# Macaulay Duration: Definition, How It Works, Formula, and Example

> **Definition**
>
> The Macaulay duration is the weighted average term to maturity of the cash flow from a bond, or the point at which the bond's value will equal its purchase price.

## What Is the Macaulay Duration?

The Macaulay duration is a formula that tells an investor the time it will take for a bond to reach profitability. It measures the [weighted average](https://www.investopedia.com/terms/w/weightedaverage.asp) [term to maturity](https://www.investopedia.com/terms/t/termtomaturity.asp) of the cash flows from the bond.

The weight of each cash flow is determined by dividing the present value of the cash flow by the price.

Macaulay duration is often used by [portfolio managers](https://www.investopedia.com/terms/p/portfoliomanager.asp) who use an immunization strategy. That is, they build a portfolio that is shielded from adverse changes in interest rates.

### Key Takeaways

*   The Macaulay duration is the weighted average number of years that a bond must be held until the present value of its bond’s cash flows equals the amount paid for the bond.
*   The bond’s price, maturity, coupon, and yield to maturity all factor into the Macaulay duration calculation.
*   The formula can be used to reveal a bond's sensitivity to changes in interest rates.

![Macaulay Duration](https://www.investopedia.com/thmb/xpr57FpI-xPCdnCB5--4q1R16WM=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/macaulayduration.asp-final-708691725ab74308bcbdc7951f92650a.png)

*— Investopedia / Julie Bang*

## Understanding the Macaulay Duration

Macaulay duration can be viewed as the economic balance point of a group of cash flows. It is the [weighted](https://www.investopedia.com/terms/w/weighted.asp) average number of years that an [investor](https://www.investopedia.com/terms/i/investor.asp) must keep the bond until the present value of the bond’s cash flows equals the amount paid for the bond.

The metric is named after its creator, Canadian economist Frederick Macaulay.

### Calculating the Macaulay Duration

Macaulay duration can be calculated as follows:

$\begin{aligned}&\text{Macaulay Duration} = \frac{ \sum_{t = 1} ^ {n} \frac{ t \times C }{ (1 + y) ^ t } + \frac{ n \times M }{ (1 + y) ^ n } }{ \text{Current Bond Price} } \\&\textbf{where:} \\&t = \text{Respective time period} \\&C = \text{Periodic coupon payment} \\&y = \text{Periodic yield} \\&n = \text{Total number of periods} \\&M = \text{Maturity value} \\\end{aligned}$  

## Factors Affecting Duration

A bond’s price, maturity, coupon, and [yield to maturity](https://www.investopedia.com/terms/y/yieldtomaturity.asp) all factor into the calculation of duration. All else being equal, duration increases as time to maturity increases. As a bond’s coupon increases, its duration decreases. As interest rates increase, duration decreases and the bond’s sensitivity to further interest rate increases goes down. Also, a [sinking fund](https://www.investopedia.com/terms/s/sinkingfund.asp) in place, a scheduled prepayment before maturity, and [call provisions](https://www.investopedia.com/terms/c/callprovision.asp) all lower a bond’s duration.

## Calculation Example

The [calculation of Macaulay duration](https://www.investopedia.com/ask/answers/051515/how-do-i-calculate-bonds-modified-duration-using-excel.asp) is straightforward. Let’s assume that a $1,000 face-value bond pays a 6% coupon and matures in three years. Interest rates are 6% per annum, with semiannual compounding. The bond pays the coupon twice a year and pays the principal on the final payment. Given this, the following cash flows are expected over the next three years:

$\begin{aligned} &\text{Period 1}: \$30 \\ &\text{Period 2}: \$30 \\ &\text{Period 3}: \$30 \\ &\text{Period 4}: \$30 \\ &\text{Period 5}: \$30 \\ &\text{Period 6}: \$1,030 \\ \end{aligned}$

With the periods and the cash flows known, a discount factor must be calculated for each period. This is calculated as 1 ÷ (1 + r), where r is the interest rate and n is the period number in question. The interest rate, r, compounded semiannually is 6% ÷ 2 = 3%. Therefore, the discount factors would be:

$\begin{aligned} &\text{Period 1 Discount Factor}: 1 \div ( 1 + .03 ) ^ 1 = 0.9709 \\ &\text{Period 2 Discount Factor}: 1 \div ( 1 + .03 ) ^ 2 = 0.9426 \\ &\text{Period 3 Discount Factor}: 1 \div ( 1 + .03 ) ^ 3 = 0.9151 \\ &\text{Period 4 Discount Factor}: 1 \div ( 1 + .03 ) ^ 4 = 0.8885 \\ &\text{Period 5 Discount Factor}: 1 \div ( 1 + .03 ) ^ 5 = 0.8626 \\ &\text{Period 6 Discount Factor}: 1 \div ( 1 + .03 ) ^ 6 = 0.8375 \\ \end{aligned}$

Next, multiply the period’s cash flow by the period number and by its corresponding discount factor to find the present value of the cash flow:

$\begin{aligned} &\text{Period 1}: 1 \times \$30 \times 0.9709 = \$29.13 \\ &\text{Period 2}: 2 \times \$30 \times 0.9426 = \$56.56 \\ &\text{Period 3}: 3 \times \$30 \times 0.9151 = \$82.36 \\ &\text{Period 4}: 4 \times \$30 \times 0.8885 = \$106.62 \\ &\text{Period 5}: 5 \times \$30 \times 0.8626 = \$129.39 \\ &\text{Period 6}: 6 \times \$1,030 \times 0.8375 = \$5,175.65 \\ &\sum_{\text{ Period } = 1} ^ {6} = \$5,579.71 = \text{numerator} \\ \end{aligned}$

$\begin{aligned} &\text{Current Bond Price} = \sum_{\text{ PV Cash Flows } = 1} ^ {6} \\ &\phantom{ \text{Current Bond Price} } = 30 \div ( 1 + .03 ) ^ 1 + 30 \div ( 1 + .03 ) ^ 2 \\ &\phantom{ \text{Current Bond Price} = } + \cdots + 1030 \div ( 1 + .03 ) ^ 6 \\ &\phantom{ \text{Current Bond Price} } = \$1,000 \\ &\phantom{ \text{Current Bond Price} } = \text{denominator} \\ \end{aligned}$

(Note that since the coupon rate and the interest rate are the same, the bond will trade at par.)

$\begin{aligned} &\text{Macaulay Duration} = \$5,579.71 \div \$1,000 = 5.58 \\ \end{aligned}$

A coupon-paying bond will always have its duration less than its time to maturity. In the example above, the duration of 5.58 half-years is less than the time to maturity of six half-years. In other words, 5.58 ÷ 2 = 2.79 years, which is less than three years.