How do duration and convexity work together to estimate bond price changes, and when does duration alone fail?

Duration and convexity are the two fundamental tools for estimating how bond prices respond to interest rate changes. Duration is the first-order (linear) approximation, and convexity is the second-order (curvature) correction. **Modified Duration:** **ΔP/P ≈ -D_mod × Δy** Where D_mod is modified duration and Δy is the change in yield. **Duration + Convexity:** **ΔP/P ≈ -D_mod × Δy + ½ × C × (Δy)²** Where C is the convexity of the bond. **Why Duration Alone Fails:** Duration assumes a linear price-yield relationship, but the actual relationship is curved (convex). For small yield changes (±25 bps), the linear approximation is fine. For larger changes, the error grows significantly. **Worked Example:** Blackstone Fixed Income holds a 10-year, 4.5% coupon bond priced at $97.50 with: - Modified duration: 7.82 - Convexity: 72.5 - Current yield: 4.85% **Scenario: Yields rise 150 bps (from 4.85% to 6.35%)** *Duration-only estimate:* - ΔP/P ≈ -7.82 × 0.015 = -11.73% - Estimated price = $97.50 × (1 - 0.1173) = **$86.06** *Duration + convexity estimate:* - ΔP/P ≈ -7.82 × 0.015 + ½ × 72.5 × (0.015)² - ΔP/P ≈ -0.1173 + ½ × 72.5 × 0.000225 - ΔP/P ≈ -0.1173 + 0.00816 = -10.91% - Estimated price = $97.50 × (1 - 0.1091) = **$86.86** *Actual price (from full repricing):* **$86.94** | Method | Estimated Price | Error | |---|---|---| | Duration only | $86.06 | -$0.88 (underestimates) | | Duration + convexity | $86.86 | -$0.08 (much closer) | | Actual repricing | $86.94 | — | Duration alone overestimated the price decline by $0.88. The convexity correction captured $0.80 of that error. **Key Properties of Convexity:** 1. **Convexity is always positive for option-free bonds** — this means the duration-only estimate always overestimates losses (when yields rise) and underestimates gains (when yields fall). Convexity is a "good" property for bondholders. 2. **Higher convexity is more valuable when rates are volatile.** Two bonds with the same duration but different convexities will behave differently: - Riverton Municipal Bond: Duration 7.8, Convexity 45 - Clearwater Treasury Bond: Duration 7.8, Convexity 85 - For the same yield change, Clearwater will lose less when rates rise and gain more when rates fall. 3. **Convexity increases with:** - Lower coupon rate (zero-coupon bonds have the highest convexity for a given maturity) - Longer maturity - Lower yield level 4. **Callable bonds have negative convexity** at low yields — as rates fall, the issuer is likely to call the bond, capping your upside. This is analogous to MBS prepayment risk. **Dollar Duration and Dollar Convexity:** For portfolio risk management, FRM also tests: - **Dollar duration** = D_mod × Price × 0.01 (price change per 1 bp yield move) - **Dollar convexity** = C × Price **Exam Tip:** FRM Part I often gives you duration and convexity and asks for the price change from a specific yield shift. Always include the convexity term if the yield change exceeds 50 bps — leaving it out will give the wrong answer. Practice more fixed income risk problems in our FRM Part I question bank.