## Expected value vs. Expected growth

“Mathematically , the formula for the Kelly stake is derived using calculus2. The actual mechanics are rather unimportant, but the result is that in order to maximize the growth of one’s bankroll when placing only one bet at a time, one should bet a percentage of bankroll equal to edge divided by decimal odds minus 1. (This is assuming the player has a positive edge. If he doesn’t his optimal bet is zero.) In other words:

Kelly Stake as percentage of bankroll = Edge / (Odds – 1) for Edge ≥ 0

Put in terms of win probability the equation becomes:3

Kelly Stake as percentage of bankroll = (Prob * Odds – 1) / (Odds – 1) for Probability * Odds ≥ 1

Let’s take a look at a few examples:

Given a bankroll of $10,000 and an edge of 5%, then on a bet at odds of +100 one should wager 5% / (2-1) = 5% of bankroll, or $500.

Given a bankroll of $10,000 and a win probability of 55%, then on a bet at odds of -110, one should wager $10,000 * (55% * 1.909091 – 1) / (1.909091-1) = 5.5% of bankroll, or $550.

Given a bankroll of $10,000 and a win probability of 25% then on a bet at odds of +350, one should wager $10,000 * (25% * 4.5 – 1) / (4.5-1) ≈ 3.57% of bankroll, or about $357.

Given a bankroll of $10,000 and a win probability of 70% then on a bet at odds of -250, one should not wager anything because edge = win prob*odds = 70%*1.4 = 98% < 1.

Let’s look at all this a little more closely. Consider a bet at even odds (decimal: 2.0000) — in this case, the bankroll growth maximizing Kelly equation simplifies to:

K(even odds) = Edge/(2-1) = Edge for Edge ≥ 0

In other words, when betting at even odds, the expected bankroll growth maximizing bet is equal to the percent edge on that bet. So if you have an edge of 5% on a bet at +100, then you should be wagering 5% of your bankroll. If your edge were only 2.5% then you should be wagering 2.5% of your bankroll. Now let’s consider a bet at -200, or decimal odds of 1.5:

K(-200 odds) = Edge/(1.5-1) = 2*Edge for Edge ≥ 0

*So this means that for a bet at -200, the expected bankroll growth maximizing bet size would be twice the edge on the bet. Similarly, for a bet at -300, one should bet three times the edge, and for a bet at -1,000 one should bet ten times the edge.*

This fits rather well with the manner in which many players size their relative bets on favorites. For a bet at a given edge if they were to bet $100 at +100, they’d bet $150 at -150, $200 at -200, $250 at -250, etc.

*Now let’s consider bets on underdogs (that is, bets on money line underdogs — bets paying greater than even odds). In the case of a bet at +200:
K(+200 odds) = Edge/(3-1) = ½*Edge for Edge ≥ 0
The optimal bet size is only half the edge. Similarly at a line of +300, the optimal bet size would be a third of edge, at +400 a quarter the edge, etc.*

Now this is quite different from the manner in which many players choose to structure their underdog bets. If they were to bet $100 on a line of +100, they might also bet $100 on a bet with the same edge at +400. For a player wanting to maximize his bankroll growth, this is inappropriate behavior because it attributes, relatively , excessively large amounts to underdog bets. Assuming constant EV an expected growth maximizing player should only bet half of his +100 bet size at +200, and only a quarter his +100 bet size at +400.” -Granchow

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