2,December by (1 +8)! = .99076. ,ane he chain Ces bave been calculated on a base of January, which was important tocorrect the compounded bias throughout the test period. The final step is to switch thecorrected chain relatives to a base of the average value. The average (97875) of line (5) isused to create line (6), taking the ratio of the corrected chain relative entries to their aver-age. The final result is the Index of Seasonal Variation. The accuracy of this result can beproved by averaging the entries of line (6), which will be 1.00. A complete study of sea-sonality using this method can be found in Courtney Smith, Seasonal Charts For Futures
Traders (Wiley, 1987).
  
The Moving Average Method
‘The moving average is a much simpler, yet very good technique for determining seasonalpatterns. Looking again at the cash corn prices in Table 7-3, take the average quarterlyprices for the years 1960-1965 rounded to the nearest cent. More practical results mayined by repeating this procedure for monthly prices.
Sane by ecpening hl roses mon Pawan moving mee callated and recorded in such a way that each value lags 2% quarters, corresponding to thecenter of the 4 points used in the calculation. Column 2 of Table 7-6 shows the Squaresmoving average positioned properly; Figure 7-3 is a plot of both the quarterly corn pr cesand the lagged moving average. By using the exact number of entries in the season,
i line is not affected by any seasonal pattern. ;a er sun even number of points in the moving average, each calculationfalls between two original data points. Column 3 of Table 76 is constructed by averagingevery two adjacent entries in column 2 and placing the results in a position correspone lingto the original data points. This avoids smoothing the initial prices. The difference 0 cotumn 1 minus column 3 is the seasonal adjustment factor (column 4) in cents per bus alithe seasonal index (column 5) is the ratio of column 1 divided by column 3. The periodicfluctuation of prices becomes obvious once these values have been recorded. A general-
~ ized seasonal adjustment factor and seasonal index are calculated by taking the average of
the quarterly entries for the five complete years (Table 7-7).
x11
‘The seasonal adjustment method X-11 (Census Method I-X-11) is most widely used forcreating a seasonally adjusted series of such information as car and housing sales, aswell as other consumer products. It is very extensive, involving both an initial estima-
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
  
+170SEASONALITY
 
TABLE 7-6 Seasonal Adjhstment by the Moving Average Method
POPULAR METHODS FOR CALCULATING SEASONALITY
71
FIGURE 7-3 Detrending corn using a moving average with a yearly period.
 
 
Average 4-Point Corresponding SeasonalQuarterly Moving 4Point Adjustment SeasonalPrice Average Values Factor Index1960 Jan-Mar 104Apr-Jun 110Jul-Sep 08 103% 103 +6Oct-Dec 90 102% | 101% -11% ‘ee1961 Jan-Mar 102 100% | 100% 41% hozAprjun 102 99% 100% 4% 1.02Jul-Sep 104 (10%, 99% 44% LosOct-Dec 95 : 99 98% -¥h ‘61962 Jan-Mar 95 90% 98% 3h 96Apr—jun 101 98 cy +3 ! 03Jul-Sep 101 98 99% 41% 1.02Oct-Dec 95 100% 102 7 31963 Jan-Mar 105 103% 106% -% 99Apr—jun 13 108% 110% +24 12Jul-Sep 122 1124 113% +8% 1.08Oct-Dec 109 113% 14, 5% “951964 Jan-Mar mW 114a% 113% -2% ‘97‘Apr-Jun 417 113 113% 4h hogJul-Sep 115 112% 112% 41% holOct-Dec 108 144% 113% -Th ‘941965 Jan-Mar u7 6 : 116% ~K 1.00‘Apr—jun 124 17% , ‘ore 124 ut 46% 1.05
Oct-Dec 109a —$—$———<———_—
———TABLE 7-7 Average Seasonal Variation Using -
 
the Moving Average Method \. SeasonatAverage of Adjustment SeasonalAllYears Factor IndexJan-Mar -11S 988Aprjun 43.53 1,052~ Jul-Sep $4.43 1.042‘Oct-Dec 7.00 932
————
tion and reestimation. Because it is widel mii i. ly used by economists, an outlin -dure follows. “ © of ts proce
1, Calculate a centered 12-month moving average (MA). Subtract this MA from theoriginal series to get an initial detrended seasonal series.
3 Amore detailed account of X-11 ind Henderson's wei
ighted moving average (step 9) can be found in Bovas Abra-ham and Johannes Ledolter, Statistical Methods for Forecasting (John Wiley & Sons, New York, pp. 178-191).‘Their book also includes a computer program for seasonal exponential smoothing,
 
