Convexity: Concept and Calculation

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title: "Convexity: Concept and Calculation" description: "Master convexity to improve bond price estimates beyond duration. Learn the calculation, interpretation, and practical application for large rate moves." slug: "convexity-concept-and-calculation" category: "Fixed Income" subcategory: "Yield Duration and Convexity" difficulty: "advanced" readingTime: "7 min" author: "Equicurious" lastUpdated: "2025-12-29"

Duration gives you the first approximation. Convexity gives you the correction that prevents costly miscalculation. For a 20-year bond with duration of 12 and convexity of 180, ignoring convexity on a 100 bps rate move creates a 0.9% pricing error—the difference between estimating an 11.1% loss versus the 12% duration-only figure (CFA Institute, 2025). The point is: convexity isn't academic refinement; it's the term that keeps your risk estimates honest when rates move significantly.

What Convexity Actually Measures (Why It Matters)

Duration captures the linear relationship between bond prices and yields—the first derivative of the price function. Convexity captures the curvature—the second derivative. The bond price-yield relationship isn't a straight line; it curves. That curvature is convexity.

For option-free fixed-rate bonds, convexity is always positive. This creates an asymmetric benefit: prices rise more than duration predicts when yields fall, and prices fall less than duration predicts when yields rise. The durable lesson: positive convexity is your friend. You gain more on the upside and lose less on the downside than a linear model suggests.

Source: CFA Institute, 2025. Yield-Based Bond Convexity and Portfolio Properties.

Convexity ranges by bond type:

• Short-term bonds (1-3 years): Low convexity (<20)
• Intermediate bonds (5-7 years): Moderate convexity (20-80)
• Long-term bonds (10-30 years): High convexity (80-200+)
• Zero-coupon bonds: Highest convexity for any given maturity

The Price-Yield Curve Explained

Picture a graph with yield on the x-axis and price on the y-axis. A straight line (duration-only estimate) shows price falling proportionally as yields rise. The actual bond price curve bows outward—it's convex to the origin.

Where duration fails:

• Small yield changes (<25 bps): Duration alone provides accurate estimates
• Moderate yield changes (25-75 bps): Duration error grows noticeable
• Large yield changes (>100 bps): Duration-only estimates can miss by 0.5-1.5%

The test: When your portfolio stress test shows a 200 bps rate shock, are you using duration-only estimates? If so, you're systematically overestimating losses.

The Convexity Formula (And What It Means)

The formal definition measures how the price-yield slope changes as yields change:

Convexity = Sum((t)(t+1)(CFt) / (1+y)^(t+2)) / (PV x (1+y)^2)

Where t = time period, CF = cash flow, y = yield, PV = present value.

The practical formula for using convexity in price estimation:

%ΔP = -ModDur x Δy + 0.5 x Convexity x (Δy)^2

Notice the convexity term is always positive (squared yield change). This is why positive convexity benefits bondholders regardless of rate direction. When yields fall, the -ModDur x Δy term is positive (price up), and the convexity term adds more upside. When yields rise, the duration term is negative (price down), but the convexity term partially offsets the loss.

Worked Example: Convexity Adjustment in Practice

You hold a bond with these characteristics (CFA Institute, 2025):

• Price: $100
• Modified duration: 7.02
• Convexity: 64.93
• Yield change: +45 bps (0.0045)

Step 1: Duration-Only Estimate

%ΔP = -7.02 x 0.0045 = -3.16%

Duration alone predicts a 3.16% price decline.

Step 2: Convexity Adjustment

Convexity adjustment = 0.5 x 64.93 x (0.0045)^2 = 0.5 x 64.93 x 0.00002025 = +0.066%

Step 3: Combined Estimate

Total %ΔP = -3.16% + 0.066% = -3.09%

New estimated price: $100 x (1 - 0.0309) = $96.91

Why this matters: The convexity adjustment reduced your estimated loss from 3.16% to 3.09%—a 7 basis point improvement. On a $10 million position, that's $7,000 less loss than duration-only would suggest. For larger rate moves, the benefit compounds.

Scaling the Convexity Effect by Rate Move Size

The convexity adjustment grows with the square of the yield change. This non-linear scaling is why convexity matters more for stress testing than for daily P&L:

The pattern: At 25 bps, convexity barely registers. At 200 bps, convexity reduces your estimated loss by 1.30 percentage points. The point is: convexity becomes consequential precisely when you need accurate risk estimates most—during rate shocks.

