How to retire a millionaire

Say that you were starting out your working career all over again and the challenge is how to retire, 50 years later, a millionaire? Let’s neglect for a moment that in 50 years from now (2061), inflation will have rendered "millionairehood" quotidian. Instead, imagine this is 1961 and you are surveying the last half of the 20^th century for your best chance to retire in 2011 as a millionaire. Which would you choose?

1. Play the lottery, $2 per day, 5 days a week ($10 per week)?
2. Invest that same $10 weekly in a Standard and Poors 500 stock index fund?
3. Marry a cop or schoolteacher?

Although there is an argument to be made for door number 3 (Forbes, "The Millionaire Cop Next Door"), let’s focus on choices 1 and 2. In order to analyze these choices, we will need a little math. (Yes, your high school algebra teacher was right – you will find this stuff useful).

Playing the Lottery

Playing the lottery is a popular choice because it is relatively painless. Spending a couple of bucks a day has little opportunity cost, i.e., you are not depriving yourself of coffee or hot dogs or paying the rent or other such daily necessities. But what are the odds of winning a million bucks? Let’s take the Massachusetts Cash Winfall lottery as an example. This game of chance costs $2 and asks you to pick 6 numbers of 46. In order to win the jackpot, you must select the proper winning numbers, each with the following odds:

1^st digit 1 out of 46 (2.1%)

2^nd digit 1 out of 45 (2.2%)

3^rd digit 1 out of 44 (2.2%)

4^th digit 1 out of 43 (2.3%)

5^th digit 1 out of 42 (2.4%)

6^th digit 1 out of 41 (2.4%)

Those odds don’t look too tough. With over 2% chance of selecting the correct number for each choice, you should be a big winner if you just play the game 50 times, right? Unfortunately, that’s not how probability works.

To calculate these odds, you must multiply the probabilities (1/46 x 1/45 x 1/44 x 1/43 x 1/42 x 1/41), which yields a vanishingly thin 1 out of 6,744,109,680. If you were required to select the numbers in the exact sequence in which they were drawn, this would be your odds of winning. Fortunately, the lottery does not require that you match the order of the numbers drawn, so the above odds can be reduced by taking into the account the number of different ways (orders) in which the winning number could be drawn.

The first winning number could be drawn as any of the 6 digits, the second as any of the remaining 5, the third as any of 4, etc. So the total number of ways that the winning numbers could be drawn is 6 x 5 x 4 x 3 x 2 x 1 = 720. We can now divide 6,744,109,680 by 720 to determine your actual odds of winning – 1 out of 9,366,819. To put this into perspective you would need to buy one ticket per week for 180,000 years before approaching certainty of winning. Did you really think it would be easy? How many million-dollar lottery winners do you know?

How about the popular Powerball game? It requires that you select only 5 numbers (and then that pesky 6^th Power Ball). This game is rigged to make your odds much worse. The first five numbers are drawn from a pool of 59:

1^st digit 1 out of 59 (1.7%)

2^nd digit 1 out of 58 (1.7%)

3^rd digit 1 out of 57 (1.8%)

4^th digit 1 out of 56 (1.8%)

5^th digit 1 out of 55 (1.8%)

Multiplying these probabilities and then dividing by the number of ways the numbers could be drawn yields a friendly 1 out of 5,006,386. But that must then be multiplied by the probability of selecting the final Power Ball number, 1 out of 39, yielding the overall odds of winning as 1 out of 195,249,054. Feeling lucky?

Bottom line, you could spend an entire lifetime squandering $10 on lottery tickets each week and your odds of hitting the big one are still much less than being hit (multiple times) by lightning.

Playing the stock market

Everyone knows that the stock market is risky. But it is orders of magnitude less risky than playing the lottery. Since 1950, the Standard and Poors 500 stock index has returned an average of 10.8% on your investment. (Yes, that includes the recent financial meltdown).

What would happen if, as an eager young worker, you resolved to put $10 each week into an S&P 500 index fund? Further assume that your account grew tax-free (e.g., an IRA or 401K), and that all earnings and dividends were reinvested. At the end of a 50 year working career, you would have accumulated a million bucks if your average stock market returns were in the neighborhood of 10.5% (see nearby chart).

Now the stock market has no guarantees and it is certainly possible to lose capital, especially in the short run. I would no more recommend that you put all of your money in the stock market as I would urge you to bet it all on the lottery. But $10 per week is a cheap gamble on the market with far greater odds of gaining you a million bucks than hitting the lottery. Here’s my advice – play the lottery for fun, play the market for retirement, and keep some money safe – at least a 6-month emergency fund. That gives you the best odds overall of retiring a millionaire.

Of course, you could always marry a school teacher.