two times ayear on average, a frequency consistent with anecdotal descriptions in thepractitioners' literature.For each cuto, the computer searches for a head-and-shoulders topfollowing each peak, and a head-and-shoulders bottom following each
268
Advanced Trading Rules
trough. To qualify as a head-and-shoulders top, a given set of fourconsecutive peaks (see Figure 9.1) ± the last three of which represent theleft shoulder, head and right shoulder ± is required to satisfy a number ofdescriptive criteria identi®ed only qualitatively in technical manuals. Thus, weformalize those criteria as follows.(1) The height of the head must exceed that of the left and right shoulders.(2) Since the pattern should presage a trend reversal, it must occurfollowing an uptrend. Thus,(a) the left shoulder must exceed the previous peak (point P)(b) the ®rst trough within the head-and-shoulders (point TL ) mustexceed the previous trough (point TLL ).(3) To avoid extreme vertical asymmetries, the pattern must be onlymoderately sloped. The right shoulder must exceed, and the righttrough must not exceed, the midpoint between the left shoulder andleft trough (TL ). Likewise, the left shoulder must exceed, and the lefttrough must not exceed, the midpoint between the right shoulder andright trough (TR ).(4) To avoid extreme horizontal asymmetries, the time between the leftshoulder and the head must be no greater than 2.5 times the timebetween the head and the right shoulder; likewise, the time between thehead and the right shoulder should be no greater than 2.5 times thetime between the left shoulder and the head.(5) Since the head-and-shoulders pattern in principle indicates an imminent trend reversal, the time required for the price to cross the necklinemust be no longer than the time interval between right and leftshoulders.These criteria apply with reverse sign for head-and-shoulders bottoms.A useless technical trading rule could by pure chance prove pro®table for aparticular price series and time period. This rule might then attract themarket's attention as a potentially valuable trading tool. If one evaluated therule using a similar price series and time period, one would be likely to ®ndpro®ts very similar to those that attracted the market's attention in the ®rstplace. However, if one tested the same rule on a completely unrelated timeseries during a completely unrelated time period, it would be extremelyunlikely to ®nd pro®ts. It is important, therefore, to test any trading ruleon a time series or for a time period that is not correlated with the data inwhich the rule was originally found to be pro®table.The rules used by chartists to enter speculative positions were originallydetermined to be important in pre-1948 equity data for head-and-shoulders(Edwards and Magee, 1966; pp. 50±72) and pre-1961 equity data for ®lters(Alexander 1961, 1964). Since we test the signi®cance of this set of signals onpost-1972 exchange rate data (that is, data from a later time period on a
Is more always better?
269
dierent set of ®nancial prices) our test data are independent from the datathat originally suggested the pattern's signi®cance. For this reason, our testsare analogous to `out of sample' tests of other predictive rules.
9.3 MEASURING PROFITS FROM TECHNICAL SIGNALSWe now evaluate the pro®ts that a market participant would earn if he or sheentered and exited the market as prescribed by the technical analysis manuals.
9.3.1 Filter rulesLike the identi®cation algorithm, the computation of pro®ts is far simpler for®lter rules than for the head-and-shoulders rule. Under ®lter rules, the traderis in the market holding either a long or short position at virtually all times. Incontrast, under a head-and-shoulders trading rule, which is much stricter, thetrader is out of the market much of the time, waiting for a new pattern toemerge before entering.Under ®lter rules, the trader begins out of the market, and assumes a longor short position depending on the evolution of prices. In particular, thetrader begins with no position at all, and keeps track of both the maximumand a minimum price observed over time. (On day one, the maximum andminimum are both set to that day's price.) As soon as the price is either atleast x per cent above the minimum or at least x per cent below the maximum,the trader enters a long or short position respectively. From that pointforward, the trader is always holding either a long or short position. As along position is closed, a short position is opened, and vice versa.One can measure pro®ts over an extended period of time by computing thedaily return from the trader's long or short position, then averaging this dailyreturn over the period. Each day's return is treated as a separate position andexpressed as a percentage of the price at entry. Thus, the daily return fromday t to day t  1 will be simply Pt1 =Pt  1, where Pt denotes the price attime t. This approach to measuring pro®ts treats pro®ts earned early in thetime period the same way as those earned later. Since the `daily averagereturn' measure does not incorporate interest, we do not give a relativelylarger weight to early gains or losses.
