I encountered this paradox a while back and there is less than meets the eye. It's just one of those tricks used to impress the gullible. As for mixing a bunch of systems - I contend that the less fancy you are the better off. You are going to have to construct and reconstruct your portfolio all the time because - guess what, markets keep changing. The key element of portfolio design is keeping the expectation as high as possible while keeping the risk of ruin as low as possible - the combination of the two being a function of your own risk tolerance. It's obvious that having markowitz's assertion about diversification is incontroversial and correct - anyone who doesn't agree and can prove it should go collect their Nobel prize (or memorial Bank prize to be pedantic about it, since there is no Nobel prize for economics). If I am playing a dice game that pays $20 on 6 and loses $10 on 1, my average expectation on each roll is $10/6. Risk of losing $10 is 1/6 If instead I can play a dice game where I roll 2 dice, each of which pay $10 on 6 and lose $5 on 1, my average expectation on each roll is still $10/6. But my risk of losing $10 is now 1/36. Substitute strategy for dice and you get the picture. We can extrapolate this - if my dice game now involves N dice, each of which pays $20/N on 6 and loses $10/N on 1. Make N arbitrarily large and your avg expectation is still $10/6. But lo and behold, your risk of losing $10 can now be arbitrarily small. Each round of your game will in fact converge to a payout distribution that tightly focuses around $10/6 ..... creating a nice steady coupon like stream. That would be really nice ..... some call that the holy grail, but I prefer to think of it as medallion.
... I'm NOT disappointed... just kiddin' I agree with most of the material you've written in terms of the basic premises you covered. I would have to add a comment about your post because I got an impression that you approach each component of the portfolio as an indepedent entity. My impression of your post would be to approach individual systems directly from the portfolio model you use. Portfolio Model(s) -> Individual Systems Before going into the details, I look at it as: Portfolio Model(s) -> System Set(s) -> Individual Systems I guess it's very close or same thing as watching the Industry Groups if I was trading a straight out equity port. I've briefly mentioned this in another post but systems are just an extraction of a specific tendency(s) of what the market does. Obviously, there are different tendencies and they can be extracted and exposed in different ways. People offset the risk of over-exposing themselves with a single tendency by running tests like correlation analysis and etc. to reduce the risk. There's quite a few of them but to provide an example. One tricky problem is when multiple tendencies are extracted as one system. These are rarely detected by running a map of correlations or other measures. Also, in most cases, systems are not tested (or testable) enough to expose and identify the specific tendency. Finally, Curve-Fitting does not happen only with optimizations. There's plenty of other ways to curve-fit even with portfolios.
1/36 : -10 8/36 : -5 16/36 : 0 2/36 : 5 8/36 : 10 1/36 : 20 Profit = 11/36 Loss = 9/36 Breakeven = 16/36 I'm not going to bother posting any charts and code stuff for this but the Std. Dev. (or... Sharpe) of the curve will be smoother. Question is what fitness you'd use to measure what's good...
http://en.wikipedia.org/wiki/Non-linearity http://en.wikipedia.org/wiki/Nonlinear_regression http://en.wikipedia.org/wiki/Artificial_neural_network http://en.wikipedia.org/wiki/Degree_of_truth http://en.wikipedia.org/wiki/Fuzzy_logic http://en.wikipedia.org/wiki/Bayesian_probability http://en.wikipedia.org/wiki/Forecasting http://en.wikipedia.org/wiki/Prediction http://en.wikipedia.org/wiki/Decision_theory http://en.wikipedia.org/wiki/Complex_adaptive_system http://en.wikipedia.org/wiki/Systems_science One thing about combining different systems, though... Remember that when you accumulate predictions and extrapolate these systems or otherwise aggregate into more complex representation - you are increasing the RMSE output and variance manyfold, just like when you extrapolate farther on continuous detail scale into time... i.e explosively increasing risk to prediction correctness. Therefore artificial neural networks are sometimes very adaptive to self-correction, through adaptive evolution and repeated training cycles for updating and self-adjusting. I used sigmoid function generation based on genetic algorithm techniques to achieve best fitting and prediction results. http://en.wikipedia.org/wiki/Root_mean_square_deviation http://en.wikipedia.org/wiki/Mean_squared_error http://en.wikipedia.org/wiki/Backpropagation http://en.wikipedia.org/wiki/Backward_induction http://en.wikipedia.org/wiki/Regression_analysis
Cobbling various strategies together I think is a very strategy collection specific challenge. There isn't a formula to it, and anyone who tells you different is trying to sell you a correlation matrix. The trouble with correlation matrices is the same reason why we divide by N -1 to obtain variance - our sample is just not big enough to inform us about the universe with great precision. Common sense is a better VaR than VaR. If you got two mo-mo strategies, one on dollar and the other on stocks, no matter what your correlation calculator says, you gotta realize that under certain conditions those two things are going to move together. The worse kind of trap to be lured into is when you plug your super duper moving correlation matrix with phase shift probability estimates blah blah blah ... and conclude that you're an ATM spitting out nickles on sharpe 3.0, so you lever up because your VaR is tiny. Well .... you're set up for a "fat tail" clobbering. The key to survival in the market is to respect the fact that there is a vast unknown territory of possibilities out there and understand the risks you are running. Keep margins of safety. Be able to repair your portfolio assembly on the fly. Single number representation of risks are for the foolhardy and recently graduated.
A better phrase in my post above would have been: "on the same continuously detailed scale over time" If your base model is "explosively way off track", the specific risk evaluation to a strategy linked to the base is close to useless - and you need to resort to general risk based on not knowing, or better, resort to larger scales and timeframes. Maybe some type of "multi-scale risk evaluation" can offset this, or like TheStudent says - use several discrete risk components before taking an investment decision.
Just to reiterate what TSGann said and to further organize. Goal: define a methodology for combining multiple systems Objectives: a. identify what separates "a system" from multiple systems. 1. differences... commonalities 2. types of simple (if there is such a thing) b. define ways to combine systems 1. define end goal for combination 2. quantify (they don't have to be old) a) define assumptions of correlation b) reinvent correlation or c) sanity check: can we have a measure that's reliable? i. some of the time (regime switching) ii. d). lit. review on ltcm, sub-prime, etc. 3... Keep in mind, as I alluded to in response to one my first interviews ever "to know where I will be in ten years is to assume all of my oppurtunities, future values and joys are fixed and known... if I made this assumption, just think think how I could stifle this company's philosophy of innovation". In other words our goals and objectives are subject to change.
Agreed. The probability of a bad event with combined models having low correlation is never zero. However, if we can achieve a 3.0 Sharpe, we don't have to leverage up if we are satisfied with consistent returns, and aren't shooting for the moon. With low leverage, the fat tail event doesn't have to wipe us out. We may experience a higher than expected max drawdown, but, we'll stay in the game. When the tail happens, we learn from it, adjust our system(s) and if necessary our money management and keep trading until the next wierd thing happens. I also agree with the premise that the markets are changing and we have to continue to adapt. However, some latent dynamics stay around for quite a while and offer a relatively high degree of predictability/profitability. When several decent systems are combined, the fat tail doesn't go away, it just may happen less frequently. In the mean time, we keep our research going and adjust our systems and combos regardless of success/failure to stay on top of things. I think we can all agree it's not a free ride. Bottom line is that we still face the fat tail problem even with a super-duper single system, and by combining models in an appropriate fashion we can significantly improve our performance.