the unexpected shocks may come from the exponentialdistribution and the chi-square distribution.5. We assume thatEj" 4t j < 1andEj 4t j < 1such that the 4th-order dierentiations of F" s and F s exist and equal4E " 4t  and E  4t  respectively, namely, F "4 0  E " 4t  and F  0  E  4t .
Some exact results for moving-average trading rules
173
6. This presumption should not aect our results because we can expect thelong-term mean of daily returns for ®nancial assets to be reasonably close tozero. In fact, Knight, Satchell and Tran (1995) ®t their model described byEquations (5.5) to (5.8) to daily returns of UK FT100 index and futures andestimate their long-term means to be 0.00042 and 0.0000 respectively. Theirresults support our presumption about the long-term mean here.7. We assume that an investor will take his or her action immediately at thebeginning of each period right after he or she observes a signal at the end ofthe previous period. We also assume that there is no price change betweenthe end of the previous period and the beginning of the present period.8. The natural logarithm of the asset price will follow a random walk onlywhen the condition p  q  12 is satis®ed.
ACKNOWLEDGMENTSThe author thanks his supervisor, Dr Stephen Satchell, for his helpfulguidance and comments.
Chapter 6The portfolio distribution ofdirectional strategiesEMMANUEL ACAR AND STEPHEN SATCHELL
6.1 INTRODUCTIONThis chapter is concerned with the rates of returns distribution generated by aset of dynamic strategies. In order to calculate the risk of a portfolio, apassive investor applying buy-and-hold strategies needs to know how relatedare the two underlying markets. This information is necessary for an activeinvestor but not sucient alone. The active investor also needs to know howthe two forecasting strategies he or she follows can dier. Very little is knownabout the exact distribution of portfolio of active strategies. Active strategiesdier from passive ones in that they include an in-built timing. Market timingaects not only the returns from a given asset, but also its volatility andcorrelation with other assets.In recent years, and particularly for banking institutions involved withunstable ®nancial markets, the need for worthwhile forecasts has beengenerally recognized by treasurers and academics alike. The choice ofwhich dynamic strategy to follow depends on the expectations one hasabout the stochastic process which drives prices. Many forecasting strategiescan be used to predict future price moves from fundamental to technicalindicators without ignoring more advanced techniques such as neural networks and genetic algorithms. Unfortunately, the study of ®nancial forecastsused for trading is relatively new.There are three reasons for investigating the distribution of portfolio oftrading rules. First, rules distributions would provide a measure of similaritybetween trading systems. With the exception of Lukac, Brorsen and Irwin(1988), rules have been merely listed rather than classi®ed on the basis of theirproperties. Many forecasting systems have been proposed and are oftenconsidered to be dierent when they are extremely similar if not identical.Second, rules distributions would permit the construction of an ecient
The portfolio distribution of directional strategies 175portfolio of rules. Until now such portfolios have been built empirically forgiven ®nancial time series (Brorsen and Lukac, 1990), but have never beenestablished theoretically for given stochastic processes. Indeed, establishingtheoretical correlations between trading rules has been considered anextremely dicult task (Brock, Lakonishok and LeBaron, 1992). Third,and perhaps more important, it will allow the joint pro®tability of a set oftrading rules to be tested. Brock, Lakonishok and LeBaron (1992), Surujarasand Sweeney (1992) and Prado (1992) have emphasized that such a test mighthave power, especially against nonlinear alternatives. The resulting tests ofnon-zero pro®tability could then be more powerful than any single test.The primary purpose of this chapter is to establish the distribution ofreturns generated by a portfolio including two active strategies for givenstochastic processes. Section 6.2 de®nes our forecasting strategies. The timingis triggered by linear forecasts for the sake of tractability. Section 6.3establishes the exact distribution of a portfolio of trading rules. Section 6.4generalizes previous results assuming elliptical processes. The extension is ofsome interest as future and foreign exchange markets are often characterizedby return distributions that are symmetric about the mean due to the abilityof investors taking long and short positions at equal costs. However, the samereturns are typically kurtotic and exhibit fatter tails than normal distributions. The elliptical family have this property of being asymmetric andkurtotic. The most important ®nding of our analysis is that conventionalportfolio theory might not apply to active directional strategies even when theunderlying assets follow a multivariate normal distribution.
