity and at the
  occurrence of certain events
• Ranking The New Notes will be unsecured and rank
  equally with any unsecured convertible notes
  previously issued by the Company. Each Ordinary
  shared issued on conversion will rank pari passu
  and form one class with the Ordinary Shares then
  on issue and be entitled to all dividends
  declared after the date of conversion

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Call Policy

• Companies can force a conversion at a preset
  price.
• If you want to maximise the shareholders value,
  you must not call the bonds if they are worth
  less than the call price.
• Similarly, you must not allow the bonds to remain
  uncalled if their value is above the call price.

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Indexed Bonds

• The interest payments and the principals of
  indexed bonds are linked with a general price
  index (e.g. CPI) or the price of a particular
  commodity.
• The main purpose of using indexed bonds is to
  hedge against inflation risk.

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Index Bonds An Example

• A 10-yr bond is indexed on the CPI. The annual
  real coupon is assumed to be 4, and the
  inflation rate is roughly around 3.

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Inflation Rate and Interest Rate

• The modified Fisher equation is
• Or, approximately,
• where p is the anticipated inflation rate, and p
  is the inflation risk premium.
• Investors require inflation risk premium because
  of unanticipated inflation.

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Duration

• A measure of the effective maturity of a bond
• The weighted average of the times until each
  payment is received, with the weights
  proportional to the present value of the payment
• Duration is shorter than maturity for all bonds
  except zero coupon bonds
• Duration is equal to maturity for zero coupon
  bonds

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Duration Calculation
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Duration Calculation Example using Table 16.3
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Duration/Price Relationship

• Price change is proportional to duration and not
  to maturity. It is the first derivative of the
  price w.r.t. the interest rate.
• ?P/P -D ?(1y) / (1y)
• D modified duration
• D D / (1y)
• ?P/P - D ?y

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Rules for Duration

• Rule 1 The duration of a zero-coupon bond equals
  its time to maturity
• Rule 2 Holding maturity constant, a bonds
  duration is higher when the coupon rate is lower
• Rule 3 Holding the coupon rate constant, a
  bonds duration generally increases with its time
  to maturity
• Rule 4 Holding other factors constant, the
  duration of a coupon bond is higher when the
  bonds yield to maturity is lower

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Rules for Duration (contd)

• Rules 5 The duration of a level perpetuity is
  equal to
• Rule 6 The duration of a level annuity is equal
  to

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Rules for Duration (contd)

• Rule 7 The duration for a corporate bond is
  equal to
• Rule 8 For coupon bonds selling at par, rule 7
  simplifies to

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Duration and Convexity
Yield
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Correction for Convexity
Correction for Convexity (second derivative)
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Portfolio Immunisation

• Portfolio immunisation insulate the portfolio
  from interest rate risk, using duration matching.
• Immunization of interest rate risk
• Net worth immunization (.e.g. banks and Insurance
  companies)
• Duration of assets Duration of liabilities
• Note the duration of a portfolio is a weighted
  average of the duration of its components. But
  this is not true for the convexity.

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Example

• An insurance company issues a 5 year Guaranteed
  Income Contract (GIC) for 10,000.
• A GIC is a zero-coupon bond.
• To insure the interest risk, the company funds
  the obligation with a 6 year 8 annual coupon
  bond.
• Verify this with Rule 8.
• There is still some small residue difference due
  to convexity.
• Need to rebalance immunised portfolios.

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Cash Flow Matching and Dedication

• Why not to simply buy a 5 year zero-coupon bond
  that covers the obligation for the GIC?
• Cash flow matching on a multi-period basis is
  referred to as a dedication strategy.
• Cash flow matching is not more widely pursued
  probably because of the constraints that it
  imposes on bond selection. Sometimes, it is not
  possible to find a good match.

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Contingent Immunization

• A combination of active and passive management.
• The strategy involves active management with a
  floor rate of return.
• As long as the rate earned exceeds the floor, the
  portfolio is actively managed.
• Once the floor rate or trigger rate is reached,
  the portfolio is immunized.