|| 66 [23 2521 63 | 81 | 80 17 [35 33 8 66 36 | 35 37 | 31 | 35 | 69 28 [83 {1422 jun 77 35 | 26 [27 | 35 35 [23 Fe 64 64 29 29 78 | 33 | 8323 33 36 30 31 37 37 33 35 33 | 33 36 | 36 33 70 37 66 3524 70 36 70 70 70 66 35 33 | 36 27 | 70 75 | 69. 75 | 6425 72 | 63 64 66 70 76 | 77 & 35 [27 29 36 3326 jul_| 66 72 [66 35, 66 75 7 35 36 | 6427 36 | 80 64 64 [77 | 66 {72 B 66 36 68 37 [30 3528 36 | 27 C 66 77 | 64 64 29 70. 35] 172 15 35 2829 35 37 64 36 | 63 66 76 2 661 7130 W31 Aug 36 66 64] | 64 | 66 36 6432 7263 76 | 66 [72 64 36 70 63 66 | 80 283B 36 70 76 | 66 64 33 7 72 66 | 63 | 37 37 7134 63 30 37 [38 64 | 83 B 66 66 | 27 36 | 64 | 64 37 6635 Sep [70 63 80 73 | 66 | 70 | 66 El 70 7 B 82 [66 76 36 | 75 36 6636 63 | 72] 70 66 33 66 70 | 66 72 66 [36 35 [35 30 69"37 70 | 37 36 63 | 33 [63 [35 [35 37 2517 _| 25 [2838 30 Bs 27 | 37 | 36 [63 | 66 63 | 23 | 37 [29 112 35 | 23 2839 27 33 63 2340 Oct | 33 70 31 29 33 | 29 | 23 81 | 82 64 {it 35 36 | 33 8i 33 2841 af 3h 29 [33 64 ai | 66 {75a2 35 36 37 | 70 76 72 | 66 | 70 | 68 [36 63 | 27 | 70 | 68 | 68 7643 22 = | 66 [33 75 72 | 64 78 [64 66 | 72 | 64 | 75 66 | 75 | 83 37 69 2844 Nov 82/83 [71 72 | 64 | 66 172 66 [70 | 29 80 72 90 | 66 | 75 | 83 | 63 | 64 | 64 66 | 66 | 71 | 76 | 7145 70 77 | 70 80. 25 | 33 | 33 63 | 63 246 70 84 a7 27 70 72 | 70 63 63 | 66 35 | 35 | 35 | 37 | 35 | 84 | 69 | 69 6447 - 66 64 63 64 | 66 6448 Dec ma 35 35 64 | 66 72 64 72 | 75 | 75 | 66 35 37.49 33 66 66 64 | 70 72 | 66 63 33 30 :50 76 37 | 66'| 64 66 | 33 72 | 83 [83 | 75 | 63 | 64 35 69 | 76 | 85 | 835 64 rz] 77176 6118 33 35, 63 76 8652
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
186
187 
(88 SEASONALITY
 
COMMON SENSE AND SEASONALITYNonseasonal Markets
Most comprehensive works on seasonality include not only agricultural markets but cur-rencies, interest rates, and indices. Common sense tells us that the primary feedgrains fol-low a clear seasonal pattern based upon a single crop year. This calendar cycle also appliesto coffee, cocoa, and oranges, although competition from South America has limited theextreme moves that resulted from a freeze in Florida's orange groves. Grain seasonality hasbeen slowly changing as well, as Brazil improves their production of soybeans. Cattle andhog prices, dependent upon the cost of feed, and still marketed more actively in the fallbecause of weather, show a strong seasonal trend even though they have a growth andfeeding cycle that is not confined to a calendar year. Livestock patterns have adjusted tolong-term competition from Australia and New Zealand.
These agricultural markets are reasonable candidates for seasonal analysis, but mayrequire careful study to identify shifting patterns in competition and storage. What about‘Treasury bonds or the Japanese yen? Should we look for seasonal patterns in the financialmarkets? While we can argue that certain stocks are highly dependent upon seasonal busi-ness, can we say that the demand on money is higher in the summer than in the winter, orthat enough tourists convert their currencies to yen when they visit Japan during the sum-mer that this effect drives the value of the yen consistently higher? Are automobile exportsfrom Japan stronger or weaker during a certain season, so that there is a predictable pat-tern? These scenarios are highly unlikely and have no assurance, such as the coming ofwinter, that cars must be sold the way grain must be harvested. In financial markets theplayers can choose their time to act; they can hedge their commitments or wait for a bet-ter opportunity. This choice makes any pattern that might exist uncertain. The study of sea-sonality should be limited to those markets that depend on weather and seasons, eitherdirectly or indirectly.
Being Too Specific about Targets
Nature is not precise and selecting an optimal day of the year to set a long position is notlikely to work. Small shifts in fundamentals, such as the building of additional storage, willallow farmers to change their selling habits slightly. The right day to buy or sell in 1975 isnot likely to be the right day this year. Even weekly data may be a little too specific. Sea-sonal patterns are best seen using monthly data. The most we might expect is to know thatthe seasonal high in corn usually occurs in July but sometimes in June or August; the har-vest lows are likely in September but could be in October or November. We must first beaware of the big picture, then study how the specific pattern develops each year.
Recognizing Nonseasonal Years and External Factors
‘The studies of bull and bear market years shown in this book should make it clear that sea-sonality is not at all consistent. It is only a few years that cause prices to show extreme highsduring the summer, while there are years in which the patterns are influenced by externalfactors and do not show normal seasonality. Seasonality, although a clear concept and a factof nature, i not a trivial analysis when applied to trading. Each year must be observed andcategorized, then a reasonable trading plan must be defined for that situation.
 
