jF  tj0
j0 ij1
wherei j  Corr Xt j ; Xt i tX
dj j1
j0
Corr Xt ; Ft 1   vuXt Xt 1Xu t 2tdi dj  i jdj  2j0
j0 ij1
Figures 4.1 to 4.7 illustrate expected rule returns for varied underlyingprocesses and trading forecasts, namely the simple moving averages, weightedmoving averages and momentum rules in their autoregressive versions givenby Table 4.1.
Expected returns of directional forecasters 149REFERENCESAcar, E. (1993) Economic Evaluation of Financial Forecasting, PhD Thesis, CityUniversity, London.Bird, P.J.W.N. (1985) The weak form eciency of the London Metal Exchange,Applied Economics, 17, 571±87.Black, F. and Perold, A.F. (1992) Theory of constant proportion portfolioinsurance, Journal of Economic Dynamics and Control, 16, 403±26.Bookstaber, R. and Clarke, R. (1987) Options can alter portfolio returns, Journalof Portfolio Management, Spring, 63±70.Brock, W., Lakonishok, J. and LeBaron, B. (1992) Simple technical rules and thestochastic properties of stock returns, Journal of Finance, 47, 1731±64.Cartinhour, J. (1990) One-dimensional marginal density functions of a truncatedmultivariate normal density function, Commun. Statist.-Theory Meth, 19(1),197±203.Cox, J.C. and Rubinstein, M. (1985) Options Markets, Prentice Hall, EnglewoodClis, New Jersey.Curcio, R. and Goodhart, C.A.E. (1992) When support/resistance levels arebroken, can pro®ts be made? Evidence from the foreign exchange market,LSE Financial markets group discussion paper series, L142, July.Davies, R. and Harte, D. (1987) Tests for Hurst eect, Biometrika, 74, 95±101.Dunis, C. (1989) Computerised technical systems and exchange rate movements.In Exchange Rates Forecasting (eds C. Dunis and M. Feeny), WoodheadFaulkner, pp. 165±205.Engle, R.F. (1982) Autoregressive conditional heteroskedasticity with estimates ofthe variance, Econometrica, 50, September, 987±1007.Hosking, J.R.M. (1981) Fractional dierencing, Biometrika, 68, 165±76.Johnson, N.L. and Kotz, S. (1972) Distributions in Statistics: Continuous Multivariate Distributions, New York: Wiley.Kaufman, P.J. (1987) The New Commodity Trading Systems and Methods, JohnWiley & Sons.Keenan, D.M. (1982) A time-series analysis of binary data, Journal of theAmerican Statistical Association, 77, 816±21.LeBaron, B. (1991) Technical Trading Rules and Regime Shifts in ForeignExchange, University of Wisconsin, Social Systems Research Institute, Working Paper 9118.LeBaron, B. (1992) Do Moving Average Trading Rule Results Imply Nonlinearities in Foreign Exchange Markets?, University of Wisconsin, Social SystemsResearch Institute, Working Paper 9222.Levich, R.M. and Thomas, L.R. (1993) The signi®cance of technical trading-rulepro®ts in the foreign exchange market: a bootstrap approach, Journal ofInternational Money and Finance, 12, 451±74.Lo, A.W. and MacKinlay, A.C. (1990) An econometric analysis of nonsynchronous trading, Journal of Econometrics, 45, 181±211.Lukac, L.P., Brorsen, B.W. and Irwin, S.H. (1988) A test of futures marketdisequilibrium using twelve dierent technical trading systems, Applied Economics, 20, 623±39.