130}
    
120(— Quarterlycorn prices
PRICE(¢/bu.)sT
 
 
100 |—90Toast 2a412a41294123412341960 «1961-1962, 1963, 19641965QUARTERS.
en LEU EEE IEEE EERE
2. Apply a weighted 5-period MA to each month separately to get an estimate of the
seasonal factors.3. Compute a centered 12-month MA of the seasonal factors (step 2) for the entire se-
ries. Fill in the six missing values at either end by repeating the first and last availableMA values. Adjust the-seasonal factors from step 2 by subtracting the centered 12-term MA. The adjusted seasonal factors will total approximately zero over any 12-month period.4, Subtract the seasonal factor estimates (step 3) from the initial detrended seasonalseries (step 1). This is the irregular component series used for outlier adjustment.5. Adjust the outliers in step 4 by the following procedure:a. Compute a 5-year moving standard deviation s of the irregular component se-ries (step 4).b. Assign weights to the series components ¢, as follows:Oifc, > 2.58linearly scaled from 0 to 1 if 2.55 2 ¢,2 1.58
Life, <1.5s -Use this weighting function to adjust the detrended series in step 1.6. Apply a weighted 7-period MA to the adjusted series (step 5) to get the prelimt-nary seasonal factor. -7. Repeat step 3 to standardize the seasonal factors.8. Subtract the series resulting in step 7 from the original series to find the prelimi-
nary seasonally adjusted series.
9. To get the trend estimate, apply a 9-, 13-, or 23-period Henderson's weighted mov-ing average’ to the seasonally adjusted series (step 8). Subtract this series from theoriginal data to find a second estimate of the detrended series.
+A specialized symmetric assignment of weighting values. A specific example can be found in Bovas Abraham andJohannes Ledolter, Statistical Methods for Forecasting (John Wiley & Sons, New York, p. 178). 
 
In SEASONALITY
10. Apply a weighted 7-period MA to each month separately to get a second estimateof the seasonal component.
11. Repeat step 3 to standardize the seasonal factors.
12. Subtract the final seasonal factors from the original series to get the final season.ally adjusted series.
Winter’s Method
Another technique for forecasting prices with a seasonal component is Winter's method,a self- generating, heuristic approach.’ It assumes that the only relevant characteristics of
price movement are the trend and seasonal components, which are represented by theformula
X,= (a+ bDS, + e,
where X, is the estimated value at time ¢(@ + 60) is a straight line that represents the trendS, is the seasonal weighting factore, is the error in the estimate at each point
If each season is represented by N data points, 5, repeats every N entries, and> Sat
The unique feature of Winter's model is that each new observation is used to correctthe previous components a, 6, and S,. Without that feature, it would have no applicabilityto commodity price forecasting, Starting with 2 or 3 years of price data, the yearly (sea-sonal) price average can be used to calculate both values a and b of the linear trend. Eachsubsequent year can be used to correct the equation a + bt using any regression analysis.Winter's method actually uses a technique similar to exponential smoothing to estimatethe next a and 6 components individually. The seasonal adjustment factors are assigned bycalculating the average variance from the linear component, expressed as a ratio, at eachpoint desired. As more observations are made, each component can be refined, and it willtake on the form of the general long-term seasonal pattern. -
\A Comparison of Seasonality Using Different Methods
Heating oil and soybeans represent two fundamentally sound seasonal markets. Evenbefore we analyze the patterns, we can expect heating oil to post higher prices during thewinter months and soybeans to be high during the summer. Heating oil is accumulatedbeginning in midsummer in anticipation of the winter season, while soybeans are subjectto speculative volatility during planting and growing seasons.
For the purposes of comparison, we will use four basic methods of calculating seasonal-ity: average price, median price, percentage change from the previous month, and the mov-ing average deseasonalizing. For our purposes, more complicated methods are unnecessary.
Figure 7-4a shows that the peak price for heating oil is most likely to come in October,well below the coldest months of the year. According fo these techniques, it is the antici-pation of winter that drives prices up, while February and March, typically colder, showlower average prices as most consumers use up the oil they had committed to buy duringthe previous September and October. Three of the methods for calculation gave similarresults, while %Change seems to lead by one month.
* Winter's method, as well as other advanced models, can be found in Douglas C. Montgomery and Lynwood A.Johnson's Forecasting and Time Series (McGraw-Hill, New York, 1976), and Bovas Abraham and JohannesLedolter’s Statistical Methods for Forecasting (John Wiley & Sons, New York, 1983).
 