6.2 PORTFOLIO RETURNS OF DIRECTIONAL STRATEGIES6.2.1 Directional strategiesLet us ®rst de®ne the trading process. A forecast Ft is used to predict the onestep ahead return Xt1 . If we assume zero transactions costs, the intuitivedecision rule is to go short if the [one-ahead] forecast is negative and go longif it is positive. That is the forecasting technique implicitly triggers a dailysignal Bt specifying a long (1) or short ( 1) position following the decisionrule:sell Bt 
1 iF Ft < 0
sell Bt  1 iF Ft > 0Returns at time t made by applying such a decision rule are called `rule
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Advanced Trading Rules
returns' and denoted Rt . Their value can be expressed as:Rt  Xt if Bt 1  1Rt  Bt 1 Xt ,Rt  Xt if Bt 1  16.2.2 Portfolio returnsNow, two trading rules are applied separately and simultaneously totwo dierent underlying assets. Both trading rules generate returnsR1;t  B1;t 1 X1;t and R2;t  B2;t 1 X2;t where Xi;t is the underlying return ofasset i. Then the investor decides to allocate a weight ! to the ®rst activestrategy applied to asset one and 1 ! to the second active strategy appliedto asset two. The corresponding portfolio returns is given by:Rt  !B1;t 1 X1;t  1
!B2;t 1 X2;t
6:1
6.3 EXACT DISTRIBUTION UNDER THE NORMAL RANDOMWALK ASSUMPTIONWe now make the following assumptions.(1) Two ®nancial series, with underlying returns X1;t and X2;t , follow azero mean bivariate normal law with variances  21 and  22 andcorrelation coecient x .(2) Two trading rules (similar or dierent) F1;t and F2;t are respectivelyapplied to the two processes fX1;t g and fX2;t g to predict one-stepahead returns. Both forecasts are zero-mean and follow a bivariatenormal law. The correlation coecient between the two trading rulesis denoted by F .(3) Forecasts and one-ahead returns are independent.Assumption (3) may seem very limiting since it implies that the forecastsused in the trading process have no value. However, the purpose of thissection is not to test the usefulness of active strategies but to derive theirstochastic properties under the random walk assumption. By doing so, newtests of the random walk hypothesis will be permitted.
Proposition 1The portfolio returns Rt speci®ed by Equation (6.1) exhibit a characteristicfunction given by:
The portfolio distribution of directional strategies 1771 1 arcsin F E exp izRt  2 z2 2 2! 1  1 exp21 1arcsin F 2 z2 2 2 exp! 1  12
! 2  22  2! 1
! 2  22
2! 1
!x 1 2 

!x 1 2 
6:2
Then from the characteristic function, we can deduce the density function.This is given by:If:! 2  21  1 ! 2  22 2! 1 !x 1 2  6 01 11 arcsin F  qg x 2 22 ! 2  1  1 ! 2  22  2! 1 !x 1 2 x2 exp2 ! 2  21  1 ! 2  22  2! 1 !x 1 2 1 11arcsin F  q2 2 ! 2  21  1 ! 2  22  2! 1 !x 1 2 x2 exp2 ! 2  21  1 ! 2  22  2! 1 !x 1 2 If:! 2  21  1
! 2  22  2! 1
!x 1 2   0
that is x  1 and !1  1 !2 1 11x21 arcsin F  pexpg x 222 28!  18!1
1arcsin F 

Proofs
E exp izRt   PrB1;t 1  1; B2;t 1  1E exp iz!X1;t  1 PrB1;t 1  1; B2;t 1  1E exp iz !X1;t
!X2;t 1
!X2;t 
 PrB1;t 1  1; B2;t 1  1E exp iz !X1;t  1 !X2;t  PrB1;t 1  1; B2;t 1  1E exp
iz!X1;t  1 !X2;t 
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Advanced Trading Rules
The probability of the truncated bivariate normal 0; 0 is given in Johnsonand Kotz (1972). This is equal to:PrB1;t 1  1; B2;t 1  1  PrF1;t 1 > 0; F2;t 1 > 0 
11 arcsin F 4 2
Similarly,1 1 arcsin F 4 21 1PrB1;t 1  1; B2;t 1  1  PrF1;t 1 < 0; F2;t 1 > 0 arcsin F 4 21 1arcsin F PrB1;t 1  1; B2;t 1  1  PrF1;t 1 > 0; F2;t 1 < 0 4 2PrB1;t 1  1; B2;t 1  1  PrF1;t 1 < 0; F2;t 1 < 0 
Then by substitution, we obtain:1 1 arcsin F E exp izRt  2 2z2 2 2 exp! 1  121 1arcsin F 2 z2 2 2 exp! 1  12
! 2  22  2! 1
! 2  22
2! 1
!x 1 2 

!x 1 2 

Active portfolio returns exhibit dierent variance than passive strategies. Thevariance of the portfolio of trading rules is given by:22 22 2Var Rt   !  1  1 !  2  2! 1 !1 2 xarcsin F This obviously diers from the variance of a passive portfolio which would beequal to:Var Rt   ! 2  21  1
! 2  22  2! 1
!1 2 x
The portfolio of active strategies will exhibit lower variance than passiveones if and only if 0 < x .Portfolio of active strategies do not exhibit any skewness under the normalrandom walk assumption. They do, however, display kurtosis. To illustratethis point, we have chosen the case where two active strategies are applied tothe same asset (1  2   and x  1) and given equal weight (!  0:5).The amount of kurtosis can be derived from the characteristic function in
