TL;DR: Duration tells you how much a bond’s price moves for a small change in yield—but only as a straight-line approximation. Convexity is the curvature duration misses. It captures the fact that bond prices rise more on a yield drop than they fall on an equal-sized yield rise. For small moves the gap is trivial; for big moves it can be the difference between a 14% loss and a 16% loss.
Duration is a tangent line, not the actual curve
A bond’s price is not a linear function of yield. Plot price on the y-axis and yield-to-maturity on the x-axis, and the line you get is curved—convex toward the origin, sloping down as yields rise. Duration measures the slope of that curve at one point: the bond’s current yield.
The duration approximation
ΔPrice ≈ − Modified Duration × ΔYield
works perfectly for an infinitesimally small yield change. As soon as the move gets meaningful, the straight tangent line drifts away from the actual price curve. Duration consistently underestimates the gain when yields fall and overestimates the loss when yields rise. The gap between the line and the curve is convexity.
This is more than a textbook nicety. FINRA’s investor guide notes that a bond’s duration “signals how much the price of your bond investment is likely to fluctuate when there’s an up or down movement in interest rates.” That is true. What FINRA doesn’t say—and what professionals add—is that the signal is a first-order approximation. Pension funds, hedge funds, and primary dealers always carry the second-order term too.
What convexity actually measures
Formally, convexity is the second derivative of price with respect to yield, scaled by price. In plain English: duration captures how the slope of the price-yield curve looks at one point; convexity captures how that slope is changing—the curvature itself.
The combined formula every fixed-income desk uses extends the duration rule with a second-order term:
%ΔPrice ≈ − Modified Duration × Δy + ½ × Convexity × (Δy)2
For a plain-vanilla bond like a U.S. Treasury note, convexity is always positive. Because (Δy)2 is also always positive, the convexity term adds to price gains when yields fall and subtracts from losses when yields rise. That is the bondholder’s best friend: the price-yield curve bows toward them.
Convexity is reported in units of years-squared. The raw number has no intuitive meaning—what matters is the product ½ × C × (Δy)2. For a 10-year Treasury, convexity near 76 contributes about 0.4 percentage points of extra price change on a 100 bp move. For a 30-year Treasury with convexity above 350, the contribution is roughly five times larger.
A worked example: today’s 10-year Treasury
On May 28, 2026, the constant-maturity 10-year Treasury yield was 4.45% per the Federal Reserve’s H.15 release. A newly issued par bond with a 4.45% semi-annual coupon and a 10-year maturity has the following properties (full bond math; no approximations):
- Macaulay duration: 8.18 years — the weighted-average time to receive the cash flows.
- Modified duration: 8.00 years — Macaulay divided by (1 + y/2). This is the price-sensitivity number.
- Convexity: 76.5 years2.
Test it with a 100 bp rate move:
- Yields fall 100 bp to 3.45%. Duration says +8.00%. Duration + convexity says +8.00 + 0.5 × 76.5 × (0.01)2 = +8.38%. Actual full reprice at 3.45% YTM: +8.40%. The convexity term captures essentially all of the missing 0.40 points.
- Yields rise 100 bp to 5.45%. Duration says −8.00%. Duration + convexity says −7.62%. Actual full reprice at 5.45% YTM: −7.63%. Again, convexity nails the difference.
Notice the asymmetry. Drop 100 bp and you make 8.40%. Rise 100 bp and you only lose 7.63%. Same magnitude of yield move, asymmetric outcome—by 0.77 percentage points. That asymmetry is convexity earning its keep.
Convexity scales fast: maturity and shock size both matter
The convexity contribution scales with the square of the yield change. Double Δy and the convexity term quadruples. For tiny rate moves convexity is a footnote; for the kind of 200–300 bp moves we saw across the 2022 inflation shock and the 2026 long-bond selloff, it is the difference between a tolerable drawdown and a margin call.
The same logic applies to long-duration bonds. Convexity grows roughly with the square of modified duration. A 30-year Treasury has roughly twice the modified duration of a 10-year but more than four times the convexity:
| Treasury par bond | YTM | Modified duration (yrs) | Convexity (yrs²) | Convexity gain on a 100 bp rate drop |
|---|---|---|---|---|
| 2-year | 3.99% | 1.90 | 4.6 | +0.02% |
| 5-year | 4.15% | 4.47 | 23.4 | +0.12% |
| 7-year | 4.29% | 5.99 | 42.0 | +0.21% |
| 10-year | 4.45% | 8.00 | 76.5 | +0.38% |
| 20-year | 4.98% | 12.57 | 211.9 | +1.06% |
| 30-year | 4.98% | 15.49 | 353.3 | +1.77% |
Why convexity is asymmetrically valuable in big moves
Re-run the 10-year scenario for ±200 bp instead of ±100 bp:
Duration alone predicts a 16.0% loss on a 200 bp rise and a 16.0% gain on a 200 bp fall—perfectly symmetric. The market doesn’t do that. The actual loss on a 200 bp rise is 14.6%; the actual gain on a 200 bp fall is 17.6%. Convexity widens the gap-to-duration by about 1.5 percentage points in each direction.
