APPENDIX TO CHAPTER 4
DV01, Duration, and Convexity

A4.1 DV01, DURATION, AND CONVEXITY OF PORTFOLIOS

Let upper P Superscript i denote the price of asset i and upper P the price of a portfolio of those assets. By definition,

Let y be the single factor generating changes in rates. Then, taking the derivative of both sides of (A4.1) with respect to y,

Then, divide both sides of (A4.2) by minus10,000 and apply Equation (4.5) of the text to see that,

(A4.3)upper D upper V Baseline 01 equals sigma-summation upper D upper V Baseline 0 1 Superscript i

Or, in words, the DV01 of a portfolio equals the sum of the individual asset DV01s.

To derive the duration of a portfolio, start from Equation (A4.2), dividing both sides by negative upper P,

(A4.4)minus StartFraction 1 Over upper P EndFraction StartFraction d upper P Over d y EndFraction equals sigma-summation minus StartFraction 1 Over upper P EndFraction StartFraction d upper P Superscript i Baseline Over d y EndFraction
(A4.5)minus StartFraction 1 Over upper P EndFraction StartFraction d upper P Over d y EndFraction equals sigma-summation minus StartFraction upper P Superscript i Baseline Over upper P EndFraction StartFraction 1 Over upper P Superscript i Baseline EndFraction StartFraction d upper P Superscript i Baseline Over d y EndFraction

Equation (A4.5) multiplies each term in the summation by one, in the form of upper P Superscript i Baseline slash upper P Superscript i. Equation (A4.6) follows from the definition of duration in the text, Equation (4.11). In words, Equation (A4.6) says that the duration of a portfolio equals the weighted average of the durations of the individual assets, where the weights are the fraction of value of each asset in the portfolio.

The proof for the convexity of a portfolio can be derived along the same lines as the duration of a portfolio. The result, given here without proof, is,

(A4.7)upper C equals sigma-summation StartFraction upper P Superscript i Baseline Over upper P EndFraction upper C Superscript i

A4.2 ESTIMATING PRICE CHANGE WITH DURATION AND CONVEXITY

Let upper P left-parenthesis y right-parenthesis be the price of an asset as a function of the single factor that describes changes in rates. Then, a second‐order Taylor approximation of the price rate function is given by,

Subtracting upper P left-parenthesis y right-parenthesis from both sides of (A4.8) and denoting the change in price, upper P left-parenthesis y plus normal upper Delta y right-parenthesis minus upper P, by normal upper Delta upper P,

(A4.9)StartLayout 1st Row 1st Column normal upper Delta upper P 2nd Column almost-equals StartFraction d upper P Over d y EndFraction normal upper Delta y plus one half StartFraction d squared upper P Over d y squared EndFraction normal upper Delta y squared EndLayout
(A4.10)StartLayout 1st Row 1st Column StartFraction normal upper Delta upper P Over upper P EndFraction 2nd Column almost-equals StartFraction 1 Over upper P EndFraction StartFraction d upper P Over d y EndFraction normal upper Delta y plus one half StartFraction 1 Over upper P EndFraction StartFraction d squared upper P Over d y squared EndFraction normal upper Delta y squared EndLayout

where (A4.11) – which is Equation (4.16) – follows from the definitions of duration and convexity, that is, from Equation (4.10) and a discrete version of Equation (4.14).

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