Macaulay duration: definition and calculation

What does Macaulay duration measure?

Macaulay duration is the average time you wait for a bond’s cash flows, where each payment counts in proportion to how much of the bond’s value it is worth today. A bond that returns most of its value early has a short duration. One that returns most of it at the end has a long one.

Two ideas build it. The time to receipt is how many years away each payment is. The present-value weights are each payment’s present value divided by the bond’s full price, so they add up to 1.

$$\text{MacDur} = \sum_{t=1}^{N} t \times \frac{PV(CF_t)}{PV^{Full}}$$

The weight on payment $t$ is its present value over the price, and the sum of $t$ times weight is the answer. Duration is measured in years once it is annualized. The previous lesson showed why it matters: it is the horizon at which price risk and reinvestment risk balance.

How do you calculate it?

You can build the table, use a one-line formula, or let a spreadsheet do it.

Macaulay durationthe same number, three routes

• Table of weighted times Memorize PV, weight, time × weight, add up; shows the idea
• Closed-form formula Use it one expression, on a coupon date; the fast exam route
• Spreadsheet DURATION function Know the idea handles a settlement date between coupons

The table shows what is going on. The closed form is faster on exam day, and on a coupon date it gives exactly the same number.

$$\text{MacDur} = \frac{1 + r}{r} - \frac{1 + r + N(c - r)}{c\left[(1 + r)^{N} - 1\right] + r} - \frac{t}{T}$$

Here $r$ is the yield per period, $c$ the coupon rate per period, $N$ the number of periods left, $t$ the days since the last coupon and $T$ the days in the period. On a coupon date $t/T = 0$. This is the closed-form duration formula .

Go deeper The spreadsheet check, and where duration goes next

A spreadsheet’s DURATION(settlement, maturity, coupon, yield, frequency) returns the annualized Macaulay duration and handles a settlement date between coupons. Use it to check a hand calculation. The next module turns this number into a price sensitivity: modified duration is the Macaulay duration divided by one plus the yield per period.

How do you handle a semiannual bond?

Run the whole calculation in periods and then annualize. Everything inside the formula (yield, coupon, number of periods) is per half-year, and the answer comes out in half-years.

$$\text{MacDur (years)} = \frac{\text{MacDur (periods)}}{m}$$

Here $m$ is the number of coupons per year, 2 for semiannual. The result is the annualized duration .

What changes between coupon dates?

Between coupon dates, every payment is closer by the fraction of the period that has passed, $t/T$, so every time to receipt drops by that fraction. The present values all rise by the same factor, so the weights do not change.

Macaulay duration between coupon dates is therefore the duration at the last coupon date minus $t/T$, measured in periods.

$$\text{MacDur}_{t} = \text{MacDur}_{0} - \frac{t}{T}$$

The fraction t/T is counted in coupon periods. For a semiannual bond, divide it by 2 before subtracting it from a duration quoted in years.

Go deeper Why duration jumps up after each coupon

Between coupon dates the duration falls by the elapsed fraction of a period. Take the 8-year, 5% annual bond at par: its duration on a coupon date is 6.786, so just before the next coupon it is 6.786 − 1 = 5.786. Once that coupon is paid, the bond is a 7-year bond with a duration of 6.076, a step up of 0.289. The payment that disappeared was the one due immediately, the earliest in the average, so what remains has a longer average wait. Over the long run duration still trends down as the maturity approaches.

What properties of Macaulay duration should you know?

Macaulay duration of a straight bondother things equal

• Zero-coupon bond Memorize equals its time to maturity (on a coupon date)
• Coupon bond Memorize always below its maturity
• Longer maturity Use it longer duration
• Higher coupon Use it shorter duration, as cash arrives sooner
• Higher yield Use it shorter duration, as late flows are discounted harder
• Time passing Use it shorter duration, with a small jump up after each coupon

A zero-coupon bond has one payment, so its weight is 1 and its duration is its time to maturity: a 5-year zero has a duration of 5.000 on a coupon date. Every other straight bond pays something earlier, which pulls the average time below the maturity. The bigger the coupon, the earlier the cash and the lower the duration; for the 5-year bond, an 8% coupon gives 4.357 years against 4.478 for 6%. A higher yield shrinks the present value of the late payments most, so it also lowers the duration.

Exam tip When you are asked which bond has the longest duration, start with the zero-coupon one, then the lowest coupon, then the lowest yield, then the longest maturity.