short position (when the position is not short, it is long).(1) Simple moving average of order m > 1sell if Pt < SMt m 
Pt  Pt 1      Pt m1m
where SMt m denotes the simple moving average over m prices up toPt .(2) Weighted moving average of order m > 1sell if Pt <
m
1Pt  m 2Pt 1      1Pt m1m m 1=2
(3) Exponential moving average of coecient 1 > a > 0sell if Pt < a Pt  1
aPt 1      1
(4) Momentum of order m > 1 3
a m 1 Pt m1 
sell if Pt < Pt m1(5) Double moving average of orders r; m; 0 < r < msell if SMt r < SMt mIt must be noted that the simple moving average is a particular case of thedouble moving average when the short moving average is the price r  1.Explicit coecients dj in Equation (4.2) for all the technical indicatorsmentioned above can be found by applying the results of Proposition 3 andare given in Table 4.1.Validity of logarithmic approximationsThe validity of the return signal was checked using Monte Carlo simulations.One hundred samples of 2500 observations have been generated from varioussimulated normal random walks with steps distributed as N ;  2 . Theequivalence between price signals Bt and return signals B~t has been tested
134
Advanced Trading RulesTable 4.1 Return/price signals equivalence
Rule
Parameter(s)
Simple order
Price sell signalsPt <
m 1X
Return sell signalsm 2X
aj Pt j
j0
Simple MA
m2
aj 
1m
Weighted MA
m2
aj 
mm m
Exponential MA
1 > a > 0; m  2
aj  a 1
Momentum
m2
aj  1 for j  maj  0 for j 6 mr 1X
Double orders
Double MA
Generalization
m>r1
j0
j1=2
dj 
a j
m 1X
aj 
aj Pt j < 0
1
j
m
j m2
j
1
see generalization1;1aj Pt j
j0
1bj  ;r
m 1X
dj  m
bj Pt j <
j0
dj Xt j < 0
j0
1m
dj  1m 2X
dj Xt j < 0
j0
dj  m
r j  1for 0  j  r 1dj  r m j 1for r  j  m 1
m 2X
dj 
j0
dj Xt j < 0; with:m 1X
ij1
ai and  
mX1
aj
j0
for the simple and weighted moving average rule. 4 As can be seen fromTables 4.2 and 4.3, signals are dierent in less than 3 per cent of all cases forsimulated series. The largest deviation comes from   0.001 and   0.03,for m  200. Even then, returns signals are identical to price signals in at least97 per cent of all cases. This case represents an upper bound in terms of bothvolatility and average returns over ten years for ®nancial series (Taylor, 1986:Tables 3 and 4, pp. 33 and 34). On the basis of the empirical results presentedin Tables 4.2 and 4.3, one can safely conclude that return signals lead toessentially the same investment strategies as price signals for values of m aslarge as 200.The nice feature of the linear rules, expressed by a linear combination ofreturns, is that it includes in a uni®ed framework autoregressive predictors(by construction) and technical systems (by reformulation). Finally, it mustbe emphasized that, although rather general, linear rules do not cover alltechnical rules used by practitioners.
Expected returns of directional forecasters 135Table 4.2 Equivalence between price fBt g and return fB~t g signals for the simple movingaverage ruleSimulated random walk N ;  2 , 100 replicaAverage (maximum) percentage of days having Bt 6 B~tOrder m No. obs
  0;   0:01   0:001;   0:01   0;   0:03   0:001;   0:03
2550100200
0.18 (0.44)0.22 (0.56)0.37 (0.68)0.48 (0.84)
2500250025002500
0.15 (0.48)0.22 (0.44)0.26 (0.60)0.30 (0.68)
0.50 (1.2)0.72 (1.40)1.08 (1.96)1.46 (2.44)
0.50 (0.96)0.19 (1.08)1.05 (1.84)1.46 (2.60)
Table 4.3 Equivalence between price fBt g and return fB~t g signals for theweighted moving average rulePrice signal Bt =return signal B~t : case of the weighted moving average ruleSimulated random walk N ;  2 , 100 replicaAverage (maximum) percentageof days having Bt 6 B~tOrder m
No. obs
  0;   0:03
  0:001;   0:03
200
2500
1.57 (2.52)
1.55 (2.92)
It is doubtful that signals from certain rules will ever accept an (almost)equivalent formulation of the type given by Equation (4.7). Rules whichmight be nonlinear are in particular rules based on intraday high and lowdata or on the maximum and minimum of certain values. An example of sucha rule is the channel rule studied by Lukac, Brorsen and Irwin (1988), Taylor(1990a, 1994), Brock, Lakonishok and LeBaron (1992) and Curcio andGoodhart (1992). These rules have not been included in this research becausethey are not easily modelled.