150
Advanced Trading Rules
Mandelbrot, B. (1963) The variation of certain speculative prices, Journal ofBusiness, 36, 394±419, reprinted in The Random Character of Stock MarketPrices (ed. P. Cootner), MIT Press, Cambridge, MA, pp. 297±337.Mandelbrot, B. (1966) Forecasts of future prices, unbiased markets, and `Martingale' Models, Journal of Business, 39, 242±55.Neftci, S.N. (1991) Naive trading rules in ®nancial markets and Wiener-Kolmogorov prediction theory: a study of technical analysis, Journal of Business, 64,549±71.Neftci, S.N. and Policano, A.J. (1984) Can chartists outperform the market?Market eciency tests for technical analysis, Journal of Futures Markets, 4(4),465±78.Peters, E.E. (1991) Chaos and Order in the Capital Markets, John Wiley & Sons.Praetz, P.D. (1976) Rates of return on ®lter tests, Journal of Finance, 31, 71±5.Schervish, M.J. (1984) Algorithm AS 195: Multivariate normal probabilities witherror bound, Applied Statistics, 33, 81±94.Silber, W. (1994) Technical trading: when it works and when it doesn't, TheJournal of Derivatives, Spring, 39-44.Surujaras, P. and Sweeney R.J. (1992) Pro®t-making Speculation in ForeignExchange Markets. The Political Economy of Global Interdependence, WestviewPress.Sweeney, R.J. (1986) Beating the foreign exchange market, Journal of Finance, 41,163±82.Taylor, S.J. (1980) Conjectured models for trends in ®nancial prices, tests andforecasts, Journal of the Royal Statistical Society, 143A, 338±62.Taylor, S.J. (1986) Modelling Financial Time Series, Chichester, England, JohnWiley & Sons.Taylor, S.J. (1990a) Reward available to currency futures speculators: Compensation for risk or evidence of inecient pricing? Economic Record (supplement),68, 105±116.Taylor, S.J. (1990b) Pro®table currency futures trading: a comparison of technicaland time-series trading rules. In The Currency Hedging Debate (ed. L.R.Thomas), IFR Publishing Ltd., London, pp. 203±39.Taylor, S.J. (1992) Eciency of the Yen futures market at the Chicago MercantileExchange. In Rational Expectations and Ecient in Future Markets (ed. B.A.Goss), pp. 109±28.Taylor, S.J. (1994) Trading futures using the channel rule: a study of thepredictive power of technical analysis with currency examples, Journal ofFutures Markets, 14(2), 215±35.Trippi, R.R. and Harri, R.B. (1991) Dynamic asset allocation rules: survey andsynthesis, Journal of Portfolio Management, Summer, 19±26.Walter, C. (1991) L'utilisation des lois levy-stables en ®nance: une solutionpossible au probleÁme poseÂ par les discontinuiteÂs des trajectoires boursieÁres(dernieÁre partie), Bulletin de l'Institut des Actuaires FrancËais, January, 350, 3±23.Zhu, Y. and Kavee, R.C. (1988) Performance of portfolio insurance strategies,Journal of Portfolio Management, Spring, 48±54.
Expected returns of directional forecasters 151NOTES1. The distribution of realized rule returns is highly skewed (Acar, 1993;Chapter 3, pp. 83±5).2. An in-depth discussion on mis-speci®ed forecasts can be found in Acar(1993; Chapter 4).3. m 1 in Kaufmann (1987).4. Return and price signals are strictly identical for the momentum rule. Thereason is that:Pt < Pt m1
,
Ln Pt =Pt m1  < 0
,
m 2X
Xt j < 0
j0
5. An example of such a strategy is:
Bt  1Bt  1
,,
Xt 1 < 0Xt 1 > 0

6. Distinguishing a fractional Gaussian process from an AR(1) model isrecognized in the literature as a dicult task (Davies and Harte, 1987)
ACKNOWLEDGMENTSI wish to acknowledge Banque Nationale de Paris plc who provided theopportunity for me to undertake this research. A special debt ofgratitude must be paid to my ex-colleague at the `technical desk', PierreLequeux. I am indebted to Dr Elias Dinenis who gave me encouragement and advice throughout the course of this research.