9.3.2 The head-and-shoulders patternTo compute pro®ts from head-and-shoulders positions, we need to de®neentry and exit criteria, including two caveats for exiting.
270
Advanced Trading Rules
EntryAccording to the technical manuals, one enters a position only after the priceline has crossed the neckline. 6;7 We use as our entry price the recorded priceon the day of the neckline's penetration. In our analysis, the reported dailyprice was used rather than the price at which the head-and-shoulders patternactually crossed the neckline. 8 Since peaks and troughs are identi®ed onlyafter they occur, the pattern could conceivably cross the neckline before theprice has declined by the cuto per cent required to identify the right shoulderex post as a peak. When this occurs, we enter only when the cuto has beenreached, thereby guaranteeing that positions do not bene®t from futureinformation. 9ExitAlthough very clear about when to enter a position, technical analysismanuals are ambiguous about when to exit. We calculate pro®ts by choosingan exit point, as might an actual market participant, when prices appear tohave `conclusively' stopped moving in the predicted direction. Like thetechnical analysis manuals themselves, the term `conclusively' is subjective.To be consistent and objective, we determine that a price has conclusivelystopped moving when a new peak or trough, as de®ned earlier, occurs. Thus,for a head-and-shoulders top, investors exit their short positions as soon as anew trough has been identi®ed, i.e. when the price has risen cuto per centabove that local minimum. Naturally, the holding period depends on theevolution of prices after one enters a position.There are two important exceptions to this exit rule.Exception 1: Stop-lossWe impose a `stop-loss' to limit losses when the theory proves wrong.Speci®cally, we close our position when the exchange rate moves by stoploss per cent in the direction opposite to that predicted. In our analysis, we setstop-loss at 1 per cent of the original price at entry. Note that the actual losscould exceed this because we exit at the day's recorded price.Exception 2: BounceAs mentioned earlier, prices may exhibit a temporary movement back towardthe neckline following a con®rmed head-and-shoulders pattern. To exit one'sposition at this point would, according to the manuals, be premature, becausethe reversal would not have completed its course and reached its trueminimum or maximum. To re¯ect this aspect of the paradigmatic headand-shoulders pattern, we allow for a possible `bounce' or brief interruptionin the reversal pattern. Speci®cally, if a downward trend turns upward before
Is more always better?
271
falling by at least 25 per cent of its `measuring objective', then investorsmaintain their short positions unchanged until one of the following twoconditions is met.(1) If the price rises to at least 1 per cent above the entry price, triggeringthe stop-loss discussed earlier.(2) A second trough (of any size) is reached in the zigzag.We chose the 25 per cent bounce parameter after discussions with practisingtechnical analysts. 10 These rules are applied with reverse sign for longpositions following con®rmed head-and-shoulders bottoms.
9.4 EMPIRICAL PROFITABILITY OF THE TECHNICAL TRADINGRULES IN FX DATA9.4.1 Data descriptionTo determine empirical pro®ts, we apply the technical trading algorithms todaily exchange rates on the dollar versus the yen, the mark, the Canadiandollar, the Swiss franc, the French franc and the UK pound from 19 March,1973, to 13 June, 1994. 11 We use the average of daily bid and ask quotesrecorded by the BIS. Data are expressed according to the market conventionof foreign currency units per dollar, except for the UK pound and theCanadian dollar, which are expressed in dollars per unit of foreign currency.9.4.2 Profitability of filter and head-and-shoulders rules 12Table 9.1 reports the average daily return on ®lter rules where the ®lter hasbeen set to 0.5 per cent, 1 per cent or 2 per cent. The column marked`Unconditional' reports the average daily return over the full sample.Columns A, B and C indicate returns as a function of the contemporaneoushead-and-shoulders trading signal, and will be discussed later. Unconditionaldaily average returns from ®lter rules for these six currencies are positive forall six currencies, and signi®cantly above head-and-shoulders returns. For themark, the yen and the French franc, the three ®lter rules generate dailyreturns in the range of 11.3±37.9 basis points, with an average of 24 basispoints.Table 9.2 summarizes the results from applying our head-and-shoulderstrading algorithm to the data. For each currency, we take on average one totwo positions per year, lasting on average a few weeks, leaving us with openpositions (either short or long) about 10 per cent of the time.For each position, we compute `per cent pro®ts' as a percentage of the priceat entry. For each currency, we sum the per cent pro®ts over all positions in
272
Advanced Trading RulesTable 9.1 Filter rule returns as function of head-and-shoulders and ®lter rules' overlapAverage daily return on ®lter rulesNumber of days
Curr.
