is critical to the success of this sjstem. The most obvious approach to finding N is by testing a isroad range of values (as will be shown in the next section). It has also been suggested that N could be variable based on the relationship of normal volatility to current volatility.

where N, is the number of days used for todays calculation

N, is the initial number of days used for normal markets V„ is the normal volatility measured over historical data

V, is the current volatility measured over a fixed period shorter than the normal volatilitji V„

As the current volatility increases, the number of dajs used in todays calculation decreases. This may also be classified as an adtive technique.

Testing the N-Day Rule

Weekly breakouts, a limited case of the N-Day Rule, were tested by Hochheimer and summarized in his report as the Weekly Price Channel Weekly breakouts produce a slow trading sjstem that depend on major moves for profits. About 75" of those futures markets tested in this 1982 shidj showed higher risk (some very Irigh), than the crossover sjstems tested earlier (also see Chapter 2 1).

The original piwpose for the Weekly Rule was to look at prices only on Friday. The close on Friday is considered in the same sense as the daily close-more important because it is ttie evening up at the end of the day Traders are athibuted with the feeling that holding a position over a weekend is the only thing worse than holding il ovemight. This evening up will prevent false signals that may never occur midday or midweek.

Tjpical trading rules for a Weekly Price Channel sjstem would be

1. Buy (and close out shorts) if the closing price on Friday exceeds the highest closing price of the past N weeks.

2. Sell (and close out longs) if the close on Friday is below the lowest closing price of the past N weeks.

Because this model is alwajs in the market, it is possible for the risk to become very high. The commitment to a new long position is the difference between the highest and lowest closing prices of the past N weeks. In addition, even if penetrated, the position is not liquidated until the close of Friday. This could be a very riskj proposition; however, that rid; is offset by the smoothing effect of only acting once each week. It may be that higher risk is better than being subjected to more frequent false signals.

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Alternate rules were tested by Hochheimer in his shidy. Buy and sell signals were taken at the time that the infraday new high or new low occurred. No comparison was available to determine whether the risk was greater or less than the conventional approach. Hochheimer also tested the Weekly Rule with the week ending on dajs other than Friday. It was not apparent that any one day was better.

Traders will find this basic breakout method is important as a benchmark study. Chapter 5 ("Trend Sjstems") inchided the N-Day Breakout as one of the basic frending methods in "Comparison of Major Trend Sjstems,- and shows 10 years of results for the Eurodollar, included with the shidy is program code to help enter this program into a computer to allow further testing.

Avoiding Problems Programming the Weekly Breakout

The Weekly Rule is often thought of having signals only on Friday; however, when programming this method, it is important to remember that a number of weeks end on Thursdajs due to holidajs. Identifjing the last day of the week in advance is a problem for a computer program. Although we can find the day of the week by simply using the

function a f W e e for Easj Language, or (fflW D A Y in Quattro, we have missed the day if the information retum ed jumps from Thursday to Monday.

The simpler and preferred method, using most intelligent market charting sjstems, is to request weekly data, or a weekly chart, rather than daily. The built-in conversion program will correctly change dajs to weeks. By looking al the high, low, and close of the weekly bar you can decide whether a signal has occurred, and executing on the close will be the last price of the week, regardless of the day on which it occurs.

WILLIAM DUNMGAN AND THE THRUST METHOD

Dunnigans work in the early 1950s is based on chart formations and is purely technical. Although an admirer of others abilities to perform fundamental analjsis, his practical approach is contained in the statements:

"If the economists are interested in the price of beans, they should, first of all, leam all they can about the price of beans. " Then, by siq3porting their observations with the fundamental elements of supply and demand, they will be "certain that the bean prices will reflect these things. "

Dunnigan did extensive research before his major publications in 1954. A follower of the Dow theory, he originally created a breakaway sjstem of trading stocks and commodities, but was forced to drop this approach because of long shings of losses even though the net results of his sjstem were profitable. He was also disappointed when his 2 3/8 Swing Method failed after its publication in A Study in Wheat Trading. But good often comes from faihire and Dunnigan had realized by now that different measurements should be applied to each market at different price levels. His next sjstem, the Percentage Wheat Method, combined a 2 I/2>o penehation and a 3-day swing, introducing the time element into his work and perhaps the first notion of thrust, a substantial move within a predefined time interval With the 2 1/2"o, 3-day swing, a buy signal was generated if the price of wheat came within 2>o of the lows, then reversed and moved up at least an additional2 I/2>ooveraperiodof atleast3 days.

For Dunnigan, the swing method of charting represented a breakthrough; it allowed each market to develop its natural pattem of moves, more or less volatile than any other

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market. He had a difficult time trjing to find one criteria for his charts that satisfied all markets, or even all grains, bul established a $2 swing for stocks where Rheas Dow Theory used only $1 moves. His shidies of percentage swings were of no help.

