The paper was first published on Jan 21, 2009. We had posted a brief profile earlier. This is the complete work.

Ralph N. Elliott1 wrote the wave principle in 1938. In 1975 Benoit B Mandelbrot2 coined the term fractal3 and in 1982 published his ideas in ‘The Fractal Geometry of Nature’. The book brought fractals into the mainstream of professional and popular mathematics. In February 1999, Benoit Mandelbrot submitted an article to Scientific American called ”A Multifractal Walk down Wall Street.” In the article, he discussed how fractal geometry can be used to model the stock market curves. 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. The continued prosperity after 1980’s was a reason why time fractals did not get researchers attention, unlike price fractal which was actively studied and researched. The fact that what we can see is what we can relate too more also made researchers focus more on price than time, which is less visible. 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 more sharp on form than on time. These were few reasons why time time fractals remained unproven. This study further connects its findings with the existing research on various economic cycles finally extending the proof to a long – short intermarket strategy on an asset pair.

The structure of the paper will be in following steps .

1: Cycles underlie fractals and Mandelbrot’s multifractals can be redrawn from a cycle perspective. This suggests that time cycles are fractals that showcase self similarity with a factor of 3. They are also more proportionate than price fractals.

2: The above mathematical proportion X, X/3, X/9, X/27…. can also be seen in the economic group of cycles (Fig. 1) viz. William Strauss and Neil Howe5, Brian Berry6, Clement Juglar7 and Joseph Kitchin8, which are also connected by 3. This hence is not a chance event but owing to time fractal nature. This means that if we isolate the Kitchin (K) cycle of 40-44 months, which is widely witnessed, we could identify lower hierarchies i.e. K/3, K/9, K/27 etc. We kept in mind the cycle characteristics before isolating the K factor.

3: Now if we assume that time fractals can be isolated, similar cycle periodicities can even be witnessed and isolated in intermarket ratio lines, which are independent of price.

4: We tabulate the cycle periodicity and test it for power law distributions.

5: We test the K/9 time fractal periodicities on intermarket9 ratio line on two assets though a long-short strategy.

**TIME FRACTALS VS. PRICE FRACTALS**

The term fractal, as Mandelbrot defined it, refers to a curve in which distinct parts are smaller scales of the whole curve. 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. A picture of his example from the article “A multifractal walk down wall street” is illustrated in Fig 2. 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. If one redraws Mandelbrot’s multifractals from a time cycle (up leg and down leg) rather than from a price trend (up leg - down leg - up leg) perspective, the same multifractals emerge out as time fractals Fig 3. Not only the iterations break up in the same proportion X, X/3, X/9, X/27 but the time fractals also are more homogeneous that the price multifractals in Fig.2.

**THE KITCHIN CYCLE ( THE K FACTOR)**

Tony Plummer10 in his book ‘Forecasting financial markets’ does give reference to time cycles as a triad of patterns11. 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. He mentions that “if the triad theory is correct, then the pattern should repeat themselves in a fractal like fashion across all genuine cyclical time”. Plummer also talks about the time aspect of the cycle along with the cycle pattern (Fig.4). The cycle pattern is represented by the move from 0 to C i.e.., 1–2–3 up and A–B–C down. 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 1923, 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 3.33 and 3.67 years. These periodicities can be found in prices.

So if we now replace the X factor witnessed from time fractals (derived from redrawing Mandelbrot’s multifractals) with the K cycle factor, which is widely seen and accepted, then the K factor should subdivide in a similar proportion as the X factor (X. X/3, X/9, X/27 ….). 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.

**ISOLATING THE K FACTOR**

CYCLE TRANSLATION AND PATTERN DISTORTION

Cycles are about pattern and periodicity. Pattern being the stronger of the two cycle characters. The focus was on identifying self similar nesting structures (Fig. 5), three smaller cycles nesting under the larger cycle. Care was also taken to identify cycle pattern distortions (Fig. 7), to illustrate potential improper cycle isolation and identification. Just like price fractals, smaller time fractals are effected by larger time fractals which drive them. The very reason for translation (Fig. 6) when the peak of a cycle shifts owing to the larger cycle above it causing cycle pattern distortion.

Once cycles have been properly categorized in the K factor and sub K factors, the cycle periodicities can be used for forecasting purposes12.

INTERMARKET STRATEGY

Owing to the easier access to information, global markets have seen an increasing interest in instruments and assets. 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. The subject’s main hypothesis is that technical analysts need to broaden their chart focus to take these intermarket correlations into consideration. 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’s Intermarket analysis also illustrated the nature of performance cyclicality irrespective of the intermarket ratio between two asset prices. 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 . 8 ) owing to its focus on trend over time just like the Elliott Wave theory fails to quantify the time element in the investment approach. The perspective signals mentioned in Murphy’s intermarket analysis rely on conventional tools like breaking of a trendline and indicative patterns on the intermarket ratios.

