distributedconditional on the ®rst, which can have an arbitrary distribution. Thus, it isplausible that our rule of holding the MO portfolio actually maximizes thehistorical Sharpe ratio; it certainly should on a period by period basis.However, whilst we have found that the empirical Sharpe ratio of the rulethat maximizes the cross-sectional Sharpe ratio is indeed maximal, the rulethat minimizes the cross-sectional variance appears to have a huge time-seriesvariance. Further research is needed to understand what sort of stochasticprocess might give us such a result.Our results are not in con¯ict with those of Harvey (1995). Harvey ®ndsthat asset pricing models based on unconditional moments are inadequate foremerging markets data. We have found outperformance from trading rulesbased on conditional, one-period ahead, moments.We describe our strategies as trading rules. They are trading rules in thesense that they re¯ect certain features of optimization, but fall short of fulloptimization. If we regard a trading rule as a compromise between anoptimizing strategy and a rule of thumb based on considerations of costand computability, then our strategies are certainly trading rules. Theyoptimize only in a narrow sense and fall short of the optimal strategy formaximizing expected intertemporal utility. They also seem to work and so areclearly examples of pro®table trading rules.REFERENCESBehaert, C. and Harvey, C.R. (1995) Time-varying world market integration,Journal of Finance, 2, June, Vol. 50, 403±444.Bloom®eld, T., Left, R. and Long, J.B. (1977) Portfolio strategies and performance, Journal of Financial Economics, 5, 201±18.Chamberlain, G. (1983) A characterisation of the distributions that imply meanvariance utility functions, Journal of Economic Theory, 29, 185±201.Dempster, A.P. (1969) Elements of Continuous Multi-variate Analysis, AddisonWesley, Reading, MA.Eekhari, B. and Satchell, S.E. (1995) Non-normality of Returns in EmergingMarkets, Mimeo, Economics Department, University of Cambridge.
Mean-variance analysis, trading rules and emerging markets 121Eekhari, B. and Satchell, S.E. (1996) A comparison of mean-variance versusmean-lower partial moment asset pricing models. In International Business andFinance, Supp. 1 (eds Doukas, J. and Lang, L.H.P.), JAI Press.Erb, C.B., Harvey, C.R., Viskanta, J.E. (1995) Country risk and global equityselection, Journal of Portfolio Management, 74±83.Frankel, J.A. (1995) The Internationalisation of Equity Markets, NBER workingpaper No. 4590.Harvey, C.R. (1995) Predictable risk and returns in emerging markets, Review ofFinancial Studies, 8, 3, 773±816.Johnston, K.H. and Shannon, D.S. (1975) Portfolio maintenance strategiesrevisited, Atlantic Economic Journal, 3, 25±35.Satchell, S.E. (1995) Elliptical Distributions and Garch Models of Volatility,Mimeo, Economics Department, University of Cambridge.Stapleton, R. and Subrahmanyam, M. (1978) A multiperiod equilibrium assetpricing model, Econometrica, 46, 1077±93.
