Convexity
The second-order effect that separates good traders from great ones.
Why Duration Isn't Enough
In March 2020, as COVID panic gripped markets, 10-year Treasury yields crashed from 1.9% to 0.5% in three weeks. A 10-year bond with 8 years duration should have gained about 11% (8 x 1.4%). Instead, it gained 14%.
That extra 3%? Convexity.
Duration tells you the first-order effect: how much price changes for a small yield move. But the bond price-yield relationship is curved, not linear. For big moves, duration understates gains when yields fall and overstates losses when yields rise.
Convexity is the curvature. And for plain vanilla bonds, it always works in your favor - you gain more than expected when yields fall, and lose less than expected when yields rise. It's like free insurance.
The traders who understood convexity in March 2020 owned high-convexity zeros and ultra-long Treasuries. They massively outperformed. Those short convexity - MBS portfolios, callable bond holders - watched their positions underperform by hundreds of basis points.
Learn From History
Convexity shows its power in big moves. Select a period to see how it mattered:
COVID Crash - The Convexity Hedge Saved Portfolios
+2-3% extra returnThe Setup
Markets panicked. 10Y yields crashed from 1.9% to 0.5% in three weeks - a 140bp move. Duration alone predicted a 12% price gain for long bonds. But convexity added another 2-3%.
What Happened
Traders long high-convexity zeros and ultra-long Treasuries saw gains of 15-18%. Those short convexity (MBS holders, callable bond portfolios) saw massive underperformance as their bonds refused to rally.
When Convexity Works (And When It Kills You)
Positive Convexity (Option-Free Bonds)
- Bullets: Non-callable Treasury and corporate bonds. Pure positive convexity.
- Zero-coupon bonds: Maximum convexity per unit of duration. All cash flow at maturity.
- Ultra-long bonds: 30Y and beyond. High convexity due to long maturity.
- Low-coupon bonds: More cash flow weighted toward maturity = higher convexity.
Result: You gain more than expected when yields fall, lose less than expected when yields rise. Convexity is your friend.
Negative Convexity (Embedded Options)
- Callable bonds: Issuer can call when rates fall. Your upside is capped.
- MBS (mortgage-backed securities): Homeowners refinance when rates drop, prepaying your principal at the worst time.
- Puttable bonds (from issuer's view): Investor can put back when rates rise.
- Short options on rates: Sold caps, floors, or swaptions.
Result: Your position extends when rates rise (you get longer when you want shorter) and contracts when rates fall (you get shorter when you want longer). Worst of both worlds.
The Curvature Explained
Think of the price-yield relationship as a curve, not a line. Duration is the slope at your current point. Convexity is how fast that slope changes.
When yields fall: The actual curve lies above the duration tangent. Your bond gains MORE than duration predicts.
When yields rise: The actual curve lies above the duration tangent. Your bond loses LESS than duration predicts.
The math: For a 100bp move, convexity of 80 adds approximately 0.4% to your return (0.5 x 80 x 0.01^2 x 100 = 0.4%). For a 200bp move, it adds 1.6%. The benefit scales with the square of the yield move.
Duration + Convexity price approximation
The first term is duration (negative because prices move opposite to yields). The second term is convexity (always positive for bullet bonds). Notice convexity scales with the SQUARE of the yield change - that's why it matters more for big moves.
Build a Convexity-Conscious Position
Trade Legs
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Trade Examples (Simple)
Here's how convexity plays out in real trades with repo-financed positions:
Example 1: Convexity as Free Insurance
You're bullish on rates falling. You can buy either: (A) a 10Y 4% coupon bond, or (B) a 30Y 3% coupon zero-like bond. Both have roughly $10M DV01 when scaled appropriately.
Both positions repo-financed with 5% margin. The 10Y costs $10M notional, the 30Y costs ~$3M notional (higher duration per dollar). Both have ~$85K DV01. The 30Y has 3x the convexity.
10Y gains ~8.5% on duration. 30Y gains ~20% on duration PLUS an extra 2% from convexity = 22% total. The 30Y outperforms by $60K on equal DV01.
10Y loses ~8.5% on duration. 30Y loses ~20% BUT convexity saves you 2% = -18% actual. Still worse than the 10Y, but better than duration alone suggested.
High-convexity bonds outperform on big rallies and underperform less on selloffs. It's asymmetric - you win more when right, lose less when wrong. That's why traders pay up for convexity.
Example 2: The MBS Trap (Negative Convexity)
You buy $10M of 30-year agency MBS yielding 6.5%, thinking you're getting 150bp pickup over Treasuries. Duration is 4 years. Convexity is NEGATIVE (-150).
Same repo financing at 5% margin. You're earning 1.5% spread over repo - looks like a carry trade. But you're short convexity. With 20x leverage, that 1.5% spread = 30% ROE annually if rates stay flat.
Duration says +4%. But homeowners refinance. Prepayments accelerate. Your duration shrinks to 2 years. Actual gain: only +2%. Treasury bullet gained 4%. You underperform by $200K.
Duration says -4%. But nobody refinances now. Your duration EXTENDS to 6+ years. Actual loss: -6%+. You lose more than expected because your position got longer at the worst time.
Negative convexity means you're short an option. When rates fall, borrowers refinance (exercise against you). When rates rise, they don't prepay (you're stuck long). You lose both ways relative to bullets. That extra yield is compensation for this risk.
Example 3: Convexity-for-Carry Trade
Sell a 10Y Treasury at 4.5%, buy an agency MBS at 6.0%. Duration-neutral. You pick up 150bp of yield. The MBS has negative convexity; you're effectively short volatility.
Long MBS financed at repo. Short Treasury delivers into futures or borrows at special rate (often cheaper). Net financing might be 20-30bp negative. You keep the 150bp yield pickup minus financing = ~120bp net carry. On $10M, that's $120K/year.
You collect your 120bp spread all year. Duration-neutral means parallel shifts don't hurt. +$120K. The trade works.
Convexity gap blows up. If rates fall 100bp: MBS underperforms by 1-2% due to negative convexity. If rates rise 100bp: MBS extends, underperforms by 1-2%. Either way: -$100-200K. Your carry gets wiped out.
This trade makes money when rates are quiet and loses when they move big. You're selling volatility (convexity) for income. Works great in calm markets. Blows up in volatile ones. Know what you're trading.