The paper was first published on Jan 21, We had posted a brief profile earlier. This is the complete work. Elliott1 wrote the wave principle in The enclosed research reworks the Mandelbrot Multifractal from a time cycle rather than trend perspective to prove that time fractal is more proportionate than the price fractal and is the real law of nature, which drives everything in nature.

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The paper was first published on Jan 21, We had posted a brief profile earlier. This is the complete work. Elliott1 wrote the wave principle in The enclosed research reworks the Mandelbrot Multifractal from a time cycle rather than trend perspective to prove that time fractal is more proportionate than the price fractal and is the real law of nature, which drives everything in nature. The case is validated by illustrating power law curves in time cycle periodicities.

Power law4 is seen across nature and in a diverse social trends. The power law in prices is a subject of extended study, but there has been no research attempt made to prove power law in time cycle periodicities. Testing cycle periodicity needs large historical data. Long-term time series are difficult to obtain and many emerging markets have seen stock market trading activity only start a decade back. Cycles are not conventionally believed to be patterns.

Patterns are understood either conventionally or as Elliott wave fractals. Even few Elliott wave practitioners have admitted the limitation of the Elliott Wave structure as being sharper on form than on time. The structure of the paper will be in following steps. This suggests that time cycles are fractals that showcase self-similarity with a factor of 3.

They are also more proportionate than price fractals. This hence is not a chance event but owing to time fractal nature. This means that if we isolate the Kitchin K cycle of months, which is widely witnessed, we could identify lower hierarchies i.

We kept in mind the cycle characteristics before isolating the K factor. A multifractal is formed by a curve pattern being repeated at smaller and smaller time scales. Mandelbrot used a 3 wave pattern, the first and last being in the direction of the general trend, the middle against the general trend. Mandelbrot multifractals focused on the price and not the time. This is the reason why price fractals and time fractals seem disconnected. We as a society can relate more to what we can see and feel.

Time is an underlying variable, which is tougher to relate compared to price. This is one reason why the debate regarding who saw it first, Elliott or Mandelbrot is inappropriate when we realize that time fractals are more proportionate than price fractals. Though Plummer comes close to the idea of a power law and self-similarity in cycles, he does not give a proof for the same.

Plummer also talks about the time aspect of the cycle along with the cycle pattern Fig. The cycle pattern is represented by the move from 0 to C i. It then consists of three lower-level sub- cycles, each of which itself contains the archetypal six-wave pattern. According to Plummer, each of these lower-level cycles will itself consist of three cycles. In other words, the cycles are nested within each other.

In all cases, significant lows can be expected to occur one-third and two-thirds along the time elapse of the next higher cycle that contains it. Similarly, important highs occur at one-sixth, one-half and five-sixths along the time elapse of that higher-level cycle.

In , Joseph Kitchin reported a short-term, three- to five-year, business cycle. There is a huge amount of evidence that the central periodicity of the short-term Kitchin cycle is somewhere between 40 and 44 months that is, somewhere between 3.

These periodicities can be found in prices. And we should see it across assets, and across any time series irrespective of the Y axis. Lack of long term data and the need for a workable investment strategy was another reason why we chose the K cycle as a workable time frame to break down.

Moreover, economic cycles research did not go below Kitchin, the very reason this study focused sub K level. The rate of change oscillator was used to illustrate the K cycle and the other K factors.

Pattern being the stronger of the two cycle characters. The focus was on identifying self similar nesting structures Fig. Care was also taken to identify cycle pattern distortions Fig. Just like price fractals, smaller time fractals are effected by larger time fractals which drive them. The very reason for translation Fig. Once cycles have been properly categorized in the K factor and sub K factors, the cycle periodicities can be used for forecasting purposes This on one side has seen a rise in trading volume, but at the same time made market relationships harder to understand.

Intermarket analysis coined by John Murphy13 has an increasing relevance in these times. Analysis of the stock market for example without consideration of existing trends in the dollar, bond and commodity markets are simply incomplete. Murphy suggests that financial markets can be used as a leading indicators of other markets and, at times, confirming indicators of related markets.

The writer of the study wanted to test time fractals on intermarket ratio between two assets, specially because they worked independent of price and were a good proxy to demonstrate fractal nature of time.

Murphy also talked about cyclicality between large asset classes like commodities and equities. This was nothing but larger time fractal K factors under action. However, intermarket analysis Fig. The Fig. The K factor is identified from the respective ratio line.

Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. A power-law distribution is also sometimes called a scale-free distribution. Because a power law is the only distribution that is the same irrespective of the scale.

This is also called as scale invariance. A closely related concept to scale invariance is self-similarity. In mathematics, a self-similar object is exactly or approximately similar to a part of itself i. The whole has the same shape as one or more of the parts. Self-similarity also means that any magnification would lead to a smaller piece of the object that is similar to the whole.

Many objects in the real world, such as coastlines, are statistically self-similar with all parts of them showing the same statistical properties at many scales. Self-similarity is a typical property of fractals.

Self similarity also appears in time cycles, as a large cycle encompasses smaller cycles, which in turn have smaller cycles nesting under them. The distribution is an exponential function, which takes a straight line form when we move on a logarithmic scale. The self similarity appeared in most cases. About 3 Kitchin cycles, nearing a decade of daily data was tested for the study. BRENT vs. WTM Brent vs. Midland GE vs. CAT General Electric vs. Caterpillar XOM vs. CVX Exxon vs. Chevron The above pairs were purposely chosen owing to their high and poor correlation.

All of the pair cycle periodicities depicted the underlying K factor hierarchy. Second part includes the distribution and tabulation of the time cycle periodicity of the respective pair.

The tables Table 1, Table 3, Table 5 carries the periodicities in days in column A. The calculations for B and C are enclosed. Third part Table 2, Table 4, Table 6 is the working of the long short strategy, where the author goes long on numerator A of the intermarket ratio under study while simultaneously selling the denominator B from the pair.

The entry number of days is the same as the time cycle periodicities carried on the second part of each working. The author has tested the strategies for an average 3 Kitchin cycles.

Underlying spot prices on the two assets making the pair are used. The strategy assumes a leverage factor of 1. The last column is the net annualized returns. A few important aspects linked with time fractals based strategies is that the fractal illustrates the performance cyclicality clearly. This was for all the time period under study. Even the other two pairs viz.

The time fractals based intermarket strategy delivers consistent returns on both the pairs. This proves that even a conventional underperformer or highly correlated assets can be traded against its sector leader performer or sector peers respectively, if the time fractal is isolated well.

This reinforces the idea of time fractal being better than the price fractal. All the three pair cycle periodicities show power law distributions. The exit number of days for the three pairs were 54, 50 and 66 in the same order.

The K factor indicator assists in this process. All strategies under study return positive gains. The intermarket ratio strategy introduced first time ever in this research redefines long-short technique as a time fractal strategy.

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This article originally appeared in the February issue of Scientific American. For a copy. We highly recommend a subscription. A Multifractal Walk Down Wall. Individual investors and professional stock and currency traders know better than ever that. Fortunes are.

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## A Multifractal Walk Down Wall Street

Mandelbrot, Scientific American , Feb. Portfolio theory is flawed. The customary theory holds that changes in prices follow a "random walk" that follows the normal distribution. Mandelbrot demonstrates convincingly that random deviations from a normal distribution do not characterize actual prices movements. With the customary theory, large fluctuations, e.

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## "A Multifractal Walk down Wall Street"

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