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.
i'm pretty sure this is correct, based on my understanding of the problem.
let me know if you have any questions.