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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: $C_{M o d} = \frac{P ^{''} ( i )}{P ( i )} = \frac{\sum _{t} t ( t + 1 ) C _{t} v ^{t + 2}}{P}$
Macaulay convexity: $C_{M a c} = \frac{\sum _{t} t ^{2} C _{t} v ^{t}}{P}$
Second-order estimate: $P (i + Δ i) \approx P (1 - D_{M o d} Δ i + \frac{1}{2} C_{M o d} (Δ i)^{2})$

Where it comes from

Take the Taylor expansion of $P (i)$ to second order: $P (i + Δ) \approx P + P^{'} Δ + \frac{1}{2} P^{''} Δ^{2}$ .
Dividing by $P$ turns the coefficients into $- D_{M o d}$ and $C_{M o d}$ .
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

Using both duration and convexity, estimate the price of the 5-year 6% par bond when the yield rises to 6.5%.

$D_{M o d} = 4.212357$ and $C_{M o d} = 22.702755$ at $i = 0.06$ .
$Δ i = 0.005$ , so the correction is $\frac{1}{2} (22.702755) (0.005)^{2} = 0.000284$ .
Estimate $= 1, 000 (1 - 0.021062 + 0.000284) = 979.22$ .
The exact price is 979.15 — the second-order estimate is out by 0.07 rather than 0.21.

Answer: 979.22

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.

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Convexity

The formulas

Where it comes from

Worked example

Memory hooks

Traps

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