π Bond Convexity Calculator
Duration's curve: price sensitivity, corrected.
Modified convexity (annual compounding)
Annual coupons, option-free bonds. Estimates for planning only, not financial advice.
Convexity measures the curvature of the bond price-yield relationship: the correction to duration when yield moves are large. A 5-year bond with a 6 percent annual coupon at a 5 percent yield prices at $1,043.29, with Macaulay duration 4.478 years, modified duration 4.265, and convexity 23.44. If yields jump 2 points, duration alone predicts an 8.53 percent price drop; adding the convexity term gives 8.06 percent, nearly half a point more accurate. For small moves the correction is trivia; for large moves it is the difference between a hedge that works and one that does not.
How to Use This Calculator
Enter the bond's face value, annual coupon rate, yield to maturity, and years to maturity (annual coupons, option-free bonds). Add a rate shock in percentage points to see the duration-only versus duration-plus-convexity price estimates side by side.
Enter face value, coupon rate, yield, and maturity. The calculator prices the bond and derives durations from the cash flows.
Enter a rate shock (try 1 and 3 points) to see how the convexity correction grows with the move size.
Compare the two estimates. The gap is duration's error; for hedging large moves, use the convexity-adjusted number. For yield and price basics, see our bond calculator.
The Formulas
Every formula weights cash flows by time: duration by t, convexity by t(t+1), which is why distant cash flows dominate convexity. The price-change approximation is a Taylor expansion: the linear term (duration) dominates small moves, the quadratic term (convexity) matters as moves grow. Because the quadratic term is always positive for option-free bonds, convexity always helps the holder relative to the duration estimate, in both directions.
Worked Example: 5-Year 6% Bond at 5% Yield
Face: $1,000. Coupon: $60 annually for 5 years plus principal at maturity. Yield: 5%. Price: $60/1.05 + $60/1.05Β² + $60/1.05Β³ + $60/1.05β΄ + $1,060/1.05β΅ = $1,043.29 (a premium bond, since coupon exceeds yield). Macaulay duration: 4.478 years. Modified duration: 4.478/1.05 = 4.265. Convexity: 23.44.
Rate shock of +2 points: duration-only estimate: -4.265 x 0.02 = -8.53%. With convexity: -8.53% + 0.5 x 23.44 x 0.0004 = -8.53% + 0.47% = -8.06%. Rate shock of -2 points: +8.53% + 0.47% = +9.00%. Notice the asymmetry the convexity term creates: up 9.00 percent on the fall, down only 8.06 percent on the rise. That asymmetry is the economic value of convexity, and it grows with the square of the move.
What Drives Convexity
| Factor | Effect on convexity | Why |
|---|---|---|
| Longer maturity | Higher | More time-weighting on cash flows |
| Lower coupon | Higher | More weight on the distant principal |
| Lower yield | Higher | Distant cash flows discounted less |
| Embedded call | Lower / negative | Upside capped when yields fall |
Zero-coupon bonds maximize convexity for their maturity: every dollar arrives at the end. That makes zeros the purest convexity trade, and also the most volatile holdings, since duration is maximized too.
Using Convexity in Practice
Portfolio managers use convexity three ways: hedging (match both duration and convexity so the hedge survives large moves), positioning (overweight convexity before expected volatility, underweight for yield when calm), and relative value (compare the yield give-up for convexity across bonds). The barbell-versus-bullet trade is the classic expression: same duration, different convexity, different performance when rates swing.
For individual investors, the practical takeaway is simpler: duration tells you the risk, convexity tells you how the risk changes, and callable bonds and MBS can have negative convexity that punishes exactly when you need protection. Check for embedded options before trusting any duration number. Our bond calculator handles price and yield, and the bond-equivalent yield calculator converts money-market yields to the comparable scale.
Trading Convexity: Barbells and Bullets
The classic convexity trade is barbell versus bullet. Two portfolios with identical duration, one split between 2-year and 30-year bonds, the other all in 10-years: the barbell has higher convexity and wins when rates move a lot in either direction, while the bullet wins when rates stay calm because it usually carries more yield. The trade is a pure volatility bet expressed in bonds, and its price is the yield give-up.
