FRM Exam Part I · Applying Duration, Convexity, and DV01

Macaulay, Modified and Effective Duration Explained

Updated 11 October 2026 · Fact-checked

Duration measures a bond's price sensitivity to yield changes. Macaulay duration is the present-value-weighted average time to cash flows, in years. Modified duration equals Macaulay ÷ (1 + y/m) and gives the % price change per unit yield change. Effective duration uses repriced bonds and suits bonds with embedded options.

Understand Macaulay, Modified and Effective Duration

Duration answers one question: if yields move, how much does the bond price move? Bond prices fall when yields rise. Duration gives you the size of that move as a percentage, to a first approximation.

Macaulay duration is a weighted average of the times at which you receive cash flows. Each time is weighted by the present value of the cash flow at that time, divided by the bond price. It is measured in years. A zero-coupon bond pays everything at maturity, so its Macaulay duration equals its maturity. A coupon bond has a shorter duration than its maturity, because some cash comes earlier.

Modified duration converts Macaulay duration into a price sensitivity. It equals Macaulay duration divided by (1 + y/m), where y is the yield and m is the compounding periods per year. If modified duration is 7, a 1% (100 bp) rise in yield lowers the price by about 7%. This is a first-order approximation, so it is accurate for small yield changes only.

Effective duration does not rely on a formula for cash flows. You reprice the bond with yields shifted down and up by the same amount, then measure the price change. It is the right measure when cash flows change with rates, as with callable, putable or mortgage-backed bonds. Modified duration assumes fixed cash flows, so it misleads for these bonds. A callable bond's effective duration is lower than that of an otherwise identical non-callable bond, and it falls as rates drop and the call becomes likely.

Special cases are common in exams. A zero-coupon bond has Macaulay duration equal to its maturity. A par floating rate bond that resets at each coupon date has duration about equal to the time to the next reset, so it is small.

Key formulas to remember

Macaulay duration: D_Mac = Σ [t × PV(CF_t)] ÷ P; t is time in years, P is the bond price (sum of PVs). Weights sum to 1.
Modified duration: D_Mod = D_Mac ÷ (1 + y/m); y is the annual yield, m is compounding periods per year. Use the yield on the same compounding basis.
Price change using modified duration: ΔP ÷ P ≈ −D_Mod × Δy; First-order approximation. Add convexity for large moves.
Effective duration: D_Eff = (P₋ − P₊) ÷ (2 × P₀ × Δy); P₋ is the price after yields fall by Δy, P₊ after yields rise by Δy, P₀ is the current price. Δy is in decimals.
Zero-coupon bond: D_Mac = T; Modified duration is then T ÷ (1 + y/m).
Par bond with annual coupon c and yield y = c: D_Mac = (1 + y) ÷ y × [1 − 1 ÷ (1 + y)^T]; A shortcut that holds for a par bond with annual coupons.
Floating rate bond: D ≈ time to next reset; Priced near par at reset dates, so duration is small.

How to solve Macaulay, Modified and Effective Duration questions

Decide first which measure the question wants and whether cash flows are fixed. Then follow these steps.

1Read what is asked: Macaulay (years), modified (% per 1% yield), or effective (option-embedded bonds).
2Check whether cash flows depend on rates. If yes, use effective duration with repriced bonds.
3For Macaulay, list each cash flow, its time t and discount it at the yield to get PV.
4Compute P as the sum of PVs, then compute Σ t × PV and divide by P.
5For modified duration, divide Macaulay by (1 + y/m). Match m to the coupon frequency.
6For effective duration, plug P₋, P₊, P₀ and Δy into the formula. Keep Δy in decimals (0.01 for 100 bp).
7Apply ΔP ÷ P ≈ −D × Δy and check the sign: yields up means price down.
8Sanity check: duration should not exceed maturity for a plain coupon bond.

Quickest way: Shortcut using special cases and the 2-sided formula

When to use it: Use when the bond is a zero, a par bond or a floater, or when prices are given and you only need effective duration.