4.4.3 Expected rule returnsOur purpose is to quantify the pro®tability of popular trading rules underplausible market conditions. More precisely, we consider the simple movingaverages, weighted moving averages and momentum rules in their autoregressive versions given by Table 4.1. Therefore, the exact expected value of
136
Advanced Trading Rules
rule returns can be established using Equation (4.5) under Gaussian processesassumptions. We will assume that a year includes 250 days and that the dailyprocess which drives underlying logarithmic returns is successively:(1)(2)(3)(4)
a random walk with driftan autoregressive process of order one without drift, AR(1)a price-trend model without drift, ARMA(1,1)a fractional Gaussian process without drift.
Random walk with driftThe underlying model isX t    "twhere  is a constant and "t is iid N 0;  2 . Figure 4.1 exhibits returns from abuy and hold strategy and from a simple moving average rule of orders 5, 20,100 as a function of the drift. Overall, the following three remarks can bemade.(1) Expected return will be a ®xed percentage of the drift. It will underperform a buy and hold strategy if the drift is positive and outperformit if the drift is negative. The expected return of the simple movingaverage rule is a positive function of the absolute value of the drift jjand a negative function of the volatility . It is a positive function ofthe order m of the simple moving average, which can be explained bythe fact that the most pro®table strategy is buy and hold if the drift ispositive. Indeed, it must be stressed that in this case technical tradingwould be useless and Figure 4.1 is really measuring how much tradingrules underperform the optimal buy and hold strategy.(2) The drift increases the instantaneous variance of return.(3) Only in the absence of any drift in the data are rule returns uncorrelated.Figure 4.2 illustrates that in decreasing order of pro®tability, we have:(1) momentum(2) simple moving average(3) weighted moving average.It means that ex ante, certain technical rules will systematically capture thedrift better than others.AR(1)The process of underlying returns is supposed to be an autoregressive modelof order one without drift, de®ned by:Xt  Xt 1  "t
Expected returns of directional forecasters 137
Rule returns %
Figure 4.1 Technical returns as a function of the drift
Rule returns %
Figure 4.2 Technical returns as a function of the order of the rule, under the random walk withdrift assumption
138
Advanced Trading Rules
Rule returns %
Figure 4.3 Technical returns as a function of the autoregressive coecient
2where is the autoregressive coecient and the "t are iid N 0; 1 2 .Figure 4.3 exhibits that for a given system (moving average type) positiveautocorrelation is required to make the investment pro®table. The relationship between expected rule returns and ®rst order autocorrelation is almostlinear for an AR(1) model without drift and small (see Equation (4.5)).Trend-following models require positive autocorrelations to be pro®table.However it is perfectly possible to create rules designed to take advantage ofnegative autocorrelations (e.g. opposite strategies). It is even possible to buildrules which display positive expected return whatever the sign of the ®rstorder autocorrelation. 5 Figure 4.4 shows that for a given order of rule, certainstrategies perform better than others. The quicker the rule responds to a newprice, the more pro®table it is. For example, a short-order system capturesbetter the autocorrelation of order one than a long-order system. In addition,the more weight given to the most recent prices, the more pro®table it is. Forinstance, a weighted moving average systematically re¯ects a new price valuebetter than a simple moving average, which in turn outperforms a momentum. Proposition 2 explains among other things why going from the randomwalk with drift process to the AR(1) process, the ranking of trading ruleshighlighted by respectively Figures 4.2 and 4.4 is inverted. Indeed, this showson the one hand that under the AR(1) process assumption, a trading rulewhich maximizes pro®ts is the short-term AR(1) forecast which is nothing elsethan the simple moving average of order 2. On the other hand, under the
Expected returns of directional forecasters 139
Rule returns %
Figure 4.4 Technical returns as a function of the order of the rule, under the AR(1) assumption
