Glossary term
Convexity
Convexity measures the curvature in a bond’s price-yield relationship, refining duration when interest rates move.
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What Is Convexity?
Convexity measures the curvature in a bond's price-yield relationship. It refines duration by showing how a bond's price sensitivity changes as yields move.
Duration is a first-order estimate: it approximates how much a bond price changes for a small change in yield. Convexity is a second-order adjustment: it helps explain why the price-yield relationship is curved rather than perfectly straight.
Key Takeaways
- Convexity measures curvature in the bond price-yield relationship.
- It becomes more important when interest-rate moves are large.
- Positive convexity usually benefits bond investors during rate volatility.
- Callable bonds and mortgage-backed securities can show negative convexity.
- Convexity should be read with duration, yield, coupon, maturity, and embedded options.
How Convexity Works
Bond prices and yields move in opposite directions. When yields fall, bond prices rise. When yields rise, bond prices fall. The price change is not perfectly symmetrical because the relationship bends. Convexity describes that bend.
A positively convex bond tends to gain more when yields fall than it loses when yields rise by the same amount. That is favorable shape. A negatively convex bond can have capped upside when rates fall because the issuer or borrower may refinance or call the bond.
Formula Context
A simplified duration-plus-convexity approximation is:
In this expression, ΔP/P is the approximate percentage price change, D is duration, Δy is the change in yield, and C is convexity. The convexity term becomes more important as the yield move gets larger.
Positive and Negative Convexity
Type | Typical source | Investor effect |
|---|---|---|
Positive convexity | Option-free bonds | More upside from falling yields than downside from equal rising-yield moves |
Negative convexity | Callable bonds and many mortgage-backed securities | Upside may be capped when rates fall |
Negative convexity is often tied to borrower or issuer optionality. When rates fall, the borrower has more reason to refinance, which can reduce the investor's benefit from lower yields.
Portfolio Use
Portfolio managers use convexity to compare bonds with similar duration. Two bonds may have the same duration but different behavior under larger rate moves. The higher-convexity bond may be more valuable in volatile rate environments, but the market often charges for that benefit through lower yield or higher price.
Convexity also affects hedging. A duration hedge can work for small yield changes but drift when yields move sharply because duration itself changes. Convexity helps explain that hedge error.
Example
Suppose two bonds both have similar duration, but one has higher positive convexity. If yields move only a few basis points, their price changes may look similar. If yields move sharply, the higher-convexity bond may perform better than the simple duration estimate suggested.
This is why convexity becomes more visible in volatile rate markets. It is a small adjustment for tiny yield changes and a more meaningful adjustment when rates move enough for curvature to matter.
What Convexity Is Not
Convexity does not measure default risk, liquidity risk, tax treatment, or inflation risk. A bond can have attractive convexity and still be a poor investment if the issuer's credit deteriorates or the price already reflects too much optimism.
Convexity is usually quoted as a model output, so assumptions matter. Yield curve shifts, option-adjusted models, prepayment assumptions, and credit spread changes can all affect the number.
For individual investors, the simple lesson is that two bonds with the same yield and duration may not respond the same way when rates move sharply.
That difference is the reason convexity belongs beside yield and duration in any serious bond comparison.
The Bottom Line
Convexity measures how bond price sensitivity changes as yields move. It is not a replacement for duration, but it is essential for understanding larger interest-rate moves, embedded options, and the true shape of fixed-income risk.