+static intents
|
173POPULAR METHODS FOR CALCULATING SEASONALITY
 
uniform %Change leadingHeating oil seasonal patterns. Results are very unif hROURE TAO Meningeal atccaa Scey srte esmegeesecur lation, with lower prices in February and March when consumers are nor ly deplesinieventor % (b) Soybean seasonal patterns. The four basic calculations show soybean pres Pein fe an reaching lows in Fear Th nt eeu nary neaerage, and higher duringiegehat there ere umestally ‘extreme moves during those months that distort the average.ing
Heating oil seasonality 1980-Mar 1994All years from continuous data
 
Percentage price change for month
 
 
 
L 1 1sl
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
—e— %Change -—O— Detrended = —t}~ Median
(a)
—m Average
Soybean seasonality 1969-1993at yeats from continuous data
 
 
 
Percentage price change for month
 
H 1 1 1 L‘Aug Sep Oct Nov Dec
 
oeApr May Jun Jul
1van Feb Mar
 
—e— %Change | —O- Detrended = —t}~ Median
)
eo
a Average 
174 SEASONALITY
 
When applied to soybeans, all four basic seasonal calculations produce patterns thatshow the expected winfer and summer cycles, with most lines rising and falling smoothlythroughout the year (see Figure 7-46). Again, it should be noted that the seasonal patternof the percentage price changes leads the normal and detrended methods because it actsin the same way as momentum, a decline in the percentage change does not mean that cur-rent prices are lower than the previous month, but that prices did not increase by as muchas the previous month.
Seasonal Volatility
Consistent seasonal patterns can be confirmed by a corresponding increase in volatility asthe season reaches its peak as seen in Figure 7-5. In this soybean chart, the peak volatilityin June, represented by one standard deviation of price equal to about 12%, shows thatthere is a 68% chance that soybeans will vary by 24% in June. For the purpose of confirm-ing the validity of the seasonal results, the steady increase and decrease in volatility sur-
- rounding the peak month of June indicates a building of the seasonal concerns that cause
wide fluctuations. If a single price shock had occurred, unrelated to seasonality, therewould be a sharp increase in volatility during one month, perhaps declining afterward, butwith no steady growth of volatility leading up to this event.
WEATHER SENSITIVITY
The effects of changes in weather, especially extremes in weather, are an inseparable partof seasonal effects. Without weather, the price of an agricultural commodity would lack thesurprises that cause them to jump around during the growing period. Once planted, youwould have a very good idea of the expected supply; one-half of the supply-demand equa-tion would be constant each year.
Each agricultural product has its own particular sensitivity to weather. Grain planting canbe delayed due to rain, causing some farmers to switch from corn to soybeans; hot weatherduring pollination will significantly reduce yields; a freeze in September can stop the ripening
 
FIGURE 7-5 Soybean monthly price variation.
 