Comparing Portfolios With Same Duration, Different Convexity

Two portfolios can have identical duration but different convexities (CFA Institute, 2025). This creates different return profiles:

Higher convexity portfolio:

• Greater sensitivity to large yield declines (bigger gains)
• Lower sensitivity to large yield increases (smaller losses)
• Typically achieved via barbell strategies (short + long maturities)

Lower convexity portfolio:

• More linear price response
• Smaller outperformance in rallies
• Smaller protection in selloffs
• Typically bullet strategies (concentrated around one maturity)

Barbell vs. Bullet Example:

• Target duration: 5 years
• Bullet portfolio: 100% in 5-year bonds, convexity = 25, yield = 4.00%
• Barbell portfolio: 50% 2-year + 50% 10-year, convexity = 60, yield = 3.85%

Scenario: Parallel shift down 100 bps

• Bullet return: +5.0%
• Barbell return: +5.3% (convexity benefit adds 30 bps)

The durable lesson: Higher convexity is valuable—but it has a cost. Barbell strategies sacrifice yield (3.85% vs. 4.00% in this example) in exchange for convexity. Whether that trade-off makes sense depends on your rate view and volatility expectations.

Detection Signals: You're Likely Miscalculating Convexity If...

• You use duration-only estimates for stress tests with rate shocks of 100+ bps
• Your actual portfolio losses consistently differ from duration-based forecasts by 0.5%+ during volatility
• You duration-match liabilities without checking whether convexity mismatch creates gap risk
• You compare barbell and bullet strategies purely on yield without adjusting for convexity value

Why Convexity Matters for Liability Matching

Pension funds and insurers match asset duration to liability duration. The point is: duration matching alone doesn't immunize against large rate moves if convexity differs (CFA Institute, 2025).

The problem: If your assets have lower convexity than your liabilities, large rate moves in either direction widen the asset-liability gap. You become underfunded even though duration was matched.

The fix: Match both duration AND convexity. Or use key rate duration matching at multiple curve points for precision hedging without convexity constraints.

Common Calculation Mistakes and Fixes

Mistake 1: Using Modified Duration for Callable Bonds

Modified duration assumes cash flows don't change with rates. For callable bonds, falling rates trigger call exercise (shorter duration), capping price gains. This is negative convexity—the price-yield curve bends the wrong way.

Consequence: Overestimate price gains by 20-40% when rates fall; hedges miscalculated.

Fix: Use effective duration and effective convexity for callable bonds and MBS (DWS Research Institute, 2024).

Mistake 2: Forgetting to Square the Yield Change

The convexity term uses (Δy)^2. Using Δy instead of (Δy)^2 produces nonsensical results.

Fix: Convert basis points to decimal (45 bps = 0.0045), then square (0.00002025).

Mistake 3: Applying Annual Convexity to Semiannual Yields

If your modified duration uses semiannual compounding, your convexity should be scaled consistently. Mixing conventions creates errors.

Fix: Use annualized duration and convexity consistently, or adjust convexity by dividing by 4 when using semiannual-yield-based duration.

Implementation Checklist

Essential (apply these first)

• Use convexity adjustment for any rate move >50 bps
• Calculate portfolio convexity as market-value-weighted average of bond convexities
• Verify duration-convexity consistency (both using same yield convention)

High-impact refinements

• Compare estimated vs. actual price changes monthly to calibrate model accuracy
• Include convexity in stress tests for ±100, ±200, ±300 bps parallel shifts
• For liability matching, verify asset convexity meets or exceeds liability convexity

When Convexity Becomes Critical

2022 Fed Hiking Cycle: The 10-year Treasury yield rose +236 bps in 2022, contributing to Bloomberg Aggregate Index losses of -13.01% (Hartford Funds, 2025). Portfolios that relied on duration-only estimates systematically overestimated losses by 0.8-1.2%—cold comfort, but material for performance attribution and rebalancing decisions.

The durable lesson: Convexity doesn't eliminate losses in rising rate environments. But it tells you the truth about how much you actually lost, preventing overcorrection at precisely the wrong moment.

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Related: Modified Duration and Price Sensitivity | Negative Convexity and Mortgage Securities | Barbell vs. Bullet Strategies Under Curve Shifts | Stress Testing Portfolios for Rate Shocks