Mortgage-backed securities are the cautionary tale. Homeowners prepay when rates fall, which caps MBS price gains and creates negative convexity: the security loses the upside it needs most. MBS investors demand extra yield as compensation, and hedgers must constantly adjust because the duration itself moves against them. Whenever someone offers you yield that looks generous, ask what convexity you are selling to get it.
For callable bonds and mortgage-backed securities, quoted convexity can be negative: prices rise less than duration predicts when yields fall, because calls and prepayments cap the upside. Always check whether a bond has embedded options before trusting convexity numbers; option-adjusted measures exist precisely for these instruments. Positive convexity is a feature you pay for through lower yield, and negative convexity is a risk you should be paid to take.
Annual coupons, option-free bonds. Estimates for planning only, not financial advice.
Frequently Asked Questions
What is bond convexity?
Convexity measures the curvature in the bond price-yield relationship: how much duration itself changes as yields move. A 5-year 6% annual bond at 5% yield has convexity of about 23.44. Positive convexity means the price rises more when yields fall than it drops when yields rise, an asymmetry investors prize.
What is the bond convexity formula?
Convexity = [Ξ£ t(t+1) Γ CF_t / (1+y)^(t+2)] / price, summed over all cash flows. Each period's cash flow is weighted by t(t+1), discounted two extra periods, and scaled by price. This calculator implements the annual-compounding version; semiannual bonds use half-periods.
Why does convexity matter?
Because duration is only a linear approximation: it predicts price changes accurately for tiny yield moves and increasingly badly for large ones. Convexity corrects the error: price change β -duration Γ Ξy + Β½ Γ convexity Γ (Ξy)Β². For a 2-point yield shock, the convexity term is material; ignoring it misprices risk.
What is the difference between duration and convexity?
Duration is the first-order sensitivity (slope): how much price moves per yield change. Convexity is the second-order term (curvature): how the slope itself changes. Duration answers "how much"; convexity answers "how wrong is duration when moves are big." You need both for large rate scenarios.
Is higher convexity better?
All else equal, yes: higher convexity means more upside when yields fall and less downside when they rise. But convexity is not free: high-convexity bonds (long maturities, low coupons) usually carry lower yields. Investors pay for convexity through yield give-up, so the question is whether the price is fair.
What is effective vs modified convexity?
Modified convexity (this calculator) comes from the formula and assumes cash flows do not change with yields. Effective convexity uses actual price changes from a pricing model, which handles bonds with embedded options like callables, where cash flows shift as yields move. For option-free bonds they agree closely.
Do zero-coupon bonds have the highest convexity?
For a given maturity and yield, yes: all cash arrives at the end, maximizing the time-weighting that drives convexity. That is why zeros are the convexity vehicle of choice for hedging and speculation on volatility. The trade is no interim cash flow and maximum price volatility.
How does convexity affect bond portfolio management?
Through barbells vs bullets: a barbell (short + long maturities) has higher convexity than a bullet (all intermediate) at the same duration, so it outperforms in large rate moves either direction. Managers buy convexity when they expect volatility and sell it (taking yield) when they expect calm.
Can convexity be negative?
Yes, for callable bonds and mortgage-backed securities: when yields fall, the bond gets called or mortgages prepay, capping the price gain, while yield rises still hurt fully. Negative convexity is the worst asymmetry: limited upside, full downside. It is why MBS investors demand extra yield.
How do I use convexity in a price estimate?
Apply both terms: ΞP/P β -D_mod Γ Ξy + 0.5 Γ C Γ (Ξy)Β². For our example bond (modified duration 4.265, convexity 23.44) with yields up 2 points: -4.265 Γ 0.02 + 0.5 Γ 23.44 Γ 0.0004 = -8.53% + 0.47% = -8.06% predicted. Duration alone would say -8.53%; convexity recovers nearly half a point of accuracy.