PhilipPirrip,
Let me try to answer the several points inyour post, if I can. Please understand, I don’t know everythign about this material, like I said, I have only scratched the surface, despite having spent decades in pursuing it.
You say:
<<This actually seems a departure from your earlier your work. If I’m not mistaken you are now saying that diversifying the portfolio and trading over the long-term would eventually lead to an even higher value than 1. In the past you said that holding a trade for shorter periods is actually more efficient. >>
No, the Sigma f, the sum of all the f components in the portfolio approaches 1 as the number of components in the portfolio increases. Let me demostrate this in rather simple terms.
Let’s say I toss a coin that pays off at 2:1. If we calculate the Optimal f in this case (or the Kelly Criterion as this is a case where the two give identical resultsO we find the expected growth optimal fraction to bet is .25. Now, let’ssay I am going to wager on two of these very same coin tosses simultaneously. Now the fraction drops to about .23 each, so my Sigma f is.46…continuing, you can see that at about 9 multiple, simultaneous coin tosses like this, your Sigma f is now>.99, and continuing, it asymptotes out at 1:
#games f Sigma f Obj Func
1 0.25 0.25 1.060660172
2 0.2302911 0.4605822 1.119119317
3 0.2108185 0.6324555 1.174520606
4 0.1915346 0.7661384 1.225875158
5 0.1725609 0.8628045 1.272072169
6 0.1543264 0.9259584 1.312023094
7 0.137522 0.962654 1.345021143
8 0.1227372 0.9818976 1.371125513
9 0.1101508 0.9913572 1.391168868
10 0.0995878 0.995878 1.406375091
11 0.0907294 0.9980234 1.417958107
12 0.0832539 0.9990468 1.426916321
13 0.0768875 0.9995375 1.433993145
14 0.0714124 0.9997736 1.439714164
15 0.0666593 0.9998895 1.444442499
16 0.0624966 0.9999456 1.448427952
17 0.0588219 0.9999723 1.451843823
So if we were playing 17 of these games simultaeously, and all ame up against us, -1, we would have an immediate drawdwon of at least 99.99723% (the obejective function is our geometric mean holding period return, what we would expect to make, on average at theend of each toss of so many simultaneous tosses)
But these are coins, so the correlation between them is nonexistent. If, say, one of the components had perfect negative correlation to the others, then our sigma f would be infinitely high.
Now, to finish going over the dynamics of this, let’s go back to playing one of these games alone. SO far, what I have mentioned is all in the asymptotic (Q->infinity) sense. If our horizon were one play (Q=1), and since there is a positive mathematical expectation, we would maximize expected geometric growth at f=1. If we were toquit at Q=2, we would have maximized growth trading at a fraction of .375. As we increase Q, it ultimately approaches (but never quite reaches) .25. Similarly, with multiple, simultaneous games, the sigma f starts at 1, and as Q increases, the peak reaches it’s asmptotic value along the given axes at the asymptotc expected growth optinal points.
The caveat to all of this is when a play cannot result in a loss, either for a single component, or, where an aggregation ofcomponents (due to correlation) does not permit a period to transpire that would be a loss. In these cases, expected geometric growth is maximized by usingf value(s) of infinity.
Those are the dynamics.
<<I’m still trying to make sense of the equation in trading terms; like what would be the horizon in my system. The components? are they the pairs traded or the number of plays. I need to time to sort of grasp it. >>
This is the crux of the matter. One must determin a criteria – what are you doing in this endeavor, what are you trying to achieve. If the answer is simply “To make a profit,” that’s too vague. Too vague to use this framework to your advantage anyhow. A lot of guys want to just survive, and make a profit. Now, I;ve said in the past, you;re always on this curve in one component, surface in N components, and likely moving about it, you jsut dont know it and the rulesof the geometry apply to you. So, if a guy is just looking to survive and make a profit and trade until his golden years, then you want to be tucked away down in the corner near f=0. trade one lots, set a maximum loss, and go from there. There’s nothing wrong with that – you’ve defined your criteria, including your time frame (trade, essentially “forever,” thus, the asymptotic curve/surface shape applies to you).
However, let’s say you are mamanging a pension or work for an insurance company, or even a hedge fund. Now you have some pretty well-defnined criteria to satisfy. As part of defining criteria, horizon is germane to that.
Suppose your criteria is to “Maximize quarterly risk adjusted returns overthe next five years.” Nowyou have a framework to pursue precisely that type of thing.
<<But you mentioned my criterion and I want to address this. I understand your work about reaching an optimal f within a predefined tolerance for a drawdown. The problem I have with that, is that the result is limited by the trader’s intolerance to drawdown. I want to do this with a drawdown of a 100%, so the actual inflection point. Not just that I want to create a situation where that inflection point is 1.
It may be impossible to do but there is no better authority to ask than yourself. I know you said its realistically impossible but bare with me… Am I crazy? >>
It’s a little long-winded of an answer here, too much so to include here. Thisis covered at-length in the Risk Opportunity book (Addendum 1 to Chapter 5 Üsing only a percentage of the stake so as to constrain risk") as well as the 1992 book where it addresses the notion of “Coninuous Dominance.” The quick answer is that you CAN do it, you pay a small price for doing so however, due to certain efficiency losses, and you then also end up with two surfaces or curves you are actually upon for your specified criteria.
Ralph Vince