# the incremental value of Kelly Criterion under difference circumstances

I know that the Kelly Criterion maximizes bankroll, but i was wondering how much value it contributes to the total return and under what circumstances. I'm trying to understand the difference between using Kelly Criterion for sizing vs. arbitrary allocations, say... every bet that I make, I allocate 10% of my bankroll, and my alternative is using Kelly Criterion. In the long run, how much more bankroll would I have if I used Kelly Criterion? I'm sure people have looked into these simulations, any pointers would be great. I'm also looking for an intuitive or conceptual answer if there are general rules of thumb.

What are you saying is not completely correct. What kelly criterion maximizes is the average growth of the capital invested. In fact, if I want to invest a fraction $f$ of my 1000 units the amount that I will have after $M$ trades will be

$1000\Pi_{i=1}^{M} (1+f\phi_i)$

What we need to maximize is expected long-term growth rate. Growth rate is given by

$\frac{1}{M} \log (1000\Pi_{i=1}^{M} (1+f\phi_i)) = \frac{1}{M} \sum_{i=1}^{M} \log (1+f\phi_i)+ \frac{1}{M}\log(1000)$

Assuming that the outcome of each trade is independent, then the expected value of this is

$\mathbb{E}[\log(1+f\phi_1)]$

$\mathbb{E}[f\phi_i - \frac{1}{2}f^2\phi^2_i+...]$

From this we find that expected long term growth rate is approximately

$f\mu-\frac{1}{2} f^2 \sigma^2$

This is maximized by choice

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

Giving an expected growth rate of $\frac{\mu^2}{2\sigma^2}$ per trade. If $\mu>0$ then $f>0$ and we can make a profit, in the long term.

Now, if given your strategy, your 10% is less than the optimal fraction $f$, your bankroll is not growing as it could, since you are playing too conservatively. Vice versa, if your 10% is higher than the optimal fraction $f$ you are playing in a risky way and increasing your odds of ruin.