That’s what you look at when you look at returns, right? Some standard unit of time? When you look at historical returns for funds you don’t look at it “per coin flip”
I agree it is arbitrary (one reason geometric mean, which is scale invariant, is better)
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ok so this question is a bit weird because the upside cases are about equal to the number of atoms in the universe so I think I object to the hypothetical?
but anyway, this will have higher average almost surely *if you run for enough years*!
In particular, something like 10^60
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So yeah, in the long run, 99% still has higher arithmetic average annual returns if you're flipping 365 times per year almost surely, as (number of years) goes to infinity!
But if you instead only have ~60 years then you'll find weird "not in the infinite limit" results.
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1) !! Now taking the limit as time approaches infinity is appropriate?
2) This is not correct. EV[annual return] is negative. As t approaches infinity, arithmetic mean of annual return will be negative
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1) idk, I think this whole thing is ridiculous! But you seem to want to insist on infinite time horizons; so be it.
2) ok so what do you think the EV of the return for the first year is: positive or negative?
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Yeah, I guess I’m wrong on the infinite timeframe. (Although I’d argue that it doesn’t make sense to do that here).
But still: look at your spreadsheet. Current wealth is zero and annual return is -99.99% no matter how much you refresh.
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er obviously "no matter how much you refresh" is false
also, though, yes, I agree, this is all ridiculous! It was ridiculous for you to write a paper premised on infinite timeframes that doesn't hold on reasonable time frames in the first place, but now we're here.
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I said no matter how much YOU refresh, in your lifetime. Didn’t say anything about your distant descendants
“Under these conditions, Uniswap pool shares have a higher expected compounding rate of growth” is true right? Not just on an infinite timescale, on any timescale?
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Hehe fair ;)
if you don't have an infinite timeframe then you have to:
a) assume the user has no other assets or valuable properties or talents (otherwise you'd be back to an essentially linear case)
b) insert "log" or "geo-averaged" in front of "rate of growth"
c) decide you want to maximize log wealth in the first place
d) decide you can't find any way to rebalance your portfolio without also opening yourself up to constant adverse selection from takes
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e) all the other assumptions the article also has to make