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Math Help - Compounded Continously is key

  1. #1
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    Compounded Continously is key

    Hello everyone!! I'm new here and I've taken up business calc this semester. It just started getting kind of tough for me, and so I've been seeking out some help.

    Here is one of the problems I've been working on.

    The Armstrongs wish is to establish a custodial account to finance their children’s
    education. If they deposit $100 monthly for 8 years in a savings account paying
    8%/year compounded continuously, how much will their savings account be worth
    at the end of this period? Round your answer to the nearest dollar.

    I've asked several people..and everyone comes forth with different methods and answers. Can someone provide me with some help on how to go about this problem. I would really appreciate it. Thanks!!
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  2. #2
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    we start off by changing whatever values we need to a common time frame, so it will be quicker if we take the interest rate and convert it into a monthly rate.

    so we would have \sqrt[12]{1.08}-1= 0.006434

    now that we have it's nothing more then plugging it into the future value annuity formula which is:

     PMT \frac{(1 + i)^n - 1}{i}, where i=interest, n is period, and since it's 8 years, we have 8*12=96

    so plugging these values in we get:

     100 \frac{(1 + 0.006434)^{96} - 1}{0.006434}
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  3. #3
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    Accumulated Value at time n of a continously compounded annuity with continous 1 dollar payments is:

    \frac{e^{n\sigma} - 1}{\sigma}

    where \sigma is the force of interest.

    The problem is your payments are monthly but your interest compounding is continuous. This sounds like a trick question, or a typo by your professor.
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  4. #4
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    Thanks to both lllll and mathceleb. My teacher told me that there aren't any typos so I'll try working it out using both methods and see which one is correct. Thanks again.
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  5. #5
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    Quote Originally Posted by rmr62353 View Post
    Thanks to both lllll and mathceleb. My teacher told me that there aren't any typos so I'll try working it out using both methods and see which one is correct. Thanks again.
    Let me know if you need the force of interest formula.
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  6. #6
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    The problem, as noted by mathceleb, is your payments are monthly but your interest compounding is continuous. I could be wrong but I believe this type of situation falls under the category of complex (or general annuity).

    Accordingly, a continuously compounded rate of 8% is equivalent to a nominal rate of approximately 8.026726025% compounded monthly since both have an effective nominal rate of approximately 8.328706767%.

    Thus, with {j = 8.026726025%, m = 12}, and since i = j/m, you simply plug the resulting value of i into the formula quoted by lllll. At the end of 8 years, their savings account should be worth about $13,402.44 if the monthly deposits were made at the end of each month. If the deposits were made at the beginning of each month, their savings account should be worth about $13,492.09 since you’d have to use the annuity due version of the future value annuity formula.
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  7. #7
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    Quote Originally Posted by jonah View Post
    The problem, as noted by mathceleb, is your payments are monthly but your interest compounding is continuous. I could be wrong but I believe this type of situation falls under the category of complex (or general annuity).

    Accordingly, a continuously compounded rate of 8% is equivalent to a nominal rate of approximately 8.026726025% compounded monthly since both have an effective nominal rate of approximately 8.328706767%.

    Thus, with {j = 8.026726025%, m = 12}, and since i = j/m, you simply plug the resulting value of i into the formula quoted by lllll. At the end of 8 years, their savings account should be worth about $13,402.44 if the monthly deposits were made at the end of each month. If the deposits were made at the beginning of each month, their savings account should be worth about $13,492.09 since you’d have to use the annuity due version of the future value annuity formula.
    Kellison has a lesson on this, and I think jonah is on the right track. Check his answer and see if it matches, and then we can provide the work.
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  8. #8
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    Hey Jonah..I appreciate the help. I think your method is correct. I have not recieved the answer from my teacher just yet..so i can't tell you the actual answer. I'll keep everyone updated. Thanks for the help.
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  9. #9
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    Reflecting a bit more on your problem, I seem to have come up with a faster way to calculate the future value of an annuity whose interest compounding is continuous without going through the trouble of converting a continuously compounded rate to a discretely compounded equivalent rate. By letting
    S = Future Value;
    R = Periodic Deposit/Payment;
    j = the continuously compounding rate of interest;
    t = time; and
    m = the number of compounding periods per year; we can then proceed with the following derivation for the future value of an annuity where the periodic deposits/payments are made at the end of the period. We proceed thus:

    S = R + Re^[j(1/m)] + Re^[j(2/m)] + … + Re^[j(tm-2)/m] + Re^[j(tm-1)/m]

    ↔ S = R(e^[j(1/m)] * e^[j(tm-1)/m] – 1)/(e^[j(1/m)] – 1)

    ↔ S = R(e^[jt] – 1)/(e^[j/m] – 1)

    For the future value of an annuity where the periodic deposits/payments are made at the beginning of the period, we get the following:

    S” = Re^[j(1/m)] + Re^[j(2/m)] + … + Re^[j(tm-2)/m] + Re^[j(tm-1)/m] + Re^[j(tm-0)/m]

    ↔ S” = Re^[j(1/m)] + Re^[j(2/m)] + … + Re^[j(tm-2)/m] + Re^[j(tm-1)/m] + Re^[j(tm)/m]

    ↔ S” = Re^[j(1/m)] + Re^[j(2/m)] + … + Re^[j(tm-2)/m] + Re^[j(tm-1)/m] + Re^[jt]

    ↔ S” = R(e^[j(1/m)] * e^[jt] – e^[j(1/m)])/(e^[j(1/m)] – 1)

    ↔ S” = R(e^[j(tm+1)/m] – e^[j/m])/(e^[j/m] – 1)
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  10. #10
    Member jonah's Avatar
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    I thought perhaps I’d add a visual touch to this curious problem by attaching four sinking fund schedules.
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  11. #11
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    Amazing Jonah, thank you for the explanation. I'll take a good look at it once my final exams are over. Are you an accounting major?
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  12. #12
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    You’re quite welcome and yes, I once was an accounting major in a galaxy far, far away. Check out Schaum's Outline of Theory and Problems of Mathematics of Finance by Petr Zima and Robert L. Brown, ca 1996. Apparently, there’s a similar problem (corresponding to your second thread) on page 101, item 6.4. For a preview, go to

    http://books.google.com/books?id=IKWKPRAu8CAC&pg=PA100&dq=Schaum's+Outline +of+Theory+and+Problems+of+Mathematics+of+Finance& source=gbs_toc_r&cad=0_0&sig=se7fC6zlpAba3d6nHM6jM QVnKoc#PPA101,M1

    This url/site seems to bear out the formula I posted on your second thread. As to the actual answer, I guess we’ll all have to wait for your professor’s confirmation.
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