The Kelly Criterion: A method to maximize expected growth of assets

In short, the Kelly Criterion is an approach for maximizing the expected growth of assets by investing a fixed percentage of wealth in a series of investments, developed by JL Kelly in 1956.

Example: Kelly criterion applied to betting

The technique is renowned in the field of gambling. To introduce the criterion, we use a simple hypothetical situation. Lets say you throw a dice, where you win if the number on the face is less than or equal to 4. This gives you a winning probability of 4/6. A friend is certain that the number on the face will be greater than 4, therefore he will allow you to bet at evens, for any number of times you like.

In theory, you expect to make a lot of money, as this friend seemingly has a hard time understanding probability theory. Lets say you have $100 as a starting budget.

The Kelly criterion determines the optimal fraction of wealth you should bet at once.

Clearly, $100 is not optimal, because if you would lose on the first try, you could lose the option to bet any further and benefit your increased probability. On the other hand, betting an amount that is too small will cause you to benefit only a small amount, and will cause good returns late in time.

\phi is the random variable taking either value 1 with probability p and value -1 with probability (1-p), where f is the fraction of wealth we bet against our friend. The growth in wealth after the dice throw is then:


From that we can derive that the expected growth is:

{\it p}ln(1+f)+(1-p)ln(1-f)

Therefore, the maximization of the expected growth rate is yielded by:

f = 2p  - 1

Applied to the example above, we would bet 2/6 as a fraction of our wealth. A lower amount would be too conservative, while higher fractions add increasing volatility to returns.

Applying the criterion to investments

Now, we’ve applied the criterion to gambling, but it applies to just any bet or investment as a management principle.

For any investment with an expected return of \mu and a coefficient of variation greater than 1, the expected return for an investment fraction f is given by:


Maximizing this expression as approximated by Taylor series, the Kelly fraction found is:

f = \frac{\mu}{\sigma^2}

However, commonly the exact \mu and \sigma are unknown. Generating a safer margin of error (betting on larger variance), the typical value for f is half Kelly.

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