Filter (%)
Unconditional(%)
DEM
0.512
0.3790.2650.146
0.3540.2100.083
0.4810.4940.443
0.377*0.2640.149
275334405
16010130
509750965065
JPY
0.512
0.2960.1930.113
0.2130.1310.096
0.3660.4310.534
0.3000.1960.113
331381420
1054516
510551055030
CAD
0.512
0.0790.0400.018
0.1030.0610.036
0.1150.0720.049
0.0760.0370.015
223211200
251263274
504449614856
CHF
0.512
0.4540.3200.180
0.2780.1630.047
0.4950.5070.662
0.4590.3240.185
177209259
1047222
525552505220
FRF
0.512
0.3570.2530.135
0.2530.1550.061
0.4100.4910.140
0.3620.2570.141*
335406440
1619056
504050355004
GBP
0.512
0.3470.2370.125
0.1090.0610.014
0.4000.2250.519
0.3650.251*0.135
404404501
11211215
502350094986
A (%)
B (%)
C (%)
A
B
C
A  days on which H&S and ®lter rules have same non-zero position.B  days on which H&S and ®lter rules have opposite non-zero position.C  days on which H&S has zero position, ®lter rules positive or negative.
Table 9.2 Pro®tability of positions following head-and-shoulders patterns, 1973±1994JPY
DEM
Average % pro®tAverage daily % pro®t
1.51580.683
0.77840.469
Average holding periodNumber of positions
22.220
16.632
CAD
CHF
FRF
GBP
0.03770.0018
0.09590.0083
0.56530.0349
0.06990.0035
18.729
11.527
16.233
20.027
*Head-and-shoulders identi®cation and pro®t-taking algorithms applied to actual exchange ratedata, assuming exit takes place when trough (or peak) has been identi®ed following entry.
Is more always better?
273
the sample period, and divide by the number of positions to reach an `averageper cent pro®t', which is reported in Table 9.1. Average per cent pro®ts arepositive for four of our six currencies, and in the range of 0.5±1.5 per cent forthe yen, the mark and the French franc for positions held on average 3±4weeks. Average per cent pro®ts are considerably less for the Swiss franc, andnegative for the UK pound and the Canadian dollar.Average per cent pro®t is divided by average holding period (in days) perposition to determine the `average daily per cent pro®t' (also reported inTable 9.2) from the head-and-shoulders trading. Average daily per cent pro®t®gures from head-and-shoulders can be compared with ®lter rules.A comparison of average per cent pro®t obtained by each of the two rulesindicates that pro®tability from ®lter rules is several times that of the headand-shoulders rules. Moreover, the ®lter rules generate these average returnsover the entire sample, whereas head-and-shoulders rules are out of themarket nearly 90 per cent of the time. Thus, ®lter rules appear to dominatethe head-and-shoulders pattern in terms of consistency (across currencies) ofpositive pro®ts, the magnitude of positive pro®ts, and the constancy (overtime) with which the pro®ts are earned.
