0, but for the three sequentialcocoa prices, 56.20, 58.30, and 57.15, the arithmetic mean is 57.2166 and the geometric is57.1871. A similar test of 5, 10, or 20 days of commodity prices will show a negligible dif-ference between the results of the two averages. If the geometric moving average is to behelpful, it would be best applied to long-term historic data with wide variance, using yearlyor quarterly average prices.
 
 
DROP-OFF EFFECT
‘Two types of trending methods can be distinguished by the drop-off effect, a common wayof expressing the abrupt change in value when a fixed-period calculation drops off a sig-nificant value. A simple moving average, linear regression, and weighted averages all use afixed period, or window, for determining the trend. When an unusually volatile piece ofdata becomes old and drops off, there can be a sudden jump in the trend value, called thedrop-off effect. For an n-period moving average, this value is the difference between thefirst and the last items, divided by the number of periods,
drop-off effect = (P, — P-n)/n
 
EXPONENTIAL SMOOTHING 75
 
A weighted average, in which the oldest values have less importance, reduces thiseffect because the high-volatility data slowly become a smaller part of the result beforebeing dropped off. Exponential smoothing, discussed next, is immune from this problemby its very nature.
EXPONENTIAL SMOOTHING
Exponential smoothing may appear to be more complex than other techniques, but it isonly another form of a weighted moving average. It has the added advantage of being sim-pler to calculate than any other method discussed; only the last exponentially smoothedvalue E,_, and the smoothing constant a are necessary to compute the new value. The tech-nique of exponential smoothing was developed during World War II for tracking aircraft andprojecting their position—the immediate past is used to predict the immediate future.
The geometric progression
a
applied to the terms of a weighted moving average
1,4,@,@,..
 
WP + WP yet + WpPron ed
 
 
Ww,Wy + Wt Wygives w, = 1, w,=4, w,=a’,..., w,=a""1. Ifa = %, also said to be 50% smoothed, theresulting sequence is .14%... COP?
This shows the rapidly decreasing importance of each older price. Substituting the geo-metric progression into the equation for the weighted moving average givesLP, + GPx) + @Py_ gt tA" "Pron t
. Ltatat--+a"?
By a lengthy arithmetic process using the formula for the sum of a geometric progres-sion, the same equation can be stated as
E,= (1—a@)P, + aE;-1where P, is the most recent price and 0 <a < 1. It can be seen that 100% of the combinedvalue of past prices is distributed such that a x 100% goes to the previous exponential mov-ing average and the balance to the most recent price. If a = .70, the current price P, willreceive a weighting of 30% of the total moving average. An equally popular form that-reverses the weighting notations and has one less calculation, but gives identical results, is:E,=8,-. +4 P,-E,-1)‘The smoothing process can be started by letting £, =P; and calculate the next value:E,=E, +a (P2,-B))
The smoothing constant, a, will determine the number of calculations necessary towind up the data. A longer-term’ smoothing, where a is closer to 0, will take more data forthe smoothed line, £,, to reach a stable value. The interpretation of this last equation canbe seen in Figure 4-3 as -
New exponential value = prior exponential value
+ some % of (today’s price — prior exponential value) 
6 TREND CALCULATIONS
eeFIGURE 4-3 Exponential smoothing.
PyPr
Pro Bent
a= .10 (P,— Ey—1)
  
