Exam FMInterest rate riskFree to read

Convexity

The second-order term. Duration alone always underprices a bond after a rate move; convexity is the correction, and it is always favourable.

The formulas

Modified convexity

$$C_{Mod} = \dfrac{P''(i)}{P(i)} = \dfrac{\sum_t t(t+1)C_t v^{t+2}}{P}$$

Macaulay convexity

$$C_{Mac} = \dfrac{\sum_t t^{2} C_t v^{t}}{P}$$

Second-order estimate

$$P(i+\Delta i) \approx P\left(1 - D_{Mod}\Delta i + \tfrac{1}{2}C_{Mod}(\Delta i)^{2}\right)$$

Where it comes from

1. Take the Taylor expansion of $P(i)$ to second order: $P(i+\Delta) \approx P + P'\Delta + \frac12 P''\Delta^2$.
2. Dividing by $P$ turns the coefficients into $-D_{Mod}$ and $C_{Mod}$.
3. Because $P'' > 0$ for any positive cash flows, the correction is positive whichever way rates move - a bondholder is always better off than duration alone predicts.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• Convexity is always your friend: the correction is +½C(Δi)² whichever direction rates move.
• Modified convexity uses v^(t+2) and t(t+1); Macaulay convexity uses v^t and t². Check which one the question wants.

Traps

• Mixing Macaulay convexity into a modified-duration estimate.
• Forgetting the factor of one half.

Related

• Modified duration and price sensitivity
• Immunisation

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