Convexity

Duration gives a straight line estimate of how a bond's price changes when yields move. But the true relationship between price and yield is curved. Convexity measures that curvature. For ordinary bonds, the curve means prices rise more when yields fall than they drop when yields rise by the same amount, which is good for investors. Convexity matters most for large yield moves, long dated bonds and securities with embedded options such as mortgages.

The price yield curve#

The formula#

Adding convexity to the duration estimate:

% price change ≈ - modified duration × Δy + ½ × convexity × (Δy)²

Because (Δy)² is always positive, positive convexity adds to the price whether yields rise or fall.

What affects convexity#

Negative convexity#

Callable bonds and mortgage backed securities can have negative convexity. When rates fall, borrowers repay early (homeowners refinance, companies call bonds), so the price rises less than a normal bond. When rates rise, prepayments slow, extending duration just as prices fall.

Convexity in portfolios#

• Barbell vs bullet: a barbell portfolio (short and long bonds) has more convexity than a bullet (intermediate bonds) of the same duration. Convexity is valuable when large rate moves are expected, but it typically comes at the cost of slightly lower yield. See Yield Curve Trades: Steepeners, Flatteners and Butterflies.
• Hedging: swaps and futures hedge duration, while options on rates hedge convexity.
• Convexity has a price: in efficient markets, more convexity usually means lower yield, because investors pay for its protection.

Convexity and options#

Positive convexity in bonds is closely related to positive gamma in options: both benefit from large moves. Negative convexity is like being short options. See Gamma.

Calculating convexity in practice#

Convexity can be computed from the bond's cash flows, but traders usually estimate it numerically by repricing the bond for a small rise and a small fall in yield:

convexity ≈ (P_down + P_up - 2 × P_0) / (P_0 × (Δy)²)

where P_down and P_up are prices after yields fall and rise by Δy. Spreadsheet functions and bond calculators do this automatically. See Bond Price, Duration and DV01 Calculator.

Frequently asked questions#

What is bond convexity?#

A measure of the curvature in the relationship between bond prices and yields, showing how duration changes as yields move.

Why is positive convexity good?#

Because prices rise more when yields fall than they decline when yields rise by the same amount.

What is negative convexity?#

A situation, common in callable bonds and mortgage securities, where price gains are limited when yields fall and losses grow when yields rise.

Next, learn the dollar measure of rate risk in DV01.