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This question is asked at a time when the Powerball jackpot is at a record high and in the news a lot.

Big jackpot games advertise their jackpot in terms of the amount that would be paid out in total if the winner chose an annuity option. If the winner chooses a lump-sum-cash option, the amount used to be about half the advertised jackpot, but now seems to be closer to the advertised jackpot (e.g. the ad is now $1.3B / $806M cash). Is there a predictable mathematical relationship between these two numbers? If so, what is it?

If the same answer applies to other big jackpot games, please feel free to note that in your answer.

Also, I realize that both numbers are estimates and that the actual value depends on ticket sales volume, but they are both published and likely to be related by some formula.

closed as off-topic by Chris W. Rea, user32479, JoeTaxpayer Jan 11 '16 at 0:01

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Is there a predictable mathematical relationship between these two numbers? If so, what is it?

As noted by Jasper, it appears from Powerball's site that the annuity is structured as one in which the payments increase 5% each year, paid out over 30 years. I'll take his word for it that insurance companies and banks compete by auction to determine the discount rate, call it r. Once that is determined, and assuming the lump-sum cash amount C is given (currently at about $589M from what I see), one can determine the first payment P via the equation

C = (P / (r - .05)) * (1 - (1.05 / (1 + r))^30

Subsequent years' payments are then successive 5% increases: P, P(1.05), P(1.05)^2, ..., and the current $950M figure I see at Powerball's site is the sum of those.

Grabbing what appear to be the current numbers, the equation above stands like

588,900,000 = (14,295,853 / (r - .05)) * (1 - (1.05 / (1 + r))^30

and I suppose r is about 2.67%.

Edit: The calculation above assumes an ordinary annuity, where the first payment would be made one year hence. As pointed out in a comment below, the first payment from the lottery will be basically right away, so we instead should treat it as an annuity due. Thus, the equation for determining the first payment should really be

C = (1 + r) * (P / (r - .05)) * (1 - (1.05 / (1 + r))^30

The only difference is the additional (1 + r) factor on the right-hand side, the reason being that the annuity-due has identical cash flows to the ordinary annuity, but all shifted one year earlier.

  • Have you adjusted for the fact that payouts start immediately, instead of being delayed by 12 months? – Jasper Jan 10 '16 at 5:48
  • @Jasper, I didn't, because I didn't know when the first payment would be made, and just went with one option. I now see your answer mentions the first payment is to be made "promptly," which does sound like it should be treated as an annuity-due. I'll make a note, thanks. – ETD Jan 10 '16 at 6:11
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The cash jackpot is 62% of the advertised jackpot.
The difference accounts for inflation over the 30 year payout period and is the numerical sum of all the dollars paid, not adjusted to present-day dollars; the present-day dollars sum is the cash option.
Mathematically, this reveals a 3.1% increase from year to year.

The Powerball FAQ states:

WHY IS THERE A DIFFERENCE BETWEEN THE CASH PAYOUTS IN THE POWERBALL AND MEGA MILLIONS GAMES?
There used to be a larger difference between the two games, but the Mega Millions game was recently changed to mirror the Powerball annuity option. The difference (spread) between the cash and annuity jackpots amounts are now about the same. Both games offer an annuity that pays out in 30 payments over 29 years (the first payment is immediate). Both annuity options are graduated. Any difference will be the result of investement choices. Mega Millions is invested in U.S. government strips. Powerball is invested in a couple dozen types of securities; all backed in some way by the U.S. government or agencies. It is important to understand that the lottery is invested in these securities; not the winner. The winner has a contract with the states to pay the annual prize payment, no matter what happens to the securities.

A fixed percentage of every Mega Millions and Powerball ticket sold goes into its CASH jackpot. The cash jackpot is all the money that the lottery has on hand from the sale of tickets in the game. If a player chooses the cash option, then the lottery will pay the entire cash amount to the winner (less income tax withholding amounts required by federal and state laws).

U.S. lotteries offer an ANNUITY jackpot option that can help reduce taxes and offers the winner a 100% guaranteed income stream over time. If a winner elects the annuity option, then the lottery will invest the entire cash amount BEFORE taxes are deducted. The difference between the CASH jackpot and the ANNUITY jackpot is the interest earnings that build over time.

At the time of writing, the Powerball ratio was tested only on very large jackpots. At the time of writing, Mega Millions was advertising a $15M jackpot / $9M cash option, which is a 60% ratio, representing an interest rate around 3.31%. It's not clear if this is a difference due to (a) rounding in advertised figures, (b) a different interest rate assumed by the different games, or (c) something wrong about this answer.

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Conceptually, the Advertised Jackpot is the total amount you would get if you take the 30-year annuity [800/30 = ~26.667 Mio/y]. As this option allows the paying company to defer payments over 30 years, today's net worth of that money stream is less than the sum of the payments.

Today's net worth can be calculated by dividing each future payment through the expected inflation for each year. If you calculate that, you will get the second amount, the Cash Option.

Of course, the details depend on the assumed inflation for each year, which you can guess, but the lottery company has their own estimate (and potentially a different one for each year).

  • It would be helpful to know what interest rate is assumed and if that's constant or not. – WBT Jan 8 '16 at 21:43
  • You could reverse calculate the interest rate that the Lottery Company used to do the math, but it is just their Estimate. Nobody knows inflation ratios for the next 30 years. - I don't have the formula from the top of my head, but it is well known. – Aganju Jan 8 '16 at 21:44
  • And it would not be constant - they have probably some qualified separate prediction for each year. Which they pay some money for, so they won't tell you. – Aganju Jan 8 '16 at 21:45
  • If someone else knows the specific statistic used to measure inflation, that'd be helpful in an answer. Also, once somebody wins, is it a set amount that they will get each year, with the schedule of payments completely known in advance, or does the amount of payment change each year based on that statistic's value? – WBT Jan 8 '16 at 21:46
  • They would need to commit to that annual payment. If the inflation ration is different than expected, it is their gain or loss. But over many winners and years, it levels out. - you can find out the market expectation for future inflation ratios by dividing future prices for various years, but that's a whole 'nother topic which needs half a book to explain. – Aganju Jan 8 '16 at 21:48
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It is common for American lotteries to design their jackpot payouts so that the advertised total payout is about 60 percent greater than the cash payout. For example, today a $ 558,000,000 cash payout is advertised as an estimated $ 900,000,000 total payout.

The exact ratio for the Powerball jackpot varies from jackpot to jackpot, and is not known until after the winning numbers are announced. The amount of money in the Powerball cash jackpot is determined based on the amount that rolled over from the previous drawing (or if no money rolled over, the amount anted up by the lottery organizers) plus a percentage of the ticket sales for the current drawing. The annuity amount is based on the results of an auction. Major insurance companies and banks compete to offer the best discount rate for the annuity. This discount rate varies from jackpot to jackpot.

In decades with declining interest rates, there is a tendency for the ratio of the advertised payout to the cash value to decline. In order to boost this ratio, every few years the lottery organizers re-design the annuity. Lottery annuities once were equal annual amounts for twenty years. Later designs have included equal annual amounts for thirty years. The current design is 30 annual payments (with the first payment occurring promptly), with the payments increasing 5 percent per year.

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