Are you in some type of a hurry now? You have been posting this error for years.. hmmm...nice way to start with someone who tries to help you and save you from becoming a ridicule in other forums. Get a probability book and read what expected value means. First of all you assume 10 draws with replacement. This is totally arbitrary notion here. I do not understand where you got that. The expected gain in you 2nd example is: E(x) = 0.7 x (-1) + 0.1 x (-5) + 0.2 x (10) = 0.8 First of all, nobody uses the simple Kelly formula for more than two outcomes. There are generalized forms for multiple outcomes, so you bring up a distracting issue here. Secondly, trading is a two outcome bet. You go either long or short (if you do nothing you are not trading). This is your probability space. It is the simplest space. You also either win or lose. Each time you go long or short, it is called a Trade. This trade has a value, like any random variable takes a value from a Prob. Dist. Obviously, some other poster was correct to point that out to you but you continuously resist in an unreasonably stuborn way. Values of random variables are not outcomes. This is a problem for you I used to give to my students in exams. Please provide an answer. If you dissappear it would mean that you are a hoax and a coward. Problem: Fair-die experiment. Each time the die is rolled you win ten times the number of the face shown. What is the expected gain of the experiment?
What other forums? And don't worry about me, slick, watch out for your own image. Your hangup is obviously semantics. If you look at my usage of "outcome", you'll see that it has been thoroughly consistent. Nitpicking at trivialities won't convey any real understanding of my message. Yeah, that lack of understanding comes from nitpicking at trivialities. Here are some clues. There are ten balls to be picked. There are ten draws with replacement. Gee, I wonder if there could possibly be any connection between ten balls and ten draws. Naw, must be totally arbitrary. Did you ever bother to read example 2 in toto, or are you just flailing at this point? Since you're picking nits here, that's the expected gain for a single bet. The formula I posted was the expected cumulative gain for your entire betting account over ten bets, because that is what we're trying to find the Kelly fraction for. Forgive my lack of total precision in my previous post, I keep forgetting how some ETers expect to be hand-held through every single detail because they're utterly incapable of filling in any gaps, no matter how elementary said gaps may be. First of all, plenty of people have been arguing for using the simple Kelly formula for more than two outcomes. That's what my posts have been about. And what I posted is the generalized form for multiple outcomes. Oh yeah, you don't like the way I'm defining "outcome". Tough; get over it. Second, going long or going short are not outcomes no matter how you define "outcome", going short or going long are decisions. And since you're picking nits, trading is actually a single outcome bet. You place a trade, you close the trade, you get a trade return: a single outcome. Dividing your returns into "average win" and "average loss" is no less arbitrary than treating each return value as a separate outcome. The difference is in how the trader chooses to analyze his trade returns to calculate his Kelly fraction, which by the way doesn't give a flying fig whether you went long or short. 35. Was that supposed to be a challenge? Do yourself a favor and read gummy's presentation. He'll almost certainly offend your nitpicking sensitivities less than I did, even though we both got the same results.
Hi kutk2k, I am trying to relate your example 2 to trading. I put on a long trade where I plan to bet R (yellow ball). My profit target is 10 R(green ball). What is the purpose of then -5R(red ball)? Is it in case my plan fails and I loose 5R instead of 1R? If this is what you mean, why stop with only these 3 outcomes? There will be guzilians of possible outcomes. I may get spooked and take 0.5R loss. I may get really greedy and successfully take 20R win. I could f*ck up in a big way and loose 50R etc...etc.. P.S. If I am correct that there will be an infinite outcomes. What will one have to bet? The sum of probability of all outcomes will be 1. What will my optimal bet be if we only work with 2 possible outcomes? 5R win and 1R loss, given that my average win rate over a statistically significant number of trades is 45% win rate when I bet R to get 5R. If the goal is to allow for different outcomes and you want to use kelly then I think you have to get your most likely loss value over the statistically significant number of trades and make that be R and your most likely win value and make that XR and use the % of wins for the same number of trades. Then you can use Kelly as usual.
Example 2 was intended to be a three-outcome betting situation to illustrate why the simple Kelly formula doesn't work for situations with more than 2 outcomes. What you're betting on is a draw of a green ball, because that is your only winning outcome. Getting a yellow ball or the red ball is a losing outcome. Because drawing a yellow ball or the red ball results in loss, the Kelly fraction limits your risk (or maximizes your expected gain). For trading you never know exactly what your trade distribution is, nor can you count on it being stationary. All you can do is (1) use the exact form and something like Excel's Solver to get the current best estimate of k, keeping in mind that the more returns you use, the better your estimate but also the more chance there is that your trade distribution has changed from start to end or (2) use the approximate Kelly fraction formula in a rolling fashion where you decide on a fixed sample number of trade returns and with each new completed trade, you add the new value and drop off the oldest, just like with a simple moving average. I prefer the latter method for estimating my Kelly. YMMV. I also prefer to use half Kelly, for the many good reasons that have been posted elsewhere.
You are equating number of trials N with number of balls. This is totally arbitrary with no justification. Ask your self: why ten draws only? Your outcome is the color, not a particular ball. You have three outcomes, not ten. Over a sufficiently long number of trials the colors will show up with relative frequencies of 70%, 20% and 10% and corresponding probabilities of 0.7, 0.2, 0.1. The expected value is the equation I gave. There is no such thing as expected value over a number of draws. You confuse expected value and expected total gain. Thus, the expected value equations you present is wrong. This is the expected total gain after 10 draws. What it means I do not know. Your outcomes are the three colors, not the ten balls Try to get over your ego and understand what you are doing wrong. The question you should ask yourself is: why just 10 trials of the game? It is the relative frequency of outcomes over the limit of trials to infinity that determines the expected value, not a number that is equal to the number of objects in the experiment.
This statement of yours is proof you do not understand Probability Theory. There is no such thing as expected value for a single bet. Expected value is an average expectation, it is how much one is expected to win or lose per trial in the sense of the law of large numbers. Each return value is not an outcome. You do not comprehend very basic concepts in probability theory. You are sampling a distribution of the random variable X, where X is the trade returns. The average win gives you a measure of how much you should expect to win each time you win (First Outcome). The average loser how much it is expected to lose each time you lose (Second Outcome). The expected gain function is derived using a property of expected values. You are not aware of these high schoool stuff it seems. I suggest to you to print a copy of these posts and go to your local community college and find a Propability professor. You confuse: 1. Random variables values with probabilities of random variable 2. Outcomes with random variable values 3. Expected values with the actual gain value In conclusion you are a total mess, yet you come out as arrogant and refuse to accept help. You ought to reconsider your behavior before it is too late for you, if it is not already. Accepting your mistakes will make you a better person.
intrabill talks out of his ass and you think I busted. Looks like you both have to go back to school. Ask yourself why he hasn't addressed the issue of overbetting, which I proved is taking place with the simple formula. He can't, so he ignores it. Q.E.D.
Just because people do not answer your stupidity that does not mean you can say Q.E.D. All of your formulas are wrong. I said you must seek professional help. You first state the Gain function where the R's are the actual trade percentage returns and then you substitute actual magnitudes for those parameters. Only a fool could do something like that. You are a nut case... bye...
Frankly, I know nothing about Kelly's size and I have no interest in finding out. Same goes for Ralph and Vince. As for you guys with all of your numeric specificity, you might do well to consider Sebastian Mallaby's words on page 231 of his book, More Money Than God: "The real lesson of LTCM's failure was not that its approach to risk was too simple. It was that all attempts to be precise about risk are unavoidably brittle."