duration and convexity

When interest rates move, the present value of any stream of future cash flows moves the other way: rates up, values down. But not everything moves by the same amount. A 30-year bond lurches when rates wobble; a bond maturing next month barely flinches. Duration is the single number that captures how sensitive a value is to a change in interest rates — roughly, the percentage change in value for a 1 percent change in rates. Convexity is the correction that tells you duration is only an approximation, because the relationship between value and rates is curved, not a straight line.

Picture duration as a kind of average waiting time for the money. A zero-coupon bond paying once in 10 years has a duration of about 10 — long, sensitive, jumpy. A bond paying lots of coupons soon has a shorter duration even if it matures at the same time, because its money arrives earlier. A useful rule of thumb: a portfolio with duration 8 will lose roughly 8 percent of its value if rates rise 1 percent. Convexity refines this: because the price-rate curve bows, the actual loss when rates rise a lot is a little smaller, and the actual gain when rates fall a lot is a little larger, than duration alone predicts. Positive convexity is therefore a friend — it cushions you on both sides.

Why it matters: liabilities have duration too. A book of long-dated annuities might have a duration of 12, meaning its value swings hard with rates. The whole art of immunization is to give your assets the same duration (and ideally more convexity) than your liabilities, so the two move together and the net position barely flinches when rates change. Duration is thus the common ruler that lets actuaries compare assets and liabilities on the same scale. The caveat: duration assumes a small, parallel shift in the whole curve; for big moves or a change in the curve's shape it is only a first guess, which is exactly why convexity and full modeling exist.