Duration and Convexity in Tandem
Duration and Convexity in Tandem
Duration is the first word in the bond price story; convexity is the second. Together, they tell you what will happen to your portfolio when the bond market moves.
Key takeaways
- Duration captures the linear relationship between yield and price; convexity captures the curvature.
- For small yield moves (under 0.5%), duration alone is sufficient. For moves 1%+, convexity becomes material.
- Portfolio managers use both metrics together to assess and manage interest rate risk.
- High duration with high positive convexity is the ideal combination: sensitivity to beneficial moves plus cushioning against adverse moves.
- Understanding both metrics prevents surprises; you can predict bond returns with accuracy.
Why both metrics matter
Consider two bonds: Bond A and Bond B, both trading at par, both yielding 4%.
Bond A: 5-year duration, positive convexity 40. Bond B: 5-year duration, positive convexity 20.
If rates stay at 4%, both bonds return 4% (their coupon/yield). But if rates move significantly:
Scenario: Rates fall to 2% (a 2% decline)
Using the formula: ΔPrice/Price ≈ −D × Δy + ½ × C × (Δy)²
Bond A: −5 × (−0.02) + ½ × 40 × (0.02)² = +0.10 + 0.008 = +10.8% Bond B: −5 × (−0.02) + ½ × 20 × (0.02)² = +0.10 + 0.004 = +10.4%
Bond A outperforms by 0.4% due to higher convexity. Over a large move, this compounds. Bond A's higher convexity is a tangible advantage.
Scenario: Rates rise to 6% (a 2% rise)
Bond A: −5 × (0.02) + ½ × 40 × (0.02)² = −0.10 + 0.008 = −9.2% Bond B: −5 × (0.02) + ½ × 20 × (0.02)² = −0.10 + 0.004 = −9.6%
Bond A loses less due to higher convexity. Again, Bond A outperforms.
If you hold a portfolio with high duration and low convexity (like many pension funds held in 2022), large rate moves become painful. If you hold high duration and high convexity (like some long-term Treasury investors), large moves become opportunities.
Duration is sensitivity; convexity is optionality
A useful way to think about the two metrics:
- Duration measures your portfolio's sensitivity to rates. Higher duration = more sensitive.
- Convexity measures whether that sensitivity is symmetrical or asymmetrical. Positive convexity = asymmetrical in your favor; negative convexity = asymmetrical against you.
A portfolio with 6-year duration and +60 convexity is like owning a 6-year bond plus a call option on bonds (an option you own, not the issuer). When rates fall sharply, you benefit from the duration move plus the option value. When rates rise, you have some downside cushion from the positive convexity.
A portfolio with 6-year duration and −60 convexity is like owning a 6-year bond minus a call option (an option you sold to the issuer). When rates fall, the issuer's option is exercised, capping your gains. When rates rise, you suffer the full duration loss without cushion.
Portfolio construction using duration and convexity
Institutional investors use both metrics in portfolio construction:
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Target duration: Decide your strategic interest rate sensitivity. "Our pension fund should have 8-year duration to match our liability horizon."
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Maximize convexity for that duration: Within the constraint of maintaining 8-year duration, tilt the portfolio to maximize positive convexity. This might mean:
- Buying longer-dated bonds with strong positive convexity (30-year Treasuries have higher convexity than 5-year Treasuries for the same duration).
- Avoiding callable bonds and mortgage-backed securities (negative convexity).
- Considering bonds with favorable coupon structures (lower coupons have higher convexity for a given duration).
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Monitor and rebalance: As rates move and duration drifts, rebalance to maintain the target duration while preserving convexity. This might mean selling bonds that have appreciated and buying new bonds—a process that keeps the portfolio aligned.
Real-world example: managing duration in 2022
In early 2022, the Federal Reserve began raising rates sharply. The fed funds rate rose from 0% to 3.25% by year-end—a historic move. Long-duration bond portfolios suffered significant losses.
A pension fund with a 7-year duration portfolio and positive convexity of 50 faced:
- Duration loss: −7 × 0.0325 = −22.75%
- Convexity gain: +½ × 50 × (0.0325)² = +0.265% (tiny, compared to the duration loss)
- Net loss: approximately −22.5%
The positive convexity provided a hair of relief but could not offset a 3%+ rate move. The lesson: convexity is valuable in volatile environments, but it cannot protect you from large duration losses if you are positioned for falling rates and rates rise instead. The fundamental risk is duration, not convexity.
