u-rate in the center of the range, they must get doubtful at the extremes at the point ofchange from one range to another. Analysts will find that they may need to study pricemovement at discrete levels, such as those shown in MPTDI, to be able to generalize aprice or volatility pattern.
The Volatility System
Another method that includes volatility and is computationally simple is the Volatility Sys-tem.’ Signals are generated when a price change is accompanied by an unusually largemove relative to average volatility. If the average volatility measured over N days is
12VAN) = > De-tes .
#21
where D is the maximum of
@) |Hi-G-1]b) [H.-L] —© [L-G-1| /
and H, is the high on day i -C; is the close on day 4 > -L, is the low on day i
Trading rules are given as
Sell if the close drops by more than K x V(N) from the previous close.Buy if the close rises by more than K x V(V) from the previous close.
The value of K is generally about 3.0.
‘Richard Bookstaber, The Complete Investment Book (Scott, Foresman, Glenview, IL, 1984, p. 231).APPLICATIONS OF SINGLE TRENDS 97
 
Note that the average volatility should not include the current day ¢ A comparison oftoday’s volatility using an average containing that value might cause inconsistent signals.
The 10-Day Moving Average Rule
The most basic application of a moving average system was proposed by Keltner in his1960 publication, How to Make Money in Commodities (The Keltner Statistical Service,Kansas City, 1960). Of three mechanical systems presented by Keltner, his choice of a mov-ing average was based on performance and experience. The system itself is quite simple: a10-day moving average based on the average of the daily high, low, and closing prices, witha band on each side formed from the 10-day moving average of the high-low range. A buysignal occurs on penetration of the upper band and a sell signal when the lower band isbroken; positions are always reversed.
The 10-Day Moving Average Rule is basic, but it does account for the fundamentalvolatility principle and serves as an example of the actual use of moving averages. Keltnerexpresses his preference for this particular technique because of its identification of minorrather than medium- or long-term trends, and there are some performance figures thatsubstantiate his conclusion. As an experienced trader, he prefers the speed of the 10-daymoving average, which follows the market prices with more reasonable risk than slowermethods. A side benefit to the selection is that the usual division required by a movingaverage calculation can be substituted by a simple shift of the decimal place; in an erabefore the pocket calculator, who knows how much impact that convenience had on Kelt-ner’s choice?
Triple Exponential Smoothing,
A triple exponential smoothing technique was described by Hutson as another approachto trend following.” Substituting the log of the price for the price itself, he applied an expo-nential smoothing three times using the same smoothing constant. A buy signal was gen-erated when the triple-smoothed series rose for 2 consecutive days; a sell signal followeda 2-day decline.
The smoothing constant selected would normally be faster (less than a 20-day equiva-lent, when 2/(n + 1) is used to convert the n days to a smoothing constant) for a triplesmoothing than for a single smoothed line. It is interesting to see what actually happenswhen a series is smoothed three times. For example, a triple smoothing of a straight 3-daymoving average gives the following:
P= (Py, Po, ***, Pr)
The single moving average at day n is“oy = Peete Pao +P)3Smoothing the-new series M gives(My -2 +My + Mn)
M,= 3.— @n-4+ 2Pn-3 + 3Pn-2 + 2Pn-1 + Pn)9and the third smoothing gives -
2 Jack K. Hutson, “Good TRIX,” Technical Analysis of Stocks & Commodities (july 1983).98 TREND SYSTEMS,
 
(My 2+ May + Mn)3— Ono t BPa_s + Pua + 7Py-3 + OPn2 + Pai + Pn)27The effect of the weighting puts increased importance on the middle values, reducing the
significance of both ends. With exponential smoothing as well, the increased emphasis willbe placed on the n — 2 value, the price which is repeated most often.
My! =
The Parabolic Time/Price System
One of the primary complaints’ about trend-following systems is that the built-in lagdestroys the trade. In the Parabolic Time/Price System,? Wilder has reduced the lag byincreasing the speed of the trend (shortening the days) whenever prices achieve new prof-itable levels. The philosophy of the Parabolic System is that time is an enemy. Once a posi-tion is entered, it must continue to be profitable or it will be liquidated.
The Parabolic Time/Price System is always in the market; whenever a position is closedout, it is also reversed. The point at which this occurs is called the Stop and Reverse (SAR).When plotted, it is similar to a trendline, although it has a decreasing lag (the distancebetween the current price and the trendline, or SAR point), as shown in Figure 5-4.
To calculate the SAR value; first assume a long or short position. If the market hasrecently moved lower and is now above the lows of that move, assume a /ong. Call the low-est point of the previous trade the SAR initial point (SIP), because it will be the startingpoint for the SAR calculation (SAR, = SIP). .
Calculate each following SAR as ‘
SAR(tomorrow) = SAR(prior) + AF x [High(today) — SAR(prior)]
This is an exponential smoothing using the prior high price. It is-unique because thesmoothing constant AF called the acceleration factor, changes. AF is set to .02 at the begin-ning of the trade. Each day that a new extreme occurs (a new high when long, or a new low
3 J, Welles Wilder, jn, New Concepts in Technical Trading Systems (Trend Research, Greensboro, NC, 1978).
 
