Question 1186712
i think i have the answer.


the starting time point is time point 0.
the current time point is time point 3.


each time point is a year.


three years ago (in time point 0), 4000 was borrowed at 12% per year compounded monthly for a period of 5 years.


the effective interest growth rate per year was (1 + .12/12)^12 = 1.12682503.


this means that 4000 * 1.12682503 ^ 5 = 7266.786794 was owed 5 years later (in time point 5).


one year ago (in time point 2), 8000 was borrowed at 16% per year compounded quarterly for a period of 5 years.


the effective interest growth rate per year was (1 + .16/4)^4 = 1.16985856.


this means that 8000 * 1.16985856 ^ 5 = 17528.98514 was owed 5 years later (in time point 7).


to determine the equal payments, i took the present value of these two amounts at 20% compounded semi-annually and found their present value in the current time period (time period 3).


20% compounded semi-annually gives an effective interest growth rate per year of (1 + .2/2) ^ 2 = 1.21.


that present value was equal to 7266.786794 / 1.21^2 (time period 5 to time period 3) + 17528.98514 / 1.21^4 (time period 7 to time period 3).


the present value of the amount owed is equal to 13140.7141.
the present value year is time period 3 (the current time period).


you want to pay off the amount owed in equal payments with the first payment now and the last payment 5 years from now.


i'll use A to represent the present value of the amount owed until the end, at which time i'll replace A with 13140.7141 to find the equal payment required.


let x = the payment to be made now (in time period 3) and in 5 years (in time period 8).


the formula is.


x / 1.21^0 + x / 1.21^5 = A


since 1.21^0 is equal to 1, the formula becomes:


x + x/1.21^5 = A


multiply both sides of this equation by 1.21^5 to get:


1.21^5 * x + x = 1.21^5 * A


factor out the x to get:


x * (1.21^5 + 1) = 1.21^5 * A


divide both sides of this equation by (1.21^5 + 1) to get:


x = 1.21^5 * A / (1.21^5 + 1)


solve for x to get:


x = .7217385466 * A


since A = 13140.7141, x = .7217385466 * that = 9484.159896.


the present value of those two payments is:


9484.159896 / 1.21^0 + 9484.159896 / 1.21^5 = 13140.7141.


this means that the amount owed is fully paid at the end of the 5 year period.


this can be seen in the following spreadsheet.


<img src = "http://theo.x10hosting.com/2021/102601.jpg" >


i'm pretty sure this is correct, based on my understanding of the problem.


let me know if you have any questions.