I found this 1926 paper “On Being the Right Size” by J. B. S. Haldane quite fascinating.

To the mouse and any smaller animal it presents practically no dangers. You can drop a mouse down a thousand-yard mine shaft; and, on arriving at the bottom it gets a slight shock and walks away, provided that the ground is fairly soft. A rat is killed, a man is broken, a horse splashes.

That reminds me of Billy Eckhardt’s comments on bet size…

If you plot system performance against bet size, you obtain a curve in the shape of a rightward-facing cartoon whale, going up in a straight line before dropping dramatically.

He said: “Trading size is one aspect you don’t want to optimize: the optimum comes just before the precipice. You want to be at the left of the optimal point, in the high zone of the straight curve.”

Not altogether true.

Expected growth-optimal bet size is a function of horizon, i.e. how many plays or periods.

For one period with a positive probability-weighted expected outcome (what most refer to as the misnomered “positive expectation”) the expected growth optimal bet size is 1, one hundred percent.

As the number of periods approach infinity, this diminishes to the asymptote at what I refer to as Optimal f (not “Kelly,” which is subset of Optimal f).

But all that is f we are discussing expected growth-optimal as criterion.

In capital markets, the criterion is often to maximize the risk-adjusted return, which occurs in the region between the inflection point less than the peak, and the point where the curve’s tangent has the highest slope, which is greater than the inflection point, but less than the peak. These two bounding point for risk-adjusted return optimality are, as with the peak itself (and, as I hope I have convinced in another, previous post, the actual “expectation”) a function of horizon.