Bond Convexity Calculator
From the price at an initial yield and at yields up/down, estimate the bond's convexity — the curvature of the price-yield curve.
Input Data
Results
At a glance:Convexity measures the curvature of the price-yield relation. Estimate: convexity = (P_down + P_up - 2 x P0) / (P0 x (Δy)^2), where P0 is the price at the initial yield and P_down/P_up are prices if the yield moves down/up by Δy. Higher convexity is good for holders: price rises more when yields fall and falls less when yields rise. It refines the duration (straight-line) estimate for small yield changes. WARNING: It is an approximation for small moves; it complements, not replaces, duration. Education only, not advice.
Formula
Convexity = Σ [t(t+1) × CF_t / (1+y)^(t+2)] / (Price × (1+y)^2).
Price change ≈ −Duration × Δy + ½ × Convexity × (Δy)^2.
$$\text{Convexity} = \dfrac{P_- + P_+ - 2P_0}{P_0 \times (\Delta y)^2}$$$$\dfrac{\Delta P}{P} \approx -D \times \Delta y + \tfrac{1}{2}\,\text{Convexity} \times (\Delta y)^2$$How to Use
- Enter the bond price at the initial yield (P0).
- Enter the estimated prices if the yield moves down and up by the same amount.
- Enter the size of the yield change (Δy).
- View the convexity.
FAQ
What does convexity mean simply?
Duration estimates price change as a straight line; convexity is the curvature. Positive convexity means the actual price falls less when yields rise and rises more when yields fall than duration alone predicts — good for holders. Most plain bonds have positive convexity.
How does convexity correct the duration estimate?
Duration alone over-estimates the fall when yields rise and under-estimates the rise when yields fall. Adding the convexity term (≈ 0.5 x convexity x (Δy)^2) corrects this. Example: duration 6.1, convexity 120, yield +1% → change ≈ -6.1 x 0.01 + 0.5 x 120 x 0.0001 = -5.5% (vs -6.1% by duration only); yield -1% → ≈ +6.7%.
Why do some bonds have negative convexity?
Bonds with embedded options, such as callable bonds (issuer can redeem early when rates fall) or MBS/prepayable bonds, can show negative convexity in part of the range — price gains are capped when yields fall. So convexity sign matters; check for call features.
Does convexity only matter for big yield moves?
The correction is small for tiny moves and grows with (Δy)^2. For a 0.1% move it is negligible; for a 1-2% move it is material. For highly curved bonds (long maturities, low coupons), even moderate moves need the convexity adjustment.
How does this relate to Hong Kong bonds?
Hong Kong dollar bonds (Exchange Fund Notes, bank issues, blue-chip bonds) are priced off HIBOR/US rates; longer maturities and lower coupons have higher convexity and bigger price swings. Use duration + convexity together to gauge interest-rate risk, and consult the HKMA and IFEC materials. This tool is educational, not advice.
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References
Content review: Calculatorism Finance Team. Results are for reference only; please refer to the relevant authorities for the official figures.