Chapter 5Some exact results for moving-averagetrading rules with applications toUK indicesGEORGE W. KUO
5.1 INTRODUCTIONTechnical analysis is commonly used by traders in ®nancial markets to detectmain turning points of the market at an early stage. The assumptionunderlying technical analysis is that investors respond to similar marketevents in very consistent ways. Such consistency implies future re-occurrencesof historical price patterns, and, therefore, makes developing trading rules toidentify certain price patterns a rewarding business. Numerous trading ruleshave been designed to provide investors and traders with trading suggestionsbased on the buying and selling signals they generate. However, the signalsfrom most trading rules are always not decisive and have to be interpreted byusers heuristically and subjectively. This makes systematic studies of alltrading rules very dicult. Thus, the studies are usually con®ned to a subsetof trading rules which can be mathematically well de®ned.According to Neftci (1991), for a trading rule to be mathematically wellde®ned it has to be able to issue signals that are Markov times. 1 Note thatMarkov times cannot depend on future information. Neftci (1991) shows thatthe moving-average rule can generate Markov times such that it is mathematically well de®ned. Because of this characteristic, the moving-average ruleis of most interest to academics and practitioners because it generates tradingsignals mechanically without depending on traders' subjective judgementswhich always implicitly incorporate future information.Recently, the feasibility of the moving-average rule in various ®nancialmarkets has been studied. The common null hypothesis of these studies is thatthe moving-average rule can produce average returns that are statisticallysigni®cant. Applying bootstrap methodology, Levich and Thomas (1993) ®ndthat a subclass of the moving-average rule does produce statistically sig-
Some exact results for moving-average trading rules
153
ni®cant average returns for futures contracts on the British pound, theCanadian dollar, the German mark, the Japanese yen and the Swiss franc.LeBaron (1991) also ®nds signi®cant weekly average returns from certainmoving-average rules in the quoted exchange rates for the British pound, theGerman mark and the Japanese yen. Furthermore, in order to identify thepossible generating processes of these major foreign exchange rates, hesimulates the returns from these moving-average rules based on severalmodels with regime shifts and persistent trends 2 and compares them withthe realized returns. He ®nds that these models cannot capture certain aspectsof the realized returns. In addition, Brock, Lakonishok and LeBaron (1992)report that the moving-average rule produces statistically signi®cant returnsin the US stock market. Using bootstrap technique, they simulate returnsfrom moving-average rules, etc., based on the random walk model, the AR(1)model, the GARCH-M model, and the exponential GARCH model, andcompare these simulated returns with those realized returns. As LeBaron(1991), they conclude that these four models still cannot fully capture thecharacteristics of the realized returns.Although LeBaron (1991) and Brock, Lakonishok and LeBaron (1992)have examined the probability distribution of realized returns from themoving-average rule empirically, it has not been investigated theoreticallyyet. Adopting Acar's (1993) general framework for studying trading rules,Acar and Satchell (1997) derive the probability distribution of realized returnsfrom a simple moving-average rule given that asset returns are Markovian. Inthis chapter, we ®rst derive some exact results for a special case of Acar andSatchell's Markovian framework and then extend these results to anothermoving-average rule in a dierent context. Finally, we apply the results forthe ®rst rule to UK FT100 index and futures.As Acar (1993) suggests, in order to investigate the probability distributionof realized returns from a trading rule, we have to explicitly specify the natureof the trading rule, the underlying stochastic process for asset returns, and theparticular return concept involved. The trading rule we study here is themoving-average rule. The return concept involved is the realized return froma sequential trading strategy based on the moving-average rule. The mostimportant part in this framework is the assumption about the stochasticprocess that generates asset returns. This process should be able to accountfor the empirical characteristics already found in ®nancial time series:asymmetry and fat tails.In the literature, many processes have been proposed to explain these twoempirical properties of ®nancial time series. Tucker (1992) categorizes theseprocesses into two types:(1) the time-independent process
