1. Introduction
This is an abridged version of a study that was conducted in 1994. The purpose of this paper is to derive a cycle relating the lunar cycle to an equity average. This cycle will then be evaluated for its profitability versus a buy-and-hold strategy. The results may be of interest to short-term traders with an interest in cyclic analysis.

Those who seek a causative link might consider the following. Serotonin is the substance in the brain of a homing pigeon that sensitizes the bird to the earth's magnetic field, allowing the pigeon to 'home in.' The field itself has been shown to fluctuate with lunar and solar influences. Nelson's work demonstrates a relationship between all of the planets and solar activity. Serotonin exists in the human body. The substance was neglected until biotechnology companies recently took an interest. Perhaps this is the link.

The Link to Markets
In John Murphy's text, Technical Analysis of the Futures Markets, he writes, "There is another important short-term cycle that tends to influence most commodity markets- the 28-day trading cycle. In other words, most markets have a tendency to form a trading low every 4 weeks. One possible explanation for this strong cyclic tendency throughout all commodity markets is the lunar cycle. Burton Pugh studied the 28-day cycle in the wheat market in the 1930s and concluded that the moon had some influence on market turning points. His theory was that wheat should be bought on a full moon and sold on a new moon. Pugh acknowledged, however, that the lunar effects were mild and could be overriden by the effects of longer cycles or important news events."

John McGinley, writing in Technical Trends, once mentioned that Arthur Merrill conducted a study of market behavior around full and new moons and found no strong correlation. More recently, the February 27,1994 issue of Mark Liebovit's Volume Reversal Survey stated that he had noted a correlation between the lunar cycle and Federal Reserve actions. Chris Carolan, noted for his work with the Spiral Calendar, has achieved some success with a lunar-based forecasting system. Indeed, many older societies utilize a lunar calendar. Our own calendar year is based upon the movement of the earth around the sun. Those technicians who rely upon the annual cycle (the average percentage change in the DJIA from January 1 to December 31) are looking at an astronomically based cycle.

2. Methodology
The lunar cycle is defined by astronomers by the period beginning and ending with the conjunction of the sun and the moon. The two bodies are conjunct when they are zero degrees apart. The faster moon then races ahead of the sun, makes a 360-degree arc, and then conjoins the sun again, completing a cycle. This process takes a mean time of 29 days, 12 hours, 44 minutes, and 2.78 seconds. This period may vary by as much as 13 hours.

The cycle study is conducted through a series of steps:

1. A list of dates of all lunar cycles from 1915 through 1994 was calculated. See table 1 as an example.

2. This database of dates is then instructed to access a daily DJIA quotes from the price database. The program then selected the DJIA price on the day of the first new moon in 1915. In the next cell, the DJIA for the following day was inserted, and so on, through to the day of the next new moon. The PC used the Friday close when it encountered a weekend. The result was a row of prices. This process was repeated for each year, 1915 through 1994. There are about 13 such cycles per year. The sample size was over 1,000 cycles.

3. The resultant array of prices was smoothed.

4. Individual cycles were then combined to obtain a composite cycle through vector addition. This depicts the average percent change in the DJIA from new moon to new moon from 1915 through 1994.

5. Fourier least squares approximation was utilized to determine the equation of the line of this cycle. This cycle line can be projected backward or forward. The result is graph 1.

6. This cycle line was tested versus a buy-and-hold strategy from 1960 through 1993 to determine its predictive value.

3. Discussion of the Results
Graph 1 summarizes the results. The horizontal axis represents the 29-day cycle. The gradations denoted by the dashed vertical lines are 10% of the cycle, or 2.9 days. The vertical axis represents the average percentage price change in the DJIA. For example, the DJIA has risen an average of 0.1% from the new moon to the cycle peak about 7 days later. The DJIA has then dropped 0.22% from this cycle peak to the cycle trough.

