Convexity Adjustment Formula
Convexity Adjustment Formula
The simple duration approximation—ΔPrice ≈ −Duration × ΔYield—is remarkably accurate for small moves. Add convexity, and you capture the second-order curvature that duration misses.
Key takeaways
- The bond price change formula is: ΔP/P ≈ −D × Δy + ½ × C × (Δy)²
- The first term (−D × Δy) is the duration effect; the second term (½ × C × (Δy)²) is the convexity effect.
- For small yield changes (0.25%–0.50%), duration alone is sufficiently accurate.
- For large yield changes (1%+), convexity becomes material and must be included for accuracy.
- The convexity adjustment is always positive; it never hurts you if convexity is positive, and it always hurts if convexity is negative.
The formula: duration plus convexity
The precise relationship between a bond's percentage price change and yield change is:
ΔPrice/Price ≈ −D × Δy + ½ × C × (Δy)²
Where:
- ΔPrice/Price is the percentage change in bond price.
- D is modified duration (in years).
- Δy is the change in yield (in decimal form; 1% = 0.01).
- C is convexity (in basis points squared, or unitless in some conventions).
- The negative sign on duration reflects the inverse relationship between prices and yields.
Let's unpack this formula piece by piece.
The duration term: −D × Δy
The first term is the familiar duration approximation. For every 1% rise in yield, the bond price falls by roughly D%. Conversely, for every 1% fall in yield, the bond price rises by roughly D%.
Example: A bond with 5-year duration yielding 4%. If yields rise to 5% (Δy = +0.01):
Duration effect = −5 × 0.01 = −0.05 = −5%
The bond price falls approximately 5%. If yields fall to 3% (Δy = −0.01):
Duration effect = −5 × (−0.01) = +0.05 = +5%
The bond price rises approximately 5%.
This is the approximation most investors use and remember. It works well for moves up to 0.5%–1.0%. Beyond that, convexity becomes important.
The convexity term: ½ × C × (Δy)²
The second term captures the curvature. It is always positive (because it is squared), meaning it always adjusts prices in the favorable direction—upward if convexity is positive, and adjusts away from favorable if convexity is negative.
The term is ½ × C × (Δy)². For a bond with convexity C = 50 (in standard units):
- If Δy = 0.01 (1% yield change): ½ × 50 × (0.01)² = 0.0025 = 0.25%
- If Δy = 0.02 (2% yield change): ½ × 50 × (0.04)² = 0.01 = 1.0%
- If Δy = 0.03 (3% yield change): ½ × 50 × (0.09)² = 0.0225 = 2.25%
Notice: the convexity adjustment is negligible for small moves (0.25% for a 1% move) but grows quadratically as yields move more. For a 3% move, it is 2.25%—very material.
Example 1: Small yield move
Bond: 5-year duration, convexity 50, current price £100, yield 4%. Scenario: Yield rises to 4.5% (Δy = +0.005).
Duration effect = −5 × 0.005 = −0.025 = −2.5% Convexity effect = ½ × 50 × (0.005)² = 0.5 × 50 × 0.000025 = 0.000625 = 0.0625%
Total price change ≈ −2.5% + 0.0625% = −2.4375%
New price ≈ £100 × (1 − 0.024375) = £97.56
The convexity adjustment is only 0.0625%, barely rounding-error territory. Duration alone would have given −2.5%, which is close enough for practical purposes.
Example 2: Large yield move
Bond: 5-year duration, convexity 50, current price £100, yield 4%. Scenario: Yield falls to 2% (Δy = −0.02).
Duration effect = −5 × (−0.02) = +0.10 = +10% Convexity effect = ½ × 50 × (−0.02)² = 0.5 × 50 × 0.0004 = 0.01 = +1.0%
Total price change ≈ +10% + 1.0% = +11.0%
New price ≈ £100 × 1.11 = £111
Duration alone would have predicted +10%. The convexity adjustment adds 1.0%, for a total of +11.0%. In a large yield move, this 1% difference is material to returns.
Example 3: Comparing positive and negative convexity
Bond A (positively convex): 5-year duration, positive convexity +60. Bond B (negatively convex): 5-year duration, negative convexity −40. Scenario: Yield rises 2% (Δy = +0.02).