154
Advanced Trading Rules
(2) the time-dependent process.Many time-independent processes have been applied to account for theasymmetry and fat tails, e.g., the stable Paretian process (Mandelbrot,1963; Fama, 1965), the symmetric Student process (Blattberg and Gonedes,1974), the mixed diusion-jump process (Merton, 1976), the lognormalnormal subordinated process (Clark, 1973), the mixed normal process(Kon, 1984) and the asymmetric stable Paretian process (Tucker, 1992).There also are some examples of the same application of the time-dependentprocesses. Akgiray (1989) uses ARCH-type processes to ®t the returns in theUS stock market, based on the fact that the independence and linearity ofstock returns are not required to achieve market eciency. In addition,recognizing the important implications of the asymmetry of stock returns insearching for an appropriate measure of risk in ®nancial economics, Knight,Satchell and Tran (1995) propose a mixed scale gamma process to capturesuch asymmetry. They assume that whether the stock return will stay aboveor below its long-term mean depends on the realization of a positive ornegative shock in the present period. Both shocks come from scale gammaprocesses with dierent parameters and their conditional switching probabilities are controlled by a Markov process. Strictly speaking, Knight, Satchelland Tran's process is a time-dependent one. The dependence comes from theMarkov process, which describes the evolution of the conditional probabilities for the stock return to stay above and below its long-term mean.Although Knight, Satchell and Tran assume in their empirical study thatthe asset return has a mixed gamma distribution, they actually propose amore general process for the asset return. Speci®cally, they assume the assetreturn to have a mixed distribution of two non-negative random variables,whose distributions satisfy certain moment conditions. This general Knight±Satchell±Tran process is the one we use for the generating process of assetreturns.The ®rst moving-average rule we analyse is a naive geometric one whichuses only the information about whether the asset return in the present periodis positive or not to issue trading signals. We show that for investorsfollowing a sequential trading strategy based on this rule to make pro®ts,either one of the following two conditions is sucient provided that thestochastic process generating asset returns is the general Knight±Satchell±Tran process. The ®rst condition is that the expected value of the positiveshocks to the asset return is much greater than the expected value of thenegative shocks. The second condition is that the positive shocks have to bevery persistent themselves and much more persistent than the negativeshocks. If we further assume that the asset return is independently identicallydistributed, we prove that either one of these conditions is still sucient for
Some exact results for moving-average trading rules
155
investors to make pro®ts. However, if conditioning the expected realizedreturn on the event that investors already hold the target asset when theydecide for the ®rst time to adopt the sequential trading strategy, we show thatthe same ®rst condition is still sucient whereas the second one changes to bethat the unexpected positive shocks must be highly persistent no matter howpersistent the negative ones are.The second case we study is a geometric moving-average rule whichproduces trading signals based on the whole past history of asset prices.We show that the corresponding sequential trading strategy always deliversnegative realized returns to investors given that the general Knight±Satchell±Tran process generates asset prices. What is more interesting is that thenegative average realized return can be minimized if the asset price actuallyfollows a random walk.Apart from deriving theoretical results, we also provide an example byusing the empirically estimated numbers from Knight, Satchell and Tran(1995) to illustrate the implications of our results for the ®rst moving-averagerule. Interestingly, the example shows that on average investors can makepro®ts by following the ®rst moving-average rule in both UK FT100 indexand FT100 futures markets if the asset return is not independently identicallydistributed. These pro®ts are around 0.3 per cent and 6.0 per cent per monthon average in the stock and futures markets respectively.The rest of the chapter is organized as follows. Section 5.2 describes themoving-average rule. Section 5.3 presents the general Knight±Satchell±Tranprocess and the trading strategy based on a naive geometric moving-averagerule and also derives the probability distribution as