Convexity and Non-Linear Effects

Why duration alone isn't enough

7 min read

Why isn't duration enough?

Duration's Limitation

Duration assumes a linear relationship between price and yield. In reality, the relationship is curved (convex).

What is Convexity?

Convexity measures the curvature of the price-yield relationship. It's the second derivative of price with respect to yield (duration is the first derivative).

Why Convexity Matters

For small yield changes: Duration alone is sufficient
For large yield changes: Duration understates price increases and overstates price decreases
Convexity "corrects" duration's linear approximation

% Price Change ≈ (-Modified Duration × Δ Yield) + (0.5 × Convexity × Δ Yield²)

Convexity Example

Bond with Duration = 8, Convexity = 75. Rates fall 1%: • Duration effect: +8% • Convexity effect: +0.5 × 75 × 0.01² = +0.375% • Total: +8.375%

Positive Convexity (Good for Investors)

Standard bonds have positive convexity. Prices rise more when rates fall than they fall when rates rise. This asymmetric benefit favors the bondholder.

Negative Convexity (Caution)

Callable bonds can have negative convexity. When rates fall significantly, issuer may call the bond. Upside is capped; you lose the asymmetric benefit.

30Y BUND EXAMPLE

For a 30Y Bund with high convexity: • 1% rate drop → Price rises ~22% (more than duration suggests) • 1% rate rise → Price falls ~18% (less than duration suggests) Convexity creates this favorable asymmetry.

KEY TAKEAWAY

Convexity is duration's partner. For large rate moves, it's the difference between a good estimate and reality.

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