Which statement best describes the relationship between Macaulay duration and modified duration?

• A. Macaulay duration equals modified duration.
• B. Modified duration equals Macaulay duration multiplied by (1 + yield).
• C. Macaulay duration is a measure of price sensitivity to small yield changes; modified duration is the same as duration.
• D. Modified duration equals Macaulay duration divided by (1 + yield) and approximates percentage price change for small yield changes.

Answer: (D)

The main idea is how to turn the time-weighted measure into a price-sensitivity measure. Macaulay duration is the weighted average time to receive the bond’s cash flows, expressed in years. To get a sensitivity to yields, you adjust it for compounding, giving modified duration by dividing Macaulay duration by (1 + yield per period). This modified duration is the actual first-order measure of how much the price is expected to change for a small change in yield (roughly the percentage price change, with the sign descending when yields rise). So the statement that modified duration equals Macaulay duration divided by (1 + yield) and that it approximates the percentage price change for small yield changes is the correct description. The other descriptions misstate the relationship or the roles of the two durations.