Revisiting Time Triads

— -K/9-K/3-K-3K-9K- – –

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

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