# What is Macaulay Duration?

The duration of a fixed income instrument is a weighted average of the times that payments (cash flows) are made. The weighting coefficients are the present value of the individual cash flows.

$D=\frac{PV(t_0)t_0+PV(t_1)t_1++PV(t_n)t_n}{PV}$

where PV(t) denotes the present value of the cash flow that occurs at time t.

If the present value calculations are based on the bond’s yield, then it is called the Macaulay duration. Duration is an important concept in bonds. It is a measure of how long in years it takes for the price of a bond to be repaid by its internal cash flows. Bonds with higher durations are more risky and have higher price volatility than bonds with lower durations.

Example:

Calculate the Macaulay Duration for 9%, 5-year bond selling to yield 9% (compounded semi-annually).

 yield 0.09 Period CF PV t*PV/P 0.5 4.5 4.31 0.02 1 4.5 4.12 0.04 1.5 4.5 3.94 0.06 2 4.5 3.77 0.08 2.5 4.5 3.61 0.09 3 4.5 3.46 0.10 3.5 4.5 3.31 0.12 4 4.5 3.16 0.13 4.5 4.5 3.03 0.14 5 104.5 67.29 3.36 100.00 4.13

The Macaulay Duration is 4.13 years.