Bond A:
- Duration effect = −5 × 0.02 = −10%
- Convexity effect = ½ × 60 × (0.02)² = 0.012 = +1.2%
- Total = −10% + 1.2% = −8.8%
Bond B:
- Duration effect = −5 × 0.02 = −10%
- Convexity effect = ½ × (−40) × (0.02)² = −0.008 = −0.8%
- Total = −10% − 0.8% = −10.8%
Bond A loses 8.8%; Bond B loses 10.8%. The 1% difference comes from the difference in convexity. Bond A's positive convexity cushions the loss; Bond B's negative convexity amplifies it.
Rearranging the formula for yield moves
Sometimes it is useful to rearrange the formula to solve for the yield move required to achieve a target price change, or to understand the break-even yield move.
If you want to know what yield move is needed to gain 5%, you rearrange:
5% = −D × Δy + ½ × C × (Δy)²
This is a quadratic equation in Δy. For a bond with D = 5, C = 50:
0.05 = −5 × Δy + 25 × (Δy)² 25(Δy)² − 5(Δy) − 0.05 = 0
Solving the quadratic: Δy ≈ −0.0099 (about −1%).
So a 1% yield drop would produce roughly a 5% gain. Duration alone says a 5% gain requires a 1% drop (5 ÷ 5 = 1%), but convexity makes the yield move slightly smaller (because convexity adds to the upside). The difference is small here but grows for larger target moves.
Convexity in bond trading
Professional traders use the convexity formula to:
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Price bonds accurately: When trading large blocks, the difference between the duration approximation and the full convexity-adjusted price can be hundreds or thousands of pounds. Traders use the convexity formula to price precisely.
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Analyze basis trades: When trading two bonds (e.g., a Treasury and a corporate), the convexity difference can create trading opportunities. If two bonds have the same duration but different convexity, the convexity gap can be exploited.
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Hedge convexity: Using options or swaptions, traders can hedge the negative convexity of callable bonds or MBS. They buy interest rate caps (which have positive convexity) to offset the negative convexity of their bond holdings.
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Portfolio risk assessment: Portfolio managers compute portfolio convexity (the weighted-average convexity of all holdings) to understand how the portfolio will behave in volatile markets. Higher convexity = more resilience to large moves.
Limits of the formula
The formula ΔP/P ≈ −D × Δy + ½ × C × (Δy)² is accurate for moderate yield moves (0–2%). For very large moves (3%+), the formula begins to drift because it uses only the first two terms of a Taylor series expansion. More sophisticated models use additional terms (gamma, sensitivity to convexity changes, etc.) for very large moves or complex bonds.
Also, the formula assumes:
- Linear convexity: Convexity is constant across the yield curve. In reality, convexity itself changes slightly as yields move (due to the changing shape of the price-yield curve).
- Parallel shift: The yield curve shifts in parallel (all rates move by the same amount). Non-parallel shifts (where long rates move differently than short rates) are not captured.
- No credit changes: The formula assumes credit quality is constant. If yields change because credit spreads widen or tighten, the effect is different from a parallel curve shift.
For most practical investment purposes—buying and holding bonds, portfolio construction, risk assessment—the formula is accurate enough.
Convexity and portfolio duration
For a portfolio of bonds, you can compute the overall duration and convexity:
Portfolio Duration = weighted average of individual bond durations. Portfolio Convexity = weighted average of individual bond convexities.
If your portfolio has duration 6 and convexity 45, and rates rise 2%, the expected portfolio loss is:
ΔPrice/Price ≈ −6 × 0.02 + ½ × 45 × (0.02)² = −0.12 + 0.009 = −11.1%
This approach allows managers to monitor portfolio-level risk. A target for portfolio convexity (say, "maintain positive convexity of at least 40") ensures the portfolio has cushion against large rate moves.
Visual representation
Next
The convexity adjustment formula is the mathematical foundation for understanding bond behavior. But in practice, bonds do not move in isolation. When you hold multiple bonds or a bond portfolio, both duration and convexity interact across maturities and bond types. The next article explores how to manage these effects together—using duration and convexity in tandem to make accurate predictions about portfolio returns.