Bond Convexity Calculator

A bond convexity calculator measures the curvature in the relationship between a bond's price and its yield, providing a more accurate estimate of price change for large yield movements than duration alone. It is used by fixed-income portfolio managers, risk analysts, and traders to assess interest rate sensitivity and compare bonds with similar durations.

How to Use the Convexity Calculator

1. Enter the bond's face value, coupon rate, coupon frequency, yield to maturity, and years to maturity.
2. Click calculate to see the bond's modified duration, convexity, and estimated price change for a given yield shift.
3. Optionally, enter a yield change scenario to see the combined duration and convexity price adjustment.

The Formula

Convexity measures the second derivative of the bond price with respect to yield:

Convexity = (1 / (P x (1 + y)^2)) x sum of (t x (t + 1) x CF_t / (1 + y)^t)

Where P is the bond price, y is the yield per period, t is the time period (in periods), and CF_t is the cash flow at time t.

The improved price change estimate using both duration and convexity:

dP/P approximately = -Modified Duration x dy + 0.5 x Convexity x (dy)^2

Where dy is the change in yield. Duration captures the linear (first-order) price change; convexity corrects for the curvature (second-order effect) that duration misses.

Real-World Example

Bond: £1,000 face value, 5% annual coupon, 10 years to maturity, yield to maturity 6%.

Modified duration: approximately 7.36 years. Convexity: approximately 63.

Estimated price change if yields rise by 1% (100 basis points):

Duration effect: -7.36 x 0.01 = -7.36% Convexity correction: 0.5 x 63 x (0.01)^2 = 0.315% Total estimated price change: -7.36% + 0.315% = approximately -7.04%

Without convexity, duration alone would predict a -7.36% price fall. The convexity correction reduces the estimated loss to -7.04% because the actual price-yield curve is convex: the bond falls by less than duration implies when yields rise, and gains more than duration implies when yields fall.

Why Convexity Is a Desirable Property

Convexity always works in the bondholder's favour. When yields rise, a more convex bond falls less than a less convex bond with the same duration. When yields fall, a more convex bond gains more. This asymmetry is beneficial: positive convexity means you benefit more from yield declines than you suffer from equivalent yield increases. As a result, higher-convexity bonds trade at a premium (lower yield) compared to lower-convexity bonds with the same duration. Investors in high-volatility rate environments place greater value on convexity because the second-order effect becomes more significant for large yield movements. Duration alone is a reasonable approximation for small yield changes (under 50 basis points), but for large moves convexity is essential for accurate risk estimation.

Frequently Asked Questions

What is the difference between modified duration and convexity? Modified duration measures the first-order (linear) sensitivity of bond price to yield changes. It provides a reasonable approximation for small yield shifts. Convexity measures the second-order sensitivity: the rate at which duration itself changes as yields move. For large yield changes, using duration alone underestimates how much a bond gains when yields fall and overestimates how much it loses when yields rise. Convexity corrects for this asymmetry.

Do all bonds have positive convexity? Most conventional bonds have positive convexity. Callable bonds can exhibit negative convexity in certain yield ranges: when yields fall significantly below the coupon rate, the probability of call increases, which limits price appreciation, making the price-yield relationship concave rather than convex in that range. Mortgage-backed securities also exhibit negative convexity (due to prepayment risk) in falling rate environments. For this reason, standard convexity calculations do not apply to callable or prepayable instruments without modification.

How is convexity used in portfolio management? Portfolio managers use convexity alongside duration to barbell or bullet structure portfolios. A barbell (combining short and long-maturity bonds) typically has higher convexity than a bullet portfolio (concentrating in medium-maturity bonds) with the same average duration. Higher convexity is generally preferred when rates are volatile because it provides more symmetrical returns. In a stable rate environment, the yield pickup from lower-convexity bonds may outweigh the benefit of higher convexity.

Does higher convexity always mean better performance? Higher convexity bonds generally outperform lower-convexity bonds when yields move significantly in either direction. However, in stable yield environments where rates remain in a narrow range, the yield premium sacrificed for convexity may not be recovered. The value of convexity depends on actual yield volatility: the more rates move, the more valuable convexity becomes.

Also try these free tools related to Bond Convexity Calculator: - Bond Price Calculator

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Convexity built from two cash flows

A two-year bond makes the arithmetic short enough to check on paper. Take a bond with a face value of 1000, a 5 percent annual coupon, and a yield to maturity of 6 percent. The cash flows are 50 at the end of the first year and 1050 at the end of the second.

