Bond Duration - Macaulay Duration

Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:

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 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 represents the PV of the payments, with the coupon circles getting smaller the further out they are and the final large circle including the final principal repayment. If these circles were put on a balance beam, the fulcrum of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.

For most practical calculations, the Macaulay duration is calculated using the yield to maturity to calculate the :

(2)

(3)

where:

• indexes the cash flows,
• is the present value of the th cash payment from an asset,
• is the cash flow of the th payment from an asset,
• is the yield to maturity (continuously compounded) for an asset,
• is the time in years until the th payment will be received,
• is the present value of all cash payments from the asset until maturity.

Macaulay gave two alternative measures:

• Expression (1) is Fisher–Weil duration which uses zero-coupon bond prices as discount factors, and
• Expression (3) which uses the bond's yield to maturity to calculate discount factors.

The key difference between the two is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield, not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.