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Lumifer comments on A hypothetical question for investors - Less Wrong Discussion
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To forestall what I expect to be a common line of advice, I'd like to point out that maximizing the expected return for the next day can lead to very poor results in the long (or even medium) term. Let me illustrate.
Your PDF for the stock price is the same every day and is pretty simple: it's a fair coin toss. You have 50% probability of earning 101% return and 50% probability of losing all your money (-100% return). The expected return is positive ($5 less transaction fees). This should cause you to invest into this stock every day.
The chances of being left penniless after merely 10 days are (1 - 1/2^10) or about 99.9%.
P.S. A somewhat related issue is the so-called St.Petersburg Paradox which is about three centuries old :-)
Sure, but I think if you look into the crystal ball and see that XYZ has a 50% chance of UTTER RUIN and a 50% chance of business as usual you turn around and ask Omega to let you go back to the real world now.
That's 50% chance of going to zero and 50% chance of doubling, not business as usual. I don't see what's so unusual about it. For example, if you buy an option (a financial instrument, a call or a put) and it expires out of the money, it's worth goes to zero. That happens all the time and no one calls it UTTER RUIN. Of course you may not want to invest your entire worth into one...
Or if you want a stock example, imagine a small biotech company with a single drug going through FDA trials. If the drug fails, the company is basically worthless, if it passes, the company is rich. That's a double-or-nothing scenario and again, not particularly uncommon in real life.
By your hypothesis the stock has a 50% chance of utter ruin and a 50% chance of doubling every day. That is very unusual.
Well, the OP hypothetical is very unusual by the same criteria :-P
To be fair though, that's only because of the strict limitations of the OP's thought experiment. If you could bet say 1% of your cash on a stock with your specifications, then assuming the transaction fees aren't a problem you should do so every day.
Well, of course the given setting limits the solutions. For example, if you can invest only a part of your wealth, the Kelly Rule comes into play.
I agree, given the daily stock fluctuations, the only way to make money would be to buy and hold over a long period of time and hope the 52 day moving average is in your favor. I would keep it all in cash.