Sometimes the real world feels insane. If you generated completely random zero-mean R and *multiplied* (1+R) by itself repeatedly - you'd have complete noise that you could become incredibly rich off of. Sorry, but it's true. Don't kill the messenger.
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I’m saying start with $100 dollars, 50 in cash and 50 in the asset. At the end of each day, observe your new allocation has shifted because of the days returns. Then, rebalance 50/50 of whatever NAV you have left into cash and the asset. This is different from your example.
That would be *exactly* the same as what I already did, except that the 'equity' variable would be multiplied by 100 to simulate starting with $100 instead of $1. It wouldn't change the overall result except that instead of ending with $0.000005 it would end with $0.0005
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If we started with a stock with a *geometric* return of zero instead of *arithmetic* return zero the story would be different and you could make money (before costs) by rebalancing every day, although not at anything like the rate that you are imagining.
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If a stock has arithmetic return of 0 you can just short it to harvest the vol decay because of the 1/2 sigma^2. Another way to make money from noise.
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