random walk with drift model, a rule which maximizes pro®ts is the long-termbuy and hold strategy.Finally, it can be noted that these results are consistent with the ®ndings ofLeBaron (1992) who performed Monte Carlo simulations to estimate theexpected returns following simple moving averages of orders twenty, thirtyand ®fty under the assumption of AR(1) models.ARMA(1,1), price-trend modelTaylor (1980) proposes an original approach to model trends in ®nancialprices. The fundamental trend idea is that several returns are in¯uenced in thesame way, either towards a positive conditional mean or towards a negativeconditional mean (Taylor, 1986). Thus, trends will cause positive autocorrelations. The impact of that current information which is not fully re¯ected inthe current price, on future returns, should diminish as time goes on. Thus,the autocorrelations should decrease as the lag increases. The simplestparametric autocorrelation functions consistent with the observations was®rst investigated in Taylor (1980), and is de®ned by:H1 : h  Ap h
A; p; h > 0
There are two parameters in H1 . Parameter A measures the proportion of
140
Advanced Trading Rules
information not re¯ected by prices within one day. Parameter p measures thespeed at which imperfectly re¯ected information is incorporated into prices.As A ! 0 or p ! 0, information is used perfectly. Credible price-trendmodels have typical parameter values A  0.03 and p  0.95 (Taylor,1986). Low values for A are inevitable whilst values for p near to one indicatetrends lasting for a long time.One simple example assumes the return Xt is the sum of an autoregressivetrend component t and an unpredictable residual et .X t   t  ett
  p t 1
  t
t are iid N 0; A 2 1
A  Var t =Var Xt 
p 2 
The returns then have autocorrelations:h  Ap hThe processes ft g and fet g are supposed to be stochastically independentand of a Gaussian nature (hence linear). Prices tend to move in one direction(the trend) for a period of time and these trends themselves change in arandom and unpredictable fashion. The mean duration of such trends isdenoted md and shown to be:md  1= 1
p
The price-trend model is in fact nothing else than a state representation of anARMA(1,1) de®ned by:Xt

p Xt 1
  "t
q"t 1
where the variance reduction A is linked to p and q via:A p
q 1
pq=f p 1
2pq  q 2 g
Consequently, this particular price-trend model is a Gaussian process.Therefore, expected rule returns can again be deduced from Equation (4.5).They are as follows, assuming a price-trend model without drift.(1) An increasing function of A for p and  ®xed ± the larger theproportion of the variance of the returns that can be explained bythe variance of the trends, the more pro®table the trading rules are.(2) An increasing function of p for A and  ®xed ± the more the trendcomponent is autocorrelated, the more pro®table the trading rules are.(3) A proportional (and positive, if trend-following, rule and positiveautocorrelations) function of the volatility for A and p ®xed.
Expected returns of directional forecasters 141
Rule returns %
Figure 4.5 Technical returns as a function of the order of the rule, under the price-trendmodel assumption
Figure 4.5 gives an example of some rule returns of orders 2 to 50 forf  0.0068, A  0.03, md  20 daysg. The most pro®table simple movingaverage corresponds to the order r  29 days. It seems logical that given amean duration of trends, a technical rule ®nds its optimal parameter aroundthis value. In the case of the moving average, it is slightly bigger (ordertwenty-nine for a mean duration of twenty days). Ranking between systems ismore complex and should be in favour of exponential moving average sinceTaylor (1986) has remarked that such representations can be very close to theoptimal forecast. Rules are no longer uniformly ranked ± that is, eithersystematically in favour of short (AR[1]) or long (see random walk with drift)strategies ± but instead depend on the mean duration of the trend.Such properties of linear trading rules might hold for nonlinear strategiessuch as the channel rule. Taylor (1994) ®nds in particular that channel rulereturns are a positive function of A for p ®xed (property a) and a positivefunction of p for A ®xed (property b). The distribution shape of channel rulereturns (Taylor, 1994) is extremely similar to the one of weighted movingaverage returns (Figure 4.3). The best order of channel rule, as is also the casefor the weighted moving average, ®nds its optimal parameter close to the truemean duration of the trend.
