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Math Help - I need to create a formula for an applied problem concerning interest in a loan

  1. #1
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    I need to create a formula for an applied problem concerning interest in a loan

    Hey guys, I used to post here under emttim or something to that effect, but anyway I'm actually posed to use the calculus I learned back in college...the only problem is I always sucked at setting up problems. I haven't much improved on that since being out of school.

    Here is what I am trying to do. Me and my wife are planning to buy a condo, so we're going to have a condo loan. I also have a student loan. The question is what combination of early payments to each loan will maximize our interest savings. It's going to be simple enough to derivate that and set the derivative equal to zero to find the maximum, however, I do need to figure out how to create the formula to derive in the first place.

    The way I figure it is that I have two, possibly three variables. I have created amortization tables for both loans and I can compute in Excel what my savings will be with various combinations. Here's a couple examples:

    If I contribute $400 to the mortgage loan, let's call that variable M, and $100 to the student loan, let's call that variable S, then the total interest savings will be $95,783.44. When M=425 and S=75, the total interest savings is $96,785.67, etc. I'm not sure if this is a simple enough formula, or if it would just be a nightmare due to have to have equalities, etc. I also don't know if you would need to designate the total combined payment as a separate variable. Anyway, let me know what you guys think...I'm curious to see if this is possible. It'd definitely be nice to apply some calculus to a real problem. Thanks!
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  2. #2
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    Over-thinking. Why isn't it just pay the maximum available amount to the highest interest until it is paid off? Make minimum required payments on everything else. Of course, this assumes that all will go well. If you get sick, that minimum payment guy will not be very sensitive to your situation.
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  3. #3
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    See, I would have thought that would be the logical approach too. However, when I actually did make the amortization tables, and played around with minimum payments and how they affect the remaining interest paid over the life of the line, the lower interest rate actually saves more. The home loan is about 4.79%. The student loan is about 6.8%. However, because the home loan is over 30 years, and on a much larger principal, it still generates more in interest. Now, you would think that would mean just put all of the extra payment toward the mortgage, but I did a few different scenarios and that actually would not yield the maximum interest savings. See below:

    Interest Savings
    M $400 SL $100 $95,783.44
    M $425 SL $75 $96,785.67
    M $450 SL $50 $97,173.09
    M $475 SL $25 $96,657.24
    M $500 SL $0 $94,601.93

    So obviously some combination thereof would be the maximum savings. Obviously, we're not talking a big difference here, since something around M $450 and SL $50 (assuming total extra payment is $500) is the best, but it'd be nice if I had a formula where I could just decide if say I want to pay 600 extra this month, what the optimal amounts for each would be. I can just keep plugging in combinations of payments, but you can see how it'd be far quicker to just use a formula if one could be created.
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  4. #4
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    Indeed, not a calculus optimization problem. Make the minimum payment on whichever loan has a lower interest rate, and pay as much as you can to the one with a higher interest rate. When that one is paid off, then do the lower interest rate one. Seriously, if it's better to put 401 dollars into A and 99 dollars into B than 400 into A and 100 into B, how could it possibly not be better still to put 402 dollars into A and 98 dollars into B? Finding a local minimum would involve some situation where it would maybe be best to pay 436 dollars to A and 64 dollars to B, but 435 and 65 or 437 and 63 costs you more, and that isn't going to happen unless the interest rate itself is dependent on how much you pay (if a penalty is considered interest, then that would be why the next dollar below the minimum amount due would create such a minimum because that would be a very expensive dollar).

    Or you can just put it all into lottery tickets, win, and pay them both off. Yeah, do that.
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  5. #5
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    You posted while I was writing that.... Ok. Student loan is higher. Pay it off first. Pay home loan after. The home loan is a lower percentage interest over a higher principle? So WHAT. That only means it will TAKE LESS MONEY to pay off the student loan. One dollar more to the student loan and one dollar less to the home loan is an IMPROVEMENT TO YOU. It is a differential improvement, all the way until the point where the home loan is the minimum payment and the student loan is EVERYTHING else. Consider a more logical extreme. Suppose you had 2 loans, 100000 dollars from your bank for which you pay 5% interest every year, and 100 dollars from your bookie for which you pay 10% interest every 10 days. Which do you pay off first? Could you possibly be so naive as to say that you're paying $13.37 interest in one day on that 100k but only 96 cents interest on the 100 dollars so you should pay the 100k off first?
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  6. #6
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    Quote Originally Posted by scubatim84 View Post
    The home loan is about 4.79%. The student loan is about 6.8%.
    Interest Savings
    M $400 SL $100 $95,783.44
    M $425 SL $75 $96,785.67
    M $450 SL $50 $97,173.09
    Impossible...you are doing something "mysteriously(!)" to end up this way.

    I have a feeling that the student loan matures quite sooner than the home loan;
    when doing your calculations, say M $400 SL $100, once the SL is repaid,
    do you then apply the full $500 to the home loan?
    Last edited by Wilmer; June 1st 2011 at 07:19 PM.
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