99.1 965 100.0 101.8 [04.4 103.5 100.9 100.9 1035 929 92.0 10091965 1.18 98.3 99.1 100.8 103.4 105.9 1068 105.1 101.7 100.0 932 881 95.81966 1.25 952 96.0 936 952 68 96.0 10/6 107.2 1080 103.2 100.8 103.21967 1.17 109.4 107.7 109.4 107.7 1068 107.7 1034 94.9 95.7 889 829 88.01968 1.04 100.0 101.9 101.9 101.9 1048 102.9 100.0 95.2 97.1 92.3 100.0 101.01969 1.12 964 97.3 97.3 100.0 106.2 105.3 96.4 105.3 1027 100.0 955 97.3{970 1.24 903 91.9 911 927. 95.2 97.6 100.0 1024 JII.3 108.1 1040 112.11956-1970 Avg 96.3 96.9 98.1 100.9° 103.9 104.7 104.0 103.9 101.5 ~94.3 90.9 96.11971 1.27 118 1126 1126 ILO. 1087 112.6 107.) 93.7 87.4 “787 76.4 85.01972 V.17 93.2 93.2 94.0 966° 983 966 97.4 983 104.3 101.7 1026 121.41973 186 747 726 736 76.3: 624 107.0 109.1 144.1 115.6 1167 117.2 12851974 292 887 945 918 825 839 88.0 99.6 1154 113.0 1181 1137 112.01975 270 113.7 1059 989 99.2 985 99.2 100.7 109.2 102.2 97.0 863 878
1971-1975 Avg 96.4 958 94.2 93.1 90.1 100.7 1028 1121 104.5 1024 99.2 106.9
 
Removing the Trend
Jake Bernstein, most well known for his seasonal studies,” uses the method of first differ-ences to remove the trend from prices before calculating the seasonal adjustment factor.
He offers the following steps for determining the cash price seasonality: .
1. Arrange the data used in a table with each row as one year. Columns can be daily,weekly, or monthly, although most analyses will use monthly. Average prices are pre-ferred for each period (see Table 7-1).
2. Compute a second table of month-to-month differences by subtracting month 1from month 2, month 2 from month 3, and so on. This new table contains de-trended values. _ .
3. Calculate the sum of the price differences in each column (month) in the new table.Find the average for that column by dividing the number of years of data (columnsmay have different numbers of entries). This is the average price change for thatmionth.
4, From the table, count the times during each month (column) that prices were up,down, or unchanged. This will give the frequency (expressed as a percent) of move-ment in each direction.
Bernstein adds the average monthly changes together, expresses the frequency of upwardprice changes, and presents the results of corn in Figure 7-2.
? Jacob Bernstein, Seasonal Concepts in Futures Trading (John Wiley & Sons, New York, 1986).FIGURE 7- Changes in volatility and variations in seasonal patterns for corn.
 
160
      
140
Mean1971-1975
CHANGE (%)
80
60 ——— 1971-1975—r- 1956-19701956-1970 Variation
40
Jan Feb Mar Apr May June July Aug Sep Oct Nov ° Dec
 
 
 
FIGURE 7-2 Seasonal price tendency in monthly cash average ‘corn prices (1936-1983).
 
Source: Jacob Bernstein, Seasonal Concepts in Futures Trading (New York: John Wiley & Sons, 1986,p, 31). Reprintedwith permission of John Wiley & Sons, Inc,168 - SEASONALITY
 
The Method of Link Relatives
Another interesting and important way of identifying the seasonal price variations and sep-arating them from other price components involves the use of link relatives. In Table 7-5each month during 1960 and 1961 is expressed as a percentage by taking the ratio of thataverage monthly price to the average price of the preceding month (found in Table 7-3) ina manner similar to an index.
After the initial calculation of 1960 and 1961 link relatives, it is necessary to find the aver-age (or the median, which is preferred if an adequate sample is used) of the monthly ratiosexpressed in rows (1) and (2), The average in row (3) represents monthly variation as a per-centage of change; each calculation is a function of the preceding month. Thus far, this is thesame as Bernstein’s average monthly price changes, expressed as a percent of the prior price.
To establish a fixed base in the manner of an index, chain relatives are constructed usingJanuary as 100; each monthly chain relative is calculated by multiplying its average link rela-tive by the average link relative of the preceding month. The March chain relative is then1.005 x 1.025 = 1.030, and February remains the same because it uses January as a base.
Aconstant trend throughout the test period can be found by multiplying the Decemberchain relative (4) by the January average link relative. If prices show no tendency for eitherupward or downward movement, the result would be 1.00; however, inflation should causean upward bias and, therefore, the results are expected to be higher. From line (4), theDecember entry multiplied by the January entry on line (3) gives .946 x 1.047 = .990, leav-ing a negative factor of 1% unaccounted. This means that the 1960-1961 years showed a
.1% downward bias; therefore, the expected rate of inflation was offset by some other eco-nomic factor, such as the accumulation of grain stocks by the U.S. government.
The chain relatives must be corrected by adding the negative bias back into the values,using the same technique as in computing compound interest. For example, from 1967—1977, the Consumer Price Index increased from 100 to 175, a total of 75% in 10 years. To cal-culate the annual compounded growth rate for that period, apply the formula:
     
valuestarting value
  
Compound rate of growth =
where AN is the number of years or the number of periods over which the growth i is com-pounded.
10/ 17510 1
= 1.05755 -1
 
