Bond Convexity Calculator

Frequently Asked Questions

What is bond convexity?

Bond convexity measures how a bond's duration itself changes as interest rates move. Duration gives a linear approximation of price sensitivity, but convexity captures the curvature in the price-yield relationship. Bonds with higher convexity gain more in price when rates fall and lose less when rates rise, compared with a purely duration-based estimate.

Why does convexity matter?

When interest rates move significantly, a duration-only estimate becomes inaccurate because it assumes a straight-line relationship. Convexity corrects for this: for large rate drops, the actual price gain exceeds the duration forecast, and for large rate rises, the actual loss is smaller. Portfolio managers use convexity to improve hedges and compare bonds with similar durations.

Is higher convexity good?

Generally yes - for the same duration, more convex bonds gain more when rates fall and lose less when they rise, a desirable asymmetry.

How is the price change estimated?

ΔP/P ≈ −Duration·Δy + ½·Convexity·(Δy)². This tool reports convexity and the convexity-adjusted price change for a given yield move.

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Estimates only. Not investment advice.

Investment Disclaimer: Estimates only. Not investment advice.

This calculator provides estimates for educational purposes only and is not investment advice. Past performance does not guarantee future results. Consult with a qualified financial advisor before making investment decisions. All investments carry risk, including potential loss of principal.

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How This Calculator Works

Duration alone gives a straight-line estimate of how a bond's price moves with yield, but the true price-yield relationship is curved. Convexity measures that curvature and supplies a second-order correction. The full estimate is ΔP ÷ P ≈ −D × Δy + ½ × C × (Δy)², where D is modified duration, C is convexity, and Δy is the yield change in decimal. This tool discounts every coupon and the final principal at the periodic yield to get price, computes Macaulay duration as the present-value-weighted average time to each cash flow, converts it to modified duration, and derives convexity from Σ t(t+1)·PV ÷ [P·(1+i)²] scaled to annual units. Because convexity is always positive for an option-free bond, it adds value when yields move sharply in either direction - this is the term a plain duration calculator omits.

Worked Numeric Example

A 10-year, 5% semiannual coupon, $1,000 bond yielding 4% prices near $1,081 with a modified duration of roughly 7.8 years and convexity around 75. For a +1% yield rise (Δy = 0.01): the duration effect is −7.8 × 0.01 = −7.80%, and the convexity effect is ½ × 75 × 0.0001 = +0.375%. The net estimated change is about −7.43%, meaningfully less painful than the −7.80% duration-only figure - convexity cushions the loss on the way up and amplifies the gain on the way down.

Real-World Use

Fixed-income desks and portfolio managers immunize liabilities by matching both duration and convexity, because a duration-only hedge drifts as rates move and must be rebalanced. Higher convexity is desirable: between two bonds with equal duration and yield, the more convex one gains more when rates fall and loses less when they rise, so it commands a slightly lower yield as "payment" for that option-like benefit. Mortgage-backed securities exhibit negative convexity from prepayment behavior - borrowers refinance when rates drop, capping the bond's upside while leaving full downside - which is why MBS investors demand extra yield. Barbell portfolios (mixing very short and very long bonds) are deliberately constructed to harvest more convexity than a duration-matched bullet, and risk reports almost always show duration and convexity side by side for exactly the reason this calculator does.

Common Mistakes

Using duration alone for large yield shocks - the linear estimate overstates losses and understates gains.

Forgetting to scale convexity to annual units when cash flows are semiannual (divide by frequency squared).

Applying option-free convexity to callable or mortgage bonds, which can have negative convexity.

Mismatching the yield convention (bond-equivalent vs. effective) between duration and convexity inputs.

Assuming a parallel curve shift - real yield curves twist and steepen, so a single Δy understates the risk of a non-parallel move.

Limitations & Related Tools

The two-term Taylor expansion is still an approximation; the next-order term grows with the cube of the yield change, so for very large moves a full reprice is more accurate than duration plus convexity. It also assumes a parallel yield-curve shift, a single discount yield for all cash flows, and no embedded options - callable, putable, and mortgage securities require option-adjusted (effective) duration and convexity computed by repricing under shifted curves rather than the closed-form approach used here. Pair this with a bond duration calculator (first-order sensitivity), a yield-to-maturity tool, and a yield-to-call calculator for callable issues to assemble a complete fixed-income workup. Outputs are educational estimates, not investment advice; consult a qualified professional before trading bonds.

Frequently Asked Questions

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