Bill, Rather than put up the entire PDF, maybe better if I just illustrate the basic idea here for the sake of perpetuating the thread (admittedly, I don't understand why you say the problem is solvable with mean-variance here, unless I am entirely misunderstanding the problem) Let's say we have a guy who can only reallocate once a month, and he wants to be expected growth optimal at the end of the quarter. Let's further stipulate he is engaged in only a single "contest" that pays 2:1 with p=.5 (so that the answer to the Kelly Criterion equals what would be the optimal fraction to trade, an instance I refer to as the 'Special Case' where that holds). Asymptotically, .25 is the optimal fraction, but at at horizon of 3 events, it is expected suboptimal. Clearly, in a contest with a positive expectation (probability-weighted mean of potential outcomes) where the horizon is only 1 outcome, the expected growth-optimal fraction is always 1. As the number of plays increases, approaching infinity, the asymptote of .25 is approached (but is never the expected growth optimal fraction, but rather a limit to it). In fact, at 3 plays the expected growth-optimal fraction to wager on each of the three consecutive contests is .37868 as it turns out. I have attached a graph of this (hope this comes through) showing the expected growth values versus fraction wagered for horizons of 1 to 8 consecutive contests. Not only does the peak migrate towards the asymptotic peak, from 1 to .25 in this particular game, but the character of the curve changes (e.g. eventually inflection points appear, etc.). Isn't this exactly the kind of thing we are discussing in this thread, or am I way off here? -Ralph Vince
On the previous graph, to clarify, the vertical axis is expected multipled made on the stake, the horizontal is fraction of the stake wagered, for campagins that terminate at 1 to 8 plays.
LivingInVol says: << A) You have calculated kelly on this to be 60%. But I only offer it once per year and it takes a year to know if you've won and get paid. You have to put up the $1000 immediately. B) You have calculated kelly to be 10%. I offer this 1000 times per year and payout is immediate if you win. >> For option A, the horizon is 1, for B, the horizon is 1000. Each has it's own curve. What he needs to measure is the returns from A vs B, not necessarily (however) at the optimal point (because for Option A, the expected growth optimal fraction is 100%, not 60%).
Actually, ralph is closer than you are... Mean-variance is to decide asset allocation and I was posing the question more along the lines of evaluating two bets that that have two different time frames in paying out. One is paid immediately and repeatable 1000 times in a year and the other takes one year and is only bet once. I think what ralph answered was what was the optimal amount bet to make in scenario B, but I'm asking for more of how to evaluate the two bets. Bet A has a much higher kelly but is played only once. Bet B has a lower kelly but can be played 1000 times. So which one is better, playing A once, or B 1000 times? It is obviously B in this example, but what if you were only allowed to play B 100 times? What about 10 times? At what point does A become more optimal? I think I've come up with a way to evaluate the two but it would be interesting to see if anyone else has another solution.
And maybe to make things a little clearer... The max bet for all games is $1000 in both scenarios and let's assume that that is an insignificant portion of your bank. It is irrelevant that kelly or optimal f tells you to wager any portion of your bankroll because the bet amount is a very low fraction of your bank anyway. So, all you are trying to do is find the best bet between the two to put your money on.
LivingInVol, Now it is a different story. That $1,000 represents a value for f on two lines. One of them is the scenario A (horizon =1) straight line function, where you are at a given point representing $1,000, and scenario B (horizon=1,000, thus, a curved line since horizon > 1) where you are at a given point between 0and 1 on that curve, as represented by that $1,000. Each point on each line will have a different horizontal value (what I refer to as "Terminal Wealth Relative,") and likely a different value for f (as represented by that $1,000 max) along the respective lines (and hence, different drawdwon characteristics, and other characteristics for being at various points on the two curves). So, you may be looking to therefore assess these based on their vertical coordinates, with the higher being the "better," but that is only iff your criterion is expected growth optimality. (yes, I got the amphibology about my being "off".) -Ralph Vince
Yep, it is clear you have no clue what allocation means, that Vince did not answer your question and that you did not make your question clear but you think Vince was closer anyway. Bye
I apologize for the initial incorrect wording of the problem and wasting your precious time. English is not my native tongue. Here is another problem for you. Who is most likely correct: 1. A poster that has over 2500 posts on ET and spends his time getting immediately annoyed and pissed at trivial things 2. A poster that obviously doesn't waste his time on message boards and has published work on the topic of discussion