Macaulay Duration on Coupon Date

[BigInt123]

My understanding is that Macaulay duration decreases between coupon dates (as time left until maturity decreases), but temporarily increases on each coupon date as there is suddenly one less future payment remaining.

I think that I understand this in an intuitive sense; thinking about the analogy of future payments on a seesaw plank, with MacDur being the point at which you would place a fulcrum to balance the seesaw, so that the fulcrum must move to the right to rebalance the seesaw each time the leftmost payment is removed.

However, I am trying to get a better sense of how this works mathematically. If MacDur is a weighted average of the time until each future payment is received (with the weights being each payment’s PV as a % of total), how does it work out that it increases temporarily when a future payment is removed? For example, do you recalculate MacDur on each coupon date? If so, what changes in the weighted average calculation from one coupon date to the next?

Thanks in advance for any insight you can provide.

[S2000magician]

@BigInt123

My understanding is that Macaulay duration decreases between coupon dates (as time left until maturity decreases), but temporarily increases on each coupon date as there is suddenly one less future payment remaining.

I think that I understand this in an intuitive sense; thinking about the analogy of future payments on a seesaw plank, with MacDur being the point at which you would place a fulcrum to balance the seesaw, so that the fulcrum must move to the right to rebalance the seesaw each time the leftmost payment is removed.

However, I am trying to get a better sense of how this works mathematically. If MacDur is a weighted average of the time until each future payment is received (with the weights being each payment’s PV as a % of total), how does it work out that it increases temporarily when a future payment is removed?

When a coupon payment is made, the present value of the remaining future payments suddenly drops, and the extremely short duration (1 day) to the payment with the highest present value disappears.