Modified duration calculator
By Jude Wallis
Modified duration is Macaulay duration divided by one plus the yield per period. A Macaulay duration of 7.5 years at a 6 percent yield paid semiannually gives 7.2816, so a one point rise in yield moves the price about 7.28 percent the other way.
Modified duration
7.282 years
Approximate percent price change for a one percentage point yield move.
- Modified duration
- 7.2816
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The formula
is Macaulay duration in years, the annual yield as a decimal and the coupon payments per year. The divisor uses the periodic yield, not the annual one.
One is a time, the other is a sensitivity
Macaulay duration is measured in years: the weighted average time until the bond's cash flows arrive. Modified duration is measured in percent per percent: how much the price moves for a one point change in yield. The formula converts between them, and the conversion is small but not optional.
At 7.5 years of Macaulay duration the modified figure is 7.2816. A yield rise of one percentage point costs about 7.28 percent of the price, and a fall of one point gains about the same.
The divisor is the periodic yield
This is where the arithmetic goes wrong most often. With semiannual coupons, a 6 percent annual yield is 3 percent per period, so the divisor is 1.03 rather than 1.06. Using the annual figure gives 7.0755 instead of 7.2816, which understates the sensitivity by about 3 percent of itself.
The second example makes the same point at different inputs: a Macaulay of 6 at a 5 percent yield with two payments a year divides by 1.025 and gives 5.8537. Macaulay against modified duration sets the two definitions side by side.
Reading it as a price move
The estimate is linear, and prices are not. Modified duration is the slope of the price and yield curve at the current yield, so it is accurate for small moves and increasingly optimistic for large ones: the true price falls less than duration predicts when yields rise, and rises more when they fall.
That curvature is convexity, and it is the second term in the same expansion. For the small yield moves that make up most days, the duration estimate alone is close enough to be the working number.
What duration is used for
Comparing interest rate sensitivity between bonds, sizing a hedge, and matching assets to liabilities by their weighted timing. The bond duration calculator builds Macaulay duration from a bond's cash flows, and the bond price calculator shows the price it applies to. Macaulay duration covers the input. This is educational material, not financial advice.
Worked examples
A Macaulay duration of 7.5 at a 6 percent yield
Macaulay duration is 7.5 years, the yield is 6 percent, and coupons are paid twice a year. What is modified duration?
- Periodic yield: percent, so the divisor is 1.03.
- Divide: .
Modified duration is 7.2816, so a one point rise in yield costs roughly 7.28 percent of the price.
A shorter bond at a lower yield
Macaulay duration is 6 years, the yield is 5 percent, and coupons are semiannual.
- Periodic yield: percent, so the divisor is 1.025.
- Divide: .
Modified duration is 5.8537, noticeably less sensitive than the 7.2816 bond.
Dividing by one plus the annual yield
With semiannual coupons the divisor is 1.03, not 1.06. Using the annual yield turns 7.2816 into 7.0755 and understates every price move it is then used to estimate. The frequency belongs in the divisor.
Common questions
What does a modified duration of 7.2816 mean?
That a one percentage point change in yield moves the price roughly 7.28 percent in the opposite direction.
Why is the estimate less accurate for large moves?
Because the price and yield relationship is curved. Duration is the straight line tangent to it, and convexity is the correction.
Is this financial advice?
No. It is educational material for the modified duration identity.
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.