 
20
 
Variance as percent of price
“10 pre
16 popJan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
 
 
 
 
Detrended —®—-+1stdev —t ~1stdev
 
WEATHER SENSITIVITY 175
 
process and damage production. Freezes are of greater concern than droughts and affectmore products. In general, patterns for crop sensitivity to weather depend upon their locationin the Northern or Southern Hemisphere. Active trade between world markets siiows thatcrops grown primarily in the Northern Hemisphere are affected by weather developments inthe Southern Hemisphere. In Table 7-8 major weather events are shown separated into South-ern and Northern Hemisphere. Figure 7-6 combines the soybean and corn weather sensitivitycharts® to show the effects of both Southern and Northern Hemisphere weather.
Measuring Weather Sensitivity
While a weather sensitivity chart, such as shown in Figure 7-6, may appear to have a strongsimilarity to a standard seasonal chart, they are actually very different. Weather sensitivity isa measurement of potential price volatility. It could simply be the highest price that the mar-ket reached during a period in which the weather event occurred. Those more adept at sta-tistics would want to record the increase in price from the average during those months inwhich weather was a factor, then show the price representing the 95% confidence level.
A thorough approach to measuring weather sensitivity is to record the temperatureeach day and find out whether it is unusually far above or below the average. Informationavailable from the U.S. weather or meteorological service will give you both regionalweather and measurements of effective heat. It is the cumulative effect of heat that is dam-aging to a crop, rather than a high ‘temperature for one day. In addition, the amount ofcrop damage is not simply a linear relationship with rising temperatures; rather, it is morelikely to start slowly and increase quickly once critical levels of time and heat are reached.
© Jon Davis, Weather Sensitivity in the Markets (Smith Barney, October 1994).
—
TABLE 7-8 Weather-Related Events in the Southern and Northern HemispheresSouthern Hemisphere Northern Hemisphere
January - ‘OJ peak for Florida freezeHeating oil cold or hot
February Corn pollination in S. Africa Heating oil cold or hot
and Argentina
Soybean pod developmentMarchApril Cotton plantingMay Soybean plantingJune Cotton boll developmentJuly Coffee freeze Brazil Sugar: heat in Russia
- Of freeze Brazil Corn pollination
‘Cocoa pod development in W. Africa
and pod rot in Brazil — ~August ~ Soybean pod developmentSeptember - _ Atlantic hurricanes affect sugar,
orange juice, heating cilCorn freeze
October Soybean freezeCotton harvest,
November Coffee in Brazil rainy season Soybean harvestTorn harvestDecember Heating oil cold or hot
ae Ee 
176 SEASONALITY
 
FIGURE 7-6 Weather sensi
 
ity for soybeans and corn.
 
           
U.S. poddevelopmentUS.pollinationS S. African, U.S.2 Argentine us. harvest> pollination, Hem. pod&
      
| developmentf
US.planting
 
253
(Very low)
 
 
Source: Smith Barney.
SEASONAL FILTERS
Seasonal filters identify periods during the year in which you would favor either a long orshort position in a market based upon a clear seasonal component. These periods are nor-mally chosen from the results of a calculation that uses detrended data; however, wherethere is a large difference between the median value and the average value, we shouldexpect the median value to be more representative of expected price movement. The largedifference between the median and average also tells us to expect extreme prices and highvolatility during those months. This can be seen in Table 7-9, showing monthly soybeanPrices as a percentage of the annual average price. The double-digit values in 1973, 1977,and 1988 are enough to cause the averages at the beginning of the year to be lower thanthe median, and June, July, and August to be higher than the median. This bias is mainlythe result of only 3 of 24 years. _
Using the examples of heating oil and soybeans, which have been discussed in the pre-vious section, we can select the months with the largest and most reliable seasonal pricemoves as the best choice for seasonal filters. We can also separate the year into upward anddownward cycles to allow for a wider range of trades. The following periods could be con-sidered for heating oil and soybeans:
 
Best Choice (Partial Year) Full-Year Seasonal Patterns
 
Heating oil Long from August | to November | Long from April | to October |Short from November | to March | Short from November | to March |Soybeans Long from March | to july { Long from March 1 to July |Short from July 1 to November | Short from July | to March 1
 
SEASONAL FILTERS- 177
 
 
TABLE 7-9 Monthly Soybean Prices as a Percentage Difference from the Average AnnualPrice, February 1969-December 1993
Year jan Feb. ~— Mar
 
