Bond Convexity Calculator

Enter your bond's face value, coupon rate, maturity, payment frequency, and YTM to calculate convexity, duration, and estimated price sensitivity to interest rate changes.

Created by Luis GonzalezLast updated: June 16, 2026

How to Use This Calculator

1
Enter your bond details
Input the face value, annual coupon rate, years to maturity, coupon payment frequency, and yield to maturity (YTM).
2
Review your results
The calculator displays Bond Convexity, Modified Duration, and Estimated Price Change for a +1% rate shock, plus an insights card with bond price, Macaulay duration, dollar convexity, and convexity benefit.

Example Calculation

An investor analyzes a $1,000 bond with a 5% annual coupon paid semi-annually, 10 years to maturity, and a 4% yield to maturity.

Face Value of Bond

1000

Annual Coupon Rate

Years to Maturity

Coupon Payment Frequency

Yield to Maturity (YTM)

Results

Bond Convexity

72.4827

Modified Duration

7.9225 yrs

Est. Price Change (+1% Rate)

-7.560%

Insights card shows bond price of $1,081.

Tips

Compare 5-Year vs 30-Year Convexity

A $1,000 bond at 5% coupon and 4% YTM has convexity of 21.0366 at 5 years but 387.2965 at 30 years. The 30-year bond's convexity benefit is 1.9365% per 1% rate move versus just 0.1052% for the 5-year, making longer bonds far more sensitive to large rate shifts.

Check the Impact of a 2% Rate Shock

For the default $1,000 bond (5% coupon, 10-year, 4% YTM), a 2% rate increase causes an estimated price drop of 14.395% or about $155.72. Duration alone would overestimate this loss -- convexity of 72.4827 provides a cushion.

Raise YTM to See Convexity Compress

Increasing the YTM from 4% to 6% on the same $1,000 10-year bond drops convexity from 72.4827 to 68.9662 and reduces the bond price from $1,081.76 to $925.61. Higher yields dampen the price-yield curvature.

Zero-Coupon Bonds Show Maximum Convexity

A 10-year zero-coupon bond at 4% YTM has convexity of 96.1169 versus 72.4827 for a 5% coupon bond at the same maturity. Without coupon payments shortening effective duration, zero-coupon bonds concentrate all cash flow at maturity, amplifying convexity.

The Mathematical Framework Behind Bond Convexity

The Bond Convexity Calculator determines the second derivative of a bond's price with respect to its yield, adjusted by the bond price itself. This captures the curvature of the price-yield relationship.

First, the bond's price is the present value of all future cash flows:

coupon payment = (face value x annual coupon rate) / frequency
bond price = sum(coupon / (1 + YTM/freq)^t) + face value / (1 + YTM/freq)^N

Then convexity is derived from the weighted sum of present values adjusted by time-squared factors:

convexity = [sum(((t^2 + t) x PV(coupon_t)) / (1 + y)^2) + (N^2 x PV(face)) / (1 + y)^2] / (price x freq^2)

For a $1,000 bond at 5% coupon, semi-annual, 10-year maturity, and 4% YTM: bond price = $1,081.76, convexity = 72.4827, modified duration = 7.9225 yrs.

How Maturity and Yield Affect Convexity

Convexity is most sensitive to maturity and yield level. Here are verified examples using $1,000 face value, 5% coupon, semi-annual payments:

Maturity	YTM	Convexity	Mod. Duration	Price Change (+1%)
5 years	4%	21.0366	4.4107 yrs	-4.306%
10 years	4%	72.4827	7.9225 yrs	-7.560%
10 years	6%	68.9662	7.6650 yrs	-7.320%
30 years	4%	387.2965	16.5995 yrs	-14.663%

A zero-coupon 10-year bond at 4% YTM has the highest convexity at 96.1169, because all cash flow is concentrated at maturity.

💡 To analyze overall bond investment returns alongside convexity, our Bond Yield Calculator computes yield to maturity, current yield, and more.

Modified Convexity vs Effective Convexity

This calculator computes modified convexity, which assumes fixed cash flows -- appropriate for option-free bonds like Treasuries and plain vanilla corporates.

For bonds with embedded options (callable, puttable, or mortgage-backed securities), use effective convexity instead. Effective convexity accounts for how cash flows change as rates move, typically requiring an option-adjusted spread (OAS) model.

Modified Convexity: Fixed cash flows -- use for Treasuries, corporate bonds
Effective Convexity: Variable cash flows -- use for callable bonds, MBS

Using modified convexity on a callable bond would overestimate positive convexity, since the call option caps price appreciation when rates fall.

💡 For a broader view of fixed income portfolio performance, our Bond Price Calculator helps you determine the fair market value of bonds given current yields.

Frequently Asked Questions

What does a high bond convexity value indicate?

A high convexity value means the bond's price-yield curve is more bowed, creating asymmetric price moves: prices rise more when yields fall than they drop when yields rise. For example, a 30-year bond at 5% coupon and 4% YTM has convexity of 387.2965 -- its convexity benefit alone adds 1.9365% for a 1% rate move, significantly cushioning downside.

How does convexity differ from duration?

Duration measures linear price sensitivity to small rate changes, while convexity captures the curvature. For a $1,000 bond (5% coupon, 10-year, 4% YTM), modified duration of 7.9225 predicts a 7.9225% loss for a +1% rate move. Convexity of 72.4827 refines this to -7.560%, adding back 0.3624% through the convexity adjustment: price change = -duration x rate change + 0.5 x convexity x (rate change)^2.

Can a bond have negative convexity?

Yes. Callable bonds exhibit negative convexity because the issuer can redeem them when rates fall, capping upside. For standard option-free bonds, convexity is always positive. This calculator computes modified convexity for option-free bonds. Use effective convexity models for callable or mortgage-backed securities.

How does maturity affect convexity?

Longer maturities dramatically increase convexity. A 5-year bond (5% coupon, 4% YTM) has convexity of 21.0366, a 10-year bond has 72.4827, and a 30-year bond has 387.2965. The relationship is roughly quadratic -- doubling maturity from 10 to 20 years more than quadruples convexity.

What is the convexity formula used here?

The calculator computes: Convexity = [sum of ((t^2 + t) x PV(coupon_t)) / (1 + y)^2 + (N^2 x PV(face)) / (1 + y)^2] / (bond price x freq^2), where t is the period number, y is the periodic yield, N is total periods, and freq is payments per year. For the default inputs, this yields 72.4827.

Why does the estimated price change not equal duration times the rate shock?

Because the price change formula includes the convexity adjustment: Price Change = -Modified Duration x rate change + 0.5 x Convexity x (rate change)^2. For the default bond, -7.9225 x 0.01 x 100 = -7.9225%, but adding the convexity term (0.5 x 72.4827 x 0.0001 x 100 = +0.3624%) gives -7.560%. The larger the rate move, the bigger this correction.