Modified Duration
In contrast to Macaulay duration, modified duration (sometimes abbreviated DM) is a price sensitivity measure, defined as the percentage derivative of price with respect to yield. Modified duration applies when a bond or other asset is considered as a function of yield. In this case one can measure the logarithmic derivative with respect to yield:
It turns out that when the yield is expressed continuously compounded, Macaulay duration and modified duration are equal.
First, consider the case of continuously compounded yields. If we take the derivative of price or present value, expression (2), with respect to the continuously compounded yield we see that:
In other words, for yields expressed continuously compounded,
- .
where:
- indexes the cash flows,
- is the time in years until the th payment will be received,
- is the present value of all cash payments from the asset.
Read more about this topic: Bond Duration
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