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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 pro-
ducing 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 forma-
tion, 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 out-
of-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 prof-
itable 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 sea-
sonal 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, com-
pared 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 dis-
placement, 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 dis-
placement 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 prof-
itability was observed out-of-sample, regardless of the order. The losses on a per-
trade 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 out-
of-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 prof-
its 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 num-
bers 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.

Tests of the Crossover Model with Confirmation
This model is identical to the basic crossover model discussed earlier, except that
entries were only taken if seasonal market behavior was contirmed by an appropri-
ate reading of the Fast %K Stochastic. Specifically, if the seasonal crossover sug-
gested a buy, the buy was only acted upon if the Fast %K was below 25%; i.e., the
market has to be declining or near a bottom (as would be expected on the basis of
the seasonal buy) before a buy can occur, Likewise, a seasonal sell was not taken
unless the market was near a top, as shown by a Fast %K greater than 75%. As
always, the standard exits were used. Entries were executed at the open (Test 7), on
a limit (Test S), or on a stop (Test 9).
Optimization for these tests involved stepping the length of the moving averages
(avglen) from 5 to 20 in increments of 5 (PI) and the displacement (disp) from 0 to
20 in increments of 1 (P2). For entry at the open, a moving average length of 15 and
a displacement of 7 were best. Entry on a limit was best with a length of 15 and a dis-
placement of 6. Entry on a stop required a length of 20 and a displacement of 9.
On a per-trade basis, the use of confirmation by the Fast %K worsened the
results for both sampling periods when entry was at the open or on a limit; with
both of these orders, trades lost heavily in both samples. When the confirmed sea-
sonal was implemented using a stop order for entry, relatively profitable perfor-
mance in both samples was seen. In-sample, the average trade pulled $846 from
the market, while the average trade pulled $1,677 out of the market in the verifi-
cation sample. In-sample, 41% of the trades were profitable and the annual return
was 5.8%. The statistical significance is not great, but at least it is better than
chance; both long and short positions were profitable. Out-of-sample, 44% of the
trades were winners, the return on account was 19.6%, and there was a better than
77% chance that the model was detecting a real market inefficiency; both longs
and shorts were profitable. Compared with other systems, the number of trades
taken was somewhat low. There were only 292 trades in-sample and 121 out-of-
sample. All in all, we again have a profitable model. Seasonality seems to have
validity as a principle for entering trades.
The equity curve for entry at open showed a gradual decline until May 1989.
It was then fairly flat until August 1993, when equity began to decline through the
remainder of the in-sample period and most of the out-of-sample period. The limit
order was somewhat similar, but the declines in equity were less steep. For the stop
order, the picture was vastly different. Equity declined at a rapid pace until May
1987. Then it began to rise at an accelerating rate, reaching a peak in June 1995,
the early part of the out-of-sample period. Thereafter, equity was close to flat. The
greatest gains in equity for the stop order occurred between June 1990 and May
1991, betweenMay 1993 and September 1993, and between January 1995 and June
1995. The last surge in equity was during the out-of-sample period.
Compared with the previous two models, there were fewer markets with con-
sistent profitability across all or most of the order types. Lumber was the only mar-
ket that showed profitability across all three order types both in- and
out-of-sample. Unleaded Gasoline yielded profits across all three order types in-
sample, and was highly profitable out-of-sample with entry at open or on stop.
Coffee and Cocoa were profitable for all order types in-sample, but profitable only
for the stop order out-of-sample. For the stop order, the NYFE, Silver, Palladium,
CHAPIFR 8 Seasanality 173




Light Crude, Lumber, and Coffee were highly profitable in both samples. Again,
there was no profitability for the wheats. There was enough consistency for this
model to be tradable using a stop order and focusing on appropriate markets.

