numvana

Bond Convexity Calculator

Measure how much a bond's price-yield relationship curves, beyond what modified duration alone captures.

Bond price
$925.61
Modified duration
7.67 years
Convexity
71.79
Est. price change for +1pp yield, duration only
-7.665%
Est. price change for +1pp yield, duration + convexity
-7.306%

How it works

Modified duration (this site's bond-duration calculator) estimates a bond's percent price change for a small yield move as a straight line: ΔP/P ≈ −modifiedDuration × Δy. But the real price-yield relationship is a curve, not a line, so that linear estimate drifts off for anything but a tiny yield move. Convexity is the curve's bend — formally, Σ CFₜ·t·(t+1)/(1+y)^(t+2) divided by price, then rescaled from periods² into years² — and adding its contribution gives a much closer second-order estimate: ΔP/P ≈ −modifiedDuration × Δy + 0.5 × convexity × Δy².

A useful sanity check: a zero-coupon bond has only one cash flow, so its convexity collapses to the closed form T(T+1)/(1+y)², where T is its maturity in years — no summation needed, since there's only one term.

Convexity is always positive for an ordinary (non-callable) bond, which means the duration-only line always underestimates the bond's actual price for any yield move, in either direction: it understates how much the price rises when yields fall, and overstates how much it falls when yields rise. That asymmetric 'convexity is a bondholder's friend' property is exactly what the two side-by-side price-change estimates below are meant to show.

FAQ

Why does convexity matter if I already have modified duration?

Duration alone is only accurate for small yield moves. For a large move — a full percentage point or more, which is common during rate-hiking or rate-cutting cycles — the gap between the duration-only estimate and the duration-plus-convexity estimate becomes large enough to matter, especially for longer-maturity bonds, which have higher convexity.

Why is convexity always positive for a plain bond?

Because the price-yield relationship is convex (curves upward, like the inside of a bowl) rather than concave — bond prices rise faster than linearly as yields fall, and fall slower than linearly as yields rise. This is a general property of discounting fixed cash flows and isn't true for every fixed-income instrument — callable bonds, for instance, can exhibit negative convexity near their call price, which is outside the scope of this calculator.

Related calculators