Project 3: Lottery Lump-Sum Payouts

When a lottery prize is announced, the present-day value of the winnings is a lot less than the announced prize. The announced prize is the total of all payments. It is similar to the mortgage example in section 3.4, where over a $240,000 mortgage at 6% over a 30-year term, the homeowner makes about $518,000 of payments over the life of the loan. If these were lottery payments, the prize would be announced as $518,000 even though the value of the payments in today’s dollars is only $240,000.

You can calculate the present-day value of a lottery prize if you know (or estimate) the interest rate that the lottery company can access by using the reverse loan payment formula from section 3.3. In this example, we will use a $1.5 million prize paid out monthly over a ten-year term.
  1. Calculate the number of payments and the amount of each payment.
    Ten years would be 10 × 12 = 120 payments.   $1.5 mil = $12,500 per month.
    120
  2. Determine (or estimate) the interest rate.

    Suppose the lottery has access to a savings account with a 5% APR for this example.

  3. Use the second loan payment formula from section 3.3 to determine what loan amount could be supported by this payment.

    Over ten years, a loan payment of $12,500 per month would support a loan amount of:

    $12,500 ×  1 – (1+  0.05 ) –12 × 10  = $1,178,517, rounded to the nearest dollar
    12
    ( 0.05 )
    12

If you have a lottery prize that is paid out weekly, replace each 12 in the formula by 52.

Calculate the present-day value of the following lottery prizes (round all calculations to the nearest dollar). Then answer the questions in the final part.

Part 1 A $10 million prize, paid out monthly over 15 years, where the lottery has access to 5.5% APR.
  1. There will be total payments, and each payment will be $.
  2. The lotter has access to an APR of %.
  3. The present-day avlue of this lottery prize is:
    ×  1 – (1+  ) –12 ×  =
    12
    ( )
    12
Part 2 A $10 million prize, paid out monthly over 25 years, where the lottery has access to 5.5% APR.
  1. There will be total payments, and each payment will be $.
  2. The lotter has access to an APR of %.
  3. The present-day avlue of this lottery prize is:
    ×  1 – (1+  ) –12 ×  =
    12
    ( )
    12
Part 3 A $1000-per-week prize, paid out over 30 years, where the lottery has access to 5% APR.
  1. There will be total payments, and each payment will be $.
  2. (don't overthink this one)
    The lottery has access to an APR of %.
  3. The present-day avlue of this lottery prize is:
When writing out the formula for the final answer, note that the payments are weekly, and since there are weeks in a year, you should replace all occurrences of 12 in the formula with this number.
Part 4 Look up the current U.S. Powerball jackpot size and payout terms (how many years and how often are the payments, ignore that the payments are “graduated”). Then search the web to see what interest rates are offered on the best long-term savings accounts. Use this data to determine the present-day value for this jackpot.

Current announced jackpot from the Powerball website (or news article): $.
  1. There will be total payments, and each payment will be $.
  2. The lottery has access to an APR of % (use the best savings account that you can find).
  3. The present-day value of this lottery prize is:
When writing out the formula for the final answer, since the number of payments each year is , you should replace all occurrences of 12 in the formula with this number.
Questions Answer the following questions:
  1. Does the relative value, compared to the announced prize, go up or down when the payout period is lengthened? (look at the answers to parts 1 and 2)
  2. Why would a lottery prefer to advertise the total amount of the payouts, rather than the present-day value of the prize?
  3. Does it make sense to reason that if you play the lottery often enough, eventually you will win back all the money you spent on lottery tickets, or is there evidence of some kind that indicates this is not the case?
  4. Bonus Question: When searching the web for interest rates on long-term savings accounts, advertised rates are usually given as APY (annual yield) rather than APR (annual rate). It is actually better to use the APY instead of the APR to estimate the present-day value of the Powerball jackpot. Can you explain why?

Once completed, this page to a pdf document; then hand it in through your course's Learning Management System.

 Neil Simonetti

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