Tests of the Crossover Model with Confirmatlon and
Inversions
This model is the same as the crossover model with confirmation. However, addi-
tional trades were taken where inversions may have occurred; i.e., if a seasonal
buy was signaled by a crossover, but the market was going up or near a top (as
indicated by the Fast %K being greater than 75%), a sell signal was posted. The
assumption with this model is that the usual seasonal cycle may have inverted or
flipped over, where a top is formed instead of a bottom. Likewise, if the crossover
signaled a sell, but the Fast %K indicated that the market was down by being less
than 25%, a buy was posted. These signals were issued in addition to those
described in the crossover with confirmation model. Entries were executed at the
open (Test IO), on a limit (Test 11). or on a stop (Test 12).
Optimization for these tests involved stepping the length of the moving
averages (avglen) from 5 to 20 in increments of 5 (PI) and the displacement
(disp) from 0 to 20 in increments of 1 (PZ). For entry at the open, a moving-
average length of 15 and a displacement of 2 were best. Entry on a limit was
best with a length of 20 and a displacement of 4. Entry on a stop again required
a length of 20 and a displacement of 9.
For entry at the open, the equity curve showed a smooth, severe decline from
one end of the sampling period to the other. There was a steeply declining equity
curve for the limit order, although the overall decline was roughly half that of the
one for entry at open. For entry on stop, the curve declined until May 1987. It then
became very choppy but essentially flat until August 1993. From August 1993
until June 1995, the curve steeply accelerated, but turned around and sharply
declined throughout the rest of the out-of-sample period. Adding the inversion ele-
ment was destructive to the crossover model. The seasonal patterns studied appar-
ently do not evidence frequent inversions, at least not insofar as is detectable by
the approach used.
Adding the inversion signals to the model dramatically worsened perfor-
mance on a per-trade basis across all entry orders, Losses were seen for every
combination of sample and order type, except for a very small profit with a stop
in the out-of-sample period. There were no markets that showed consistently good
trading across multiple order types in both samples, although there were isolated
instances of strong profits for a particular order type in certain markets. The NYFE
again traded very profitably with a stop order, as did IO-Year Notes, in both sam-
ples. Platinum and Silver also traded well in- and out-of-sample with a stop order.
Soybeans traded very profitably with a stop order in-sample, but only somewhat
profitably out-of-sample. There were consistent losses, or no profits at all, across
all orders and samples for the wheats.


Summary Analyses
Results from the different tests were combined into one table and two charts for
overall analysis. Table 8-3 provides performance numbers broken down by sample,
entry order, and model. For each model, there are two rows of numbers: The first
row contains the figures for annualized return-on-account, and the second row con-
tains the average dollar profit or loss per trade. The two rightmost columns contain
averages across all order types for the in-sample and out-of-sample performance.
The last two rows contain the average across all models for each type of order.
Of the three types of orders, it is evident that the stop performed best. The
limit and entry at open performed about equally well, but much worse than the stop.
In fact, for the stop, the average across all four models actually showed a positive
return and small profits on a per-trade basis both in-sample and out-of-sample.
When examining models averaged across all order types, the in-sample per-
formance was best using the crossover-with-confirmation model and worst for
the crossover-with-confirmation and inversion model. Out-of-sample, the basic
crossover model showed the best performance, while the crossover with confir-
mation and inversion had the worst.
As can be seen, in-sample the best performance was observed when the stop
order was used with the crossover-with-confirmation model. This is also the com-
bination of order and model that performed best out-of-sample.

8-3
TABLE

Performance of Seasonal Entry Models Broken Down by Model,
Order, and Sample
In earlier chapters, other types of entry models were tested and the limit
order was usually found to perform best. In the case of seasonality, the stop order
had a dramatic, beneficial impact on performance, despite the additional transac-
tion costs. Previously it appeared that countertrend principles may perform better
when combined with some trend-following or confirming element, such as an
entry on a stop. lt seems that, with seasonality, the kind of confirmation achieved
by using something like a stop is important, perhaps even more so than the kind
achieved using the Fast %K. In other words, if, based on seasonal patterns, the
market is expected to rise, confirmation that it is indeed rising should be obtained
before a trade is entered.
Overall, it seems that there is something important about seasonality: It has
a real influence on the markets, as evidenced by the seasonality-based system with
the stop performing better, or at least on par, with the best of the entry models.
This was one of the few profitable entry models tested. Seasonal phenomena seem
to have fairly strong effects on the markets, making such models definitely worth
further exploration.
It would be interesting to test a restriction of the seasonality model to mar-
kets on which it performed best or to markets that, for fundamental reasons, would
be expected to have strong seasonal behavior. From an examination of the market-
by-market analysis (discussed earlier in the context of individual tests), there is
quite a bit ofprimafacie evidence that certain markets are highly amenable to sea-
sonal trading. When breakout models were restricted to the currencies, dramatic
benefits resulted. Perhaps restricting seasonal models to appropriate markets
would be equally beneficial.
When looking over all the tests and counting the number of instances in
which there were significant, positive returns, an impression can be obtained of
the markets that lend themselves to various forms of seasonal trading. Across all
tests, Coffee had one of the highest number of positive returns in-sample and no
losses-true also for its out-of-sample performance. Coffee, therefore, appears
to be a good market to trade with a seasonal model, which makes sense since
Coffee is subject to weather damage during frost seasons, causing shortages and
thus major increases in price. For Coffee, I1 out of the 12 tests in-sample and 8
out of 12 tests out-of-sample were profitable. Unleaded Gasoline was another
commodity that had a large number of positive returns across all tests in both
samples: in-sample 8 and out-of-sample 11. Light Crude had 8 in-sample but
only 4 out-of-sample positive returns. Live Hogs was another market showing a
large number of positive returns in both samples.
Figure 8-l illustrates equity growth broken down by type of order and aver-
aged across all models. As can be seen, the stop order performed best, while entry
at open performed worst, and the limit order was in between the two.
Figure 8-2 shows equity growth broken down by model, with averaging done
over orders. The crossover-with-confirmation model had the overall best perfor-
mance, especially in the later half of the testing periods. The basic crossover model
FIGURE R-1

