Confusion with payment calculation for fixed payments on fixed interest loan

Question

I'm trying to understand how to calculate the monthly payments on a fixed interest loan with a given term. I understand this can be done with the PMT excel function.

Whenever I see this explained online, such as in this video, it goes something like this.

1. to pay a loan in fixed amounts, each payment should be a bit bigger than the sum of the principal divided by the amount of total payments + the interest accrued between the time you took the loan and the first payment, so that your total balance decreases more and more until reaching zero at the end of the term. How much bigger is not important for my question.

2. Here comes the tricky part I don't understand: when calculating the interest accrued before the first payment, calculations are done as follows. If you have monthly payments and a X% yearly rate, multiply your borrowed amount by (X/12)% and that's your interest due. But I certainly don't owe that amount of money after a month, correct? If you compound it over a year, that gives higher than X%.

Since at each point in time I pay interest based on my outstanding principal, and (assuming no monthly payment obligations existed) if I didn't repay a single $ during the first year my total due amount would be X% higher than the borrowed amount, it can't be that the amount due after the first month is X/12%? Any help understanding this would be greatly appreciated, thanks!

Answer 1

It's simply how interest rates are quoted - they're quoted as an annualized rate but they are applied monthly, hence the X/12%. It does not mean that you accrue exactly X% over the year if you do not make any payments. In that case, the monthly effective rate would be (1+X)^(1/12) - 1 which is not nearly as intuitive for consumers.

Take a simpler example:

You borrow $1,000 at a quoted 12% annual interest, which actually means 1% per month (quoted rates generally do not take compounding into effect). That means that each month, you have accrued 1% of the remaining balance in interest. For the first month, that's $10.

Suppose your monthly payment is $15. That means that $10 of that payment goes to accrued interest, and the rest ($5) goes to paying down the loan balance, which is now $995.

The next month, your accrued interest is 1% of $995, or $9.95. Your payment is still $15, but now $5.05 goes to reducing your balance, which is now $989.95.

Repeat until the principal is paid off. Each month, the interest portion decreases and the principal portion increases.

But I certainly don't owe that amount of money after a month, correct?

Yes, as shown above

If you compound it over a year, that gives higher than X%.

If you didn't pay the interest, yes it would compound to more than X% over a year (specifically, (1+X)^12 - 1. But since you pay down part of the principal each month, your interest goes down over the year and you actually end up paying less than X% of the original balance over the year.