9.5 THE INCREMENTAL PROFITABILITY OF THEHEAD-AND-SHOULDERS PATTERNAlthough resulting in far lower pro®ts on a stand-alone basis, head-andshoulders signals could nonetheless prove to be incrementally valuable byreinforcing or contradicting signals from ®lter rules. At the margin, head-andshoulders rules would add value if, for example, days in which ®lter and headand-shoulders rules coincided proved more pro®table than days in which theydid not. Since the pro®tability of ®lter rules is already well known, we focuson the incremental rather than absolute value of head-and-shoulders rules.To test the incremental value of head-and-shoulders rules, we comparedaily positions from head-and-shoulders and ®lter rules, classifying them intoone of three categories:(A) head-and-shoulders and ®lter rules have the same non-zero position(B) head-and-shoulders and ®lter rules have the opposite non-zeroposition(C) head-and-shoulders has a zero position and the ®lter rule has apositive or negative position.We then compare the average daily returns for days of type (A), (B) and (C).If head-and-shoulders rules do provide useful information incrementally, then
274
Advanced Trading Rules
average daily returns on the ®lter rule should be ranked:Adaily return > Cdaily return > Bdaily returnAn empirical comparison leads to the opposite conclusion: head-andshoulders technical signals do not add value to ®lter rules. Table 9.1 liststhe ®lter rules' average daily returns as well as the number of days of type(A) ± same sign, (B) ± opposite sign and (C) ± no head-and-shouldersposition, for all six currencies and ®lters of size 0.5 per cent, 1 per cent and2 per cent. The average daily return on type (A) days (when head-andshoulders and ®lter rules have the same sign) is never higher than on type (B)days (when head-and-shoulders and ®lter rules have the opposite sign).Furthermore, average daily returns are higher on type (C) days (when thereis no head-and-shoulders signal and only the ®lter rule indicates a non-zeroposition) than type (B) days in only three out of eighteen cases, indicated by *in Table 9.1. Both these results indicate that the head-and-shoulders rule doesnot contribute information to an existing ®lter rule. 13;14Nonetheless, the overall pro®tability of the head-and-shoulders rule can beexplained through its positive correlation with ®lter rules, whose averagedaily return is positive on days of types (A), (B) and (C). As Table 9.1indicates, type (A) days consistently outnumber type (B) days by a substantialmargin, except for the Canadian dollar, for which the head-and-shouldersrule is in any case not pro®table.Table 9.3 summarizes more rigorous measures of correlation between headand-shoulders positions and the ®lter rules. We represent a long position by1, a short position by 1, and no position by 0. We can, thus, compute a`position correlation' between head-and-shoulders positions and those basedon ®lter rules. As Table 9.3 indicates, head-and-shoulders positions arepositively correlated with those from all three ®lter rules. We also computea `return correlation' using returns from head-and-shoulders positions andthose based on ®lter rules. This would dier signi®cantly from `positioncorrelation' if head-and-shoulders systematically diered from ®lter ruleswhen returns were abnormally high or low (although such a phenomenon isnot suggested by Table 9.1). A casual comparison of these two conditionsdoes not suggest a systematic dierence of this type, echoing the returnsrankings reported in Table 9.1.Since it generates smaller pro®ts, one can think of the head-and-shoulderstrading rule as a noisy but, nonetheless, positively correlated version of ®lterrules, adding no incremental value of its own. Overall, a comparison with®lter rules indicates that head-and-shoulders rules are both more dicult toimplement and less reliable than either ®lter rules or moving averages.Moreover, a technical analyst already relying on ®lter rules would not gain
Is more always better?
275
Table 9.3 Correlation between head-and-shoulders rules and ®lter rulesPosition correlationFilter size
DEM
JPY
CAD
CHF
FRF
GBP
0.5% Filter1% Filter2% Filter
0.2400.0790.157
0.2640.1430.225
0.1340.1260.167
0.1890.0640.114
0.1430.2250.264
0.2850.1670.166
Return correlationFilter size
DEM
JPY
CAD
CHF
FRF
GBP
0.5% Filter1% Filter2% Filter
0.0570.0920.199
0.1470.1910.228
0.1260.1670.134
0.0160.0410.127
0.1470.1920.228
0.0390.0690.153
Position correlation: correlation of daily positions of two dierent trading strategies, with longpositions represented by 1, short by 1, and neutral positions by 0.Return correlation: correlation between daily returns of two dierent strategies.
incrementally (and may even lose) by combining ®lter rules with head-andshoulders signals. The pro®tability of head-and-shoulders-based rules can beunderstood through the positive correlation of head-and-shoulders-basedpositions with those based on ®lter rules.
