What is the intuition behind the fact that Modified duration = Macaulay Duration / (1+r)?

Question

I understand the derivation of both:take dP/dR and divide by P which will give you both 1) modified duration OR 2) macaulay duration / (1+r)

(notice the weighted average time built into the function from taking the derivative - the math makes sense)

My question is about intuition: how can discounting the weighted average time to maturity by an extra period be equal to the sensitivity of %price to %yield? Perhaps using continuously compounded returns can help in the intuition?

Should I memorize the fact and move on and not require intuition behind it

Answer 1

The intuition behind Macaulay Duration is the average time it takes to get all the cash flows from a bond. Think of it as computing the centre of gravity for a see-saw.

You can find the image depicting the same here:

This should immediately tell you that Macaulay Duration for Zero coupon bond is the maturity of the bond.

In continuous discounting Macaulay Duration equals Modified duration. The extra period discounting in discrete discounting is to get the math right for Modified Duration.