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Tests are performed on two seasonality-based entry models: the crossover model (both with and without confirmation and inversions) and the momentum model. Each model is examined using the usual three entry orders: enter at open, on limit, and on stop. Tables 8-l and 8-2 provide information on the specific commodities that the model traded profitably, and those that lost, for the in-sample (Table 8-l) and outof-sample (Table 8-2) runs. The SYM column represents the market being studied. The rightmost column (COUNT) contains the number of profitable tests for a given market. The numbers in the first row represent test identifiers. The last row (COUNT) contains the number of markets on which a given model was profitable. The data in these tables provide relatively detailed information about which markets are and are not profitable when traded by each of the models: One dash (-) indicates a moderate loss per trade, i.e., $2,000 to $4,ooO, two dashes (--) represent a large loss per trade, i.e., $4,ooO or more; one plus sign (+) means a moderate profit per trade, i.e., $1,000 to $2,000, two pluses (+ +) indicate a large gain per trade, i.e., $2,000 or more; and a blank cell means that the loss was between $0 and $1,999 or the profit was between $0 and $1,000 per trade. (For information about the various markets and their symbols, see Table II-1 in the Introduction to Part II.)
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Tests of the Basic Crossover Model
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A simple moving average (maI) of a specified length (av&n) was computed for the integrated, price-like seasonality series. A second moving average (ma2) was taken of the first moving average. A buy signal was generated when ma1 crossed above ma2. A sell signal was generated when maI crossed below ma2. (This is the same movingaverage crossover model discussed in the chapter on moving averages, except here it is computed on a predicted, seasonal series, rather than on prices.) The entries were effected by either a market at open (Test l), a limit (Test 2), or a stop order (Test 3). Optimization for these tests involved stepping the length of the moving averages (avglen) from 5 to 20 in increments of 5 and the displacement (disp) from 0 to 20 in increments of 1. For entry at the open, the best performance (in terms of in-sample risk-to-reward ratio) was achieved with a moving average length of 20 and a displacement of 5. Entry on a limit was best with a length of 20 and a dis-
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T A B L E
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In-Sample Performance Broken Down by Test and Market
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placement of 8. Entry on a stop required a length of 20 and a displacement of 6. The model seemed to prefer longer moving averages (more smoothing), and somewhat earlier signals were required by the limit order when compared with the stop or market-at-open orders. In-sample, none of the results were profitable when both long and short positions were traded. However, the losses were much smaller on a per-trade basis than those observed in many of the other tests presented in earlier chapters. The stop order performed best. The market-at-open order performed worst. The limit order came in not too far behind the stop. For both the stop and limit orders, trading was
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Out-of-Sample Performance Broken Down by Test and Market
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actually profitable for long positions. In all cases, ZO-bar moving averages provided the best results, but the displacement varied depending on the order. A look-ahead of 5 bars was optimal for entry at tbe open, 8 bars for entry on a limit, and 6 bars for entry on a stop. This makes sense, in that one would want to post a limit order earlier than a market-on-open order so as to give the limit some time to be filled. Out-of-sample, the results showed the same ordering of overall performance as measured by the average profit per trade ($TRD), with the stop order actually producing a positive return of $576 per trade, representing an 8.3% annual retam on account; this system, although not great, was actually profitable on recent data. For
the stop entry, both in- and out-of-sample trading was profitable when only the long trades were considered, but the short side lost in both samples. This is a pattern that has been observed a number of times in the various tests. The percentage of wins for all orders and samples was between 40 and 43%. It is interesting that, even for the losing variations, the losses here were much smaller than what seems typical for many of the models so far tested. With entry at the open, equity declined until November 1988. It retraced approximately 50% of the way until July 1989, making a small U-shaped formation, with the second of a double-top around November 1990. Equity then declined rather steeply until November 1992 and, in a more choppy fashion, throughout tbe remainder of the in-sample period and the first third of the out-of-sample period. The decline ended in April 1996 when equity gradually climbed throughout the remainder of the out-of-sample period. With entry on a limit, equity was fairly flat until January 1987, rose very rapidly to a peak in May 1987, and then declined until November 1992. From then through July 1994, equity rose steeply. Afterward, choppy performance was observed with no significant trend. The stop order produced strong equity growth until June 1988. Equity then declined in a choppy fashion through most of the in-sample period and about the first quarter of the out-of-sample period. It reached bottom in December 1995 and then rose sharply through the end of the out-of-sample period in February 1999. Across all three entry orders, the best-performing market was Unleaded Gasoline, in which strong, consistent profits were observed in both samples. Palladium was also a strong market for this model: Both the entry at open and entry on limit produced profits in- and out-of-sample, with the entry on limit having strong profits in both samples, and the entry on stop producing strong profits outof-sample and neutral results in-sample. Live Hogs was another good market to trade seasonally: Every order type yielded profits in-sample, while two of the three order types yielded profits out-of-sample; both the limit and stop orders were profitable in both samples. Yet another good market to trade with this model is Coffee: All three orders produced in-sample profits, while the market at open and stop orders produced strong profits out-of-sample. Finally, Cotton did not do too badly: The stop order yielded strong profits in both samples, and no order resulted in strong losses in either sample. Finding good performance for Unleaded Gasoline is totally in accord with expectations. What is moderately surprising is that Heating Oil, for which there is a strong seasonal demand characteristic, was only profitable in both samples when using the limit order. Coffee also traditionally has strong seasonal patterns caused by, e.g., recurrent frosts that damage crops, create shortages, and drive up prices. Surprisingly, the wheats did not produce many profits in-sample. The only exception was a small profit for Minnesota Wheat with a limit order. More profits in the wheat group were seen out-of-sample, where the limit order led to profitable trading in all three wheat markets and the stop order in Kansas Wheat.
