SOLUTION: A loan of $1000 is made at an interest rate of 10% compounded quarterly. The loan is to be repaid with three payments: $200 at the end of the first year $800 at the end of the fift

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Question 1201066: A loan of $1000 is made at an interest rate of 10% compounded quarterly. The loan is to be repaid with three payments: $200 at the end of the first year $800 at the end of the fifth year and the balance at the end of the tenth year. Calculate the amount of the final payment.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
200 is paid at the end of the first year.
800 is paid at the end of the fifth year.
balance is paid at the end of the tenth year.
interest rate is 10% compounded quarterly.
interest rate per quarter = 10% / 4 = 2.5% per quarter.
the end of the first year is 4 quarters away.
the end of the fifth year is 20 quarters away.
the end of the tenth year is 40 quarters away.
the present value factors are:
4 quarters away = 1/1.025^4 = .9059506448.
20 quarters away = 1/1.025^20 = .6102709429.
40 quarters away = 1/1.025^40 = .3724306237.
the present value of the first payment is 200 * .90..... = 181.190129.
the present value of the second payment is 800 * .61.... = 488.2167543.
total presdent value so far is 669.4068832.
remaining present value to account for is 1000 minus that = 330.5931168.
future value of that for 40 quarters is that * 1.025^40 = 887.663623.
that would be last payment in 40 quarters.

the procedure was to make the present flow of the future payments equal to the initial loan..
the initial initial loan was 1000.
since that is in the present, it is also equal to the present value of the investments.
the present value of the first investments was able to be calculated.
what was left was how much the present value of the last investment had to be.
that was calculated by taking the loan amount and subtracting the present value of the known payments.
what was left was the presnt value of the remaining payment.
to find the future value of that, it had to be multiplied by 1.025^40.
that gave the value of the last investment in the laqst time point.

i also checked to make sure this was done correctly by finding the internal rate of return of the cash flows from the loan (shown as negative) to the future payments (shown as positive).
the internal rate of rturn was 2.5%, as it should have been if the last payment was entered correctly, which it was.

your solution is that the value of the last payment is equal to 887.663623 in time period 40 (40 quarters from the loan).
let me know if you have any questions.
theo
Answer by ikleyn(52748)   (Show Source): You can put this solution on YOUR website!
.
A loan of $1000 is made at an interest rate of 10% compounded quarterly.
The loan is to be repaid with three payments: $200 at the end of the first year
$800 at the end of the fifth year and
the balance at the end of the tenth year.
Calculate the amount of the final payment.
~~~~~~~~~~~~~~~~~~~~ 

At the end of the 1st year the debt is   = 1103.81  dollars.

After paying $200, the unpayed balance at the end of the 1st year is  $1103.81 - $200 = $903.81.



At the end of the 5th year (four years later), the debt is   = 1341.72 dollars.

After paying $800, the unpayed balance at the end of the 5th year is  $1341.72 - $800 = $541.72.



At the end of the 10th year (five years later), the debt is   = 887.67 dollars.

The amount to pay at the end of the 10th year is 887.67.     ANSWER

Solved.



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