Bond Convexity Calculator

Compute a bond's convexity, modified duration, price, and convexity-adjusted price change from its face value, coupon rate, yield to maturity, and term.

Quick Facts

Formula
Convexity = (1/P) × Σ CFt · t(t+1) / (1+y)^(t+2)
P is price, y is the per-period yield; divide by payments-per-year squared for years².
Positive convexity
Option-free bonds always have positive convexity
Price gains from a yield fall exceed losses from an equal yield rise.

Your Results

Calculated
Convexity
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Annual, in years²
Bond price
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Present value of all cash flows
Modified duration
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Years; first-order rate sensitivity
Est. price change
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For the entered yield shift

Ready

Enter the bond's terms and yield, then calculate.

How to use the Bond Convexity Calculator

Convexity measures the curvature of a bond's price-yield relationship. Duration alone assumes that relationship is a straight line, but it is actually curved: as yields fall, a bond's price rises at an accelerating rate, and as yields rise, it falls at a decelerating rate. Convexity quantifies that curvature so you can refine the price-change estimate duration gives you.

What the calculator computes

  • Bond price: the present value of every coupon plus the face value, each discounted at the yield to maturity. Coupons are paid at your chosen frequency (annual, semiannual, quarterly, or monthly).
  • Modified duration: the first-order, straight-line sensitivity of price to yield, in years. A modified duration of 7 implies roughly a 7% price drop for a 1-percentage-point rise in yield, before the convexity correction.
  • Convexity: the second-order term, in years squared. It captures how duration itself changes as yields move, and it is the number this tool is built around.
  • Convexity-adjusted price change: the estimated dollar change in price for the yield shift you enter, combining the duration and convexity terms.

The formula

For a bond with periodic cash flows CFt discounted at the per-period yield y, convexity is the price-weighted sum of t(t+1)/(1+y)t+2 across all periods: Convexity = (1/P) × Σ CFt · t(t+1) / (1+y)t+2. Because t counts periods, the raw figure is in periods squared; dividing by the number of payments per year squared converts it to the more familiar years squared. The tool assumes fixed coupons, a flat yield curve, and no embedded options such as call or put features.

Estimating a price change

Duration and convexity combine to approximate the percentage price change for a yield move Δy (in decimal form): %ΔP ≈ −ModDur × Δy + ½ × Convexity × Δy². The duration term is a straight line that always overstates the price drop when yields rise and understates the gain when yields fall; the convexity term corrects for that, and the correction grows with the size of the yield move.

Frequently Asked Questions

What is bond convexity?
Convexity measures the curvature of a bond's price-yield relationship - how its duration changes as yields move. It refines duration's straight-line estimate, capturing that a bond's price rises more when yields fall than it drops when yields rise by the same amount. Option-free bonds always have positive convexity.
How is bond convexity calculated here?
The tool discounts each coupon and the final face value at the yield to maturity to get the price, then sums each cash flow times t(t+1) divided by (1+y) raised to the (t+2) power, where t is the period number and y is the per-period yield, and divides that sum by the price. Dividing by payments-per-year squared states the result in years squared.
How do duration and convexity estimate a price change?
The convexity-adjusted estimate is: percent price change equals minus modified duration times the yield change, plus one-half times convexity times the yield change squared, with the yield change in decimal form. Convexity corrects duration's tendency to overstate losses and understate gains, and the correction matters most for large yield moves.
Is higher convexity good or bad?
For a bond buyer, higher convexity is generally desirable. For two bonds with the same duration, the more convex one gains more when rates fall and loses less when rates rise. That asymmetric benefit is why investors often accept a slightly higher price, and thus a lower yield, for more convex bonds.

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