Cycle Analysis
sonality, but it is more difficult to evaluate; therefore, it is often overlooked. Cyclescome in many forms—seasonality, production start-up and shutdown, inventory orstocks, behavioral, and astronomical. Seasonality is a special case of a calendar or annualcycle. Seasonality was covered in the previous chapter and its special features are not con-sidered here. Some of the cycles are clearly periodic, having regular intervals betweenpeaks and valleys; others are more uniform in their amplitude or height but irregular inperiod. The most definitive and regular cycle remains the seasonal, which is determined byperiodic physical phenomena. ; /This chapter will discuss the major commodity cycles that result from business deci-sions, government programs, long-term market characteristics, and phenomena. Short-termcycles are usually attributed to behavior and will be covered in a later chapter on patternrecognition. There are a few important ways to find the cycle, the most common beingtrigonometric curve fitting and Fourier (spectral) analysis, Both will require a computerand will be explained in the following sections. Examples of solutions will be included inthe explanation of the methods and applications will follow. Computer programs that solvethe trigonometric problems can be found in Appendix 4 along with additional examples.
T= cycle is another basic element of price movement, along with the trend and sea-
 
CYCLE BASICS
Even when the seasonal pattern is eliminated, most cycles are still based on the periodiceffects in our solar system. After the 1-year orbit of our planet around the sun, there is the28-day lunar cycle; converted to business days, this gives the very familiar 20-day reference,which remains overwhelmingly popular among all analysts. Other planetary effects, whichshould by no means be discarded offhand, can be found in Chapter 14 under the topic“Financial Astrology.” Throughout this chapter, there should be the undercurrent that plan-etary motion may account for the consistency of any cycle that repeats with a fixed period.
Cycles can be complex and difficult to see because there are often larger and smallerpatterns, and cycles within cycles, all acting at the same time. Nevertheless, they-exist andthey are real. The cycles that appear to be most important are either long-term or the sumof a number of subcycles that come together at peaks or bottoms. This gives us a way toidentify one point on a cycle; we must remember that, when the individual componentsare found, there may be a number of smaller patterns that cause this effect. A reference toharmonics, just as in music, means that a smaller cycle is a fraction of the larger (for exam-ple, its cycle length is %, %, %, .. . of the larger). When two cycles are synchronized, theirpeaks or valleys occur at the same time. Any price series can be decomposed into individ-ual cycles, although there may be many cycles needed.
Observing the Cycle
 
Before selecting a market for cycle analysis, it is necessary to observe that a dominant cycleexists; it is also useful to know why it exists to avoid uncovering spurious patterns. This is 
192 CYCLE ANALYSIS
 
FIGURE 8-2 Cycles in Swiss franc futures.
UNCOVERING THE C¥CLE 193
ee
FIGURE 8-3 The Kondratieff Wave.
 
Weekly High, Low—Friday Close
COMODETY PIRPECTIVI/CHCACO, KLINT Smt
        
wr sere mawoe = weneesT ruruees
 
0.680——— 0-660s V Is S R C 0-640
 
 
SOHANS JAE OMDUFRAR JIN ECHOLS IRAN SIR SO MDIFRAN JA
 
WOU PRANII REO KOEN AR.a8 1978 ser se ro
 
Source:
 
Bernstein, The Handbook of Commodity Cycles (New York: John Wiley & Sons, 1982).
price pattern will never correspond exactly to the predicted peaks and valleys, which,because they are a mathematical cycle, must come at regular intervals.
Terminology
Before getting very technical about the measurement and calculation of cycles, there are afew terms that describe most of the concepts discussed throughout this chapter. Note thatthe use of wave and cycle is interchangeable.
 
