Lesson 1 of 6 · All lessons in this module

Free lesson: the first lesson of every module is open to everyone.

Lesson 1 of 6 · 14 min

From Macaulay duration to modified duration

Modified duration is Macaulay duration divided by one plus the yield per period, and it tells you roughly what percentage of its full price a bond loses or gains when its yield moves.

Video lesson · 7 min. Watch it first, then use the text below to go deeper.

In short

Yield duration measures price sensitivity to a change in the bond's own YTM, with cash flows assumed certain. Curve duration (later modules) uses a benchmark curve instead.
Macaulay duration = PV-weighted average time to receipt of the cash flows, with weights = each cash flow's share of the full price.
Modified duration = MacDur ÷ (1 + yield per period). It is the slope of the price-yield curve expressed as a percentage of price.
Estimated % change in full price ≈ −AnnModDur × ΔYield. Higher modified duration → steeper curve → more interest rate risk.
The estimate is linear, so it gives the same size of move for a rise and a fall in yield; the true curve is convex.
With a negative yield, 1 + r < 1, so modified duration is larger than Macaulay duration.

Measuring price risk with one number

Every option-free bond's price falls when its yield rises, but not by the same amount. A one-year zero barely moves when yields rise by 1 percentage point; a 30-year coupon bond can lose a sixth of its value. Investors need a number that ranks bonds by this sensitivity.

The natural number is the slope of the price-yield curve. A flat curve means little risk; a steep one means a lot. Duration is that slope, rescaled so it is easy to use.

This module covers yield duration statistics: Macaulay duration, modified duration, money duration and the price value of a basis point. They all assume the cash flows are certain and that only the bond's own yield-to-maturity changes, instantly, with everything else (including time to maturity) held fixed. Curve duration, which allows for uncertain cash flows and shifts in a benchmark yield curve, is not a yield duration and comes later.

Macaulay duration: a weighted average time

Macaulay duration (MacDur) is the average time until the investor receives the bond's cash flows, where each date is weighted by the share of the bond's full price that its cash flow represents. You met it in the previous module as the holding period at which reinvestment risk and price risk offset each other.

To compute it: discount each cash flow at the YTM, divide each PV by the full price to get its weight, multiply each weight by its time (in periods), and add up. The result is in periods; divide by the number of periods per year to annualize.

For a coupon bond, MacDur is always shorter than the maturity, because some value arrives early as coupons. The final cash flow usually carries most of the weight.

3-year, 6% annual-coupon bond priced at a 5% YTM (Example 1)
Year t	Cash flow	PV at 5%	Weight	t × weight
1	6	5.7143	0.0556	0.0556
2	6	5.4422	0.0530	0.1060
3	106	91.5668	0.8914	2.6742
Total		102.7232	1.0000	2.8358

Almost 90% of the price arrives in year 3, so the weighted average time is pulled close to maturity: MacDur = 2.84 years, just short of 3 because the two coupons carry a little weight early.

Modified duration: the slope as a percentage of price

If you differentiate the bond price formula with respect to the yield per period r, every term picks up its time t and an extra factor 1/(1 + r). Divide by the price and the sum that appears is exactly Macaulay duration. So the percentage slope of the price-yield curve is $-MacDur/(1+r)$ .

Dropping the minus sign gives modified duration: $ModDur = MacDur/(1+r)$ , where r is the yield per period. You do not need the calculus for the exam, only the result and what it means.

Like Macaulay duration, modified duration comes out in periods. Annualize it by dividing by the number of coupon periods per year. Equivalently, take the annualized MacDur and divide it by (1 + yield per period).

Spreadsheets and calculators give it directly: MDURATION in Excel or Google Sheets, and DUR in the BA II Plus Bond worksheet (which shows the annualized modified duration).

Macaulay duration is a time (years). Modified duration is a sensitivity (% price change per 1.00 = 100% change in yield). They are close in size because 1 + r is close to 1, but they answer different questions.

Estimating the price change, and its limits

Multiply both sides by the change in yield and you get the working rule: $\%\Delta PV^{Full} \approx -AnnModDur \times \Delta Yield$ . A bond with a modified duration of 7 loses about 7% for a 100 bp rise in yield and gains about 7% for a 100 bp fall.

