finance charge of 632 is also due with the first payment.
i think you might be looking for the internal rate of return.
i used the internal rate of return function of excel to find that.
excel says the internal rate of return is 0.9993862%.
i used excel to verify that's the correct interest rate.
the results of the excel analysis are shown below.
what happens is that you borrow 12000.
you need to pay the first monthly payment right away and you need to pay 632 finance charge right away.
the end result is that you receive 10368.
that's the 12000 you requested minus the 1000 payment minus the 632 finance charge.
you then pay 1000 at the beginning of the remaining 11 months of the loan.
the internal rate of return interest rat is .009993862.
at the beginning of the first month, you have 10368.
at the end of that month (the same time period is also the beginning of the next month), 10368 is multiplied by 1.009993862 to get 10471.61636 and then 1000 payment is subtract from it to get a remaining balance of 9471.61636 in time point 1.
note that the loan starts in the beginning of the first month, which is time point 0.
time point 1 is the end of the first month and the beginning of the second month.
time point 2 is the end of the second month and the beginning of the third month.
this continues throughout the loan period.
the last payment of 1000 is made in time point 11.
that's the end of the 11th month and the beginning of the 12th month.
the loan ends at the end of the 12th month.
since the loan payments are made at the beginning of each month, there are no loan payments due at the end of the 12th month, as shown in the excel spreadsheet.
the remaining balance at the end of the loan period is 0 as it should be if all the payments were made on time as required.
the excel printout confirms the calculations are correct and that the interest rate required to service the loan is .9993862%.
i also did an analysis using the arachnoid financial calculator at https://arachnoid.com/finance/
the results of that analysis are shown below.
the interest rate in that analysis is .9993862% which the display truncates to .999386%.
the present value is the 12000 loan minus the up front payment of 632 for the finance charge.
the 1000 payment is not subtracted because the calculator is set to pay the maintenance charge of 1000 at the beginnning of each month, so it's automatically taken care of.
the analysis shows that the loan is fully paid off in 12 months because the future value is 0.
both the results from the financial calculator and from excel support the conclusion that the interest rate of the loan is .9993862%, rounded to 7 decimal digits.
that's the effective monthly interest rate.
you were asked for the effective annual interest rate.
the effective annual growth rate is 1.009993862 ^ 12 = 1.126742857.
the effective annual interest rate is that minus 1 = .126742857 * 100 = 12.6742857%.
that's your solution.