That seems modest until you scale it. A primary dealer carrying $5 billion of 10-year notional during a 200 bp shock is looking at roughly $75 million in benefit from positive convexity versus a purely duration-hedged book. For Treasury-curve traders, capturing convexity is a profit center, not an academic footnote.
Negative convexity: when the curve bows the wrong way
Plain-vanilla Treasuries always have positive convexity. Mortgage-backed securities (MBS) and callable corporate bonds usually do not.
Callable bonds. A callable bond can be redeemed by the issuer at a pre-set price, usually at par. When yields fall enough that the bond would otherwise trade at a meaningful premium, the issuer is highly likely to call it and refinance at the lower rate. This caps the bond’s upside near the call price. As yields drop, the price-yield curve flattens and can even bow downward—the opposite of positive convexity. The bondholder gave up that upside in exchange for a higher coupon at issuance; the call option belongs to the issuer.
Mortgage-backed securities. The same problem arrives from a different source. Homeowners who borrowed the underlying mortgages can prepay—refinance their mortgage when rates fall. Each prepayment returns principal to the MBS holder at par, who must then reinvest at the new, lower yields. As rates drop, prepayments accelerate, the MBS’s duration shortens, and the price barely budges above par. As rates rise, prepayments slow, duration lengthens, and the price drops faster than a comparable Treasury. That’s negative convexity working against the holder in both directions—and it’s why MBS hedging is structurally harder than Treasury hedging.
The U.S. agency MBS market is the largest population of negative-convexity instruments in the world—a multi-trillion-dollar block tracked by SIFMA’s U.S. MBS statistics. When mortgage rates move, MBS desks rebalance Treasury hedges in size; the resulting Treasury flows are a well-documented source of curve volatility on big rate days.
Common mistakes
Stress-testing with duration only
For everyday risk reporting on small yield moves, duration alone is fine. For a stress scenario—say, a 200 bp parallel shift in either direction—omitting convexity systematically overstates losses on a long Treasury portfolio. A risk manager reporting a duration-implied 16% drawdown when the actual reprice is 14.6% is not being conservative; they are misallocating capital. The same mistake in reverse—ignoring the convexity gain on rate rallies—underestimates expected returns in a duration-positive book.
Assuming convexity is always your friend
Some hedge fund pitch decks describe “long convexity” as an upside-without-downside trade. For Treasuries the math is real. For MBS, callable corporates, and structured credit, the convexity sign can be wrong—and a generic positive-convexity intuition can produce outsized losses in a rate rally. Always check whether the bond’s convexity is positive or negative before leaning on it.
Confusing modified duration and effective duration
For bonds without embedded options, modified duration and effective duration are nearly identical. For callable and putable bonds and MBS, they can differ substantially. Effective duration uses an option-adjusted spread (OAS) model that re-prices the bond under explicit up- and down-yield scenarios and lets the embedded option behave the way it actually will. It is the right metric for any bond with an embedded option, and it is also the input to effective convexity—the version of convexity that turns negative for callables and MBS.
What to learn next
- Bond duration: the first-order companion to convexity, with the full Macaulay-versus-modified breakdown and the long-bond selloff math (covered in our duration explainer).
- Effective duration and OAS: the option-adjusted versions used for callable bonds and MBS, where modified-duration math breaks down.
- Key-rate duration: how a bond reacts to twists in the yield curve rather than parallel shifts—essential for managing barbells versus bullets.
- The yield curve itself: the shape of yields across maturities and what changes in that shape say about the economy.
- Treasury market mechanics: see the TreasuryDirect explainer on Treasury notes for how the 2-, 3-, 5-, 7-, and 10-year notes that anchor this article are auctioned and priced.
Sources
- Federal Reserve — H.15 Selected Interest Rates (Treasury constant maturity yields, May 28, 2026)
- FINRA — Bonds and Interest Rates (duration as a measure of price sensitivity)
- SEC Office of Investor Education — Bonds (interest rate risk explainer)
- TreasuryDirect — Treasury Notes (instrument specifications)
- SIFMA — U.S. Mortgage-Backed Securities Statistics
- ECMSource — Bond Duration Explained: Why Yields Move Bond Prices
Disclosure: This article was produced with AI assistance and reviewed before publication. It is for informational purposes only and is not investment advice.