That's not strictly true, no? The only statement is that if you trade forever (in the true sense of the word) and you have finite capital, then you will face ruin for certain at some point. Optimal exit time is quite different, no?
Quite possibly it's my misunderstanding of the question! What I meant was that if you have a target, or an exit time, then after the target is hit, or the exit time is passed, the rest of time doesn't matter ... because you are out. You can't be ruined after you are out. But if you never get out (because you don't have a target, or a stop time) then eventually you will be ruined. I may have misunderstood what was being asked (which wouldn't be the fist time it's happened!).
That's why I figured the question is for laughs (intradaybill feel free to correct if I misunderstood too). If it is, then the statement is true without any need to qualify what the strategy is or what the returns are like (so long as they follow some pretty wide basic criteria); If not, then the whole problem depends on a nature of the strategy returns; And then it's a really really really difficult problem.
OK guys, let me try to state the problem better: Define: 1. A trading system that generates entries and exits with some frequency F > 0 per unit time and starts with bankroll B. 2. A Cummulative profit objective P 3. A time to quit for good objective T 4. The probability of ruin R such that bankroll B reaches 0. The questions (problems) are: (1) Given P = Pquit and/or T = Tquit are there conditions that may lead to R =1? In other words, even if there is a quit time or quit profit, under which conditions ruin R = 1 is certain? Please exlude the case where expectancy is negative. (2) Alternative, if there are no P and T constraints is ruin certain (R = 1)? I think (2) has been answered already. (1) remains.
Oh boy.... now we are getting into some actually interesting stuff... But we need to make it less interesting first and specify that nature of the price series and trading rules - the answer will be dependent on the those. In fact, the results are so dependent on the specification of the price series dynamics, that even simulations are likely very biased by your apriori assumptions.
So are you saying that it depends on the distribution of returns of the specific trading system in relation to underline market returns?
Absolutely. You can convince yourself with a simple experiment (you can do this in Excel if you don't have matlab or something similar): Trading strategy: Long/short when Price > MA(50) (doesn't matter), Exit with time stop or simple stop loss. Simulate the drawdowns over 1000 days under: A. Purely normal random returns: r(t) = e(t) where e(t) = norm(0,1) B. Random returns with a long memory: r(t) = 0.8*r(t-1) + 0.2*e(t).
Thanks. But the expectancy of system A will be negative due to design. I excluded negative expectancy systems.
Also, drawdowns and expected returns (can we please move on from using silly casino probabilities terms? like risk or ruin?) are not monotonic functions of each other.