lately I am busy with working through the Paper of Kelly concerning "A New Interpretation of Information Rate" for the purpose of moneymanagement in Trading. http://pnaelvlinux.net/brokerage/kelly.pdf
In his Paper he introduces the following formula (on page 919) for the exponential capital growth $G$ of a Trader's capital:
$$ G = q\cdot ld(1+l) + p \cdot ld(1-l) $$
$l$ is the fraction of your capital, that you bet. $q$ is the probabilit of winning and $p = 1-q$ the probability of losing. Thus if you win (and since the odds are 2 with probability one), you multiply your capital by $(1+l)$ and if you lose, you lose the betted amount of your capital.
To maximize G you should chose $l$ as:
$$ l = 2q-1 $$
Now I found the following formula on the Internet (http://www.dummies.com/how-to/content/kelly-criterion-method-of-money-management.html):
$$ \text{Kelly} \% = W – [(1 – W) / R] $$
Where W is the percentage of winning trades, and R is the ratio of the AVERAGE gain of the winning trades relative to the AVARAGE loss of the losing trades.
So there is just an expected value if the gain of a winning trade.
Does anyone know how to find a derivation for this formula? I searched the internet, but can't find a derivation of the kelly criterion for expected wins and losses.
Here is my try, which is mathematically incorrect, when it comes to solving the limes:
It is: \begin{equation} \sum_i p_i = p \ \wedge \ \sum_j q_j = q \ \wedge \ p+q = 1 \end{equation}
We look at the capital $V_N$ after N Trades. Random variables $A_i$ for losses und $B_i$ the gain in the winning case. Number of winning trades $W$ and number of losing trades $L$, where $N = L + W$.
\begin{align} V_N = (1-A_1\cdot a)\cdot ... \cdot (1-A_L\cdot a) \cdot (1+B_1\cdot a)\cdot ... \cdot (1+B_W \cdot a) \cdot V_0 \end{align}
For the expected exponential growth $G$ we get: \begin{align} G & = \lim\limits_{N \rightarrow \infty}{\frac{1}{N} ld\left(\frac{V_N}{V_0}\right)}\\ & = \lim\limits_{N \rightarrow \infty}{\frac{1}{N}(ld(1-A_1\cdot a)+...+ld(1-A_L\cdot a)+ld(1+B_1\cdot a)+...+ld(1+B_W\cdot a))}\\ & = \lim\limits_{N \rightarrow \infty}{\frac{1}{N}\left(\frac{L}{L} \sum_{i=1}^L ld(1-A_i\cdot a)+\frac{W}{W} \sum_{i=1}^W ld(1+B_i \cdot a)\right)} \\ &= p \cdot E(ld(1-A_i\cdot a)+q \cdot E(ld(1+B_i \cdot a))\\ & \stackrel{\text{Jensen's inequality}}{\ge} p \cdot ld(E(1-A_i \cdot a)) + q \cdot ld(E(1+B_i\cdot a)) \\ & = p \cdot ld(1-E(A_i) \cdot a + q \cdot ld(1+ E(B_i) \cdot a) \end{align}
Now we look for the maximum $G$:
\begin{align} & \frac{dG}{da} = - E(A_i)p \frac{1}{1-E(A_i\cdot a)} + E(B_i)q \frac{1}{1+E(B_i)\cdot a} \stackrel{!}{=} 0 \\ & \implies E(A_i) \cdot a = q - \frac{p}{E(B_i)/E(A_i)} \end{align}
There are two problems regarding this derivation:
First $\lim ab \neq \lim a \lim b$ in general.
Second if I maximize a function that is smaller than the target function, then I don't find the maximum of the target function.
