How modified duration works
By Jude Wallis
Modified duration is Macaulay duration divided by one plus the yield for one period. With a Macaulay duration of 7.5 years, a 6 percent yield and two coupon periods a year, the divisor is 1.03 and modified duration is 7.2816, which reads as a percent price move per point of yield.
Modified duration
7.282 years
Approximate percent price change for a one percentage point yield move.
- Modified duration
- 7.2816
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In short
- Modified duration is Macaulay duration over one plus the periodic yield: 7.5 divided by 1.03 is 7.2816.
- The periodic yield is the annual yield divided by the number of coupon periods a year, so 6 percent semiannual is 3 percent a period.
- Read the result as a percentage price change for a one point change in yield, in the opposite direction.
- With annual coupons the divisor is larger: a Macaulay duration of 4.55 at a 5 percent annual yield gives 4.3333.
- It is a first-order estimate, and it understates the price rise and overstates the fall for large moves.
A wait becomes a sensitivity
Macaulay duration answers when the money arrives, on average, weighting each cash flow by its present value. That is measured in years, and it is not a price sensitivity by itself. Dividing by one plus the periodic yield converts it into one.
With a 6 percent yield paid twice a year, the periodic yield is 3 percent, so the divisor is 1.03 and the 7.5 year wait becomes a modified duration of 7.2816. The units have changed as well as the number: this one is a percentage move per point of yield.
The compounding frequency is inside the divisor
The most common slip in this calculation is dividing by one plus the annual yield when coupons are semiannual. The divisor has to use the yield for one coupon period, which is the annual yield divided by the periods per year.
Get it wrong and the answer is close enough to look right, which is what makes the error persistent. On a bond with annual coupons the divisor genuinely is one plus the annual yield, so a Macaulay duration of 4.55 at 5 percent gives 4.3333. Same formula, different period.
What it estimates and where it fails
A modified duration of 7.2816 says a one point rise in yield moves the price down by roughly 7.28 percent, and a one point fall moves it up by roughly the same. Roughly is doing real work in that sentence: the price to yield relationship is curved, and duration is the straight line tangent to it.
So for small moves it is a good estimate, and for large ones it understates gains and overstates losses. Convexity is the second-order correction, and how bond duration works covers where the straight-line approximation starts to matter.
What it is used for
Hedging and comparison. Two bonds with the same modified duration respond similarly to a small parallel shift in yields, whatever their coupons and maturities look like. That is what makes it the number a hedge is built on rather than the Macaulay figure. Macaulay against modified duration sets the two out together, how DV01 works converts the percentage into money per basis point, and the modified duration calculator makes the coupon frequency an explicit input. This is educational material, not financial advice.
Worked examples
A 7.5 year Macaulay duration at a 6 percent semiannual yield
A bond has a Macaulay duration of 7.5 years, a yield of 6 percent, and pays coupons twice a year. What is its modified duration?
- The periodic yield is 6 percent divided by 2, which is 3 percent, so the divisor is 1.03.
- Divide 7.5 by 1.03, which gives 7.2816.
Modified duration is 7.2816, so a one point rise in yield implies a price fall of about 7.28 percent.
An annual-pay bond
A bond with annual coupons has a Macaulay duration of 4.55 years and a yield of 5 percent.
- With one period a year the periodic yield is the full 5 percent, so the divisor is 1.05.
- Divide 4.55 by 1.05, which gives 4.3333.
Modified duration is 4.3333. The divisor is larger than in the semiannual case, so the gap from Macaulay is wider.
Common questions
Why divide by one plus the periodic yield?
It converts a present-value-weighted wait into the derivative of price with respect to yield, which is a sensitivity.
Is modified duration always smaller than Macaulay?
Yes, whenever the yield is positive, because the divisor is greater than one.
How accurate is the estimate?
Good for small yield moves. For large ones the curvature of the price to yield relationship needs a convexity adjustment.
Is this financial advice?
No. It is educational material about a bond risk measure.
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.