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

Question

AcadiFi · Accepted Answer

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.

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

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