Expanding on RomeoStevens’ comment… Maths time! Suppose that he has now 10,000 dollars and 500 bitcoins, each bitcoin now costs $100, and that by the end of the year a bitcoin will cost $10 with probability 1⁄3, $100 with probability 1⁄3, and $1000 with probability 1⁄3. Suppose also that his utility function is the logarithm of his net worth in dollars by the end of the year. How many bitcoins should he sell to maximize his expected utility? Hint: the answer isn’t close to 0 or to 500. And I don’t think that a more realistic model would change it by that much.
Khoth suggests modeling it as starting with an endowment of $60k and considering the sum of the 3 equally probable outcomes plus or minus the difference between the original price and the closing price, in which case the optimal number of coins to hold seems to be 300:
Of course, your specific payoffs and probabilities imply that one should be buying bitcoins since in 1⁄3 of the outcomes the price is unchanged, in 1⁄3 one loses 90% of the invested money, and in the remaining 1⁄3, one instead gains 1000% of the invested money...
I’ve fiddled around a bit, and ISTM that so long as the probability distribution of the logarithm of the eventual value of bitcoins is symmetric around the current value (and your utility function is logarithm), you should buy or sell so that half of your current net worth is in dollars and half is in bitcoins.
Expanding on RomeoStevens’ comment… Maths time! Suppose that he has now 10,000 dollars and 500 bitcoins, each bitcoin now costs $100, and that by the end of the year a bitcoin will cost $10 with probability 1⁄3, $100 with probability 1⁄3, and $1000 with probability 1⁄3. Suppose also that his utility function is the logarithm of his net worth in dollars by the end of the year. How many bitcoins should he sell to maximize his expected utility? Hint: the answer isn’t close to 0 or to 500. And I don’t think that a more realistic model would change it by that much.
Khoth suggests modeling it as starting with an endowment of $60k and considering the sum of the 3 equally probable outcomes plus or minus the difference between the original price and the closing price, in which case the optimal number of coins to hold seems to be 300:
Of course, your specific payoffs and probabilities imply that one should be buying bitcoins since in 1⁄3 of the outcomes the price is unchanged, in 1⁄3 one loses 90% of the invested money, and in the remaining 1⁄3, one instead gains 1000% of the invested money...
I’ve fiddled around a bit, and ISTM that so long as the probability distribution of the logarithm of the eventual value of bitcoins is symmetric around the current value (and your utility function is logarithm), you should buy or sell so that half of your current net worth is in dollars and half is in bitcoins.
Nevermind, Gwern posted it before me.