Duration and Convexity: Measuring Bond Interest Rate Risk
What Is Bond Duration?
Duration answers a simple question: how much will my bond’s price change if interest rates move by 1%? It’s expressed in years, but think of it as a sensitivity measure. A bond with a duration of 7 means a 1% rate increase causes roughly a 7% price decline.
Duration depends on three factors: maturity (longer = higher duration), coupon rate (lower coupon = higher duration), and yield to maturity (lower yield = higher duration). A zero-coupon bond has the highest duration for its maturity because all cash flow arrives at the end.
Types of Duration
| Type | What It Measures | When to Use |
|---|---|---|
| Macaulay Duration | Weighted average time to receive cash flows (in years) | Academic analysis; understanding the concept |
| Modified Duration | Price sensitivity to a 1% yield change (as a %) | Estimating price impact of rate changes |
| Effective Duration | Price sensitivity for bonds with embedded options | Callable and putable bonds |
| Key Rate Duration | Sensitivity to specific points on the yield curve | Portfolio risk management; curve positioning |
Modified Duration Formula
Where YTM is the yield to maturity and n is the number of coupon periods per year. The price change estimate is then:
Duration Example
You hold a bond worth $1,000 with a modified duration of 6.5. If yields rise by 0.50% (50 basis points):
| Input | Value |
|---|---|
| Current Price | $1,000 |
| Modified Duration | 6.5 |
| Yield Change | +0.50% |
| Estimated Price Change | −6.5 × 0.005 × $1,000 = −$32.50 |
| New Estimated Price | $967.50 |
What Is Convexity?
Duration gives you a straight-line approximation — but the actual price-yield relationship is curved. Convexity measures that curvature. For large rate moves, duration alone underestimates price gains (when rates fall) and overestimates price losses (when rates rise). Convexity corrects this.
Positive convexity is desirable because it means your bond gains more when rates fall than it loses when rates rise. Most standard bonds have positive convexity. Callable bonds can exhibit negative convexity — their upside is capped because the issuer can call them when rates drop.
Duration in Practice
Portfolio management: Fund managers target a specific portfolio duration to match their interest rate outlook. Bullish on rates falling? Extend duration. Expecting rate hikes? Shorten duration. The average duration of a bond ETF tells you its rate sensitivity at a glance.
Asset-liability matching: Insurance companies and pension funds match the duration of their bond portfolios to the duration of their future obligations. This immunization strategy ensures that rate changes affect assets and liabilities equally.
Risk comparison: Duration lets you compare rate risk across very different bonds. A 2-year Treasury note and a 10-year corporate bond may have similar durations if the corporate has a high coupon, meaning similar rate sensitivity despite different maturities.
Duration Rules of Thumb
| Factor | Effect on Duration |
|---|---|
| Longer maturity | Higher duration (more rate-sensitive) |
| Higher coupon rate | Lower duration (cash flows arrive sooner) |
| Higher yield | Lower duration (future cash flows discounted more heavily) |
| Zero coupon bond | Duration equals maturity (maximum sensitivity) |
Key Takeaways
- Duration measures a bond’s price sensitivity to interest rate changes — higher duration means more risk.
- Modified duration gives the approximate percentage price change for a 1% yield shift.
- Convexity adjusts for the curved (non-linear) relationship between price and yield, improving estimates for large rate moves.
- Positive convexity benefits bondholders; negative convexity (in callable bonds) limits upside.
- Duration is essential for portfolio management, risk comparison, and asset-liability matching.
Frequently Asked Questions
What is a good duration for a bond portfolio?
It depends on your rate outlook and risk tolerance. Conservative investors often target 2-4 years of duration. Moderate portfolios might target 4-6 years. Aggressive rate bets push duration to 7+ years. Match duration to how long you plan to hold the bonds.
What is the difference between duration and maturity?
Maturity is simply when the bond expires. Duration is a measure of interest rate sensitivity that accounts for all cash flows (coupons and principal), not just the final payment. A 10-year bond with high coupons has a duration well below 10 years.
Why do zero-coupon bonds have the highest duration?
Because all their cash flow comes as a single payment at maturity. With no interim coupons to provide earlier cash flows, the entire value is concentrated at the end — maximizing sensitivity to rate changes. A 10-year zero-coupon bond has a Macaulay duration of exactly 10 years.
How does convexity affect bond prices?
Convexity means bond prices don’t change linearly with yields. For a bond with positive convexity, price gains when rates fall are larger than price losses when rates rise by the same amount. This asymmetry benefits bondholders and is worth paying a premium for.
Can duration be negative?
Standard duration is always positive. However, some instruments — like interest-only mortgage strips or certain derivative positions — can exhibit negative duration, meaning they gain value when rates rise. These are specialized tools typically used by institutional investors.