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 niajoroil companies, such as Exxon, may not show a seasonal pattern similar to crude oi 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- 
* 462 SEASONALITY
 
 
 
 
   
     
   
 
     
    
  
   
  
 
SRLSPLS AIRGAS STS SSS RTA SSR Bazs3 Fai a Oaracna zn = sysrE/SFSESSSSASRESSSSSSRBSSSSS [ETuous contract. This contract is gap-adjusted to remove the price jumps in futures when BER SLRARREARSSSSRRSRERSSS [o'sdelivery 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 the : SLShRERASTHRIEARSRAARSIVZARY RGSSmid-1970s are about $2 per bushel higher. In some cases, such as interest rates, prices that i 3 SSSSSSSARSSSESABSSSESSSRS [STFare 25 years in the past can actually become negative because of gap adjusting. A precise SAS essrsshs =seasonal analysis will need to use continuous cash prices, rather than a constructed series, o nonneun <to have valid percentages. Results of these examples, however, show seasonality similar to SRSABSSHRRR SSR AT SR YTIAASA Sarsstudies that use cash data. The following terminology and concepts will also be of help. Z/S|SESRSSRRRSSLSASSTRESSLIES [Aras5Basic Calculati mi i 2 = o wo =jon Components i £| |BASEASSSISERSSLRASRRIRATAB (RST7 Bl=-we sarod wnnimandnr=~xn—H ad =Average Prices | 7S sSSREZRSESSSTRESSLRSERSERS [ersFinding the seasonal pattern does not need to be complicated; however, some basic rules { 3 Sareea eae - -must be followed to get sound results. For most analysts, it is easiest to begin with a spread- { >1 BEMLLASARSSKSSSeSrByesesege jassheet, where the months are recorded in each column and the rows represent years (see Table | EF ~ Bee eee ee eee soe eacs e2887-1). The average monthly price, placed in each cell, can give a good indication of seasonal §(<|)SSRRSRRARSTRLR LRT 8 Sar sRas gs 8patterns by simply averaging each column and plotting the results as shown in the first of four 3 TTS aesummary lines at the bottom. The major criticism of this technique is that it ignores the chang- S| levoneeazsseg-agensegetaaag [Sana: ing price levels over time. For example, a 25-year study of soybeans will use prices that vary S|SI\BR ARE ESSSS ean eReneSgeese |g-#F' from $6 to $15 per bushel; price changes at the $15 level could overwhelm other years. B/ | SSSSSERARLERASSSRRR SSR S a &. 3 ZRAREE s‘ 2Indexing the Data 2 SkeS=ess eeseertaneaang |geraA simple way to adjust for price differences over time is by indexing data, where each new |s|\Seeagans ¥¢ z gage g 3 Re [BtF-entry is based on a percentage change from the previous value. This method will also work g| FSR SR RAS SGeRR ROR YSS Iswith seasonal studies, but must use unadjusted cash data, Because the monthly average 8Prices for these examples were created from TradeStation continuous data, they cannot be B| JSSESALSTRASRSRIKRSRSRTRTA [HVSBi 2) Slaesoaanag TROnHwvunr~ ww - rr am g-seused for this purpose. 2/2/8582 582 ESSSSSSlskkxeeeSSS jesPercentage Change g Toe2 nOver the mariy years needed for a seasonal study, prices may increase by a factor of two or ©) |SSSRSqSSSVSSERSSSSRTVMrRSe [LRRS> s y °y el) 3 RR RRATAR OMEN KH ARS Yad SVY Hoagthree, and the magnitude of volatility at these different levels will vary significantly, distort- Bs) S/SESARTSESSSSSSSSSRESERESS [BORing the results. To correct for the problems associated with a long-term increase or decline 2 Sita srassees =in price, a monthly percentage change can be substituted for the average price. The use of 2<2 - pens-aeiseeepercentages adjusts for most of the volatility changes and, at the same time, removes any 8| .|$32=238s FESS SAR SST SRST BBlERSEJong-term trend bias. It is a simple and very credible method for finding seasonality, but its s\2)sSe28e8 SRSEZESSRASSRS SSIS gS"results can be more erratic than the sophisticated calculations found in the next section, a -- see 7g uwDetrending 2] (anne SRgSR Rac eargsjaggs. SlalSduas 2 SAnSSAegeslevsRemoving the trend using a moving average, then comparing the percentage differences g|2|/S8s8s RSRSERRSRASSH S/S TSTbetween the price and the corresponding trend value, is a classic approach to evaluating = - 7 ,seasonality. =u 2~ 3 3R884 Se SRSLARLAIAR TRA Sage; _ _ -22 2 wo Sa-s¥¥HS wo air-Median Price _ . &| 5) S8855 S83 SSSLRFESETSS|STEThe median is the middle price of a series of values that have been sorted in numerical 2 77 77 ~order. Instead of using an average monthly price, or the average of, for example, all June E ae XOMCNROLSHSSBSRSESSLEHREIBEEEa r e, q iz RIS zee BSS SLE SEaverage prices, using the median value should improve results. The median ignores the a|2|2e8 ERRRRRRRRERRRRRRARRERES Z52zextreme prices that occur during some unusual years and gives a typical price. For exam- 2s ~ple, if the average September price of heating oil was 50 cents per gallon for four years, but z - si3sreachetl $1.00 per gallon during one year, then the average would be 60 cents and the wy - <g>2median would be 50 cents. Later in this chapter you might note that a large difference 3 =between the median and the average prices indicates that there were a few volatile years = <
 
