As others have already said, this is way underspecified. But I think the following is at least a fairly decent answer for most plausible ways of filling in the details:
Let’s suppose you are aiming for wealth in the longish term (clearly you aren’t much interested in the short term or else you would be spending some of this money) and let’s suppose your utility is proportional to log(wealth), which is (1) empirically at least semi-plausible, (2) quite nice mathematically, and (3) traditional (it goes all the way back to Bernoulli’s treatment of the St Petersburg paradox).
Then every day you get to choose to multiply your wealth by either about 1 (if you put your assets in cash) or some random factor with known pdf (if you put your assets in the stock). In other words, you get to add to log(wealth) either about 0 or some random addend with known pdf.
If those random things are reasonably well behaved, then with very high probability after a while your log(wealth) is approximately log(original wealth) + the sum of E(delta log wealth). Which suggests that, ignoring horizon effects when you know the game will be ending soon, you always want to choose the outcome that maximizes the expectation of delta log wealth.
(Of course you should include the effects of the transaction fee in this. Since we have neglected the impact of future transaction fees, it might be a good idea to compensate by adding a little extra “friction” and, say, pretending that the transaction fee is $3 instead of $2 when doing the calculation.)
Worked example #1: Consider Lumifer’s example where every day the stock either goes up by 2.01x or goes all the way down to zero. The expectation of delta log wealth, ignoring transaction costs, is 1/2(log 2.01 + log 0) = -infinity, so unless your current wealth is barely more than the transaction cost and you’re already invested in the stock you want to be in cash. (So you will never invest in the stock, so you will never get into the crazy situations where the transaction cost might change your decision.)
Worked example #2: suppose on the first day when you have $1000 you know that the stock will either go to 0.9 or 1.2 of its previous value, each with probability 1⁄2. And suppose what you currently have is cash. Then your options are to stay in cash, with E(delta log wealth) = 0 because this is a no-op, or to buy shares, with E(delta log wealth) = 1/2[log(1198/1000)+log(898/1000)] ~= 1/2(0.181-0.108), which is positive. So in this case you should get into the market.
Worked example #3: same as #2 but now you only have $32. So now if you buy you have $30 in stock and it will move to $27 or $36 with equal probability. So the expectation is 1/2[log(27/32)+log(36/32)] which you can easily check is negative; so in this case you sit on the cash and hope for a better PDF next time.
Nope. And if what you’re after is the best long-run result and your utility is anything like logarithmic in wealth, this is exactly what you want.
(Although if Pr(lose everything) is small enough then the observation that you almost always get approximately the expectation in the long run is irrelevant unless the run is infeasibly long. So you might want to truncate your return distributions somehow, if you’re prepared to accept a tiny probability of ruin for doing better almost all the time.)
Notice that if you have a fixed time horizon the situation changes and you can optimize for how large a probability of ruin should you be prepared to ignore.
This doesn’t seem right. Let’s assume that the stock gives double or nothing, with 51% probability of double. The Kelly Criterion suggests giving 1% of the total payroll in stock. Yes, this neglects the balancing fee. Your argument seems to suggest that we should be all in cash. But the Kelly bet outperforms this.
I don’t understand: the situation here is one where your only option is to be all in cash or all in the stock. The Kelly criterion only makes sense when you can choose an arbitrary fraction to be in each.
(And the Kelly criterion amounts to maximizing E(delta log wealth), which is exactly what I’m proposing. If you have to wager your entire bankroll, any gamble with a nonzero chance of bankrupting you has E(delta log wealth) = -infinity and just sitting on your cash is better.)
As others have already said, this is way underspecified. But I think the following is at least a fairly decent answer for most plausible ways of filling in the details:
Let’s suppose you are aiming for wealth in the longish term (clearly you aren’t much interested in the short term or else you would be spending some of this money) and let’s suppose your utility is proportional to log(wealth), which is (1) empirically at least semi-plausible, (2) quite nice mathematically, and (3) traditional (it goes all the way back to Bernoulli’s treatment of the St Petersburg paradox).
Then every day you get to choose to multiply your wealth by either about 1 (if you put your assets in cash) or some random factor with known pdf (if you put your assets in the stock). In other words, you get to add to log(wealth) either about 0 or some random addend with known pdf.
If those random things are reasonably well behaved, then with very high probability after a while your log(wealth) is approximately log(original wealth) + the sum of E(delta log wealth). Which suggests that, ignoring horizon effects when you know the game will be ending soon, you always want to choose the outcome that maximizes the expectation of delta log wealth.
(Of course you should include the effects of the transaction fee in this. Since we have neglected the impact of future transaction fees, it might be a good idea to compensate by adding a little extra “friction” and, say, pretending that the transaction fee is $3 instead of $2 when doing the calculation.)
Worked example #1: Consider Lumifer’s example where every day the stock either goes up by 2.01x or goes all the way down to zero. The expectation of delta log wealth, ignoring transaction costs, is 1/2(log 2.01 + log 0) = -infinity, so unless your current wealth is barely more than the transaction cost and you’re already invested in the stock you want to be in cash. (So you will never invest in the stock, so you will never get into the crazy situations where the transaction cost might change your decision.)
Worked example #2: suppose on the first day when you have $1000 you know that the stock will either go to 0.9 or 1.2 of its previous value, each with probability 1⁄2. And suppose what you currently have is cash. Then your options are to stay in cash, with E(delta log wealth) = 0 because this is a no-op, or to buy shares, with E(delta log wealth) = 1/2[log(1198/1000)+log(898/1000)] ~= 1/2(0.181-0.108), which is positive. So in this case you should get into the market.
Worked example #3: same as #2 but now you only have $32. So now if you buy you have $30 in stock and it will move to $27 or $36 with equal probability. So the expectation is 1/2[log(27/32)+log(36/32)] which you can easily check is negative; so in this case you sit on the cash and hope for a better PDF next time.
Would this system ever invest in stock when the probability of losing all the money is non-zero?
Nope. And if what you’re after is the best long-run result and your utility is anything like logarithmic in wealth, this is exactly what you want.
(Although if Pr(lose everything) is small enough then the observation that you almost always get approximately the expectation in the long run is irrelevant unless the run is infeasibly long. So you might want to truncate your return distributions somehow, if you’re prepared to accept a tiny probability of ruin for doing better almost all the time.)
[EDITED to add a missing right-parenthesis.]
Notice that if you have a fixed time horizon the situation changes and you can optimize for how large a probability of ruin should you be prepared to ignore.
That’s why I said, in my original comment, “ignoring horizon effects when you know the game will be ending soon” :-).
This doesn’t seem right. Let’s assume that the stock gives double or nothing, with 51% probability of double. The Kelly Criterion suggests giving 1% of the total payroll in stock. Yes, this neglects the balancing fee. Your argument seems to suggest that we should be all in cash. But the Kelly bet outperforms this.
I don’t understand: the situation here is one where your only option is to be all in cash or all in the stock. The Kelly criterion only makes sense when you can choose an arbitrary fraction to be in each.
(And the Kelly criterion amounts to maximizing E(delta log wealth), which is exactly what I’m proposing. If you have to wager your entire bankroll, any gamble with a nonzero chance of bankrupting you has E(delta log wealth) = -infinity and just sitting on your cash is better.)
Ah, I missed that part of the OP. So then I think your argument is correct.