Written by Dan Abrams from Playing Not To Lose
Most knowledgeable bettors are familiar with the concept of expected value (EV) and how to use it. If you aren’t, stop. Read this Trademate Sports explainer first. Far fewer bettors are familiar with the concept of expected growth (EG). Well, at least they don’t THINK they are. They’ve heard of the Kelly Criterion (if you haven’t, Trademate explains that here too!), and they know it’s supposed to maximise something. That “something” is expected growth.
In short, EG is the percentage by which your entire bankroll grows (in the long run) when doing whatever it is you’re doing. That “something” can be making an independent bet (one that has no connection or correlation to any other bet you have open), but it can also be making several bets that are correlated somehow, or betting in a correlated way with some of your existing bets. The Kelly Criterion formula of “edge over odds” only applies to that first case, where the bet you’re making has no relation to any other current or future bet. How can you figure out your expected growth for those other cases? Well, first let’s figure out why we should want expected growth above all else.
We all know that the goal of betting on sports is to make money. But, how much money do you want to make? This is not a trick question. If you said “a million dollars,” then more power to you! If all you have is $1,000 to start, then it’s going to be a tough road. Say you can find someone who’s willing to flip a coin with you 10 times for a 1:1 payout (2.0 in odds), and at any stake you like. You could flip 10 times with them, going double or nothing each time you win, and have over $1 million if you win all 10. Of course, if you lose even once you’re broke. No million for you, not even your original $1k.
Now, say that this mystery person is feeling generous and offers you odds of 2.16 instead. Wow, that would give you an 8% edge! Do you see why? Because you still have a 50% chance to win the flip, and the formula for calculating your mathematical edge is:
b= net fractional odds of the bet (for American odds +200 → +200/100 = 2, -200 → -100/-200 = 0.5)
p= probability that the bet wins
q= probability that the bet loses, or 1 -p
Now you only need to win 9 flips in a row to have over $1 million! Hopefully, you’re snickering by now because you realise that trying to make a million this way is foolish. Thing is, once your benefactor ups the odds to 2.16 (or anything more than 2.0), the fact that you have a positive edge means that the amount for you to bet that maximises your EV is always 100% of your bankroll. So, if you subscribe to the theory that says your goal should be to maximise your EV, then you’d happily try to hit 9 flips in a row. And, most likely, you’d go home very sad.
The problem isn’t that +EV is bad, the problem is that variance is. And simply calculating your edge only predicts your mathematical EV, but doesn’t account for variance. Expected growth, however, accounts for both. It seeks to balance the goodness of +EV with the badness of variance, so that your median expected bankroll at the end is the highest. Even though staking at full Kelly on independent bets can be a wild ride, it theoretically maximises your EG if you accurately know your edge. For our independent coin flips example, your full Kelly fraction is:
So if you bet $69 on the first flip, and adjust your stake according to your new bankroll after each one, after 2,500 flips you would have a median bankroll of just under $1 million. So if you want to win a million dollars, you should want to maximise your EG.
The answer is: very, very carefully. Because your EG represents the amount that your entire bankroll will grow (in percent), rather than the amount you theoretically win on the amount your bet (like ROI does), your EG on each independent bet will be incredibly small. In fact, I prefer to talk about it in terms of “basis points,” which is a financial term for 1/100 of 1%. A couple of basis points here and there will add up. If you can get 40 or 50 basis points, you’ve hit the jackpot!
For our benevolent coin flipper example above, your EG for each flip is about 27 basis points. So, after your first flip, your median bankroll will be about $1,002.75. Great! Seriously, this is a great result because 8% edges are hard to find (though Trademate does a great job of finding them for you). This is the first step on your journey to winning your theoretical million. The last step will be when your bankroll is at $997,250 and you expect to win a median of $2,750 on your flip. Of course, your million dollar roll will also have some variance. It could be more, it could be less, just like when you’re measuring your actual results vs. ROI, but there is no quicker way (in terms of the number of bets you make) to grow your median bankroll to $1 million than by optimizing your EG. That’s what John Kelly proved all those years ago.
As for most complex things, the answer is a little complicated. But, to simplify it, here are the top three steps to take to help you grow your bankroll the quickest:
Figuring out just how to maximise your EG in spots like this has become a passion of mine. Hopefully, you want to win long term just as much as I do, and you’ll stick around on this blog to find out more about how to undertake these methods.
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