Chapter 4Expected returns of directional forecastersEMMANUEL ACAR
4.1 INTRODUCTIONAn important area not yet covered in the literature is an assessment of thepro®tability of directional forecasters. Can non-zero pro®t be expected in®nancial markets from such methods, and if so, what are the parameters ofthe underlying price process which make the rule pro®table? The aim of thispaper is to specify the theoretical relationship between rule returns andstandard statistical measures of serial dependency. Such a speci®cation,although not pursued in previous research, is useful because rule returnsprovide a measure of economic signi®cance for serial dependencies in®nancial returns that otherwise might not be readily interpretable. Gaugingthe economic signi®cance of observed daily asset return autocorrelations isdicult, because the relationship between the magnitude of observed serialcorrelation coecients and pro®ts is not clear. This chapter attempts toresolve this issue by examining how trading rule returns are related to thestatistical characteristics of the underlying series. The goal is to show thatusing stochastic modelling, it is possible to establish the parameters of theunderlying price process which generate any non-zero expected return fromtrading rules.Many previous studies have used historical returns to explore the pro®tability of ®nancial forecasting (Bird, 1985; Sweeney, 1986; Dunis, 1989;Surujaras and Sweeney, 1992; Levich and Thomas, 1993; LeBaron, 1991,1992; Brock, Lakonishok and LeBaron, 1992; Taylor, 1986, 1990a, 1990b,1992, 1994; Silber, 1994). These studies play an important role in suggestingthe historical behaviour of forecasting strategies. However, they may notconstitute an appropriate guide because their results are highly dependent onthe asset and time period covered by the research. To get a broader view, wewill turn to theoretical or stochastic modelling. Very little is known in theliterature about the expected returns of dynamic strategies. Theoretical resultsexist only under the assumption of a random walk with drift for the ®lter rule
Expected returns of directional forecasters 123(Praetz, 1976; Bird, 1985; Sweeney, 1986), the option strategy (Cox andRubinstein, 1985), the stop-loss and constant proportion insurance strategies(Black and Perold, 1992). However, no other results exist for stochasticprocesses which exhibit dependencies. We believe the random walk to be anunreasonably narrow view of reality and will subsequently establish theexpected return of directional forecasts for any Gaussian stochastic process.Two reasons can be given for restricting the study to such processes. First,Gaussian processes describe numerous classes of models and, therefore,monitor a wide range of possible market conditions. So a Gaussian processmay be the preliminary step to more complex models. Second, it isquestionable whether complicated nonlinear models will bring much additional support to our arguments.In order to model rule returns, restrictions must also be placed on thenature of the rule used. This is why the set of trading rules investigated in thischapter will be restricted to autoregressive models and linear technical ruleswhich are well de®ned in the Neftci (1991) sense. This constitutes aconsiderable improvement on past studies since it covers a wider range ofpossible market conditions and trading rules. Particular emphasis is given tothe expected return and variance of trading rules by providing exactanalytical formulae.Section 4.2 de®nes the trading rule process. Section 4.3 de®nes autoregressive forecasts and rule returns, and establishes their expected values underGaussian processes assumptions. The forecasts which maximize pro®ts areexplicited. Section 4.4 shows that many technical indicators can be reformulated as autoregressive models. Section 4.5 measures the eect of conditionalheteroskedasticity on linear rule returns. Finally, section 4.6 summarizes ourresults and conclusions.
4.2 TRADING RULES4.2.1 Rule signalsSuppose that at each day t, a decision rule is applied with the intention ofachieving pro®table trades. It is the price trend which is based on marketexpectations that determines whether the asset is bought or sold. When theasset is bought, the position initiated in the market is said to be `long'. Whenthe asset is sold, the position initiated in the market is said to be `short'. Aforecasting technique is assessed as useful if, in dealer terms, it can `makemoney'. For achieving this purpose, market participants use price-basedforecasts. Therefore, the predictor Ft is completely characterized by a
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function f of past prices:Ft  f fPt ; Pt 1 ; Pt 2 ; . . . ; P0 gThe only crucial feature which is required from the forecasting technique is itsability to accurately predict the direction of the trend in order to generatepro®table buy and sell signals. Trading signals, buy (1) and sell ( 1), canthen be formalized by the binary stochastic process Bt :sell: Bt 
1
iF
Ft  f fPt ; Pt 1 ; Pt 2 ; . . . ; P0 g < 0
Note that the signal of a trading rule is completely de®ned by one of theinequalities giving a sell or buy order, because if the position is not short, it islong. Only in the trivial case of a buy and hold (sell and hold) strategy, thesignal Bt is deterministic and is 1 ( 1) irrespective of the underlying process.Otherwise, trading signals Bt are stochastic variables. They are time series ofbinary data generated by an underlying time series of continuous data. Thefamily of discretization mechanisms is broad since it is the one of tradingrules. However, in all cases, discretizations arise by a truncation of acontinuous-valued process which is a special case of Keenan (1982). Bynature, the signal is a highly nonlinear function of the observed price series Pt(Neftci and Policano, 1984; Neftci, 1991) and, therefore, it can be highlydependent through time. Bt remains constant for a certain random period,then jumps to a new level as Pt behaves in a certain way. Trading in the assetoccurs throughout the investment horizon at times that depend on a ®xed setof rules and future price changes.
