Correcting the Curve - Convexity Explained

In Chapters 75 and 76, we used Duration to estimate how bond prices change.¹ However, duration has a major flaw: it assumes the relationship between prices and yields is a straight line.² In reality, this relationship is a curve.

Convexity is the mathematical measure of this curvature.³ It acts as a "correction factor" that makes your price predictions much more accurate, especially when interest rates make large moves.⁴

1. The "Linear Error" of Duration

Duration provides a linear approximation.⁵

• The Problem: If you use only duration, you will underestimate price gains when rates fall and overstate price losses when rates rise.⁶
• The Reality: Because of the curve, bond prices actually rise faster than they fall.⁷ Convexity accounts for this extra "boost" or "cushion".⁸

2. Positive vs. Negative Convexity

In the 2026 market, professional managers categorize bonds by their "shape":

I. Positive Convexity (The Standard)

• What it is: Most standard (non-callable) bonds have positive convexity.⁹
• The Benefit: When yields fall, the price rises more than duration predicts.¹⁰ When yields rise, the price falls less than duration predicts.¹¹
• The Catchphrase: "Gains accelerate, losses decelerate".¹²

II. Negative Convexity (The Risk)

• What it is: Common in Callable Bonds and Mortgage-Backed Securities (MBS).¹³
• The Danger: As rates fall, the issuer is likely to "call" the bond, capping your price gains.¹⁴ Conversely, if rates rise, the price can fall much faster than expected.
• The Catchphrase: "Gains are capped, losses are amplified".

3. The Math: Adding the Convexity Adjustment

To get a precise prediction of a bond's price move, you add the Convexity Adjustment to the Duration estimate:

%∆Price ≈ (-ModD x ∆y) +

Where,

Duration Effect = (-ModD x ∆y)

Convexity Adjustment =

Example: You have a bond with Duration = 9 and Convexity = 115. Interest rates rise by 1% (0.01).

1. Duration Effect: 9 x 0.01 = -9.00%
2. Convexity Adjustment: 0.5 x 115 x (0.01)² = +0.575%
3. Total Estimated Move: -9.00% + 0.575% = -8.425%

Without convexity, you would have wrongly predicted a 9% loss.

4. Why Convexity is Valuable in 2026

Investors often pay a "premium" for bonds with high convexity because they are more stable in volatile markets.¹⁵

• Maturity: Longer-maturity bonds have much higher convexity.¹⁶
• Coupon: Lower-coupon bonds (especially zero-coupons) have higher convexity.
• Yield: Convexity is more important and "valuable" when market interest rates are low.

Summary: Duration vs. Convexity