179
Kurtosis
The portfolio distribution of directional strategies
Forecast correlationFigure 6.1 Kurtosis  F (forecast correlation)
Equation (6.2). This is equal to:K
310:5  arcsin F 
6:3
Kurtosis of active portfolio returns is a negative function of the correlationcoecient between forecasts F . The more negatively correlated the forecasts,the higher the amount of kurtosis (see Figure 6.1) In principle, one couldcalculate K for an arbitrary portfolio using Equation (6.2), the result is longand cumbersome and we therefore omit it.
6.4 GENERALIZATIONWe now generalize proposition 1 to consider examples of processes where: X1;t 21 x 1 22st  N 0; s t S where S X2;t x 1 2 22and likewise the forecasts are: F1;tqt  N 0; q 2t SF F2;t
where SF 

 21;FF 1;F 2;F
F 1;F 2;F 22;F

and st and qt are independent. This characterization consists of a large familyof possible processes, indeed these are a subclass of the elliptical class.Dierent choices for st (or qt ) lead to a mixture of normals if t is Bernoullior a multivariate t with  degrees of freedom, if t is distributed as the
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Advanced Trading Rules
reciprocal of a chi-squared with  degrees of freedom divided by , other lessfamiliar examples can be calculated by choosing any positive distribution forst . Let the mgf of s 2t be de®ned by s   E exp s 2  then the followingresults hold.Proposition 2The portfolios returns have a characteristic function given by:1 1 arcsin F E exp izRt  2  z2 2 22 2 s!  1  1 !  2  2! 1 !x 1 2 21 1arcsin F 2  z2 2 22 2!  1  1 !  2 2! 1 !x 1 2  s2
6:4
ProofThe steps are exactly the same as in proposition 1 except that in Equation(6.4) we would have:1 1 arcsin F Eexp izRt =s 2  2 2s z exp! 2  21  1 ! 2  22  2! 1 !x 1 2 21 1arcsin F 2  2 2s z2 22 2 exp!  1  1 !  2 2! 1 !x 1 2 2The general form of this expression is for constants C1 ; C2 ; C3 ; C4Eexp izRt =s  C1 exp s 2 C2   C3 exp s 2 C4 Using the relationshipEexp izRt =s  Es E exp izRt =s
and noting that s Cj   Es exp s 2 Cj j2;4 gives us the result. Q.E.D.It is not possible to advance this argument further without specifying thepdf of s 2t . For the case of a Bernoulli where s 2t   21 with probability p ands 2t   22 with probability 1 p, we see thats   Eexp s 2t   p exp  21   1
p exp  22 
The portfolio distribution of directional strategies 181and thus replacing t byz2 2 2! 1  12
! 2  22  2x 1 2 
will give us the mgf of a mixture of normals. It is apparent that the returnswill have a pdf which is the mixture of four normals with the followingprobability weights and variances.Weights1 1 arcsin F  p2 1 1 arcsin F  12 1 1arcsin F  p2 1 1arcsin F  12 
Variances 21 ! 2  21  1
! 2  22  2x 1 2 
p  22 ! 2  21  1
! 2  22  2x 1 2 
 21 ! 2  21  1
! 2  22
2x 1 2 
p  22 ! 2  21  1
! 2  22
2x 1 2 
6.5 CONCLUSIONSUnder the normal random walk assumption, both buy and hold and activestrategies exhibit zero returns. However, when more than one directionalstrategy is used, the distribution is no longer normal. When two forecasts areused to time the markets, the distribution is a mixture of two normal laws.The mixture coecient depends on the correlation coecient between the twoforecasts. A portfolio of directional strategies introduces a substantialamount of kurtosis which is a decreasing function of the correlationcoecient between forecasting strategies. The more negatively correlatedthe forecasts, the higher is the kurtosis. This is due to the fact that mostoften there will be no position taken in the market. One forecast will trigger along position whereas the other will trigger a short position. Consequently,the return distribution will have many zero values with sparse non-zeroobservations. This, in turn, explains the positive kurtosis. Results have beengeneralized for a subclass of the elliptical family of distributions. Theknowledge of theoretical active portfolio returns has many potential applications. First, this might help to build an ecient portfolio. Second, it mightallow the building of new tests of a random walk from the joint pro®tabilityof active strategies.Finally, our results could be extended to consider multiple assets and rulesalthough expressions for the probabilities as de®ned in section 6.3 mayrequire numerical solutions.