Graph 1 reveals that the DJIA has, on average, risen from the new moon for about 7 days. The DJIA then has bottomed about 4 days before the next new moon. The price slide seems to accelerate after the occurrence of the full moon. (This would explain why Arthur Merrill did not find turning points near the actual lunations; the top and bottom of the cycle tend top fall between the two phenomena.)

Graphs 2 through 9 depict the same relationship broken into time segments. Graph 3 shows the same relationship from 1915 to 1920 only. Graph 4 represents the cycle for the decade 1920 to 1930 only. Graphs 5 through 9 depict the cycle by decade through 1990. The period 1920-1930 (graph 3) shows the greatest difference from the average in graph 1. The 1960 decade in graph 7 is similar to the average, but shows a higher peak 1 to 2 days after the full moon. In the 1970s (graph 8) the cycle bottom occurred much earlier then in the average cycle in graph 1. In the remaining decades, the relationship was fairly consistent with the average overall cycle. The cycle in the 1980s was consistent with average.

Graph 10 is the same study applied to the S&P 500 from 1950 through 1994. The shape of the curve is roughly the same. This study was added to demonstrate that there was little variation in the effect of the cycle in relationship to these two popular averages.

Buy and Sell Test Versus a Buy and Hold Strategy
The cycle was tested as a short-term timing aid. The program was instructed to buy the DJIA at every cycle bottom and to sell (move to cash) at every cycle high. The program bought at every "long" arrow (marked by 'lo' on the graph).

Table 2 summarizes the results for 1993.

The test began in 1960 and concluded with 1993. The yearly results depicted in table 2 are summarized in annual form in table 3.

1. The signals derived from the cycle turned an initial $1,000 into $2,138. A buy-and-hold strategy returned $5,526.

2. Of the 421 buy signals, 228 or 54% were profitable.

3. Trading by the cycle exceeded the buy-and-hold in 11 of the 34 years tested.

4. Cycle trading yielded the best returns in 1987 (21% versus 2.9%) and 1988 (15.4% versus 11.9%).

5. Cycle trading returns were poorest in 1973 (23% loss versus a 16.6% loss).

Three Attempts to Improve the Results
Attempts were made to improve the batting average of the cycle by confirming the buy signals with a 14-day oscillator such as an RSI or a stochastic. These tests did not significantly improve the results.

1. This strategy underperformed both the first strategy and the buy-and-hold strategy. It returned only $1,416.

2. Of the 420 buy signals, 239 or 57% were profitable.

3. Trading by the cycle exceeded the buy-and-hold in 11 of the 34 years tested.

4. Cycle trading yielded the best return in 1975 (19%), but underperformed a buy-and-hold (38.3%).

5. Cycle trading returns were poorest in 1990 (18% loss versus a 4.5% loss).

One more attempt was made to improve the results. The buy-sell test was repeated as in the first test. That is, the cycle lows were bought and the cycle highs were sold. However, this time the buy signals were accepted only if the annual cycle pointed up.

The annual change in the DJIA was computed on a daily basis. (The annual cycle is based upon the calendar, which is derived from the relationship of the earth and the sun. So, a solar cycle was calculated. The methodology for the determination of the annual cycle was the same as that for the lunar cycle.) The relationship is shown as graph 11, and will likely be familiar to any technician who employs the seasonal cycle. This cycle rises, on average, in the following time periods every year:

Jan. 26-Feb. 9
Feb. 23-March 12
April 1-18
May 28-June 12
June 24-July 15
July 29-Sept. 5
Sept. 30-Oct. 5
Oct. 26-Nov. 6
Nov.24- Dec.3
Dec.18- Jan.11

So a lunar cycle buy signal was accepted if it fell in one of these time periods. These were times when both the lunar and the annual cycle pointed up. Buy signals that fell 1 day before any of the above time periods were accepted. I felt that the annual cycle upturn only 1 day later would be sufficient reason to initiate a long position. Buy signals that occurred 1 day before the end of any of these time periods were rejected. This was done because the shorter lunar cycle would have to 'swim upstream' versus the stronger annual cycle which was only 1 day away from topping. One possible criticism is that the annual cycle may have had a different shape in the 1960s or the 1970s. This would then change the time periods above. But seasonality appears to be consistent enough, especially in the post-WW2 years, that the analysis was conducted.

