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Lecture_16

Lecture_16 - InterestRate Management The Price-Rate...

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Interest Rate  Management Professor Ruslan Goyenko
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Fall 2010 Investment  2 The Price-Rate Function for a Zero 100 bp At a rate of 5%, the price is 0.2273 If rates fall to 4%, the price is 0.3048 The actual change is 0.077 Using a linear approximation, the change is about 0.0665
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Fall 2010 Investment  3 Computing dollar duration for a zero. .. Recall By this definition, the dollar duration of the zero is directly related to the slope of the price-rate function. Example: The dollar duration of $1 par of a 30-year zero at an interest rate of 5% is 6.65, as illustrated in the last slide: -0.0665/(-0.01)=0.0665/0.01=6.65. We can use calculus to compute the slope of the price-rate function and get an explicit formula for the dollar duration of any zero.
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Fall 2010 Investment  4 Duration Duration approximates the percent change in price for a 100 basis point change in rates:
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 Price Approximation Using Duration Price Yield y 3 p * Actual Price Tangent Line at y* (estimated price) y * y 1 y 2 y 4 Error in Estimating Price Based only on Duration Error in Estimating Price Based only on Duration
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Fall 2010 Investment  6 Taylor Series A theorem from calculus says that the value of a function can be approximated near a given point using its "Taylor series" around that point . Using only the first two derivatives, the Taylor series approximation is: 2 0 0 0 0 0 ) ( ) ( ' ' 2 1 ) ( ) ( ' ) ( ) ( : x x x f x x x f x f x f or - × + - × -
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Fall 2010 Investment  7 The Convexity Correction Change in price = ΔP = - $D x Δr + 0.5 x $C x (Δr)2 = - dollar duration x change in rates + (1/2) x dollar convexity x change in rates squared 2 0 0 0 0 0 ) ( ) ( ' ' 2 1 ) ( ) ( ' ) ( ) ( x x x f x x x f x f x f - × + - × -
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Lecture_16 - InterestRate Management The Price-Rate...

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