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CFA® Level I · Fixed Income · free lesson
Yield-Based Bond Convexity and Portfolio Properties
Duration draws a straight line through a curved price-yield relationship, and the bigger the yield move, the more that shortcut misleads. This module measures the curvature (convexity), combines it with duration to estimate price changes accurately, and extends both measures from single bonds to portfolios, including the assumption about parallel yield-curve shifts that every portfolio duration figure quietly relies on.
What you will learn
1. Calculate and interpret convexity, including approximate convexity and money convexity, and identify the bond and portfolio characteristics that increase it.
- ApproxCon = (PV- + PV+ - 2 x PV0) / ((ΔYield)^2 x PV0): squared yield change, no factor of 2.
- Option-free bonds have positive convexity: gains from falling yields exceed losses from equal rises.
- Money convexity = annual convexity x full price; currency adjustment = 0.5 x money convexity x (ΔYield)^2.
- Convexity rises with maturity, falls with coupon and yield, and rises with cash-flow dispersion.
- For the same duration, a barbell is more convex than a bullet.
2. Estimate the percentage and currency change in a bond's full price for a given yield change by combining modified duration with the convexity adjustment.
- %ΔPV ≈ -AnnModDur x ΔYield + 0.5 x AnnConvexity x (ΔYield)^2.
- For option-free bonds the convexity adjustment is positive for both rises and falls in yield.
- The adjustment grows with the square of the yield change: negligible for small moves, material for large ones.
- Currency version: -MoneyDur x ΔYield + 0.5 x MoneyCon x (ΔYield)^2.
- Check the convexity scaling convention before plugging a quoted figure into the formula.
3. Calculate portfolio duration and convexity as market-value-weighted averages and explain the limitations of these portfolio measures.
- Practical portfolio duration = market-value-weighted average of bond durations.
- Portfolio convexity is weighted the same way; money durations add across positions.
- Weights use market value (full price x par), never par alone.
- The weighted average assumes a parallel shift: all yields change by the same amount.
- Equal-duration portfolios (bullet versus barbell) can perform differently when the curve twists.
Module at a glance
- Convexity measures the curvature of the price-yield relationship; option-free bonds have positive convexity.
- ApproxCon = (PV- + PV+ - 2 x PV0) / ((ΔYield)^2 x PV0).
- %ΔPV ≈ -AnnModDur x ΔYield + 0.5 x AnnConvexity x (ΔYield)^2; the convexity term is added in both directions.
- Money convexity = convexity x market value; the currency adjustment is 0.5 x money convexity x (ΔYield)^2.
- Convexity rises with maturity, falls with coupon and yield, and rises with cash-flow dispersion: barbells beat bullets.
- Portfolio duration and convexity are market-value-weighted averages of the individual bonds' measures.
- Portfolio duration assumes a parallel shift; curve twists make equal-duration portfolios behave differently.
This free module has 3 lessons, 14 practice questions with full explanations.