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 澳洲幸运5官方开奖结果体彩网:weighted average 澳洲幸运5官方开奖结果体彩网:term to maturity of the cash flows from the bond.

The weight of each cash flow i✤s determined by dividing the present value of the cash flow by the price.

Macaulay duration is often used by 澳洲幸运5官方开奖结果体彩网:portfolio managers who use an immunization strategy. That is, they build a por🐼tfolio that is shielde🌠d from adverse changes in interest rates.

Understanding the Macaulay Duration

Macaulay duration can be viewed as the economic balance point of a group of cash flows. It is the weighted average number of years that an investor 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 Maca⭕ulay.

Calculating the Macaulay Duration

Macaulay duration can be calculated as follows:

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

Factors Affecting Duration

A bond’s price, maturity, coupon, and 澳洲幸运5官方开奖结果体彩网:yield to maturity 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 澳洲幸运5官方开奖结果体彩网:sinking fund in place, a scheduled prepayment before maturity, and 澳洲幸运5官方开奖结果体彩网:call provisions all lower a bond’s duration.

Calculation Example

The 澳洲幸运5官方开奖结果体彩网:calculation of Macaulay duration is straightforward. Let’s assume that a $1,000 face-value bond pays a 6% coupon and matures in t🔴hree 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:

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

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

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

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

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

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

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

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

A coupon-paying bond will always have its duration less than its time to maturity. In the example above, the dura🍨tion 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.