Thursday, March 29, 2012

Money for (almost) Nothing

Five Hundred Mega Dollars, to be precise.
(Image from Wikipedia)
I am not typically interested in lotteries.  They seem silly and I am seriously beginning to question their usefulness in bringing about a good harvest.  But this morning I read in the news that the Mega Millions lottery currently has a world record jackpot up for grabs.  In fact, the jackpot is so big...

Tonight Show Audience:  HOW BIG IS IT?

It is so big that I decided to do a little bit of analysis on the expected returns.  Zing!

Some Background


First, a little background.  The Mega Millions lottery is an aptly named lottery in which numbered ping pong balls are pulled from a giant rotating tub of randomization.  Five of these are drawn from one tub of 56 balls, with no replacement.  The sixth ball (the so-called "Mega Ball") is drawn from a separate tub of 46 balls.

To play, one picks 5 different numbers (1-56) for the regular draws and one number (1-46) for the Mega Ball.  The first five can match in any order, but the last ball has to match with the Mega Ball.  Prizes are given out based on how many numbers you match.

Stolen from the Mega Millions website, the prizes and odds are given in the table below.  The current jackpot is listed at $500 million (if taken in annuity) or $359 million if taken in an up-front lump sum.  It costs $1 dollar to play.

Don't worry about the asterisk.  It just says CA is lame.
(Source:  Mega Millions)
Hot diggity daffodil, we're ready to get going!

Expected Winnings


Alright, so it costs $1 to play and we could potentially win $500 million.  It sure feels like it is worth it to play (what's the harm?).  But we can do better than feelings, we have... MATH!

Since we have an exhaustive list of outcomes and their probabilities (which is just the inverse of the big number in the "chances" column), we can calculate the expectation value for our winnings.  The expectation value is just the sum over all the possible prize values times the probability of winning that prize.  In other words,

\[\langle W \rangle = \sum_i W_i \times p_i, \]

where we denote our expected winnings in angled brackets. 

In essence, this value represents the average prize you would win if you played this lottery over and over and over again (or played all the combinations of numbers).  

Setting the jackpot to $500 million, we can now compute the expected winnings as

\[ \langle W \rangle = \frac{\$ 500,000,000}{175,711,536} + \frac{\$ 250,000}{3,904,701} +\frac{\$ 10,000}{689,065} + \frac{\$ 150}{15,313} + \frac{\$ 150}{13,781} \]
\[+ \frac{\$ 10}{844} + \frac{\$ 7}{306} + \frac{\$ 3}{141} + \frac{\$ 2}{75}\]

A few flicks of the abacus later, we find that the expectation value of our prize is

\[\langle W \rangle = \$ 3.02,\]

which means that after we subtract the dollar we paid for the ticket, our expected return is $2.02.  

But what if we had chosen to take our winnings as a lump sum of $359 million instead of the $500 million paid out over a span of 26 years?  In that case we find

\[\langle W \rangle = \$ 2.22,\]

which results in a $1.22 gain when we subtract the dollar we paid for the ticket.  

At least in a statistical sense for this particular jackpot, one is better off playing than not playing.  But are we forgetting anything?

The Taxman


If you win a $500 million jackpot, do you really get a $500 million jackpot?  Well, no.  For winnings in a lottery over $5000, the IRS withholds 25% in federal income taxes.  Additionally, the winnings are subject to state taxes as well.  For example, if I were to win, the great state of New York would be entitled to about 6.8% (apparently also just for winnings above $5000).  

After applying federal and state taxes to the prizes above $5000, we now have an expected winnings of

\[ \langle W \rangle = \left[1-(0.25 + 0.068)\right]\times\left(\frac{\mbox{Jackpot}}{175,711,536} + \frac{\$ 250,000}{3,904,701} +\frac{\$ 10,000}{689,065}\right) \]
\[+ \frac{\$ 150}{15,313} + \frac{\$ 150}{13,781}+ \frac{\$ 10}{844} + \frac{\$ 7}{306} + \frac{\$ 3}{141} + \frac{\$ 2}{75},\]

which gives an expected net win (minus the $1 for the ticket) of $1.10 for the $500 million annuity prize and $0.55 for the $359 million up-front lump sum.  

We're still in the black, but it's slowly slipping away.  Is there anything else we need to factor in?  Well, yes.  For one thing, winning the jackpot qualifies us for the top tax bracket, so most of the winnings would be taxed at the top marginal tax rate of 35%.  Welcome to the 1%, kids! [1].


Changing the federal tax rate on the jackpot from 25% to 35% and recalculating, we find net expected winnings of $0.81 for the $500 million annuity and $0.34 for the $359 million lump sum.  Surprisingly, it is still worth it in a statistical sense.

