We can also express Modified duration as follows:
...Eq. (3)
The expression in the brackets of the modified duration formula is given by equation (3) and is a measure created by Frederick Macaulay in 1938. This measure is known as Macaulay duration. Actually Macaulay duration is nothing but the weighted-average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price. Rewriting the formula of modified duration, we get: