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Let's say that someone gets his profit (interest payment) from the bank on the first day of each month. A second person gets is profit on the 15th or whatever the day is.

Is this correct and clear in meaning?

The period or the cycle of the first person starts on the beginning of the month. The monthly period or cycle of the second person starts from a date that he specified.

  • What if the 1st or 15th is a Saturday or Sunday? – Michael Harvey May 30 '19 at 21:07
  • @MichaelHarvey In my country, It's an automated process. so It's not affected by holidays or weekends. – user2824371 May 31 '19 at 11:57
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It is very unclear what you are saying. I don't think there is any need to use the word "period" or "cycle":

You use the phrasing "The period of the first person". This phrasing that you used made me think at first that you were talking about a womans menstral cycle (her period).

Now I think you mean the "interest period". I would not express this with the word "period" but either using an adverb like "monthly" or a phrase like "on ... of each month".

Both people receive interest monthly.

The first person is paid on the first of each month. (or at the start of the month)

The second person is paid on a specified day each month

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A big-name bank in the US says this:

When you’re figuring interest on a savings account, keep in mind that it will be paid every time interest is calculated depending on the agreement you have with your bank. It may be daily, monthly, semiannually or annually.

With that in mind, I think the best way to describe your situation is to leverage the word monthly:

Interest is paid monthly. The first person's interest payment is calculated on the first day of each month. The second person's payment is calculated each month on the day specified by the customer.

By using the word monthly, there is no need to use a word like period or cycle.

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