Duration, Convexity and Interest Rate Risk

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Duration, Convexity and Interest Rate Risk
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Summary

• Variations of Duration
• Macaulay Duration
• Modified Duration
• Effective Duration (option-adjusted)
• Interpretations of Duration
• Weighted average time where CFs from a FIS are
  received (Macaulay D)
• First derivative of P-Y relationship of FIS
  (Modified D)
• Measure of sensitivity of bond price to small
  changes in interest rates (Effective D)

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Variations on Duration (I)

• Macaulay Duration weighted average
  time-to-maturity of the cash flows of a bond.
  (the weight of each cash flow is based on its
  discounted present value)

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Example

• Calculate the Macaulay Duration for 9, 5-year
  bond selling to yield 9. (compounded
  semi-annually)

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Macaulay Duration
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Variations on Duration (II)

• Modified Duration an adjusted measure of
  Macaulay duration that produces a more accurate
  estimate of bond price sensitivity.
• Modified Duration (MD)
• m is the of compounding period per year.

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Variations on Modified Duration (A)

• Dollar Duration measure the dollar change in a
  bonds price for a given change in its yield.
• Dollar duration P0MD

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Variations on Modified Duration (B)

• The price value of a basis point (PVBP)
• Dollar duration of a bond for a 1 basis point
  change in yield.
• PVBP0.0001 P0MD0.0001 Dollar D

?PPVBP (-?r), where ?r measured in b.p.
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Example

• Bonds with a par value of 100,000 are priced at
  103.25. they have a modified duration of 4.84
  and a yield of 5.15. If yield increased by 1
  basis print, the price would decrease to 103.20.
• Compute the bonds PVBP per 100 of par.
• Compute the bonds PVBP for the full 100,000
  position.
• Compute the dollar duration of the portfolio if
  yield change by 75 basis points.

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Answer

• PVBP0.01x(103.25)x4.840.05, or
  PVBP103.2-103.250.05
• PVBP0.05x100k50
• Dollar DP x MD x ?r 103.25x4.84x0.753748

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Variations of Duration (III)

• Effective duration and convexity (option
  adjusted)
• A duration/convexity measure that includes the
  effect of embedded option on a bonds price
  behavior.

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Effective Duration and Convexity

• Effective Duration
• Effective Convexity
• Bond price to interest rate change

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Example

• A 10-year, 8 coupon bond is selling at 93.5,
  with a YTM of 9.0. If the yield on bond is
  increased by 50 b.p., its price falls to 90.452,
  and if the yield on bond falls by 50 b.p., the
  bonds price will increase to 96.6764. What is
  the effective duration and convexity of this bond
  at the current yield of 9.0? If yield should
  increase 100 b.p., what is the estimated
  percentage change in the price of bond?

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Answer
If ?r-1, ?P6.79
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Dedicated Portfolios
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Example of Duration matching
We need 1850 8 years from now. (1) Buy 8
8-year bond.
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Example of Duration matching (continues)
Buy 8 14-year bond (invest too long net price
risk)
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Example of Duration matching (continues)
Buy 8 12-year bond (price risk reinvestment
risk D8)
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Discussion
What if reinvestment rate first goes down (to
7), causing the interest payments to be
reinvested at lower rate, then goes up (to 9)
just before year 8, causing the bond to be sold
at a lower price? (i.e., what if we lost on both
reinvestment and price risks?)