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I was recently on a quest to find certain formulas that would solve for the different variables in investments.  This discussion gets a little technical.  You'll need to have some advanced Algebra skills to fully comprehend the formulas:

First, the variables:

FV = future value
A = one-time investment (not for annuities)
p = investment per compound period
i = interest rate
c = # of compound periods per year
n = # of compound periods

To get p, I simply take the amount I want to invest per month, multiply it by 12 to get a yearly investment amount, then divide by c to get the investment per compound period.  To get n, I take the number of years I wish to invest and multiply it by c to get the number of compound periods.

First, let's deal with simple compound interest with one-time investments.  Here's the formula that will let us know the future value (FV) of our investment after n years if we invest A at i interest compounded c times per year:

FV = A (1 + i/c)⁽ⁿ⁾

OK, now let's say we want to find out what we have to invest today (A) to have FV in the future if we get i interest compounded c times per year for n years:

           FV
A = -----------
       (1 + i/c)ⁿ

Finally, I want to find out how long it will take me (n) to have FV in the future if I invest A initially at i interest compounded c times per year:

       ln(FV) - ln(A)
n = ------------------
       ln(c + i) - ln(c)

NOTE: ln is the natural logarithm function.

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Now, let's go on to annuities.  Annuities are similar to one-time investments in all respects, except that you invest at regular intervals instead of just a one-time sum of money.  For instance, investing $150.00 per month in a mutual fund.

Here's the formula that tells us how much we will have (FV) after n years if we invest p per compound period at i interest compounded c times per year:

         p [(1 + i/c)ⁿ - 1]
FV = --------------------
                  (i/c)

Wouldn't it be interesting to find out how much we need to invest per month (p) to reach $1 million (FV) at i interest compounded c times per year for n years?

              FVi
p = -------------------
      c [(1 + i/c)ⁿ - 1]

And finally, I think it would be great to figure out how long it would take me (n) to reach $1 million (FV) if I make p monthly investments at i interest compounded c times per year:

      ln(FVi + cp) - ln(cp)
n = -------------------------
          ln(c + i) - ln(c)

NOTE: ln is the natural logarithm function.

I designed some forms for annuities as well:

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