The Kondratieff wave
 
 
 
 
 
|Ub swap
 
1900 te20,-= 840860 18801900 1920 1940 19601980et 1
Source: Jeff Walker,“"What K-wave?” Technical Analysis of Stocks & Commodities, 8, no. 7 (July 1989). © 1989Technical Analysis, Inc. Used with permission.
Amplitude—the height 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 360°Phase—a measurement of the separation of tops or bottoms of two waves with thesame period
Left and right translation—the tendency for a cycle peak to fall to the left or right ofthe center of the cycle
UNCOVERING THE CYCLE
Before resorting to the highly mathematical methods for finding cycles, there are somesimple approaches that may serve many traders. For example, if you believe that there is adominant 20-day cycle, then you simply create a new price series by subtracting the currentdata from 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 Cycle by Removing the Trerid_
The cycle can become more obvious by removing the price trend. The use of two trend-lines seems to work very well in most cases.” First, smooth the data using two exponentialmoving 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
+ tn his article, “Finding Cycles in Time Series Data” (Technical Analysis of Stocks & Commodities (Apgust 1990),A. Bruce Johnson credits John Ehlers for his work in the use of two exponential trends. See John Ehlers, “Moving‘Averages, Part 1” and “Moving Averages, Part 2,” Technical Analysis of Stocks & Commodities (1988), 
194 CYCLE ANALYSIS
UNCOVERING THE CYCLE 195
 
MACD index by subtracting the value of one exponential trend from the other; the result-ing synthetic series avoids the lag inherent in most methods.
‘A technique that enhances the cycle is the use of triangular weighting instead of expo-nential smoothing. This method creates a set of weighting factors that are smallest at theends and peak in the middle of the weighted average. For example, if you wanted a 10-period triangular average, the peak weighting factor would always be 2.0 and is assignedto the price just before the middle (there is no middle value for an even number of peri-ods; therefore, the first value is eliminated), then weighting factors would begin andincrease by 2.0/(P/2), where P = 10. This creates the series of weighting factors, beginningat t - P + 2 (to get a center value you need to have an odd number of entries), .4, .8, 1.2,1.6, 2.0, 1.6, 1.2, .8, .4. Once the weighting factors, w, are known, the sum of the weight-ing factors is P (the same as the period), and the triangular average is
TMA, = (Wi X Prep a + WX Prope to + We X PP
When applying a triangular average, each of the two averages used (one of which is half theperiod of the other) get triangular weights and the difference is taken. The smooth curveof the triangular MACD in Figure 8-4 shows what appears to be a regular cyclic pattern inthe price of IBM.
Trigonometric Price Analysis
Most cycles can be found using the trigonometric functions sine and cosine. These func-tions are also called periodic waves, because they repeat every 360° or 2 radians (where %= 3.141592). Because radians can be converted to degrees using the relationship
2n360all work that follows will be in degrees. Some other necessary terms are:
1 degree =
Amplitude (a)—the height of the wave from the center (x = axis)
Frequency (w)—the number of wavelengths that repeat every 360°, calculated as@=UT
Period (T)—the number of time units necessary to complete one wavelenigth (cycle)
Asimple sine wave fluctuates back and forth from +1 to —1 (0, +1, 0, —1, 0) for eachcycle (one wavelength) as the degrees increase from 0° to 360° (see Figure 8-5). To relatethe wavelength to a specific distance in boxes (on graph paper), simply divide 360° bythe number of boxes in a full wavelength, resulting in the box size (in degrees). Forexample, a 100-box cycle would give a value of 3.6° to each box. The wavelength can bechanged to other than 360° by using the frequency as a multiplier of the angle of the sinewave
sin ooIf w> 1, the frequency increases and the wavelength shortens to less than 360°; if @ <1, the frequency decreases and the wavelength increases. Because @ is the frequency, itgives the number of wavelengths in each 360° cycle. To change the phase of the wave (thestarting point), the value 6 is added to the anglesin (0 +B)
If 8 is 180°, the sine wave will start in the second half of the cycle; b serves to shift thewave to the left. The amplitude can be changed by multiplying the resulting value by a con-
    
