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Rate of return witrh logs

  1. Sep 11, 2010 #1
    1. The problem statement, all variables and given/known data

    In 1903, a Picasso painting was purchased for $600. The family of the original owner sold the painting in 1995 for $29,152,000. What rate of return (interest) did the family receive on the investment?



    2. Relevant equations

    Single Payment Compound Interest Formula:

    F = P(1+i)n

    where,

    F= a future some of money (future value)
    P= a present sum of money
    n= number of interest periods
    i = interest rate per interest period.

    3. The attempt at a solution

    solve for i:

    F = P(1+i)n
    F/P = (1+i)n
    log(F/P) = n log (1+i)

    This is as far as I get. I know that n log (1+i) does not equal n log 1 + n log i, but I don't know what to do to isolate the i. I can't just divide both sides by n log because there is no such thing as n log. I thought about moving n log (1+i) to the left side and setting the equation to zero, but I didn't get anywhere with that either.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited: Sep 11, 2010
  2. jcsd
  3. Sep 11, 2010 #2

    eumyang

    User Avatar
    Homework Helper

    What is the base of "log"? When I look at it, I assume common log (base 10), but IIRC in other places the natural log (base e) is assumed. Whatever it is, you can raise a number to both sides. In other words,
    [tex]x = y \rightarrow a^x = a^y[/tex]
    The number you choose for a should probably be the base of the logarithm you were using earlier. :wink:
     
    Last edited by a moderator: Sep 15, 2010
  4. Sep 11, 2010 #3
    The problem says nothing about the base, so I can only assume it is base 10. So then, if I understand you correctly, I would end up with:

    10log(F/P) = 10n log(1+i)

    Please excuse my ignorance, but I've always had a problem grasping logs.

    so, if the above is correct, then I should be able to use the alog u = u property and get:

    F/P = (1+i)

    i = (F/P) - 1

    Is this correct?
     
    Last edited: Sep 11, 2010
  5. Sep 11, 2010 #4
    nope. That's not correct. That gives me an answer of 48,585.67. Seems a bit high for an interest rate. Plus it doesn't work when I plug the numbers into the original formula.
     
  6. Sep 11, 2010 #5

    Mark44

    Staff: Mentor

    When you get to this step,
    F/P = (1+i)n,
    instead of taking the log of both sides, take the n-th root of both sides (n is known). That will isolate 1 + i.
     
  7. Sep 11, 2010 #6
    You are right, n is known. Even so, if I am doing this right I still end up with i = F/P - 1

    (F/P)92 = (1+i)92

    i = (F/P) - 1

    = ($29,152,000/$600) - 1

    = 48,586.67%

    Ok, this works out if I take $600 x 48,586.67%, however, when I try to plug $48586.67% into the original F=P(1+i)n formula, I get an overflow error when I should get $29,152,000.

    F = P(1+i)n

    = $600(1+48,586.67%)92 = overflow error
     
  8. Sep 11, 2010 #7
    I just thought of something. I assumed that n = 1995 - 103 = 92 years. This might be wrong. The painting was not compounded yearly. It was only sold once. Therefore, n=1.

    Now it works

    F = $600(1+48,586.67%)1
    = $29,152,600.00

    Thank you everyone for your help.
     
  9. Sep 12, 2010 #8

    Mark44

    Staff: Mentor

    No, this isn't right.
    Assuming that n = 92 is correct, if you start with F/P = (1 + i)92, and then take the 92nd root (not power) of each side, what do you get?
    $48586.67% doesn't make any sense. Is it a dollar amount or a percentage? It can't be both. Note that 48,586.67% = 485.8667 as a decimal value.
     
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