Macaulay and modified duration
Two distinct measures often confused, one expressing the weighted average maturity of cash flows in years, the other the relative price change for a one percent move in rates.
Macaulay duration is a length of time: the average of the cash flow dates weighted by present value, in years. It tells you when the money comes back on average, and it is the number you set against liability duration when matching. Modified duration is a sensitivity. It equals Macaulay duration divided by one plus the yield, and gives the relative change in market value for a change in rates directly. When rates are low the two nearly coincide; when rates rise the gap shows, which made the distinction operational again after 2022. Two limits govern their use. The relationship is linear, so it understates value for large rate moves, a gap convexity corrects. And it assumes a parallel shift of the whole curve, which fails as soon as the curve steepens or flattens; then only key rate durations give the real sensitivity.
Ten-year bond, 3% annual coupon, 3% yield. Macaulay duration 8.8 years, modified duration 8.5. A fifty basis point rate rise costs roughly 4.3% of market value according to modified duration. At the 0.5% yield that prevailed in 2021, the same bond showed a modified duration of 9.3, hence a 4.7% loss for the same shock: sensitivity itself depends on the level of rates.
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