“Powerball Tip #1: Only pick winning numbers.” – Stephen Colbert
Unless you have been living under a rock, all of our readers are probably aware that last week’s Powerball jackpot swelled to an all-time high of over $1.5 billion. News channels and water cooler conversations were abuzz all week with talk of the massive lottery jackpot. Last week I was asked by a client whether buying a lottery ticket made sense from an odds perspective since the jackpot was so large. Having never played the lottery, there was a bit of a learning curve for me to answer this question. What I discovered along the way became the genesis for this weeks’ Insight.
A Powerball lottery ticket cost $2 to purchase and has exactly a 1 in 292,201,338 chance of winning the jackpot. These odds are based on the number of different combinations five white balls ranging from 1 to 69 and one red ball ranging from 1 to 26 can produce. These odds remain the same no matter how many people buy tickets or the size of the jackpot. As such, a quick assumption might be made that the way to figure out whether it makes sense to buy a Powerball ticket from an odds perspective (we call this having a positive expected value) would be to divide the jackpot by 292,201,338 and compare it to the $2 investment. In last week’s lottery that would have meant each ticket had an possitive expected value of little over $3 after accounting for the $2 cost. Unfortunately, this quick and dirty analysis fails to account for several key factors.
Time Value of Money
The first thing to understand about the lottery is that the jackpot number is not the same as the payout. If a winner wants to receive the advertised amount of money, they have to accept it in annual installments over a 30 year time horizon. In the history of Powerball, only two winners have ever selected this option. The more common alternative is to take the lump sum payment which recently has equated to around 62% of the stated jackpot. For last week’s jackpot, that would mean a lump sum payment of around $982 million.
How is this discount factor determined? I can’t say for sure, but my guess would be it is based on what the lottery thinks it can earn on the funds by holding them over a 30 year period versus paying them out in a lump sum today. Based on last week’s jackpot, the return assumption works out to 3% a year. In other words, if you think you can earn more than 3% a year on your money, it is better to take the lump sum than the annuity all else equal (e.g. taxes, spending habits, etc.)
The Perpetual Winner
Even though the odds of any single person winning the lotter is extremely low, there is one entity that is always a winner and that is the IRS. The IRS always wins because lottery winnings are considered income and as such are taxed at the current, highest marginal income tax rate of 39.6%. Additionally, depending on where you live, state taxing authorities will want a bite at the apple as well. In fact, for New York City residents, additional taxes will have to be paid at the municipal level (NYC residents pay an additional 8.82% in state and 3.876% in municipal taxes on lottery winnings). On the flip side, some states colored in grey in the map below have no state taxes on lottery winnings. Since we are a Colorado based firm, we used the Colorado state tax rate of 4% on lottery winnings for our analysis, which drops the lump sum payment down to $554 million after federal and state taxes are paid.
Kissing Your Sister
Navy football coach Eddie Erdelatz is the first person credited with saying “a tie is like kissing your sister” after a scoreless outing against Duke in 1953. Having grown up without any sisters, I’ll have to take his word for it. When it comes to the lottery, tying obviously isn’t as bad (we are still talking about tens to hundreds of millions of dollars here), but it makes a big difference when calculating the expected value of a single Powerball ticket. Unlike the odds of a single ticket winning the Powerball jackpot (1 in 292,201,338), the odds of a tie are very much dictated by the number of tickets in play. The way to estimate the number of tickets in play is to measure the difference in the previous lump sum payout versus the current lump sum payout and multiply that number by 1.5. For last week’s lottery, that number comes out to 590 million tickets in play. The math behind how to calculate the odds of tying are beyond the scope of this post, but the table below summaries the results.
From the table we see that, assuming a winner, the odds of a two person tie (31.2%) in last week’s Powerball lottery were actually greater than a single winner (31.0%) and the chance of a tie between any number of people (summing up all the probabilities for 2 to 8 winners = 69.0%) was much greater than a single winner. In order to calculate the expected value of a lottery ticket with the potential of a tie, we simply add up all the payouts multiplied by their probability of occurring ($554M*31.0% + $277M*31.2%...), which works out to be just under $318 million.
At this point we now have a realistic, expected value for a winning lottery ticket which accounts for the lump sum discount, taxes paid, and the chance of splitting the pot with other winners. Therefore we take this number and multiply it by our chance of winning (1/292,201,338) in order to calculate the expected value of our Powerball ticket. For last week’s jackpot, this worked out to be $1.09. In addition to the jackpot, there are smaller payouts for tickets that match some of the winning numbers. These payouts add an extra $0.22 in value per ticket as summarized in the embedded table. Therefore, the expected value of a single Powerball ticket in last week’s lottery was $1.31 ($1.09 + $0.22). In other words, an expected value decision maker (like a computer or Spock from Star Trek) would gladly trade a Powerball ticket last week for any amount of money greater than $1.31 and would happily buy a Powerball ticket for any amount of money less than $1.31.
Once we factor in the initial cost of $2 per ticket, the expected value of a Powerball ticket drops to a negative 69 cents. In other words, even at the historic jackpot of over $1.5 billion, each Powerball ticket was still a bad bet. In fact, the stated jackpot would need to be twice as large at $2.6 billion in order for a Powerball ticket to have a breakeven expected value (this assumes the same odds of tying we used above, which would probably be higher with a jackpot that large so $2.6 billion actually understates the amount). So why were people flocking to buy Powerball tickets in droves last week? The answer is obviously that humans are emotional beings that are not purely dictated by logic (yet another plug for why we use a quantitative approach to investing which takes “gut feel” and emotions out of the equation). But allow me to go even one level deeper to explore why buying a Powerball ticket may actually make sense for a lot of people.
Unlike computers or Spock from Star Trek, human beings derive value from emotional experiences. For example, Spock would never choose to pay $10 to go see a movie because he wouldn’t derive $10 worth of enjoyment out of the experience, but millions of people consciously make that decision and willingly fork over their $10 to enjoy a movie every day. As such, an argument could be made that as long as one could derive $0.69 of value out of the watercooler banter and allowing oneself to dream about the “what if” scenarios, then purchasing a single Powerball lottery ticket makes complete sense. That being said, pressing the bad bet by buying multiple Powerball tickets such as the group which spent $146,000 on lottery tickets, makes no sense at all since the marginal value of the all the additional lottery tickets is close to nothing in comparison to the first (e.g. you can still banter and dream with 1 ticket as well as you can with 73,000 tickets).
I’m sure some of you may be wondering at this point whether I purchased a Powerball ticket last week or not. The answer is I did, and my losing numbers (along with my two buddies) will forever be memorialized in the thumbnail image of this post. I too enjoyed allowing myself to briefly dream about the “what if” scenarios before it all went up in smoke once the winning numbers were read. My conclusion through this brief journey was that human emotions can often have a much stronger influence than logic and reason when it comes to decision making, but we can’t completely discount our emotions as they are an integral part of what makes us human and drives us to pursue what makes life truly rich.
Author Elliott Orsillo, CFA is a founding member of Season Investments and serves on the investment committee overseeing the management of client assets. He spent nearly ten years as a financial analyst and portfolio manager working primarily with institutional clients prior to co-founding Season Investments. Elliott earned a bachelor's degree in Engineering from Oral Roberts University and a master's degree from Stanford University in Management Science & Engineering with an emphasis in Finance. Elliott and his wife Gigi have three children and like to spend their time outdoors enjoying everything the great state of Colorado has to offer.
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