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Imagine that you are in the finance profession. You know that x% of the time when you pick a stock it ends up positive. Conversely, 1-x% end up negatively. With this in mind is there an ideal portion you should be betting to achieve maximum return? How would you do this type of problem mathematically?

For example, if you know you get 100% of the time correct. You would want to invest all your all the firm's money. If its 99% of time correct, you would probably still do most of money. If your 50/50, you can set aside half of your money just in case you will lose or any number between 0-100%. If its zero, then you probably don't want to bet at all. This problem has been knawing at me for some time now. Please help.

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  • $\begingroup$ This is too general. If nothing else, "positive" is very weak information. Suppose one stock has a $99\%$ probability of going up a very small amount, while another can go to $0$. Are the stocks correlated? What Utility Function should be assumed? And so on. $\endgroup$ – lulu Aug 28 '17 at 17:48
  • $\begingroup$ If you know that $99\%$ of the time stock grows a positive of one dollar and the other $1\%$ of the time it drops negative of one million dollars... $\endgroup$ – Graham Kemp Aug 29 '17 at 0:52
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It doesn't just matter whether the stock is up or down, but also by how much. The first thing is to check that the expectation is positive. If not, you shouldn't buy any stock at all. You also have to define a utility function that you are trying to maximize the expectation of. If every dollar is equally valuable you should invest all your money if the bet is in your favor, as I discuss here. Most of the time you will be broke, but the times you are not will make up for it in expected value. More common is to have a utility function that rises more slowly than linearly, so going broke is very bad. In that case you can compute the proper amount to invest to maximize the expected utility. You might read about the Kelly criterion, which is one approach to this question.

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