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Formula for annuity payments

  1. Jan 13, 2016 #1
    So, unless you've been living under a rock, you should know the jackpot for the powerball is at 1.5 billion dollars.

    I was looking up the distribution of annuity payments and the website said that the payments are not equally distributed. They are incremented by 5% each year.

    Like the nerd I am, I tried figuring out the math but didn't know where to start.

    I know that 1.5 billion / 30 payments = 50 million a year (before taxes)

    How would you go about figuring out 30 payments equating to 1.5 billion but the condition is that each payment is 5% more than the next?
     
  2. jcsd
  3. Jan 13, 2016 #2

    Krylov

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    Is the amount increasing or decreasing in time?

    If the total amount is ##p## (= 1.5 billon dollars) and the rate of increase / decrease is ##\lambda## (= 1.05 or 0.95) and your amount in the ##k##th year is ##a_k##, then ##a_k = \lambda^{k-1}a_1## for ##k = 1,\ldots,n## where ##n## is the amount of years. Now set ##\sum_{k=1}^n{a_k} = p## (geometric sum) and solve for ##a_1##.
     
  4. Jan 13, 2016 #3
    Increasing.

    Year 1 = x
    Year 2 = x +(x* 0.05)
    and keeps incrementing until 1.5 billion
     
  5. Jan 13, 2016 #4

    Krylov

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    Ok, so then you set ##\lambda = 1.05##, take the geometric sum and solve the equation for ##a_1##. Once ##a_1## is known, use the formula for ##a_k## to compute the amount in year ##k##.
     
  6. Jan 13, 2016 #5
    Thanks a lot!
     
  7. Jan 13, 2016 #6

    Krylov

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    Don't mention it. Just be sure to let me know if you win the jackpot :wink:
     
  8. Jan 13, 2016 #7
    I just plugged it in year one

    I think I'm doing it wrong.

    1.5 billion = (1.05) ^ (1-1) * a1

    and I solve for a1?

    It seems like I get 1.5 billion
     
  9. Jan 13, 2016 #8
    Nvm I get it.
     
  10. Jan 13, 2016 #9

    Krylov

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    No, that is not correct. You need to solve
    $$
    a_1\sum_{k=1}^n{\lambda^{k-1}} = p
    $$
    First you need to evaluate the sum, using the standard formula for the geometric sum. I leave that up to you as a challenge. Once that is done, you can solve for ##a_1##.
     
  11. Jan 13, 2016 #10

    Krylov

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    Ok, very well!
     
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