May jun jul Sep Oct Now
   
69 O17 -O.15 0.02 -0.48 -047 O11 050 0.6070-183) -1.62 -1.61 -1.04 O11 151 0.28 1,82 2.2471 -0.31 -042 -088 -1.60 0.34 2.20 052 0.99 0.0772 4.10 -3.42 ~1.56 0.78 -1.38 -1.12 0.47 0.57 3.2373 -25.18 -1667 -13.26 6.23 25.10 15.97 3.93 -0.42 -1.8574 -1.87  -3.20 -9.14 -881 1.80 5.63 11.83 6.1075 1.69 0.19 -2.71 -285 1.22 223 -1.55 ~5.4576 -7.00  -7.60 4260 3.74 7.14 5.90 4367 O41 TAB 16.24 7.35 -6.01 ~10.31 S7678 665 -741 —O5I 31S 1.25. -0.73 1.20 2.6479-010 3.44 3.78 0.80 383 232 0.90 49280-094 -1.24 4.79 A711 -648 1.39 5.98 10.40Bl 10.25 7.30 5.54 475 0.67 0.84 6.58 8.9682 626 531 3.49 5.80 237 0.66 6.54 5.5383 832 -7.86 -7.86 -7.08 -9.78 4.59 16.19 7.3884 791 4.09 9.25 14.64 7.31 4.92 9.44 9.2985 6.79 5.38 5.31 2.00 2.07 -041 A73 6.6286 282 046 = 0.52 0.81 0.09 -0.16 1.26 1.0087 446 -5.68 | -5.37 281 3.28 -0.55 0.12 47488-1245 1292-1234 2.36 1832 1291 859 -L7S89 11.07 6.75 8.28 250 171 0.36 6.03 6.7690 1.410 0.21 2.39 490 til 089 0.68 6.66a 433 442 (4.56 149 -0.15 4.66 146 -54792 132. 232 9 4.03 478 602 -151 3.99 -2.7393 -634 -7.79 609 4.82 -3.49 -3.33 12.00 1.70 5.07 8.03
  
Avg -0.94 -1.29 0.42 0.16 125 206 1.44 0.75 0.20 -0.92 -0.96 -1.38
Mdn -063 0.16 0.15 os! Lil 0.66 047 -0.96 -1.75
     
One practical note is needed here. Although there may appear to be a statistical benefit to sell-ing heating oil in the later part of winter, there is also extreme risk. Prices jumped 40% fromMarch to April 1982, and 20% during the same period in 1983. We must also consider that theGulf War, which pushed prices higher from August 1990, conformed by chance to the seasonalpattern; in reality, it could have happened any time during the year. These special situationsdo not always cause prices to rise. In 1986, Saudi Arabia decided to dump oil in a politicallymotivated effort to drive prices down, successfully pushing crude oil under $10 per barrel
 
Years with Similar Characte
Seasonal studies often yield results that are not as clear as desirable, and these results maybe rejected because of the obvious lack of consistency. Often, this is caused by a few yearsthat conflict with the normal seasonal patterns due to special events. David Handmakershows that separating the data into analogous years can give strikingly better results.”
 
? David Handmaker, “Low-Frequency Filters in Seasonal Analysis,” Journal of Futures Markets, Vol. 1, No. 3 (ohnWiley & Sons, New York, 1981). 
178 SEASONALITY
 