The results did not enhance the trading record. The number of trades dropped from 420 to 182. The number of profitable trades was 102, or 56% of the total. The theoretical portfolio of $1,000 increased to only $1,875.

DJIA Highs and Lows in Relation to the Cycle
Another test was devised in order to determine if there is any consistency to the cycle. A list of highs was generated utilizing a 10% filter rule from Arthur Merrill's books, Behavior of Prices on Wall Street and Filtered Waves. That is, all moves of less than 10% were filtered out of the DJIA from 1885 through 1994. This produced a list of 249 highs and lows. These dates were then sorted to determine where they occurred in the 29-day cycle. For these purposes, the cycle was divided into its 8 astronomic phases as in chart 1. These 8 divisions are marked on the cycle graph as 8 vertical solid lines in graph 12. The name of the phase appears at the bottom of the graph. The percentages represent the percent of 10% filter highs that fell in that phase historically. For example, 8.9% of all highs determined by the 10% filter method from 1885 to 1994 fell in the new moon phase.

If the highs were evenly distributed, one would expect an average of 12.5% of the highs to fall in any one phase. If the cycle is indeed operative, then the highs would tend to cluster around the cycle high, the crescent, 1st quarter, and gibbous phases. Fewer cycle highs would be anticipated at the cycle bottom, the 3rd quarter phase.

The results reveal a somewhat higher probability for 10% highs in the crescent and gibbous phases (2 of the 3 phases around the cycle top) and a lower probability of highs in the cycle bottom, or 3rd quarter phase.

This process was repeated for 10% lows (see graph 13). Few lows (16.8%) fell in the 2 phases around the projected cycle high. Most of the lows (29.6%) fell in the last 2 phases, near the projected cycle low.

The same test was conducted for a 5% filter set of highs and lows from 1885 to 1994. This produced 851 turning points. This was done because the 29-day cycle is a short one, and the use of a 10% filter produced an average of only 2.5 turning points per year. Graph 14 depicts the distribution of 5% highs. There has been a greater percentage of highs in the second, third, and fourth (crescent, 1st quarter, gibbous) phases, the high phases of the cycle line. Graph 15 demonstrates the same graph for the 5% lows. This gave a less definitive picture of than did that for the highs. The lows tended to be somewhat more evenly distributed than the highs. There tended to be more lows in the crescent and the balsamic phases, the latter phase being the bottom in the cycle line.

Big One-Day Rises and Declines in Relation to the Cycle
A list of the 100 largest one-day rises and the 100 largest declines (in terms of DJIA points and in terms of percentage change) was obtained from Delafield, Harvey, and Tabell. The list was updated before this study, so the total numbers 102 in each case. As with the highs and the lows in the previous test, the lunar phase in which these changes occurred was determined. The results are plotted in graphs 16 and 17 at the bottom of the graph, in the same fashion as that for the highs and lows in graphs 14 and 15.

The greatest percentage of rises occurred in the crescent and full moon phases. The cycle is rising in the crescent phase, so the large number of increases here is in agreement with the cycle. The large number in the full moon phase differs from the cycle, which is declining in that phase. Perhaps this reflects the "blow off" nature of tops.

The distribution of 1-day declines was more closely in agreement with the cycle line. Most of the drops (35.3%) fell in the disseminating and 3rd quarter phase, at the bottom of the cycle. Perhaps this reflects the occurrence of selling climaxes at lows.

This analysis was repeated utilizing the 100 biggest up and down days in terms of percentage change, rather than points. This method yields many days in the 1930s. The points method yields many days in the 1980s and the 1990s. The results are plotted in graphs 15 and 16 at the top of each graph.

