Bond Duration - Convexity

Convexity

Duration is a linear measure of how the price of a bond changes in response to interest rate changes. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates. Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes. Specifically, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question, and the convexity as the second derivative.

Convexity also gives an idea of the spread of future cashflows. (Just as the duration gives the discounted mean term, so convexity can be used to calculate the discounted standard deviation, say, of return.)

Note that convexity can be positive or negative. A bond with positive convexity will not have any call features - i.e. the issuer must redeem the bond at maturity - which means that as rates fall, both its duration and price will rise.

On the other hand, a bond with call features - i.e. where the issuer can redeem the bond early - is deemed to have negative convexity as rates approach the option strike, which is to say its duration will fall as rates fall, and hence its price will rise less quickly. This is because the issuer can redeem the old bond at a high coupon and re-issue a new bond at a lower rate, thus providing the issuer with valuable optionality.

Mortgage-backed securities (pass-through mortgage principal prepayments) with US-style 15- or 30-year fixed rate mortgages as collateral are examples of callable bonds.