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Math Help - Interest Compounded Monthly for 30 Years

  1. #1
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    Interest Compounded Monthly for 30 Years

    Hey everyone, I've got this problem from my calc course and I've been getting some numbers that don't look realistic to me.

    A $100,000 loan is to be paid off after 30 years. The annual interest rate is 0.09 (9 percent).
    A) The total amount in the loan after n months is given by P(1+i)n, where P is the starting amount for the loan, i is the monthly interest rate, and n is the total months the loan is being paid off. How much in total will have to be paid off for the loan mentioned above?
    B) Assume you pay a fixed monthly amount D to pay off the loan (the payment is done at the end of the month). After the first month, $D dollars will be paid off of the loan. With the annual interest rate of 0.09 stated above, what is the total paid off after 2 months? What about 3? Determine a general formula for the total paid off after 30 years, which will be in terms of D.

    For part A I used the given formula and got $1,473,057.61 (which seems kinda reasonable given that it's compounded monthly for 30 years.
    For part B, I calculated using the formula D=Pi/[1-(1/(1+i))^n] and got $804.62 per month. This value seems way too low to me because at a fixed $804.62 per month for 360 months (30 years) is only $289,664.14 and not the $1mil+ number calculated in part A. What am I doing wrong here? Thanks!
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  2. #2
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    Re: Interest Compounded Monthly for 30 Years

    For part A, your calculation is correct. That is what the amount of the loan would be if no payments were made. Of course, the bank would not allow that.

    For part B, you got the correct answer for the payment to pay off the loan over 30 years. The payments are smaller because they are paid earlier.

    I think they're asking for something different in part B. They say the amount paid off after one month is D dollars - I think they want you to calculate what the payments would be worth if they were in a separate account. Otherwise they should have said D-750, where $750 is the principal times the monthly interest rate ($100,000 * 0.0075). So after two months, it would be 1.0075D + D. The first month's payment grew by 0.75%, and then the second month's payment was added in. After n months, you would have D times 1 + 1.0075 + ... + (1.0075)^(n-1).

    This is a finite geometric series - have you learned how to calculate the sum?

    - Hollywood
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  3. #3
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    Re: Interest Compounded Monthly for 30 Years

    Yea, we've learned how to calculate the sum but I'm still wondering why he would state that it's a "fixed monthly amount" but have it change every month. Wouldn't a fixed monthly amount mean that it's the same amount every month? Thanks for the response!
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  4. #4
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    Re: Interest Compounded Monthly for 30 Years

    I don't understand your question. By "fixed" they mean the amount D is the same for each month.

    So I think the question is asking you to calculate D times (1 + 1.0075 + ... + (1.0075)^(n-1)) for n=360. This will be some function of D.

    If you set it to $1,473,057.61 and solve for D, you should get D=$804.62, but that's beyond the scope of the problem.

    - Hollywood
    Thanks from kayodoc
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  5. #5
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    Re: Interest Compounded Monthly for 30 Years

    I guess I had just read your response incorrectly last time. I understand it all and it was correct! Thank you so much for all the help! I really appreciate it
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