i-tion be entered contrary to the direction of price movement. Although chart interpreta-tions attempt to remedy this, unusually large losses may result.
It is not likely that an indicator can be found that will identify only the trades that willbe profitable. It is far more likely that an indicator will add more patterns to the already dif-ficult combinations that are seen in price; therefore, it is not clear whether an indicator willsimplify trading or make it more complex. If you need to interpret the indicator becauseeach situation creates new decisions, then you've got the wrong one.SOME FINAL COMMENTS ON MOMENTUM 459
 
Indicators can be designed to reach extremes when a specific situation occurs. Duringother times it tells us nothing. One practical use of any oscillator is to show when a marketis relatively overbought or oversold. If you are using a trending approach, and each tradeis held about 20 days, an oscillator can be tuned to provide entry timing. For example, ifyou are willing to wait up to 2 days to enter a long position based on the daily close, thenconstruct a 10-period oscillator of 5-, 10-, or 15-minute data. Test the oscillator to see if itgenerates at least one, but preferably two, oversold signals during each 2-day period. If so,use it to time your entry and you are likely to be buying dips rather than rallies. 
Seasonality
hapter 3 introduced prices as a time series and identified its four components as the
trend, the seasonal pattern, the cycle, and the chance movement, it included various
ways of finding the trend using statistical analysis and forecasting techniques. Chap-
ter 4 then showed various ways to calculate trends. Of all techniques, the trend is over-whelmingly the most popular foundation for trading systems. In this and Chapter 8 we willturn our attention to two other principal components, the seasonal and cyclic movements.Seasonality is a cycle that occurs yearly. It is most often associated with the planting andharvesting of crops, which can directly affect the feeding and marketing of livestock. Nor-mally, prices are higher when a product is not as readily available, or when there is a greaterdemand relative to the supply, as often occurs with food during the winter months. For grain,the season is dominated by planting, harvesting, and weather-related events that occur inbetween. Most crops have been produced in the Northern Hemisphere, but South Americansoybeans and orange juice have become significant factors since the early 1980s, as have Aus-tralian and New Zealand beef and lamb, resulting in a structural change in seasonal patterns.Globalization has not only affected financial markets, but nearly everything we purchase... ,Consumer habits can cause a seasonal pattern in metals and stocks as weather does foragricultural products. Passenger airline traffic is much heavier in the summer than in thewinter, and profits of that industry, especially companies not diversified, reflect seasonal-ity. Eastman Kodak once had a classic pattern caused by much more active picture takingduring the summer months, when most workers take vacations. This seasonal activity alsoincreased demand on silver, a major ingredient of film coating, which developed a co-
dependent pattern before a substitute was found. ~\
A CONSISTENT FACTOR
Even when the impact of seasonality on agricultural products is not clear from the pricepatterns, it is still there. Consider the following factors that do not change:
1. More corn, wheat, and soybeans are sold during harvest than at any other time of year
~ because there is not enough available storage to hold all of the new crop. Rental stor-age, when available, requires a minimum three-month charge. Lack of storage and theneed for immediate income result in greater sales and cause prices to decline.
2. Because feedgrains are“harvested only once each year, forward contracts include astorage cost as part of the total carrying charge. Therefore, each forward deliveryprice should be higher within the same crop year.~ ~
Sometimes the price pattern of forward months does not seem to reflect the addedcosts of carry. Occasionally these markets will even invert and the nearest delivery will tradeat a price higher than the deferred months, a situation familiar to crude oil and copper. Thecost of carry, however, still exists in an inverted or backwardation market. Extreme short-term demand pushes the nearest delivery much higher, while the events causing price dis-ruption are expected to be temporary. The normal carry is still there, it is just overwhelmedby temporary demand.POPULAR METHODS FOR CALCULATING SEASONALITY 161
 