FIGURE 5-4 Parabolic Time/Price System.
 
SIP (prior high)
      
féLec*K\ SAR ong” (uses highs)
sip -(prior low)APPLICATIONS OF SINGLE TRENDS 99
 
when short), AF is increased by .02. In moving average days, AF begins at 99 days andincreases in speed to a maximum of a 9-day moving average, but not in a linear fashion.
In the SAR calculation, High(today) is used when a long position is held, andLow(today) is used when a short is held. An additional rule also prevents premature rever-sal by not allowing the SAR to enter the price range of the most recent 2 days:
If long, the SAR may never be greater than the low of today or the prior day. If it isgreater than this low, set the SAR to that low value. A reversal will occur on a new intra-day low that penetrates the SAR.
If short, the SAR is calculated in the same way based on the highs.
Wilder’s Parabolic Time/Price System was a new and interesting idea—varying theexponential smoothing constant. This could also be accomplished by varying the movingaverage days, but not as effectively. The constant increment of .02 and the limitations of theTange, .02 to .20, are likely to be shortcomings. Traders might want to relate the changingspeed to more basic price phenomena such as volatility or momentum. For those readersinterested in the further development of this method, the Parabolic Time/Price System iscombined with Directional Movement (described in Chapter 23) to form the DirectionalParabolic System discussed in Chapter 6.
The Master Trading Formula
Another method of variable speed is Mart’s The Master Trading Formula.‘ It uses an expo-nential smoothing formula in which the smoothing constant and band are calculated daily,based on market volatility. Mart averages the ‘extended daily range (using the maximum ofyesterday's close and today’s high and low) and the net price movement over the sameinterval to define a correlated volatility factor. This value is then used to look up asmoothing constant in a modified log-scale table, in which an 8%-smoothing (24-day equiv-alent) is used for the lowest volatility and a 33% smoothing (2-day equivalent) for the great-est volatility, with the median value at 14% (13-day equivalent).
A band is also placed around the trendline, based on an inverse relationship to the cor-related volatility factor. The band is widest when there is low volatility and the smoothingconstant produces a slow trend; it is narrowest in fast, highly volatile markets.
Rules for trading Mart’s Master Formula are centered around the basic band tradingrules. The system is always in the market, taking long positions when the upper band isbroken and short positions when the lower band is crossed.
Raschke’s First Cross
Although we usually trade a trend from the beginning to end based on some smoothingmethod, Linda Bradford Raschke has shown that a selected piece of the trend move can bea very reliable trade. Her First Cross system enters a trend trade on the first pullback afteran initial trend signal based on a faster momentum indicator. She did not offer any exitrules, but there are some that can be suggested. To create this program we need
sTowMA Slow-moving average, part of oscillator calculation
fastMA Fast-moving average, part of oscillator calculation
osc Moving average oscillator, fastMA - slowMA
trend Moving average of osc
The oscillator is the difference between the fast- and slow-moving averages
osc = fastMA - slowMA -
* Donald S. Mart, The Master Trading Formula (Winsor Books, Brightwaters, NY, 1981).too TREND SYSTEMS
 
To generate a buy or sell entry signal, use the following steps (comments on the right):
Bl. osc{1]> trend and osc <= trend osc crosses trend moving down
B2. low > low{1] low of bar rises
B3. buy
S1. osc{1] < trend and osc >= trend osc crosses trend moving up$2. high < high[1] high of the bar declines
$3. sell
A likely exit rule for the First Cross system should maintain the short-term nature of theselection. You might simply consider the entry on the first pullback as better entry timingfor a longer-term trend. Or, you might want to trade only the short period isolated by thefirst pullback until this initial impulse is exhausted. If you chose the second method, youmight try the following exit condition:
Exit for a long position
X1. close[1] > close{2] close of bar turns up (confirms trend)X2. close < close[1] after going up, the close turns downX3. exit Jong
Raschke’s idea is an excellent example of selectivity. First, you recognize that the beginningof a trend is different from other times. As traders recognize a change of direction there isan increasingly strong move. Because most systems lag the market, they are often too lateor miss the first part of the turn, which could create large profits. As an alternative, thistechnique waits for the first move to be exhausted, then enters in anticipation of anotherfast surge as the new trend reasserts itself, Once these early moves are over, the generaltrend move may not be as easy to work with, and you may find yourself trying to enter asthe trend comes to an end.
COMPARISON OF MAJORTREND SYSTEMS
xA section on trends cannot be complete without looking at the comparative performance
of the six most popular approaches: -
. Simple moving average ‘
. Exponential smoothing. Linear regression slope. N-day breakout
. Swing breakout
. Point-and-figure
AVYAWN
Trading Rules
To see the characteristics of each system, the trading rules will be as simple as possible. Allsix systems aré always in the market. That-is, once they enter a long or short position, theyalways reverse on a fiew signal, going from long to short, or short to long. There are nostop-losses or other risk control rules; therefore, each system has its own, natural risk pro-file. A $100 round-turn charge was applied to each trade to cover commissions and slip-page. Table 5-2 summarizes the type of system, calculation method, and trading rules.
Varying the Speed of the Trend
Each system must be tested with a wide range of fast and slow trends. For systems 1through 4, the period, or number of days, n, determines the trend speed. For the swingCOMPARISON OF MAJOR TREND SYSTEMS tor
 