A number of other markets showed profitable, but less consistent, trading across sample and order type. Again, it is impressive to see the great number of markets that trade well across both samples, especially when compared with many of the other models that have been tested in the preceding chapters. It is also interesting that there is a discrepancy between the performance of the seasonality model in the current tests and in our earlier tests of the S&P 500 (Katz and McCormick, April 1997). The differences are probably due to such factors as tuning. In the earlier tests, the moving averages were specifically toned to the S&P 500; in the current tests, they were tuned to the entire portfolio. Moreover, compared with other markets, the seasonal behavior of the S&P 500 appears to involve fairly rapid movements and, therefore, requires a much shorter moving average to achieve optimal results. Finally, the earlier tests did not use separate exits and so a seasonal trend lasting several weeks could be captured. In the current test, only the first 10 days could be captured, after which time the standard exit closes out the trade. It is likely that the performance observed, not just on the S&P 500, but on all the markets in the portfolio, would be better if the standard exit were replaced with an exit capable of holding onto a sustained trend. Tests of the Basic Momentum Model
For the momentum model, the unintegrated seasonal price change series was smoothed with a centered simple moving average of a specified length (avglen). The centered average introduces no lag because it examines as many future data points, relative to the current bar, as it does past data points. The use of a centered moving average is legitimate, because the seasonality estimate at the current bar is based on data that is at least 1 year away. For this series of smoothed seasonal price changes, a series of average absolute deviations was computed: To produce the desired result, the absolute value for each bar in the smoothed seasonal series was computed and a loo-bar simple moving average was taken. A buy signal was issued if the seasonal momentum, at the current bar plus some displacement (disp), was greater than some multiple (thresh) of the average absolute deviation of the seasonal momentum. A sell signal was issued if the seasonal momentum, at the same displaced bar, was less than minus the same multiple of the average absolute deviation. Entries were executed at the open (Test 4), on a limit (Test 5), or on a stop (Test 6). Optimization was carried out for the length of the moving averages, the displacement, and the threshold. The length was stepped from 5 to 15 in increments of 5; the displacement from 1 to 10 in steps of 1; and the threshold from 1.5 to 2.5 in increments of 0.5. The best in-sample performance was observed with a length of 15 and a threshold of 2.5, regardless of entry order. For the market at open and for the stop a displacement of 2 was required. The limit worked best with a displacement of 1. In agreement with expectations, these displacements are much
smaller than those that were optimal for the crossover model where there was a need to compensate for the lag associated with the moving averages. Overall, the results were much poorer than for the seasonal crossover model. In-sample, profitability was only observed with the stop order. No profitability was observed out-of-sample, regardless of the order. The losses on a pertrade basis were quite heavy. Interestingly, the long side performed less well overall than the short side. This reverses the usual pattern of better-performing long trades than short ones. With both entry at open and on limit, equity declined in a choppy fashion from the beginning of the in-sample period through the end of the out-of-sample period. The decline was less steep with the limit order than with entry at open. With the stop order, equity was choppy, but basically flat, until May 1990, when it began a very steep ascent, reaching a peak in September 1990. Equity then showed steady erosion through the remaining half of the in-sample and most of the out-of-sample periods. The curve flattened out, but remained choppy, after April 1997. The model traded the T-Bonds, IO-Year Notes, Japanese Yen, Light Crude, Heating Oil, Unleaded Gasoline, Live Hogs, and Coffee fairly well both in- and outof-sample. For example, T-Bonds and lo-Year Notes were very profitable in both samples when using either the limit or stop orders. JapaneseYen performed best with the stop, showing heavy profits in both samples, but was profitable with the other orders as well. The same was true for Light Crude. Heating Oil showed heavy profits with the entry at open and on stop in both samples, but not with entry on limit, This was also true for Unleaded Gasoline. Live Hogs performed best, showing heavy profits in both samples, with entry at open or on stop. This market was profitable with all three orders in both samples. Coffee was also profitable with all three orders on both samples, with the strongest and most consistent profits being with the stop. For the most part, this model showed losses on the wheats. In-sample, 15 markets were profitable to one degree or another using a stop, 13 using a market order at the open, and 9 using a limit. Out-of-sample, the numbers were 15, 16, and 16, respectively, for the stop, open, and limit. Even though the momentum model performed more poorly on the entire portfolio, it performed better on a larger number of individual markets than did the crossover model. The momentum model, if traded on markets with appropriate seasonal behavior, can produce good results.
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