It is important to be able to identify seasonal patterns. The methods for finding themare simple, and made more so by the use of a spreadsheet and a computer. These will bediscussed in this chapter along with some practical applications.
THE SEASONAL PATTERN
Seasonal patterns are easier to find than the longer-term cycles or economic trends,because they must repeat each calendar year. Although any 12-month period can be usedto find the seasonal pattern, academic studies usually begin with the new crop year forgrains, just following harvest, when prices tend to be lowest. This approach will make thecarrying charges, which increase steadily throughout the new crop year, more apparent.For uniformity, the examples in this chapter will always begin with a calendar year, whichassumes no knowledge of where the season starts, and can be equally applied to stocks. Itwill always include carrying charges as an integral part of the market price.
United States agricultural production is considered to be the standard for seasonal,even though a wheat crop is harvested continuously throughout the year in different partsof the world. Prices are expected to be lower during the U.S. harvest and highest duringthe middle of the growing season. The influence of world stocks and anticipated harvestfrom other major producers in South America or Russia will cause an overall dampeningor inflating of prices, rather than change the seasonal pattern. While the costs of trans-porting grain to the United States is not practical, except for special products, each pur-chaser in the world market will select a supplier at the best price. This fungibility has adirect impact on the U.S. market, which must alter its local price based on export demand.
Industrial commodities have seasonal price variation based on demand. Silver, althoughincreasingly used in electronics as a conductor, is also consumed for jewelry and photogra-phy, and has served as a general hedge against inflation. It still shows a seasonal pattern ofgreater demand during the summer months. Almost half of all copper is used in electricaland heat conductivity, with much of it in the form of an alloy with nickel and silver. Its sea-sonality is heavily related to the housing industry, where it is required for both electrical andwater systems. New sources of ore are introduced infrequently, and the possibility of discov-ery or expansion is rarely seen in price movement as short-term anticipation. The primarysupply problems in copper are related to labor as well as social and political changes in pro-ducing countries.
There are many businesses with finished products that have seasonal demand andtheir publicly traded stock prices will reflect that tendency. Because the shares in a com-pany are far removed from buying and selling the raw materials that they use, even majoroil companies, such as Exxon, may not show a seasonal pattern similar to crude oil or itsrefined products. As with Eastman Kodak, these firms have thoroughly diversified, and theimpact of a $1.00 increase in a barrel of oil or a $.50 increase in an ounce of silver will haveonly a small effect on the profitability of the firm, and may be the cause of stock pricesadvancing or declining. Yet some industries, such as airlines, still show seasonal patterns,and the same procedures given here can be used to find them.
POPULAR METHODS FOR CALCULATING SEASONALITY
Seasonal patterns are most often calculated using monthly data, although some studieshave tried to pinpoint their periodic turns to specific days. As with most other analysis,closer observation or shorter time periods also bring more noise and erratic results. Withthis in mind, the seasonal studies in this chapter will use monthly data and keep an eyetoward the big picture.
There is one important caveat about the prices used in some of these examples: theywere created using a computer program that averaged the prices of a TradeStation contin-162 SEASONALITY
 
uous contract. This contract is gap-adjusted to remove the price jumps in futures whendelivery months change; therefore, in the earlier years prices may be higher or lower thanthey actually were at the time. In the case of soybeans you will see that the prices in themid-1970s are about $2 per bushel higher. In some cases, such as interest rates, prices thatare 25 years in the past can actually become negative because of gap adjusting. A preciseseasonal analysis will need to use continuous cash prices, rather than a constructed series,to have valid percentages. Results of these examples, however, show seasonality similar tostudies that use cash data. The following terminology and concepts will also be of help.
Basic Calculation Components
Average Prices
Finding the seasonal pattern does not need to be complicated; however, some basic rulesmust be followed to get sound results. For most analysts, it is easiest to begin with a spread-sheet, where the months are recorded in each column and the rows represent years (see Table7-1). The average monthly price, placed in each cell, can give a good indication of seasonalpatterns by simply averaging each column and plotting the results as shown in the first of foursummary lines at the bottom. The major criticism of this technique is that it ignores the chang-ing price levels over time. For example, a 25-year study of soybeans will use prices that varyfrom $6 to $15 per bushel; price changes at the $15 level could overwhelm other years.
Indexing the Data :
A simple way to adjust for price differences over time is by indexing data, where each newentry is based on a percentage change from the previous value. This method will also workwith seasonal studies, but must use unadjusted cash data. Because the monthly averageprices for these examples were created from TradeStation continuous data, they cannot beused for this purpose.
Percentage Change
Over the mariy years needed for a seasonal study, prices may increase by a factor of two orthree, and the magnitude of volatility at these different levels will vary significantly, distort-ing the results. To correct for the problems associated with a long-term increase or declinein price, a monthly percentage change can be substituted for the average price. The use ofpercentages adjusts for most of the volatility changes and, at the same time, removes anylong-term trend bias. It is a simple and very credible method for finding seasonality, but itsresults can be more erratic than the sophisticated calculations found in the next section.
Detrending
Removing the trend using a moving average, then comparing the percentage differencesbetween the price and the corresponding trend value, is a classic approach to evaluatingseasonality.
Median Price _
The median is the middle price of a series of values that have been sorted in numericalorder. Instead of uSing an average monthly price, or the average of, for example, all Juneaverage prices, using the median value should improve results. The median ignores theextreme prices that occur during some unusual years and gives a typical price. For exam-ple, if the average September price of heating oil was 50 cents per galion for four years, butreached $1.00 per galion during one year, then the average would be 60 cents and themedian would be 50 cents. Later in this chapter you might note that a large differencebetween the median and the average prices indicates that there were a few volatile yearsmixed with mostly normal patterns. 
 
