Macaulay vs modified duration
Macaulay duration is when the cash flows arrive, on average, in present-value terms. Modified duration is how far the price moves. On a 5-year 5 percent par bond of $1,000 they are 4.55 years and 4.33 years, and DV01 is $0.43. A hedge built on the wait is the wrong size.
| Macaulay duration | Modified duration | |
|---|---|---|
| What it measures | A weighted-average wait. | Price sensitivity to a yield change. |
| 5-year par bond | 4.55 years, on a $1,000 5 percent annual coupon at a 5 percent yield. | 4.33 years, which is 4.55 / 1.05. DV01 is $0.43. |
| 10-year premium, semiannual | 7.76 years, on a $1,077.95 price. | 7.57 years. DV01 is $0.82. |
| 5-year zero at 5 percent | 5.00 years, equal to maturity, on a $783.53 price. | 4.76 years. DV01 is $0.37. |
| When they match | They do not, except in the limit of continuous compounding. | The gap is the factor. On the par bond that factor is 1.05. |
| What a hedge needs | Not this number. Using 4.55 as a hedge ratio is 5 percent too large on this par bond. | This number, or DV01, which is this number times price over 10,000. |
On this page
A wait is not a sensitivity
On the par bond, each coupon is $50 and the price is $1,000. Macaulay duration is 4.55 years because some of that $1,000 arrives as $50 coupons before year 5. Modified duration is 4.33 years. DV01 is $0.43.
The 1.05 in the denominator is why they disagree. A hedge that needs how far the price moves and is built on 4.55 is using the wait. Macaulay duration is the wait. How bond duration works is the long form.
A zero is the one case the wait equals maturity
A 5-year zero at 5 percent prices at $783.53. Macaulay duration is 5.00 years, equal to maturity, because nothing arrives before then. Modified duration is still 4.76 years, because the sensitivity still divides by 1.05. DV01 is $0.37, less than the par bond's $0.43, because the price being shocked is $783.53 rather than $1,000.
The bond duration calculator prints all three. How bond pricing works is the price in the DV01. 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?
- Each coupon is , so $50. At a 5 percent yield the bond is at par: price $1,000.
- Macaulay duration, the present-value-weighted wait, is 4.55 years.
- Modified duration: years.
- DV01: , 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?
- Coupon each half-year: , so $30.
- Price, discounting 20 coupons and the face at 2.50 percent a half-year: $1,077.95.
- 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.
- Price is .
- The only cash flow is at year 5, so Macaulay duration is 5.00 years.
- Modified duration: years. DV01: .
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
Can I use Macaulay duration as a hedge ratio?
No. On the par bond, Macaulay 4.55 years is the wait. Modified 4.33 is the sensitivity. Using 4.55 is the 1.05 factor too large. DV01 of $0.43 is the money version of the sensitivity.
Why is a zero's Macaulay duration equal to maturity?
Because the only cash flow is the face, at year 5, so the weighted-average wait is 5.00 years. Modified duration is still 4.76, and DV01 is $0.37 on the $783.53 price.
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.