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Suppose I've raised some initial capital, $C$. I would like to invest it according to three different trading rules, $T_1$, $T_2$, and $T_3$.

Each of these rules will yield several trades over the course of a given year. Imagine I am relatively confident in my estimates of the edge for each of these rules. It is positive in all cases, but some rules are better than others.

My question: what is the proper use of the Kelly criterion?

  • Do I divide $C$ among the trading rules in advance (i.e., Kelly at the "rule level")? For example, $T_1$ gets to wager at most $p_1C$ from now into eternity ($p_1$ ranging from 0 to 1).
  • Do I allocate portions of $C$ to an individual trade (i.e., at the "trade level")? For example, $T_1$ might return higher or lower expected returns from trade to trade. Intuitively, it makes sense to allocate more capital to cases where $T_1$ is more optimistic.
  • Are both of these the same? Am I totally misunderstanding the problem in the first place?

A caveat: I know Kelly isn't the only game in town for things like position sizing. I'm open to alternatives, but can we stick to Kelly on this one for the sake of simplicity. I'm trying to understand the general nature of the problem.

Additional assumptions: As pointed out in the comments, the trading strategies can be assumed to produce moderately correlated (r = .50) predictions about price movements. Moreover, their output will be continuous probability distributions about expected price movements.

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    $\begingroup$ My guess is that you need more assumptions. Are those trading rules correlated and are they continuous/discrete strategies? Besides, under some assumptions, you may also consider merging the three strategies into a grand mean-variance optimized strategy. $\endgroup$ Sep 17 at 16:00
  • $\begingroup$ Thanks, for the context. In this case, I am imagining strategies with moderately correlated predictions (e.g., r = .50). I'm also imagining a each trading strategy will output a Bayesian posterior distribution of price movement, so the strategies are probably best understood as continuous (although they could be collapsed to something binary). $\endgroup$
    – Ian Cero
    Sep 17 at 21:09
  • $\begingroup$ Assume that investment returns are distributed in a multivariate normal distribution, you can first use the mean-variance optimization technique to find the optimal mixes, then apply Kelly Criterion for such a strategy. Make sure you don't have to feed your family if you really implement it as you will likely bankrupt from over leveraging and numerous things that can go wrong. $\endgroup$ Sep 21 at 2:57
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As correctly pointed out in the first comment, the core question is about the covariance of your strategies (or trading rules if you prefer) in addition to their mean rates of return. Kelly criterion is not a tool for combining a set of strategies; it is about an optimal betting (leverage) level for a single strategy using a log wealth utility. Even then the Kelly-optimal leverage gives a simplistic result which tends to be too high. This point, as well as ways of combining strategies (involving less trivial math), are discussed in this book.

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