Convexity
Convexity describes how a bond’s duration changes as yields change. Because of convexity, an ordinary bond’s price rises more when yields fall than it drops when yields rise by the same amount.
Duration gives a straight-line estimate of how a bond’s price reacts to yield changes. Real bond prices curve. Convexity measures that curve: for ordinary bonds it works in the investor’s favour, adding a little extra gain when yields fall and cushioning losses when they rise. For callable bonds and some mortgage securities it can work against you. This entry explains convexity without heavy maths.
- Category
- Bonds and Interest Rates
- Reading time
- 4 min
- Also called
- Bond convexity, Positive convexity, Negative convexity
- Updated
Convexity in plain words
Duration says roughly how much a bond’s value changes when rates move — the higher the duration, the more sensitive the bond.1 See Duration.
But the relationship between price and yield is curved, not straight. Bond prices and market rates move in opposite directions,2 and for an ordinary bond the price gains from a fall in yields are a little bigger than the losses from an equal rise. That extra is convexity.
How convexity fits in, step by step
Start with duration
Price change ≈ −duration × yield change.
Add the convexity adjustment
A positive term that grows with the square of the yield change.
Small moves
Duration alone is usually close enough.
Large moves
Convexity becomes important.
Check bond features
Calls can create negative convexity. See Callable and Putable Bonds.
Compare portfolios
Two portfolios with the same duration can differ in convexity.
Worked examples
Example 1 — Asymmetric moves
CurvatureA long-term bond has a duration of about 15. Yields move 2 percentage points either way.
| Duration estimate | ±30% |
| Actual: yields fall 2 points | More than +30% |
| Actual: yields rise 2 points | Less than −30% |
Positive convexity makes gains a bit larger and losses a bit smaller than duration suggests.
Example 2 — Negative convexity
NegativeA callable bond’s price rises as yields fall, but stalls near its call price.
| When yields fall | Limited gains |
| When yields rise | Full losses |
The call feature caps the upside — the opposite of positive convexity.
Example 3 — Same duration, different convexity
ComparisonPortfolio A holds bonds spread evenly from 1 to 20 years; portfolio B holds only 10-year bonds. Both have similar duration.
| Portfolio A (spread out) | Higher convexity |
| Portfolio B (concentrated) | Lower convexity |
Duration alone does not capture every difference in rate risk.
Positive vs. negative convexity
| Aspect | Positive | Negative |
|---|---|---|
| Typical bonds | Ordinary fixed-rate bonds | Callable bonds, many mortgage securities |
| When yields fall | Gains exceed duration estimate | Gains capped |
| When yields rise | Losses below duration estimate | Losses in full |
| For the investor | Helpful | Harmful |
Where convexity matters
Large rate moves
When duration estimates break down.
Callable and mortgage bonds
Spotting negative convexity.
Portfolio construction
Comparing bond portfolios beyond duration. See Portfolio Allocation.
Hedging
Hedges based on duration alone drift as rates move. See Hedging.
Advantages and disadvantages
Advantages
- Improves price estimates for large rate moves.
- Highlights hidden risks in callable bonds.
- Helps compare portfolios with equal duration.
Disadvantages
- Harder to calculate and explain than duration.
- Changes as yields and prices change.
- Small effect for small rate moves.
Going deeper
Read as far as you need. Each level builds on the one before it.
A curved line
Beginner
Plot bond price against yield and you get a curve that bends upward. Duration is the straight line touching it; convexity is the bend.
What increases convexity
Intermediate
Longer maturities and lower coupons usually mean more convexity, just as they mean more duration. See Zero-Coupon Bond.
Convexity has a cost
Advanced
Because positive convexity is valuable, bonds with more of it tend to offer slightly lower yields, all else equal.
Common mistakes
Relying on duration for big moves
Add convexity.
Ignoring negative convexity
Callable bonds behave differently.
Treating convexity as fixed
It changes with yields.
Overcomplicating small moves
Duration is usually enough.
Check your understanding
Test yourself before moving on. Each answer comes with an explanation.
Key points to remember
- Convexity measures the curve in the price–yield relationship.
- It refines duration estimates, especially for large moves.
- Ordinary bonds have positive convexity.
- Callable bonds can have negative convexity.
- Portfolios with equal duration can differ in convexity.
Frequently asked questions
Is high convexity good?
Do I need convexity for small rate changes?
Why do mortgage bonds have negative convexity?
Further learning
- Duration
The first-order measure convexity refines.
- Callable and Putable Bonds
The source of negative convexity.
- Zero-Coupon Bond
High duration and convexity.
- Interest Rate
What moves bond prices.
Sources
This article is written in our own words. The sources below support the specific facts, rules and figures marked with a number in the text. Links open the original publisher’s page.
Educational content, not financial advice. Examples use hypothetical prices and simplified costs; actual fees, margin rules and regulations vary by broker, market and country. We are not affiliated with any publisher linked above. Last reviewed . Report an error or request a source credit.