Macaulay Duration

What Is the Macaulay Duration?

The Macaulay duration is the weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by partitioning the current value of the cash flow by the price. Macaulay duration is as often as possible utilized by portfolio managers who utilize an immunization strategy.

Macaulay duration can be calculated as follows:
$\begin &\text = \frac{ \sum_ ^ \left ( \frac{ t \times C }{ (1 + y) ^ t } + \frac{ n \times M }{ (1 + y) ^ n } \right ) }{ \text } \ &\textbf \ &t = \text \ &C = \text \ &y = \text \ &n = \text \ &M = \text \ &\text = \text \ \end$

Grasping the Macaulay Duration

The measurement is named after its maker, Frederick Macaulay. Macaulay duration can be seen as the economic balance point of a group of cash flows. One more method for interpretting the statistic is that it is the weighted average number of years that a investor must keep a position in the bond until the current value of the bond's cash flows equals the amount paid for the bond.

Factors Affecting Duration

A bond's price, maturity, coupon and yield to maturity all factor into the calculation of duration. All else being equivalent, duration increases as maturity increases. As a bond's coupon increases, its duration diminishes. As interest rates increase, duration diminishes and the bond's sensitivity to additional interest rate increases goes down. Likewise, a sinking fund in place, a scheduled prepayment before maturity, and call provisions all lower a bond's duration.

Model Calculation

The calculation of Macaulay duration is clear. We should expect that a $1,000 face-value bond pays a 6% coupon and develops in three years. Interest rates are 6% per annum, with semiannual compounding. The bond pays the coupon two times per year and pays the principal on the last payment. Given this, the accompanying cash flows are expected throughout the next three years:
$\begin &\text{Period 1}: $30 \ &\text{Period 2}: $30 \ &\text{Period 3}: $30 \ &\text{Period 4}: $30 \ &\text{Period 5}: $30 \ &\text{Period 6}: $1,030 \ \end$
With the periods and the cash flows known, a discount factor must be calculated for every period. This is calculated as 1 \u00f7 (1 + r)ⁿ, where r is the interest rate and n is the period number being referred to. The interest rate, r, accumulated semiannually is 6% \u00f7 2 = 3%. In this way, the discount factors would be:
$\begin &\text{Period 1 Discount Factor}: 1 \div ( 1 + .03 ) ^ 1 = 0.9709 \ &\text{Period 2 Discount Factor}: 1 \div ( 1 + .03 ) ^ 2 = 0.9426 \ &\text{Period 3 Discount Factor}: 1 \div ( 1 + .03 ) ^ 3 = 0.9151 \ &\text{Period 4 Discount Factor}: 1 \div ( 1 + .03 ) ^ 4 = 0.8885 \ &\text{Period 5 Discount Factor}: 1 \div ( 1 + .03 ) ^ 5 = 0.8626 \ &\text{Period 6 Discount Factor}: 1 \div ( 1 + .03 ) ^ 6 = 0.8375 \ \end$
Next, increase the period's cash flow by the period number and by its relating discount factor to find the current value of the cash flow:
$\begin &\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 = 1} ^ {6} = $5,579.71 = \text \ \end$
$\begin &\text = \sum_{\text = 1} ^ {6} \ &\phantom{ \text } = 30 \div ( 1 + .03 ) ^ 1 + 30 \div ( 1 + .03 ) ^ 2 \ &\phantom{ \text = } + \cdots + 1030 \div ( 1 + .03 ) ^ 6 \ &\phantom{ \text } = $1,000 \ &\phantom{ \text } = \text \ \end$
(Note that since the coupon rate and the interest rate are something similar, the bond will trade at par.)
$\begin &\text = $5,579.71 \div $1,000 = 5.58 \ \end$
A coupon-paying bond will constantly have its duration not exactly its chance to maturity. In the model over, the duration of 5.58 half-years is not exactly the opportunity to maturity of six half-years. At the end of the day, 5.58 \u00f7 2 = 2.79 years, which is under three years.