The beautifully simple formula for the Kelly criterion calculates the optimal proportion of your bankroll to bet in order to maximize the geometric growth rate of your wealth. But not only does it promise you maximum profit from effectively leveraging your opportunities; it also promises you safety from gambler’s ruin.

*Primary Origin: [John Larry Kelly Jr.](https://en.wikipedia.org/wiki/John_Larry_Kelly_Jr.)*

f is the proportion of your bankroll that you should bet which is the function of the probability of winning, the probability of losing, and the odds you have—i.e. the payoff ratio.

The Kelly criterion has three prerequisites:

1. You must know the exact odds and probabilities to input.
2. If only one of them is in your favor, it must more than offset the other, i.e. there must be a positive expected return.
3. You must scale the Kelly output so that the amount you bet is equal to the potential loss.

The last point is vital and it’s where I see a lot of people go wrong when using the formula. I’ve found many websites that don’t scale the output correctly when dealing with a situation where you can lose “some” but not all. It’s actually amazing how far up the academic ladder this goes. Seeing how so many practicians of the Kelly criterion get this wrong brings home a quote of Ed Thorp’s from his early days in the stock market that he was both surprised and encouraged at how little was known by so many.

The Kelly criterion must be used in such a way that what is bet must equal the potential loss. It’s inherent in the word “bet”: What we bet is what we put on the line. In our leveraged investment example, the base loss was 10%, so if we were to put 100% of our capital into the bet, we could only lose 10%. Thus, the right thing to do in this case is to scale the output by 10 which leads to a leveraged bet.

Let’s take our examples so far and put them into the Kelly formula.

In the rigged coin-tossing game we had a 51% win probability with equal payoffs. Inserting these inputs in the Kelly criterion formula shows that the optimal betting proportion of our bankroll is 2%.

In our investment example, we had a 50% win probability with unequal payoffs of 2-for-1 (20% win vs. -10% loss). The Kelly criterion, therefore, suggests betting with a maximum loss of 25% of the bankroll which, as we found out, is equal to a 2.5x leverage from the base loss of 10%.

Let’s take a more normal investment example that wouldn’t require leverage in the terms of borrowed money. Say you found a company that you believe is worth at least $100 dollars per share and it’s currently trading at $80 per share. You are only 50% certain that the company is worth your intrinsic value estimate.

The odds in this case is 1.25 (100 divided by 80) and the Kelly criterion thus suggests to bet 10% of the bankroll on the investment.