 
 
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Effects of a Few Volatile Years
All seasonal calculations can be distorted by a few very volatile years that show large per-centage price changes, especially during a time when there is not normally a strong sea-sonal bias. For example, if heating oil prices were stable for four years, or tended slightlylower, then jump 100% in one unusual period, the result of an OPEC announcement cou-pled with low inventories, the average will show a seasonal increase of 20%. In fact, that iswhat happens in many markets in which one or two unrelated events cause a price changethat can be interpreted as seasonal. In a computerized environment, the use of the median(discussed above) or more data can prevent mistakes; however, common sense will alsoshow that a single odd year will erroneously distort the average.
Yearly Averages
The most basic way of measuring or describing seasonality is by the monthly variation fromthe yearly average or crop average (for agricultural products), usually calculated as a ratioor percentage. The results of this technique using the 1975 corn prices can be seen in Table7-2. During this one year, the highest prices occur in January and August and the lowest atthe end of the year after harvest, confirming what we might expect of the corn season, withthe exception of the January high. The average price for 1975 was 2.92, and the extent ofvariation throughout the year ranges, coincidentally, 13.7% above and below the average. Byapplying this method to the 20 years of data for corn shown in Table 7-3, the percentagevariation can be shown in the corresponding Table 7-4. For those who prefer using negativepercentages to show when values are below the average, simply subtract 100 from eachnumber. ~
The long-term seasonality, called the seasonal adjustment factor, is the monthlyaverage of the percentages shown in Table 7-4 and appears at the bottom of Table 7-4 andin Figure 7-1. These values must be divided by 100 before they can be correctly used aspercentages. In both cases, the analysis has been separated into the two periods,1956-1970 and 1971-1975, to indicate the changing volatility and slight change in pat-tern. The shaded area in Figure 7-1 represents the range in the percentage variation forthe 1956-1970 period, and the broken lines show the corresponding 1971~1975 results.Figure 7-1 shows that the traditional seasonality can also have sharp fluctuations,reduced to intervals of short duration. The more recent period reflects the traditionalseasonal highs and lows, but shows that there is a considerable wide range of other pos-sibilities. All methods of determining seasonality will be influenced by those years inwhich unusual, overwhelming factors caused prices to move counter to normal seasonalpatterns. A later section will discuss the advantages of distinguishing seasonal and non-seasonal years.
 
‘TABLE 7-2 Percentage of Monthly Corn Prices to the Average -
 
 
 
 
Avg Jan Feb Mar 7 Apr May jun
Price* 2.70 3.07 2.86 2.67 2.68 2.66 2.68Percent 100.0 113.7 105.9 98.9 99.2 98.5 99.2jul Aug Sep Oct Noy Dec
Price — 2.72 2.95 - 2.76 2.62 2.33 _ 237Percent 100.7 109.2 102.2 97.0 86.3 878
 
*Midmonth U.S. farm price, 1975.POPULAR METHODS FOR CALCULATING SEASONALITY 165ee
aTABLE 7-3 Corn Cash Prices
Year Ayg jan Feb Mar Apr May fun ful Aug = Sep = Oct’ «3S Nov Dee
 
 
 
