[RockyMcNuts]

Suppose you restate it. You own a $100 stock. It has a 50% chance of doubling, and a 50% chance of going down 50%.

Should you sell it, or hold onto it? Expected value of holding is ($200 + $50)/2 = $125 . So it seems like you should hold on!

If you repeat that wager indefinitely, the standard deviation of the net wins (number of ups-downs) goes up according to a square root law.

And the value of your stock goes up exponentially as the number of net wins goes up.

So over time the EV goes to infinity like (2 ^ (n/2)). The EV of say the -1SD outcome goes to zero. The EV of the +1SD outcome goes to 1/that. So the average EV overall goes to infinity.

And yet the expected value of the growth rate is 0. For every 16x win there's a loss down to 1/16th. But the average of those 2 outcomes diverges to infinity.

All this to say, when you're looking at exponential returns (or other processes), you need to measure growth rates, not average outcomes. And it has nothing to do with log utility.

[olalonde]

Javascript simulation (might freeze your browser so would be a good idea to run it in Node):

    var t = 100; for (var i = 0; i < 1e6; i++) { if (Math.random() > 0.5) { t = t * 2; } else { t = t / 2; } console.log(t); }

It's interesting how quickly the return goes from extremely small to extremely big amounts.

[RockyMcNuts]

yeah. Now I wonder what happens when you have $10,000, and always put 1% of your portfolio in that stock? I think that turns it into a positive growth expectation. You do it 10 times and outcomes are 50/50, 5 times you made about $100, 5 times you lost about $50.

On the other hand, if you bet your whole stack each time, you doubled up 5 times and lost half your stack five times, you're even.

if you bet your whole portfolio every time, I think your long run growth rate is 0. if you bet a small amount each time, I think your growth rate is positive.

The Kelly Criterion or gambler's curse in action. In the first case you're taking a positive EV bet and turning into a long run no-growth situation by overbetting.