duration (Macaulay, modified)
/ MAK-uh-lay /
If you own a bond or hold a stream of future cash flows, two natural questions arise: on average, how long until you get your money back, and how much will the value swing if interest rates move? Duration answers both. In its first sense, it is the average time you wait for your money, weighting each payment by how much of the total present value it represents — so a payment of 100 in year ten counts more in the average than a payment of 1 in year two.
Macaulay duration is exactly that present-value-weighted average time, measured in years. Modified duration is a closely related number that measures price sensitivity: it is approximately the percentage change in value for a one-percentage-point change in interest rates. Modified duration equals Macaulay duration divided by (1 + i). So a bond with a modified duration of 7 will lose roughly 7 percent of its value if rates rise by one percentage point, and gain roughly 7 percent if they fall by one. Longer-dated, lower-coupon cash flows have higher durations and so are more interest-rate sensitive.
Duration is the workhorse measure of interest-rate risk in actuarial and investment practice. By matching the duration of assets to the duration of liabilities — duration matching, a core piece of immunization — an insurer or pension fund can make its surplus largely insensitive to small parallel shifts in rates. But duration is only a first-order, straight-line approximation: it is accurate for small rate changes and gets worse for large ones, and it assumes the whole yield curve shifts up or down together. The correction for larger moves is convexity, the next concept.
A bond has Macaulay duration 8 years at a 6 percent yield. Its modified duration is 8 divided by 1.06, about 7.55, so a 1-percentage-point rate rise drops its price by roughly 7.55 percent.
Macaulay duration is an average waiting time; modified duration turns it into a price-sensitivity figure.
Duration assumes a small, parallel shift of the whole yield curve; it underestimates the gain when rates fall and overestimates the loss when rates rise, because the true price-yield relationship is curved, not straight.