4.2.2 Rule returnsThe study of the binary process of signals is of limited interest for tradingpurposes. The focus should be the economic consequence, i.e. the returnsprocess implied by the decision rule, rather than on the generating process ofthe signal. Let us recall the investment strategy. Assume a position is taken inthe market for a given period t 1; t and that no dividends are paid duringthe period. The logarithmic return during this time is Xt  Ln Pt =Pt 1 . Thenature of the position (long or short) is given by the signal triggered at timet 1, Bt 1 following a given technical rule. Returns at time t made byapplying such a decision rule are called `rule returns' and denoted Rt . Theirvalue can be expressed as:Rt  Xt if Bt 1  14:1Rt  Bt 1 Xt ,Rt  Xt if Bt 1  1The following two important remarks should be made.
Expected returns of directional forecasters 125(1) Rule returns are the product of a binary stochastic signal and acontinuous-returns random variable. Except in the trivial case of abuy and hold strategy, the signal Bt is a stochastic variable and so rulereturns are conditional on the position taken in the market (longBt  1 or short Bt  1). This is the main feature of rule returns. Upto this point, little attention has been paid to the rule returns process.Earlier studies have mainly focused on the price-change process orunderlying returns. The fact is that when evaluating forecasting abilitythe mean squared error criterion has been used to evaluate theirperformance rather than any economic evaluation. So their measureshave been unconditional to the position taken in the market.(2) Our rule return de®nition clearly corresponds to an unrealized return.By unrealized, we mean that rule returns are recorded every day even ifthe position is neither closed nor reversed, but is simply left open.4.3 AUTOREGRESSIVE MODELS4.3.1 DefinitionA linear forecast is used to predict the one-step ahead return Xt1 , given by:Ft   
tX
4:2
dj X t j
j0
with  and the dj being constants.We will denote by F the expected value of Ft and by  2F the variance of Ft .This type of forecasting technique is referred to as an autoregressivepredictor. The predictor is normally de®ned such that it minimizes themean squared error between the forecast value and the one-step ahead returnto be estimated. If the true process of returns is linear, autoregressiveforecasts must yield the best forecasts of a stochastic process in the meansquared error sense. Autoregressive models do not generate explicit tradingsignals. However, if we assume zero transactions costs, the intuitive decisionrule derived from autoregressive models is to go short if the `one-ahead'forecast is negative and go long if it is positive. That is the forecastingtechnique implicitly triggers a daily signal Bt specifying a long (1) or short( 1) position following the decision rule:sell: Bt 
1
iF
Ft   
1X
dj Xt j < 0
4:3
j0
4.3.2 Rule returns process(Unrealized) rule returns are the product of a binary stochastic signal Bt 1
126
Advanced Trading Rules
and a continuous return random variable Xt . Equation (4.1) represents thetrading rule return equation assuming discrete trading in markets where theunderlying asset is lognormally distributed.If we assume that the underlying process Xt is Gaussian, and the rule linear,the forecast Ft is Gaussian. It can be seen from Equations (4.1) and (4.3) thatin this case, the rule return function is a mixture of marginal density functionsof a truncated bivariate normal density. Such a distribution has been studiedby Cartinhour (1990). He derived it in a form that can be evaluated using acomputer algorithm developed by Schervish (1984). He showed that themarginal density function is a truncated normal density function multipliedby a `skew function'. In general, the greater the degree of truncation, the moresevere the skewing eect will be.A truncated distribution is a common feature of portfolio insurancestrategies. As shown in Trippi and Harri (1991), the terminal returndistribution of dynamic asset allocation rules is highly asymmetric beingeither left-truncated or positively skewed. Bookstaber and Clarke (1987)showed that the put option strategy truncates the lower tail and maintainsthe upside potential. Zhu and Kavee (1988) showed, using Monte Carlosimulations, that two strategies, namely the synthetic put approach and theconstant proportion strategy, have the ability to reshape the return distribution so as to reduce downward risk and retain a certain part of upward gains.There is, however, a