# Kelly Betting and the Law of Large Numbers.

Kelly optimal solves for the fraction of capital one needs to bet in an unfair game to maximize its asymptotic (in a long run) rate of capital growth. It was first derived in the 1956 paper [Kelly J.L., “A new interpretation of the information rate”, Bell System Technical Journal, 1956] for the case of a biased coin. Since then, many academics generalized this result in many different settings: horse racing, black jack, stock market, etc. In a series of blog posts I will try to generalize some of these results to solve the following problem: given an open portfolio of trades, if I want to open a new position what is the fraction of my capital I should bet?

I will start with the simplest case of the biased coin. You start with a capital of \$100 and you are offered to play in a following game: there is a flawed coin which comes up “tails” with a known probability $p>0.5$. You make a bet and then flip a coin. If it comes up “tails” you win and in this case you get one dollar per each one dollar bet. If you lose banks takes your bet. You can make bets any number of times, how much should you bet to maximize your capital?

First, we need to understand what does this mean to maximize the capital. Reasonable criterion would be to maximize your capital after one or several bets relative to an extra risk constraint. This is a subject of mean-variance portfolio optimization. Here, we will follow another route and maximize the rate of growth of my capital.

Denote by $\xi$ the random variable which is “tails” with the probability $p$ and “heads” with the probability $(1-p)$. Assume, each time we bet we bet a fixed fraction of our current capital. This is a strong assumption but in what follows we will see that we can in fact prove that it has to be constant. By the definition, the growth after $T$ bets is defined as $G(T) = (\prod_{t=1}^T (1 + f \xi_t))^{\frac{1}{T}}$. Our goal is to maximize it with respect to $f$ when $T$ goes to infinity.

To maximize $\lim_{T \to \infty} G(T)$ is the same as to maximize the logarithm $\lim_{T \to \infty} \log(G(T)) = \lim_{T \to \infty} T^{-1} \sum_{t=1}^T \log (1+f \xi_t).$ It is a direct application of the Law of Large Numbers that this limit is equal to the expected value $\mathbb{E} \log (1+f\xi)$.

Finally, in order to find the value of $f$ we can write $\mathbb{E} \log (1+f\xi) = p \log(1+f) + (1-p)\log(1-f)$ since $\xi$ take only on two values. By differentiating with respect to $f$ and setting the result to zero we obtain $\frac{d}{df} \mathbb{E} \log(1+f\xi) = \frac{p}{1+f} - \frac{1-p}{1-f} = 0$. This equation has a unique solution $f = 2p-1$.

In the next post I will explain why we can always assume the $f$ is constant and how this generalizes to a more general situation.

## 2 thoughts on “Kelly Betting and the Law of Large Numbers.”

1. Rob says:

Nice work, very succinct. After you introduce xi in the fourth paragraph perhaps you could add the values that xi takes when it is tails and also when it is heads. The values can be represented as a variable of course. That will flow nicely into the growth formula. There seems to be an assumption the gain/loss on heads/tails are equal (but opposite) in your unique solution. If so I think it’s worth stating that as well. Perhaps you could show the solution when the gain/loss is asymmetrical. Well done!

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2. vladimirkrouglov says:

Hi Rob, thanks, all great comments.

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