‘An important feature of the exponentially smoothed moving average is that all datapreviously used are always part of the new result, although with diminishing significance.In general: .
E,=a (P+ A-@)P,-1 + (L-@)Pyea to + (1a)
For example, if the smoothing constant @ = .10, add 10% of the new difference to theold average:
E,=E,-1+-10 P,- Er-1) -
Each new calculation reduces all data from points 1 through t ~ 1 by 10%; the next cal-culation for ¢ + 1 will cause the data from f through 1 to be reduced again by 10%. There-fore, at any time ¢ the impact of data used at a previous time & is based on the number ofdays elapsed, t — &, and the smoothing constant a. Let the significance Py — E,-1=# = De.Then on day kwe have -
k=a-D,k+1l=a'D,-a-a-Dy
\
ktn=a-Dy-(@*Dyta@-Dyt + +a") - Dy)net s /=a-Dy- >, @D, (written in summation form)
=d(a-> @)70 as nt
= aThis shows that the significance of the data on day k goes to zero as n gets infinitely large.Consider the following example. An investor makes a deal to buy 10% of the outstandingshares of stock in a corporation in which there are t — 1 investors. The corporation decidesto take in another investor and gives that investor 10% of the total outstanding shares.There are now f + 1 investors, and the original ¢ share owners are all diluted by 10%; theinvestor ¢ who bought 10% now owns 9%. Another investor buys in at 10% of the total;there are now f + 2 investors, and the investor ¢ has 8.1% of the stock (10% less). As more
 
 
EXPONENTIAL SMOOTHING 7
 
investors are added, t's stock holding dwindles to 7.29%, 6.561%, and 5.9049%. Eventhough the number of shares that are held remain the same, each previous owner holds aless significant part of the whole. No matter how many investors are added at 10%, theoriginal shareholders will always have some small percent of the whole. In exactly the sameway, the original price used in an exponentially smoothed moving average always retainssome relevance; with a standard moving average of n days, the (n+ 1)th day is dropped offand ceases to have any impact.
Smoothing and Restoring the Lag
As a trend continues in its direction, the exponentially smoothed moving average will lagfarther behind. By selecting a smoothing constant nearer to 1, the magnitude of this lagwill be lessened but it will still increase, just as using a smaller number of days, or shorterperiod, will limit the lag of a simple moving average. If the lag is considered the predictiveerror é, in the exponential smoothing calculation, then
e,=P,—E,where £, is the exponential smoothing approximation of the price P;. The same exponen-
tial smoothing technique can be applied to the pattern of increasing or decreasing errorsto get
ERR, = ERR, -, + @(e; —ERR,_1)
and then add the difference between the original smoothing value and this second-ordersmoothing back into the approximation:
EE, = E,+ ERR,
By measuring the new error, the difference between the first-order exponential £, and thesecond-order ZE,, it can be seen whether there has been an improvement in the forecast.This process can be continued to third-order smoothing by applying the same method tothe difference between the current price and the second-order trend EE,. A comparison ofthis method, restoring the forecast error, is compared with other exponential smoothingtechniques in Figure 4-4 and Table 4-4. .
Double Smoothing
‘To make a trendline smoother, the period of a moving average or the exponential smooth-ing constant is normally increased. This succeeds in reducing the short-term market noiseat the cost of increasing the lag. It has been suggested that each of these trending methodscould be double-smoothed, that is, the trend values can themselves be smoothed. This willslow down the trendline, but gives weight to the previous values in a way that may beunexpected. _
A double-smoothed simple 3-day moving average, MA, would take the original threemoving average values and average them to get a double-smoothed moving average,DMA:
MAs = (P, + P2 + P3)/3MA, = (P, + Ps + P/3MA, = (Ps + Py + P.)/3thenDDMAs = (MA; + MAy + MAs)/3 
78
 
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TABLE 4-4 Comparison of exponential smoothing techniques applied to Swiss franc fromTradeStation continuous daily data
 
Exp
 
 
   