key dierence between option and technical rulereturns. On the one hand, when using a put option, the left truncation is ®xedat a deterministic level, the exercise price for an option. On the other hand,when applying a mechanical system downside risk reduction still occurs, 1 butthe left or right truncation is a random one, due to the signal eect. By nature,a trading rule generates random infrequent trading. The signal of a ruleremains constant for a certain random period, then jumps to a new level asthe price behaves in a certain way.This point highlights that rule returns are in fact closely related to theliterature of infrequent trading and in particular with the Lo and MacKinlay(1990) approach. The stochastic model of nonsynchronous asset prices theydeveloped is based on sampling with random censoring. They give explicitcalculation of the eects of infrequent trading on the time series properties ofasset returns. Contrary to Lo and MacKinlay (1990), we will have to considerexplicitly two situations they only mentioned. First, our nontrading process isby its nature dependent, trading tomorrow (reversal of signal) depends on thesignal of today. Second, we will relax their assumptions of independent andidentically distributed underlying returns.The expected value of rule returns can be established analytically, assumingthat underlying returns follow a Gaussian process, although the exact
Expected returns of directional forecasters 127distribution cannot. This is the most important statistic for trading purposes.In addition, the one-period variance can be deduced from the expected valueusing the relation:Var Rt   E R 2t 
E Rt  2  E B 2t 1 X 2t 
E Rt  2
We know that by de®nition:B 2t 1  1andE X 2t    2   2
where  is the expected value or drift of Xt , and  2 is the variance of Xt .Therefore:Var Rt   E X 2t 
E Rt  2   2   2
E Rt  2
4:4
An important observation is that the variance of rule returns is a negativefunction of its squared expected value. Then for given underlying marketparameters, the more pro®table the trading rule, the less risky this will be ifrisk is measured by the volatility of the asset. Consequently, maximizingreturns will also maximize returns per unit of risk. This implies that attentionneed only be focused on the expected rule return.
4.3.3 Expected rule returnsProposition1If the underlying process of returns fXt g follows a linear Gaussian processwith drift, the expected value of linear rule returns Rt is given by:r2E Rt   Corr Xt ; Ft 1  exp  2F =2 2F    1 2F F =F  4:5where F is the cumulative function of a normal distribution N 0; 1,  and  2are the mean and variance of the underlying series fXt g, F and  2F the meanand variance of fFt 1 g and Corr Xt ; Ft 1  the linear correlation coecientbetween the future underlying return fXt g and the forecaster fFt 1 g.ProofsE Rt   E Bt 1 Xt   E Bt 1 X t    E Bt 1 X t   E Bt 1 
where X t designs a unit normal variate,   E Xt  and  2  Var Xt .E Bt 1   Pr Ft 1 > 01
2F
Pr Ft 1 < 0  1
F =F 
2 Pr Ft 1 < 0
128
Advanced Trading RulesE Bt 1 X t  
X t
X tX t F t 1 > F =F
X t F t 1 < F =F
where F t 1 designs a unit normal variate, F  E Ft 1  and  2F  Var Ft 1 .Then, using the truncated bivariate moments given by Johnston and Kotz(1972, p. 116), it follows that:r2E Bt 1 X t   exp  2F =2 2F with  Corr Xt ; Ft 1 Therefore, Equation (4.5) results from the weighted summation of the twoprevious terms:E Rt   E Bt 1 X t   E Bt 1 r2 exp  2F =2 2F    1
2F F =F 
4:5
The expected value of rule returns is composed of two components. Onecomes from the general Gaussian process without drift and the other from thedrift component. Equation (4.5) generalizes and re®nes results from previousresearches. The only theoretical formulae to which Equation (4.5) can becompared to are given by Praetz (1976) under the assumption of a randomwalk with drift. Having noted f , the frequency of short positions, Praetz(1976) has established that:E Rt    1Var Rt   
2
2f 
Under the random walk with drift assumption, there is no correlationbetween the forecast and the one-ahead return, then   Corr Xt ; Ft 1   0.Therefore, Equation (4.5) can be rewritten as:E Rt    1
2F F =F 
Then using the relationships between variance and expected value given byEquation (4.4), it follows that:Var Rt    2   2
f 1
2F F =F g 2
  2  4 2 F F =F  1
F F =F 
Equation (4.5) uses the probability of being short given by F F =F  ratherthan the ex-post frequency of short positions, f , derived by Praetz (1976) andused by Bird (1985), Sweeney (1986) and Surujuras and Sweeney (1992).