The Thrust Method

Dunnigans final development of the Thrust Method combined both the use of percentage measurements with the interpretation of chart pattems, later modified with some mathematical price objectives. He defines a downswing as a decline in which the current days high and low are both lower than the correaponding high and low of the highest day of the prior upswing. If currently in an upswing, a higher high or higher low will continue the same move. The reverse effect of having both a higher high and low would result in a change from a downswing to an upswing. The top and bottom of a swing are the highest high of an upswing and the lowest low of a downswing, respectively. It should be noted that a broadening or consolidation day, in which the highs and lows are both greater or both contained within any previous day of the same swing, has no effect on the direction.

In addition to the swings, Dunnigan defines the five key buy pattems:

1. Test of the bottom-where prices come within a predsermmed percentage of a prior low

2. Closing-price reversal-a new low for the swing followed by a higher close than the prior day

3. Narrow range-where the current days range is less than half of the largest range for the swing

4. Inside range-where both the high and low fall within the prior range

5. Penetration of the top-by any arnount, contbrming to the standard Dow theory buy signal

Aii of these conditions can be reversed for the sell pattems. An entry buy signal was generated by combining the pattems indicating a preliminary buy, with a thrust the next day confirming the move. The thrust was defined as a variable price gain based on the price level of the maiket (for 1954 wheat, this was from 1/2 to 1 l/2c Dunnigans sjstem attempted to enter the maiket iong near a bottom and short near a top, an improvement on the Dow theory. Because of the rid, the maiket was ad;ed to give evidence of a change of direction by satisfjing two of the first four pattems followed by a thrust on the next day. Any variance wouid not satisfy the conditions and an entry near the top or bottom wouid be passed.

The same buy and sell signals applied to changes in direction that did not occur at prior tops and bottoms bul somewhere within the previous trading range. In the event ali the conditions were not satisfied and prices penetrated either the top or bottom, moving into a new price area, the fifUi pattern satisfied the preliminary signal and a thrusl couid occur on any day. This was not restricted to the day following the pendration. So that if nothing else happened. Dunnigan followed the rules of the Dow Theory to insure that a major move wouid not be missed.

It has been said by followers of Dunnigans method that his repeat signals are the strongest part of his sjstem. even Dunnigan states that they are more reliable although they limit the size of the profit by not taking fiiii advantage of the frend from its start. Repeat signals use relaxed riUes not requiring a new thrust because the frend has been identified. Two key situations for repeat buy signals are:

1. A test of the bottom followed by an inside range (interpreted as maiket indecision)

2. A closing price reversal followed by an inside range

A double thrust occurs when the first thrust is followed immediately by a second thrust; or, after the first thrust, a congestion area develops, followed by a second thrust in the same direction as the first. Although Dunnigan used a fixed number of points to define his thrust, todays traders may find the standard deviation of the daiiy price changes or another volatility measure as a more practical basis for identifjing significant price moves.

One-Way Formula

Dunnigan worked on what he hoped wouid be a generalized version of his successful Thrust Method and called it the One-Way Formula. Based on his conclusions that the Thrust Method was too sensitive, causing more false signals than he was prepared to accept, he modified the confirmation aspect of the signal and made the thrust into the preiiminarj signal. He also emphasized longer price frends which smooth performance and reduce signals.

With the upswing and downswing rules remaining the same, Dunnigan modified the thrust to require its entire range to be outside the range of the prior day. For a preiiminarj buy, the low of the day must be above the higli of the prior day. This is a stronger condition than his original thrust, yet oniy constitutes a preiiminarj buy. The confirmation occurs oniy if an additional uptlirust occurs after the formation of, or test of a previous bottom. There must be a double bottom or ascending bottom followed by a thrust to get a buy signal near the lows, if the confirmation does not occui after the first bottom of an adjustment, it may stiii be valid on subsequent tests of the bottom.

For the One-Way Formula, repeat signals are identical to original confirming signals. Eadi one occurs on a pullback and test of a previous bottom, or ascending bottom, fclowed by an uptlimst. Both the initial and repeat signals allow the trader to enter after a reaction to the main frend. The Dow approach to penetration is atill aliowaUe in the event ali else fails. The refinement of the original thrust method satisfied Dunnigans problem of getting in too soon.

The Square-Root Theory

The two previous methods show a conspicuous concenfration of entry techniques and an absence of wajs to exit. Although it is valid to reverse positions when an opposite entrj condition appears, Dunnigan apends a great efforl in portfolio management and rida-eward conditions that were linked to exits. By his own definition, his tedinique wouid be considered trap forecasting, taking a quick or calculated profit rather than letting the frend run its course (the latter was called continuous forecasting).

A fascinating calculation of risk evaluation and profit objectives is the Square-Root Theory. He strongly supported this method, thinking of it as the golden key, and claiming siport of numerous esoteric sources such as The joumal of the American Statistical Association, The Analjsis joumal, and Econometrica. The theory claims that prices move in a square root relationship. For example, a maiket trading at 81 (or 9) wouid move to 64 (8) or 100