Bankroll Management: How Bet Sizing Impacts Your Sports Betting

Proportional or Flat stake sizing

Bet sizing plays an extremely important role in any profitable betting strategy, and in this article I'm gonna try to give some insight into exactly how important it is. 

In the sports betting community, there are mainly two general staking strategies bettors use. A flat stake, and a proportional stake. With a flat bet size, you either put the same wager on every single game, or you put the same wager on games that have the same odds and edge. Flat bet sizing is fairly easy to use, but it's hard to select a proper size. A size too big will increase the chance of going broke, while a size too low will not yield big enough profits.

As an alternative to this article, you can watch this video. Or you can do both:

 

The Kelly Criterion: A Proportional Stake Sizing Strategy for bankroll management

A proportional strategy is where you place a certain percentage of your current bankroll on each bet. Kelly's Criterion is a formula that maximizes the growth rate of your bankroll. The formula for the Kelly's Criterion is

 Kelly Criterion Formula

Kelly Criterion Formula

where P is the probability of success and G is the odds you're given. Let's take an example.

If you were to bet on a coinflip - in a fair world you'd get an odds of 2. If you were to get odds of 2.1, you would've had an edge (you can read more about how edges occur here) of 2.12=0.05=5%2.12=0.05=5% and kelly would suggest

 Percentage of bankroll you should stake according to Kelly

Percentage of bankroll you should stake according to Kelly

This means you should bet 4.545% of your current bankroll. Let's take the scenario where this is ran 2 times with a starting bankroll of $100, you win one and lose one.

  1. You bet 0.04545∗100=4.5450.04545∗100=4.545 and lose, now you've got $100−$4.545=$95.455$100−$4.545=$95.455 left.
  2. You bet 0.04545∗95.455=4.3380.04545∗95.455=4.338 and win, now you've got $95.455+$4.338∗(2.1−1)=$100.227$95.455+$4.338∗(2.1−1)=$100.227 left. (4.338 * 1.1 is the net return of the bet)

The interesting thing, is that the order of the bets does not matter. You would've ended up with $100.227 if you'd won the first one and lost the second one.

example of Results Distribution with Different Kelly Percentages

Now over to a practical scenario. The example written over is not exactly representative for the common bettor, because it assumes that you know the outcome of the previous bet before placing the next one. In the sports betting world, one would usually have between 5-50 open bets at the time. This obviously affects the bet sizing. If you were to place 10 bets in a row with similar odds and edge, the bet sizing would decrease as your current bankroll decreases with each bet. If every single bet were with the odds of 2.1 and edge of 5%, you would expect to win 50% of the bets. This gives you the possibility of losing all your biggest bets and winning all your smallest bets - hence ending up in the negatives after placing nothing but profitable bets. It's worth noting that the other way around could occur, you could win all your biggest bets hence flipping a huge profit.

Here is a distribution of all possible combinations of bets with odds of 2.1 and edge of 5%, giving a win rate of 50%. There are 20 open bets, and it randomly picks 10 winners and 10 losers, so it's running perfectly every time - and only a matter of how the wins and loses are distributed. The x-axis shows the bankroll after the bets are settled relative to the starting bankroll, and the y-axis shows the number of occurences (out of the 184756 possible combinations). It follows the kelly bet sizing, so the bet size decreases for each bet as it's always placing 4.545% of its current bankroll. The blue shows the results for 100% of kelly, the orange for placing 50% of kelly, and the green for placing 30% of kelly.

 Distribution of simulation outcomes with different Kelly Criterion Percentages

Distribution of simulation outcomes with different Kelly Criterion Percentages

As we can see from the plot, the 100% of kelly (blue line) yields the highest possible return, and the highest average (~1.03). However the spread is huge. One could end up with everything from 88% to 117% of your starting bankroll. As you decrease the fraction of kelly to 50% and 30%, the spread gets alot smaller. The probability of ending up in profit skyrockets, where as the average return gets lowered ever so slightly. It's very important to note that by betting a lower fraction of kelly, you risk less money. By betting 100% of kelly, you risk ~60% of your bankroll to grow it to an average of 104%. By betting 50% of kelly, you risk ~37% of your bankroll to grow it to an average of 102%. By betting 30% of kelly, you risk only ~24% to grow your bankroll to an average of 101%.

You can read more about what you can expect from your results as your sample size increases in this article on the Law of Large Numbers and how it relates to sports betting. 

Stats for 3 Different Kelly Settings 

Table showing Expected Value, Variance, Standard Deviation, best and worst case scenarios for 3 different Kelly Criterion Percentages

Betting less than full kelly to reduce variance

There is no doubt that following full kelly is the strategy that would maximize the growth of your bankroll. Many people will tell you to bet less than the Kelly formula tells you to. Two reasons are generally given for this.

  1. The edge may change. If you place a bet 10 hours before kick off, the odds can swing both in and out of your favor. By placing a fraction of kelly, let's say 50% you have more rooms for error if the edge slightly decrease. If it increases, you can simply place more money on that bet.
  2. The Kelly formula leads to extreme volatility, the probability of being badly down for an unacceptable long strech may occur (You can read more about variance here, and how to reduce variance here).

Additionally, placing a lower fraction of kelly would generally decrease your bet size, making the probability of getting limited by the bookmakers smaller. You can read more about how bookmakers profile winning players here, and  how to avoid bookmaker limitations here

Do not bet more than Kelly suggests, so if you chose to go high risk (100% of kelly), do not over-bet. That will actually decrease the growth of your bankroll - as the full kelly value is a maximizer.

I really hope this gave some insight into how important bet sizing is. Additionaly, I want to point out that there are no perfect bet sizing that fits all. The fraction of kelly you're comfortable with placing depends on your personal risk profile. Do whatever you're comfortable with. In Trademate Sports - you can now select between 3 different betting sizes, 100%, 50% and 30%. Just go to your profile, and it'll be an option.

Source: http://blog.edgebet.net/betsize-and-simula...