Expected returns of directional forecasters 129Another dissimilarity with Praetz (1976) is that directional trading mustincrease the variance of returns. In fact, the latter result shares one propertywith nonsynchronous trading (Lo and MacKinlay, 1990). Namely, infrequenttrading increases the variance of individual security returns with a non-zeromean. To the best of the author's knowledge, the expected value of rulereturns for a general Gaussian process has not been derived before. It is notsurprising that an exact analytical formulae for the expected value of linearrule returns can be established for any Gaussian process, since linear rules arewell de®ned (Neftci, 1991). A ®rst comment is that a biased forecast might besuboptimal. 2 This can be simply noted by considering a Gaussian processwithout drift   0. Assuming that F 6 0 gives an expected return equalto:r2 Corr Xt ; Ft 1  expE Rt  
 2F =2 2F 
This is of course below the expected return of a similar but unbiasedforecaster having F  0.
4.3.4 Maximizing returnsProposition 2If the underlying process of logarithmic returns Xt is Gaussian, a forecasterFt 1 maximizes expected returns if and only if:(1) it maximizes:   Corr Ft 1 ; Xt (2) F =F  = :ProofsExpected returns given by Equation (4.5) are functions of the following twovariables: x  F =F , the ratio mean/standard deviation of the forecast.  Corr Ft 1 ; Xt , the correlation between forecast and one-ahead return.Consequently, Equation (4.5) can be rewritten as:r2 expE Rt  
x 2 =2   1
2F x
4:6
The two variables  and x are independent. So the forecast which maximizesexpected returns must ®rst maximize . Then for a given , the forecast whichmaximizes pro®ts is obtained by deriving formula (4.6) as a function of x. It
130
Advanced Trading Rules
follows that the second condition is given by:rr22dE Rt =dx  0 , x exp x=2  exp,
x  
,
,
F =F  = 
x 2 =2  0
x  = 
A ®rst remark is that the optimal forecast in terms of mean squared error,FMSE satis®es criteria (1) and (2). So if the Xt process is Gaussian, no lineartrading rule obtained from a ®nite history of Xt can generate expected returnsover and above vector autoregressions. A similar ®nding is attributable toNeftci (1991). It has been proven here that minimizing the mean squared erroris a sucient but not necessary condition to maximize returns. The forecastwhich maximizes pro®ts need not be the forecast which minimizes the meansquared error. For instance, any forecast proportional to FMSE , a  FMSE (witha > 0) still maximizes pro®ts. Therefore, there are cases for which a forecastcan be highly pro®table, although not accurate. When the true model isknown, choosing the forecast which minimizes the mean squared error willmaximise returns. However, when the true model is unknown, decreasing themean squared error may be an inappropriate procedure to increase returns.Acar (1993) shows that the relationship between mis-speci®cations in terms ofmean squared error and returns is highly nonlinear. There are cases for whicha forecast can be badly mis-speci®ed in terms of mean squared error butalmost optimal in terms of returns and vice versa. The striking fact is that onecannot deduce from an increase in mean squared error a decrease in returnsand vice versa.4.4 TECHNICAL INDICATORSThe majority of traders forecast price changes using technical analysis, eventhough autoregressive techniques should yield better forecasts if the underlying process is linear (Neftci, 1991). Financial market players often prefertechnical rules to