Kelly Sizing and Expected Long-Run Logarithmic Growth
Summary
The answer clarifies that the Kelly criterion maximizes expected logarithmic growth of capital, rather than bankroll in an unrestricted sense or a guaranteed return. If a fixed fraction of capital is invested on each trade, wealth compounds as a product of trade returns. Taking the logarithm turns that product into a sum, making expected per-trade log growth the quantity to maximize under the stated assumption that outcomes are independent and identically distributed.
A Taylor approximation for small returns gives expected growth of roughly the position fraction times mean return minus half the squared fraction times return variance. In this approximation, the maximizing fraction is mean divided by variance, with maximum growth proportional to squared mean over variance. A fixed allocation such as ten percent can therefore underinvest relative to this fraction or take excessive risk if it is larger, potentially raising ruin risk. The approximation is not a universal sizing formula: it depends on the return distribution, independence assumption, and truncation of higher-order terms, and it does not quantify the advantage over a particular fixed allocation.
Key ideas
- Kelly sizing maximizes expected logarithmic growth under the assumed trade-return model.
- Compounded wealth becomes a sum of log returns when expressed in logarithmic form.
- For small returns, approximate growth is linear in mean return and penalized by position-size squared times variance.
- A fixed allocation may be too small or too large relative to the model’s optimal fraction.
- The approximation depends on assumptions and does not guarantee profits or quantify performance against a chosen allocation.
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Full text
# the incremental value of Kelly Criterion under difference circumstances
# 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.
## Answer by james42 (score 1, accepted)
https://quant.stackexchange.com/a/18045
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
$\mathbb{E}[\log(1+f\phi_1)]$
Expanding the argument of the log in Taylor series we get
$\mathbb{E}[f\phi_i - \frac{1}{2}f^2\phi^2_i+...]$
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.