In my last article, we explored the concept of EV and how to apply it to the evaluation of some common NFL prop bets. In this, the second in the series of 3 articles, we’ll discuss what to do when you identify that something is a good bet. Proper bet sizing can be the difference between the penthouse and the poor house!
+EV Revisited
Let’s return to an example from my last article.
If you had the opportunity to bet $1 on a coin flip at +108 odds, should you take it?
(Correct answer: Yes!)
Now, what if instead of $1 you could bet any amount you wanted. How much should you bet?
(Correct answer: Somewhere between $1 and your entire life savings, depending on your personal preference for risk. Remember, you have a 50% chance of losing this bet.)
Now, suppose that instead of a one-time offer, you were able to make this bet over and over again, choosing your bet size each time, once every minute for the rest of eternity. How much money could you possibly make, and how likely are you to make it?
Correct answer: If you choose your bet sizes carefully…as much money as there is in the world, and 100% likely. This is the power of +EV! Now, a bet opportunity that repeats forever is obviously a fictional construct. However, it can be used as a model to help us optimize our long-term bankroll growth.
Good decisions can have bad outcomes
When you identify a +EV bet, your first inclination might be to “bet the house”. However, gambling is still gambling – even when the odds are in your favor. Overbetting and losing can be financially devastating and – more importantly for our purposes – can impact your ability to take advantage of future +EV opportunities.
The most important piece of money management advice for any serious bettor is to have a bankroll – a pot of money used exclusively for betting, that grows with every win and shrinks with every loss. Treat your bankroll like an investment portfolio – your goal is to grow it as large and as quickly as possible, while minimizing the risk of severe (or total) losses.
Starting Simple
To evaluate different bet sizing strategies, we’ll use the +108 coin flip example, repeat it every day for a simulated year, and then simulate the year 100,000 times. We’ll start each simulated year with a $1,000 bankroll.
Let’s start with the simplest possible strategy – the flat bet. Bet $100 each day until the year ends or you run out of money, whichever comes first. What do the results of the 100,000 simulated years look like?
Ending Bankroll
Flat $100 Bets
Best Case = $11,300
Median = $1,940
Worst Case = $0
Probability of Profit = 59%
Probability of Bust = 39%
The good news is that your median result is a $1,940 ending bankroll (for a $940 year-end profit) and you can end up with up to $11,300. The bad news is that your year ends with a profit only 59% of the time, and 39% of the time your bankroll doesn’t even survive to the end of the year. We can do better.
Live to Bet Another Day
The worst thing one can do as a +EV bettor is to put oneself at risk of going bust. Why? Because not only do you lose your entire bankroll, you shut yourself out from every +EV opportunity that comes after your bankroll hits zero.
There is a very simple way to ensure that you never go bust – instead of betting a fixed dollar amount, bet a fixed percentage of your bankroll. That way, your bet size moves up or down as your bankroll gets larger or smaller. Growing bankroll? Bet more! Shrinking bankroll? Bet less!
But what percentage of bankroll to bet? I use something called the Kelly Criterion. I’ll spare you the detailed history and mathematics behind the Kelly Criterion (but the internet is full of great articles if you’re so inclined), but the formula is as follows:
Bet amount = Bankroll x Edge ÷ (Decimal Odds – 1)
In our coin flip example,
Bet amount = Bankroll x 0.04 ÷ (2.08 – 1) = 3.7% of bankroll
So you’d start by betting 3.7% of 1,000 = $37 on day 1. If you win on day 1, your new bankroll is $1,040 and your bet on day 2 would be 3.7% of $1,040 = $39. If you lose on day 1, your new bankroll is $963 and your bet on day 2 would be 3.7% of $963 = $36. Your result on day 2 would influence your bankroll for day 3, etc, etc.
So how does this strategy perform?
Ending Bankroll
Kelly Criterion
Best Case = $34,502
Median = $1,362
Worst Case = $58
Probability of Profit = 67%
Probability of Bust = 0%
We’ve lowered our median amount of profit, BUT we’ve tripled our best case, increased our likelihood of ending the year with a profit and most importantly we never go bust!
At this point, I should point out that there are two major downsides to the Kelly Criterion:
- It’s not for the faint of heart. You can see the volatility – you could end the year anywhere from $58 to $34,502. You’ll end the year with a loss one third of the time. While it’s true that you won’t lose your entire bankroll, you could lose up to 94% of it…and if you’re lucky enough to hit a winning streak and grow your bankroll quickly, you could find yourself making larger bets than you’re comfortable with.
- It’s very important that you calculate your edge correctly. It’s easy in our example, but sports betting is not like coin flips or roulette wheels where the probabilities are precisely known. If you overestimate your edge, you could be in for a world of trouble. We’ll explore this in more detail in my next article.
For both of these reasons, some bettors prefer to use something called “fractional Kelly” where they calculate the Kelly Criterion amount and then bet a fraction of it, such as a half or a third.
Let’s see how half Kelly performs:
Ending Bankroll
1/2 Kelly
Best Case = $6,288
Median = $1,248
Worst Case = $258
Probability of Profit = 70%
Probability of Bust = 0%
You can see the typical tradeoffs of a reduced-risk system – smaller upside for smaller downside.
At this point you might be thinking “ok, I buy the idea of betting a percentage of bankroll, but the Kelly Criterion value of 3.7% doesn’t get me excited. Let’s kick it up a notch!” Ok cowboy (or cowgirl), you asked for it…
Ending Bankroll
10% of Bankroll
Best Case = $4,124,748
Median = $665
Worst Case = $.13
Probability of Profit = 42%
Probability of Bust = 0%
This is +EV betting taken to the extreme, for the serious thrill seeker. You will have lost money at the end of the year, more often than not. Your bankroll still will not go to zero, but it could go as low as thirteen cents. However, the upside speaks for itself!
Conclusion
We now know how to use the concept of +EV to identify good bets, and we’ve explored the ins and outs of some of the most common bet sizing systems. In the next and final article of the series, we’ll dive deeper into how sportsbooks set odds, and we’ll learn how the same market dynamics that Billy Beane exploited in the book/movie “Moneyball” can help you find good bets.
Tweet me with any comments or questions at @PlusEVAnalytics. Until next time, may your EV be positive and may your variance be favorable!