well as the expected valueof the realized return. Section 5.4 extends the results in section 5.3 to ageometric moving-average rule utilizing the past history of asset prices toissue trading signals. Section 5.5 applies the theoretical results in section 5.3to UK stock and futures markets. Finally, section 5.6 concludes the results.5.2 THE MOVING-AVERAGE TRADING RULEThe moving-average rule is one of the trend-determining techniques intechnical analysis. The feasibility of these techniques hinges on the assumption that crowd psychology makes the asset price move in discernible trends.However, the trends can be very volatile, almost haphazard at times. Amoving average serves to smooth down the ¯uctuations of asset prices so thatdistortions can be reduced to a minimum.In practice, a moving-average rule usually consists of two moving averageswith dierent lengths. It delivers a buying or selling signal depending onwhether a valid penetration exists. A valid penetration happens when the
156
Advanced Trading Rules
moving average with a shorter length penetrates the one with a longer lengtheither from the below or from the above respectively. To be precise, we denotea moving-average rule, which consists of two moving averages with n and mlags respectively n > m, as moving-average n; m. Mathematically, amoving-average n; m rule can be de®ned as:Bt 
m 1XPt j1=mj0
n 1XPt j1=n
5:1
j0
where fPt g is a sequence of asset prices. Note that Bt  0 implies a validpenetration from the below of the m-lag moving average into its n-lagcounterpart and that Bt < 0 implies the opposite. In other words, amoving-average n; m rule will generate a buying signal for the next periodwhen Bt  0 and a selling signal for the next period when Bt < 0.The validity of a moving-average n; m rule depends mainly on its timelags, m and n. These time lags should be determined according to the length ofthe market cycle to be detected. Since market cycles have been found to benonperiodic, there does not exist a universal principle for choosing the timelags. This makes deciding optimal time lags for a moving-average n; m rulemuch more a science of art. Thus, many experiments are needed to search foroptimal moving-average rules in dierent ®nancial markets.Academics and ®nancial practitioners have devoted much eort in searching for optimal moving-average n; m rules. Among the optimal rules foundso far, the moving-average n; 1-type rules, such as MA 150; 1 andMA 200; 1, are the simplest and most popular. Although these rules havepassed standard empirical tests and passed them well, their feasibility has notyet been theoretically justi®ed.In general, there are two classes of moving-average rules: the arithmeticmoving-average rule and the geometric moving-average rule. The movingaverage n; m rule that can be described by Equation (5.1) belongs to theformer. Speci®cally, an arithmetic moving-average n; 1 rule, denoted asAMA n; 1, can be de®ned as:Bt  0; iF Pt 
n 1XPt j ;1=n
Bt < 0; otherwise
5:2
j0
whereas a geometric moving-average n; 1 rule, denoted by GMA n; 1 canbe de®ned as:Bt  0; iF Pt 
Yn 1j0
Pt j
1=n
;
Bt < 0; otherwise
5:3
Some exact results for moving-average trading rules
157
Taking the natural logarithm of the inequality in Equation (5.3) andrearranging, we have:Rt 
n 2Xnj1
n
j  1Rt j1
5:4
where fRt g is a sequence of asset returns. Equation (5.4) shows that as long asthe asset return at present time t is greater than or equal to the negative of theweighted average of the asset returns in the past n 2 periods, a geometricmoving-average n; 1 rule will suggest investors to hold the asset for the nextperiod. Otherwise, it will issue a selling signal suggesting investors to closetheir positions at the beginning of the next period. Under a speci®cassumption, 3 we can also simplify the arithmetic moving-average n; 1 rulein Equation (5.2) as the weighted average of the past asset returns expressedas Equation (5.4). The primary results derived in the next two sections aremainly based upon the geometric moving-average n; 1 rules.5.3 THE STOCHASTIC PROCESS FOR ASSET RETURNSKnight, Satchell and Tran (1995) assume daily asset returns to follow thestochastic process:Rt  l  Zt "t
1
Zt t
5:5
where l is the long-term mean of daily asset returns; "t and t are two nonnegative independently identically distributed random variables, e.g., twodierent scale gamma processes with probability density functions:81< 1 1 x 1exp 1 x if x  05:6pd f" x G 1:0otherwise812< 2 x 2exp 2 x if x  0pd f x 5:7G 2:0otherwisewhere G  is a gamma function; the random variable Zt is a stationary twostate Markov process with the transition probabilities:PrZt  1 j Zt 1  1  pPrZt  0 j Zt 1  1  1
p
PrZt  1 j Zt 1  0  1
q
PrZt  0 j Zt 1  0  qwhere 0  p; q  1.