 
TABLE 5-2 Summary of Six Major Trending Systems
 
 
 
System Type Formulas Rules1. Simple moving average MA = @Sum(close,n) / n BUY if MAchange > 0MA is the average of previous MA = @Average(close,n)} SELL if MAchange < 0n-day closing prices. MAchange = MA - MALI] {the moving average turns
up or down)
2. Exponential smoothing EXP= EXPL1]+s*(close-EXP[1]) BUY if ExPchange > 0EXP is a weighting of the recent EXPchange = EXP - EXP[1] SELL if EXPchange < 0closing price by a percentage, s {the trend turns up or down)(called the smoothing constant).3. Linear regression slope LRS = BUY if LRS > 0LRS is the slope (angle) of a SELL if LRS < 0straight line found using a least- n*Sum(x*c) -Sum(x)*Sum(c)squares regression. Buy wher the LRS is up (slope is
n*Sum(x*x) ~Sum(x)*Sum(x) — positive), and self when the LRSis down (slope is negative).where c is the ci ose,and x are .the integers, |,2,3,....n,representing the time intervals.
 
4. N-day breakout nhigh = @high- BUY if high > nhigh
Signals occur when the current est(high(1],n-1) and close’> close[1]
price reaches a new high or low (a SELL if low < nlow
“breakout”) for both intraday and nlow = and close < close[1]
end of day, based on the previous @lowest(low[1],n-1)
a-days. - | Buy-when the current high price isgreater than the highest price of
the past n days, and the close ishigher than the previous close[1].
 
 
5. Swing breakout swhi gh is the highest price that BUY if high > swhigh
Signals occur when the current occurs before a price decline of SELL if low < swlow
high or low price exceeds the p*swhigh, where p is a percent
previous swing high or low price. Buy when the current high
A swing is a price move that sw] ow is the lowest price that exceeds the previous swing high.
begins with a reversal of a occurs before a rally of p*swlow, Sell when the current low falls
minimum percentage value. where p is a percent. below the previous swing low.
6. Point-and-figure Same rules as Swing Breakout, BUY if high > PFhigh + 1 box
Signals occur when the current except that new highs and lows SELL if low < PFlow - 1 box
high or low price exceeds the are recorded only if they exceed a 5
previous high or low, which is preset box size. A reversal is Buy when the current high
formed by filling boxes of a preset recorded if a minimum of 3 boxes exceeds the previous PF swing
point size. ~ can be filled, not including the high by at least one box. Selfcurrent box when the current low falls below
the previous PF swing low by atleast | box.
  
 
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breakout, a percentage swing makes it more or less sensitive to price movements. Point-and-figure uses a constant box size, which is then multiplied by 3 to get the reversal size.Tests shown in Table 5-3 for the Eurodollar confirm a broad range of trading from veryshort-term to very long-term. The number of trades varies from over 500 in 10 years (about2,500 days) to 6 trades in 10 years. Some showed a fairly high minimum number of trades
for the longest periods.
Different Trend Philosophies
The moving average, exponential, and linear regression are time-series processes; that is,each day is a small part of the total calculation. This can make it very difficult for a singlelarge price jump to cause a change of signal. For very fast trends, each price will have moreimportance than for slower trends.
The N-day breakout, swing breakout, and point-and-figure systems all depend on aprice breakout. A signal occurs when the market makes a new high or low. Therefore, a sin-gle big move often causes a new signal. Breakout systems can be considered event driven,while a moving average system requires a trend to evolve.
Different Risk Profiles
Each of the six systems fall into particular categories of risk profiles.
Moving Averages and Exponential Smoothing
Both the moving average and exponential smoothing return a weighted average. The sim-ple moving average uses equal weighting, and the exponential gives more importance tothe most recent value, according to the smoothing constant, s.
Both systems are considered conservation-of-capital methods because they keeplosses small. When current price levels are trading near the n-day average, it takes very lit-tle to change the trend from up to down and back to up. Although this is annoying totraders, it is part of the conservation-of-capital benefit. Table 5-3 shows that the minimumnumber of trades were 42 to 44 for these systems, far greater than the other four strategies.The whipsaws seem to increase as the trends become very slow, which is the reason forusing a filter,-or band, in most trend systems. In Table 5-3 the number of trades doubledwhen the trend slowed from 150 to 200 days.
Breakout Systems
The swing breakout, point-and-figure, and N-day breakout systems can all be grouped ashigh-risk strategies. Once a buy signal has occurred because of new high prices, the tradeis reversed only when new low prices occur. The initial risk is —
breakout risk = highest high - lowest low
for the N-day period. The rate at which risk declines is different for each of the systems. Theswing breakout and point and figure systems reduce risk each time a new swing low occursonly if that low is higher than the previous swing low (see Figure 5-5). The N-day breakoutreduces risk if the oldest days, which are dropped off, were also the lowest prices. Other-wise, its risk is the same as the other breakout systems.
Linear Regression Risk
The linear regression method, which creates a straight-line forecast, is very different fromother trending systems. It does not change direction when prices move up or downthrough the trendline (as in the smoothing systems). It is possible that a very large, but fastprice drop, which recovers quickly, will not cause a long position signal to reverse.COMPARISON OF MAJOR TREND SYSTEMS. $05
 