However, investors who held very long-duration bonds (20+ year duration) with strong positive convexity fared somewhat better than those holding intermediate-duration bonds with similar convexity. The reason: when a 3%+ rate move occurs, the quadratic convexity term becomes more significant relative to the linear duration term. A 20-year duration bond with convexity 100 fell roughly:
−20 × 0.03 + ½ × 100 × (0.03)² = −0.60 + 0.045 = −55.5%
A 7-year duration bond with convexity 50 fell roughly:
−7 × 0.03 + ½ × 50 × (0.03)² = −0.21 + 0.0225 = −18.75%
The long-duration bond fell more in absolute terms, but the convexity effect was 4.5% vs. 2.25%, a relative advantage. This is why some "long bond" investors held their positions in 2022; the massive rate move meant convexity actually mattered.
Duration and convexity in portfolio risk assessment
Portfolio managers often compute two key metrics:
Effective Duration = −(Price_up − Price_down) / (2 × P₀ × Δy)
This measure captures the actual price sensitivity of the portfolio to yield moves, accounting for all convexity and option effects.
Effective Convexity = (Price_up + Price_down − 2 × P₀) / (P₀ × (Δy)²)
This measures the curvature of the portfolio's price-yield relationship.
For a portfolio with 6-year effective duration and 50 effective convexity:
- A 1% yield move produces roughly 6% price change, adjusted by convexity.
- A 2% yield move produces roughly 12% change plus 2% convexity benefit.
- A 3% yield move produces 18% change plus 4.5% convexity benefit.
These metrics allow managers to stress-test portfolios. They ask, "If rates spike 3%, what happens?" and compute the expected loss using duration and convexity. If the loss is unacceptable, they reduce duration or increase convexity.
The ideal portfolio: high duration, high convexity
In an ideal world, every portfolio would have high duration (interest rate sensitivity) AND high positive convexity (asymmetrical gains). But trade-offs exist:
- Low-coupon, long-maturity bonds (like 30-year zero-coupon bonds) have very high positive convexity but are illiquid and highly volatile.
- High-coupon bonds have lower convexity and are less appealing to investors seeking convexity.
- Callable bonds offer higher yield but negative convexity (a bad trade).
The best practical approach is to maximize convexity subject to other constraints (yield, credit risk, liquidity, tax efficiency). A portfolio of:
- Non-callable government bonds (3% of convexity per duration unit).
- Investment-grade non-callable corporates (2–3% of convexity per duration unit).
- Minimal MBS or callable bonds.
would have strong positive convexity while maintaining reasonable yields and liquidity.
Duration-convexity scatter plots
Professional managers sometimes plot portfolios on a scatter:
- X-axis: duration.
- Y-axis: convexity.
A portfolio in the upper-right region (high duration, high convexity) is ideal for long-term investors expecting volatile rates. A portfolio in the upper-left (low duration, high convexity) suits defensive investors. A portfolio in the lower-right (high duration, low convexity) is risky; it is sensitive to rates but offers no convexity cushion.
The limits of duration and convexity
Despite their power, duration and convexity have limits:
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Non-parallel yield curve shifts: If short rates rise 2% but long rates rise only 1%, a portfolio of long bonds will behave differently than duration and convexity alone predict.
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Credit spread changes: If yields rise because credit spreads widen (not due to a parallel shift in rates), bonds of different credit quality will behave differently.
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Embedded options complexity: Callable bonds and MBS have option values that shift as rates move. The effective duration and convexity can change rapidly.
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Very large moves: For yield moves exceeding 3–4%, the convexity formula's higher-order terms become material. Even more sophisticated models using gamma and other measures may be needed.
For most practical investing purposes—deciding how much interest rate risk to take, comparing portfolios, predicting returns in typical scenarios—duration and convexity are sufficient and powerful.
Flowchart: Using duration and convexity together
Next
Duration and convexity are powerful tools for predicting bond prices. But in practice, portfolio managers face a more complex challenge: adjusting duration constantly as market conditions change. The next article explores portfolio duration management—how to actively adjust a portfolio's duration via fund swaps, futures, or tactical allocations to respond to changing market conditions and rate forecasts.