Exam FMInterest rate riskFree to read

Macaulay duration

The present-value-weighted average time to the cash flows — the single number that says how long a bond really is.

The formulas

Definition: $D_{M a c} = \frac{\sum _{t} t C _{t} v ^{t}}{\sum _{t} C _{t} v ^{t}}$
Zero-coupon bond: $D_{M a c} = n$
Level annuity: $D_{M a c} = \frac{( I a ) _{\overline{n} ∣}}{a _{\overline{n} ∣}}$
Perpetuity: $D_{M a c} = \frac{1 + i}{i}$

Where it comes from

Each cash flow contributes a weight equal to its share of the total present value; duration is the mean payment time under those weights.
A zero-coupon bond has one cash flow, so its duration is its term — the anchor every other case is compared to.
For a coupon bond some value arrives early, so the duration is strictly less than the term.

Worked example

Find the Macaulay duration of a 5-year 1,000 par bond with 6% annual coupons at a yield of 6%.

Cash flows are 60 at times 1–4 and 1,060 at time 5; the price at a 6% yield is 1,000 (it prices at par).
$D = \frac{\sum t C _{t} v ^{t}}{1 , 000}$ with $v = 1/1.06$ .
Working the sum gives $D_{M a c} = 4.4651$ years.
Less than 5, because four coupons arrive before maturity.

Answer: 4.4651 years

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

Duration is a weighted AVERAGE TIME, measured in years. If your answer is not in years, you computed something else.
Zero-coupon: D = n. Perpetuity: D = (1+i)/i. Everything else is in between.

Traps

Weighting by the cash flows themselves rather than by their present values.
Reporting modified duration when the question asked for Macaulay (they differ by the factor 1 + i).

Drill this: the Exam FM question bank has original questions on this topic, and today’s free round is open to everyone.

Macaulay duration

The formulas

Where it comes from

Worked example

Memory hooks

Traps

Related