Understand the curve behind a bond’s price.
Bond convexity describes how a bond’s sensitivity to yield changes itself changes as yields move. Duration gives a useful first estimate of a price change; convexity accounts for the curve that a straight-line estimate misses.
For a conventional fixed-rate bond without embedded options, that curve is generally favorable: an equal-sized yield decrease produces a larger price gain than the price loss from a yield increase, with everything else held constant. Callable bonds and mortgage-backed securities can behave differently because their future cash flows may change.
You do not need calculus to understand the practical question: if market yields move significantly, how much confidence should you place in a duration-only price estimate?
What is bond convexity?
Convexity measures the curvature of the relationship between a bond’s price and its yield. It helps explain why the price response does not remain constant across different yield levels.
Charles Schwab’s fixed-income glossary describes convexity as a calculation used alongside duration to improve estimates of how prices respond to interest-rate changes. Think of duration as the slope at today’s point on the curve, and convexity as how that slope changes as you move along it.
This is a sensitivity measure, not a payment you receive. A convexity figure of 30 does not mean a 30% yield or a promised return.
Bond convexity vs. duration vs. maturity
| Measure | Question it answers | Practical use |
|---|---|---|
| Maturity | When is the principal scheduled to be repaid? | Helps match a bond to a spending date. |
| Duration | How sensitive is the price around current yields? | Provides an initial price-change estimate. |
| Convexity | How does that sensitivity change as yields move? | Improves the estimate for larger movements. |
Maturity is a calendar endpoint. Duration is a measure of sensitivity even though it is often expressed in years. A bond with ten years until maturity need not have a duration of ten.
FINRA’s duration guide explains that higher duration means greater sensitivity to interest-rate changes and distinguishes modified duration from effective duration. Modified duration is used for fixed cash flows; effective measures are particularly useful when interest rates can alter those cash flows.
Yield and coupon are also different. A fixed coupon can stay unchanged while the market price and yield move. Our guide to yield to maturity vs. coupon rate explains that distinction.
A bond convexity example with real price calculations
Consider a hypothetical option-free zero-coupon bond that pays $1,000 in five years. Assume annual compounding, no default, and no fees. At a 5% annual yield, its calculated price is approximately $783.53.
Reprice the same promised payment at 4% and 6%, keeping the five-year term unchanged:
| Annual yield | Calculated price | Change from the 5% price |
|---|---|---|
| 4% | $821.93 | +4.90% |
| 5% | $783.53 | Starting point |
| 6% | $747.26 | −4.63% |
A decline from 5% to 4% is one percentage point, or 100 basis points. The rise to 6% is the same-sized change in the other direction. Yet the dollar gain of about $38.40 exceeds the loss of about $36.27.
At the starting yield, the bond’s modified duration is approximately 4.76 years. A duration-only estimate would suggest a gain or loss of about 4.76%. The exact prices above show the curvature: the gain is slightly larger and the loss slightly smaller.
These are illustrative calculations, not market quotes or forecasts. The example deliberately isolates a yield change and does not include the passage of time, credit deterioration, transaction costs, or an issuer’s right to repay early. A real security can have different compounding and payment conventions.
For a related example of securities that separate principal and interest payments, read our explanation of Treasury STRIPS.
Positive convexity: why the curve can help
An option-free fixed-rate bond has positive convexity under the conventional price-yield model. The convexity adjustment improves on the linear estimate: it adds to the estimated gain when yields fall and reduces the estimated loss when yields rise.
The CFA Institute’s reading on yield-based convexity explains why this adjustment matters more for larger yield moves and longer-maturity bonds. Other things equal, lower coupons, longer maturities, and lower starting yields generally increase convexity.
That does not make a long-term bond safe. High duration can still produce a substantial price decline when yields rise. Convexity refines that decline estimate; it does not cancel it.
Negative convexity: when the upside is limited
Callable bonds
A callable bond gives the issuer a contractual ability to redeem it before the scheduled maturity, subject to the call terms. Falling interest rates can make refinancing attractive to the issuer.
As a call becomes more likely, the bond’s price appreciation may slow because investors expect repayment near the call price. In that region, the bond can exhibit negative convexity. It does not mean every callable bond has negative convexity at every yield level.
