Ernest Chan (blog) writes in his book Quantitative Trading - How to Build Your Own Algorithmic Trading Business:

Here is a little puzzle that may stymie many a professional trader. Suppose a certain stock exhibits a true (geometric) random walk, by which I mean there is a 50-50 chance that the stock is going up 1 percent or down 1 percent every [day]. If you buy this stock, are you most likely--in the long run and ignoring financing costs--to make money, lose money, or be flat?
Most traders will blur out the answer "Flat!," and that is wrong. The correct answer is that you will lose money, at the rate of 0.005 percent (or 0.5 basis points) every [day]! This is because for a geometric random walk, the average compounded rate of return is not the return μ, but is g = μ - σ^2/2.

For this very reason, geometric Brownian Motion is often written as
where "μ - σ^2/2" is the expected return and "μ" is the return of the expected prices, i.e. ln(E[S_t]/E[S_t-1]).

If you lose 50 percent of your portfolio, you have to make 100 percent to get back to even... that's what everybody knows. But it's also interesting to see how mild volatiltiy hurts over time. Here are ten (random - no cherry picking) realizations of a geometric Brownian Motion with a daily volatility of 1% (i.e. a yearly volatility of 16% when having 252 trading days) over the period of 100 years:










R Development Core Team (2008). R: A language and environment for statistical computing.

PS: Hey, this looks promising: