Why do zero-coupon bonds have higher convexity than coupon bonds with the same duration?

Question

AcadiFi · Accepted Answer

This is a subtlety that many Level I candidates miss. The key insight is understanding what convexity measures and how cash flow dispersion affects it. **First, the Correction:** Actually, zero-coupon bonds have **lower** convexity than coupon bonds with the same **duration**, not higher. Let me explain why this common confusion arises and then clarify the correct relationship. **The Confusion:** - Zero-coupon bonds have the highest **duration** for a given maturity (because all cash flow is at maturity) - Long duration means more price sensitivity, which people confuse with high convexity - But for the same duration, a zero-coupon bond has **less** convexity than a coupon bond **Why? Cash Flow Dispersion:** Convexity is related to the dispersion (spread) of cash flows around the duration point. - **Zero-coupon bond**: All cash flow at one point (maturity). Zero dispersion. - **Coupon bond**: Cash flows spread across multiple dates. Positive dispersion. Greater dispersion = greater convexity (for the same duration) **Mathematical Intuition:** Convexity measures the curvature of the price-yield relationship. A bond with cash flows at many different times has a price-yield curve that bends more (more convex) than one with a single cash flow. **Practical Demonstration:** Consider two bonds, both with Macaulay duration = 8 years: Bond A: 8-year zero-coupon bond - All $1,000 arrives in year 8 - Dispersion of cash flows = 0 - Convexity ≈ 64 (duration squared / (1+y)^2, approximately) Bond B: 15-year, 7% coupon bond (duration also ~8 years) - Cash flows from year 1 to year 15 - Wide dispersion around the 8-year duration point - Convexity ≈ 95 **Why This Matters for Portfolio Management:** ```mermaid flowchart LR A[Same Duration
Same Rate Sensitivity] --> B[Zero-Coupon
Low Convexity
Focused Cash Flow] A --> C[Coupon Bond
High Convexity
Dispersed Cash Flows] A --> D[Barbell Portfolio
Highest Convexity
Most Dispersed] B --> E[Best for Immunization
Minimizes structural risk] C --> F[Moderate convexity
Some structural risk] D --> G[Maximum convexity
Most structural risk] style B fill:#c9a84c,color:#000 style E fill:#c9a84c,color:#000 ``` **The Convexity Hierarchy (for same duration):** 1. Barbell portfolio (highest convexity, highest dispersion) 2. Coupon bonds (moderate convexity) 3. Zero-coupon bonds (lowest convexity, zero dispersion) **Immunization Implication:** For liability immunization, you want to minimize convexity (match duration with minimum dispersion). This is why zero-coupon bonds are the ideal immunization vehicle — they eliminate both reinvestment risk and structural risk. **Exam Tip:** Remember the relationship: dispersion drives convexity. For immunization, minimize convexity. For speculation on rate changes (in either direction), maximize convexity. Deepen your understanding in our CFA Level I fixed income course.

Why do zero-coupon bonds have higher convexity than coupon bonds with the same duration?

Master Level I with our CFA Course

Related Questions

Practice Questions