Convexity  Lecture

# Convexity  Lecture - Debt Instruments and Markets...

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Debt Instruments and Markets Professor Carpenter Convexity 1 Convexity Concepts and Buzzwords Dollar Convexity Convexity Curvature Taylor series Barbell, Bullet Readings Veronesi, Chapter 4 Tuckman, Chapters 5 and 6

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Debt Instruments and Markets Professor Carpenter Convexity 2 Convexity Convexity is a measure of the curvature of the value of a security or portfolio as a function of interest rates. Duration is related to the slope, i.e., the first derivative. Convexity is related to the curvature, i.e. the second derivative of the price function. Using convexity together with duration gives a better approximation of the change in value given a change in interest rates than using duration alone. Price Interest Rate (in decimal) Example: Security with Positive Convexity Price‐Rate Func:on Linear approximation of price function Approximation error
Debt Instruments and Markets Professor Carpenter Convexity 3 Correc:ng the Dura:on Error The price‐rate function is nonlinear. Duration and dollar duration use a linear approximation to the price rate function to measure the change in price given a change in rates. The error in the approximation can be substantially reduced by making a convexity correction. Taylor Series The Taylor 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: f ( x ) f ( x 0 ) + f '( x 0 ) × ( x x 0 ) + 1 2 f ''( x 0 ) × ( x x 0 ) 2 Or, f ( x ) f ( x 0 ) f '( x 0 ) × ( x x 0 ) + 1 2 f ''( x 0 ) × ( x x 0 ) 2

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Debt Instruments and Markets Professor Carpenter Convexity 4 Dollar Convexity Think of bond prices, or bond portfolio values, as functions of interest rates. The Taylor Theorem says that if we know the first and second derivatives of the price function (at current rates), then we can approximate the price impact of a given change in rates. The first derivative is minus dollar duration. Call the second derivative dollar convexity . Then change in price ≈ ‐\$duration x change in rates + 0.5 x \$convexity x change in rates squared f ( x ) f ( x 0 ) f '( x 0 ) × ( x x 0 ) + 0.5 × f ''( x 0 ) × ( x x 0 ) 2 Dollar Convexity of a PorBolio If we assume all rates change by the same amount, then the dollar convexity of a portfolio is the sum of the dollar convexities of its securities .
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• Spring '15
• Jennifer
• Finance, Bond convexity

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