than 10%. Similarly, a two-year coupon bond will have a Macaulay duration of somewhat below 2 years and a modified duration of somewhat below 2%. Macaulay duration Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows, in which the time of receipt of each payment is weighted by the present value of that payment. The denominator is the sum of the weights, which is precisely the price of the bond. Consider some set of fixed cash flows. The present value of these cash flows is: The Macaulay duration is defined as: (1) where: indexes the cash flows, is the present value of the th cash payment from an asset, is the time in years until the th payment will be received, is the present value of all future cash payments from the asset. In the second expression the fractional term is the ratio of the cash flow to the total PV. These terms add to 1.0 and serve as weights for a weighted average. Thus the overall expression is a weighted average of time until cash flow payments, with weight being the proportion of the asset's present value due to cash flow . For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if cash flows are, in order, , then: with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond. Macaulay duration has the diagrammatic interpretation shown in figure 1. This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance