# Effective Interest Rate Interpretation

Assume a \$1,000 loan for 24 months, with monthly interest rate 2.5%, and monthly payments (\$55.91 as calculated by any online tool, e.g. here).

This will give us annual interest rate (air) of 30%, and effective interest rate 34,4889%, calculated as `eir = (1 + air/n)^n - 1`.

Is there any intuitive interpretation of 34.4889%? I have hard time understanding how to interpret this number as payments are coincident with interest rate calculation and in fact there is no compounding (interest calculated from interest) at all.

wikipedia on effective interest rate:

The effective interest rate is calculated as if compounded annually.

wikipedia on compound interest:

Compound interest is interest added to the principal of a deposit or loan so that the added interest also earns interest from then on. This addition of interest to the principal is called compounding.

The effective annual interest rate is the amount by which your loan will increase in a year if you don't make any payments. If you don't make any loan payments in the first 12 months, then your loan amount will be `\$1000*(1+0.025)^12 = \$1344.89` which is 34.489% higher than your original loan amount.

Another way to think of it: if you had a loan that compounded annually at a rate of 34.489% and didn't pay it for a year, then the loan amount after one year will be `\$1000*(1+0.34489) = \$1344.89`, the same amount as above.

R = I ^ P

1.80872594958 = 1.025 ^ 24

I = R ^ (1/P)

1.34488882425 = 1.80872594958 ^ (1/2)

2.5% per month = 34.5% per year