Convexity

Before you rely on it

What to check

Small yield move, duration is fine. Large move, add convexity or the estimate is knowingly wrong.

The common error

Where people go wrong with convexity

Stress-testing a portfolio for a two-point rate shock using duration alone. The whole point of a stress test is the large move, and that is precisely where the straight-line approximation stops working.

Worked through

Convexity in plain numbers

A bond has a modified duration of 5. Duration alone says a one-point rise in yields cuts the price about 5%, and a one-point fall lifts it 5%. In practice the fall lifts the price a little more than 5% and the rise cuts it a little less. Say convexity is 40. The extra price effect is roughly ½ × 40 × (0.01)², which is 0.2%. So a one-point drop adds about 5.2% and a one-point rise subtracts about 4.8%; the curve bends upward. That asymmetry favors the holder, and it grows with the size of the move.

Easily confused

Convexity and its neighbours

Convexity is the second-order companion to duration. Duration is the slope of the price-yield curve and convexity is its curvature; the two together give a much better estimate of price changes than duration alone, especially for large moves; it is not the same as volatility or credit risk. Bonds with call options can have negative convexity, where price gains flatten as yields fall, because the issuer can redeem early. Mortgage-backed securities show this often. Positive convexity is generally desirable, and investors sometimes pay for it.

In real decisions

Where convexity turns up

Bond managers compare convexity across holdings, and it matters more when rates could move a lot; a barbell portfolio of short and long bonds tends to have higher convexity than a bullet portfolio with the same duration. Risk systems include it when stress-testing for large rate shocks. Traders of interest-rate derivatives price it into options and swaps. For an individual holding a single bond to maturity, it matters little, but for funds that trade it can add up.

Know the edges

Where convexity stops being reliable

Like duration, convexity is an approximation, and it assumes parallel shifts in yields and fixed cash flows. Real curves twist and bonds can be called or prepaid. Formulas differ by convention, so the scale of a quoted number varies by provider. Very large moves need full repricing rather than the approximation, and treat convexity as a refinement to duration, useful for understanding the shape of risk, not as a forecast of returns.

Quick answers

What people ask about convexity

Is higher convexity better? Usually yes for the holder, since price gains from falling yields exceed losses from rising ones. The market often charges for it through a lower yield.

Which bonds have negative convexity? Callable bonds and many mortgage-backed securities. When yields fall, borrowers refinance or the issuer calls the bond, so the price gain is capped.