9.6 CONCLUSIONSThis chapter assesses the incremental value of the head-and-shoulderspattern, consistently cited by technical analysts as particularly frequent andreliable, relative to ®lter rules, which have already been proven pro®table inearlier studies. Previously, non-linear visual chart patterns such as the headand-shoulders, though strongly recommended by technical analysis manuals,remained largely untested empirically. To evaluate the pattern's incrementalcontribution to pro®tability, we ®rst de®ne a head-and-shoulders pattern,then describe algorithms to recognize such patterns and to measure pro®tsderived from following these trading rules in daily foreign exchange data.On a stand-alone basis, ®lter rules clearly dominate the head-and-shoulderspattern as a technical trading signal. Head-and-shoulders rules generatepositive pro®ts for four out of six currencies, but the average daily return isseveral times greater using ®lter rules. Furthermore, ®lter rules generate theseaverage pro®ts over the entire sample period. In contrast, head-andshoulders-based trading takes a market position only about 10 per cent ofthe time for a given currency, remaining completely out of the market
276
Advanced Trading Rules
otherwise. In sum, pro®t from ®lter rules is more consistent, much higherand sustainable for longer holding periods than pro®ts from head-andshoulders.On an incremental basis, we show that the head-and-shoulders tradingrules add noise but no value. Days on which ®lter rules and head-andshoulders indicate the same long/short position are no more pro®table, andsometimes less so, than days on which the two rules indicate oppositepositions or days on which head-and-shoulders rules are out of the market.Thus, a trader would do no better, and possibly worse, by following bothhead-and-shoulders and ®lter rules instead of ®lter rules only. However, onecan successfully explain head-and-shoulders-based pro®ts in terms of positivecorrelation with ®lter rules, which are simpler and more reliable. In sum, ouranalysis strongly favours relying only on ®lter rules and ignoring the headand-shoulders pattern.The economic justi®cation for pro®ts from any of these technical rulesremains unclear. Such pro®ts may indicate the presence of market ineciencies or, as suggested by LeBaron (1994) and Silber (1994), the presenceof non-pro®t-maximizing agents such as central banks. Much more researchis needed before we will understand the underlying economic forces behindpro®ts from any technical trading rules ± the simpler algorithms such as®lter rules, or more complex, visually based rules such as head-andshoulders.
REFERENCESAlexander, S.S. (1961) Price movements in speculative markets: trends or randomwalks, Industrial Management Review, II, May, 7±26.Alexander, S.S. (1964) Price movements in speculative markets: trends or randomwalks, No. 2, Industrial Management Review, Spring, 25±46.Arnold, C. and Rahfeldt, D. (1986) Timing the Market: How to Pro®t in Bull andBear Markets with Technical Analysis, Chicago: Probus Publishing.Brock, W., Lakonishok, J. and LeBaron, B. (1992) Simple technical trading rulesand the stochastic properties of stock returns, Journal of Finance, 48, 5,December, 1731±64.Chang, P.H.K. and Osler, C.L. (1996) Head and shoulders: just a ¯aky pattern?,Federal Reserve Bank of New York Working Paper.Dooley, M.P. and Shafer, J.H. (1984) Analysis of short-run exchange ratebehaviour: March 1973 to November 1981. In Floating Exchange Rates andthe State of World Trade and Payments (eds Bigman and Taya), BallingerPublishing Company, Cambridge, MA, pp. 43±70.Edwards, D. and Magee, J. (1966) Technical Analysis of Stock Trends, Fifthedition, Boston: John Magee Inc.
Is more always better?