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Advanced Trading Rules
REFERENCESBrock, W., Lakonishok, J. and LeBaron, B. (1992) Simple technical rules and thestochastic properties of stock returns, Journal of Finance, 47, 1731±64.Brorsen, B.W. and Lukac, L.P. (1990) Optimal portfolios for commodity futuresfunds, Journal of Futures Markets, 10(3), 247±58.Lukac, L.P., Brorsen, B.W. and Irwin, S.H. (1988) Similarity of computer guidedtechnical trading systems, Journal of Futures Markets, 8(1), 1±13.Prado, R. (1992) Design, Testing and Optimization of Trading Systems, John Wiley& Sons.Surujaras, P. and Sweeney, R.J. (1992) Pro®t-making speculation in foreignexchange markets, The Political Economy of Global Interdependence, WestviewPress.
Chapter 7The pro®ts to technical analysis in foreignexchange markets have not disappearedJOHN OKUNEV AND DEREK WHITE7.1 INTRODUCTIONThe relative ecacy of foreign exchange trading rules has been examinedintensively during the previous two decades. Most would argue that in theabsence of a risk-based explanation the pro®tability for simple trading rulesshould have disappeared during this time if foreign exchange markets areecient. However, numerous papers have shown many trading strategies forindividual currencies would have generated signi®cant returns (Sweeney,1986; Schulmeister, 1988; Levich and Thomas, 1993; Taylor, 1994; Kho,1996; Neely, Weller, and Dittmar, 1997; LeBaron, 1999; Marsh, 2000). Whilethese studies, among many others, have clearly documented the pro®tabilityto technical trading rules in the foreign exchange market, we must be carefulto ensure that the results are not due to an ex post selection of the successfulmethods (Brock, Lakonishok, and LeBaron, 1992; Sullivan, Timmerman, andWhite, 1999). Some studies justify the analysis of only a few strategies byclaiming to examine the trading rules that `are most commonly used in themarket'. Unfortunately, this defence is invalid as the trading rules that areused in the market are those that have performed well ex post. While somewould counter that the exact speci®cation of the trading rule does not matter,many studies are further hamstrung in that they only consider the performance relative to one base currency of reference. This is perhaps a moreinsidious form of data-snooping and ex post selection bias. It is not at all clearthat a pro®table trading rule using the US dollar as a base currency wouldalso earn excess returns from a Japanese yen perspective. The stability oftrading rules needs to be examined not only over time, but also across asmany base currencies as possible. Anything less may leave the study subject toex post selection critiques.In this chapter, we wish to analyse the performance of several classes offoreign exchange trading rules. That is, instead of analysing individual
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Advanced Trading Rules
trading rules, we examine multiple speci®cations of a given trading rule andthen average the returns across those speci®cations to determine the returnfor that class of trading techniques. For example, one individual movingaverage speci®cation may involve a short-run moving-average length of fourmonths and a long-run moving-average of 36 months. The moving-averagerule class returns are calculated by averaging the returns of many individualshort-run, long-run moving-average combinations.Okunev and White (2002) examined the class of short-run, long-runmoving-average rules that identify a most attractive and least attractivecurrency and then initiate the appropriate long±short positions in thosecurrencies. Speci®cally, 354 individual moving-average rules were examinedacross eight base currencies of reference for the period January 1980 to June2000. Okunev and White (2002) found the long±short strategies acrosscurrencies generate returns of about 7 per cent per year on average andthat the results are robust to the exact speci®cation of the moving-averagerule and the base currency of reference. Moreover, neither a static nor a timevarying risk premium could explain the results.In this chapter, we wish to extend the analysis to include multiple classes oftrading rules. Broadly speaking, we will examine trading rules that maintaineither long or short positions in all individual non-domestic currencies withthose that allocate exposures to a subset of the non-domestic currencies (e.g. along exposure to one currency and a short exposure to another currency). Wewill identify two divisions of