Is it always like this?


One thing to keep in mind as we make these estimates is that this is a historically large jackpot.  So even though it may be favorable to play this time, this will not always be the case.  In fact, we can find the minimum jackpot value for which this is the case.

The condition in which our expected return is a gain (rather than a loss) is

\[ \langle W \rangle - \$1.00 > 0. \]

For simplicity, let's ignore the top marginal tax rate and just factor in the 25% withholding and the 6.8% state tax.  Solving for the minimum jackpot using the expression for we found in the last section, we see that 

\[ \mbox{Jackpot}_{min} = \$217~\mbox{million}.\]

Technically, this would have to be the amount actually awarded by the payment method of your choice.  The stated jackpot is always the annuity method (because it looks higher).  The lump sum offering is at most about 70% of the stated jackpot.  So if you want to take the lump sum offering the stated jackpot will need to be 

\[ \mbox{Jackpot}_{min} = \$217~\mbox{million} / 0.7 = \$310~\mbox{million}.\]

In fact, these values are likely a bit low, since we have not included the increase to the marginal tax rate, nor have we included other effects like having to split a prize (which seems to happen a lot) or inflation effects if you take the prize in yearly installments.

In any case, a quick look through the jackpot history shows that these threshold values are only met occasionally.  An eyeball estimate puts about one jackpot per year that exceeds the (absolute) minimum $217 million threshold.

So am I going to win?


No.  No, you will not.  BUT if you played record setting lotteries hundreds of millions of times, you might see decent (~10%) returns.  Although, it may just be easiest to, you know, invest that money.



Only One Useless Footnote


[1] Although, to be fair, the top marginal tax rate is currently at historical lows.  It could always be worse... [back]


12 comments:

  1. Another thing that can trip you up is multiple winners, which would cut your prize down in half (or more). If the number of tickets sold is in the tens of millions this is actually a likely occurrence.

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  2. You also need to consider a stochastic factor known as the Megaplier which allows the jackpot to be multiplied by the factor if it is in fact purchased.

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    Replies
    1. I saw this and really didn't want to have to factor it in. BUT, the Megaplier does NOT apply to the jackpot, just the rest of the prizes. Since the jackpot is main factor here (the rest of the terms add up to an expected value of maybe a dozen or so cents, I think), it should't really change the results by that much.

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  3. There's another bit of analysis you should apply, which is very revealing.

    Granted that you have a positive expectation... how many tickets should you buy? Do you sell the house and invest everything you have in the lottery? No... because that gives a 97.5% chance of being wiped out; and then you couldn't try again next time opportunity knocks.

    The mathematical solution here is called the "Kelly Criteron". Assuming you have a certain amount of funds, you should invest the fraction of that which gives you the most rapid rate of increase of your bankroll, on the assumption you keep repeating a bet of that fraction of your bank every time you get this opportunity. Simply put, you bet the maximize the logarithm of your bank roll.

    So let me ask this; what amount of assets should you have to make buying a single ticket a good investment?

    The answer to THIS question is .... you should have a bit over 120 million dollars in assets before even buying 1 ticket is a worthwhile investment.

    Wikipedia has a good introduction to the Kelly criterion for those who want to dig into why it works.

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    Replies
    1. Sounds neat, I'll take a look at it. Thanks for the info!

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  4. How are the odds of getting the last ball (1-46) only 1/75? Shouldn't the odds be 1/46?

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    Replies
    1. If the prize were for JUST getting the last ball (and didn't care about matching any of the other numbers) then the probability would be 1/46. But in this case we have to match the last ball AND not match the others. This reduces the probability.

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    2. That threw me at first, too. But, if you add up the probabilities of all outcomes with a matching final ball, it does indeed work out to 1/46.

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  5. Shouldn't you also try and factor in the opportunity cost of physically buying the ticket? That's going to be almost identical no matter how many tickets you buy (although keep increasing if you have to go to more than one location). That's going to be prohibitive for single, or low numbers of tickets.

    And of course if you personally want to enter, you need to factor in the cost of doing the maths, which is already sunk into the project. Assuming the expected outcome remains positive after accounting for possible jackpot splitting, then doesn't that dramatically increase the number of tickets that you would have to buy for this to be optimal? :-p

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  6. I'm not sure why you applied the tax factor to prizes that bring in less than $5000, if that is the only time they are taxed.

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  7. See http://www.durangobill.com/MegaMillionsOdds.html for an analysis of the expected value after taking into account the chance of having to split a prize. Conclusion: "even for a huge Jackpot similar to the quoted $640 million for 3/30/2012, your after tax expected return is less than $0.73 for every $1.00 ticket that you buy."

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    ReplyDelete