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(L05)BondsSens(f)2

# (L05)BondsSens(f)2 - Lecture 5 Bond Price Sensitivity...

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Primbs/Investment Science 1 Lecture 5: Bond Price Sensitivity Reading: Luenberger Chapter 3, Sections 5 – 8 Chapter 4, Sections 8 - 10

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Primbs/Investment Science 2 Macaulay Duration Connections Bond Price Bond Price Sensitivity Sensitivity Modified Duration (yield) Duration of a Portfolio Immunization Fisher-Weil Duration Quasi-Modified Duration General Principles and Convexity Yield Sensitivity Spot Rate Curve Sensitivity Duration and Portfolios
Primbs/Investment Science 3 Question : I would like to invest my money in a bond, but I am worried that bond yields may change in the future. Should I invest in a short or long maturity bond to reduce the effects of changes in yield on the value of my bond? What about coupons? Should I choose a bond with large coupons or small coupons? Answer : We need a way to compute the sensitivity of bond prices to yield.

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Primbs/Investment Science 4 Duration Maturity time is related to the sensitivity of the bond price to its yield. When there are coupons, maturity time does not exactly correspond to sensitivity. We will define something called duration to try to “generalize” the idea that for a zero coupon bond the time to maturity captures its sensitivity to yield.
Primbs/Investment Science 5 Macaulay Duration Let’s compute the weighted average of the times to the cash flows, where we weight each cash flow by its present value (computed with the yield) divided by the total present value. t n 0 C C C C C F t 1 t 2 t n 0 t 1 t 2 PV(t 1 ) PV(t 2 ) PV(t 3 ) PV(t n ) Replace by present values computed using the yield. The average time is called Macaulay Duration

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Primbs/Investment Science 6 Macaulay Duration Defn : Macaulay Duration: total n n PV t t PV t t PV t t PV D ) ( ... ) ( ) ( 2 2 1 1 + + + = PV(t i ) is the present value of the cash flow at time t k computed using the yield as the interest rate!. is the present value of the entire cash flow. n t t t t w t w t w n + + + = ... 2 1 2 1 where total i t PV t PV w i ) ( = = = n i i total t PV PV 0 ) (
Primbs/Investment Science 7 Macaulay Duration Macaulay duration is a weighted average of times . It is quoted in years . In practice, when people refer to duration, they often mean Macaulay duration.

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Primbs/Investment Science 8 Macaulay Duration and Maturity Zero-Coupon bond: Macaulay Duration = Maturity Date Coupon bond: Macaulay Duration < Maturity Date You should think : A bond with Macaulay Duration D has the same yield sensitivity as a zero coupon bond with maturity D .
Primbs/Investment Science 9 Macaulay Duration Formula The Macaulay duration for a bond with: coupon rate/year = C yield = λ periods per year = m periods remaining = n λ λ λ λ λ λ + - + - + + - + = ] 1 ) 1 [( ) ( 1 1 n m m m c m m c n D t n 0 C/m C/m C/m C/m C/m F t 1 t 2

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Primbs/Investment Science 10 Macaulay Duration Connections Bond Price Bond Price Sensitivity Sensitivity Modified Duration (yield) Duration of a Portfolio Immunization Fisher-Weil Duration Quasi-Modified Duration General Principles and Convexity Yield Sensitivity Spot Rate Curve Sensitivity Duration and Portfolios
Primbs/Investment Science 11 Principal 1000 Coupon

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