Bond Convexity Calculator
Duration tells you roughly how much a bond's price moves for a small change in yield, but the price-yield curve is not a straight line. Convexity captures that curvature so your estimate stays accurate for larger yield moves. This calculator discounts each coupon and the final principal at the bond's yield, computes the price, modified duration, and annual convexity from first principles, then uses them to estimate the price change for a yield shift you specify. Coupon, yield, and term are user inputs reflecting the specific bond.
Bond convexity formula
y = periodic yield = YTM / frequency
Price = sum of cash flow t / (1 + y)^t
Macaulay duration (periods) = sum of t * PV(t) / price
Modified duration (years) = Macaulay / (1 + y) / frequency
Convexity (annual) = [sum of t*(t+1)*PV(t) / price] / (1 + y)^2 / frequency^2
Price change % = -ModDur * dy + 0.5 * convexity * dy^2
Each period t runs from 1 to years times frequency; PV(t) is the discounted cash flow. The dy term in the estimate is the yield change expressed as a decimal.
Understanding convexity
- Convexity is the second-order term that refines the duration-based price estimate.
- Option-free bonds have positive convexity, which benefits holders during large yield moves.
- Longer maturity and lower coupon both tend to raise convexity.
- The duration estimate alone understates price gains and overstates price losses.
- This tool prices the bond exactly from its cash flows rather than using a shortcut.
Bond convexity: frequently asked questions
What is bond convexity?
Convexity measures how a bond's duration itself changes as yields move, capturing the curvature of the price-yield relationship. Duration alone gives a straight-line estimate of price change; convexity adds a second-order correction that improves accuracy for larger yield moves, especially because bond prices fall less than duration predicts when yields rise and rise more when yields fall.
How is convexity calculated?
Convexity is the second derivative of price with respect to yield, scaled by price. For a standard bond it equals the present-value-weighted sum of t*(t+1) over each cash flow, divided by price and by (1 + y per period) squared, all expressed in periods then converted to annual terms by dividing by the number of periods per year squared.
How do duration and convexity estimate price change?
The combined estimate is percent price change is about minus modified duration times the yield change plus one half times convexity times the yield change squared. The duration term handles the linear part; the convexity term corrects for curvature and is always positive for an option-free bond.
Why does convexity matter to investors?
Higher convexity is generally desirable because it means prices rise more when yields fall and fall less when yields rise, for a given duration. Investors compare bonds with similar durations by their convexity to judge how they will behave under large interest-rate swings.
What inputs does this calculator need?
Enter the annual coupon rate, face value, the bond's annual yield to maturity, the number of years to maturity, and the number of coupon payments per year. The calculator builds the cash flows, prices the bond, and computes modified duration and annual convexity from first principles.
Official sources
- U.S. Securities and Exchange Commission: Duration.
- U.S. Securities and Exchange Commission: Bonds and fixed income.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.