Convexity

What Is Convexity?

Convexity describes the curve in the relationship between a bond's price and its bond yield. It is a second-order measure of interest rate risk, built on top of duration, which is the first-order measure. Duration estimates how much a bond's price moves for a small change in yield by treating that relationship as a straight line. In practice, the price-yield relationship is curved, and convexity quantifies that curve.

How It Works

For bonds with the same duration, a bond with higher convexity gains more in price when yields fall than it loses when yields rise by the same amount. Duration alone would predict equal gains and losses in either direction, so convexity corrects for that difference. Features such as embedded options can change the picture. A callable bond, for example, may be redeemed by the issuer when rates drop, which limits price gains and can produce negative convexity over some ranges of yields.

A common approximation for the percentage price change is:

ΔP/P ≈ −(Modified Duration × Δy) + ½ × Convexity × (Δy)²

Here Δy is the change in yield, expressed as a decimal. The first term captures the duration effect, and the second captures the convexity adjustment.

Example

Imagine two bonds with the same duration but different convexity. If yields move by the same amount in either direction, the bond with higher convexity will generally show a better price outcome, because its gain in a falling-rate scenario is larger than its loss in a rising-rate scenario. The difference becomes more noticeable for large yield moves, since the squared term in the formula grows quickly. A zero-coupon bond is a simple case where convexity is easy to see, because it has no interim payments that could shift its sensitivity.

Why It Matters to Investors

Convexity helps explain how a bond or bond portfolio may respond to large rate moves that duration alone would misjudge. Higher convexity is generally viewed as a cushion against rate swings, while negative convexity signals that price gains may be capped. Convexity estimates are less precise for very large yield changes, and convexity is a risk concept to understand rather than a guarantee of any particular return.