Frequency Analysis of MER

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From the Update 11/21/2008: The strongest frequency, according to classical frequency analysis, is an oscillation with a length of 16 trading sessions.

Frequency Analysis based on Monthly Comb

MER:  Frequency Analysis based on Monthly Comb

It is obvious from inspection of price lines, that many seem to contain cycles that may persist in repeating for a long time. Through the years, technicians have expended much effort to measure and exploit these cycles. It would appear simple enough to do a frequency analysis on the time series. But the cycles or periods that seem so easy to see in the price data, have proven to be surprisingly difficult to measure by classical methods. The classical methods apply to stationary cycles of exactly regular period. We can easily remove the trends from historical price data to make the series stationary. However, the fluctuations seen in financial data are almost never exactly periodic, a fact that has played havoc with most schemes to exploit the market cycles.

One factor in the behaviour of market fluctuations is "Anticipation". When speculators detect a cycle in prices, they will buy earlier before the peak, and sell earlier before the dip. This phenomena causes an increase in frequency, and eventually a breakup of the established cycle. Traditional frequency analysis methods such as the Fourier Transform, break down badly attempting to deal with these irregularities.

Wilder estimated that there was an important period of about 28 trading sessions. The half cycle of this period (14 sessions) is a default factor in many of the technical indicators. We have found this to be a reasonable figure, even in today's markets, but there is significant variation between issues. Some companies just seem to have faster or slower cycles for one reason or another, and the indicators may be more accurate when fitted to the situation.

New methods had to be invented to deal with the seeming mathematical absurdity in the notion of non-periodic cycles.




General Market Conditions

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More than 70 Hi-Rez charts daily for Merrill Lynch:

Frequency from Monthly Comb Filter
Representative Cycle (Trading Sessions) = 31.00
Sawtooth = 49.84
Peak TypeAvg. CycleMedian CycleCycle ModeCycle Amplitude
High16.5831.0029.0014.82 %
Low16.2931.0015.0018.21 %
Recent HighPriceRecent LowPrice Cycle
Tuesday, November 04, 200819.9932
Thursday, October 09, 200813.3219
Friday, September 19, 200829.5014
Friday, September 12, 200817.0517
Friday, August 29, 200827.8928
Tuesday, August 19, 200823.4316
Monday, July 28, 200823.9384
Tuesday, July 22, 200833.6255


Fourier Transform after Digital Filter M

Cycle LengthFourier Amplitude
90.78
140.53
160.49
200.32
230.31
260.28
320.22
290.21
350.16
410.16
390.13



Frequency Analysis based on Quarterly Comb

MER:  Frequency Analysis based on Quarterly Comb

Frequency from Quarterly Comb Filter
Representative Cycle (Trading Sessions) = 97.00
Sawtooth = 50.71
Peak TypeAvg. CycleMedian CycleCycle ModeCycle Amplitude
High48.0688.0070.0027.19 %
Low50.1484.0062.0024.11 %
Recent HighPriceRecent LowPrice Cycle
Friday, September 19, 200829.5097
Friday, March 28, 200838.8655
Friday, May 02, 200851.3063
Tuesday, January 08, 200846.47100
Friday, February 01, 200856.3481
Wednesday, August 15, 200765.68107
Friday, October 05, 200773.4597
Wednesday, March 14, 200775.14187


Fourier Transform after Digital Filter Q

Cycle LengthFourier Amplitude
160.26
2050.17
1780.16
200.15
490.14
850.13
240.13
1640.13
260.13
410.13
530.12



Frequency Analysis based on Yearly Comb

MER:  Frequency Analysis based on Yearly Comb

Representative Cycle (Trading Sessions) = 0.00
Sawtooth = 46.94
Peak TypeAvg. CycleMedian CycleCycle ModeCycle Amplitude
High0.000.000.000.00 %
Low0.000.000.000.00 %
Recent HighPriceRecent LowPrice Cycle
Wednesday, January 24, 200792.05202
Wednesday, June 14, 200661.73467
Tuesday, April 04, 200675.81273
Friday, August 06, 200444.86461
Friday, March 04, 200558.09251
Monday, October 07, 200226.83262
Friday, March 05, 200460.64540
Thursday, September 20, 200133.99525


Fourier Transform on Raw Closing Prices

Cycle LengthFourier Amplitude
81910.85
5856.12
4555.50
6834.93
2414.05
123.74
4103.71
2733.69
2933.49
1783.11
172.73




Breaking News:

Friday, March 19, 2010: We have news on L-3 Communications Holdings, Inc., ticker symbol LLL. Signs of an over-bought condition have become noticable. Also, there are breaking events concerning American Tower Corporation and TECO Energy, Inc..

From the News Archive: (3/18/2010 ) Bad news came from Allergan, Inc., Sempra Energy, MicroSoft, and Mylan Inc..



