About the Bond Convexity Calculator
This bond convexity calculator measures how curved a bond’s price–yield relationship is and uses it to improve the duration estimate of a price change. Enter the face value, coupon, yield to maturity, years to maturity, coupon frequency and a yield shift in basis points. It returns the convexity, modified duration, the duration effect, the convexity adjustment, the combined estimate and the exact repriced change for comparison.
Duration alone treats price and yield as a straight line, so it overstates losses when yields rise and understates gains when they fall. Convexity corrects that error, and the correction grows with the square of the yield move. Portfolio managers, CFA and FRM candidates and bond investors weighing long maturities use convexity to compare how bonds will behave in a large rate move.
The model assumes a fixed-rate, option-free bond valued on a coupon date, with yield compounded at the coupon frequency and a flat yield shift. Callable bonds and mortgage-backed securities can have negative convexity, which needs an effective-convexity model with option pricing.
With the default inputs, the convexity is 226.61. Change any value above to recalculate instantly.
How to use the bond convexity calculator
- 1Enter the face value, annual coupon rate and current yield to maturity.
- 2Enter the years to maturity and the coupon frequency.
- 3Enter the yield shift you want to test in basis points (negative for a fall).
- 4Compare the duration effect, the convexity adjustment and the exact repriced change.
- 5Use the chart to see how the straight duration line drifts from the true price curve.
Formula and method
Convexity is the second derivative of the bond price with respect to yield, divided by the price. Each cash flow CFₜ in period t contributes CFₜ × t(t + 1) ÷ (1 + y/f)^(t+2); the sum is divided by the price and by f² so that convexity is expressed in years², the raw figure used in most textbooks and CFA curriculum material (not divided by 100).
The price change for a yield change Δy (in decimal, so 100 bp = 0.01) is approximated by a second-order Taylor expansion: the duration effect −D_mod × Δy plus the convexity adjustment ½ × C × Δy². Because Δy is squared, the adjustment is always positive for an option-free bond, so convexity helps whether rates rise or fall. The calculator also reprices the bond exactly at the new yield so you can see how close the approximation is.
- C
- Convexity (years²)
- D_mod
- Modified duration = Macaulay duration ÷ (1 + y/f)
- CFₜ
- Cash flow in period t
- y, f
- Annual yield and coupon payments per year
- Δy
- Yield change in decimal (1 bp = 0.0001)
Worked examples
20-year 4% bond at a 5% yield, +200 bp
At a 5% yield the bond prices at $874.49 with modified duration 13.14 and convexity 226.6. A 200 bp rise gives a duration effect of −26.29%, but the convexity adjustment adds back ½ × 226.6 × 0.02² = +4.53%, for an estimate of −21.76% — close to the exact −22.28%.
10-year zero-coupon bond, yields fall 100 bp
A zero pays only at maturity, so its convexity of 10 × 11 ÷ 1.04² = 101.7 is high for its maturity. A 100 bp fall lifts the price by an estimated 9.62% + 0.51% = 10.12%, almost exactly the true 10.14%.
3-year 6% annual bond at par, +100 bp
A short bond has little curvature: convexity is only 9.89, so for a 100 bp rise the adjustment adds just 0.049% to the −2.673% duration effect, giving −2.624% — virtually identical to the exact reprice.
Frequently asked questions
What is bond convexity?+
Convexity measures how the duration of a bond changes as yields change — the curvature of the price–yield curve. A bond with higher convexity gains more when yields fall and loses less when yields rise than a lower-convexity bond with the same duration.
How do you use convexity to estimate a price change?+
Add the convexity adjustment to the duration estimate: %ΔP ≈ −modified duration × Δy + ½ × convexity × Δy², with Δy in decimal. For a 200 bp rise on a bond with duration 13.14 and convexity 226.6, that is −26.3% + 4.5% = −21.8%.
Why do some sources show convexity divided by 100?+
Some data providers quote convexity scaled by 100 so that it can be multiplied by Δy in percentage points rather than decimals. This calculator shows the raw value in years²; divide by 100 to match a scaled quote.
What makes a bond more convex?+
Longer maturity, lower coupons and lower yields all raise convexity. For a given duration, a barbell of short and long bonds is more convex than a single bullet bond, which is why convexity matters when constructing bond portfolios.
Can convexity be negative?+
Yes. Callable bonds and mortgage-backed securities can show negative convexity because when yields fall the issuer or homeowners refinance, capping price gains. Those securities need an effective-convexity calculation with an option model rather than this cash-flow formula.
Results are estimates for educational purposes and are not financial advice. Rates, fees and terms vary — confirm with your lender or a licensed advisor before making decisions.