Bond Convexity — Smart Financial Analysis
Calculate bond convexity, modified duration, and price sensitivity to yield changes. The duration correction factor for fixed income.
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Bond convexity measures the curvature of the price-yield relationship — the second-order sensitivity that duration misses. Convexity = (1/P) × Σ [t(t+1) × CF_t / (1+y)^(t+2)]. Duration is the first derivative (linear slope); convexity is the second derivative (curvature). Positive convexity (plain vanilla bonds): prices rise more when rates fall than they drop when rates rise — always beneficial.
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Why: Bond convexity measures the curvature of the price-yield relationship — the second-order sensitivity that duration misses. Duration gives a linear approximation; convexity captu...
How: Enter Face Value ($), Coupon Rate (%), Yield to Maturity (%) to get instant results. Try the preset examples to see how different scenarios affect the outcome, then adjust to match your situation.
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📋 Examples — Click to Load
Bond Parameters
Price-Yield Curve: Convexity Curvature
Duration vs Convexity Comparison
Price Change: Duration Only vs With Convexity
Convexity by Maturity
For educational purposes only — not financial advice. Consult a qualified advisor before making decisions.
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Bond Convexity — The Duration Correction
Bond convexity measures the curvature of the price-yield relationship — it captures what duration misses. Duration gives a linear approximation of price change; convexity adds the second-order correction. Price Change ≈ -Duration × Δy + ½ × Convexity × (Δy)². For a 30-year bond with duration 18.5 and convexity 420: a 1% rate increase → Duration says -18.5%, Convexity correction +2.1% → actual ≈ -16.4%. Positive convexity means bond prices rise MORE when rates fall and fall LESS when rates rise — it's always beneficial. MBS have negative convexity due to prepayment risk — a unique and dangerous characteristic.
Key Takeaways
- Convexity measures the curvature of the price-yield relationship
- Higher convexity = price rises MORE when rates fall and falls LESS when rates rise (good!)
- Price change ≈ -Duration × Δy + 0.5 × Convexity × (Δy)²
- Convexity matters most for large rate movements (100+ bps)
Convexity Formula
Convexity = (1/P) × Σ [t(t+1) × CF_t / (1+y)^(t+2)]. It measures the second derivative of price with respect to yield. The convexity adjustment to price change is ½ × Convexity × (Δy)² × Price.
Convexity vs Duration
Duration is the first derivative (linear slope); convexity is the second derivative (curvature). Duration alone underestimates price gains when rates fall and overestimates price losses when rates rise. For large rate moves, the convexity correction can be 2-5% of the total price change.
Positive vs Negative Convexity
Plain vanilla bonds have positive convexity — always beneficial. Callable bonds exhibit negative convexity at low yields (price ceiling from call option). MBS show negative convexity due to prepayment risk when rates fall — homeowners refinance, cutting your cash flows short.
Convexity and Interest Rate Risk
Higher convexity reduces downside when rates rise and amplifies upside when rates fall. Zero-coupon and long-maturity bonds have the highest convexity. Barbell portfolios (short + long) have higher convexity than bullet portfolios at the same duration.
Convexity Adjustment
The convexity adjustment is ½ × Convexity × (Δy)² × Price. For a 30-year bond with convexity 420 and 1% rate increase: Duration says -18.5%, Convexity adds +2.1% correction → actual ≈ -16.4%. The adjustment is always positive for plain bonds.
Sources
- CFA Institute — Fixed income and convexity methodology
- Bloomberg Fixed Income — Bond market data and analytics
- PIMCO — MBS and negative convexity
- Fabozzi Fixed Income — Bond valuation and convexity
⚠️ Disclaimer: This calculator provides convexity estimates for educational purposes. Actual bond prices depend on market conditions, liquidity, and issuer-specific factors. Not financial advice. Consult a professional for investment decisions.
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