5:8
158
Advanced Trading Rules
As Zt is stationary, we can compute the stationary probability of state one,P, by:P
12
qp
q
Therefore, in the stationary state, the asset return will stay above its ownlong-term mean with probability P and below its long-term mean withprobability 1 P, namely:Rt  l  "t  l with probability PRt  l
with probability 1
t <  l
P
We also assume that "t , t and Zt are independent to one another for 8t. Notethat although, in their empirical study, Knight, Satchell and Tran (1995) usescale gamma distributions for the unexpected shocks ± "t ; t ± they actuallypropose a more general process for the asset return. We call it the generalKnight±Satchell±Tran process. This process, as described by Equation (5.5),just requires the unexpected shocks be two non-negative independentlyidentically distributed random variables with characteristic functions F" sand F s 4 and to satisfy certain moment conditions. 5 In addition, wepresume the long-term mean of asset return to be zero. 6After assigning the general Knight±Satchell±Tran process as the generatingstochastic process of the asset return, we still need to clarify the concept of therealized return from the geometric moving-average n; 1 rule whose probabilistic properties we want to examine. We assume that an investor follows asequential trading strategy according to the trading signals generated by thegeometric moving-average n; 1 rule. Suppose that an investor decides tohold an asset A for whatever reason at the very beginning of period oneaccording to a buying signal he receives at the end of period zero, that is,B0  0. Since then, he has to decide sequentially at the beginning of everysubsequent period whether he is going to keep holding asset A or sell it basedon the signal generated by the geometric moving-average n; 1 rule at the endof the previous period. Note that the only information he has when he makesthe decision is either a buying or a selling signal he receives at the end of theprevious period. He has no prior information other than the knowledge of thereturn generating process about what will happen during the period after hetakes his action. Thus, according to the nature of this sequential tradingstrategy, if an investor decides to keep holding asset A until the end of periodT, it means that he or she observes buying signals from Equation (5.3) in the®rst T 1 periods, i.e.Bt  0; 1  t  T
1
and receives a selling signal at the end of period T 7 , namely, BT < 0. Note
Some exact results for moving-average trading rules
159
that T here is actually a random variable. Now we can de®ne the realizedreturn for T periods as:RRT 
TXt1
Rt  IfBt 1 0g
5:9
where IfBt 0g is an indicator function whose value equals one when Bt  0and zero when otherwise. We next present a proposition, Proposition 1,which gives a particular form for the characteristic function of realizedreturns. Our proof mainly follows Acar and Satchell (1997). We specializeour results to the case where the asset return is generated by Equation (5.5).
Proposition 1Assume that the general Knight±Satchell±Tran process is the underlyingstochastic process of asset returns. The probability distribution of the realizedreturn in Equation (5.9) resulting from the geometric moving-average 2; 1rule can be described by the following characteristic function.FRRT s  1
PF
s  P 1
p
F s1 pF" s
5:10
ProofFrom Equation (5.4) we see that the geometric moving-average 2; 1 rule canbe expressed as:Bt  0; iF Rt  0;
Bt < 0; otherwise
Moreover, we can also see from Equation (5.5) that:Rt  0; iF Zt  1;
Rt < 0; iF Zt  0
Therefore, the information contained in the sequence of trading signals,fB0 ; . . . ; Bt g, is identical to that contained in the sequence of Markovianrandom variables, fZ0 ; . . . ; Zt g.To derive the distribution of the realized return, we have to compute theprobability for an investor to hold the target asset until the end of period Tconditional on the event that a buying signal is generated by the geometricmoving-average 2; 1 rule at the end of period zero.
160
Advanced Trading RulesPr N  T  Pr BT < 0; BT 1  0; . . . ; B1  0; B0  0 Pr ZT  0; ZT 1  1; . . . ; Z1  1; Z0  1 Pr ZT  0; ZT 1  1; . . . ; Z1  1 j Z0  1 Pr Z0  1(Pp T 1 1 p if T  11 Pif T  0
where P is the stationary distribution of state one of the Markov process inEquation (5.8).Now we can derive the conditional characteristic function of the realizedreturn for T periods, RRT , from the sequential trading strategy based on thegeometric moving-average 2; 1 rule as follows.PNFRRT jNT s  Ee fi t1 Rt IfBt 10 g sg j N  TPT Ee fi t1 Rt IfB