277
Hardy, C. Colburn (1978) The Investor's Guide to Technical Analysis, New York:McGraw-Hill.Kaufman, P. (1978) Commodity Trading Systems and Methods, New York:Ronald Press.LeBaron, B. (1991) Technical Trading Rules and Regime Shifts in ForeignExchange, Mimeo, University of Wisconsin.LeBaron, B. (1994) Technical Trading Rule Pro®tability and Foreign ExchangeIntervention, Mimeo, University of Wisconsin.Levich, R. and Thomas, L. (1993) The signi®cance of technical trading-rulepro®ts in the foreign exchange market: a bootstrap approach, Journal ofInternational Money and Finance, 12, 5, 451±74.Logue, D., Sweeney, R. and Willett, T. (1978) The speculative behaviour offoreign exchange rates during the current ¯oat, Journal of Business Research, 6,May, 159±74.Murphy, J.J. (1986) Technical Analysis of the Futures Market: A Comprehensive Guide to Trading Methods and Applications, Prentice Hall, NewYork.Neftci, S. (1991) Naive trading rules in ®nancial markets and Wiener-Kolmogorov prediction theory: a study of `technical analysis', Journal of Business, 64,4, 549±71.Pring, M. (1985) Technical Analysis Explained: The Successful Investor's Guide toSpotting Investment Trends and Turning Points, Third Edition, New York,McGraw-Hill.Silber, William L. (1994) Technical trading: when it works and when it doesn't,The Journal of Derivatives, Spring, 39±44.Sklarew, A. (1980) Techniques of a Professional Commodity Chart Analyst, NewYork, Commodity Research Bureau.Sweeney, R.J. (1986) Beating the foreign exchange market, Journal of Finance, 41,1, 163±82.
NOTES1. Those manuals are Arnold and Rahfeldt (1986), Edwards and Magee(1966), Hardy (1978), Kaufmann (1978), Murphy (1986), Pring (1985) andSklarew (1980).2. In our analysis, we exclude duplicates when examining the aggregate of allhead-and-shoulders positions found for a range of cutos.3. In this sample, standard deviations of daily per cent changes were as follows:DEM 0.697 per cent, JPY 0.615 per cent, CAD 0.274 per cent, CHF 0.794per cent, GBP 0.657 per cent and FRF 0.698 per cent.4. Increasing the cuto beyond 4.50 times the standard deviation ofdaily exchange rate changes resulted in very few additional positionstaken.5. If a head-and-shoulders pattern using one cuto suggested entering aposition within two days of a previously identi®ed entry date, we do not
278
Advanced Trading Rules
include the new position in our analysis. Our results are unaected bywhether we identify such positions with successively higher cutos orsuccessively lower cutos.6. Entry always entails going long or short one unit in the priced currency, i.e.the denominator in a conventional quote, with pro®ts measured in terms ofthe numeraire currency, i.e. the numerator in the quote. Thus, for allcurrencies but the UK pound, positions involve a long or a short positionin one dollar. For the UK pound, positions involve one UK pound. This hadno eect, however, on the interpretation of our results, which are alwaysexpressed in terms of `per cent return'.7. In practice, we ®nd that 25±40 per cent of all head-and-shoulders patterns are con®rmed by penetration of the neckline, depending on thecurrency.8. Manuals giving detailed instructions on this point stress that the penetration must be `decisive' and, furthermore, we do not observe pricesintraday.9. Neftci (1991) asserts that acting on the head-and-shoulders pattern necessarily requires future information. This assertion could be correct only in thecontext of his particular representation of the head-and-shoulders pattern,which does not correspond to representations in the technical analysismanuals we studied. The assertion also relies on the requirement that theright and left trough be very nearly identical, which is not required in themanuals and which we do not require in our algorithm.10. In practice, we ®nd that for most currencies the bounce possibility wasinvoked about 15 per cent of the time.11. Japanese data were sampled as the `Tokyo market closing middle rate'.Mark data were taken at the ocial ®xing, 1:00 pm Frankfurt time. The UKpound and Swiss franc rates represent `current market rates at 2:15 pm Swisstime'. French franc rates are `indicate rates at 2:15 pm'. Canadian dollarrates represent `London market middle rates at around noon Swiss time'.(Source: BIS Data Bank, Courtesy of the Federal Reserve Bank of NewYork.)12. In this chapter, we report returns from technical trading rules withoutdiscussing their statistical signi®cance. For a discussion of the statisticalsigni®cance or riskiness of these rules' pro®ts, see Chang and Osler (1996),which is more technical and more detailed.13. In fact, for the majority of cases, it appears that head-and-shoulders signalsreduce the pro®tability of the ®lter rules. We have not computed thestatistical signi®cance of this asymmetry. In any case, the evidence doesnot support the hypothesis that head-and-shoulders rules add value