Question 1193794: Please help me with the homework.
Paul owes Winston wealth R1000 due in three years from now and R8000 due in five years from now. He wishes to reschedule his debt so as to pay two sums on different dates, one say X, in one year from now, and the other, which is twice (i.e 2X),five years later, Winston agrees, provided that the interest rate is 18% p. a. Compounded quarterly.what are Paul's payments?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe that, if they have the same present value, then they will be equivalent.
the present value of 1000 in 3 years = 1000 / 1.045^12 = 589.6638649.
the present value of 8000 in 5 years = 8000 / 1.045^20 =3317.142877.
their sum is 3906.806742.
the present value of x in 1 year is x / 1.045^4.
the present value of 2x in 5 years is 2x / 1.045^20
their sum is x / 1.045^4 + 2x / 1.045^20
your equation is x/1.045^4 + 2x / 1.045^20 = 3906.806742.
multiply both sides of that equation by 1.045^4 * 1.045^20 to get:
1.045^20 * x + 1.045^4 * 2x = 3906.806742 * 1.045^20 * 1.045^4
factor out the x to get:
x * (1.045^20 + 1.045^4 * 2) = 3906.806742 * 1.045^20 * 1.045^4.
simplify to get:
4.796751226 * x = 11236.03024.
solve for x to get:
x = 11236.04024 / 4.796751226 = 2342.425051.
2x is equal to 4684.850104
confirm the present value are equivalent by replace x and 2x in the equation of x/1.045^4 + 2x / 1.045^20 = 3906.806742 to get:
2342.425051 / 1.045^4 + 4684.850104 / 1.045^20 = 3906.806742 which becomes:
3906.806742 = 3906.806742.
this confirms the present values are the same which means the two cash flow streams are equivalent at 18% per year compounded quarterly.
the present value of the loan at 18% compounded quarterly is 3906.806742.
you can pay it off by giving 1000 in 3 years and 8000 in 5 years or by giving 2342.425051 in 1 year and 4684.850104 in 5 years.
i did the remaining balance comparison in excel.
here are the results.
the procedure for remaining balance is:
the remaining balance in the preceding quarter is multiplied by 1.045 and then any payment in that quarter is subtracted.
for example:
in the first two columns.
the remaining balance at the end of quarter 12 = 6340.17 * 1.045 - 1000 = 5625.47765.
the excel spreadsheet shows 5625.48.
any differences between the two values will be due to differences in rounding.
as you can see, the remaining balance at the end of the 20th quarter from the first two columns and the remaining balance at the end of the 20th quarter from the second two columns are both equal to 0, as they should be, if the loan has been completely paid off.
let me know if you have any questions.
theo
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