# How do compound interest calculations interact with penny round-off?

[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. Nov 26, 2019 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.

• 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). Nov 26, 2019 at 14:53
• 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. Nov 26, 2019 at 20:27

it would actually be \$1159.9 per month because you don't divide the interest rate by your months, this is the equation. Y= 1000(1.1599)^(months) For instance, if you were paying interest for 4 months then it would be Y= 1000(1.1599)^4 and your answer would be \$1810.01 (never round-up for money because you can't have more than is available). Hope this helps!

• This seems incorrect to me. Interest rates are typically given as an "APR": this is the rate per compounding period multiplied by the number of periods in a year. Thus you need to divide by the number of periods in a year to see the actual rate per compounding period. See investopedia.com/terms/a/apr.asp for example. In any case, this answer is not addressing the main question in the OP, which is about possible discrepancies between the formula and the actual amount due to rounding. Apr 17, 2021 at 3:08

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.

True, but look at what happens next month. Your balance is \$1,013.33, your interest rate is still 1.3325% per month, and your interest charged is \$13.5026, which is rounded to \$13.50 and you give half of that half a penny back! Your balance with rounding is now \$1,026.83. Without rounding, your balance would be `\$1000 * (1+.1599/12) ^ 2` = \$1,026.8276, so now the error is only \$0.0024.

Is there a systematic reason why this error can't accumulate as n grows?

Yes, because (statistically) half of the time you are rounded down, and half of the time you're rounded up. The effect of rounding is that on average all of the rounding effects cancel each other out. Some periods you get half a penny, some months you give it, and on average it should be a wash (or at least a much smaller error than \$0.005).

• What gives you confidence that "(statistically) half the time you are rounded down, and half the time you're rounded up" in every scenario? It makes intuitive sense to me that that's a reasonable thing to expect most of the time; nonetheless, I reiterate from the OP the concern 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." How do you rule this out? Apr 17, 2021 at 3:00
• Yes, there might be specific situations where you're rounded down (or up) more than 50%, but if you look at the range of possible values at any point, half are rounded down and half are rounded up. I'm not saying that half of all values in a given schedule are rounded down, but on average (over lots of loans) the ups and down cancel each other out. That's also why the final payment is adjusted to balance out (as other answers explained). Apr 19, 2021 at 14:15