142
Advanced Trading Rules
Fractional ARIMA (0, d, 0)An ARIMA (0; d; 0) process or fractional Gaussian process is formallyde®ned by Hosking (1981) as:H d Xt
  et
whereHd  11
B d dB
1  Xdk0
12d 1
k
d B 2
B k16d 1
d 2
d B 3

and B is the backward operator de®ned by B Xt   Xt 1 ;  the mean returnand fet g the white noise process. In this chapter, the fet g consists ofindependent identically distributed (normal) random variables with meanzero and variance  2e .As in the ®nancial literature, the fractional Gaussian process is interpretedhere as a function of the Hurst exponent rather than the parameter d. It isrecalled that H is related to d by the relation H  d  0.5.Financial fractional Gaussian processes usually fall in the range 12 < H < 1(Walter, 1991; Peters, 1991). They are characterized by a tendency to havetrends and cycles, as in the price-trend model. In contrast, however, thefractional Gaussian process exhibits abrupt discontinuous changes because ofan in®nite or unde®ned variance. Cycles are no longer regular but erratic andaperiodic.Expected rule returns assuming a fractional Gaussian process can also beestablished using Equation (4.5). Figures 4.6 and 4.7 exhibit that when thereis no drift, expected rule returns are very similar to the ones corresponding toan autoregressive process of order one (Figures 4.3 and 4.4). This is becausetechnical indicators do not exploit the main feature of a fractional Gaussianprocess, namely the long-term dependence (for H > 0.5). They only extractthe short-term dependence, which is very much that of an AR(1) process. 6Nevertheless, there does exist a major dierence with more usual Gaussianprocesses: the maximum possible gain is in®nite. Indeed, it appears that theoptimal forecast de®ned by Hosking (1981) displays both in®nite expectedreturn and variance because the autocorrelations are not summable. Therefore, technical predictors might produce returns very far from the maximumachievable gain. However, it has been claimed (Mandelbrot, 1966) that thebest linear forecast is useless for predicting the time series since it relies onparameter estimation.Consequently, the fractional Gaussian process might constitute a casewhere technical trading rules might be preferred to the best linear forecastand would contradict Mandelbrot's (1963) opinion that expected gains from
Expected returns of directional forecasters 143
Rule returns %
Figure 4.6 Technical returns as a function of the order of the rule, under the fractional Gaussianprocess assumption
Rule returns %
Figure 4.7 Technical returns as a function of the fractional parameter
144
Advanced Trading Rules
the `®lter method' depend entirely on the assumption that price is continuous.Mandelbrot (1966, p. 242) stated: `It is also possible to conceive of modelswhere successive price changes are dependent so that prices do not follow apure random walk, but where the nature of the dependence is such that itcannot be used to increase expected pro®ts'. This does not apply to thefractional Gaussian process. Indeed, in the latter case, technical rules arequite pro®table and do not rely on parameter estimation.
4.5 CONDITIONAL HETEROSKEDASTICITY AND LINEARRULE RETURNS4.5.1 ARCH(p) modelIt has been recognized that models for returns should have either nonstationary variance or, conditional upon past observations, a variance dependent on such observations and additional variables. The pioneering work ofEngle (1982) de®nes a -mean, autoregressive conditional heteroskedasticityARCH(p) process, Xt , by:vupXu2 "XXt    t 0 it iti1
there being p  1 non-negative parameters i with 0 > 0 and "t Gaussianwhite noise, with "t  N 0; 1. This model has a very complicated unconditional distribution and it is dicult to establish conditions for stationarity orto subsequently ®nd the moments.
Proposition 4If the underlying process of returns fXt g is a zero-mean autoregressiveconditional heteroskedasticity ARCH(p) process, the expected value of linearrule returns Rt is zero.ProofsE Rt   EfE fXt 1 g Bt 1 Xt gwhere E fXt 1 g means the expected value of Xt 1 conditional to the knowledgeoffXt 1 g  fXt 1 ; Xt 2 ; . . . ; Xt m ; . . .g
Expected returns of directional forecasters 145vupXu2 "E Rt   E Bt 1 t 0 Xi t i t
i1
vupXufXt 1 g2 E Bt 1 t 0 "t iX t iEi1
Since the stochastic process "t is independent of fXt 1 g, it follows thatE fXt 1 g t   0
ThenvupXu2tE Rt   E Bt 1iX t i0  00i1
Proposition 4 has just established the expected value of rule returns for one ofthese alternatives, namely the ARCH(p) model, still assuming process without drift. As long as the Xt are uncorrelated, no non-zero rule returns can beexpected. So mean nonlinearities might be necessary to generate non-zeropro®ts as in the case of the fractional Gaussian process.