Exp Smoothed Price DbI ExpPrice Exp Dbl Exp ExpErr_~—Error_—Error~— Change Change Change930316 6459 6459 64.59 64.59 0.00 60930317 64.23 6455 64.59 64.52 0.32 0.03 0.36 = 0.04 ~0.04930318 65.26 6462 64.59 64.66 0.64 = 0.03 1.03 0.07  -0.03930319 65.24 64.69 64.60 64.77 055 0.09 0.02 0.06 = -0.02930322 65.07 64.72 64.61 64.84 035 0.11 = -O.17 0.04 = -0.01930323 65.14 64.77 64.63 64.90 037° (0.14 0.07930324 64.98 64.79 64.64 64.93 019 014  -0.16930325 64.76 64.78 6446 = GA9E = 0.02, 0.19 -0.22930326 65.11 64.82 64.67 64.96 029° «(0.14 0.35930329 65.46 64.69 65.07 058 = 0.19 0.35930330 65.94 6472 65.25 095 0.26 0.48930331 66.10 64.76 65.44 1000.34 0.16930401 66.87 6481 65.74 159 0.46 0.77930402 66.56 6487 65.94 116 053 -0.31930405 66,71 64.94 66.13 1.18 0.60 0.15930406 66.19 65.00 66.20 059 0.60 © -0,52930407 66.14 65.07 66.24 049. 0.58 = -0.05930408 66.64 65.14 66.37 089 0.62 0.50930412 67.33 65.21 66.61 142 0.70 0.69930413 68.18 6531 © 66.97 204 = 0.83 08s930414 67.48 6540 67.14 121 087 -0.70930415 67.19 65.50 67.23 083 086 = -0.29930416. 66.46 65.59 67.16 0.09 0.79 -0.73930419 67.20 65.67 (67.24 074 0.78 0.74930420 67.62 65.76 67.38 105 0.8! 0.42930421 67.66 65.86 67.51 098 = (0.83 0.04930422 67.89 65.95 67.65 109° (0.85 0.23930423 69.19 66.06 68.02 215 0.98 1.30930426 69.68 66.18 68.43 2380-112 0.49930427 69.31 66.32 68.69 180 1.19 -0.37930428 69.11 6645 68.88 144 1.21 -0.20930429 69.27 66.59 69.06 144124 0.16930430 68.97 66.72 69.16, «1.03.22 0.30930503 69.11 66.86 69.26 105 1.20 0.14930504 69.50 66.99 69.41 13000 121 039930505 69.70 67.13 _ 69.58 135 1.22 0.20930506 69.94 6727 69.76 14300 1.24 0.24930507 69.11 6740 69.74 054 117-083930510 67.64 6750 6945) 0.84 097-147930511 - 67.75 67.59 065 0.81 O11930512 67.47 67.67 084 .- 065 = -0.28930513 67.50 67.72 073° 05) 0.03930514 68.18 6777 0.05 0.45 0.68930517 67.35 67.81 079 «033° -0.83930518 66.74 (87.83 -126 017-061930519 67.00 67.84 0.90 0.06 0.26930520 67.46 67.84 -039 0.02 0.46
 
 
 
 
 
 
 
  
 
