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Financial Instruments, Markets and Conventions / Fixed Income: Bonds, Yields, and Interest Rate Risk

Derivation — Duration and Convexity from the Taylor Expansion of Bond Price

The duration and convexity lesson stated the key formulas without full derivation. This note carries out that algebra explicitly, starting from the bond pricing equation and expanding P(y+Δy)P(y + \Delta y) as a Taylor series in Δy\Delta y. The two leading terms produce precisely the duration and convexity approximations — no more, no less.

Two goals: (1) see where the 1/(1+y)1/(1+y) factor in the denominator of modified duration comes from; (2) understand why both terms in the Taylor expansion are necessary for large yield moves, and why higher-order terms can often be safely neglected.

In this lesson

  1. Setup: the bond price as a function of yield
  2. First derivative: dollar duration
  3. Second derivative: dollar convexity
  4. The Taylor expansion
  5. Verification against Fabozzi exhibits
  6. Why both terms are necessary

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