autoregressive models, mainly because they are not lookingfor the forecast which minimizes the mean squared error but that whichmaximizes pro®ts. Technical analysts have claimed that contrary to autoregressive models, technical indicators are able to capture the complexnonlinearity observed in ®nancial prices.Although technical analysis and autoregressive models might have dierentobjectives, they both use the same information, that is, historical prices.Technical analysis covers a broad category of forecasting rules. However,certain of these are highly subjective and ill-de®ned. To be objective, buy andsell signals should be based on data available up to the current time t and
Expected returns of directional forecasters 131should be independent of future information. Using the theory of Markovtimes, Neftci (1991) shows that rules based on mathematical formulae usingpast prices fPt ; . . . ; Pt m ; . . .g are well de®ned and objective in the sense thattheir performances can be assessed. It must be emphasized however, thatthere does not exist any theory or `research algorithm' to design technicalrules. A current practice among traders is to measure the pro®ts and lossesgenerated by an arbitrary set of trading rules and to select the rule whichmaximizes pro®ts.4.4.1 Technical indicators as autoregressive modelsTechnical indicator signals are usually expressed by an inequality in terms ofpast prices (`price' signal). An equivalent formulation in terms of (logarithmic) returns should be sought whenever possible (`return' signal). There aretwo reasons for this.(1) Ability to model rule returns. It has been shown in the previous sectionthat when the signal is expressed by a linear combination of returns,the expected value of rule returns can be easily found for any underlying Gaussian process.(2) Purpose of comparison with autoregressive models. Autoregressivemodels are expressed in terms of returns. So if technical indicatorssignals were to stay a function of price, direct comparison withautoregressive models would be dicult.4.4.2 Technical linear rulesDefinitionA rule is said to be `linear' if it can be expressed in the form of Equation (4.2).Proposition 3Any mechanical system triggering a sell signal from a ®nite linear combination of past prices of the form:sell Bt 
1
,
m 1X
aj P t j < 0
j0
where m being an integer larger than one and aj constants admits an (almost)equivalent linear return formulation of the form:sell B~t 
1
,
where Xt  Ln Pt =Pt 1 

m 2Xj0
dj Xt j < 0
4:7
132
Advanced Trading Rules

m 1X
m 2X
dj 
aj ;
j0
ai
i j
The only required assumption is that rates of returns can be approximated bytheir logarithmic versions. That is:1
Pt j =Pt  Ln Pt =Pt j 
for j  1, m
1
Proofssell Bt 
with b0  1
1
,
m 1X
,
Pt
aj Pt j < 0
j0
m 1X
bj Pt j < 0
j0
a0 and bj 
aj for j  1, mm 1X
,
Pt
,
m 1X
,
m 1X
j0
bj P t j 
bj P t
j1
bj 1
j1
1m 1X
m 1X
bj Pt
bj Pt < 0
j0
j0
Pt j   1
m 1 Xbj P t < 0
Pt j =Pt   1
mX1
j0
bj < 0
j0
Let us assume that1
Pt j =Pt  Ln Pt =Pt j  
for j  1, mmX1j1
bj
j 1Xi0
j 1X
4:8
Xt i
i0
1. It follows thatXt i  1
m 1X
bj < 0
j0
,
1
,

2 jm 1 m 2  mXXXbj bm 1 i Xt j < 0j0
m 2Xj0
j0
dj Xt j < 0
i0
Expected returns of directional forecasters 133with1
mX1j0
bj 
m 1X
aj
j0
anddj 
mX2 ji0
mX2 j
bm 1 i 
i0
am 1 i 
m 1X
ai
ij1
Consequently, many popular technical systems are implicitly linear rules. Thisis especially the case for moving-average type and momentum indicators. Letus recall their de®nition through the necessary and sucient conditions whichtrigger a