For example, crop production is primarily determined by weather. Poor weather willcause sharp rallies during the growing season, whereas good weather results in a dull,sideways market. Bad weather develops slowly. A drought is not caused by the first hot daybut prolonged days of sunshine and no rain. Similarly, delayed planting due to wet fieldsor a late winter will set the stage for an underdeveloped crop. A trader can see the charac-teristics of a weather market in time to act on it.
In Figure 7-7a, the seasonal corn pattern has been separated into those years withgood weather and those with bad weather. In a later study by ContiCommodity, soybeanseasonality was separated into bull and bear years (Figure 7-76). Bull years include all badweather years but also years with such incidents as the 1973 Soviet grain sale. Becauseother events are not confined to the growing months, they may distort rather than clarifythe patterns. Bear would represent mostly good weather patterns; however, markets thatdecline steadily during the year following a bull market would also be included.
During the bad weather years, the corn chart shows a prolonged rise during the pri-mary growing months, May through August. The study of bull years for soybeans shows alonger sustained rise from the beginning of the new crop in October into the followingJune. In the case of good weather and bear markets, corn shows a rise from May to June,some sideways hesitation from June to July (possibly the last anticipation of bad weather),then a return to previous price levels. For bear years soybeans have steady declinesthrough April and then a short rally through the summer.
‘The corn analysis shows a critical point around July 1, when the market decides thatthe weather is not a factor. The soybean analysis is completely different showing nearly theopposite patterns for bull and bear markets. Because the resulting pattern of combiningboth bull and bear markets looks very similar to the bull pattern, the magnitude of the bullmoves must overwhelm the overall picture. In fact, the magnitude of the bull moves areoften three to four times greater than the price change in bear markets.
Updating Bull, Bear, and Nonseasonal Market Patterns
Both the Handmaker and ContiCommodity studies ended in the early 1980s and while grainfundamentals have essentially remained the same, the use of more data and recent years canonly help understand seasonality better. Using the soybean average prices expressed as apercentage of average annual price in Table 7-9, we separated the years from 1969 to 1993into bull and bear categories, where bull years (and bear years) had significantly higher (orlower) prices at the end of the year than at the beginning. Table 7-10, parts (a) and (), whichalso give the prices as a percent of the average annual price, show that there were about thesame number of years in each category. The averages at the bottom of those tables indicatethat both bull and bear years had a price swing of about 10% over the year.
Bull and bear years are nearly opposite in pattern, as seen in Figure 7-8, and cross atabout the same price in June. This makes the seasonal pattern using all years peak in June.As interesting as it is to know that there are an equal number of years. going the oppositedirections within a seasonal pattern, the bull and bear tables were created after the factand can not be traded unless you know in advance that a bull or bear year will occur.Instead, it is possible to look at nonseasonal years, those in which the- price of soybeanswas higher during the harvest months of September and October than during the previ-ous summer months.
One important quality of crops is that each year we begin again; a shortage one yeardoes not mean a shortage the next year. This is seen in the chart of nonseasonal years in Fig-ure 7-8, After a nonseasonal year, prices rally early, nervous about planting and shortages,
® ContiCommodity, Seasonality in Agricultural Futures Markets (ContiCommodity Services, Chicago, 1983)
SEASONAL FILTERS - 179
 
FIGURE 7-7 (a) Corn price seasonality separated into all years, years with good weather, andyears with bad weather. (b) Soybeans (10 years) shown with separate bull years (5 years) andbear years (5 years).
 
 
ns — tt
not \ 4{\
os. Yew good weather 7 V\ 4
    
‘Au years
 
 
 
 
 
95 4NKFo) 17 Ff MAM S GAS nD@
120
TT TT wa TT TT TT100}- / 4
 
 
Bear years Nee
—100{- 4
 
 
120Jan Feb Mar Apr May June July Aug Sep Oct Nov Dec (Jan)
©
 
but the reality of a good crop and normal demand drive prices steadily lower during the sec-ond half of the year. The pattern is typically seasonal and very well defined by large swings.
Seasonality for Trading
Whatever method is used to find the seasonal pattern, it is equally important to apply itproperly-when trading. Select a market that has a dominant seasonal pattern. The cropsalways qualify, with the exception of potatoes, which are fully stored and do not react tothe harvest depression. Livestock tends to be more complex than other commodi
  
180 SEASONALITY
—_TABLE 7-10 Bull, Bear, aid Nonseasonal Market Patterns
(a) Bull years begin at relatively low levels, peak during the summer but remain high at the end of the year. (b) Bearyears decline steadily after April and do not reflect the concerns that often drive prices higher during the summer.
(c) Nonseasonal years follow years in which the harvest months of September-October posted higher prices than thes