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well-known relationship between modified duration and Macaulay duration if yield is periodically compounded (not continually):

where:

- is the compounding frequency per year (1 for annual, 2 for semi-annual, 12 for monthly, 52 for weekly, etc.),
- is the yield to maturity for an asset, periodically compounded

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**Bond duration - Wikipedia, the free encyclopedia**

uously compounded. Then expression (2) becomes: To find modified duration, when we take the derivative of the value with respect to the periodically compounded yield we find [5] Rearranging (dividing both sides by -V) gives: which is the <span>well-known relationship between modified duration and Macaulay duration: where: indexes the cash flows, is the compounding frequency per year (1 for annual, 2 for semi-annual, 12 for monthly, 52 for weekly, etc.), is the cash flow of the th payment from an asset, is the time in years until the th payment will be received (e.g. a two-year semi-annual would be represented by a index of 0.5, 1.0, 1.5, and 2.0), is the yield to maturity for an asset, periodically compounded is the present value of all cash payments from the asset. This gives the well-known relation between Macaulay duration and modified duration quoted above. It should be remembered that, even though Macaulay duration and modified duration are clo

uously compounded. Then expression (2) becomes: To find modified duration, when we take the derivative of the value with respect to the periodically compounded yield we find [5] Rearranging (dividing both sides by -V) gives: which is the <span>well-known relationship between modified duration and Macaulay duration: where: indexes the cash flows, is the compounding frequency per year (1 for annual, 2 for semi-annual, 12 for monthly, 52 for weekly, etc.), is the cash flow of the th payment from an asset, is the time in years until the th payment will be received (e.g. a two-year semi-annual would be represented by a index of 0.5, 1.0, 1.5, and 2.0), is the yield to maturity for an asset, periodically compounded is the present value of all cash payments from the asset. This gives the well-known relation between Macaulay duration and modified duration quoted above. It should be remembered that, even though Macaulay duration and modified duration are clo

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