# What formula would I use to calculate the percent of an APR if I knew the payment, principal, and interest?

I'm writing a blog article about potential scenarios in which I might lend someone money and explain how interests work. In my example, I am lending someone \$400 and agreed to pay back \$100. With no interest, the payment would be done in four months. But if I said that \$50 would go towards the principal balance, and the other \$50 would go towards the interest, then you would be paying \$800 over eight months.

I heard that some short term loans and stuff like that have triple-digit APRs that are hardly found in traditional loan terms, like anywhere from 200% to 900% APR as opposed to 10%-25%.

• If the interest in the first month is \$50, that is an interest rate of 150% APR (12.5% per month). With a regular payment of \$100 per month, the loan would be paid off in 6 months not 8, because the interest is less each month as the balance is reduced. (It wouldn’t make sense to pay \$50 interest on a loan balance of \$50; that would be an interest rate of 1200%.)
– prl
Mar 1 at 1:09

The way most installment loans work is that they have a fixed interest rate and a fixed monthly payment. As you pay down the loan, the amount of each payment that goes to interest declines.

For example, say you borrow \$1000 at 12% interest. I say 12% because that works out neatly to 1% per month. The payments are \$100 per month.

So the first month, the borrower owes 1% of \$1000 = \$10 interest. So he pays \$100. \$10 goes to interest and \$90 goes to principal.

The second month he only owes \$910, because he's now paid off \$90. 1% of \$910 is \$9.10. So he pays \$100. \$9.10 goes to interest and the remaining \$90.90 goes to principal.

The third month the principal is now \$910.00 - \$90.90 = \$819.10. 1% of that is \$8.19 (and a fraction that gets rounded off). So off his \$100 payment, \$8.19 goes to interest and \$91.81 goes to principal.

Etc.

Usually when you get a loan, they start with the amount of the loan, the interest rate, and the repayment time and calculate the payment amount, rather than starting with the payment amount and calculating how long it will take to pay it off.

The formula is a bit complicated. If:

``````P is the principal
t is the number of months to repay
r is the monthly interest rate (i.e. annual rate divided by 12)
``````

then the amount of the monthly payment is:

``````m=P x ((r+1)^t x r) / ((r+1)^t - 1)
``````

For example, suppose the principal is \$1000, the monthly interest is 1% (12% per year), and the repayment time is 120 months (10 years). Then the monthly payment is:

``````m=\$1000 x (1.01^120 x .01) / (1.01^120 - 1)
=\$1000 x (3.300 x .01) / (3.300 -1)
=\$1000 x (.0330 / 2.300)
=\$1000 x .01435
=\$14.35
``````

See how simple?

A loan with a fixed interest and principal amount is not a realistic scenario for an amortized loan.

A more standard loan structure is one where the interest is calculated on the remaining principal balance each month, and as the principal is paid down, the amount of interest in each payment is reduced until the payment is almost all principal. That type of loan does have a formula for the payment amount, but there's not a closed-form formula for the rate - it has to be solved for iteratively.

One thing you could calculate for your scenario is the Internal Rate of Return (IRR) of the loan, which you also have to solve for iteratively.

Just for fun, I calculated the IRR of that loan in Excel (an initial outflow of \$400 followed by 8 inflows of \$100), and the "rate" over 8 months would be 19% per month. In other words, it would be equivalent to an amortized loan with a 19% monthly interest rate. That would be an equivalent annual percentage rate (APR) of 676%.

If you're writing a blog article about how interest works, I'd use a more realistic example, rather than fabricating a loan structure and backing out an exorbitant interest rate.