Equity Growth as a Function of Order Type




started out the best, but after 1990, performance deteriorated. However, the equity
trend seemed to reverse to the upside after 1995 in the out-of-sample period, a time
when the other models tended to have declining equity when considered across all
order types.

CONCLUSION
These explorations into seasonality have demonstrated that there are significant
seasonal effects to be found in the markets. Decisions about how to trade can be
made based on an examination of the behavior of the market at nearby dates for a
number of years in the past. The information contained on the same date (or a date
before or a date after) for a number of years in the past is useful in making a deter-
mination about what the market will do in the near future. Although the seasonal
effect is not sufficient to be really tradable on the whole portfolio, it is sufficient
to overcome transaction costs leading to some profits. For specific markets, how-
ever, even the simple models tested might be worth trading. In other words, sea-
sonal phenomena appear to be real and able to provide useful information. There
are times of the year when a market rises and times of the year when a market falls,
FIGURE 2-2

Equity Growth as a Function of Model




and models like those tested in the chapter can capture such seasonal ebbs and
flows in a potentially profitable manner.
Seasonality, as defined herein, has been demonstrated to be worthy of seri-
ous consideration. If the kinds of simple entry models illustrated above are elabo-
rated by adding confirmations and by using an exit better than the standard one,
some impressive trading results are likely to result.

WHAT HAVE WE LEARNED?

a Recurrent seasonal patterns appear to have real predictive validity and are
definitely worthy of further study.
n The usefulness of seasonal patterns for trading varies from market to

market, with certain markets being particularly amenable to seasonal trad-
ing. Trading a basket of seasonally reactive markets could be a highly
lucrative endeavor.
n To obtain the best results, raw seasonal information should be combined

with some form of confirmation or trend detection. Making use of additional
information can improve the performance of an unadorned seasonal model.
CHAPTER S


Lunar and Solar Rhythms




I n the previous chapter, seasonal rhythms were defined as recurrent phenomena
connected to the calendar. Seasonality involves the orbital position and tilt of the
earth in relation to the sun. Every year, on approximately the same date, the earth
reaches a similar phase in its movement around the sun. Other solar bodies gener-
ate similar rhythms, most notably the moon, with its recurrent phases, and
“sunspot” phenomena. In this chapter, market cycles determined by external forces
will again be analyzed, but this time the relationship between market behavior and
planetary and solar rhythms will be examined.

LEGITIMACY OR LUNACY?
Discussion of planetary influences conjures up images of what some regard as
astrological nonsense. However, it seems arrogant to dismiss, a priori, the™possible
infhrence of lunar, solar, and other such forces, simply because of the pervasive
belief that astrology (which Webster defines as “the pseudo-science which claims
to foretell the future by studying the supposed influence of the relative positions of
the moon, sun, and stars on human affairs”) is ludicrous. Such beliefs are often
based on the absence of knowledge. But what is already known about so-called
astrology, especially in the form of planetary and solar effects on earthly events?
Scientists have demonstrated some correlation between personality and plan-
etary positions. In one study (Mayo, White, and Eysenck, 1978), published in the
Journal of Social Psychology, researchers tested the claim that introversion and
extroversion are determined by the zodiac sign a person is born under. A personal-
ity test was given to 2,324 individuals to determine whether they were introverts or
extroverts. The hypothesis was that individuals born under the signs Aries, Gemini,
Leo, Libra, Sagittarius, and Aquarius were likely to be extroverts, while those under
Taurus, Cancer, Virgo, Scorpio, Capricorn, and Pisces, tend to be introverts, The
results of the study were significantly consistent with the hypothesis, as well as
with other related studies (Gauquelin, Gauquelin, and Eysenck, 1979).
It is also known that the moon™s gravitational force influences the aquatic,

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