The price is 981.666073. Macaulay duration is the present-value-weighted average time: one times 47.169811 plus two times 934.496262, divided by the price. That numerator is 1916.1623 and the quotient is 1.951949 years. Modified duration divides Macaulay by one plus the yield, so 1.951949 divided by 1.06 gives 1.841462.

Convexity weights each cash flow by t times (t plus 1). The numerator is two times 47.169811 plus six times 934.496262, which is 94.3396 plus 5606.9776, or 5701.3172. The denominator is the price times 1.06 squared, and that simplifies. Multiplying the price by 1.06 squared cancels the discounting on both cash flows except where the exponent turns negative, so the denominator reduces to 50 times 1.06 plus 1050, which is exactly 1103. Divide and convexity is 5.168919.

That shortcut is worth keeping. Any arithmetic slip in the price shows up immediately, because the two sides of the simplification have to agree.

Duration alone against duration plus convexity

The improved estimate adds half the convexity times the square of the yield change to the duration term. For this bond the duration term moves the price by 1.841462 percent for each 1 percent move in yield, and the convexity term is small but consistent.

Every figure in the error column is negative in the sense that the duration-only estimate is too pessimistic: it understates the gain when yields fall and overstates the loss when they rise. On this two-year bond the miss runs to about 0.026 percentage points at a 100 basis point move and about 0.006 points at 50 basis points. Dividing the two gives a ratio of about four, which is what a quadratic correction predicts for a halving of the shift.

Convexity takes almost all of that back. The remaining error at a 100 basis point move is 0.0003 percentage points, on a bond whose annual price move is nearly 1.9 percent. A reader who wants a rule of thumb: duration alone is adequate under about 50 basis points and convexity earns its place above it.

Coupon, price and duration on a ten-year bond

The same arithmetic on a ten-year bond shows how much the coupon moves the answer.

Two bonds of the same maturity at the same yield carry different durations because the coupons differ. The 5 percent bond returns less of its value before maturity, so more weight sits on the final payment and the weighted average time is longer. Duration is a re-expression of one price-yield curve at one coupon and one price, and a duration figure quoted without those two numbers cannot be checked against anything.

That matters when a figure looks low. A modified duration of 7.36 years belongs to a ten-year bond priced at par, which is a bond whose coupon equals its yield. The same bond with a 5 percent coupon against a 6 percent yield runs to 7.568428 years, and its convexity to 72.56926 against 69.740393 for the par bond. A coupon below the yield lengthens duration, and the two cases sit about 0.21 years apart on a ten-year bond.

Annual and semiannual units

Convexity carries the square of the time unit, so a per-period figure is not comparable with an annual one until it is divided by the frequency squared. The same two-year bond priced on a semiannual basis, with a 2.5 percent coupon each half year and a 3 percent yield each half year, shows the effect.

Two readings follow. A per-period convexity of 17.939657 divided by four gives 4.484914 per year squared, and using the per-period figure against an annual yield change would inflate the correction from 0.022425 to 0.089698 percent, a factor of four. And the two annualised durations differ, 1.841462 against 1.871103, because a semiannual bond pays its cash earlier and that shortens the weighted average time.

Method and assumptions behind the two measures

The formulas assume a fixed set of cash flows discounted on a single yield. Five assumptions follow from that, and each one fails on some instruments.

• The yield curve is flat and shifts in parallel. A real curve moves by different amounts at different maturities, and a steepening curve changes the price in a way one yield cannot express.
• The cash flows do not change. A callable bond can be redeemed early and a mortgage-backed security prepays, so the cash flows depend on the very rate change being measured.
• No default and no credit migration. Both measures describe interest rate sensitivity, not credit risk.
• The shift is small enough for the second-order term to carry the approximation. Convexity itself changes as yields move, and a large shift needs a full reprice.
• Coupons are reinvested at the same yield. That assumption sits inside the yield-to-maturity figure rather than inside the duration arithmetic, and it is the one most often overlooked.

For callable bonds and mortgage-backed securities, convexity turns negative over part of the yield range. The price-yield curve bends the other way because falling rates accelerate prepayment or exercise, which caps the price. The standard formula above does not capture that, and an analyst who applies it unchanged will overstate the gain.

A note on the two measures

Frederick Macaulay introduced duration in his 1938 study for the National Bureau of Economic Research, Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United States since 1856. He built it as a better measure of a bond's length than its maturity date, and the present-value weighting has not changed. Frank Redington, an actuary at the Prudential, added the second-order term in a 1952 paper for the Journal of the Institute of Actuaries, Review of the Principles of Life-office Valuations, where he set out the idea now called immunisation. Duration matches a portfolio's sensitivity to a small rate move and convexity handles the rest.