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Suppose we have a loan worth 10000 that is being repaid late as a lump sum on a given day 30 days after the due date. Suppose the original interest rate is 5% so that the amount owed before late interest is 10500. Suppose late interest accrues at .1% per day so that the total amount is 10543.15.

Now suppose instead that we split the loan into two payments: one 15 days after the due date and another 30. In this case the total is 10532.19, which is lower. Has the late interest rate effectively changed due to the multiple payments? What is the difference?

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The actual interest rate has not changed. The interest charged has, and the effective interest rate calculated after the fact would, but the actual rate itself hasn't. You just end up paying less interest, because the basis for interest calculation changed halfway through.

Compound interest is calculated on a (in your case, daily) basis based on the current amount owed. In your case, your amount is solely reflecting the 5% annual rate; there isn't a penalty amount in that as far as I can tell (and 0.1% doesn't really make sense, that's either too low or too high depending on what you're saying 0.1% is per). 5% true APR is equivalent to around a 0.0134% daily rate. (IE, if you on a daily basis charge around 0.0134% interest on that day's loan balance, the effective interest rate ends up being 5% if the loan is paid off in full on the day one year from the start of the loan.)

So your first day's new balance is 10,500+0.000134*(10,500) = 10,501.42. The next day's balance (day 2) is 10,501.42+0.000134*(10,501.42) = 10502.84. The amount's actually going up a tiny bit - just less than a penny through the first few days.. (That's why by the end of this you are at $10,543 and change not $10,542 and change).

However, if you pay half off on day 15, you suddenly aren't adding $1.42 or so per day anymore, because now the multiplier is not 10,500 or so, but is actually 5,250 or so times the interest rate (or whatever you paid off) - which means you're now adding 70-75 cents per day in interest for the last 15 days. Subtract 15 days times around 75 cents, and you have a bit over 11 dollars off the ultimately due interest - around 10,532.

That does mean the effective interest rate on the entire loan is now lower - as that's calculated by summing total interest paid divided by initial loan value - but the actual rate does not change.

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