The formula I will leave to more knowledgeable people, but intuitively the way I think about it that there's really only one metric we ultimately care about, which is the "modified duration". The duration tells you something along the lines of how long it takes until you get your money back. That's it.Hi Everyone! What is the intuition behind modified duration being Macaulay duration divided by (1+Yield/K)?
Now the problem is, yields, which is something like your return, actually come in many different forms - annually, semi-anually, continuously. So you can have the exact same return, written in different ways. And as such, the duration formula needs to account for that as well. It does so by dividing the "macaulay duration" of the portfolio, which ignores in way way the yield is written, by yield / k, to get a summary which does no longer ignore how the yield is quoted. And you do that every single time you want to get the "(modified) duration" of the portfolio to approximate changes in portfolio value.
The macaulay duration, at least as far as I have seen, is never relevant, it is merely a step to get to what we care about, the modified duration. Now, the one twist here is that macaulay duration is actually equivalent to the modified duration, for continuous compounding.
Edit: I think the formula itself simply comes from the taking the first derivative of a bond value w.r.t. a change in yield