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Math Help - Amortization of Loans and Annuities - Need Help

  1. #1
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    Amortization of Loans and Annuities - Need Help

    Hey guys. I need some help on a few questions. I really appreciate the help.

    1) A owes B the sum of $5000 and agrees to pay B the sum of $1000 at the end of each year for five years and a final payment at the end of the sixth year. How much should the final payment be if interest is at 8% compounded annually?

    2) A debt of $18,000 is being repaid by 15 equal semiannual payments, with the first payment to be made in six months from now. Interest rate is at the rate of 7% compounded semiannually. However, after two years, the interest rate increases to 8% compounded semiannually. If the debt must be paid off on the original date agreed upon, find the new annual payment. Give your answer to the nearest dollar.

    3) The federal gov has a program to aid low-income owners in urban areas. This program allows certain qualified homwoeners to obtain low-interest home improvement loans. Each loan is processed through a commercial bank. The bank makes home improvement loans at an annual rate of 9 1/4% compounded monthly. However, the gov subsidizes the bank so that the loan to the homeowner is at the annual rate of 4% compounded monthly. If the monthly payment at the 4% rate is x dollars (x dollars is the homeowner's monthly payment) and the monthly payment 9 1/4% rate is y dollars (y dollars is the monthly payment the bank must receive), then the gov makes up the difference y - x to the bank each month. The gov does not want to bother with monthly payments. Instead at the beginning of the loan, the gov pays the present value of all such monthly differences at an annual rate of 9 1/4% compounded monthly.

    If a qualified homeowner takes out a loan for $5000 for five years, determine thegov's payment to the bank at the beginning of the loan.

    Once again, I greatly appreciate the help guys. Thanks.
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  2. #2
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    Re: Amortization of Loans and Annuities - Need Help

    Quote Originally Posted by HAL1993 View Post
    1) A owes B the sum of $5000 and agrees to pay B the sum of $1000 at the end of each year for five years and a final payment at the end of the sixth year. How much should the final payment be if interest is at 8% compounded annually?
    Owing at end of 5th year:
    5000(1.08^5) - 1000(1.08^5 - 1)/.08 = 1480.04

    Owing 1 year later: 1480.04(1.08) = 1598.44
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    Re: Amortization of Loans and Annuities - Need Help

    Quote Originally Posted by Wilmer View Post
    Owing at end of 5th year:
    5000(1.08^5) - 1000(1.08^5 - 1)/.08 = 1480.04

    Owing 1 year later: 1480.04(1.08) = 1598.44
    Awesome, you verified my answer (the book doesn't have the solution and/or answer). Thanks!

    The other two, I barely know how to begin though.
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    Re: Amortization of Loans and Annuities - Need Help

    Quote Originally Posted by HAL1993 View Post
    2) A debt of $18,000 is being repaid by 15 equal semiannual payments, with the first payment to be made in six months from now. Interest rate is at the rate of 7% compounded semiannually. However, after two years, the interest rate increases to 8% compounded semiannually. If the debt must be paid off on the original date agreed upon, find the new annual payment. Give your answer to the nearest dollar.
    Required payment initially:
    18000(.035) / (1 - 1/1.035^15) = 1562.85

    Owing 2 years later (or 4 payments later):
    18000(1.035^4) - 1562.85(1.035^4 - 1)/.035 = 14068.08

    Required payment to pay off above at 8% (11 payments):
    14068.08(.04) / (1 - 1/1.04^11) = 1605.86
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    Re: Amortization of Loans and Annuities - Need Help

    Quote Originally Posted by HAL1993 View Post
    3) The federal gov has a program to aid low-income owners in urban areas. This program allows certain qualified homwoeners to obtain low-interest home improvement loans. Each loan is processed through a commercial bank. The bank makes home improvement loans at an annual rate of 9 1/4% compounded monthly. However, the gov subsidizes the bank so that the loan to the homeowner is at the annual rate of 4% compounded monthly. If the monthly payment at the 4% rate is x dollars (x dollars is the homeowner's monthly payment) and the monthly payment 9 1/4% rate is y dollars (y dollars is the monthly payment the bank must receive), then the gov makes up the difference y - x to the bank each month. The gov does not want to bother with monthly payments. Instead at the beginning of the loan, the gov pays the present value of all such monthly differences at an annual rate of 9 1/4% compounded monthly.

    If a qualified homeowner takes out a loan for $5000 for five years, determine thegov's payment to the bank at the beginning of the loan.
    Monthly payment @ 9.25%:
    (i = .0925/12))
    5000(i) / (1 - 1/(1+i)^60) = 104.40

    Monthly payment @ 4%:
    (i = .04/12))
    5000(i) / (1 - 1/(1+i)^60) = 92.08

    Difference = 12.32; present value (i = .0925/12):
    12.32[1 - 1/(1+i)^60] / i = 590.04
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