Zero-coupon: Macaulay equals maturity. Divide by (1 + y/m) for modified.
Floater at reset: duration equals time to next reset.
Prices given: compute (P₋ − P₊) ÷ (2 × P₀ × Δy) directly, no cash flow tables.
Estimate the price change as −D × Δy and pick the option closest to it.
Eliminate options with the wrong sign or a modified duration above Macaulay.

Common mistakes in Macaulay, Modified and Effective Duration

Treating modified duration as equal to Macaulay duration
Both are called duration and are close in value.
Fix: Divide Macaulay by (1 + y/m). Modified is always smaller for positive yields.
Using the wrong m in the divisor for semiannual bonds
Students divide by (1 + y) instead of (1 + y/2).
Fix: Match m to the yield quoting basis. Semiannual yield quotes use y/2.
Using modified duration for callable bonds
The formula is familiar, and the option is overlooked.
Fix: Use effective duration with repriced bond values that include the option.
Entering Δy as 1 instead of 0.01 in the effective duration formula
Confusing percent with decimal.
Fix: Write 100 bp as 0.01 and check the result is a sensible number of years.
Forgetting the negative sign when estimating price change
Duration is quoted as a positive number.
Fix: Use ΔP ÷ P ≈ −D × Δy. Yields up, price down.
Assuming a floating rate bond has duration equal to its maturity
Confusing it with a fixed-rate bond.
Fix: Its coupon resets, so duration is about the time to the next reset.

Worked examples

Example 1

A 2-year bond pays a 6% annual coupon on face value 100 and yields 6% (annual compounding). Find the Macaulay and modified duration.

Show the solution

Cash flows: year 1 = 6, year 2 = 106. Yield 6%.
PV1 = 6 ÷ 1.06 = 5.6604. PV2 = 106 ÷ 1.1236 = 94.3396.
Price P = 5.6604 + 94.3396 = 100.
Σ t × PV = 1 × 5.6604 + 2 × 94.3396 = 5.6604 + 188.6792 = 194.3396.
Macaulay = 194.3396 ÷ 100 = 1.9434 years.
Modified = 1.9434 ÷ 1.06 = 1.8334.

Answer: Macaulay duration ≈ 1.943 years; modified duration ≈ 1.833.

Example 2

A callable bond is priced at 102.00. If yields fall 50 bp its price is 103.20, and if yields rise 50 bp its price is 100.60. Find the effective duration and the approximate price change for a 25 bp rise.

Show the solution

P₋ = 103.20, P₊ = 100.60, P₀ = 102.00, Δy = 0.005.
Effective duration = (103.20 − 100.60) ÷ (2 × 102.00 × 0.005).
Numerator = 2.60. Denominator = 2 × 102 × 0.005 = 1.02.
D_Eff = 2.60 ÷ 1.02 = 2.549.
For a 25 bp rise: ΔP ÷ P ≈ −2.549 × 0.0025 = −0.00637, about −0.64%.

Answer: Effective duration ≈ 2.55; price falls about 0.64% for a 25 bp rise.

Exam tips

Identify bonds with embedded options at once and go to effective duration.
Learn the zero-coupon and floater special cases. They give fast, free marks.
Check units: Macaulay is in years, modified is in % per 1% yield change.
Expect the effective duration formula with prices given. Keep Δy in decimals.
Use a financial calculator's PV and cash-flow functions to speed up the Macaulay table, but verify P first.

Practice questions from Applying Duration, Convexity, and DV01

Macaulay, Modified and Effective Duration in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Macaulay, Modified and Effective Duration: frequently asked questions

What is the difference between Macaulay and modified duration?

Macaulay duration is a weighted average time to cash flows, in years. Modified duration is Macaulay divided by (1 + y/m) and measures the percentage price change for a unit yield change.

When do I use effective duration instead of modified duration?

Use effective duration when cash flows change with interest rates, as with callable, putable or mortgage-backed bonds. It reprices the bond after small yield shifts, so the option is reflected.

What is the duration of a zero-coupon bond?

Its Macaulay duration equals its time to maturity, because the only cash flow comes at the end. Modified duration is that maturity divided by (1 + y/m).

What is the duration of a floating rate bond?

A floater that resets to market rates is priced near par on reset dates. Its duration is about the time until the next reset, so it is small.