Positive Convexity

A bond exhibits positive convexity when its price increases more rapidly for a given fall in yield than it decreases for an equivalent rise in yield. This is the typical behavior of standard bonds. The intuition: as yields fall, duration lengthens (creating larger gains), and when yields rise, duration shortens (limiting losses).

The math of positive convexity

Bond prices move inversely to yields. A 100-basis-point (1%) drop in yield raises a bond price by more than a 1% rise in yield lowers it. The precise magnitude of these moves depends on both duration and convexity.

Mathematically, the price change is:

• Approximation 1 (linear): ΔPrice ≈ –Duration × ΔYield
• Approximation 2 (convexity-adjusted): ΔPrice ≈ –Duration × ΔYield + 0.5 × Convexity × (ΔYield)²

The second term is always positive (whether yield rises or falls). If convexity is 100 and yield changes by 1%, the convexity contribution is +0.5%. This means:

• Yield down 1%: Price up ≈ Duration + 0.5% convexity benefit = larger gain.
• Yield up 1%: Price down ≈ Duration, offset by 0.5% convexity cushion = smaller loss.

This asymmetry — larger gains on declines, smaller losses on rises — is the definition of positive convexity.

Why standard bonds have positive convexity

A bond with positive convexity has no embedded options. The issuer cannot force redemption, and the bondholder cannot force early prepayment. Each promised cash flow will be paid in full.

When yields fall, the bondholder benefits twice:

1. Existing coupons are worth more (discounted at lower rates).
2. The entire cash flow stream extends further into the future (because duration lengthens), amplifying the rate-decline benefit.

When yields rise, the bondholder loses, but the second effect reverses: duration shortens (the cash flow stream seems less distant), so the loss is contained.

This is a free option for the bondholder: you get the good asymmetry without paying for it (beyond the general bond yield).

Positive convexity quantified

A bond with duration of 5 years and convexity of 75 faces:

• Yield down 1%: Price change ≈ –5 × (–0.01) + 0.5 × 75 × (0.01)² = 5% + 0.375% = 5.375%
• Yield up 1%: Price change ≈ –5 × (0.01) + 0.5 × 75 × (0.01)² = –5% + 0.375% = –4.625%

The asymmetry is visible: a 1% yield move gains 5.375% but loses only 4.625%.

Convexity values for typical bonds range from 50 to 200+. Longer-duration bonds have higher convexity; shorter-duration bonds have lower convexity. A 30-year Treasury might have convexity of 600+, while a 2-year Note might have convexity of 3.

Positive convexity vs negative convexity

Positive convexity is the standard case and favors bondholders. Negative convexity (or “negative gamma”) occurs when the bond contains embedded options.

Callable bonds and negative convexity

A callable bond gives the issuer the right to redeem the bond before maturity, usually when interest rates have fallen (making refinancing attractive for the issuer).

When yields fall sharply:

• A standard bond gains substantially.
• A callable bond gains less, because the issuer is likely to call it (redeeming it at par, capping the bondholder’s upside).

This is negative convexity: the upside is capped. The bondholder paid for this disadvantage via a higher yield at issuance, but the benefit is real: when yields fall, the callable bond underperforms a standard bond.

Mortgage-backed securities and negative convexity

Mortgage-backed securities carry an implicit call option: homeowners can refinance (prepay) when interest rates fall. As yields decline, prepayments accelerate, shortening the effective duration and limiting the bondholder’s price appreciation.

The convexity premium

Investors who want positive convexity must “pay” for it by accepting lower yields. A callable bond with negative convexity typically yields 50–200 basis points more than a comparable non-callable bond to compensate investors for the missing upside.

The trade-off:

• Buy a non-callable bond at 3.5% yield: You get positive convexity.
• Buy a callable bond at 4.2% yield: Higher current income, but negative convexity limits upside if rates fall.

Which is better depends on your yield forecast. If you expect yields to fall, the non-callable bond is better. If you expect yields to stay flat or rise, the extra current yield from the callable bond is attractive.

Convexity and portfolio management

Investors managing bond portfolios use convexity as a portfolio lever:

• Bullish on rates falling: Overweight longer-duration, non-callable bonds to maximize positive convexity benefit.
• Neutral or bearish on rates: Overweight shorter-duration or callable bonds to reduce convexity drag.

High-convexity positions are especially valuable in low-volatility environments, where yields are unlikely to move dramatically in either direction. In high-volatility environments, the convexity asymmetry pays off more often.

Practical hedging with convexity

A bond manager who expects interest rates to become more volatile might buy longer-duration, high-convexity bonds as a hedge. The positive convexity acts like a free option that benefits from volatility regardless of direction.

Conversely, a manager who expects stable rates might accept negative convexity (callable bonds) to capture the higher yield without fear that the missing upside will be realized.

Cross-links and further reading