 
mixed with mostly normal patterns. :4 163
   
lea SEASONALITY
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 ratiooF 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 pattern. 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 thé traditionalseasonal highs and lows, but shows thar 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 Jon Feb Mar-Apr May JonPrice* 2.70 3.07 2.86 267 2.68 2.66 2.68Percent 100.0 113.7 105.9 98.9 99.2 98.5 992
jut Aug Sep Oct Nov DecPrice 27 2.95 276 2.62 2.33 2.37Percent 100.7 109.2 102.2 97.0 86.3 378SB
*Midmonth US.farm price, 1975.
 
ttii
 
POPULAR METHODS FOR CALCULATING SEASONALITY 165
TABLE 7-3 Corn Cash Prices
 
 
   
Year Avg jan_—Feb Apr Jun Jul Aug Sep Oct_ «Nov. Dec19s6r 1.30 141.16 132 144 145 146 145 112 1.22 1.231957 125 122 1.16 tis 120 121 1.22 113 1.04 98 1.011958 110-9798 118 123° 123 1.24 LIS 1.02 94 1.041959 1.09 1.05 1.06 117 119 117 117) 10996 1.00991960 1.03 1.03 1.04 1.09 Lit Lib 109 107.97) 8292.1961 1ol 991.03 98 1.04 1.06 1.04 1.02 1.00 91 941962 98 94 95 98 103 1.02 101 100 94 91 1.001963 Hub 1,03 1.05 1.09 119 12t 1.21 123 1.04 1.02 1.201964 1130 112 1.09 Las WIT 14 114 LIZ 050441965 118 4.16 LIZ 122 126 1.24 (1.20 1IB 1.10 104113West 1.25 191.20 1.19 120 127 134 135 (129 1.26 1.291967 117 128 1.26 1.26 126 421 LUE 12 1.0497 031968 104 1.04 1.06 1.06 107 1.04 «99 1.01 961.04 1.051969 112 1,08 1.09 112 Hi8 108 1.18 LIS 1.12" 1.07 1.091970 124112 1.14 1.15 12t 124 127) 138 134 129 1.391971 127° 1421.43 (at 143° 136 119 LIP 1.0097 5.081972 117 1.09 1.09 113 113 114 LIS 122 119 4.20421973 186 139 1.35 142 199 203 268 215 217% 218 2381974 292 259 276 241 257 291 337 330 345 332 3.21975 2.70 307 286 268 272 295 276 262 233 237Average 130 1.30 131 139 145) 140 132 128 [1341976 244 248
 
*1956-1965 midmonth Iiinois farm prices (U of I).11966-1975 midmonth US. farm prices (CRB Yearbook).
‘A general formula for the monthly average is:*
1f,,% 2APP, = 100 fal > (Pa! > *) |}ANwhere. APP, is the average percentage price in monthdis the calendar month from 1 to 12
Nis the number of years in the analysisP,, is the average price for month j of year n
 
