[As I have very little experience on money.SE, I welcome feedback about the following question, in addition to any answers of course.]

The balance on a loan or deposit is always reported as an integer multiple of $0.01. Typically, at any given point, the interest rate multiplied by the balance will include fractional pennies, so the actual interest paid must be rounded off to the nearest penny. I assume that subsequent interest payments are on the reported balance. But if this is so, then the famous formula A = P(1+i)^n (where A is the present value, P is the principal, i = annual nominal interest rate divided by number of compounding periods per year, and n = total number of compounding periods elapsed) is potentially off by some rounding error. For example:

Suppose I borrow $1000 at a 15.99% interest rate compounded monthly. Now 15.99% divided by 12 is 1.3325%, and 1.3325% of $1000 is $13.325, so (I presume - please correct me!) $13.33 is added to the balance, and the new balance is $1013.33. Then the formula A = P(1+i)^n (with P=1000, i=0.013325, and n=1) is off by half a penny.

Now evidently as long as n is small, the error is very small. And it seems plausible to me that usually the error will stay small even for large n because there's no (obvious to me, anyway) systematic reason for the rounding error to always go in the same direction (up or down), thus perhaps the errors will tend to cancel out. But it seems at least plausible to me that there might be some specific values of the principal and the rate such that the error is usually in the same direction for some significant stretch of values of n, and then at the end of this stretch, the formula might be off by a substantive amount.

Is there a systematic reason why this error can't accumulate as n grows? Or is it just that even if it accumulated it would remain small in comparison with the balance, and therefore it can be ignored in the types of situations in which the formula A=P(1+i)^n is used? Or, am I thinking about this completely wrong somehow and the error is illusory?

  • You can determine the rounding error by comparing A=P(1+i)^n to n rounded P(1+i) calculations in Excel. – RonJohn Nov 26 '19 at 7:27

Every mortgage or car loan I have ever considered had the last payment be slightly different than the rest of the payments. That last payment was different because the formula for the the monthly payment required payments to the fraction of a penny, which is of course impossible.

That would mean that the error could be at most 99% of the penny each month, thus after the first 359 payments of a 30 year mortgage, the last payment would have to differ from the others by at most $3.60.

This would also apply to revolving credit, but revolving credit doesn't have amortization tables that make this obvious. Plus minimum payments, constantly changing balances, and possibly changing interest rates would make it even harder to notice.

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  • Speaking as someone who looks at lending data all day long, it's pretty shocking the number of people who seem unprepared for the last payment to be different (and either over-pay or otherwise cause an operational problem). – dwizum Nov 26 '19 at 14:53
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    This answer deals with the rounding of the required payment to the nearest cent. The OP dealt with the rounding of the monthly interest to the nearest cent. – DJohnM Nov 26 '19 at 20:27

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