In case you’ve been on the weird part of the internet today, you probably already know that the Mega Millions drawing’s jackpot tonight is for a prize of $636,000,000 and a cash option of $341,000,000.
I’ll admit shedding a single tear on behalf of math when reading Business Insider’s piece today on the Mega Millions having a positive expected value. Folks, listen: there is no positive expected value in the Mega Millions tonight, nor will there be next time there is a massive prize.
We originally covered this in a post calculating the expected value for the last huge Mega Millions drawing, and we even followed it up with a calculator so you can determine what the expected value is for the Mega Millions yourself. (Click it, it won’t hurt).
Here’s Where The Math is Wrong
Our calculation? Here you go, from our Mega Millions expected value calculator:
We estimate that with every dollar you spend (and a 39.6% federal marginal tax rate and 5% state tax rate), you’re throwing 56 cents down the drain. Get that? EV = 44 cents per ticket. That’s well below $1.00.
Even in a Randian fantasy with $0 federal and $0 state taxes (where do you live? Sealand?) you’d only have an expected value of 78 cents per ticket – essentially throwing away 22 cents in that fantasy land.
So I Don’t Have to Go to That Page and Read the Methodology, Explain it to Me Simply?
(This math is simplified – scroll to the end for the advanced [read: more correct, kinda] math).
Sure, here’s the TL;DR: if you’re going to use the $636,000,000 number you’re not comparing like quantities with the $1 you are spending on the ticket.
The Mega Millions pays you immediately then for 29 years after (0, 1, 2 …. 29 – not 30 years, basically), so to get from $341,000,000 to $636,000,000 we have to do some compounding – the math is:
rate = ( (future/present) ^ (1/ number of periods) ) – 1
rate = ( (636 / 341) ^ (1/29) ) – 1
rate = 2.1726%
Okay, but what does that mean?
It means you need to either use the $341,000,000 number when you do your calculations, or apply the compounding rate to your $1 – making each of your tickets actually cost $1.87 today.
Well, It’s Too Late Now. I won. Now What?
Take the cash. I know that we used a 10% discount rate on our stock fundamental value calculator, but there’s a way easier trade to beat 2.17% – buying 30 year treasuries, which closed today at 3.88%.
Now: stop thinking you have a positive expected value.
(End of article)
Still Reading? The Advanced Math.
If you are already convinced, here’s the minutiae on why the ‘take the cash’ option is better, but not by as much as estimated. It’s also math on why your EV is even lower when you do the math with the $636M – your discount rate will end higher; you’ll see.
In the section above I did the math by ‘buying’ $636,000,000 to be paid out with $341,000,000 today. That’s not correct, as our friend Jason Hull pointed out on Twitter (his site here). He later sent me the page explaining how the payouts work, found here.
So, we can model this and find the internal rate of return to figure out what our ‘return’ would be. To do this, we set up a spreadsheet like the following, with ‘$-341,000,000′ our initial ‘investment’ since we are assuming we’ll win (ha) and payments increasing 5% a year, including a payment which we get “today”:
So, in your favorite spreadsheet program, do the XIRR (or IRR, since it is uniform) on that and get: 3.80%. A closer match to the 30 year bond, but it still reigns supreme (more on that in a second). Also, you would do the original math using 3.80% versus $1 to come up with an actual discounted cost of $2.95 a ticket(!) today… so all the articles saying there is positive EV should use $2.95. Isn’t finance fun?
Also, the annuity and the Treasury bond don’t just “wait” 30 (or 29) years to pay you off (that would be a ‘no coupon’ bond, if you’re paying attention) – they give you cash flows every year. If you’re reading this site, I guarantee you’ll reinvest that money. That means you’ll see somewhere north of 3.88% if you buy the 30 Year and reinvest when it pays coupons. Please see our ten year Treasury coupon reinvestment calculator for details on why reinvested yields will be higher than effective yields.