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set of markets than the in-sample behavior of the model. This does not mean that the out-of-sample performance was bad while the in-sample performance was good, but rather that most markets simply did not trade in one sample if they did in the other. The low number of trades observed with the genetic models was due to the specific namre of the particular rule templates and the ways in which the individual rules were combined to obtain buy and sell signals. With some changes in the rule templates, especially in the number of rules used and in how they are combined, the pattern of rare event trading can be entirely altered. There were times when the preferred kind of model was not available for a given market. In such cases, models were examined that performed poorly on a whole-portfolio basis, but that did trade one or two difficult markets acceptably. For example, the RSI overbought/oversold model with entry on limit was a poor performer on a portfolio-wide basis. However, this model traded Gold and Silver reasonably well. It pulled returns of 27.3 and 3.9%, annualized, on the in-sample data, with average trades of $9,446 and $4,164, respectively. Out-of-sample, the system pulled 23.6% out of the Gold and 51.7% out of the Silver markets, with average trades yielding $12,194 and $24,890, respectively. One of the large neural networks that appeared to be highly over-optimized was used for the three wheat markets-markets that did not trade at a statistically
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significant level with any of the other models. The large, long-side, turning-point network with entry on limit, however, had high statistical significance when trading each of the wheats, pulling more than 40% annually from each, and more than $15,000 per trade. The amazing thing is that, out-of-sample, despite the size of the net and the degree of curve-fitting seen on its portfolio performance, the model pulled in no less than 24%, with $5,000 per trade, from each of the wheats. The cycle model, which worked well on hardly any market, did trade the S&P 500 profitably-returning 15.3%, with an average in-sample trade of $4,613, and 21.4% with $4,698-per-trade profit out-of-sample. It should be noted that a cycle model was found to trade the S&P-500 successfully in the tests reported in our earlier study (Katz and McCormick, May 1997). Once each market was paired with a good model-order combination, the performance data were analyzed, both in- and out-of-sample, for each of the markets. An equity curve was prepared that covered both periods (see Figure C-l). Returns and statistical significance were calculated for the multiple-model portfolio, both in-sample and out-of-sample. It was surprising to discover that the out-of-sample performance data revealed a return-on-account of 625% annualized! A manifestation of the Holy Grail Because model-market combinations were selected on the basis of their in-sample statistical significance, the 544% annualized in-sample return was not unexpected. The probability of obtaining an in-sample profit as large as that is less than 1 in 3,000,000,000,000,000,000 (i.e., 3 X lo *). Even if massive amounts of optimization, with tests of tens of thousands of combinations, took place, the results would still be extremely significant, in a statistical sense. Out-of-sample, the probability of finding a risk-to-reward ratio or annualized return as good as that observed is less than 1 in 40 million. Again, even corrected for extensive optimization, the results would still be of extreme statistical significance. In fact, no out-of-sample optimization took place. In-sample, the systems were only optimized on the entire portfolio. Model parameters were never adjusted for the selected markets on which the models were to be traded. And only the minimal standard exit strategy was used. Performance could be very substantially improved using the best of the exits found in Part III. These findings demonstrate that while most systems do not work and most tests show losses, a sufficiently extensive search (as conducted in this book) can discover enough that do work to put together a portfolio trading strategy capable of producing nothing less than stellar results.
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