1,12
‘This formula may be applied to weekly or quarterly average prices by changing the 12 to” “Tne use of ae aval average price can bias a clear seasonal pattern hecause of isinability to account for a long-term trend in the price of the commodity. If the rate of infotion in the United States is 6%, there will be a tendency for each month to be ighes,resulting in a trend toward higher prices at the end of the year. Longer trent ds, such 25 thesteady rise in grain prices from 1972-1975, followed by a longer decline in the .obscure or even distort the seasonality unless the trend is removed.
' David Handmaker, “Low-Frequency Filters for Seasonal Analysis," in Handbook of Futures Markets, Percy J.Kaufman (ed.) John Wiley & Sons, New York, 1984). 
166
TABLE 7-4 Corn
SEASONALITY
 
ice as Percentage Average (Annual)
Year Ave Jan Feb Mar Apr May jun ful Aug —‘Sep. Oa’ Nov Dec
 
1956 130 876 89.2 915 1015 1085 1108 1115 1123 1115 861 938 9461957 1.25 976 928 944 944 96.0 96.0 68 976 869 869 754 808tose 110 882 89.1 945 1073 109.1 1118 1118 1127 885 927 a54 9451959 1.09 96.3 97.2 99.1 107.3 108.2 109.2 107.3 107.3 100.0 881 91.7 9011960 103 (00.0 01.0 101.0 [058 1078 1078 1078 1058 1039 942 796 8931961 1.01 98.0 102.0 103.0 97.0 103.0 103.0 1049 1030 101.0 990 901 9311962 98 95.9 96.9 97.9 100.0 105.1 105.1 104.1 103.1 1020 959 928 10201963 Il 928 946 95.5 982 101.0 1072 1090 1090 1108 937 91.9 9911964 1.13 99.1 96.5 100.0 101.8 1044 1035 1009 1009 1035 929 920 100.91965 118 98.3 99.1 100.8 103.4 105.9 1068 105.1 101.7 1000 932 881 95.81966 1.25 95.2 96.0 936 95.2 968 960 1016 1072 1080 1032 1008 10321967 1.17 109.4 107.7 109.4 107.7 1068 107.7 1034 949 957 989 829 88.01968 1.04 100.0 101.9 101.9 101.9 1048 1029 1000 952 97.1 923 1000 101.01969 1:12 964 97.3 973 100.0. 1062 1053 964 1053 1027 1000 955 97.31970 124 903 919 91 927° 95.2 97.6 100.0 1024 1113 1081 1040 112)
1956-1970 Avg 96.3 96.9 98.1 100.9" 103.9 1047 1040 1039 Jo15 -943 90.9 96.1
1971
1972i97319741975
127 1118 1126 1126 111.0 1087 1126 1071 937 874 787 764 850117 932 932 940 966° 983 966 974 983 1043 1017 1026 | 214186 747 726 736 763. 624 1070 109.1 144.1 1156-1 16.7 1172 1285292 887 945 918 825 839 880 996 1154 1130 lial 1137 ) 12.0270 1137 105.9 989 99.2 985 992 1007 1092 1022 970 863 878
[971-1975 Avg 964 95.8 942 93.1 90.1 100.7 1028 112.1 1045 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-tended 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 thatmonth.
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).
 
Changes in volatility and variations in seasonal patterns for corn.
 
  
FIGURE 7-1160v0120a | ter aors3 00 eonZ *S25_0703j
80
60 ~—— 1973-1975—— 1956-1970[1956-1970 variation
40
‘Nov Dee
Yan Feb Mar Apr May June July Aug Sep Oct
 
 
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). Reprinted
with permission of John Wiley & Sons, inc.
167 
16a SEASONALITY
as O/T
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.
A constant 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:
     
Compound rate,of growth =
 
starting value
where N is the number of years or the number of periods over which the growth is com-pounded. ‘
10/ 175100
1.05755 -1
R
 
05755,
TABLE 7-5 Corn Prices Expressed as Link Relatives
Jon 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 .957ee
 
POPULAR METHODS FOR GALCULATING SEASONALITY 189
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 vnultiplying the February entry by (1 +R) = .99916, March by (1 +R)? = 9983