EXPONENTIAL SMOCTHING 79TABLE 4-4 (Continued)Exp Smoothed Price Exp DbIExpPrice Exp _DbIExp — ExpErr —Error__—Error_~— Change Change Change930521 67.36 67,80 - 67.83 67.78 0.44 0.03 0.10 0.05 —0.00930524 67.37 «67.76 «67.83 67.70. 039 -0.07 001 =-0.04 0.01930525 6778 67.76. 67.82 67.71 0.02 © -0.06 ot 0.00 © -0.01930526 67.96 = 67.78 = 67.82_—««67.75 0.18 -0.03 0.18 0.02 -0.00930527 68.97 67.90 67.82, 67.98 1.07 0.08 1.01 012 0.01930528 69.79. 68.09 67.85 68.33 1.70 0.24 0.82 0.19 0.03930601 69.53 68.23 67.89 68.58 1.30 0.34 0.26 O14 0.04930602 69.52 68.36 67.94 68.79 116 043 -0.01 0.13 0.05930603 69.17 68.44 67.99 68.90 0.73 0.46 0.35 0.08 0.05930604 67.41 68.34 68.02. BGG -0.93 032-176 = 0.10.04930607 67.96 68.30», 68.05 6855 | 034 0.25 055 0.04 0.03930608 67.62 «68.23. | 6807 6840 0610.16 = 0.34 «= 0.07. 0.02930609 67.18 68.13" . 6808 =. 6818 = 0.95005, 044-0. 0.01930610 67.65 «68.08, 68.08 += 68.09 0430.01 047 , -0.05 0.00930611 68.08 68.08" * 68.08 68.09 = -0.00 0.00 0.43 -0.00 0.00930614 68.06 68.08 68.08 = 68.08 = ~-0.02 S000 0.02. «0.00 0.00930615 67.16 6791 083-008 | -0.90 -009 -0.01930616 66.59 6765 | 126-020-057 0.14 0.02930617 66.77 097-028 0.18  -O11 0.03930618 65.95 -16l 041-082" -0.18 0.05930621 65.89 1.50 0.52 0.06 0.17 0.06930622 65.48 172-064-041 — -0.19 0.07930623 65.69 }-136  -071 021 0.15 0.08 ©930624 65.30 -158 0.80 0.39 0.18 —0.09930625 65.37 1.36 0.85 0.07 0.15 0.09930628 65.70 -0.92 0,86 033° -0.10 0.10930629 66.12 | 0.45 0.82 0.42 0.05 0.09930630 65.37 —1.08 0.85 0.75 0.12 0.09930701 65.64 0.73 0.83 027° 0.08 = -0.09930702 65.54 -0.75 083 -0.10 -0.08 -0.09930706 65.47 0.74 -082 007 008  ~0.09930707 65.45 : 068 -080 0.02 -008 -0.09930708 65.50 | 057 0.78 0.05 0.06 -0.09930709 64.70 ~1.23 0.82 0.80 0.14 ~0.09930712 64.82 —1.00 0.84 O12 O11 —0.09930713 65.27 049-081 045 -005 —-0.09930714 65.62 0.13 0.74 0.35 0.01 0.08930715 65.36 035 0.70 026 004 0.08930716 65.85 0.12 0.62 0.49 0.01 ~0.07930719 66.09 0.33 0.52 0.24 0.04 = -0.06930720 66.28 047 -0.42 0.19 0.05 -0.05930721 66.07 6:23 0.36 0.21 0.03 0.04930722 66.03 O17 031 0.04 002  -0.03930723 65.22 057 0.33 0.81 0.06 0.04"930726 65.19 054 -035 0.03 0.06 -0.04;930727 65.19 049-037 0.00 = -0.05 +004 ©930728 65.59 0.08 0.34 0.40 -0.01 0.04 =930729 64.80 0.78 038 -079  -0.09 —-0.04
 
  
80 TREND CALCULATIONS
i
i i i iques.A simple .10 smoothingFIGURE 4-4 Comparison of exponential smoothing techniques. : :(Exp) is compared with a double-smoothed series (Db! Exp), which applies the smoothing con-stant .10 first to prices, then to the smoothed series, and also to a smoothed series with the
error restored (Exp Err).
 
Comparison of exponentiat smoothing
 
 
 
 
 
 
Price —-— Exponential= Double exponential - Exponential error
   
 
If we substitute the original prices, P;, . . . , Ps, into the equation for DMA, we find out that
DMAs = (Pi + 2P2 + 3P3 + 2P4+Ps)/9This shows that double smoothing results in weighting the middle values more than theend values, rather than the original equal weighting of the simple moving average. Forexponential smoothing the result is different. Because the most recent value receives theweight of the smoothing constant, a, the double smoothing -
DE, = DE,-, + a(B; — DE,-1)
i Ider values as well.causes the nearby value t to be smoothed twice by 4, or a x a, and older values |Therefore, the net effect of using a smoothing of a = .10 for double smoothing will resultmuch more closely tovusing the square root of a, approximately .031. Figure 4-4 and Table -4-4 give an example of this method.”
_Biau’s Double Smoothing - :
To avoid this compounded lag, William Blau has substituted the price changes, P; — P;-»,for the price itself‘ The smoothing is then performed’on an accelerated price series, and
the smoothed result restores the speed of the series back to normal. When 7 = 1, the price -”
changes are called the first difference, or speed. In effect, the smoothed series does nothave a lag, Blau has found this to be a successful proxy for long-term trends, in which thefirst smoothing may be as long as 300 periods and the second a much shorter 5 periods
{also see Table 4-4). .
‘William Blau, Momentum, Direction, and Divergence Jjobn Wiley & Sons, New York, 1995).
shine
a=
&
 
