Maximizing Kelly is Equivalent to Maximizing Win Rate

My recent post on optimal trading using the Kelly formula has generated a lot of responses, questions and even challenges. A few readers have asked questions that, as they think, hide the key to the “holy grail” of investing and trading. However, as I have tried to explain before, the “holy grail” is a combination of common sense, knowledge of basic algebra and a couple of good tools to analyze the market. Here I will show how a seemingly complex idea gets transformed into a trivial statement via the use of high school algebra.

A user a Price Action Lab contact me last night, asking me whether I would consider adding a pattern search criterion to the software based on the maximization of the Kelly ratio. My reply to him was to use his common sense; if the maximum value of the Kelly ratio is 1 and that is obtained for a win rate equal to 1 (100%) then the optimum value for the ratio is obtained for investment and trading strategies with the highest possible win rate for the maximum achievable or desirable risk/reward parameters. Math is not even needed when common sense is properly used in this case but I am going to do the math here because it is less disputed nowadays than words. (However, please note that we are entering a very sophisticated era of crankiness when even math is disputed because the loss of common sense is accelerating due to the on-going destruction of basic human brain capacity and the alarming declining of IQ).

I will use here different symbols than in the paper and blog. I will use W for the win rate instead of P and a plain R for the average win to average loss ratio. Then the Kelly formula becomes:

Kelly = f = W – (1-W)/R        (1)

which is equivalent to:

f × R = R×W – (1-W) = R×W – 1 + W

and after solving the above for R we obtain:

R = (1-W)/(W-f)                   (2)

The above ratio tells us that since W, f, R > 0 then it is necessary that

f ≤ W

i.e. the Kelly ratio f cannot be greater than the win rate W and that the maximum it can get is equal to 1. Also, as f gets close to W, the ratio R gets extremely large, something that in all practical cases is not achievable. Thus, the optimum Kelly ratio is constrained not only by the above inequality, i.e. its upper bound is the win rate, but also by the path the system takes. In the limiting case where W = 1, R is equal to infinity because the average loser is 0. Thus, the optimal investment or trading strategy is one that has the highest possible win rate for maximum achievable or desirable value of R.

About R

Values of R are often predetermined (desirable) and that parameter is fixed. For example, often a strategy is designed with a stop-loss and a profit target of certain number of points, ticks or pips. In case R is not fixed,  the maximum achievable value is determined within the context of overall strategy objectives but a detailed discussion is outside the scope of this post. The point is the R should not be optimized in the same sense as W because different values of it may reflect a significant change in the nature and dynamics of the system. For example, values of R between 1 and 3 are realistic in the case of position systems where much larger values, like above 5 or 6, indicate trend-following capability. Thus, it is often the case that the trading system developer sets an upper bound on R to remain within the domain of operations of a certain strategy type. If both W and R are free parameters then the theoretical optimum is W = 1 and R = ∞, i.e. a 100% winning system. The practical optimum is the highest W for the highest achievable R. The trade-off between the two should depend on other objectives, like on the type of strategy desired (position vs. trend), for example. But other than in the case of some automated strategy development tools that utilize indicators, genetic programming and fitting functions to come up with strategies that are not constrained in both W and R, and thus quite often fitted well to the data and unlikely to perform in the future, most traders use preset values of R to what they think is desirable or achievable for their market and style.

Conclusion

The highest possible equity growth of any investment or trading strategy is achieved when losses are minimized in both number and size and profits are let to run, resulting in highest possible W and R. You don’t have to hire a Harvard or MIT graduate to tell you that. Just use your common sense.

Just one clarification regarding the above analysis: by maximizing Kelly I did not mean actually using the Kelly ratio to size positions. That would lead to a disaster. The maximization treated Kelly as a performance criterion only. In my opinion, nobody should ever invest more than 5% of net worth in any particular opportunity. Even that is very high in the case of frequent trading. Values below 1% are recommended. The formulas for the position sizing can be found here.

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