Convexity is a measure of how a bond’s price reacts to changes in yield once you account for the fact the relationship is curved, not straight. It captures how a bond’s duration itself shifts as yields rise or fall.
In plain terms, duration gives you the first step in estimating a price move for a given yield change. Convexity adds the bend in the line. Together they produce a better forecast of interest rate risk for bonds and bond portfolios.
How convexity sits alongside duration
Duration is a linear approximation. If a bond has a modified duration of 6, a 1% rise in yield points to about a 6% price fall, and a 1% drop suggests about a 6% rise. Real bond prices do not move on a straight line though. The price–yield curve is bowed. That bow is convexity.
Positive convexity means the price rises a bit more when yields fall than it declines when yields rise by the same amount. Most conventional fixed coupon bonds show this. Convexity is essentially the second derivative of price with respect to yield. If that sounds abstract, think of it as the measure that tweaks duration up or down as yields move, improving your estimate, especially for larger rate changes.
Why most bonds have positive convexity
The present value of a bond is the sum of discounted cash flows. When yields fall, you discount those cash flows at a lower rate, pushing up today’s value. Because payments are spread across time, and the discounting is nonlinear, the price–yield curve has curvature. Longer maturities and lower coupons usually increase that curvature, so long-dated and low-coupon bonds tend to exhibit higher convexity.
Some quick patterns:
- Longer maturity usually means higher convexity, all else equal.
- Lower coupon usually means higher convexity, because more value sits in the distant principal payment.
- Zero-coupon bonds often have high convexity for their maturity, as the entire value is a single distant cash flow.
- Floating-rate notes tend to have very low convexity, since coupons reset with rates and price stays near par.
When convexity turns negative: embedded options and prepayments
Not all instruments have positive convexity. If a bond can be redeemed early by the issuer, the price upside can be capped when yields fall. That causes negative convexity over parts of the curve. Two common cases:
- Callable bonds: The issuer holds a right to repay early. This is economically similar to the issuer being long a call option on the bond. When yields drop, the chance of a call rises, limiting price gains and creating negative convexity near the call region.
- Mortgage-backed securities: Homeowners can prepay. When yields fall and refinancing is attractive, cash flows return sooner, again muting price rises and causing negative convexity.
Negative convexity matters for risk management. As yields fall, effective duration of such securities can rise, so hedges that worked before may under- or over-compensate. Managers often need to adjust interest rate hedges dynamically to reflect this changing profile.
Estimating a price move using duration and convexity
For small yield changes, duration alone can be acceptable. For larger moves, adding convexity improves the estimate. A common approximation is:
Percentage price change ≈ [− modified duration × change in yield] + [0.5 × convexity × (change in yield)^2].
Example with simple numbers. Suppose a bond has modified duration 7 and convexity 60. Consider a 100 basis point move in yield, which is 0.01 in decimal terms.
- If yield falls by 1.00%: estimate ≈ +7.00% + 0.5 × 60 × 0.0001 = +7.00% + 0.30% = +7.30%.
- If yield rises by 1.00%: estimate ≈ −7.00% + 0.30% = −6.70%.
The asymmetry comes from positive convexity. Price gains for a given fall in yields are slightly larger than losses for the same rise.
Two cautions. First, different sources define and scale convexity slightly differently, so units can vary by provider. Second, for securities with embedded options, both effective duration and convexity depend on interest rate levels and volatility, so the numbers can shift as conditions change.
Where you see convexity used in practice
- Risk reports: Bond funds and dealers track portfolio duration and convexity to summarise interest rate exposure. They may also show key-rate durations to pinpoint sensitivity at different maturities, with convexity providing the overall curvature measure.
- Hedging and immunisation: Liability-driven investors often target a duration and convexity match between assets and liabilities. Matching both reduces the risk that a parallel rate shift or a curve twist knocks the plan off course.
- Security selection: Some managers seek more convexity for a given yield, preferring portfolios that lose less in rate sell-offs and gain more in rallies. They might tilt to longer, higher-convexity bonds if they expect larger rate swings.
- Structured products and mortgages: Markets with negative convexity often require active hedging. Mortgage investors, for example, may adjust interest rate swaps or futures more aggressively as refinancing risk changes.
What drives convexity levels
- Coupon and maturity: Lower coupons and longer maturities usually raise convexity. High coupons and short maturities produce flatter price–yield curves.
- Yield level: At very low yields, prices are more sensitive to changes in discount rates, which can increase convexity in percentage terms.
- Cash flow structure: Amortising schedules, sinking funds or prepayment features reshape convexity by moving cash flows forward or backward.
- Embedded options: Calls, puts and prepayment rights can flip convexity positive or negative across different yield ranges.
Related but different: the “convexity adjustment” in rates
In interest rate derivatives, you may also hear about a convexity adjustment when converting between futures-implied rates and forward rates. That usage refers to how daily margining and the correlation between rates and discount factors create a small bias between the two. It is connected to curvature in pricing models but is distinct from bond price convexity discussed above.
Convexity adds nuance to duration. It explains why two portfolios with the same duration do not always behave the same when yields swing, and it gives you a better yardstick for price moves when rates shift by more than a token amount.