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开奖号码历史查询:weighted average 澳洲幸运5开奖号码历史查询:term to maturity of the cash flows from the bond.

The weight of each cash flow is determined by dividing the presen🔯t value of the cash flow by the price.

Macaulay duration is often used by 澳洲幸运5开奖号码历史查询:portfolio managers who use an immunization strategy. Tha🔯t is, they build a portfolio that is shielded 𒁃from 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 f𝓰lows equals the amount paid for th🌄e bond.

The metric is named after it൩s creator, Canadian economist Frederick Macaulay.

Calculating the Macaulay Duration

Macaulay duration can be calculated as follows:

$\begin{aligned}&\text{Macaulay Duration} = \frac{ \sum_{t = 1} ^ {n} \frac{ t \times C }{ (1 + y) ^ t } + \frac{ n \times M }{ (1 + y) ^ n } }{ \text{Current Bond Price} } \\&\textbf{where:} \\&t = \text{Respective time period} \\&C = \text{Periodic coupon payment} \\&y = \text{Periodic yield} \\&n = \text{Total number of periods} \\&M = \text{Maturity value} \\\end{aligned}$​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开奖号码历史查询: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开奖号码历史查询:sinking fund in place, a scheduled prepayment before maturity, and 澳洲幸运5开奖号码历史查询:call provisions all lower a bond’s duration.

Calculation Example

The 澳洲幸运5开奖号码历史查询:calculation of Macaulay duration is straightforward. Let’s ass🦩ume that a $1,000 face-value bond pa♎ys 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:

$\begin{aligned} &\text{Period 1}: \$30 \\ &\text{Period 2}: \$30 \\ &\text{Period 3}: \$30 \\ &\text{Period 4}: \$30 \\ &\text{Period 5}: \$30 \\ &\text{Period 6}: \$1,030 \\ \end{aligned}$​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. The interest rate, r, compounded semiannually is 6% ÷ 2 = 3%. Therefore,🌜 the discount factors would be:

$\begin{aligned} &\text{Period 1 Discount Factor}: 1 \div ( 1 + .03 ) ^ 1 = 0.9709 \\ &\text{Period 2 Discount Factor}: 1 \div ( 1 + .03 ) ^ 2 = 0.9426 \\ &\text{Period 3 Discount Factor}: 1 \div ( 1 + .03 ) ^ 3 = 0.9151 \\ &\text{Period 4 Discount Factor}: 1 \div ( 1 + .03 ) ^ 4 = 0.8885 \\ &\text{Period 5 Discount Factor}: 1 \div ( 1 + .03 ) ^ 5 = 0.8626 \\ &\text{Period 6 Discount Factor}: 1 \div ( 1 + .03 ) ^ 6 = 0.8375 \\ \end{aligned}$​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 flow by the period number and by its corresponding discount factor to find the present value of the🔜 cash flow:

$\begin{aligned} &\text{Period 1}: 1 \times \$30 \times 0.9709 = \$29.13 \\ &\text{Period 2}: 2 \times \$30 \times 0.9426 = \$56.56 \\ &\text{Period 3}: 3 \times \$30 \times 0.9151 = \$82.36 \\ &\text{Period 4}: 4 \times \$30 \times 0.8885 = \$106.62 \\ &\text{Period 5}: 5 \times \$30 \times 0.8626 = \$129.39 \\ &\text{Period 6}: 6 \times \$1,030 \times 0.8375 = \$5,175.65 \\ &\sum_{\text{ Period } = 1} ^ {6} = \$5,579.71 = \text{numerator} \\ \end{aligned}$​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 r♋ate and the inter🐼est rate are the same, the bond will trade at par.)

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

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