Other Companies
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Average Sawtooth based on Monthly Combo

Average Sawtooth based on Monthly Combo of Merrill Lynch

Average Sawtooth based on Bi-weekly Combo

MER:  Average Sawtooth based on Bi-weeklyly Combo



The Sawtooth statistic measures the time from each periodic high to the next periodic low, in comparison to the time from that low to the next high point . The mid-point, corresponding to a perfect sine wave, is 50 percent. Values less than this indicate the sell slope is steeper than the buy slope, while values over 50 percent show the opposite.

The Sawtooth Statistic gives a clue to the general mood of the markets, functioning as a sort of sentiment indicator. The chart represents the sawtooth percentage of (approximately) monthly cycles over the past year, and of the (approximately) bi-weekly cycles, averaged over the past six months.

.
Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25Signaled Appreciation minus non-signaled AppreciationAnnualized %
Buy Signals - 3 instances
Future30SesAvgClose 11.82 % minus 2.73 % = 9.08 %76.89 %
Future5SesAvgClose 4.72 % minus 0.55 % = 4.17 %211.87 %
Future2SesAvgClose 1.56 % minus 0.28 % = 1.28 %162.65 %
Sell Signals - 6 instances
Future30SesAvgClose 3.62 % minus 2.73 % = 0.88 %7.47 %
Future5SesAvgClose -0.02 % minus 0.55 % = -0.57 %-29.06 %
Future2SesAvgClose -0.15 % minus 0.28 % = -0.43 %-54.61 %
.
Semi-Annual Avg. of Bi-weekly Sawtooth Signals middle CrossingSignaled Appreciation minus non-signaled AppreciationAnnualized %
Buy Signals - 48 instances
Future30SesAvgClose 1.68 % minus 2.73 % = -1.05 %-8.93 %
Future5SesAvgClose 0.22 % minus 0.55 % = -0.33 %-16.90 %
Future2SesAvgClose -0.01 % minus 0.28 % = -0.29 %-36.57 %
Sell Signals - 49 instances
Future30SesAvgClose 2.82 % minus 2.73 % = 0.09 %0.73 %
Future5SesAvgClose 0.18 % minus 0.55 % = -0.37 %-19.03 %
Future2SesAvgClose 0.31 % minus 0.28 % = 0.03 %3.24 %
.
Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 70 and 30Signaled Appreciation minus non-signaled AppreciationAnnualized %
Buy Signals - 11 instances
Future30SesAvgClose 4.90 % minus 2.73 % = 2.17 %18.37 %
Future5SesAvgClose 1.35 % minus 0.55 % = 0.80 %40.64 %
Future2SesAvgClose 0.13 % minus 0.28 % = -0.15 %-18.91 %
Sell Signals - 16 instances
Future30SesAvgClose 7.87 % minus 2.73 % = 5.14 %43.51 %
Future5SesAvgClose 1.02 % minus 0.55 % = 0.47 %24.03 %
Future2SesAvgClose 0.27 % minus 0.28 % = -0.02 %-2.13 %
.
Projection of Representative Monthly CycleSignaled Appreciation minus non-signaled AppreciationAnnualized %
Buy Signals - 170 instances
Future30SesAvgClose 1.89 % minus 2.73 % = -0.84 %-7.12 %
Future5SesAvgClose 0.26 % minus 0.55 % = -0.29 %-14.83 %
Future2SesAvgClose 0.14 % minus 0.28 % = -0.14 %-17.85 %
Sell Signals - 183 instances
Future30SesAvgClose 2.93 % minus 2.73 % = 0.20 %1.66 %
Future5SesAvgClose 0.78 % minus 0.55 % = 0.23 %11.82 %
Future2SesAvgClose 0.48 % minus 0.28 % = 0.20 %25.37 %
.
Projection of BiWeekly Modal CyclesSignaled Appreciation minus non-signaled AppreciationAnnualized %
Buy Signals - 376 instances
Future30SesAvgClose 2.85 % minus 2.73 % = 0.11 %0.96 %
Future5SesAvgClose 0.66 % minus 0.55 % = 0.10 %5.31 %
Future2SesAvgClose 0.31 % minus 0.28 % = 0.03 %3.24 %
Sell Signals - 384 instances
Future30SesAvgClose 2.32 % minus 2.73 % = -0.41 %-3.47 %
Future5SesAvgClose 0.49 % minus 0.55 % = -0.06 %-3.24 %
Future2SesAvgClose 0.19 % minus 0.28 % = -0.09 %-11.95 %
.
The best buy signal for 30 future sessions was 9.08 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal for was -0.41 % from the Projection of BiWeekly Modal Cycles.
For 5 future sessions the best buy signal was 4.17 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal was -0.57 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25.
The best buy signal for 2 future sessions was 1.28 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25,and the best sell signal was -0.43 % from the Semi-Annual Avg. of Bi-weekly Sawtooth Signals at 75 and 25.



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