Convexity

The duration estimate \(\Delta P/P \approx -D_{mod}\Delta i\) is just a tangent line — it captures the slope of the price–yield curve but not its bend. Because that curve curves upward (is convex), adding a second-order correction term makes the estimate track the true curve much more closely, especially for large yield moves.

By the end you'll be able to compute the two-term Taylor estimate \(\Delta P/P \approx -D_{mod}\Delta i + \tfrac12 C(\Delta i)^2\) and explain why the convexity term always corrects the duration-only estimate toward the true price.

Predict: for a large yield change, does the duration-only estimate over- or under-state the true price change? Drag \(\Delta i\) out toward either edge and check against the curve.

This is the price–yield curve of a 20-year, 6% annual-coupon bond priced to yield 6% today, plotted as \(\Delta P/P\) against a yield change \(\Delta i\). The true curve bends upward; the duration-only tangent line only matches its slope at \(\Delta i = 0\), so it drifts below the curve on both sides; the duration+convexity parabola adds the curvature term back and hugs the true curve far longer.

  • True \(\Delta P/P\): +0.00%
  • Duration-only estimate: +0.00%
  • Duration+convexity estimate: +0.00%

Error vs. true: duration-only 0.00 pp · duration+convexity 0.00 pp

\(\Delta P/P\) vs. \(\Delta i\) — true curve, duration-only tangent, and duration+convexity parabola
true \(\Delta P/P\) duration-only duration+convexity

Duration is the price–yield curve's slope; convexity is its curvature. Together they turn a one-term tangent-line guess into a much better two-term estimate.

Formal

Expanding \(P(i+\Delta i)\) about \(i\) to second order and dividing by \(P\) gives \(\Delta P/P \approx -D_{mod}\Delta i + \tfrac12 C(\Delta i)^2\), where convexity \(C = (1/P)\cdot d^2P/di^2 = \Sigma\, t(t+1)\cdot PV_t\cdot v^2 / P\) is the present-value-weighted second-order sensitivity of price to yield. \(C\) is always positive for an ordinary fixed-income stream, so the \(\tfrac12 C(\Delta i)^2\) term is always positive too — it depends on \((\Delta i)^2\), never on the sign of \(\Delta i\) — and it grows fastest for large yield moves, exactly where the duration-only line drifts furthest from the curve.

Applied

A portfolio manager stress-testing a bond position against a sudden 200 or 300 basis point rate shock can't trust the duration-only number — it was calibrated for small moves near today's yield. Adding the convexity term costs one extra calculation and meaningfully tightens the estimate, which is exactly why risk desks quote both numbers together rather than duration alone.

Worked example

A bond has \(P = 100\), \(D_{mod} = 6\), and \(C = 55\). For \(\Delta i = +0.02\): duration-only gives \(-6(0.02) = -0.1200\), or \(-12.00\%\). Adding convexity: \(-0.1200 + \tfrac12(55)(0.02)^2 = -0.1200 + 0.0110 = -0.1090\), or \(-10.90\%\) — the convexity term adds back \(+1.10\%\). For the symmetric \(\Delta i = -0.02\): duration-only gives \(+12.00\%\), and with convexity \(+12.00\% + 1.10\% = \) \(+13.10\%\). The price gains more on a rate drop than it loses on an equal-size rate rise — that asymmetry is exactly what positive convexity buys.

Your turn

Same two-term formula, different numbers: a bond has \(D_{mod} = 8\) and \(C = 90\). For \(\Delta i = -0.03\): duration-only gives \(-8(-0.03) = 0.2400\), or \(+24.00\%\). The convexity term is \(\tfrac12(90)(0.03)^2 = 0.0405\), or \(+4.05\%\). Now finish it: the duration+convexity estimate is \(24.00\% + 4.05\% = \) ____

Reveal the answer

\(24.00\% + 4.05\% = \) \(+28.05\%\). Notice that dropping the convexity term (duration-only, \(+24.00\%\)) would have understated the price gain by over 4 percentage points — the same underestimate pattern you can watch in the diagram above as you drag \(\Delta i\) toward either edge.

More info — why the convexity term never flips sign

The duration term \(-D_{mod}\Delta i\) is linear in \(\Delta i\), so it flips sign with \(\Delta i\): a rate rise costs you, a rate drop pays you. The convexity term \(\tfrac12 C(\Delta i)^2\) is quadratic — squaring \(\Delta i\) erases its sign entirely, so the term is positive whichever direction the yield moves, so long as \(C > 0\) (true for any ordinary bond or cash-flow stream). That's why a bond with higher convexity gains more from a rate drop than it loses from an equal-size rate rise: the favorable asymmetry that Redington immunization deliberately exploits by requiring asset convexity to be at least as large as liability convexity. A bad \(D_{mod}\), though, isn't something convexity can rescue — see the pitfalls in the Dive deeper links below.

Check your understanding

Question 1 of 4

A bond has modified duration \(D_{mod} = 10\) and convexity \(C = 120\). Estimate \(\Delta P/P\) for \(\Delta i = +0.015\) using the two-term Taylor approximation.

Question 2 of 4

A bond has positive convexity. Compared to the true price change, is the duration-only (tangent-line) estimate of \(\Delta P/P\) too high or too low, and for which direction(s) of \(\Delta i\)?

Question 3 of 4

Which best describes how convexity relates to modified duration when estimating a bond's price change?

Question 4 of 4

What must both the duration-only estimate and the duration+convexity estimate equal when \(\Delta i = 0\)?

Recap

  • Convexity \(C = (1/P)\cdot d^2P/di^2\) is the PV-weighted second-order sensitivity of price to yield — the curvature the duration tangent line misses.
  • Two-term estimate: \(\Delta P/P \approx -D_{mod}\Delta i + \tfrac12 C(\Delta i)^2\). Don't drop the \(\tfrac12\), and don't flip the sign of the convexity term.
  • \(C > 0\) for ordinary bonds, so the convexity adjustment is always positive — it corrects the duration-only estimate toward the true price for both rate rises and rate drops, and the correction grows with \((\Delta i)^2\).
  • Convexity refines a correct duration estimate; it cannot fix a wrong one — the duration term still dominates.

Dive deeper

Sources

  • Convexity and the Second-Order Price Approximation