# The Kelly Formula (or Kelly Criterion)

There are bold traders that are risking 10% of their  capital in a single trade (that is insane) and others that put a single digit percent of their account on a single trade and a fraction of that is their risk. With those both extreme situations in mind, the question that one may ask is which is the right amount to risk in each single bet to maximise the profit.

From a pure theoretical point of view, the amount can be calculated with the Kelly formula.

John Kelly was a scientist that worked at Bell Labs and that assisted AT&T with its long distance telephone signal noise issues. His method was publicated in 1956 on the “Bell System Technical Journal ” in a paper titled “A New Interpretation of Information Rate“. Those of you who are math geeks and willing to dive deep into the formulas can find it here.

After this publication, the Kelly criterion soon become know in the mainstream professional gambling community as a tool to determine the maximum amount to bet in games to maximise profit.

## The Kelly formula

The Kelly formula takes into account the distribution of events within a given system. It considers how many time the betting system wins, how much it wins, how many times it loses and how much it loses. Combining these variables, it determines which is the optimal risk to take in order to maximise the capital gain.

Calling W the probability of winning and R the average win/average loss ratio, the fraction of the bankroll to bet, let’s call it f* is calculated as follow:

f* = W – (1 – W) / R = [(R + 1 ) * W – 1] / R

The simpler, but also very interesting, experiment that one can do using the Kelly formula is the coin tossing. In this simple practice we know that the probability of winning (W) are 50%.

The average win/average loss ratio, on the other hand, depends on the rule of the games. Let’s consider the case where one either wins \$1 or loses \$1.

In such this example R would be equal to 1/1 or 1 and the formula become:

F* =[(1+1) * 0.5 – 1] / 1 =[2*0.5 – 1] / 1 = 0

What Kelly is practically is saying is…to stay out of the game (bet \$0).

Let’s now consider the same coin tossing system, but with different conditions: W remains equal to 0.5, but R is now equal to \$1.25: in case of a win we gain \$1.25, but in case of a loss we lose only \$1.

Let’s do the math:

F* =[(1.25+1) * 0.5 – 1] / 1.25 =[2.25*0.5 – 1] / 1 = 0.1 = 10%

With this scenario, the Kelly criterion tells us that to maximize our gain we should risk 10% of our capital on each bet.

The above results are valid only with systems that generate exactly the percentages that are used to compute f*, in our case 50% of probabilities that the outcome is a tail and 50% of probability that the outcome is a head.

In real the real word that exact percentage can be obtained only with an infinite number of repetition of the experiment.

To get close “enough” to a 50-50 split needs at least more 100 repetitions and possibly more than 1000. But even with such a high number, there will be still some cases in which one can get slightly better or slightly worst percentage with brutal deviations of gains from the theoretical case.

## Conclusions

Kelly’s formula cannot be used “out of the box”. The gap between theoretical and practical application is just too huge.

Expectations from probabilistic calculations are never met in real life. Sometimes they are better, sometimes are worst and for this reason, rules and formulas shall never be blindly followed without use good sense and without a deep understanding of the assumptions on which they built.

Given a big enough sample of real trades (or backtested) data, a trader should calculate the Kelly formula as reference point after which he should not dare to go. If one’s trading strategy brings to a computation of an f* equal to 3%, a trader should always stay south of that percentage. Betting behind this risk *for sure* is not going to improve returns.

Various risk management books and authors belives that “half Kelly”, that is f*/2 would allow a system to achieve 2/4 of its maximum gain with much less volatility.