Convexity — when duration isn't enough

The curvature duration misses

Duration is a linear approximation. %ΔP ≈ −Duration × Δr works well for small yield changes (say, 25 bp). For larger moves, the approximation breaks down — duration overestimates losses when rates rise a lot, and underestimates gains when rates fall a lot. The correction is convexity.

The actual price-yield relationship is curved, not linear. Duration is the slope of the curve at a single point (the first derivative — Track 1 lesson 8 callback!). Convexity is the curvature (the second derivative). Together they give a much better approximation:

The convexity term is always positive (the price-yield curve bends upward), so it adds to gains when rates fall and softens losses when rates rise. Bond holders get a small bonus from convexity, both ways. It's "the bond holder's friend."

Why the curve bends

Bond price as a function of yield is P = sum of CF/(1+r)^t. The denominator is exponential in t. Plotting P against r gives a curve that drops as r rises but flattens out at high yields (price can never go below zero). That curvature is what duration misses and convexity catches.

The shape is asymmetric in a way that helps you: for the same change in r, the price gain when r falls is bigger than the price loss when r rises. That's the convexity bonus.

How much does convexity matter in practice?

For small rate moves (under ~50 bp), duration alone is fine. The convexity correction is in the second decimal place. For large moves (200+ bp like 2022), convexity matters meaningfully — it's why long Treasury ETFs lost "only" 30% in 2022 instead of the 50%+ that pure duration math would predict.

Higher-convexity bonds have:

• Longer maturities (more cash flows being affected by curvature)
• Lower coupons (cash returns later, more time-value impact)
• Zero-coupon bonds have the highest convexity for a given maturity

Lower-convexity bonds: short-maturity, high-coupon, or callable bonds (where the issuer's option caps your upside).

Practical decisions

For most retail bond holders, knowing duration is enough. Convexity is a second-order correction — it matters for fund managers, traders, and large rate moves, but not for everyday allocation decisions.

The one practical takeaway: convexity is your friend. When considering bond positions, the curvature works in your favor (asymmetric — you gain more from rate drops than you lose from rate rises). It's a small structural advantage in holding bonds.

The takeaway

Convexity = curvature of the price-yield relationship. The second derivative correction to duration's linear approximation. Always positive, always works in the bond holder's favor. Matters for large rate moves; small ones are fine with duration alone. Higher convexity for longer-maturity, lower-coupon bonds. Most retail investors can stick with duration; convexity is a refinement.