CFA Institute’s analysis of embedded options explains the negative convexity of callable bonds when the option is near the money and the usefulness of effective risk measures. Check the call schedule and price before treating a maturity date as a guaranteed holding period.
Mortgage-backed securities
Homeowners can repay or refinance mortgages early. When rates fall, faster repayments can shorten the expected life of mortgage-backed securities and limit price appreciation. When rates rise, slower repayments can extend the expected life and leave investors exposed to rate risk for longer.
The CFA Institute’s explanation of mortgage convexity discusses this changing repayment pattern. The actual outcome depends on the security’s structure and borrower behavior, rather than rates alone.
How analysts use the convexity adjustment
Analysts start with a duration estimate and then add a correction for curvature. In the five-year zero-coupon example, an annual convexity value of approximately 27.21 gives a correction of about 0.14 percentage points for a one-percentage-point yield move.
Adding that correction to the duration-only estimate produces an approximate 4.90% gain for the yield decline and a 4.63% loss for the increase—close to the repriced values in the table.
The correction depends on the square of the yield movement, so doubling the movement increases this part of the estimate approximately fourfold if the starting measures remain fixed. Very large moves still require caution because duration and convexity change too.
Use consistent units. A one-percentage-point yield change is entered as 0.01 in calculations that use decimal yields, not as 1. Annual and semiannual conventions can also produce different reported figures. Ask the data provider how its numbers were calculated before comparing them.
What convexity means for bond funds and ETFs
A fund’s risk depends on its holdings, including maturity ranges, coupon rates, call features, and mortgage exposure. Portfolio convexity helps summarize those exposures, but the portfolio can change as managers trade or securities repay.
Do not assume a conventional bond ETF becomes insensitive to rates because it owns many bonds. Diversification and interest-rate exposure address different questions.
Likewise, a fund with a target maturity year is not the same as a single bond promising a fixed face-value payment. Our guide to building a ladder with bond ETFs explains the distinction.
A single portfolio duration and convexity estimate is most useful for simplified yield shifts. Actual yields at different maturities can move by different amounts, and changes in credit spreads can affect corporate bonds independently.
What to check before comparing two bonds
- Purpose and time horizon: Can you hold the investment until the intended date, or might you need to sell earlier?
- Duration: Compare interest-rate sensitivity before judging convexity in isolation.
- Options: Read call, put, and prepayment provisions.
- Yield and price: Confirm the yield measure and whether a quote includes accrued interest.
- Credit and liquidity: Review the issuer’s ability to pay and the likely cost of selling.
- Methodology: Compare like-for-like effective or modified measures and consistent conventions.
Convexity does not assess every risk. Investor.gov’s bond overview describes risks and features that should be considered alongside a price-sensitivity measure. Taxes, inflation, fees, and reinvestment opportunities also affect the result you receive.
If you are comparing a bond with an insured deposit product, see our guide to bonds vs. CDs before treating the two as equivalent.
Frequently asked questions
Is higher bond convexity always better?
Positive convexity is favorable for the price response under comparable conditions, but it is not a complete investment ranking. Two securities can have different duration, credit quality, prices, yields, and costs.
Can a positively convex bond still lose money?
Yes. A yield increase can reduce its market price even with positive convexity. Other risks, including default and inflation, can also affect the investment.
Is convexity the same as duration?
No. Duration estimates sensitivity around current yields. Convexity measures how that sensitivity changes as yields move.
Does every callable bond have negative convexity?
No. The sign and magnitude depend on market conditions and how likely the call option is to be exercised. Negative convexity is especially relevant when the call meaningfully limits upside.
Does holding a bond to maturity remove convexity risk?
Holding an option-free bond to maturity avoids having to realize an interim market price, assuming the issuer pays as promised. It does not remove default, inflation, opportunity-cost, or reinvestment risk, and callable securities can repay earlier.
Bond convexity is most useful when paired with duration and an understanding of the security’s cash flows. It explains why equal yield changes can produce unequal price changes—and why call or prepayment features can change that relationship. Use it to ask better questions about rate exposure, rather than as a standalone signal to buy a bond.