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# Loan with decreasing payments....

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A loan for \$1,000,000 is granted by the Bank to ABC Importing Company. The loan is to be repaid in 10 years and bears an interest rate of 5% compounded monthly. The annual payments will start with P for the first year and decrease by 2% annually thereafter. What is the initial payment that ABC Company must pay to the Bank? And how much interest will cost the Company over the 10-year period? Thank you for any help.

May 24, 2017

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It is little ambiguous about the 2% decrease in payments. For this purpose, I shall assume that the payments will decrease by 2% from the previous payment. In other words the payments will decrease as follows: \$1, \$0.98, \$0.9604(0.98 x 0.98), \$0.941192(0.9604 x 0.98), 0.98^4....etc. to the 10th payment.

The interest rate is 5% compounded monthly. Will adjust it to compounded annually to match the annual payments, which comes to =5.11618978817%.

Now, we just find the Present Value of the above payments, which comes to =7.08110996994. And then the annual, or initial payment, will be =\$1,000,000 / 7.08110996994 =\$141,220.80. This payment will decrease by 2% annually from the previous payment.

To calculate the interest paid by the ABC Company, we simply have to add the 10 payments which come to a total of =\$1,291,656.23. By subtracting the loan of \$1,000,000 we get =\$291,656.23 total interest paid by the Company over a period of 10 years.

P.S. There is a specific formula used for such problems, which is rarely used, but comes in as very handy in calculating such problems. It is written like this: PV = P*(((1+G) / (1+R))^N - 1) / (G-R), where G =% increase or decrease in payment, R=Interest rate per period, N=Number of periods, P=Periodic payment, PV=Present Value.

May 24, 2017