trading rules: those that trade in all individualnon-domestic currencies every month as Long±Short Within and those thattrade in only a subset of the non-domestic currencies as Long±Short Across.Long±Short Within trading rules will not consider other non-domesticcurrencies when setting a long or short position in an individual currency.Long±Short Across trading rules will evaluate whether to trade in a given nondomestic currency and the type of position (long or short) by comparingtechnical performance across other non-domestic currencies. Within eachdivision, we will examine the following classes of trading rules: momentum,buying/selling based upon relative interest rates, and multiple parameterizations for moving-average rules.In earlier work, Surajaras and Sweeney (1992) examine the Long±ShortAcross trading strategies using moving-average rules based upon daily foreignexchange movements. In addition, they also conduct portfolio type testssimilar to that used in this chapter. A number of dierences between theirmethodology and ours may be identi®ed. First, the parameters for technicaltrading rules were optimized for each currency in their paper. We take a moreconservative approach by averaging the performance of individual parameterizations rather than by attempting to ®nd the best speci®cation. Second,
The pro®ts to technical analysis in foreign exchange markets 185their tests focused on moving averages using daily returns and assumedtrading in spot rather than futures markets. Our approach is to use monthlyreturns and to assume futures trading (realised returns are interest adjusted).Third, their tests used only the US dollar as the base currency of reference.We conduct our tests across eight base currencies. This is especially importantwhen attempting to benchmark performance. In addition, their tests examined a much earlier time period (1978±1986). Finally, they ®nd performanceto be particularly weak during the mid 1980s. We ®nd our results to beparticularly strong during this time period. If technical trading rules can bearbitraged away, they should have long since disappeared.In general, we ®nd that Long±Short Across foreign exchange trading rulesoutperform the Long±Short Within division. 1 The moving-average class inboth divisions tends to outperform the momentum and interest rate basedstrategies, however, signi®cant bene®ts may accrue through the combinationof these strategies. Moreover, on a risk-adjusted basis, the moving-averagestrategy in the Long±Short Within class compares very favourably with themoving-average rule in the Long±Short Across class. All tests are conductedusing eight base currencies of reference: the Australian dollar, the Britishpound, the Canadian dollar, the French franc, the German mark, theJapanese yen, the Swiss franc, and the United States dollar. We show ourresults are robust to the base currency of reference and we analyse the resultsfor the period January 1980 to June 2000.The results of our chapter are especially relevant for a corporation withmultiple foreign currency exposures or an international funds institution.These entities have a natural foreign exchange exposure that they can hedgecompletely, leave unhedged, or tilt towards currencies they expect toappreciate while their exposure persists. Hedging and tilting currencyexposures can be easily accomplished through the use of futures positions.For an international funds institution an extra 1 per cent gained throughastute foreign currency management may mean the dierence betweenoperating at the median or the upper quartile of performance. For acorporation with multiple currency exposures, shrewd foreign currency riskmanagement may produce a signi®cant positive revenue impact.The methods we advocate are very simple, involving minimal trading andtransactions cost. We examine monthly returns and our holding period is atleast one month. Perhaps a shorter time horizon would increase the pro®tability of the methods we analyse ± but that is outside the scope of thischapter. Prior studies have advocated simple strategies over more complexones and have shown them to perform just as well (Taylor, 1992; Ntungo andBoyd, 1998). While the chapter does focus on classes of foreign exchangetrading strategies, the positions need not be executed this way in practice.
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Advanced Trading Rules
Many of the individual foreign exchange trading strategies within a givenclass will outperform other speci®cations within that