# Understanding effective interest rate and compounding period

An article I'm reading says "An effective monthly rate [on a loan] is quoted over a one-month period. Since it is effective, it is compounded once per month." Is that still true if there's no compound interest on the loan, for example if you pay all the interest due each month?

• What happens if you don't pay the interest in a month? Jan 2, 2019 at 23:09
• Most loans in the US do not allow compounding - so unpaid interest is not added to the unpaid balance. It's just set aside. Jan 3, 2019 at 14:11
• Can you link to the article source? Jan 3, 2019 at 16:52
• Here it is: www.jstor.org/stable/41948835 Jan 4, 2019 at 21:42

If you pay all the interest due at the end of each month, then the interest can't compound because it isn't there (regardless of whether it is a compounding loan or not). It your loan is a compounding loan, it also matters how often your loan compounds.

# Example with monthly compounding

If you have a \$100 loan at a monthly effective interest rate of 5%, and the interest compounds monthly, and you make a \$5 payment at the end of each month, then there is nothing to compound, since you are paying off the interest. Your loan will stay at \$100 for as long as you pay it.

If you pay nothing over the first year and your loan is a compounding loan, at the end of the first year you will owe \$179.59 (`loan_amount = initial_loan*(1+monthly_rate)^12`).

If your loan is not a compounding loan, and you don't pay anything, then you will owe \$160.

So there is only a difference between a compounding loan and a non-compounding loan if you don't pay down the interest each time the interest is applied.

# Same example, but with annual compounding

If your \$100 loan instead compounds annually, with an effective monthly interest rate of 5%, then your effective annual interest rate is 79.59% (`annual_rate = (1+monthly_rate)^12 - 1`). If you don't pay back any of the loan over the first year, you will owe \$179.59 (just like in the compounding example above.)

However, if you pay \$5 at the end of each month, then after 11 months of payment your loan amount will be \$45 (`= \$100 - 11*\$5`) and then after compounding at 79.59% your loan will be \$80.81 before you make your 12th payment. In this case, you would eventually pay off your loan by making \$5 payments every month, even though the effective monthly interest rates are the same.