RELATING EXPONENTIAL AND STANDARD MOVING AVERAGES ar
 
Comparing Exponential Smoothing Methods
Figure 4-4 and Table 4-4 itlustrate the differences between a simple exponential smoothing(Exp) with smoothing constant a = .10, smoothing with the lag (or forecast error) restored(Exp Err) using the same a = .10, and double smoothing (Dbl Exp) with the same smooth-ing constant. In the following example, a trend was defined as the direction of the unfiltered trendline. When the change in the trendline value was Positive, the trend was up;when it was negative, the trend was down.
The single exponential, Exp, appears as the medium-term trend in the sample periodof about 4% months. It had 9 distinct trend Periods and a net profit, without any transac-tion costs, of 2.59 points. Double smoothing was predictably slower, showing a muchsmoother curve and only 4 trend changes. It netted only .07 points, but this particular datasample had a change of direction midway, which does not allow for long-term profits.Restoring the error, Exp Err, only increased the number of trends to 12 and gave a profit of-83 points, despite its appearance as a very fast trend.
Blau’s double smoothing of the price changes was especially interesting, It performeddifferently from the other smoothing methods, yet posted 10 trends, making it similar inspeed to the simple exponential, rather than the double-smoothed. Profits of 1.67 pointswere also high for this short sample, and the trend patterns that it identified were signifi-cantly different. .
These alternatives all offer good possibilities for identifying the trend and should notbe judged by the small sample used in the example.
RELATING EXPONENTIAL SMOOTHING AND STANDARD MOVING AVERAGES
Itis much easier to visualize the amount of smoothing in a 10-day moving average thanexponential smoothing in which a = .10 (which can be called 10% smoothing). Althoughwe try to relate the speed of both techniques, the simple moving average is equallyweighted, and the exponential is front-weighted; therefore, they will produce a very dif.ferent pattern. Because of the inclusion of old data, a 50% smoothing appears slower thana 2-day moving average, a 10% smoothing is slower than a 10-day moving average, and a5% smoothing is slower than a 20-day moving average. The important factor is that for anyspecified smoothing constant, the exponential moving average includes all prior data. Ifa5-day moving average was compared with an exponential with only 5 total days included,the relationship would be closer to a straight moving average than if the exponential had10 or 20 days of elapsed calculations.
Instead of using prices, a series of numbers from 1 through 15 and back to 1 will beused to compare a 5-day moving average with an exponential moving. The exponential iscalculated two ways: once using only the last five prices (a modified approach for ourexample); the other using all prices from the beginning (the standard method).
Figures 4-5 and 4-6 and Table 4-5 show the relationship between the standard expo-nential and the modified exponential using 5 points. During the period of constant priceincrease and decrease of at least 5 consecutive days, both the 5-day moving average andexponential with five points stabilize; those 5 days represent their entire set of calculationvalues. At the peak, the standard moving average reacts more quickly than the other meth-‘ods in staying closer to the current price; the 5-point exponential gives 20% of its weightto the most recent price, and less to prior prices, causing it to react more slowly than thestandard moving average.
The standard exponential smoothing is different, lagging farther behind each day butincreasingly approaching the value of one, as thé data increase by one, for long time peri-ods. The weighting of the near values is offset by the retained significance of the oldestdata, which is never fully lost, causing the exponentially smoothed moving average to lag 
82
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FIGURE 4-5 Daily price with moving averages.
S-Day Moving Average
‘Standard 20% Exponential
   
Tes 67 SO Wi isl4isisisiziilo 987654325
 
ind the current prices. Although there are 14 days of constant decline, thewaerd oxponeatal has not yet stabilized, still reflecting the turning of prices from up tosoe fon the specific relationship between exponential smoothing and standard mov:ing aver