Bond Convexity Calculator
Calculate the convexity of a fixed-rate bond to measure the curvature of its price-yield relationship and improve duration-based price change estimates.
How to use this tool
- Enter face value, annual coupon rate, yield to maturity (ytm), years to maturity and coupon frequency in the fields above.
- Results update instantly as you type — or click Calculate.
- Read your convexity and the full breakdown beneath it.
⚠ This tool provides general estimates for education only and is not financial, tax or legal advice. Figures may not reflect your situation — verify with a qualified professional.
Formula
Convexity = Σ[t(t+1) · PV(CFt)] / [P · (1+y/m)2 · m2]
Where t is the period number, PV(CFt) is the present value of the cash flow in period t, P is the bond price, y is the annual yield, and m is the coupon frequency per year.
How it works
Convexity measures the curvature of the bond's price-yield curve. When yield changes are large, duration alone underestimates price decreases and overestimates price increases; convexity provides a second-order correction: ΔP/P ≈ −Dmod·Δy + ½·Convexity·(Δy)².
This calculator uses the standard textbook formula for full-period (not between-coupon) settlement and applies the same period rate and period numbering for both Macaulay duration and convexity. Accrued interest and fractional first periods are not modeled.
Worked example
2-year annual 5% coupon bond at par (YTM = 5%)
- Cash flows: CF₁ = $50, CF₂ = $1,050; period rate = 5%
- PV(CF₁) = 50/1.05 = 47.619; PV(CF₂) = 1050/1.05² = 952.381
- Price = 47.619 + 952.381 = $1,000.00
- Conv numerator = 1×2×47.619 + 2×3×952.381 = 95.238 + 5,714.286 = 5,809.524
- Convexity = 5,809.524 / (1000 × 1.05² × 1²) = 5,809.524 / 1,102.5 = 5.2694 yrs²
Convexity = 5.2694 yrs²
Common mistakes to avoid
- Relying on duration alone for large yield changes — duration is a linear approximation; convexity captures the curvature that makes duration underestimate price gains and overestimate price losses.
- Using annual periods in the convexity formula for a semi-annual coupon bond without adjusting by m^2, producing a figure that is off by a factor of 4.
- Confusing positive and negative convexity — callable bonds can exhibit negative convexity at low yields because the price is capped by the call price.
Key terms
- Convexity
- A measure of the curvature of the relationship between a bond's price and its yield; higher convexity means the bond price rises more when yields fall and falls less when yields rise.
- Macaulay Duration
- The weighted average time (in years) until a bond's cash flows are received, with weights equal to the present value of each cash flow.
- Modified Duration
- Macaulay Duration divided by (1 + periodic yield); it approximates the percentage change in bond price for a 1% change in yield.
- Yield to Maturity (YTM)
- The total annual return an investor would earn if the bond is held until it matures, assuming all coupon payments are reinvested at the same rate.
Frequently asked questions
- Why is positive convexity desirable in a bond?
- A bond with positive convexity gains more in price when yields fall than it loses when yields rise by an equal amount. This asymmetry benefits the holder and is reflected in the bond's price.
- How do I use convexity to improve a duration-based price estimate?
- Approximate price change = -Duration x dy + 0.5 x Convexity x (dy)^2, where dy is the change in yield. The convexity term adds the curved correction to the linear duration estimate.
- Do zero-coupon bonds have high or low convexity?
- High convexity, because their duration equals their maturity (the longest possible for a given term) and convexity increases with duration. A 20-year zero has substantially more convexity than a 20-year coupon bond.