How this instrument works
Duration approximates a bond's price-yield relationship with a straight line tangent to today's price — useful for small yield moves, but the real relationship is a curve, and convexity is the number that measures how much that curve bows away from the line duration draws. For an ordinary fixed-coupon bond the true curve sits above the straight-line estimate on both sides, which is why duration alone understates how much price rises when yields fall and overstates how much it falls when yields rise.
The people who reach for a convexity figure are usually running a bond portfolio or a pension fund's fixed-income book, not pricing a single retail purchase. Duration and convexity together feed the standard two-term estimate of price change for a yield shift, and skipping the second term misprices anything larger than a small move. Convexity also separates bonds that look identical on duration alone: given two bonds with the same duration, the one with higher convexity gains more when yields drop and loses less when yields rise, which is why it can trade at a slightly lower yield — investors pay for that asymmetry.
This instrument computes convexity numerically rather than from a textbook second-derivative formula: it reprices the bond at the yield shifted up by the amount in Yield shift for numerical convexity (decimal), reprices it shifted down by the same amount, and combines those two prices with the price at today's yield. That bump-and-reprice method works for any bond a pricing formula can value, but the plain-coupon formula behind this calculator has no call feature or prepayment option built in, so it will only ever return a positive number — real callable bonds and mortgage-backed pools can turn negative once the issuer's option to redeem early starts capping the price gain.
- Enter Face value, $ and Annual coupon rate, % — together they set the dollar coupon this bond pays each year.
- Set Yield to maturity, % to the market's current required return and Years to maturity to the years remaining.
- Leave Yield shift for numerical convexity (decimal) at 0.0001, one basis point, unless you have a reason to widen or narrow the bump.
- Read Bond price at current yield first — it is the baseline the up-and-down repricing compares against.
- Read Convexity — the curvature measure that corrects the straight-line estimate duration alone would give.
Worked example — the 5%-coupon, 10-year bond priced at 6%
Set Face value, $ to 1000, Annual coupon rate, % to 5, Yield to maturity, % to 6, Years to maturity to 10, and Yield shift for numerical convexity (decimal) to 0.0001, one basis point. Bond price at current yield reads $926.40: at a 6% required return, the 5% coupon bond trades below its $1,000 face value. Convexity reads 72.5693, built from three prices — at 6.01%, 6%, and 5.99% — combined through the bump formula above.
That positive number is the point: because Δy is squared in the denominator, the curvature term comes out positive whichever direction the yield moves, so this bond's price gains more from a one-point yield drop than it loses from a one-point yield rise of the same size. Combined with duration in the standard two-term estimate, ΔP/P ≈ −D·Δy + 0.5·Convexity·Δy², this convexity figure is the correction that keeps a large yield-swing estimate from running too pessimistic on the downside and too stingy on the upside.
Questions
What does convexity measure that duration does not?
Duration measures the slope of the price-yield line at today's yield — a straight-line estimate. Convexity measures how much the real price-yield relationship curves away from that line, the second-order term duration leaves out. Adding convexity to a duration estimate corrects for the fact that price actually rises faster than duration predicts when yields fall, and falls slower than duration predicts when yields rise.
Why is convexity for an ordinary bond always positive?
Because a plain fixed-coupon bond's price-yield curve bows upward at every point — nothing caps how high the price can climb as yields fall. The formula behind this calculator prices only that plain coupon structure, so Convexity here always comes out positive. Bonds with an embedded option, like a callable bond or a mortgage-backed pool, can turn negative once the option starts limiting the price's upside.
Why does this calculator bump the yield up and down instead of using a closed-form formula?
A textbook second-derivative formula exists for a plain coupon bond, but it only works because that bond's price has a clean algebraic shape. Repricing the bond at yield plus and minus a small shift and combining the three prices gives the same curvature number without needing that closed form, and the same bump-and-reprice method still works for bonds — callable issues, mortgage pools — whose price has no tidy formula at all.
What yield shift, Δy, should I use?
One basis point, 0.0001, is the market convention and this calculator's default — small enough that the curvature estimate is accurate, large enough that it does not get lost in floating-point rounding. Shifting by a much larger amount, like 0.01, starts to pick up higher-order curvature the two-price bump was not designed to capture; shifting by a much smaller amount can make the result noisy.
Why would I choose a bond with higher convexity over one with equal duration?
Because convexity is asymmetric in the bondholder's favor: given two bonds with matching duration, the one with higher convexity gains more when yields fall and loses less when yields rise. Bond portfolio managers building a fixed-income book weigh that asymmetry directly, and a bond with meaningfully higher convexity can be worth accepting a slightly lower yield to hold, since the extra curvature is itself a form of protection against large rate moves.
Does convexity tell me whether to buy this bond?
No — it describes how the bond's price would respond to yield changes of different sizes, nothing more. Whether that response fits a portfolio depends on liabilities, time horizon, and risk tolerance that this arithmetic cannot see; convexity is one input a fixed-income decision uses, not a verdict on the bond itself.
References
- SEC Investor.gov — Bonds or fixed income products
- Federal Reserve — Selected interest rates (H.15 release)
Read this first: This instrument shows arithmetic, not advice. Real offers add fees, taxes and terms that vary by lender and place — verify the figures against your actual paperwork before deciding anything.