The Fig. 8 depicts the price of asset A, asset B and the ratio between them. The K factor is identified from the respective ratio line. This addition of the time fractal to the intermarket ratio line gives the intermarket strategy. (Fig .9)

** **

**POWER LAW**

A power law is a mathematical formula which states that as a phenomenon increases in scale, it also decreases in frequency. Time cycles also reduce in number as the time scale increase (K/9 to K/3 to K). Power laws appear widely in physics, biology, earth and planetary sciences, economics and finance, computer science, demography and the social sciences. For instance, the distributions of the sizes of cities, earthquakes, solar flares, moon craters, wars and people’s personal fortunes, stock indices and prices all appear to follow power laws.

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.e. K= 3*K/3). 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 power law can be described as…P(x)= cx-α

Here, α is the scaling exponent. The distribution is an exponential function, which takes a straight line form when we move on a logarithmic scale. lnP(x)= lnc-αlnx

Formally, this sharing of dynamics is referred to as universality, and systems with precisely the same critical exponents are said to belong to the same universality class. Working on the assumptions that time cycles belonged to the same universality class and were self similar fractals, we pulled out the k factor intermarket ratio cycle periodicities to test for power law distribution.

**THE K TREE**

Many emerging market index pairs and top Dow Jones components were paired to isolate the K factor for cycle periodicities. The self similarity appeared in most cases. About 3 Kitchin cycles, nearing a decade of daily data was tested for the study. The author has illustrated the detailed workings of the following three intermarket ratio lines.

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. BRT-WTM correlation was 0.99, XOM-CVX correlation was 0.97 and GE – CAT correlation was at 0.12 for the period under study. All of the pair cycle periodicities depicted the underlying K factor hierarchy.

**INTERMARKET STRATEGY**

The strategy has three parts. First the author has carried a visual of three iterations of the K factor on the intermarket ratio line (Fig 12, Fig. 14, Fig. 16).

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.

B = Periodicity in days*STDEV + Mean

C= NORMDIST (B,STDEV, MEAN, FALSE)

Where STDEV is standard deviation and NORMDIST is the normal distribution functions.

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 exit number of days are taken as half of the K/9 cycle. The author has tested the strategies for an average 3 Kitchin cycles. Underlying spot prices on the two assets making the pair are used. There is also a stop factor of 10% put to see how many times the pairs lose more than 10%. 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. For example the GE-CAT conventionally showcased a secular underperformance of GE against CAT. This was for all the time period under study. However, despite such an underperformance of GE against CAT, the K factor allowed us to trade long GE vs. Short CAT strategy successfully over the K/9 time frame (102-131 days) with exits on an average of 54 days. Even the other two pairs viz. BRT-WTM, and XOM-CVX are highly correlated pairs that even from a conventional long short strategy are not easy to trade. 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 average entry number of days for the three pairs were 111, 102, 131 for GE-CAT, BRT-WTM and XOM-CVX respectively. The exit number of days for the three pairs were 54, 50 and 66 in the same order. The stop loss of 10% was hit twice in 80 readings, once on both GE-CAT and XOM-CVX pair. The average annualized non leveraged return was at 54%.

**CONCLUSION**

The proof of the superiority of the time fractal over the price fractal clearly emerges when we redraw Mandelbrot’s multifractals and put to test intermarket ratios cycle periodicities for power law and as an investment strategy . The K factor indicator assists in this process. The author relies on both high and low correlated pairs to showcase the performance cyclicality over the K/9 factor time frame. 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. The strategy can be used by fund mangers across different assets and time frames, aggressively or passively by altering the K factor. The study has assumed a leverage of 1, but real market leverage can change the profile of the strategy. Overall, time fractals is a subject which traverses beyond capital market forecasting and can be utilized in many areas of scientific research.

FOOTNOTES

1.Ralph N. Elliott, Father of the Elliott Wave Principle and author of the Nature’s Law.

2.Benoit B. Mandelbrot, is a French mathematician, best known as the father of fractal geometry. The Mandelbrot set, named after him is a set of points in the complex plane, the boundary of which forms a fractal.

3.A fractal is generally “a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole, a property called self-similarity.”

4.A power law is a special kind of mathematical relationship between two quantities. If one quantity is the frequency of an event, the relationship is a power-law distribution, and the frequencies decrease very slowly as the size of the event increases.

5.William Strauss and Neil Howe presented a strong case for an intergenerational, 85- to 99-year cycle.

6.Brian Berry pointed to the presence of a generation-length, 25- to 35-year, cycle.

7.Clement Juglar reported a medium term, seven- to 11-year, cycle.

8.Joseph Kitchen reported a short-term, three- to five-year, business cycle.

9.Intermarket ratio is a line built from price series of two different assets.

10.Tony Plummer is a time cyclist and author of the book ‘Forecasting Financial markets’.

11.Triad of Patterns are the classic six wave pattern structure found in cycles with 1-2-3-a-b-c legs.

12.Forecasting purposes can include price, macro and micro economic, social trend and scientific forecasts.

13.John Murphy is a market technician considered as the father of modern technical analysis.

Download the paper from the SSRN…