FIGURE 8-2 Cjcles in Swiss franc futures

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S..urce J Ben.Eiein,TheHan.tt:.o..k..fC.Jumodiiy?7cleEilTewYork J..L„ wf,ley & ?i,e, 19BJ.

price pattern will never correspond exactly to the predicted peaks and vallejs, which, because they are a mathematical cycle, must come at regular intervals.

Terminologj

Before getting very technical about the measurement and calculation of cycles, there are a few terms that describe most of the concepts discussed throughout this chapter. Note that the use of wave and cycle is interchangeable.

FIGURE 8-3 The KondratiefF Wave.

e: e«Vllter.-Whi«

•• Tfinical Analysis of Stocks Con

7 (July 9S9><e> 1989

Amplitude-the heieht of the wave (cycle) Period-the time needed to finish one wave (one cycle)

Frequency-(for the more mathematical) the number of cycles that repeat every 3600

Phase a measurement of the separation of tops or bottoms of two waves with the same period

Left and right translation-the tendency for a cycle peak to fall to the left or right of the center of the cycle

UNCOVERING THE CYCLE

Before resorting to the highly mathematical methods for finding cycles, there are some simple approadies that may serve many traders. For example, if you believe that there is a dominant 20-day cycle, then you simply create a new price series by subtracting the current data fran a 20-day moving average. This removes the trend that may obscure the cycle. Most oscillators, such as a stochastic or RSI, also serve to identify a price cycle.

Enhancing the Cjcle by Removing the Trend

The cycle can become more obvious by removing the price trend. The use of two trendlmes seems to work very well in most cases .(J)First smooth the data using two exponential moving averages, in which the longer average is half the period of the dominant cycle (using your best guess), and the shorter one is half the period of the other. Then create an

5 In his article. "Findius in Time Sanes Data (TectiHC.iu-.ualysis ..fSock.-& Ciiiuodnk.- (Ausust 199ii.), A Bruce J-liuson creOits Liii. Ehierr for his w.dc in the use -i tn- eiTonential tren.t See J-lii, Ehiers, M..vii,s Aver.iges. Part 1" and M..vii,s Aver.iges. Part 2," Technical Analysis ..f .- & CmuoditiesdEI)

MACD index by subtracting the value of one exponential trend from the other., the resulting sjnthetic series avoids the lag inherent in most methods.

A technique that enhances the cycle is the use of friangular weighting instead of exponential smoothing. This method creates a set of weighting factors that are smallest at the ends and peak in the middle of the weighted average. For 1 , if you wanted a lOperiod friangular average, the peak weighting factor would alwajs be 2.0 and is assigned to the price just before the middle (there is no middle value for an even number of periods; therefore, the firsl value is eliminated), then weighting factors would begin and increase by 2.0/(P/2), where P = 10. This creates the series of weighting factors, beginning at t - P + 2 (to get a center value you need to have an odd number of enfries). A, .8,1.2, L6, 2.0,1.6, 1.2, .8, .4. Once the weighting factors, w, are known, the sum of the weighting factors is P (the same as the period), and the friangular average is

tma,= (w, Xp,.p2 + w2 xp, P. J + +Wp , xP,)/P

When applying a triangular average, each of the two averages used (one of which is half the period of the other) get triangular weights and the difference Is taken The smooth curve of the triangular MACD in Figure 8-4 shows what appears to be a regular cyclic partem in the price of IBM.

trigonometric price analysis

Most cycles can be found using the trigoixMnetric functions sine and cosine. These functions are also called periodic waves, because they repeat eveiy 360° or 2 radians (where n = 3 141592). Because radians can be convened to degrees using the relationship

all work that follows will be in degrees. Some other necessary terms are: Amplitude ( )-the heil of the wave from the center {x = axis) freguency (to)-the number of wavelengths thai repeat every 360°, calculated as

-1 -Period (7)-

Ihe number of time units necessary to complete one wavelengjh (cyde)

A simple sine wave fluctuates back and fonh from+1 to-1 (0. +1.0, -1,0) for each cycle (one wavelngth asthe egrees ncrease from0" to360° (see Figure 8-5). To relate the wavelengjh to a specific di taiKc in boxes on graph paper), simply divide 360° by the number of boxes in a lull wavelengjh, resulting in box size {in degrees). For example, a lOO-box cyde would give a value of 3 6" to each box. The wavelength can be changed to other than 360° by using the frequency as a multiplier of the angle of the sine

sinto*.

If to > 1, the frequency increases and the wavelength shortens to less than 360°; if < 1, the frequency deceases and the wavelength increases Because (o is the frequency, it ves the number of wavelengths in each 360° cyde. Tochange thepftaseof thewave (the staning point), the value badded to the angle

81 (1» + 6)

If l> is 180°, the sine wave will start in the second half of the cyde; b serves to shift the wave to the left. The cmifilUude can be changed by muteplying the resulting value by a con-

FIGURE 8-4 A triangular MACD shows an apparent cycle in IBM. Johnson indicator on IBM %/1/88-3/30/90.

IBM /88- / /9

Triangular MACD vs. MADL

Sourcc.A. Bruce jolmson, "Finding cycles in time series data," Technical Analjsis of Stocks & Commodities, 8 no. 8 (August 1990). m 1990 Technical Analjsis, Inc. Used with permission.

slant a. Because the sine ranges from+1 to -1, the new range will be+ato -a (Figure 8-6). This is written asin(c* + b).

There are few exanples of price movement that can be represented by a single wave; thus, two sine waves must be added together to form a compound wave