R
= .05755
 
TABLE 7-5 Corn Prices Expressed as Link Relativesjon Feb Mar Apr May Jun “Jul Aug Sep Oct Nov Dec
 
 
(I) 1960 1.040 1.010 1.000 1.048 1.018 1.000 1.000 .982 .982 .907 .845 1.122(2) 1961 1.053 1.040 1.010 .942 1.061 1.000 1.019 .981 .981 .980 .910 1.031(3) Average 1.047 1.025 1.005 .995 1.040 1.000 1.010 .982 .982 .944 878 1.077(4) Chain relatives 1,000 1.025 1.030 1.000 1.035 1.040 1.010 .992 .964 .927 .829 .946
(5) Corrected chain relatives 1.000 1.024 1,028 .997 1.032 1.036 1.005 .986 .958 .920 822 .937(6) Indices of seasonal variations 1.022 1.046 1.050 1.019 1.054 1058 1.027 1.007 .979 .940 840 .957POPULAR METHODS FOR GALCULATING SEASONALITY 169
 
This indicates a compounded rate of inflation equal to 5.75% per year. In the case of corn,if the trend had been positive, that is, greater than 1.00 instead of .990, the growth ratewould be subtracted from each month to offset the upward bias. In this case, the resultsare added back into the chain relative to compensate for the negative influence. A .1%decline, compounded over 12 consecutive entries gives:
 
= .99916-1
= —.00084
This is a compounded deflation of about 8/100 of 1%. The corrected chain relative wasfound by multiplying the February entry by (1 + R) = .99916, March by (1 + R)? = .99832,and December by (1 + 8)" = 99076.
The chain relatives have 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 FuturesTraders (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 may beobtained by repeating this procedure for monthly prices.
Because every four entries completes a season, a 4-quarter moving average is calcu-lated 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 4-quartermoving average positioned properly; Figure 7-3 is a plot of both the quarterly corn pricesand the lagged moving average. By using the exact number of entries in the season, themoving average line is not affected by any seasonal pattern.
Because there was an even number of points in the moving average, each calculationfalls between two original data points. Column 3 of Table 7-6 is constructed by averagingevery two adjacent entries in column 2 and placing the results in a position correspondingto the original data points. This avoids smoothing the initial prices. The difference of col-umn 1 minus column 3 is the seasonal adjustment factor (column 4) in cents per bushel;the 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 ofthe quarterly entries for the five complete years (Table 7-7).
X-Hi
The seasonal adjustment method X-11 (Census Method II-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-170 SEASONALITY
 
TABLE 7-6 Seasonal Adjistment by the Moving Average Method
 
 
Average 4-Point Corresponding SeasonalQuarterly Moving 4-Point Adjustment SeasonalPrice Average Values Factor Index1960 Jan-Mar 104Apr—jun (10JubSep 109 103% | 103 +6 1.06Oct-Dec 90 102% | 101% -IN%K 881961 Jan-Mar 102 100% 100% +h 1.02Apr-Jun 102 994 100% +h 1.02Jul-Sep 104 : 100%, 99% +4 1.04Oct-Dec 95 99 98% -3% 961962 Jan-Mar 95 98% 98% -3% 9%Apr-Jun lol 98 98 +3 1.03JubSep lol 98 99% 42h 1.02Oct-Dec 95 100% 102 7 931963 Jan-Mar 105 103% 106% “lk 99Apr-Jun 113 108% 110% 424A 12Ju-Sep 122 W124 113% 48% . 1.08Oct-Dec 109 V3 1144 5A 51964 Jan-Mar ut 4% 113% -2% 7Apr—jun u7 113 113% +44 1.04Jut-Sep WS 112% mo V12% +14 LolOct-Dec 108 N44 : 1134 -Th 941965 Jan-Mar "7 16 = 6% “4 1.00Apr-Jun 124 N74 117% +6% 1.05Jul-Sep 121 .Oct-Dec 109
 
 
TABLE 7-7 Average Seasonal Variation Using ~
the Moving Average Method \SeasonalAverage of Adjustment SeasonalAllYears Factor IndexJan-Mar —1AS 988Apr-Jun +353 1,052- Jul-Sep +443 1.042Oct-Dec -7.00 -932
 
tion and reestimation. Because it is widely used by economists, an outline of its proce-dure follows.?
1, Calculate a centered 12-month moving average (MA). Subtract this MA from theoriginal series to get an initial detrended seasonal series.
3A more detailed account of X-11 Sind Henderson's weighted 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 aiso includes a computer program for seasonal exponential smoothing.POPULAR METHODS FOR CALCULATING SEASONALITY 171
 
eeFIGURE 7-3 Detrending corn using a moving average with a yearly period.
 