1956* 130 L14 b16 LI9 132 1L4l 144 145 146 145) 112 122 1.231957 1250 1.22 116 118 118 120 120 121 122 LI3 1.04 98 [011958 110.97) 98 (1.04) F.18 1200 -1:230«:42.23 24 OLS) 1.02 941.041959 109 1.05 1.06 108 LI7) LI8 LIS) LIZ) LIZ) 109 96 100 991960 103 1.03) 1.04 «1.04 (1.09) LIE Lt Lit 1.09) 107) 997) ga 921961 1.01 99 103 104 -.98 £04 [04 106 1.04 102 1.00 91 941962 3% 94 95 96 .98 1.03 103 102 101 1.00 94 91 1.001963 Lil 103 1.05) 1.06 1.09 1.12 119 2b 121 1.23) 1.04 102 1.201964 1130412) 809 LIS LIS IBIZA 4 T7005 0441965 118 616 Lt7 btg 1220 1.25 4:26 124 «120 «IB LO L043l966t 125 119) 120 LIZ) Lt9 121 1200 1.27) 1340 «135129 1261.291967 VI7 128 1.26 61.28 1.26 125 126 121 Lt oLf2) 1.04 971.031968 1.04 1.04 1.06 1.06 1.06 1.09 107 1.04 99 1.01 96 104 1.051969 £12 108 109 109) 1.12) 119 118 108 LIB LIS 112" 1.07 1.091970 124 112 4.14 113 LIS LIS L2t 124 127) 138 134 129839\971 127, 142 143 143 414i 138 143) 136 LI9 Lit 100° 97 5.081972 Hi7 1.09) 1.09 EO 13 IS 13 4S 229 20.42.1973 186 1.39 135 137 142) 1.16 199 203 268 215 217 218 239
1974 292 259 276 268 241 245 257 291 337 3.30 345 332 3.271975 270 3.07 286 267 268 266 268 272 295 2.76 262 233 237Average 130 1.30 129 131 132 138 139 145 140 132 1.28 (1341976 244 248 2.50
 
*1956-1965 midmonth Itinois farm prices (U of ).11966-1975 midmonth US. farm prices (CRB Yearbook).
A general formula for the monthly average is:*
Nn wzAPP, = 100 {2 > (a1§:P.) i=1,12net yawhere - APP, is the average percentage price in month 7
iis the calendar month from 1 to 12
N is the number of years in the analysis
P,, is the average price for month j of year 7
This formula may be applied to weekly or quarterly average prices by changing the 12 to52 or 4, respectively. - :
The-use of an annual average price can bias a clear seasonal pattern because of itsinability to account for a long-term trend in the price of the commodity. If the rate of infla-tion in the United States is 6%, there will be a tendency for each month to be 4% higher,resulting in a trend toward higher prices at the end of the year. Longer trends, such as thesteady rise in grain prices from 1972-1975, followed by a longer decline in the 1980s, willobscure or even distort the seasonality unless the trend is removed.
1 David Handmaker, “Low-Frequency Filters for Seasonal Analysis,” in Handbook of Futures Markets, Perry J.Kaufman (ed.) (John Wiley & Sons, New York, 1984).166 SEASONALITY
 
LS
TABLE 7-4 Corn Price as Percentage Average (Annual)
 
Year Avg Jan Feb Mar Apr May Jun Jul Aug = Sep = Oct’ «Nov Dec
 
 
1956 130 87.6 89.2 91.5 1015 1085 110.8 1115 1123 ILS 861 938 94.61957 (25 976 928 944 944 960 960 98 976 869 869 754 8081958 (10 88.2 89.1 94.5 107.3 109.1 1118 1/18 1127 885 927 854 94.51959 1.09 96.3 97.2 99.1 107.3 108.2 109.2 107.3 107.3 100.0 881 91.7 90.11960 1.03 100.0 [0/.0 101.0 [05.8 107.8 1078 107.8 105.8 103.9 94.2 796 989.31961 1.01 98.0 102.0 103.0 97.0 103.0 103.0 104.9 1030 101.0 99.0 901) 93.11962 98 95.9 96.9 97.9 100.0 105.1 105.1 1041 103.1 102.0 95.9 92.8 102.01963 (11 928 946 95.5 98.2 101.0 107.2 109.0 109.0 1108 93.7 91.9 99.11964 1.13 