 
FIGURE 8-4 A triangular MACD shows an apparent cycle in IBM.
Johnson indicator on {BM 2/1/88-3/30/90,IBM 2/1/88-3/30/30
 
13012612011511010510095908580
5.004.003.002.001.000.00-1.00-2.00-3.00-4.00“5.00
es
   
 
fronnson indicator
  
 
6667777789898957913579135791
135 7911111222229333944444551357913579135 791357913
Triangular MACD vs. MADL
~—-Triangular MACD
 
 
| MADL
 
Source: A. Bruce Johnson, “Finding cycles in time series data,” Technical Analysis of Stocks & Commodities, 8,n0.8 (August 1990). © 1990 Technical Analysis, Inc. Used with permission.
stant a. Because the sine ranges from +1 to ~1, the new range will be +a to —a (Figure8-6). This is written asin (wo + 6). ~
There are few examples of price movement that can be represented by a single wave;thus, two sine waves must be added together to form a compound wave:
Y= 4 Sin (O19 + B,) + @2 Sin (O26 + b2)
Each set of characteristic variables, a1, 001, by, and @2, @2, b, can be different, but bothwaves are measured at the same point. at the same time. Consider an example that letsthe phase constants b, and b, be zero: 
196 CYCLE ANALYSIS
 
Sinusoidal wave.
 
 
y= asin (wo + bd
y-axis
 
 
peezo axis380"yi =3sin 4992 =5 sin 66 .YrM tI:
Figure 8-6 shows the individual regular waves y, and yz, and the compound wave y overthe interval 0° to 180°. Note that both y, and y; began the normal upward cycle at 0°; butby 180°, they are perfectly out-of-phase. During the next 180°, the two waves come backinto phase.
When combining periodic waves, it is useful to know the maximum and minimumamplitude of the resulting wave. Because the peaks of the two elementary waves do notnecessarily fall at the same point, the maximum amplitude of either wave may not bereached. A mathematical technique, called differentiation, is used to find the maximumand minimum amplitudes. The first derivative, with respect to angle 6, is written dy/do or9’, where y is the formula to be differentiated. The rules are:
 
FIGURE 8-6 Compound sine wave.
-
i
          
YH +9
JN
dh #3 sin ag
 
2 = 6 sin Se
 
AMPLITUDE
 
 
 
UNCOVERING THE GYCLE 197
 
 
qo! .
we (sin 9) = cos 6;
d
Jp SiN OH) = @ cos oH
“ (sin (@6 + )) = @ cos (@p + 6)
$s (a, Sin (@, + by) + a2 sin (29 + b2))
= G0; COS (01 + by) + 4202 COS (026 + 2)Applying this method to the previous example,y =3 sin 40 + 6 sin 59
yGi 77 = 12.608 40 + 30 cos 5¢
The points of maximum and/or minimum value occur when 3’ = 0. For yi = 12 cos 46,the maxima and minima occur when 49 = 90° and 270° (@ = 22° and 67%°) (Figure 8-6).For y; = 30 cos 56, the maximum and minimum values occur at 59 = 90° and 270° ( = 18°and 54°). It must be pointed out that the first derivative identifies the location of theextreme highs and lows, but does not tell which one is the maximum and which is the min-imum. The second derivative, y”, calculated by taking the derivative of y’, is used for thispurpose as follows:
Ify’(@) = 0 and y”(x) > 0, then y(x) is a minimum.If y’(x) = 0 and y”(x) < 0, then y() is a maximum.
Then, y, = 22%° and y2 = 18° are maxima and y, = 67¥° and y; = 54° are minima.
‘Anyone interested in pursuing the analysis of extrema will find more complete discus-sions in a text on calculus. Rather than concentrating on these theoretical aspects ofcurves,® consider a practical example of finding a cycle in the price of scrap copper, shownin Table 8-1 and charted in Figure 8-7. The price peaks seem evenly spaced, occurring atmid-1966, January 1970, and January 1974, about 4 years apart. The solutions to theseproblems are tedious; therefore, calculations will be performed using the computer pro-grams in Appendix 4.
The results obtained by using actual copper prices will not be as clear as using ficti-tious data. It is important to be able to understand the significance of practical results andapply them effectively.
Because trigonometric curves fluctuate above and below a horizontal line of valuezero, the first step is to detrend the data using the least-squares method. This results in theequation for a straight line representing the upward bias of the data. The value of thedetrending line is then subtracted from the original data to produce copper prices thatvary equally above and below the line from positive to negative values.
The straight line y = a + 6x, which best represents the trend, can be found by solvingthe least-squares equations:
© A more specific presentation of trigonometric curve fitting can be found in Claude Cleeton, The Art of Indepen-dent Investing (Prentice-Hall, 1976, Chapter 8). The material covered in this section is carried further in thatwork. 
198 CYCLE ANALYSIS
 
