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46

[□ ,956-1S70V,

Frb Mir Apr May June July Aug Sep Ocl Nov Dec

7-2 Seasonal price tendency in monthly ca average com prices (1936-1983)

urce Jacob Benistein, Seasonal Concepts in Futures Traliiie

wY.dc J..hiiWiley&?.ns, 19

) 3 1) Eefnntedwithpf

)n..f 1..1 Wiley & Sons,

The Method of Link Relatives

Another interestmg and important way of identifjing the seasonal price variations and separating them from other price components involves the use of link relatives. In Table 7-5 eadi month during 1960 and 1961 is expressed as a percentage by taking the ratio of that average monthly price to the average price of the preceding month (found in Table 7-3) in a manner similar to an index.

After the initial calculation of 1960 and 1961 link relatives, it is necessary to find the average (or the median, which is preferred if an adequate sample is used) of the monthly ratios expressed in rows (1) and (2). The average in row (3) represents monthly variation as a percentage of change, eadi calculation is a function of the preceding month. Thus far, this is the same as Bemsteins average monthly price changes, expressed as a percent of the prior price.



establish fixed base in the manner of an index, chain relatives are constructed using January as 100; each monthly chain relative is calculated by multipljing its average link relative by the average link relative of the preceding month. The March chain relative is then 1.005 x 1.025 = 1.030, and February remains the same because it uses January as abase.

A constant trend throughout the test period can be found by multipljing the December chain relative (4) by the January average link relative. If prices show no tendency for either upward or downward movement, the result would be 1.00; however, inflation should cause an upward bias and, therefore, the results are expected to be higher. From line (4), the December entrjmultiplied by the January entrjon line (3) gives .946 x 1,047 = .990, leaving a negative factor of Po unaccounted. This means that the 1960-1961 years showed a . Po downward bias; therefore, the expected rate of inflation was offset by some other economic factor, such as the accumulation of grain stocks by the U.S govemment

The chain relatives must be correaed by adding the negative bias bad; into the values, using the same technique as in computing compound interest. For exanple, from 1967-197, the Consumer Price Index increased from 100 to 175, a total of 75>o in 10 years. To calculate the annual compounded growth rate for that period, apply the formula:

Compound rate of growth = 1 --- -- i \ starting value

starting v

where N is the number of years or the number of periods over which the growth is compounded.

/ 175 TABLE T-5 Con Prices Eiprewd a> Unk Relatives

~\00 "" M Mo< fipr }un Ju[ A.g Stp Qg WW D«

(i) IW LOW 1 0 lAOO 1 4 IXIIE lAOO AOO 9S2 9S2 907 45 I 117

= 1ft>7SS-l 1.053 i.(HO 1 10 94! 1.061 IBOO .0 9 MIM WO 91111 31

l.vj/srj (JjAverage 1 47 1 25 1 05 99S l.0« IBOO 1.0 .982 982 9 7 IjOTI

{4)ChamrSatives IBOO 1 25 IBJO IBOO J03S l.0« I.OlO .991 9 927 19 .9«

(1 7= (S)Ccmcledchairrda«vcs IBOO IB24 IBIB 997 J032 I.03S IBOS 986 958 920 22 9J7

AJilJ (6)lndice50fjeasorulv>raiion! IB22 1.046 IBSO 1.019 IBS4 1058 \mi 1 07 979 940 «40 «7

This indicates a compounded rate of inflation equal to 5.75>o per year. In the case of com, if the frend had been positive, that is, greater than 1.00 instead of .990, the growth rate would be subtracted from each month to offset the upward bias. In this case, the results are added back into the chain relative to compensate for the negative influence. A .Po decline, compounded over 12 consecutive enfries gives:

=-.00084

1 5 is a compounded deflation of about 8/100 of \%. The corrected chain relative was found by multiplying the February entry by (1 + J?) = .99916, March by (1 + *>- - .99832, and December by (1 = .99076

The chain relatives have been calculated on a base of January, which was important to correct the compounded bias throughout the lest period. The final step is to swiich the corrected chain relatives tn a base of the average value. The average { 97875) of line (5) is used to create line (6), takirigth rati of the corrected chain relative ntn lotheiraver-age. The final result is the/» /5 5 / ( The ura ihis result can be proved by averaging the entries of 1 ne which wili be A complete smdy of seasonality using this method can be found in Counnev Smith. Seasonal Charts For Futures Thaders (Wiley, 1987).

The Moving Average Method

The moving average is a much simpler, yet very good technique for determining seasonal pattems. Looking again at the cadi corn prices in Table 7-3, take the average quarterly prices for the years 19601965 rounded to the



nearest cent. More practical results may he obtained by repeating this procedure for monthly prices.

Because every four entries completes a season, a 4-quarter moving average is calculated and recorded m such a way that eadi value lags 2 1/2 quarters, corresponding to the center of the 4 points used in the calculation. Column 2 of Table 7-6 shows the 4-quarter moving average positioned property; Figure 7-3 is a plot of both the quarterly com prices and the lagged moving average. By using the exact number of entries in the season, the moving average line is not affected by any seasonal pattem.

Because there was an even number of points in the moving average, each calculation falls between two original data points. Column 3 of Table 7-6 is constmcted by averaging every two adjacent entries in column 2 and placing the results in a position corresponding to the original data points. This avoids smoothing the initial prices. The difference of column 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 periodic fluctuation of prices becomes obvious once these values have been recorded. A generalized seasonal adjustment factor and seasonal index are calculated by taking the average of the quarterly entries for the five complete years (Table 7-7).

X.-11

The seasonal adjustment method X-U (Census Method U-X-U) is most widely used for creating a seasonally d series of such infonnation as car and housing sales, as well as other consumer products. It is very extensive, involving both an initial estimation

TABLE 7-6 Seasonal Adjustment by the Moving Average Method

4-Pom

Quamttf

1-Poin,

Price

Averose

Vo*.es

fijaor

I960

Jan-Mar

104 MO

Jul-Sep

103/.

1.06

Oc-Dec

102 .11

101 v.

1961

Jan-Mar

lOOV. i

100

Apr-Jun

99/ 1

100/

1.01

lOOV.

Oc-Dec

98i4

1962

Jan-Mar

Apr-Jun

9S 101

W/. 98

-3A -3

96 1

Jul-Sep

-1 -

Oc-Dec

100/

1963

Jan-Mar

10

106

Apr-Jun

110/.

1 tl

Jut-Sep

112/.

113/.

1.08

Oci-Dec

113/

1964

Jan-Mar

luy.

113/.

*r-Jm

;*

Jul-Sep

112*/.

112/

01

Oc-Dec

-7..

I96S

Jan-Mar

116/.

Apr-Jun

ItTA

1174

Jul-Sep

Oc-Dec

TABLE 7-7 Average Seasonal Variation Using the MovingAverage Method

Average of AJfVeors

Seosonal Mjustmeni foctor

Seasonal Index

an-Mar Apr-Jun lul-Sep Oct-Dec

-LIS +3.53 +4.43 700

1052 I 042 932

and reestimation. Because it is widely used by economists, an ouUine of its procedure follows.

1 Calculate a centered 12-month moving average (MA). Subtract this MA from the original series to get an



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