Quartertycorn prices
PRICE(¢/bu.)
 
1234123412341234123412341960 1961 1962 1963 1964 “1985,QUARTERS
 
2. Apply a weighted 5-| period J MA to each month separately to get an estimate of theseasonal 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-ties (step 4).b. Assign weights to the series components c; as follows:Oifc, > 2.5slinearly scaled from 0 to 1 if 2.5s 2 c,2 1.5slifc,< 15s _Use this weighting function to adjust the detrended series iri step 1.
6. Apply a weighted 7-period MA to the adjusted series (step 5) to get the brelimi-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).
 172 SEASONALITY10. 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 ofprice movement are the trend and seasonal components, which are represented by theformula
X,= (4 + b1)S, + &
where 4X, is the estimated value at time ¢(a + bt) 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, $, repeats every N entries, and
»SSst
iL
The unique feature of Winter’s model is that each new observation is used to correctthe previous components 4, b, 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 of the linear trend. Eachsubsequent year can be used to correct the equation 4 + bt using any regression analysis.Winter's method actually uses a technique similar to exponential smoothing to estimatethe next a and b 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 will
take 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 to 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 JohannesLedokter’s Statistical Methods for Forecasting (John Wiley & Sons, New York, 1983).POPULAR METHODS FOR CALCULATING SEASONALITY 173
 
———FIGURE 7-4 (a) Heating oil seasonal patterns, Results are very uniform with %Change leadingthe other methods by one month. Peak prices come in October, during the time of greatestaccumulation, with lower prices in February and March when consumers are normally depletinginventory. (b) Soybean seasonal patterns.The four basic calculations show soybean prices peak-ing in June and reaching lows in February. The use of the median gives results in July that aremuch lower than the average, and higher during the period November through January, indicat-ing that there were unusually extreme moves during those months that distort the average.
Heating oil seasonality 1980—Mar 1994All years from continuous data
15
 
 
Percentage price change for montha
 
 
i }5Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
 
—™— Average  —®— %Change  —O— Detrended —C Median(a)
Soybean seasonality 1969-1993All years from continuous data
 
Percentage price change for month
 
 
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Average  —®—%Change —O~ Detrended — Median(b)174 SEASONALITY
 
When applied to soybeans, all four basic seasonal calculations produce patterns thatshow the expected winter 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 causewide 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.
 
 
Variance as percent of price
 
 
 
 
5Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
—— Detrended —@- +1stdev —{} -1stdevWEATHER SENSITIMITY 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 Hemispheres
 
 
Southern Hemisphere Northern HemisphereJanuary - OJ peak for Florida freezeHeating oil cold or hotFebruary Corn pollination in S. Africa : Heating oil cold or hot
and ArgentinaSoybean pod development
MarchApril Cotton plantingMay Soybean plantingJune - Cotton boll developmentJuly Coffee freeze Brazil Sugar: heat in Russia~ O} freeze Brazil Corn pollinationCocoa pod development inW. Africaand pod rot in Brazil ~August ™ Soybean pod developmentSeptember - _ Atlantic hurricanes affect sugar,orange juice, heating oilCorn freezeOctober Soybean freezeCotton harvestNovember Coffee in Brazil rainy season = Soybean harvest
Corn harvestDecember Heating oi! cold or hot176 SEASONALITY
 
 
FIGURE 7-6 Weather sensitivity for soybeans and corn.
 
                  
U.S. poddevelopmentpollination
S S. African, p U.S.= Argentine US. harvesta pollination s Hem. pod freeze= development
US.2 planting=3 \25oD=2 ’2=
 
Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Jan
 
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 |Short from November | to March |Soybeans Long from March | to July |
Short from July | to November |
Long from April | to October |Short from November | to March |Long from March | to July |
Short from July | to March |SEASONAL FILTERS- \77eS
———_TABLE 7-9 Monthly Soybean Prices as a Percentage Difference from the Average AnnualPrice, February |969~December 1993
 
Year fan Feb Mar Apr May jun jul Aug Sep Oct Nov Dec
 
69 017° -O.15 -0.07 0.02 -048 -047 -0.26 -O11 050 0,60 0.2570 1.83 1.62) -1.61  -1.25 -1.04 O11 151 0.33 0.28 1.82 2.2