Every once in a while, I buy a Powerball lottery ticket for the dream of being rich. Earlier tonight I was at a gas station and decided to buy one. I was somewhat shocked to find out that the price had gone from $1 to $2. I asked the attendant about this, and he said that the price had gone up at the start of the year. I was curious to see if the value of a ticket had also climbed. This page is an attempt to calculate the approximate market value of a single Powerball ticket given the rules in place as of April 2012..
I assumed the market value to be where the odds of winning multiplied by the amount won would be the market value. For example, if there was a 1 in 5 chance of winning $10, the value of a ticket would be (1/5)*$10 = $2. Given the multiple ways of winning, this is a good approach, but the calculations are a little more complex.
The first part of calculating the value of the ticket is the easiest part. According to the Powerball web site, there are nine ways to win. The first eight of these are easy to see how they contribute to the value of the ticket:
· 1 in 55.41 odds of winning $4 -> $0.0722 of value
· 1 in 110.81 odds of winning $4 -> $0.0361 of value
· 1 in 706.43 odds of winning $7 -> $0.00991 of value
· 1 in 360.14 odds of winning $7 -> $0.0194 of value
· 1 in 12,244.83 odds of winning $100 -> $0.00817 of value
· 1 in 19,087.53 odds of winning $100 -> $0.00524 of value
· 1 in 648,975.96 odds of winning $10,000-> $0.0154 of value
· 1 in 5,153,632.65 odds of winning $1,000,000 -> $0.194 of value
I’m left looking at the large percentage of cost going into the one million dollar prizes in compared to the other prizes; I’d bet J that there is a good marketing reason behind that. However, I digress.
Before calculating the total value the non-jackpot prizes add to the value of the ticket, I need to include an often forgotten factor, which is you need to pay taxes on winnings. Because I play with post-tax money, I’ll calculate winnings in post-tax terms. For this calculation I’ll assume that taxes are only paid in amounts greater than or equal to $10,000. For tax rates, I’ll assume that the total tax burden is 40% (35% federal and 5% state – although I live in Minnesota, so 5% is too low.)
Note: If I win, I got to remember to see if it is possible to become a Florida resident before claiming the winnings in order to decrease the tax burden)
With taxes factored in, the last two non-jackpot entries become:
· 1 in 648,975.96 odds of winning ($10,000 * 60%) -> $0.00925 of value
· 1 in 5,153,632.65 odds of winning ($1,000,000 * 60%) -> $0.116 of value
Thus, the after-tax non-jackpot value of a lottery ticket is roughly:
$0.0722 + $0.0361 + $0.00991 + $0.0194 + $0.00817 + $0.00524 + $0.0108 + $0.136 = $0.276
The next part of the calculation is to determine the value of the jackpot. Given what I have mentioned, it might seem that you just figure the after-tax value of the jackpot and then divide by the odds of winning it. While this approach would provide an upper bound for the value, it does not account for the chances of multiple winners splitting the jackpot. In order to get a sense of the value of the portion of the jackpot you are likely to win, we need to have some way to estimate the number of tickets being sold, and I’m going to start with an assumption that more tickets are sold when the jackpot is larger…
For this bit, I’m borrowing some math from this site. (Not surprising, someone else has done this type of thing before using a straight-forward approach to figure out the number of tickets in play. I’ll simplify the math, and state that you can roughly assume that you will obtain an average of 75% of the jackpot if you match all the numbers. Given that estimate, the value of the jackpot is:
([Jackpot value] * 60% * 75%)/175,223,510.00 = (45% * [Jackpot value])/175,223,510.00
I have assumed here that I take the jackpot as a lump sum. I don’t want to get into the math of the future value of money given inflation. Besides, everyone takes the lump sum. What I end up with is a graph of the approximate value of a Powerball ticket. The graph is:
So, for the ticket that I bought for tomorrow night’s lottery of 81.4 million dollars, the market value was 49 cents. (The remaining $1.51 was just paying for the dream.)
As an interesting side note, it looks like the break-even point on the graph is somewhere around a $700,000,000 cash payout, which is the cash payout you should expect when annuity (the big number in the advertising) to be at 1.13 Billion Dollars.
Note: This page has lots of assumptions, and should not be assumed to be financial advice. If you are here looking for financial advice on random web pages, you might consider reflecting on the wisdom of those choices. This page was just written for my entertainment.