Modified duration and Macaulay duration are two ways to measure the sensitivity of a bond’s price to changes in interest rates. Both measures are used to estimate the price change of a bond in response to a change in the yield to maturity, which is the expected return on a bond if it is held until it matures.

Modified duration is a measure of the bond’s price sensitivity to changes in yield, taking into account the impact of periodic coupon payments. It is calculated by dividing the Macaulay duration by 1 + the yield to maturity. Modified duration is expressed in terms of years, and a higher modified duration indicates a greater price sensitivity to changes in yield.

Macaulay duration, on the other hand, is a measure of the average time it takes for a bond to pay back the investor’s initial investment, in terms of the bond’s cash flows. It is calculated by weighting each cash flow by the length of time until it is received, and then summing these values. Macaulay duration is expressed in terms of years, and a longer Macaulay duration indicates a longer expected payback period for the investor.

While both measures are used to estimate the price sensitivity of a bond to changes in yield, modified duration takes into account the impact of periodic coupon payments, while Macaulay duration does not. As a result, modified duration is generally considered to be a more accurate measure of a bond’s price sensitivity to changes in interest rates.