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22 FIGURE 4.4 Comfnrison of exponential smoothing techniques.A simple. 10 smoothing (Exp) Is compared with a doublesmoothed series (Dbl Exp), which applies the smoothing constant . 10 first to prices, then to the smoothed series, and also to a smoothed series with die error restored (Exp Err).
If we subsaicute tbe original prites, F„ . . , P„ into the equation for DMA, we find nut that DMA, . (P, + 2Pj + /*! + 2Pi 1 / 9 This shows that double smoothing results in weighting the middle values more than the end values, rather than the original equal weighting of the simple moving average. For exponeniial smoothing the result is different Because the most recent value receives the weight of the smoothing constsm. a, the double smoothing causes the nearby value / to be smoothed twice by a, ot a . a, and older values as well. Therefore, the net effect of using a smoothing oYa  .IO for double smoothing will result much mort closely to usitig the siuarc root of a. approximately .03 J. Figure 44 and Table 44 give an example of this method Blaus Double Smoothing lo avoid this compounded lag, Willism Blau has substituted the price changes. P.  for th price tself The smoothing then performed or\ an accelrated price series, and the smoothed result restore th peed of the series back to normal When  I the price changes are called theirsf tierience, or speed In effect, the smoothed senes does not have a lag. Dlau has found this tn be a succe.sslul proxy for longterm trends, in which the first smoothing may be as long 300 periods and the second a much shorter 5 periods (also see Table 44>. William Mau. 1 ««« , . and Dn KTg™ (John Wdcy & Sons. New Vork. 1»95> Comparing Rxponential Smoothing Methods Figure 44 and Table 44 illustrate 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 Rxp) with the same smoothing constant. In the following example, a trend was defined as the direction of the unfiltered trendline. AVhen the change in the trendline value was positive, the trend was up; when it was ;, the trend was down. The single exponential, Rxp, appears as the mediumterm trend in the sample period of about 4 l/2months. II had 9 distinct trend periods and a net profit, without any transaction costs, of 2.59 points. Double smoothing was predictably slower, showing a much smoother curve and only 4 trend changes. It netted only.07 points, but this particular data sample had a change of direction midway, which does not allow for longterm profits. Restoring the error, Rxp Err, only increased the number of trends to 12 and gave a profit of .83 points, despite its sppearance as a very fast trend. Blaus double smoothing of the price changes was especially interesting, it performed differently from the other smoothing methods, yet posted 10 trends, making it similar in speed to the simple exponential, rather than the doublesmoothed. Profits of 1.67 points were also high for this short sample, and the trend pattems that it identified were significantly different. These altemstives all offer good possibilities for identifjing the trend and should not be jutted by the small sample used in the example. RELATING EXPONENTIAL SMOOTfflNG AND STANDARD MOVING AVERAGES It is much easier to visualize the amount of smoothing in a 10dsy moving average than exponential smoothing in which a = . 10 (which can be called 10°o smoothing). Although we trjto relate the speed of both techniques, the simple moving average is equally weighted, and the exponential is frcntweighted; therefore, they will produce a very different pattem. Because of the inclusion of old data, a 50°o smoothing sppears slower than a 2dsy moving average, a 10°o smoothing is slower than a 10dsy moving average, and a 5% smoothing is slower than a 20dsy moving average. The important factor is that for any specified smoothing constant, the e«ponential moving average includes all prior data, if a 5dsy moving average was compared with an exponential with only 5 total dsjs included, the relstionship would be closer to a straight moving average than if the exponential had 10 or 20 dsjs of elspsed calculstions. Instead of using prices, a series of numbers from 1 through 15 and bad; to 1 will be used to compare a 5dsy moving average with an exponential moving. The exponential is calculated two wajs: once using only the last five prices (a modified spproadi for our example); the other using all prices from the beginning (the standard method). Figures 45 and 46 and Table 45 show the relstionship between the standard exponential and the modified exponential using 5 points. During the period of constant price increase and decrease of at least 5 consecutive dsjs. both the 5dsy moving average and exponential with five points stabilize; those 5 dsys represent their entire set of
calculation values. At the peak, the standard moving average reacts more quickly than the other methods in stajing closer to the current price; the 5point exponential gives 20° of its weight to the most recent price, and less to prior prices, causing it to react more slowly than the standard moving average. The standard exponential smoothing is different, lagging farther behind eadi day but increasingly approadiing the value of one, as the data increase by one, for long time periods. The weighting of tiie near values is offset by the retained significance of the oldest data, which is never fully lost, causing the exponentially smoothed moving average to lag, FIGLIRE 45 Daily price witii moving the fartiiest behmd the current prices. Although there are 14 days of constant dechne, the standard exponential has not yet stabilized, still reflecting the turning of prices from up to down at 15. To form the specific relationship between exponential smoothing and standard moving averages, create a table showing the significance of eadi oldest day in the exponential calculation. In Table 46, the .50 smoothing constant gives 50°o of the total value to the current price, 25oto the prior day, 12.5>oto the next oldest until theth oldest day adds only .8°o to the total value of the exponential moving average. Table 47 accumulates these weights to show how much of the calculation has been completed by the elapsed dajs printed across the top. Table 47 is plotted in Figure 47. The most recent dajs (on the left) receive the bulk of the significance; the oldest prices are of litde impact. The . 10 smoothing calculation is 90.Po complete by day 22; the total remaining dajs added togetiier only account for 9.9% of the value. FIGLIRE 46 Comparative changes in straight and exponential moving averages. TABLE 45 Comparison of Lag between Standard and Exponentially
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