What is Convexity?
Convexity is duration's fine-tuning — it captures what duration misses. For large yield moves, duration alone underestimates price gains and overestimates losses. Convexity corrects this, and positive convexity is always in your favor.
Convexity measures the curvature of the price-yield relationship for a bond. It's the second-order correction to duration's linear estimate of price changes. Positive convexity means bond prices rise more when yields fall than they fall when yields rise — a beneficial asymmetry for investors.
Contents
Why Duration Isn't Enough
Duration assumes a linear price-yield relationship, but the actual relationship is curved (convex). For small yield changes (±0.25%), duration works fine. For larger moves (±1% or more), duration's estimate becomes inaccurate — convexity provides the correction.
Positive Convexity
Most bonds have positive convexity: prices rise more than duration predicts when yields fall, and fall less when yields rise. This is favorable — you gain more on the upside than you lose on the downside. Zero-coupon bonds have the highest convexity.
Price Change Formula
Full price change ≈ -Duration × Δyield + ½ × Convexity × (Δyield)². The convexity term is always positive (squared), so it always adds to price for any yield move — improving your position regardless of direction.
Convexity of Different Securities
| Security Type | Convexity | Why |
|---|---|---|
| Zero-coupon long bond | Highest | All cash flow at maturity |
| Long-term coupon bond | High | Distant cash flows dominate |
| Short-term bond | Low | Little time for curve effects |
| Callable bond | Negative (at low yields) | Call option caps upside |
| Mortgage-backed securities | Often negative | Prepayment risk |
Practical Example: Convexity in Action
You hold a 30-year Bund with duration 20 and convexity 400. Yields fall 1%. Duration predicts: +20% price gain. Convexity adjustment: ½ × 400 × (0.01)² = +0.02 = +2%. Actual price gain: about 22%, not 20%. Yields rise 1%? Duration predicts -20%. With convexity: -20% + 2% = -18%. You lose less than duration predicted.
Frequently Asked Questions
What is convexity in simple terms?
Convexity is a measure of how 'curved' a bond's price response is to yield changes. Positive convexity means you benefit more from falling yields than you suffer from rising yields — a win-win.
Why should I care about convexity?
For small rate changes, you don't need to. But for larger moves (1%+), convexity significantly affects returns. High-convexity bonds outperform low-convexity bonds in volatile rate environments.
How do I get more convexity?
Buy longer-maturity bonds, lower-coupon bonds, or zero-coupon bonds. Avoid callable bonds (negative convexity). Barbell strategies (short + long bonds) have more convexity than bullet strategies (all medium-term).
What is negative convexity?
Some bonds (callable bonds, MBS) have negative convexity at certain yield levels. This means they gain less when yields fall than they lose when yields rise — the opposite of what you want. Callable bonds exhibit this because low yields trigger the call option.
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