The percentage method reveals many more big up days in the first 3 phases, more in line with the cycle graph. The biggest difference was in the full moon phase where the percent of big 1-day moves fell from 20.6% to 9.0%.

In terms of the percentage of 1-day declines, the major difference was, again, in the full moon phase where the percentage almost doubled.

Support from a Previous Study
Frank Guarino conducted a study entitled Relationship of the Stock Market to the Lunar Cycle as a requirement for an MBA at Pace University in the late 1970s. Guarino tested the rate of change in the DJIA between lunar phases from 1950 through 1973.

He also computed the number of price increases and decreases and the averages of these changes between phases. The actual rates of change were also compared to the average rates of change. The ranges and the average deviations were also calculated. The average deviation was the arithmetic mean of the absolute values of the deviations of the rates of change from the arithmetic mean.

The findings were:

1. The period from the balsamic (last) phase to the new moon (first) phase showed the largest average rate of increase and the smallest average rate of decrease. The period from the full moon to the balsamic (last) phase had the largest average rate of decrease and the smallest average rate of increase.

2. The highest average rates of change occurred in the 2 phases around the new moon.

3. DJIA increases (in terms of the number of increases) were more prevalent between the new moon phase and the 1st quarter phase.

4. Analysis of the ranges revealed that the period between the 1st quarter and the full moon had the widest limits. This period also had the largest average deviation. The period from the 3rd quarter to the new moon had the most narrow range and the smallest average deviation.

Guarino concluded that the period from the full to the balsamic phase (from the 50 gradation through the 100 gradation on graphs of the cycle) were the least favorable for the trader who is long. The period beginning with the balsamic and new moon phases (90 and 0 gradations on the graphs) is the most favorable for the bull.

This study, conducted along different lines and for a much shorter time period, supports the relationship that has been demonstrated in Graph 1. Note that the highest average rates of change and the largest number of price increases fell in the phases at the bottom of the derived cycle. Also, prices tended to have the smallest deviation around the cycle bottom and the highest around the cycle top. In other words, price action at cycle bottoms was more descriptive of a bottoming or basing process. The Guarino numbers show that prices fluctuate more around the projected cycle top. Volatility is known to increase around market tops.

4. Summary of Findings
The study indicates that there is, on average, an upmove in the DJIA commencing in the days prior to the new moon and ending about 6 to 7 days afterward. The breakdown of the cycle by decade demonstrates this as does the Guarino study. In addition, the buy-and-sell tests show that buying the lows outpeforms buying the highs. Whereas the 'batting average' of profitable trades did not decrease when the highs were used as buy points, the magnitude of the profits shrank while the magnitude of the losses grew.

This cycle is too weak to be relied upon solely as a trading timer. The buy-and-sell study shows that such a strategy does not keep pace with a simple buy-and-hold strategy. Only 54% of the purchases timed by the cycle were profitable. This percentage is approximately in line with the percentage of rising days (52%) in the DJIA as calculated by Arthur Merrill. The two percentages are not comparable on an apples-for-apples basis, but the scant excess of the cycle-generated trades over 52% does not seem encouraging. Methods designed to enhance the returns did not succeed. The addition of technical oscillators as a confirming mechanism did not improve the results, nor did selling cycle highs or confirming buys with the annual cycle. Traders who are tempted by the sale of such trading systems are advised to think twice before purchasing any system based upon this one cycle. These findings should not discourage further attempts to link price series cycles to phenomena outside of the marketplace.

OEX traders with short-term time horizons who rely upon cycles may wish to take note of certain findings. By itself, the lunar cycle does not outperform. But the DJIA does demonstrate more upside volatility from the phase prior to the new moon to the phase immediately after. There also is a slight tendency for the DJIA to show more downside volatilty after the cycle peak. This may be useful knowledge to shorter-term players who employ leverage.