Alright. Read it a bit more carefully. Makes sense to me, even though I'm not completely sure if the original Kelly formula doesn't cover his case (outcomes other than double-or-none and risk size greater than 1) as well, by relatively straightforward modification, in which case I'm not sure if the rename was completely appropriate, but that's maybe nitpicking. It was an interesting read, for sure.

Here's a little observation how to apply the 'generalized Kelly'/'Sanden criterion' for different trading assumptions:

Starting from the assumption you're 100% in USD, and that your goal is to make a USD profit, let's say

**you give it equal chances that USD/BTC will go up by P% or down by P%**. Applying the criterion we non-surprisingly get

(0.5)/(P) - (1-0.5)/P

which reduces to 0. In other words, don't trade at all if you don't know where price is going.

But now assume for a moment you're 100% in BTC, and

**your goal is to maximize your total of BTC**. This only makes sense of course if you're willing to accept a (short term) USD loss, and is based on the assumption that in the very long run, USD/BTC will go much higher and you want to increase your coin stash as much as possible.

Then the calculation becomes one of "selling BTC now to rebuy cheaper later or not".

**Say again we assume with equal likelihood that USD/BTC goes up P% or down P%**. If price goes up, you made a BTC loss, more precisely: you rebuy for 1/1+P, which means your new BTC total is 1-(1/1+P) as percentage of your previous BTC total. Similar for the case where your trade is BTC profitable and price goes down.

But you can already see where I'm going with this: for example, price going down 50% and rebuying means your BTC total

**doubles**, while price going up 50% and rebuying (at a BTC loss) means your BTC total is reduced by

**only 1/3.**Using the criterion formula, we get

(1/2)/(1-(1/(1+P))) - (1-1/2)/((1/(1-P))-1)

which reduces to... 1.100% of your position.

^{[1]}So, the conclusion would be that, if you are a long term BTC maximizer (what user Rampion once called the "land grabbing" strategy), then it makes sense to sell your **entire** coin stash relatively quickly, on a "whim" so to speak, as long as you're conviced that the possible **upside** potential of USD/BTC (on a short/medium term swing) is identical on to the potential **downside** potential during that swing.I'll leave it up to the reader if that really makes sense as a strategy. Note please that this doesn't mean the criterion itself is mathematically wrong: the thing to keep in mind is that Kelly (and Sanden) define the

**maximum** size you should risk to maximize profit, but for independent reason

^{[2]} it might well be more sensible to stay below that maximum.

* * *

[1] If I made a thinking mistake somewhere, let me know please. Although I went through it again, and it looks right to me: it's basically a mildly surprising (to me at least) consequence if you don't measure profit in the unit that undergoes change P% (USD, in the first example), but in BTC.

[2] Such as: It's possible that when we think chances are 50/50 that price goes up or down by equal percent, you're actually systematically misjudging the market. It's certainly a thought I have quite often ("to me it looks like it could go up 10% or down 10% with equal likelihood"), but that doesn't mean I am

**justified** in that belief. Could be a systematic cognitive bias I need to take into account.