Calculating expected return Assume I have a trading system that I'm evaluating over a three-year period. The returns are 25%, -40% and 25%. Empirically, I can see that this system loses because at the end of three years, I have less than when I started. 
Wikipedia defines expected return as follows:
E(R)= Sum: probability (in scenario i) * the return (in scenario i)

If we insert our values into this formula, we get the following:
E(R) = (.33 * .25) + (.33 * -.40) + (.33 * .25) = .033

So a positive expected return for a system that loses every time, no matter if you re-arrange the order of the returns. 
What's wrong here?
To further explain how the game works, consider an initial start value of 100. What can you expect at the end of the game? There are no re-investments, no withdrawls, no dividends, no broker fees nor SEC fees. It is a simple game. Here is some R code to illustrate the game.
first  <- c(.25, .25, -.4)
second <- c(-.4, .25, .25)
third  <- c(.25, -.4, .25)

Pass any of the above sequence of returns into this function:
game <- function(x){

start <- 100

for(i in 1:NROW(x))

start <- start + start*x[i]

return(start)   
}

NOTE: I asked a similar question on quantexchange, but I'm interested here in the math behind the expected return equation. 
 A: Wikipedia is right. So are your calculations: this system should win.
However: it also assumes that your investments in each part are equal in size (because you give equal probability 1/3 to each). If this is not true, that may explain the difference with your empirical observations (perhaps you should share the numbers with us on those). e.g. If you invested twice as much in the part that has a negative return than in the two other parts, the probabilities become 1/4,1/2,1/4 and you'd have a losing system.
This is also ignoring any extra costs, so if these extra costs are bigger than your meager (?) 1/30 return, this could be another explanation (I'm not familiar with the practicalities of trading).
A: The difference between the two ways to look at the return is whether you are reinvesting your gains. If you start with 100 dollars and reinvest all of the money the next year, your balances will be: 125, 75, 93.75 dollars - you lost money. However if you invest 100 dollars every year, then you get +25 - 40 +25 = +10 dollars gain. That's one way you could think about the Wikipedia formula. A better way to interpret it, though, is that you are investing in one of the three years chosen at random, and it gives you the expected return.
