Convexity
If you’ve ever priced a bond using duration alone and noticed the estimate was off — especially for large rate moves — convexity is the missing piece. It’s the second-order correction that turns a rough straight-line estimate into something far more accurate.
Why Duration Alone Isn’t Enough
Duration gives you a first approximation: for a 1% change in yield, the bond’s price changes by roughly the duration percentage. But that relationship isn’t linear — it’s curved. For small yield changes (say, 10-20 basis points), duration works well. For large moves (100+ basis points), the straight-line estimate diverges meaningfully from the actual price path.
That curvature is convexity. It tells you how much extra price gain you get when rates fall — and how much less you lose when rates rise — relative to what duration alone would predict.
The Convexity Formula
Where:
| Variable | Meaning |
|---|---|
| P | Current bond price |
| P₊ | Bond price if yield increases by Δy |
| P₋ | Bond price if yield decreases by Δy |
| Δy | Change in yield (in decimal form) |
This is the “effective convexity” approach — it works for any bond, including those with embedded options like callable bonds or convertible bonds, because it uses actual price changes rather than assuming fixed cash flows.
The Full Price Change Estimate
When you combine duration and convexity, the total estimated percentage price change for a yield shift of Δy is:
The first term is the duration effect (linear). The second is the convexity adjustment (curved). Notice that the convexity term is always positive for a standard bond — whether rates go up or down, convexity adds value.
Convexity in Practice: A Quick Example
Suppose you hold a bond with a modified duration of 7.0 and a convexity of 60. Interest rates spike by 1% (Δy = 0.01).
| Estimate | Calculation | Result |
|---|---|---|
| Duration only | −7.0 × 0.01 | −7.00% |
| Convexity adjustment | ½ × 60 × 0.01² | +0.30% |
| Combined estimate | −7.00% + 0.30% | −6.70% |
Duration alone would overstate the loss by 30 basis points. That gap gets wider as the rate move gets bigger — exactly when accuracy matters most.
Positive vs. Negative Convexity
| Type | What Happens | Typical Bonds |
|---|---|---|
| Positive convexity | Price gains accelerate as rates fall; price losses decelerate as rates rise. The bondholder benefits in both directions. | Most standard coupon bonds, zero-coupon bonds, Treasuries |
| Negative convexity | Price gains slow down as rates fall (because the issuer is likely to call the bond). The bondholder is disadvantaged. | Callable bonds, mortgage-backed securities |
What Drives a Bond’s Convexity
Several factors determine how much convexity a bond has:
| Factor | Effect on Convexity |
|---|---|
| Longer maturity | Higher convexity — more time for compounding effects to magnify price sensitivity |
| Lower coupon rate | Higher convexity — more cash flow is weighted toward maturity, amplifying curvature |
| Lower yield environment | Higher convexity — discount factor changes are more powerful at low yields |
| Embedded call option | Reduces (or reverses) convexity — the call caps upside when rates fall |
Zero-coupon bonds have the highest convexity for a given maturity because all their cash flow sits at the end. That’s why long-dated zeros are the most volatile fixed-income instruments — and the most convex.
Convexity and Portfolio Management
Portfolio managers use convexity in several concrete ways:
Immunization. When building a portfolio to match a liability’s duration, adding a convexity constraint ensures the portfolio tracks the liability more closely across a range of rate scenarios — not just for small parallel shifts.
Barbell vs. Bullet strategies. A barbell portfolio (short-term and long-term bonds, nothing in the middle) typically has higher convexity than a bullet portfolio (bonds clustered around one maturity), even when both have the same duration. During volatile markets, the barbell benefits from that extra convexity.
Relative value. If two bonds offer similar yield and duration but different convexity, the higher-convexity bond is the better risk-adjusted pick. The spread difference may compensate for the convexity gap — or it may not, which creates a trade opportunity.
Convexity for Callable Bonds
Callable bonds deserve special attention. As interest rates fall, the probability that the issuer will redeem the bond early increases. This caps the bond’s price upside around the call price, compressing its price-yield curve and creating negative convexity in that region.
You can think of a callable bond as: a standard bond minus a call option sold to the issuer. That embedded short call is what strips away convexity. The lower the rate drops below the coupon, the more negative convexity dominates. This is why callable bonds typically offer a higher credit spread — investors demand compensation for that convexity disadvantage.
Related Terms
| Term | Relationship |
|---|---|
| Duration | First-order price sensitivity to rates; convexity is the second-order correction |
| Yield to Maturity | The discount rate used in convexity calculations |
| Callable Bond | Exhibits negative convexity due to the embedded call option |
| Zero-Coupon Bond | Highest convexity for a given maturity |
| Bond | The underlying instrument convexity applies to |
Key Takeaways
- Convexity measures the curvature of the bond price–yield relationship and corrects duration’s linear approximation.
- Positive convexity benefits bondholders — prices rise more than expected when rates fall and decline less than expected when rates rise.
- Negative convexity, found in callable bonds and MBS, hurts bondholders by capping upside.
- Longer maturity, lower coupon, and lower yields all increase convexity.
- The full price change estimate requires both duration and convexity: %ΔP ≈ (−Duration × Δy) + (½ × Convexity × Δy²).
Frequently Asked Questions
What is convexity in simple terms?
Convexity describes how the price sensitivity of a bond changes as interest rates move. Duration gives you a straight-line estimate of price change; convexity accounts for the fact that the actual relationship is curved. Higher convexity means the bond gains more when rates fall and loses less when rates rise, compared to what duration alone predicts.
Is higher convexity always better?
For a bondholder — yes, generally. Positive convexity is a desirable property because it means asymmetric payoffs in your favor. However, higher-convexity bonds often trade at lower yields (higher prices) because other investors value that property too. You’re effectively paying for the benefit.
What is negative convexity?
Negative convexity occurs when a bond’s price gains become smaller as rates fall. This typically happens with callable bonds and mortgage-backed securities, where the issuer or borrower can refinance at lower rates, effectively capping the investor’s upside. Investors demand a higher yield — a wider spread — to compensate.
How does convexity relate to duration?
Duration is the first derivative of the price-yield function — it tells you the slope. Convexity is the second derivative — it tells you how the slope changes. In practice, duration handles most of the price estimate, while convexity becomes increasingly important for larger yield moves (50+ basis points).
Why do zero-coupon bonds have the highest convexity?
Zero-coupon bonds concentrate all cash flow at maturity, meaning their entire value is a single discounted payment far in the future. This makes their price-yield curve more sharply bowed than coupon bonds of the same maturity, which spread their cash flows across multiple dates.