Convexity

What is it?

Convexity measures the curvature of the relationship between a bond's price and its yield. Modified duration describes that relationship as a straight line, but the true relationship bends: prices rise faster when yields fall than they drop when yields rise. Convexity quantifies that bend and corrects the duration estimate.

The following formula is used:

Convexity = ∑ ( CashFlow × Time × (Time + 1) ÷ (1 + Yield) ^ Time ) ÷ ( BondPrice × (1 + Yield) ^ 2 )

Combined with duration, the price change is estimated as:

Price Change ≈ ( – Modified Duration × Yield Change + ½ × Convexity × Yield Change ^ 2 ) × BondPrice

BondPrice – current market price of the bond, including accrued interest.

CashFlow – coupon payment, or coupon plus redemption amount for the final one.

Time – time from today until the cash flow, in years.

Yield – annual rate used for discounting, taken as the bond's yield to worst.

Yield Change – change in yield, expressed as a decimal (0.01 for one percentage point).

Convexity can be ignored for small rate movements, but not for large ones, where duration alone overstates losses and understates gains. It always works in the holder's favor, so among two bonds with the same duration the one with greater convexity is preferable.

It is measured in years squared, ranging from single digits on short issues to several hundred on thirty-year ones.