Betting Size and Kelly's Formula

After several weeks of highly volatile results, I decided to go out and search for a mathematical formula to compute betting size for each game. This is a topic that I would like to dedicate more time to and even come up with a more sophisticated method to decrease my chances of ruin and increase my profitability in the long run. For what is left of this year at least, I will use Kelly's formula.

Kelly's formula determines the percentage of your bankroll to use on a bet. In our case of the weekly NFL spread, it will determine the amount of bankroll to use each week. The formula is simple and takes into consideration the odds received on the wager and the probability of winning.

Fraction of Bankroll = [(odds)*(prob) - (1 - prob)]/odds,

where prob=probability of covering the spread and odds=odds given for the wager

I will make the following assumptions and then compute the fraction of the bankroll to use each week:

  • Odds = 10 to 11. The typical -110, for every $11 you bet you receive 10. Most bookies change these odds depending on the game and which team you choose. To simplify the computation, we assume the odds are always -110.
  • Prob = 60%. The probability that my system makes correct picks is assumed to be 60%. This is where we stand today and this is where we finished last year. We assume, regardless of the confidence measure, that every game has equal chance of success.
  • Picks are independent of each other. Kelly's formula assumes independent events. The fact is that NFL picks are not like rolling a ball on a roulette, but it is not that far off to assume they are independent. Here we are assuming that last week's New England pick is not correlated with this week's New England pick because the Vegas spread adjusts to avoid trends.

Therefore, the formula simplifies to: [(.909)(.6) - (.4)]/.909 = 16%

Now, this does not mean that for every game you will spend 16%. Kelly's formula applies to sequential gambles. Because on Sundays, bets are mainly made simultaneously, I would use 15% (a bit less to go on the safe side) of your bankroll every week (including Thursday, Sunday, and Monday night). Although Thursday, Sunday, and Monday night games you are able to make bets sequentially, I do not want to give more bankroll to these games just because they are the night games.

For example, if you have a bankroll of say $1,000 then you can risk around $150 per week. If there are 3 games you choose to bet one, then (for now) bet $50 on each game. Next week, if your bankroll increases or decreases, you adjust the total per week accordingly. At the end of the year you will not be ruined and hopefully you have a decent profit.

To put this in perspective, suppose you start investing in week 4 with a bankroll of $1,000. At $150 a week and 60% success rate you would average about $35 profit a week. Even if you do not increase your weekly bet as your bankroll increases, which you should, you would end up (13 weeks plus 2 playoff weeks) with a total profit of $35*15 = $525 or about a 53% return on investment. Not bad considering there seems to be no bottom in the stock market. Kelly's formula is only valid as you make more and more gambles, that is, it is not a short-term solution it is an optimal solution for the long run (think 10-20 years).

I know it is tempting to bet it all or larger amounts, especially when you want to make up ground. Betting it all or deciding on a size without a strategy will make you act like a gambler and not like an investor in the NFL point spread market. Stay calm, optimize betting size, play it safe, and use 15% of your bankroll every week. Even if you think Buffalo is a given cover this week, remember that in any given Sunday anything can happen.

In the future, I would like to relax the second assumption, the one that states that every game has equal chance of winning. I have been publishing a confidence measure that although is detecting opportunities well, withing those opportunistic games, it is not clear-cut that one game is better than the other. If this measure becomes more accurate, then of the $150 for the week, the distribution of these $150 can be done more optimally according to the refined confident measure.