Expected Geometric Growth Rate (Kelly's Criterion)

The Wikipedia article for Kelly Criterion establishes its main formula using the expected geometric growth rate $r = (1 + fb)^p * (1 - fa)^q$, where $f$ is the fraction of an account (that starts with unit capital) allocated per trade, $b$ is the profit earned by a winning trade as a fraction of capital allocated to it, $a$ is the capital forfeited by a losing trade as a fraction of capital allocated to it, $p$ is the probability that a trade wins, and $q$ is $p^c$. Thus an expected geometric growth rate has the form...

$\text{account capital after first trade if it wins}^{\text{probability of it winning}} * \text{account capital after first trade if it loses}^{\text{probability of it losing}}$

I have noticed similarity between this form and the probability mass function of the Bernoulli Distribution. In both cases, the exponents are complements, but in the Bernoulli Distribution, the bases are also complements, which does not hold here. I have also noticed similarity between the form here and the geometric mean, but at best it seems the two could only be harmonized in the special case where $p = q = \frac{1}{2}$.

What is a good intuition for this notion of expected geometric growth rate to use as a stepping stone toward understanding Kelly Criterion? I am much more experienced in calculus than statistics, so if it results from a differential equation or recurrence relation that is easier to internalize than the solution itself, this may be helpful to know.

Solution 1:

I believe an easy way to understand the Kelly's criterion is the following:

Think about the following growth process: you have a discrete-time process where in each step you invest and amount/proportion $f$ of your current capital, and you start with an amount of cash equal to $W_0$. Each time-step, if you win, your wealth will increase by the product $(1-f)+f(1+b) = (1+fb)$ with $b>0$ being equal to the return rate each time you win, and conversely, if you loose at the actual stage your wealth will decrease by the product $(1-f)+f(1-a)=(1-fa)$ with $a>0$ being equal to the return rate reduction each time you loose (so, if my return is $-6\%$ the reduction in my wealth is going to be given by the product $(1-f\cdot 6\%)$). All of these, assuming that your probability of winning are loosing at each stage are independent, lets say, you could be winning at each stage with probability $p$ or loosing with probability $q = 1-p$.

With these, note that the order of winning and looses doesn´t matter, so my current wealth at any stage $t=T$ will be given by the number of winning $N$ and the number of looses $M$ (where $N+M = T$): $$ \begin{array}{r c l} W_t & = & W_0 \cdot (1+fb) \cdot (1+fb) \cdot (1-fa) \cdots (1-fa) \cdot (1+fb) \\ & = & W_0\cdot(1+fb)^N \cdot (1-fa)^M \end{array}$$

Now, let try to calculate the equivalent Compound Growth rate $\tau$ for the process at stage $T$ $$\begin{array}{rcl} (1+\tau)^T & = & (1+fb)^N \cdot (1-fa)^M \\ & = & \left( (1+fb)^\frac{N}{T} \cdot (1-fa)^\frac{M}{T}\right)^T \\ & = & \left( (1+fb)^\frac{N}{N+M} \cdot (1-fa)^\frac{M}{N+M}\right)^T \\ & \overset{\text{behave as when}\,T\to\infty}{\approx} & \left( (1+fb)^p \cdot (1-fa)^q\right)^T \\ \end{array}$$ so if you think as the time-expected-average path of the process could be defined as $\left< W_t \right>_t = W_0 \cdot (1+\tau)^t$ you can now figure out the meaning of the proportion $1+\tau = (1+fb)^p \cdot (1-fa)^q$ which is maximized through the Kelly's proportion $f^* = \frac{pb-aq}{ab}$, and understanding also the relation of the exponents that will tend to be the winning/loosing probabilities at each stage.

I believe that John Larry Kelly Jr. made one of the most important discoveries in Information Theory through this, founding that exist an optimal investing proportion which make a random process to growth exponentially, and since it doesn´t arise directly from Markowitz Theory it could be saying that important improvements could be done to optimal portfolios theories (maybe, maximizing the mode instead of the mean, but it just speculation right now). An interesting review is given by Edward O. Thorp here.

Additional review:

Think about the required return at each stage (betting my whole wallet each time): if I am betting in the casino, at each bet I could loose the whole amount I am investing so $1-a=100\%$, so, if each bet gives a prize of $\nu = 1+b$, at each bet I will win or loose with an outcome given by the expected value $E_t = (\nu p -1\cdot(1-p)) > 1$ if I want to be making money in the long term, so it requires that $\nu > 2/p-1$ which is always $\nu >1$ and highly restrictive, as example, for $p=0.5$ you need a prize of at least a $\nu >3$ times your bet.

But now, thinking in investing, if I loose (go bankrupt), I still can sell my assets to pay the doubt and keep some of my money (if I was and responsible manager), so the loosing proportion $\eta = (1-a)$ will be less than one (different from making a bet), so now the outcome given by the expected value $E_t = \nu p -\eta(1-p) > 1$ if I want to be making money in the long term, so will require that $\nu > 1/p+\eta(1-p)/p$, which could be even $\nu < 1$ and having an $E_t > 1$ given some values from $\eta$, so is sensefull to be doing investing, and is also different from making bets.

Right now coming back to Kelly's, if I invest a proportion $f$ on each trial/stage, the expected outcome by stage will be $E_t = f(\nu p -\eta(1-p)) > 1$ to be winning in the long term, so it will require that $f<1/(p(b-a)+2p+a-1)$ so doing bets as martingales are actually bad strategies even if I have residual value in my investing.

Solution 2:

Consider a one period setting. Next period wealth is $W_0R$, where $W_0$ is initial wealth and $R$ is simple return. Here $R=1+fb$ if you win and $R=1-fa$ if you lose. Kelly maximizes the expected logarithm of this:

$$\mathbb{E}\log(W_0R)$$

which is equivalent ($W_0$ is a constant) to maximizing expected logarithmic return

$$\mathbb{E}\log(R)=p\log(1+fb)+(1-p)\log(1-fa)$$

because $\exp$ is a monotonic function this is also equivalent to maximizing

$$\exp\mathbb{E}\log(R)=p\log(1+fb)+(1-p)\log(1-fa)=(1+fb)^p(1-fa)^{1-p}\equiv r$$

This $r\equiv \exp\mathbb{E}\log(R)$ is sometimes called expected geometric growth rate. The intuition is that over multiple periods the geometric average return is

$(1+fb)^{p^s}(1-fa)^{1-p^s}$

Here $p^s$ is the fraction of wins that in the long run tends to $p$.