# Convexity > Convexity measures how a bond's duration changes as yields move, refining price estimates for big moves. Learn the formula, positive and negative convexity and uses. Source: https://learn.tradelabsai.com/bonds-credit/convexity/ Track: Bonds, Rates and Credit · Level: Advanced · Updated: 2026-10-03 Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice. Cite as: TradeLabs Learn, "Convexity", https://learn.tradelabsai.com/bonds-credit/convexity/ Duration gives a straight line estimate of how a bond's price changes when yields move. But the true relationship between price and yield is curved. Convexity measures that curvature. For ordinary bonds, the curve means prices rise more when yields fall than they drop when yields rise by the same amount, which is good for investors. Convexity matters most for large yield moves, long dated bonds and securities with embedded options such as mortgages. ## The price yield curve *Figure: Actual bond prices sit above the duration line on both sides: that gap is convexity.* ## The formula Adding convexity to the duration estimate: ``` % price change ≈ - modified duration × Δy + ½ × convexity × (Δy)² ``` Because (Δy)² is always positive, positive convexity adds to the price whether yields rise or fall. **Example: Duration plus convexity** A 30 year Treasury bond yields 4% with a modified duration of about 17.4 and convexity of about 420. Yields rise by 1 percentage point (Δy = 0.01): - Duration estimate: minus 17.4 × 0.01 = minus 17.4%. - Convexity adjustment: ½ × 420 × 0.0001 = +2.1%. - Total: about minus 15.3%. Yields fall by 1 point: - Duration: +17.4%. Convexity: +2.1%. Total: about +19.5%. Repricing the bond exactly gives minus 15.5% and +19.7%, very close to the estimates. The bond gains about 20% on a fall in yields but loses only about 15% on an equal rise. For a 0.10 point move, the convexity adjustment is only 0.02%, which is why convexity matters mainly for large moves. ## What affects convexity | Factor | Effect on convexity | |---|---| | Longer maturity | Higher convexity | | Lower coupon | Higher convexity | | Lower yield | Higher convexity | | Spread out cash flows (barbell) | Higher convexity than concentrated cash flows (bullet) with the same duration | | Embedded call options | Lower, possibly negative | ## Negative convexity Callable bonds and mortgage backed securities can have negative convexity. When rates fall, borrowers repay early (homeowners refinance, companies call bonds), so the price rises less than a normal bond. When rates rise, prepayments slow, extending duration just as prices fall. **Example: Mortgage securities in a falling rate market** A mortgage backed security with duration 5 might gain only 3% when mortgage rates fall 1 point, because refinancing returns principal at par, but lose 6% when rates rise 1 point as prepayments slow and duration lengthens. Investors who hold such securities, and banks that hedge them, often have to buy or sell duration in a hurry, which can amplify bond market moves. ## Convexity in portfolios - **Barbell vs bullet:** a barbell portfolio (short and long bonds) has more convexity than a bullet (intermediate bonds) of the same duration. Convexity is valuable when large rate moves are expected, but it typically comes at the cost of slightly lower yield. See [Yield Curve Trades: Steepeners, Flatteners and Butterflies](https://learn.tradelabsai.com/bonds-credit/yield-curve-trades/). - **Hedging:** swaps and futures hedge duration, while options on rates hedge convexity. - **Convexity has a price:** in efficient markets, more convexity usually means lower yield, because investors pay for its protection. ## Convexity and options Positive convexity in bonds is closely related to positive gamma in options: both benefit from large moves. Negative convexity is like being short options. See [Gamma](https://learn.tradelabsai.com/options/gamma/). ## Calculating convexity in practice Convexity can be computed from the bond's cash flows, but traders usually estimate it numerically by repricing the bond for a small rise and a small fall in yield: ``` convexity ≈ (P_down + P_up - 2 × P_0) / (P_0 × (Δy)²) ``` where P_down and P_up are prices after yields fall and rise by Δy. Spreadsheet functions and bond calculators do this automatically. See [Bond Price, Duration and DV01 Calculator](https://learn.tradelabsai.com/tools/bond-calculator/). ## Frequently asked questions ### What is bond convexity? A measure of the curvature in the relationship between bond prices and yields, showing how duration changes as yields move. ### Why is positive convexity good? Because prices rise more when yields fall than they decline when yields rise by the same amount. ### What is negative convexity? A situation, common in callable bonds and mortgage securities, where price gains are limited when yields fall and losses grow when yields rise. Next, learn the dollar measure of rate risk in [DV01](https://learn.tradelabsai.com/bonds-credit/dv01/). ## Continue learning - Next lesson: [DV01](https://learn.tradelabsai.com/bonds-credit/dv01/) - Previous lesson: [Duration](https://learn.tradelabsai.com/bonds-credit/duration/) - Related: [Duration](https://learn.tradelabsai.com/bonds-credit/duration/): Duration measures how sensitive a bond's price is to interest rate changes. Learn Macaulay, modified and effective duration, how to calculate them and their uses. - Related: [DV01](https://learn.tradelabsai.com/bonds-credit/dv01/): DV01 measures how many dollars a bond or portfolio gains or loses for a one basis point change in yield. Learn the formula, hedge ratios and how traders use it. - Related: [Gamma](https://learn.tradelabsai.com/options/gamma/): Gamma measures how much an option's delta changes for a $1 move in the underlying. Learn why gamma peaks at the money near expiry and how it drives risk. - Related: [Yield to Maturity](https://learn.tradelabsai.com/bonds-credit/yield-to-maturity/): Yield to maturity is the total return of a bond held to maturity. Learn how YTM is calculated, current yield, yield to call and worst, and YTM's assumptions. - Related: [Interest Rate Swaps](https://learn.tradelabsai.com/bonds-credit/interest-rate-swaps/): An interest rate swap exchanges fixed interest payments for floating ones on a notional amount. Learn how swaps work, SOFR, swap rates, valuation, uses and risks.