Bankroll Management – Part II
Continuing our journey through bankroll management, now is a good time to discuss the Kelly Criterion in further detail.
The Kelly Criterion
The Kelly Criterion is a betting strategy that will maximum median wealth expectation over the long term. Betting more than the Kelly Criterion suggests will lead you to suboptimal results. The Kelly bet size is found by maximizing the expected value of the logarithm of wealth and was formulated by scientists John L. Kelly and Claude Shannon in 1956.
We have a workbook to calculate the Kelly Criterion yourself. For American odds, the following formulas are used:
For example, if you are getting +200 on a bet where your winning percentage is 40%, your Kelly Bet Allocation would be calculated as:
And if you’re getting -250 on a bet that you win 75% of the time, your Kelly Bet Allocation would be calculated as:
What Happens If We Bet More Than the Kelly Criterion?
In the example introduced in Part I (1,000 bets, -110 odds with a true 55.0% win probability), the Kelly Criterion states that a bet of 5.5% of your bankroll would maximize wealth expectation in the long run.
What happens if you want to bet more than the Kelly Criterion? Well, if you bet 1.5x Kelly Criterion (or 8.25%), 66% of the time you will wind up worse off than if you bet 1.0x Kelly (Full Kelly). Starting with a bankroll of $1,000, the distribution of potential outcomes based on bet size is displayed below.
After 10,000 bets, you will be worse off 90.6% of the time.
As the number of wagers grows, the less likely it is to have an outcome where betting more than 1.0x Kelly would yield better returns.
Phrased differently, over a 10,000-game sample size, you would only be better off betting 1.5x Kelly if you exceed your true win percentage and win at least 55.7% of your bets. The likelihood of this happening (if your true win % is 55.0%) is approximately 9% and shrinks as you increase your sample size.
In the long run, your winning percentage converges to the true win probability, which is the median of the distribution of potential outcomes. As a result, the Kelly Criterion is determined as the percent of bankroll wagered to maximize median wealth in the long run. Unless you’re not planning to bet for very long, there’s no reason to wager more than the Kelly Criterion.
The Kelly Criterion relies on one very important assumption: that you know your true win probability with absolute certainty.
In a casino game like blackjack or craps, the true probability of an outcome is known (assuming fair dice/deck). We know that the probability of rolling a 12 (2 sixes) in craps is 1/36.
The challenge when betting sports: When someone says that the Dodgers have a 67.0% probability of beating the Diamondbacks, that person is estimating this probability using historical data and mathematical models. There is no way of knowing with absolute certainty what the win probability is for the Dodgers. This presents a challenge because it leaves you vulnerable to overbetting.
Uncertainty of True Winning Percentage
Due to the uncertainty around true win probability, many people argue that a Full Kelly betting strategy is too aggressive. We tend to agree.
Let’s assume that you have an NBA model that you believe predicts every regular season game (1,230 games) against the spread at a 55.0% winning percentage (highly unlikely that you can find this much value in every game, but let’s use this as a simple example). You start with $1,000 and bet all 1,230 games at Full Kelly allocation of 5.5% of your bankroll. Let’s say, however, that you were slightly overconfident in your model, and the actual true win probability is 54.0% (still very good). Despite having a winning model, your Full Kelly bet allocation should be only 3.4%, and you have therefore been betting the equivalent of 1.62x Kelly.
How does overbetting affect your expected results? We’ve summarized the results below.
You have a situation here where, despite having a winning model, you have a higher chance of losing money because of an imprecise estimate of your winning percentage. Therefore, we generally suggest that you either 1) estimate your winning percentage more conservatively, or 2) practice using a “Fractional Kelly” approach towards bankroll management.
Fractional Kelly Betting
Fractional Kelly betting is just as it sounds – betting a fraction of what the Kelly Criterion suggests as optimal. Below we’ve plotted various bet allocations for your NBA model with a 54% true win percentage.
Yes, there is always a chance of losing money. However, we want to only accept that risk if we are being fairly compensated with the expectation of making money. If you’re not receiving incremental returns, you should not take on more risk.
For instance, let’s compare the bankroll management strategies of 0.5x Kelly betting vs 1.5x Kelly betting. Both strategies have a median expected final bankroll of ~$1,600. By betting 0.5x Kelly, however your chances of losing 20% or more of your bankroll are only 10%, while betting 1.5x Kelly would result in you losing 20% or more of your bankroll around 33% of the time. Therefore, the 1.5x Kelly betting strategy is dominated and is never recommended.
Let’s borrow a term from modern portfolio theory and say only bet allocations between zero and Full Kelly are on the efficient frontier. Therefore, these are the only bankroll management strategies that should be considered. We’ve plotted the efficient frontier of bet allocations below.
Personal Risk Appetite
It’s worth noting that even with a precise estimate of true winning percentage, using a Full Kelly bankroll management system might be too aggressive for your personal preferences. That’s fine – your bankroll management system should be tailored to your desired to risk-return profile, and everything at or below a Full Kelly strategy is a valid bankroll management strategy.
Let’s say that you’re risk averse, and your objective is to minimize the chances of losing 20% of your bankroll. You decide that you don’t want greater than a 5% chance of losing 20%, so you can choose a 0.25x Kelly Fraction as your desired bankroll management strategy.
The above calculations assume that all bets are independent of one another and that no bets are occurring simultaneously. If you have simultaneously occurring wagers, such as betting on the Lakers and the Patriots at the same time, you should consider betting a little less than the Kelly Criterion suggests.
Example: Consider a situation in which you have 20 bets that you want to bet 5% of your bankroll on. If you bet them all simultaneously and lose all 20 bets (hey, it can happen!), you’ve lost your entire bankroll. If those wagers had happened sequentially, the Kelly Criterion would have told you to bet less money after each loss and you still would have 36% of your bankroll remaining.
The calculations are little trickier and beyond the scope of this article. For further reading, you can check out “Algorithms for optimal allocation of bets on many simultaneous events” by Chris Whitrow. (Reader beware, unless you have a good handle on multivariable calculus, the paper may seem like a foreign language.)
What If I Don’t Know My Winning Percentage?
This Kelly Criterion discussion builds the framework around the upper bound of your bet sizing with a known winning percentage. If you don’t have a reasonable estimate of your winning percentage on a particular bet, I urge you to be very conservative with your bet size. Until you are confident in your estimated win percentage of a bet, I would suggest limiting your bet size to 0.5% to 1.0% of your bankroll.