Skip to content

How bond duration works

Macaulay duration is the present-value-weighted time until a bond's cash flows arrive. A 5-year 5 percent annual coupon bond at a 5 percent yield, priced at par of $1,000, has a Macaulay duration of 4.55 years, a modified duration of 4.33 years, and a DV01 of $0.43.

Macaulay duration

4.55 years

Modified duration 4.33 years. A one basis point fall in yield lifts the price by about $0.43.

Price
$1,000.00
Coupon each period
$50.00
Macaulay duration
4.55 years
Modified duration
4.33 years
DV01
$0.43
$
%

Annual coupon as a percent of face. Paid in equal instalments at the frequency below.

%
yr
On this page

Next on Models and deals

DV01

In short

  • A 5-year 5 percent annual coupon bond at a 5 percent yield prices at $1,000. Each coupon is $50. Macaulay duration is 4.55 years, shorter than maturity, because some value arrives as coupons before the face.
  • Modified duration is Macaulay divided by 1+y/k1 + y/k. On this par bond that is 4.55/1.05=4.334.55 / 1.05 = 4.33 years. DV01 is modified duration times price over 10,000, so $0.43 per basis point.
  • A 10-year 6 percent coupon bond at a 5 percent yield, two payments a year, prices at $1,077.95. Each half-year coupon is $30. Macaulay 7.76 years, modified 7.57, DV01 $0.82.
  • A zero-coupon 5-year bond at 5 percent prices at $783.53. Macaulay duration is 5.00 years, equal to maturity, because nothing arrives before then. Modified 4.76 years, DV01 $0.37.
  • Macaulay duration is a wait. Modified duration is a percent sensitivity. DV01 is money. They are not interchangeable.

A weighted-average wait, then a sensitivity

Macaulay duration is the present-value-weighted time until a bond's cash flows arrive:

DMac=∑tt PV(CFt)PDMod=DMac1+y/kD_{Mac} = \frac{\sum_t t\, PV(CF_t)}{P} \qquad D_{Mod} = \frac{D_{Mac}}{1 + y/k}

On a 5-year 5 percent annual coupon bond at a 5 percent yield, the price is par, $1,000. Each coupon is $50. Macaulay duration is 4.55 years, shorter than 5, because the coupons pull weight forward. Modified duration is 4.33 years. DV01 is $0.43: the dollar change for a one basis-point fall in yield.

The bond duration calculator on this page prints all three. How bond pricing works is the PP in the denominator. How bonds work is the cash-flow schedule being weighted.

A hedge that needs 'how far the price moves' and is built on 4.55 rather than 4.33 is using the wait as a sensitivity. The names exist because the two numbers are not interchangeable.

Duration says how far one bond's price moves when its yield moves. The yield curve explorer is the line those yields sit on: a steepener and a flattener are two different shocks to two different points on that line.

Macaulay is a wait. Modified is a percent sensitivity. Macaulay against modified duration is the two numbers on one sheet, and why a hedge built on 4.55 rather than 4.33 is using the wait as if it were the slope.

Modified duration times price over 10,000 is money. How DV01 works owns the $0.43 on this par bond. This page owns the 4.55-year wait and the 4.33-year slope.

A premium coupon bond, semiannual

Face $1,000, 6 percent coupon, 5 percent yield, 10 years, two payments a year. Each half-year coupon is $30. The bond prices at $1,077.95, a premium because the coupon sits above the yield. Macaulay duration 7.76 years, modified 7.57, DV01 $0.82.

Duration is still shorter than maturity. The extra coupons, and the extra years of them, pull more weight into the middle of the schedule than a 5-year par bond had, but they do not pull it all the way out to year 10.

A zero is the one case where duration equals maturity

Face $1,000, coupon 0, yield 5 percent, 5 years, annual. Price $783.53. The only cash flow is at year 5, so Macaulay duration is 5.00 years. Modified duration is 4.76 years. DV01 is $0.37, less than the par bond's $0.43, because the price being shocked is $783.53 rather than $1,000.

Higher coupons pull duration in. Higher yields pull duration in, because they put more weight on the earlier cash flows. A zero has neither pull, so the wait equals the maturity.

What this page is not doing

It is not default risk, not convexity, and not a hedge recipe. Duration measures interest-rate sensitivity of a promised cash-flow schedule. A high-yield bond can have a short duration and a high chance of default at the same time. The three sheets are a 5-year par bond (Macaulay 4.55, DV01 $0.43), a 10-year premium (7.76, DV01 $0.82), and a 5-year zero (5.00, DV01 $0.37). This is educational material, not financial advice.

Worked examples

A 5-year par bond, annual coupons

Face $1,000, 5 percent annual coupon, 5 percent yield, 5 years, annual payments. Price, Macaulay duration, modified duration, DV01?

  1. Each coupon is 1000×0.05=501000 \times 0.05 = 50, so $50. At a 5 percent yield the bond is at par: price $1,000.
  2. Macaulay duration, the present-value-weighted wait, is 4.55 years.
  3. Modified duration: 4.55/1.05=4.334.55 / 1.05 = 4.33 years.
  4. DV01: 4.33×1000/10000=0.434.33 \times 1000 / 10000 = 0.43, so $0.43 per basis point.

The bond prices at $1,000. Macaulay duration is 4.55 years, modified duration 4.33 years, and DV01 is $0.43. Coupon each year is $50.

A 10-year premium bond, semiannual coupons

Face $1,000, 6 percent coupon, 5 percent yield, 10 years, two payments a year. Price and duration?

  1. Coupon each half-year: 1000×0.06/2=301000 \times 0.06 / 2 = 30, so $30.
  2. Price, discounting 20 coupons and the face at 2.50 percent a half-year: $1,077.95.
  3. Macaulay duration 7.76 years, modified duration 7.57 years, DV01 $0.82.

The bond prices at $1,077.95, a premium because the coupon sits above the yield. Macaulay duration is 7.76 years, modified duration 7.57 years, DV01 $0.82. Each half-year coupon is $30.

A zero-coupon 5-year bond at 5 percent

Face $1,000, coupon 0, yield 5 percent, 5 years, annual. Confirm Macaulay duration equals maturity.

  1. Price is 1000/1.055=783.531000 / 1.05^{5} = 783.53.
  2. The only cash flow is at year 5, so Macaulay duration is 5.00 years.
  3. Modified duration: 5/1.05=4.765 / 1.05 = 4.76 years. DV01: 4.76×783.53/10000=0.374.76 \times 783.53 / 10000 = 0.37.

Price $783.53. Macaulay duration is 5.00 years, equal to maturity, because nothing arrives before then. Modified duration is 4.76 years and DV01 is $0.37. The coupon each period is $0.

Common questions

Why is duration shorter than maturity?

Because some of the value arrives as coupons before the face is paid. On the first sheet, Macaulay duration is 4.55 years on a 5-year bond that pays $50 a year. Only a zero-coupon bond has duration equal to maturity, 5.00 years on the third sheet.

Is DV01 the same as duration?

No. DV01 is the dollar change in price for a one basis point move, which is modified duration times price over 10,000. On the par bond that is $0.43. Duration is a time or a percent. DV01 is money.

Does duration measure default risk?

No. It measures interest-rate sensitivity of a promised cash-flow schedule. A high-yield bond can have a short duration and a high chance of default at the same time.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.