Bond pricing theorems

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Bond pricing theorems
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Bond convexity

• The mathematical relationship between bond yields
  and prices

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Duration

• A measure of the average maturity of the stream
  of payments generated by a financial asset
• D (1)CF1/(1 ytm)   (2)CF2/(1 ytm)2 
  ....... (t)CFt/(1 ytm)t /(Price)
• Very often used
• D (modified duration) D/(1ytm)

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Exemplification A 6 coupon bond
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Exemplification A 6 coupon bond
Observation Bond prices and yields move
inversely.
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Exemplification A 6 coupon bond
2.6 decrease in price
2.5 increase in price
Observation Dollar changes in bond prices are
not symmetrical for a given basis point
increase/decrease in YTM, other things constant
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Exemplification A 6 coupon bond
Observation The longer the maturity, the longer
the duration, other things held constant.
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Exemplification A 6 coupon bond
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Exemplification A 6 coupon bond
3.4 decrease in price
0.94 decrease in price
3.2 increase in price
0.95increase in price
Observation Longer maturity bonds are more
sensitive to yield changes than shorter maturity
bonds, other things held constant
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Exemplification A 6 coupon bond
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Exemplification A 6 coupon bond
Observation As maturity approaches, bond prices
converge towards their face value at an
increasing rate, other things held constant.
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A 6 coupon bond
A 5 coupon bond
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A 6 coupon bond
A 5 coupon bond
Observation The lower the coupon rate the longer
the duration
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A 6 coupon bond
3.4 decrease in price
3.2 increase in price
A 5 coupon bond
3.43 decrease in price
3.6 increase in price
Observation Lower coupon bonds are more sensitive
to yield changes than higher coupon bonds
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Bond Pricing Theorems A Summary

• I. Bond prices and yields move inversely.
• II. As maturity approaches, bond prices converge
  towards their face value at an increasing rate,
  other things held constant.
• III. Dollar changes in bond prices are not
  symmetrical for a given basis point
  increase/decrease in YTM, other things constant.
• IV. Lower coupon bonds are more sensitive to
  yield changes than higher coupon bonds, other
  things held constant.
• V. Longer maturity bonds are more sensitive to
  yield changes than shorter maturity bonds, other
  things held constant.

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Duration Theorems A Summary

• I. The duration of a zero coupon bond always
  equals its time to maturity.
• II. The lower the coupon rate the longer the
  duration, other things held constant.
• III. The longer the maturity, the longer the
  duration, other things held constant.
• IV. The lower the yield to maturity, the longer
  the duration, other things held constant

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Using duration to approximate bond price changes

• The following formula approximates the change in
  bond prices for small changes in yields
• (P1 - P0)/P0 - D (ytm1- ytm0)  
• A better approximation is given by the following
  formula (P1 - P0)/P0 - D(ytm1- ytm0)
  (0.5)(Convexity)(ytm1- ytm0)2
• Convexity
• The rate of change of  the rate of change of the
  bond price.