Bond Convexity Calculator
Bond convexity calculator. Second-derivative interest-rate sensitivity beyond duration. Estimates the price change for large yield moves.
Bond Convexity Calculator
Convexity is the second-order measure of a bond's interest-rate sensitivity — the curvature that duration alone misses. For large yield moves, duration overstates losses and understates gains; convexity corrects both. Enter a yield shift to see the duration-only estimate versus the more accurate duration-plus-convexity estimate.
How to Use the Convexity Calculator
Enter the bond's terms
Face value, coupon, years to maturity, and yield to maturity — the same inputs as the duration calculator. Convexity is computed from the identical cash flows.
Choose a yield change to model
Enter the yield shift you want to stress-test — e.g. +2% for a sharp rate rise. This is where convexity earns its keep: the bigger the move, the more it matters.
Compare the two estimates
The result shows the duration-only price change versus the duration-plus-convexity estimate. The gap between them is the convexity correction.
Read convexity as a cushion
Higher convexity means a bond loses less when yields rise and gains more when they fall, versus a same-duration bond. It's a desirable, value-adding property.
Why Convexity Matters
The Curvature Duration Misses
The relationship between a bond's price and its yield isn't a straight line — it's a convex curve. Duration is the slope of that curve at one point; it's an excellent local approximation but a poor one for large moves. Convexity measures how much the curve bends. The combined estimate, ΔP/P ≈ −Duration·Δy + ½·Convexity·(Δy)², captures both the slope and the bend, and is materially more accurate when yields move a percentage point or more.
Convexity Is Always Your Friend (for plain bonds)
For an ordinary option-free bond, convexity is positive, and positive convexity is always beneficial: when yields fall, the price rises more than duration predicts; when yields rise, the price falls less than duration predicts. That asymmetry is why two bonds with identical durations can have different values — the more convex one is worth more, and investors will pay up for it.
"Two bonds, same duration, different convexity: the higher-convexity bond loses less when rates jump and gains more when they fall. Convexity is the free lunch duration can't see."
When Convexity Turns Negative
Callable bonds and mortgage-backed securities can exhibit negative convexity at low yields: as rates fall, the issuer's option to refinance (call the bond or prepay the mortgage) caps the price upside, bending the curve the wrong way. This calculator models plain option-free bonds, which always have positive convexity. For callable or MBS instruments, the embedded option requires a more complex, scenario-based model.
10 Facts About Bond Convexity
Convexity is the second derivative of price with respect to yield — duration is the first.
The price-yield curve bends; duration is a straight-line approximation to it.
Positive convexity is always good: less downside, more upside for the same duration.
The combined estimate: ΔP/P ≈ −D·Δy + ½·C·Δy².
Convexity matters most for large yield moves — for tiny moves, duration alone suffices.
Longer maturity and lower coupon raise convexity, just as they raise duration.
Callable bonds and MBS can have negative convexity at low yields.
Investors will pay a premium for higher convexity — it's a real, priced property.
A barbell portfolio (short + long bonds) has more convexity than a duration-matched bullet.
Convexity is quoted in years² and is always positive for plain vanilla bonds.
Frequently Asked Questions
- Convexity measures how much a bond's price-yield curve bends. Duration is the slope of that curve; convexity is its curvature. Because the curve bends in the bondholder's favour, accounting for convexity means your price-change estimate is more accurate — and shows that prices rise faster than they fall for equal yield moves.
- Duration assumes a straight-line relationship between price and yield, which is only accurate for small moves. For a 2% or 3% yield swing, duration alone overstates the loss and understates the gain. Convexity adds the second-order term that corrects this, so the combined estimate tracks the true curve much more closely.
- For plain option-free bonds, yes. Higher convexity means smaller losses when yields rise and bigger gains when they fall, for the same duration — a strictly favourable asymmetry. That's why investors pay a premium for it. The exception is bonds with embedded options (callables, MBS), which can have negative convexity.
- Callable bonds and mortgage-backed securities can show negative convexity at low yields: as rates fall, the issuer's right to call or prepay caps the price upside, so the curve bends the wrong way. This calculator models plain option-free bonds (always positive convexity). Option-embedded instruments need a scenario/option-adjusted model.
- Annualised convexity is quoted in years-squared. It only becomes intuitive inside the price-change formula, where the ½·Convexity·(Δy)² term produces a percentage. The calculator reports both the raw convexity number and the resulting price-change correction so you don't have to interpret the units directly.
- Below about 0.25%, duration alone is fine. From roughly 0.5% upward the convexity correction becomes visible, and at 2–3% it can be worth a meaningful fraction of a percent of price — especially for long-duration bonds. Model a few different yield shifts here to see the effect grow with the size of the move.
- Yes — identical inputs and identical cash-flow modelling. It computes the bond price and modified duration the same way the Bond Duration Calculator does, then adds the convexity term. Use them together: duration for the headline risk number, convexity for accurate large-move estimates.
- A bullet portfolio concentrates maturities around one point; a barbell splits between short and long maturities. For the same duration, a barbell has more convexity — it benefits more from large rate moves in either direction. That extra convexity is a key reason managers choose barbells when they expect volatility.
- No. It's an analytics tool that computes convexity and an estimated price change from the inputs you provide, for option-free bonds. It doesn't account for credit risk, liquidity, embedded options, or whether a bond suits your objectives. Consult a licensed adviser for investment decisions.
- Use duration for the headline rate-risk number and the first-order price estimate; add convexity for accuracy on large moves. The combined formula ΔP/P ≈ −Duration·Δy + ½·Convexity·(Δy)² uses both. In practice, run the Bond Duration Calculator for the duration figure, then this one to see how much the convexity term changes the estimate for the yield shift you care about.
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Method & sources
How it computes
Convexity measures how much a bond's price/yield relationship curves, which is what duration alone misses. Each cash flow contributes CF x t(t+1) / (1+y/f)^(t+2); the sum is divided by the price and then by the square of the coupon frequency to express the result in years squared. The estimated price change combines the first-order duration term with the second-order convexity correction: -modified duration x dy + 0.5 x convexity x dy^2.
What this tool implements
- The result is annualised by dividing the period-based figure by f squared, so convexity is in years squared and is directly comparable across bonds of different coupon frequency.
- The price estimate shows the duration term and the convexity correction SEPARATELY as well as combined, because the point of the tool is that the correction is small for small yield moves and matters for large ones.
- Verified 2026-08-31 against the closed form for a zero-coupon bond, T(T + 1/f)/(1 + y/f)^2. A 10-year zero at 5% semiannual gives 99.94 both ways.
- ⚠️ This is a two-term Taylor expansion, not the true price. It is far better than duration alone for a large yield move but it is still an approximation, and it will not match a full reprice. As with duration, it assumes a flat yield curve and option-free cash flows.
Sources
- Fabozzi FJ. Bond Markets, Analysis, and Strategies — the standard derivation of the second-order term and the convention of expressing convexity in years squared.
- Macaulay FR. Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856. National Bureau of Economic Research, 1938 — the duration measure the convexity correction is applied on top of: https://www.nber.org/books/maca38-1
What can make this go out of date
- Nothing is fetched. The output depends entirely on the yield and the modelled yield shift the user enters.
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