Convexity
Convexity measures curvature in the bond price-yield relationship and refines duration-based rate-risk estimates.
Convexity measures the curvature in the relationship between a bond’s price and its yield. It matters because a bond’s price-yield relationship is not a perfectly straight line.
Duration gives the first-order, straight-line estimate. Convexity adds the second-order adjustment that becomes important when yield moves are larger.
Core Formula
A common duration-plus-convexity price approximation is:
Where \(D\) is duration, \(C\) is convexity, and \(\Delta y\) is the yield change. The duration term gives the linear estimate; the convexity term adjusts for curvature.
Worked Example: Duration Plus Convexity
Assume an option-free bond has modified duration of 5.0, convexity of 40, and a current price of $1,000. Use decimal yield changes in the approximation.
For a 1-percentage-point yield increase, \(\Delta y = 0.01\):
For a 1-percentage-point yield decrease, \(\Delta y = -0.01\):
| Yield scenario | Duration-only estimate | Convexity-adjusted estimate | Estimated price |
|---|---|---|---|
| Yield rises 1 percentage point | -5.00% | -4.80% | $952 |
| Yield falls 1 percentage point | 5.00% | 5.20% | $1,052 |
Positive convexity adds 0.20 percentage point in both calculations. It reduces the estimated loss when yields rise and increases the estimated gain when yields fall. The approximation still omits credit-spread moves, liquidity, transaction costs, and any change in expected cash flows.
Convexity conventions can differ by pricing system and compounding basis. Use a duration and convexity pair produced under compatible assumptions rather than combining unrelated vendor figures.
Why It Matters
Convexity matters because duration is most reliable for small yield changes. As yields move farther, the straight-line estimate becomes less accurate.
For many option-free bonds with positive convexity:
- price gains from falling yields are larger than a straight-line duration estimate
- price losses from rising yields are smaller than a straight-line duration estimate
- the price-yield curve bends in a way that can help the investor when rates move sharply
This is why investors often prefer more positive convexity when the yield, credit risk, liquidity, and cost are otherwise comparable.
Positive vs. Negative Convexity
| Pattern | Typical structure | Rate-move behavior | Main caution |
|---|---|---|---|
| Positive convexity | Option-free government or corporate bond | More upside when yields fall and less downside when yields rise than the duration line suggests | Often comes at a richer price or lower yield |
| Negative Convexity | Callable bond or mortgage-linked security | Upside can be capped when yields fall because calls or prepayments become more likely | Lower apparent duration may hide lost upside |
| Low convexity | Shorter maturity, higher coupon, or structures with limited curvature | Duration may capture most of the price move for small shocks | Still needs spread, liquidity, and credit review |
Convexity is not automatically “good” or “bad” without price. Investors often pay for favorable convexity through a lower yield or higher purchase price.
Public Source Checks
Useful public references include:
- MSRB Evaluating a Municipal Bond’s Interest Rate Risk: Municipal-bond education that discusses duration and convexity in rate-risk evaluation.
- FINRA Brush Up on Bonds: Interest Rate Changes and Duration: Investor education on duration, rate changes, and the limits of simple price-sensitivity estimates.
- OCC Interest Rate Risk Comptroller’s Handbook: Supervisory reference that discusses duration, convexity, and interest-rate-risk measurement.
These sources help frame the public risk concept. A security-specific convexity decision still requires the bond’s pricing model, cash-flow assumptions, and rate scenario.
This example is educational and is not an investment recommendation or interest-rate forecast.
When Convexity Misleads
Convexity can mislead when:
- it is read without duration, yield, spread, and price
- positive convexity is assumed to be free
- a callable or prepayable bond is analyzed with option-free convexity
- a small-shock convexity number is applied to a large stress without model review
- spread duration and credit risk move at the same time as interest rates
- portfolio convexity hides concentrated exposure to a few securities or curve points
- liquidity costs dominate the theoretical price benefit
Use convexity to refine duration, not to replace it. The full rate-risk view usually needs duration, convexity, key-rate exposure, spread risk, optionality, and liquidity.
Related Terms
- Duration: The first-order bond price-sensitivity measure convexity refines.
- Modified Duration: Linear price-sensitivity measure used before adding convexity.
- Effective Duration: Duration measure for bonds where cash flows can change.
- Negative Convexity: Less favorable convexity pattern often seen in callable and mortgage-linked securities.
- Callable Bond: A structure where convexity can become negative as rates fall.
- Yield Curve Risk: Risk from nonparallel changes in curve shape.