Library/CFA (Chartered Financial Analyst)/FIXED INCOME: CFA® Program Curriculum, 2026 • Level I • Volume 6/Learning Module 12 Yield-Based Bond Convexity and Portfolio Properties

Learning Module 12 Yield-Based Bond Convexity and Portfolio Properties

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Definition and Rationale for Convexity5 min

Convexity complements duration by capturing the second-order, nonlinear relationship between bond price and yield. For an option-free fixed-rate bond the true price-yield relationship is curved (convex), and duration alone (a linear approximation) systematically misestimates price change for larger changes in yield. The convexity adjustment increases estimated price gains when yields fall and reduces estimated price losses when yields rise. Annualized convexity (AnnConvexity) can be computed from cash-flow present values using a spreadsheet: for each cash flow, compute weight = PV of cash flow / PV, then sum (time)(time+1)(weight)(1+periodic YTM)^(−periods per year) across all cash flows and divide by (periods per year)^2 to obtain AnnConvexity. An accurate approximation for annual convexity uses PV+ and PV− (full prices after raising and lowering yield by the same small delta): ApproxCon = [PV− + PV+ − 2PV0] / [(ΔYield)^2 PV0]. Convexity is always positive for option-free fixed-rate bonds and increases with longer maturity, lower coupon, lower yield, and greater dispersion of cash flows. Money convexity (MoneyCon) expresses AnnConvexity in currency units: MoneyCon = AnnConvexity PVFull (or market value of position).

Key Points

Convexity is the second-order (nonlinear) price–yield effect and complements duration.
AnnConvexity can be computed from cash-flow present values or approximated via PV+ and PV−.
Convexity is positive for option-free fixed-rate bonds and rises with longer maturity, lower coupon, and lower yield.
Money convexity converts AnnConvexity into currency units via multiplication by PV.

Formulas and Practical Calculation5 min

Key formulas: ModDur = MacDur / (1 + r per period); AnnModDur ≈ [PV− − PV+] / [2 ΔYield PV0]; ApproxCon = [PV− + PV+ − 2PV0] / [(ΔYield)^2 PV0]; %ΔPVFull ≈ (−AnnModDur ΔYield) + 0.5 AnnConvexity (ΔYield)^2; MoneyCon = AnnConvexity PVFull; ΔPVFull ≈ −MoneyDur ΔYield + 0.5 MoneyCon * (ΔYield)^2. In practice, approximate modified duration and convexity are often calculated by computing PV+ and PV− for a small change in yield (e.g., 5 bps or 1 bp) and applying the formulas above. Ensure units are consistent: if convexity is reported as a small decimal, rescale to the same order of magnitude as duration squared when using percent or decimal yield changes. For option-containing securities, PV+ and PV− should be derived from a model that captures option exercise behavior; for option-free bonds the Excel PRICE function may be used. Choose ΔYield small enough for accuracy but mindful that for option-laden instruments scenario size and model inputs matter.

Key Points

Use PV+, PV− and PV0 to approximate AnnModDur and ApproxCon.
Combine AnnModDur and AnnConvexity to estimate percentage price change including the convexity adjustment.
MoneyCon = AnnConvexity * PVFull; use money measures for currency change estimates.
Maintain unit consistency and select appropriate ΔYield for approximations.

Application: Single Bonds and Portfolios6 min

Examples show convexity materially improves estimates for large yield moves and long maturities: duration-only estimates can be off by many basis points, whereas adding convexity yields estimates much closer to exact PRICE-derived results. Portfolio duration and convexity can be computed as market-value-weighted averages of individual bond durations and convexities: portfolio AnnModDur = sum(w_i AnnModDur_i); portfolio AnnConvexity = sum(w_i AnnConvexity_i). This weighted-average method is commonly used but implicitly assumes parallel shifts in the yield curve; the theoretically correct approach would sum cash flows across the portfolio and compute duration and convexity from the aggregate cash flows, which is rarely practical. Money convexity is also aggregated by weighting individual MoneyCon values (or by computing portfolio AnnConvexity * portfolio market value). Convexity has beneficial attributes for investors: for two bonds with identical duration, the bond with greater convexity will gain more for a decline in yields and lose less for an equal increase, making higher convexity valuable (though typically priced into yields). For bonds with embedded options convexity behavior may be complex (negative convexity is possible), but for option-free fixed-rate bonds convexity is positive. Practical considerations: choose an appropriate ΔYield for approximations (small changes for yield-derived convexity; for option-containing instruments the model choice and scenario size matter), ensure convexity units are rescaled consistently with the yield change (e.g., convert reported convexity to same order of magnitude as duration squared if necessary), and be aware of limitations of portfolio-weighted measures when the yield curve moves non-parallel.

Key Points

Adding convexity to duration estimates substantially improves price-change accuracy, especially for large yield moves and long maturities.
Portfolio duration and convexity are usually computed as market-value-weighted averages but assume parallel yield curve shifts.
Higher convexity is beneficial: more upside when yields fall and less downside when yields rise (for option-free bonds).
Be cautious with convexity in optioned instruments and be aware of unit scaling and scenario selection.

Questions

Question 1

Which statement best describes bond convexity for an option-free fixed-rate bond?

Learning Module 12 Yield-Based Bond Convexity and Portfolio Properties

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