Three details. The minus sign says price and yield move in opposite directions. The change applies to the full price (flat price plus accrued interest), not the flat price. And the ≈ sign is there because duration draws a straight line through a curved relationship.

Because the line is straight, the estimate for +80 bp and −80 bp has the same size. The true price-yield curve is convex: the real gain from a fall in yield is larger than the real loss from an equal rise. A later module adds a convexity adjustment to fix this.

Negative yields. When r < 0, the denominator 1 + r is below 1, so modified duration is greater than Macaulay duration. With a positive yield, modified is always the smaller of the two.

The 3-year, 6% bond from Example 1. Modified duration (2.70) is the slope of the dashed tangent at 5%, as a share of price. The straight line sits below the curve on both sides, so it overstates losses and understates gains.

Formulas

Macaulay duration (in periods): $MacDur = \sum_{t=1}^{N} t \times \dfrac{CF_t/(1+r)^t}{PV^{Full}}$
r = yield per period. Divide by periods per year to annualize. Weights are shares of the full price.
Modified duration: $ModDur = \dfrac{MacDur}{1+r} \qquad AnnModDur = \dfrac{AnnMacDur}{1+r}$
r = yield per period (annual YTM ÷ periods per year). Annualize by dividing by periods per year.
Modified duration as a slope: $\dfrac{dPV/dr}{PV} = -\dfrac{MacDur}{1+r} = -ModDur$
The percentage change in price per unit change in yield. Shown for intuition; no calculus is tested.
Estimated percentage price change: $\%\Delta PV^{Full} \approx -AnnModDur \times \Delta Yield$
ΔYield as a decimal (50 bp = 0.0050). Linear estimate: same size for rises and falls.

Worked examples

Example 1 Macaulay and modified duration of an annual-pay bond

A 3-year bond pays a 6% annual coupon and is priced at a 5% YTM on a coupon date. What are its Macaulay and modified durations, and what price change does duration predict for a 100 bp rise in yield?

Example 2 Semiannual bond: annualizing and estimating a price change

A 4-year bond pays a 4% coupon semiannually and yields 5%, settling on a coupon date. Its Macaulay duration is 7.4602 semiannual periods. Find the annualized modified duration and the estimated price change for a 50 bp rise in yield.

Example 3 Interpreting duration when yields are negative

A 4-year CHF zero-coupon bond yields −0.50% (annual compounding). Is its modified duration above or below its Macaulay duration, and what is the estimated price change if the yield rises to −0.30%?

On the exam

Expect a direct ModDur calculation from a given Macaulay duration (watch the per-period yield), a percentage price estimate for a given bp change, and concept items on full price, linearity and negative yields.

Dividing by 1 + the annual yield for a semiannual bond. Use 1 + the yield per period.
Using Macaulay duration in the price-change formula. The percentage estimate needs modified duration.
Forgetting the minus sign: a yield rise gives a price fall.
Applying the estimated change to the flat price. Duration statistics refer to the full price.

Check yourself

Q1An annual-pay bond has a Macaulay duration of 6.40 years and a YTM of 4%. Its modified duration is closest to:
Q2A semiannual-pay bond has an annualized Macaulay duration of 5.20 and a YTM of 6%. For a 25 bp rise in yield, the estimated change in its full price is closest to:
Q3A bond's annualized modified duration is 7. The most accurate interpretation is that for a 100 bp change in its YTM, its full price will change by approximately:
Q4The percentage price change estimated with modified duration refers to the bond's:
Q5A EUR bond trades at a YTM of −0.20%. Compared with its Macaulay duration, its modified duration is most likely:
Q6Duration estimates the price change for a 50 bp rise and a 50 bp fall in yield as equal in size. This is because duration:

0/6Check yourself

Answer the 6 questions: +10 XP for each one you get right first time.

Liked this lesson? Get the rest free for 7 days

Create a free account to open the other 5 lessons in this module, with videos, flashcards that come back before you forget them, the question bank and full mocks. No card needed.

Start your 7-day free trial Log in