Macaulay Duration

What Is Macaulay Duration?

Macaulay duration is a measure of how long, on average, an investor must wait to recover the cost of a bond through its payments. It is named after the economist Frederick Macaulay, who introduced the concept in 1938. Unlike a bond's maturity, which only marks the date of the final payment, Macaulay duration accounts for every payment, including the coupon payments made along the way.

How It Is Calculated

The calculation weights the time of each cash flow by its share of the bond's total value. Each payment is discounted to its present value, and that amount is then multiplied by the number of years until it arrives. The results are added together and divided by the bond's current price:

Macaulay Duration = (Σ t × PV of cash flow at time t) ÷ Bond Price

Here, t is the time until each payment in years, and the discounting uses the bond's yield to maturity.

Example

Consider a 3-year bond with a face value of 100, a 5% annual coupon, and a yield of 5%. Its price is 100. The three cash flows are 5, 5, and 105. Their present values are about 4.76, 4.54, and 90.70. Weighting each by its year (1, 2, and 3) and dividing by the price gives a Macaulay duration of about 2.86 years, slightly less than the 3-year maturity because the coupons arrive before the final payment.

Why It Matters to Investors

• It describes how long a bond's cash flows take to arrive, which helps compare bonds with different coupons and maturities.
• A zero-coupon bond has a Macaulay duration equal to its maturity, since its only payment comes at the end.
• Higher coupons generally shorten Macaulay duration, because more cash arrives early.
• Dividing Macaulay duration by one plus the yield per period gives modified duration, which estimates how much a bond's price changes for a given change in yield.
• Institutions sometimes match a portfolio's duration to a future liability in a strategy called immunization, which is intended to reduce the effect of interest rate risk on the portfolio's value.

Macaulay duration is a simplified model. It assumes a single, flat discount rate and does not capture every feature of a bond, such as embedded options.