Learning Module 12
Yield-Based Bond Convexity and Portfolio Properties
Key Outcomes Summary & Practice Problems
What you must be able to do
Curriculum Year: 2026
Calculate and interpret convexity and describe the convexity adjustment — the second‑order (non‑linear) effect of yield changes on bond price for option‑free fixed‑rate bonds.
Calculate the percentage price change of a bond for a specified change in yield, given the bond's duration and convexity — combining the linear duration estimate with the convexity adjustment.
Calculate portfolio duration and convexity and explain the limitations of these measures — weighted‑average method and the assumption of a parallel shift in the yield curve.
1 · Bond Convexity and Convexity Adjustment
Convexity: A complementary risk metric that measures the second‑order (non‑linear) effect of yield changes on price for an option‑free fixed‑rate bond.
Positive convexity: For option‑free bonds, convexity is always positive — price increases for yield decreases are greaterthan price decreases for yield increases of the same magnitude.
Convexity adjustment: The term added to the linear duration estimate:
½ × AnnConvexity × (ΔYield)²Annualized convexity: Calculated from the sum of convexity contributions divided by periods per year squared.
Approximate convexity:
ApproxCon = (PV− + PV+ − 2PV0) / [(ΔYield)² × PV0]Money convexity:
MoneyCon = AnnConvexity × PVFullFactors increasing convexity: Longer maturity, lower coupon, lower yield, greater dispersion of cash flows.
%ΔPV = −AnnModDur × ΔYield + ½ × AnnConvexity × (ΔYield)²
ApproxCon = (PV− + PV+ − 2PV0) / [(ΔYield)² × PV0]
MoneyCon = AnnConvexity × PVFull
2 · Bond Risk and Return Using Duration and Convexity
Duration alone is a linear approximation of the price‑yield relationship. For small yield changes, it's accurate; for larger changes, convexity improves the estimate.
Combined estimate:
%ΔPV ≈ −AnnModDur × ΔYield + ½ × AnnConvexity × (ΔYield)²Currency change:
ΔPV ≈ −MoneyDur × ΔYield + ½ × MoneyCon × (ΔYield)²Convexity is valuable to investors: It reduces downside risk and increases upside potential. Investors pay for convexity (higher price, lower yield).
Duration estimates a price decline. Convexity adjustment reduces the decline (price falls less than duration suggests).
Duration estimates a price increase. Convexity adjustment adds to the increase (price rises more than duration suggests).
3 · Portfolio Duration and Convexity
Weighted‑average method: Portfolio duration = Σ (weight × bond duration). Portfolio convexity = Σ (weight × bond convexity).
Weight: Market value of the bond position / total portfolio market value.
Limitations: Weighted‑average duration and convexity implicitly assume a parallel shift in the yield curve (all yields change by the same amount in the same direction). In reality, yield curve shifts are rarely parallel (steepening, flattening, twisting).
Alternative (theoretically correct): Use the weighted average of time to receipt of aggregate cash flows — but this is difficult in practice.
Weighted‑average method is commonly used by portfolio managers despite its limitations.
Portfolio Duration = Σ (wᵢ × Durᵢ)
Portfolio Convexity = Σ (wᵢ × Conᵢ)
wᵢ = Market Valueᵢ / Total Market Value
Easy to calculate and apply; commonly used by practitioners.
Assumes parallel shift in the yield curve — rare in practice.
Weighted average of aggregate cash flow timings — but difficult to implement.
4 · Quick Reference — Convexity Formulas
Measure | Formula | Description |
|---|---|---|
Annual Convexity |
| Second‑order price sensitivity |
Approx Convexity |
| Estimation using price changes |
Money Convexity |
| Convexity in currency units |
% Price Change |
| Duration + convexity adjustment |
Portfolio Duration |
| Weighted average of individual durations |
Portfolio Convexity |
| Weighted average of individual convexities |