Signal to noise

In fact, suppose Bob really was right more often than not, but barely so, so he was making a 20% return a year on average. How long would it take to realize that this was so and he wasn't just random? You often see stock prices fluctuating 30% in one year. So you expect with constant buying and selling a return of 20% $\pm$ 30%. The standard deviation swamps the mean. He might lose 10% the first year and the second year, but over 20 years, be making a profit.

To understand how long Bob has to wait, to prove that he is making a sizable return, we use what we learned about statistics. The more data points, the lower the standard deviation of the mean. If with one years trading, there is a 30% standard deviation in the stock price, then after years its $30\%/\sqrt{N}$ . For statistical significance, the mean has to be at least two standard deviations. So to attain statistical significance, Bob will have to wait several years before being able to reject the null hypothesis.

That's how long you'd have to wait to see if you were or weren't an investment charlatan. Several years? That's right. Most of our investment stud friends have decided they're great with far less experience than that.

Also I'm assuming that Bob isn't allowed just to hold a bunch of stocks, like an index fund, which make around 10% a year. If this background drift is subtracted off the 20%, you'd have to wait even longer!

josh 2010-10-20