TABLE 8-1 Dealer’s Buying Price, No. 2 Heavy Copper Scrap at New York*
 
Average Quarterly Price (¢/lb}
 
 
Year Ist 2nd 3rd 4th1963 22.42 22.46 2217 22.001964 2318 24.56 25.57 30.591965 28.23 33.77 35.90 40.051966 46.22 51.48 40.76 40.161967 36.51 29.30 30.36 36.421968 39.75 30.07 29.08 32.431969 38.94 42.95 43.38 46.231970 47.70 46.98 35.78 27.351971 25.40 29.45 27.15 28.481972 32.74 33.53 30.0! 29.251973 36.82 45.07 55.13 65.511974 66.56 70.06 47.30 35.621975 32.06 31.46 35.75 36.461976 38.22 43.24 45.46 38.961977 37.08 38.72 34.01 33.00+ 1978 35.07 40.23 4163 “44.951979 SLI2 63.71 59.56 63.38
 
“Based on prices from the American Metal Market.
Nd dsDdyvae-(54)H(d27-6 Ss)
To do this, let x be the date and y be the price on that date. For convenience, instead of let-ting x = 1967, 1967%, 1967%, ..., letx = 1, 2, 3,.... The solution, using a computer pro-gram in Appendix 2 (also an integral part of the programs in Appendix 4) or thehand-calculation method, is
YY = 28.89 +.267x
Figure 8-7 displays the original copper prices and the regression line. The original pricescan now be detrended using the equation above, subtracting the line values from the cor-responding prices. Complete step-by-step results for this example can be found in Appen-dix 4, The detrended data is now used in the general trigonometric single-frequencywave:
a
 
Jr=@ Cos Ot +b sin ot — ueThe variable f replaces to express the angle in integer units rather than in degrees. Thiswill be more convenient to visualize and to chart.
To find the frequency ©, it will be necessary to first solve the equation:
1cos @ ~ > Or= 0 -2
using the system of equations,
UNCOVERING THE GYCLE 199
 
————
FIGURE 8-7 Copper prices 1963-1979.
(No. 2 heavy scrap)
Price (¢/ib.)
 
Year
OY2 =i + V3Os =J2+ Hs
OYn-1=Yn-2t In
‘This is expressed as a summation (similar to least-squares) in which the values for ¢ and dmust be found: -
ad e= Sedwhere ¢ = y_ and d = yx-1 + Just. Summing the detrended values c* and cd gives Ec =6338.4 and >. cd = 9282.2, resulting in a = 1.464. The value for o: is substituted into theintermediate equation and solved for the frequency @:
cos @ 5 0.464) =0
cos @ = .732w= 42.9
The period T is 360/42.9 = 8.4 calendar quarters. The last step in solving the equationfor a single frequency is to write the normal equations:
a > cos? wt + > cos of sin wt = >” y,cos wt
a> sin wt cos wt + b >) sin’wt = >) y, sin oF 
200 CYCLE ANALYSIS
 
and solve for a and 6, where t= 1, ... , 40, and w = 42.9. As in the other solutions, a com-puter program is best for finding the sums (using detrended data) necessary to solve theequations. The sums are
a>dicosot SY sin@tcosat — >’ y, cos we
Ssinr-or Sy, sin wt
‘Then, a and 6 can be found by substituting in the following equations:
b S 9x sin ot cos’ wt ~ Sy, cos ot© S) sin? ot 5. cos? wt — S. cos wt
 
ot
> 9: cos wt — bY’ cos wt sin wtqa-a e
2© cos? wt
The results a = -.603 and b = 1.831 give the single-frequency curve as:J: = -.603 cos 42.9¢ + 1.831 sin 42.16
‘Taking f = 1 to be 1967 and ¢ = 68 to be 1979% and adding back the trend, the resultingperiodic curve is shown in Figure 8-8. —
The single-frequency curve shown in Figure 8-8 matches seven out of the eight peaksin copper; however, it is not much more than could have been done using the Ehrlich
 
FIGURE 8-8 Copper prices 1963-1979: single-frequency copper cycle manually scaled toapproximate amplitude.
Price (cents/pound)
 
63 6465 66 67 68 69 70 71 72 73 74 7