I have been studying about Kelly fractions.
Background
I have wealth $W$. There is a game which I win with probability $p$ and lose with $1-p$. If I win the game then I get back whatever I invested plus $o$ times whatever I invested. If I lose I lose what I invested and get nothing. What is the fraction $x$ of wealth $W$ should I invest each time over say $N$ times so that my expected wealth is maximised in the end. This is what is given by Kelly fraction as
$$ x = \frac{p(o+1)-1}{o} $$
My question
I need the expression for expected wealth which I can then differentiate w.r.t to $x$ and arrive at the above formula. The following formula for expected winnings seems intuitive to me:
$$E(Wealth) = (1-x)^{(1-p)N} (1+ox)^{Np}W...(1)$$
I differentiate the above and arrive at the formula . But the following also seems correct to me
$$E(Wealth) = (p(1+ox) +(1-p)(1-x))^NW...(2)$$
But $(2)$ gives me something completely different. But I cant find anything wrong with $(2)$ ?
The Kelly formula will tell you the correct amount to bet in order to maximise the final MEDIAN expected value. If you want to maximise the final expected simple average value then you should bet all of your wealth each time if the odds are in your favor.