4.5.2 Price-trend model with conditional heteroskedasticityThe ARCH process is constructed from strict white noise and, therefore, willalways be uncorrelated. Extensions have been proposed to introduce smallautocorrelations (Taylor, 1986). An example of this is given by the price-trendmodel with conditional heteroskedasticity:Xt  t  etwheret  pt 1  t ;
Corr Xt ; Xt   Ap 
and
A  Var t =Var Xt 
The conditional heteroskedasticity is added through:et  Vt "twhereLn Vt 
 Ln Vt 1 
  t
with "t iid N 0; 1 A and t iid N 0; 2 1  2 . It is not possible to deriveanalytical expected rule returns for such models. However, expected returnscan be established using Monte Carlo simulations. We generated 500 samplesof 1500 rates following a price-trend model with conditional heteroskedasticity. The parameters considered are typical of exchange rates models
146
Advanced Trading RulesTable 4.4 Conditional heteroskedasticity and annualized returns from linear rules
Price-trend model with conditional heteroskedasticity
S(5)
S(10)
S(20)
S(40)
Simulated returnExpected linear return
7.337.61
9.389.69
10.7811.00
10.6310.85
AR(2)-GARCH(1,1) model
S(5)
S(10)
S(20)
S(40)
Simulated returnExpected linear return
24.0624.56
17.9018.66
12.7713.57
9.399.72
(Taylor, 1994). They are A  0.03, p  0.95,  5.15,  0.422,  0.973.Average returns were measured for the simple moving average of orders®ve, ten, twenty and forty (Table 4.4). Conditional heteroskedasticity makesthe rule marginally more pro®table than without conditional heteroskedasticity. The expected rule return resulting from the simulations is slightlyhigher than the expected return assuming a pure linear models given byEquation (4.5). However, the dierences are rather small and the rankingbetween strategies remains unchanged.4.5.3 AR(2)-GARCH(1,1) modelAnother popular model including both conditional heteroskedasticity andserial dependencies is the AR(2)-GARCH(1,1) model (LeBaron, 1991)de®ned by:Xt  a  b1 Xt 1  b2 Xt 2  "twith"t 
pht z t ;
ht 
0
21"t 1 
ht 1
and
zt iid N 0; 1
As for the price-trend model with conditional heteroskedasticity, it is notpossible to derive analytical expected rule returns for the AR(2)GARCH(1,1) model. However, expected returns can be established usingMonte Carlo simulations. Five hundred samples of 1500 rates following anAR(2)-GARCH(1,1) model were generated. The parameters considered aretypical of exchange rates models (LeBaron, 1991). They are:a  0;
b1  0:0832;
 0:77 287;
b2  0:0324;
1  0:1680
0  2:294E
5;
As before, average returns were measured for the simple moving average oforders ®ve, ten, twenty and forty (Table 4.4). This time, conditional
Expected returns of directional forecasters 147heteroskedasticity makes the rule marginally less pro®table than withoutconditional heteroskedasticity. The expected rule return resulting from thesimulations is slightly lower than the expected return assuming a pure linearmodels given by Equation (4.5). The dierences are again rather small and theranking between strategies also continues unchanged.To conclude, we do not believe that the presence of conditional heteroskedasticity will drastically aect returns generated by linear rules. That is,the presence of conditional heteroskedasticity, if unrelated to serial dependencies, may be neither a source of pro®ts nor losses for linear rules.
4.6 CONCLUSIONSUnder the assumption that underlying asset returns follow a Gaussianprocess, the linear rule returns distribution is a mixture of marginal densityfunctions of a truncated bivariate density function. Exact expected values canbe obtained and are of importance because for given underlying marketparameters, maximizing returns will also minimize risk.The expected return following a linear trading rule is zero if the underlyingprocess is a random walk without drift. This is non-zero if the underlyingprocess exhibits a drift and/or autocorrelations. If the underlying process is arandom walk with drift, the expected return of a trend-following trading ruleis a positive function of the drift and a negative function of the volatility. Ifthe underlying process exhibits positive (negative) autocorrelations but nodrift, the expected return of a trend-following (contrarian) strategy is apositive function of the volatility.Under the assumptions of an underlying Gaussian process and a linearforecast, minimizing the mean squared error is a sucient but not necessarycondition to maximize expected returns. Consequently, maximizing returnsover and above autoregressive models is not possible. However, when the trueprice model is unknown, a non-optimal forecast has to be used. In thesesituations, the degree of mis-speci®cation can be quite dierent in terms ofmean squared error and returns.Many popular technical trading rules can be expressed as autoregressiveforecasts. Such re-formulations permit the application of both technical andstatistical predictors in a uni®ed framework called `linear rules'.To conclude, the purpose of this research has not been to test marketeciency, in itself a dicult task, rather to provide an understanding of theperformances of some forecasting strategies relative to plausible models of®nancial prices.
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Advanced Trading Rules
4.7 APPENDIXThis appendix writes down Equation (4.5) for the dierent stochasticprocesses and trading rules investigated in this paper.Proposition 1 has shown that if the underlying process of returns fXt gfollows a linear Gaussian process with drift  and volatility , the expectedvalue of linear rule returns Rt is given by:r2E Rt   Corr Xt ; Ft 1  exp
 2F =2 2F    1
2F F =F 4:5
whereFt   
tX
dj X t j
j0
with  and the dj being constants, F the expected value of Ft and  2F thevariance of Ft . Then it follows that:F    
tX
dj
j0
andvu tt Xt 1XuXd 2j  2di dj  i