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Math Help - Amortized loan

  1. #1
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    Amortized loan

    On May 1, 1988, the Shaws purchased a home for $308,000. Their down payment was $46,000, and the remaining $262,000 was financed with a standard 15-year amortized loan at a nominal interest rate of 5.55% compounded monthly. Assuming monthly payments beginning June 1, 1988, how much interest did they pay in 1993? (Answer - $10,922.22)

    I'm not sure how you would set this up to find the interest in a specific year ... the only thing I can think of is to set it up to get the amount of money they have up until 1993 and then just calculate that year's interest (i.e. there would be 67 monthly payments before 1993 so part of it would be 262000/(a-angle-67).004511?) but I think I'm overthinking it again ... any help on how to set it up would be great thank you!
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  2. #2
    Member jonah's Avatar
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    Quote Originally Posted by tbl9301 View Post
    On May 1, 1988, the Shaws purchased a home for $308,000. Their down payment was $46,000, and the remaining $262,000 was financed with a standard 15-year amortized loan at a nominal interest rate of 5.55% compounded monthly. Assuming monthly payments beginning June 1, 1988, how much interest did they pay in 1993? (Answer - $10,922.22)
    I would have to disagree.
    Using Sir Wilmer's highly enviable and clever (and may I say ingenious) formula (simplified) for determining the amount of interest paid between payment x = 56 and payment y = 68 (interest paid from 57th/January1993 up until 68th/December1993), I get

    <br />
R\left( {56 - 68} \right) + \left[ {\frac{R}{{0.004625}} - A} \right]\left[ {\left( {1.004625} \right)^{56}  - \left( {1.004625} \right)^{68} } \right] \approx \$ {\rm{10,853}}{\rm{.5365678635}}...<br />

    where

    <br />
R = \frac{{308,000 - 46,000}}{{a_{\left. {\overline {\, <br />
 {180} \,}}\! \right| 0.004625} }}<br />

    and

    <br />
A = 308,000 - 46,000<br />

    Your given answer is really more like interest paid between payment x = 55 and payment y = 67 (interest paid from 56th/December1992 up until 67th/November1993). Thus

    <br />
R\left( {55 - 67} \right) + \left[ {\frac{R}{{0.004625}} - A} \right]\left[ {\left( {1.004625} \right)^{55}  - \left( {1.004625} \right)^{67} } \right] \approx \$ {\rm{10,922}}{\rm{.2195698802}}<br />

    For more of Sir Wilmer's/Denis' original expanded formula, go here.
    See my Excel spreadsheet proof.
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