Bond Convexity Calculator
Enter your bond parameters and this calculator instantly finds the bond price, Macaulay duration, modified duration, convexity, dollar value of a basis point (DV01), and the estimated price change for a chosen yield shift. Results update as you type, and a price-vs-yield chart shows the curvature that convexity captures beyond the linear duration estimate.
What is bond convexity?
Convexity measures the curvature of the relationship between a bond's price and its yield. Duration gives a linear first-order approximation of how much a bond's price changes when yields move, but that approximation becomes less accurate as the yield shift grows larger. Convexity captures the second-order (curvature) effect, improving price-change estimates for larger moves. Mathematically, convexity is the second derivative of the bond's price with respect to yield, divided by price. A bond with higher convexity gains more in price when yields fall than it loses when yields rise by the same amount, all else equal, which is why investors often prize convexity in volatile rate environments.
How this calculator works
Enter the face value, annual coupon rate, yield to maturity, years to maturity, and coupon frequency. The calculator discounts each future cash flow at the periodic YTM to find the bond price, then accumulates the weighted and double-weighted present values needed for Macaulay duration and convexity. Modified duration is derived from Macaulay duration by dividing by one plus the periodic yield. DV01 (dollar value of a basis point) is the modified duration times price times 0.0001. The price-change estimate uses the standard second-order Taylor expansion: change = price times (-modDuration times dy + 0.5 times convexity times dy squared), where dy is the yield shift converted from basis points to a decimal. The price-vs-yield chart plots the exact price curve, the duration-only tangent line, and the duration-plus-convexity parabola so you can see how well each approximation tracks the true price.
Duration vs. convexity - when does each matter?
For small yield moves of up to 25 basis points, modified duration alone is usually accurate enough for risk management. For larger moves of 50 basis points or more, the convexity adjustment becomes meaningful, particularly for long-duration bonds. Convexity also determines how symmetric a bond's price response is: a bond with high positive convexity benefits from rate volatility because its price rises more than it falls for equal-sized moves. Portfolio managers who expect higher rate volatility therefore prefer high-convexity bonds, even if they must accept a slightly lower yield. Negative-convexity instruments like callable bonds or mortgage-backed securities behave the opposite way: price appreciation is capped near the call price, so their effective duration shortens as yields fall.
The price-change formula
The combined duration-and-convexity price-change formula is: percent price change = -ModDuration * dy + 0.5 * Convexity * dy squared. Adding the convexity term corrects the over-stated loss (and under-stated gain) from the linear duration model. For example, if a bond has a modified duration of 8 and a convexity of 80, and yields rise by 1% (dy = 0.01), the duration-only estimate is -8%, while the convexity adjustment adds 0.5 times 80 times 0.0001 = +0.4%, giving a better estimate of -7.6%. This matters because for a bond priced at $1,000, the difference is $4 per $1,000 face value. For institutional portfolios holding millions of notional, the convexity correction is critical for accurate hedging.
Convexity and duration characteristics by bond type
| Bond type | Typical convexity | Duration range | Notes |
|---|---|---|---|
| Short-term government (2 yr) | 0.05 - 0.10 | 1.8 - 2.0 yrs | Very low interest rate risk |
| Medium-term government (10 yr) | 0.70 - 1.20 | 7.5 - 9.0 yrs | Benchmark duration |
| Long-term government (30 yr) | 4.0 - 7.0 | 15 - 20 yrs | Highest convexity benefit |
| Investment-grade corporate (10 yr) | 0.60 - 1.10 | 7.0 - 8.5 yrs | Credit spread adds yield |
| Zero-coupon bond (10 yr) | 1.10 - 1.30 | 10.0 yrs | Duration = maturity |
| Callable corporate bond | Negative to 0.5 | 3 - 6 yrs | Negative convexity near call price |
| Mortgage-backed security | Negative to 0.3 | 2 - 7 yrs | Prepayment risk causes negative convexity |
General ranges based on typical market conditions. Callable and mortgage-backed bonds can exhibit negative convexity.
Frequently asked questions
What is a good convexity for a bond?
Higher convexity is generally better for the investor because it means the bond's price will rise more than duration predicts when yields fall and fall less than duration predicts when yields rise. Typical investment-grade 10-year bonds have convexities in the range of 0.6 to 1.2. Long-duration zero-coupon bonds can have convexities above 1.0, while 30-year Treasuries can exceed 4.0. Negative convexity, as seen in callable bonds and some mortgage-backed securities, is undesirable from a pure interest rate risk perspective.
What is the difference between Macaulay duration and modified duration?
Macaulay duration is the weighted average time (in years) to receive a bond's cash flows, weighted by each cash flow's present value. Modified duration equals Macaulay duration divided by one plus the periodic yield. Modified duration is directly useful: it tells you the approximate percentage price change for a 1% change in yield. For example, a modified duration of 7 means a bond's price will fall about 7% if yields rise by 1%.
What is DV01 and how is it used?
DV01 (dollar value of a basis point, also called PVBP) is the change in a bond's dollar price for a 1 basis point (0.01%) rise in yield. It equals modified duration times price times 0.0001. Traders use DV01 to size hedges and measure interest rate risk in dollar terms. For example, a DV01 of $0.75 per $1,000 face value means a 10 bps yield rise costs $7.50 per bond in price terms.
Does convexity always improve a price estimate?
For option-free (non-callable, non-putable) bonds, convexity is always positive and always improves the price-change estimate for large yield moves in either direction. However, callable bonds can have negative convexity because the issuer may call (redeem) them early when yields fall, capping price appreciation. For callable bonds, effective convexity, which requires an option-adjusted model, is more appropriate than the cash-flow convexity computed here.
Why does convexity increase with lower coupons and longer maturities?
Lower coupons mean more of the bond's value comes from the distant face-value payment. That deferred cash flow is more sensitive to yield changes, stretching duration and convexity. Similarly, longer maturities push cash flows further into the future, amplifying curvature. Zero-coupon bonds have the highest convexity relative to their duration because all cash flow arrives at a single point. Two bonds with identical duration but different coupon structures will have different convexities, with the lower-coupon bond having greater convexity.
How accurate is the convexity-adjusted price estimate?
The second-order Taylor expansion using duration and convexity is quite accurate for yield shifts up to about 200 basis points. Beyond that, higher-order terms become non-trivial and the estimate diverges from the true price. The chart in this calculator shows the exact price curve alongside the duration estimate and the duration-plus-convexity parabola so you can visually assess accuracy for any yield range.