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, in percentage points — e.g. 2 for a sharp rise from 6% to 8%, −1 for a fall. This is where convexity earns its keep: the correction grows with the square of the move.
Compare the two estimates
The result shows the duration-only price change, the duration-plus-convexity estimate, and the exact price at the new yield. The gap between the first two is the convexity correction; the gap between the last two is what even a two-term estimate still misses.
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 with the same duration are not the same bond. The more convex one loses less when yields jump and gains more when they fall — which is exactly why the market makes you pay for it.
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.
Duration is the straight line; convexity is the bend
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, a lower coupon and a lower yield all 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.
- The correction grows with the square of the move, so it is invisible for a few basis points and material for a percentage point or more. On this tool's default bond (20-year, 5% coupon, 6% yield, semi-annual) a +2 point shift gives a duration-only estimate of −24.0%, a convexity correction of +4.0% and a combined −20.1%, against an exact reprice of −20.5% — the page now prints that exact figure beside the estimates. Try 0.25, 1 and 2 points and watch the gap open.
- Yes. Both pages price the bond with the same formula as the Bond Yield Calculator — including a fractional term (say 20.25 years), where the part-period to the next coupon is priced with its accrued interest rather than rounded — so the price and modified duration here match the Bond Duration Calculator's on the same inputs. Use them together: duration for the headline risk number, convexity for large-move estimates, and the exact reprice here as the check on both.
- 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.
Method & sources
How it computes
Convexity (1/P)·d²P/dy² and modified duration −(1/P)·dP/dy of the 31 CFR 356 Appendix B price function — on a coupon date exactly Σ CF·t(t+1)/(1+y/f)^(t+2)/(P·f²) and the PV-weighted time ÷ (1+y/f). Estimate ΔP/P ≈ −D·Δy + ½·C·Δy², shown beside an exact reprice at the shifted yield (BondKernel.analyseConvexity).
What this tool implements
- Annualised in years² (the period figure divided by f²), so convexity is comparable across coupon frequencies.
- The duration term and the convexity correction are shown separately and combined, and beside the EXACT price at the shifted yield, so the reader sees both what convexity adds and what a two-term Taylor expansion still misses.
- Option-free bond on a flat yield curve. A fractional term is priced with the part-period to the next coupon and its accrued interest, by the same Appendix B formula as the bond yield calculator, rather than rounded to whole periods.
- Checked against the zero-coupon closed form T(T + 1/f)/(1 + y/f)²: a 10-year zero at 5% semi-annual gives 99.9405 both ways (to 1e-9 in the kernel suite).
- Page figures, recomputed by the kernel and an independent Python sum: the default bond (USD 1,000, 5% coupon, 20 years, 6% yield, semi-annual) prices at 884.43 with modified duration 12.011 and convexity 198.36; a +2 point shift gives −24.02% from duration alone, +3.97% from convexity, −20.06% combined, against an exact reprice of −20.50% (703.11).
- Corrected 2026-10-16: the term was rounded to whole coupon periods; a blank box read as 0; error messages were never visible (shown with display '' on a box the stylesheet hides); the exact reprice the FAQ cited was not on the page; the pull-quote was set as an unattributed quotation and called convexity a "free lunch" although the page says it is priced.
Sources
- Fabozzi FJ. Bond Markets, Analysis, and Strategies. 10th ed. MIT Press; 2021.
- 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. https://www.nber.org/books/maca38-1
- 31 CFR Part 356, Appendix B — Formulas for Determining Prices, Yields and Accrued Interest on Treasury Notes and Bonds (the price function differentiated here). https://www.govinfo.gov/content/pkg/CFR-2024-title31-vol2/pdf/CFR…
What can make this go out of date
- None at runtime — the output depends entirely on the bond terms, yield and modelled yield shift the user enters.
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