How is modified duration (Dmod) calculated from Macaulay duration (Dmac)?

Modified duration is a measure of a bond's price sensitivity to changes in interest rates, reflecting how much the price of a bond is expected to change for a 1% change in yield. It is derived from Macaulay duration, which is a weighted average time until cash flows are received and measures the time value of a bond’s cash flows.

The relationship between modified duration and Macaulay duration incorporates the bond's yield to maturity (y) and the number of periods per year (m) to adjust Macaulay duration for changing interest rates. Specifically, the formula for modified duration is:

[ D_{mod} = \frac{D_{mac}}{(1 + y/m)} ]

Here, the term ( (1 + y/m) ) adjusts the Macaulay duration downwards, accounting for the bond’s yield. This reflects how the present value of cash flows changes as interest rates rise or fall. When you divide Macaulay duration by this term, you account for the price sensitivity of the bond to changes in yield, thereby converting Macaulay duration into a measure that expresses how much the bond's price would change in response to shifts in interest rates.

In summary, the correct calculation of modified duration from