aFIGURE 5-5 Swing breakout risk pattern. As prices move lower, the swing highs become thepoint at which signals reverse. This pattern is the same for point-and-figure.
 
reversal 1
 
—reversal 2sell‘ —reversal 3
 
Least squares puts the greatest importance on those prices closest to the straight-lineestimate. It is very sluggish to change direction when a larger number of days is used.Because of this pattern, it can have a mixture of large and small losses.
Frequency of Trades
You might notice that the number of trades varies considerably across systems. This reflectstheir risk. The longer you hold a trade, the greater the risk exposure. The two smoothingsystems (moving average and exponential smoothing) cut losses short; therefore, theyshow the largest number of trades for a slow trend.
Breakout systems can trade very little if they enter on a new high after a volatile mar-ket period. A long position can be held for months waiting for the swing low to move up.An expanding range, one that makes a new high, followed by a new low and another newhigh, can be devastating.
Expectations
We form expectations so that we can recognize when test results are wrong. It is the sameas an engineer estimating the length of a bridge before a computer program produces thefinal numbers. An accidental shift of a decimal point may give you plans for a 10-foot bridgeover a 100-foot river. :
Similar Systems Give Similar Results
We expect the first two and last two systems to be very similar. In each case the rules arenearly the same. We expect the N-day breakout to have the highest risk because volatilityincreases over time. The n-day high and low should create a much larger range than themost recent swing high and low. Of course, we expect slower systems to be more profitablethan faster ones, because longer periods often reflect fundamental price movements.
Robust Testing -
To get the most value from these tests (an approach covered in detail in Chapter 21), thereneeds to be as much consistency as possible between tests. The following standards are set:
1. Test periods. All markets and all tests covered the same 10-year period, endingMarch 31, 1994 (except the OEX, which began in 1986 and totaled 9 years).
2. Range of trend speeds. Those systems that used days were tested from 4 to 200. Sys-tems 5 and 6, swing breakout and point-and-figure, used values that produced asimilar number of trades.TREND SYSTEMS
 
3. Adjusted rate of return. Returns were specified as the 10-year return divided by themaximum intraday drawdown, :4, Test averages. Below each test group are the averages of all tests, the standard devi-ation of the total returns, and the average minus one standard deviation. The bestresult is the system with the highest average return and the smallest standard devi-
ation.
Which System Is Best?
Table 5-4 gives only the average returns by system and market. It then averages thoseresults down and to the right, giving the average of averages. Results show that Eurodollarswere profitable for all systems and that the V-day breakout had the best returns for all mar-kets. In trying to understand the results, it is useful to remove those tests that resulted invery few trades or too many trades to be practical. Table 5-5 presents the same results forall results with the number of trades from 24 through 260 over the 10-year period (from2.5 per year to 1 every 2 weeks). The table shows that the simple moving average was muchbetter and the N-day breakout much worse. Total results were better, showing that the fasttrends were big losers. This is particularly true of the smoothing methods.
A Different View of Results
The average results give a reasonable picture of performance, but it is clear that each typeof system has a different distribution of trades. For example, doubling the swing percent-
 
TABLE 5-4 Average Returns
 
 
MA EXP LRS SWG PRE NDAY Avg
ED 2 140 Hg 160 105 27 169us 108 72 17 -24 ~6 123 48DM 133 95 11 16 8I 106 95sP -57 4 66 14 -27 49 44OEX -35 -B ~29 -12 -I7 -7 2cL 135 112 47 18 30 152, 66sy 16 -14 108 39 47 78 45Avg