forecasting and trading strategies based on a price trend ... · jvtirncd of furecu.sting j....

Jvtirncd of Furecu.sting

J. Forecast. 19, 485 498 (2000)

Forecasting and Trading StrategiesBased on a Price Trend Model

JOSEPHINE W. C. KWAN,^ K. LAfVl,^ MIKE K. P.and PHILIP L. H. YU^*^ The tJniversity of Hong Kong, Hong Kong

Kong Baptist University. Hong KongHong Kong University of Science and Technology. Hong Kong

ABSTRACT

In this paper, we consider the price trend model in which it is assumed thatthe time series of a security"s prices contain a stochastic trend componentwhich remains constant on each of a sequence of time intervals, with eachinterval having random duration. A quasi-maximutn likelihood method isused to estimate the model parameters. Optimal one-step-ahead forecastsof returns are derived. The trading rule based on these forecasts is con-structed and is found to bear similarity to a popular trading rule based onmoving averages. When applying the methods to forecast the returns of theHatig Seng Index Futures in Hong Kong, we find that the performance ofthe newly developed trading rule is satisfactory. Copyright '' 2000 JohnWiley & Sons. Ltd.

KEY WORDS forecasting; moving averages: price trend model; trading rules

INTRODUCTION

Of great interest to forecasters in speculative markets is the prediction of the one-step-aheadreturn in a financial time series. These forecasts can be made use of in order to construct technicaltrading rules which can profit from the price movement in a financial asset. However, the earlierefforts do not seem to be fruitful because of the highly random character inherent in speculativemarket prices. In 1910. Bachelier proposed the random walk model for stocks' price move-ments, and Kendall (1953). Alexander (1964) and Cootner (1964) gave empirical support tothis random walk hypothesis. In 1970. Fama proposed the efficient market hypothesis (EMH). Inits weak form, the EMH states that all information has been reflected in (he current price andhence historical prices are not useful in predicting future prices. Many empirical studies in earlierliterature support the EMH (see. for example, Jensen. 1978). Under an efficient market.no trading strategy wbicb is Markov in nature (see Neftei. 1991), can beat the naive 'buy-and-hold" strategy.

•Correspondence lo: Philip L. H. Yu, Deparlment of Statistics atid Aciuariai Science. The University of Hong Kotig,Pokfiilam Road. Hong Kong.

Received November 1996

Copyright r. 2000 John Wiley & Sons, Lid. Accepted August 1999

486 ./, IV. C Kmtn et al.

Although earlier hterature concltidcd that technical trading may not be able to make extraprofit over the naive buy-and-hold strategy, recent studies reveal that there are examples in whichfuture returns are predictable from past returns. For example. Lo and MacKinlay (1988) foundpositive autocorrelations of weekly returns on portfolios of NYSE stocks. Fama and French(1988) found negative serial correlation in returns of individual stocks and various portfolios ofsmall and large firms. Gencay (1996) found strong evidence of non-linear predictability in dailyreturns of the Dow Jones Industrial Average Index by using the past buy-and-sell signalsgenerated from the moving average rules. Moreover, more and more researchers find evidence ofmarket inefficiency. For example. Farrell and Olszewski (1993) discovered that the S&P 500futures market exhibits slight inefficiencies and there exist, in principle, better trading strategiesthan the benchmark buy-and-hold strategy. Brock, Lakonishok and LcBaron (1992) studiedsome simple trading rules including moving averages and trading range breaks rules, and foundthat they have predictive power for the Dow .lones Index. Taylor (1994) found that the channeltrading rule can make net profits at currency futures markets. Also, see LeRoy (1990) andFortune (1991) for the literature review of inefficiencies in stock prices.

These evidences of predictability indicate that price movements in a financial market maydepart from the random walk model. Alternatives lo the random walk model like the fads model(Summers. 1986). the mean reversion model (Poterba and Summers. 1988) and the price trendmodel (Taylor. 1980) have been proposed.

This paper is devoted to the estimation and forecasting under the Taylor's price trend model.Unlike the fads model and the mean reversion model which focus on the mean revertingbebaviourofsecurity prices measured over a long horizon. Taylor's price trend model focuses onthe short-temi pattern of the price trend and Taylor (1980) demonstrated that the daily sugarprices can be modelled by fitting a basic price trend model. Taylor (1992) gave more empiricalevidence that the price trend model is important and useful in analysing prices in the futuresmarkets and foreign exchange markets.

In view of the successful application of the price trend model to real financial data, moretheoretical attention should be given to the modelling and forecasting of the trend component inthe price trend model, in the literature, the Generalized Methods of Moment (GMM) methodbased on the autocorrelation function was used to estimate the model parameters. Forecasts offuture returns were generated as if the trend model is an ordinary ARM A( 1.1) model (see Taylor.1986. 1992). However, optimal estimation and forecasting methods bave not been developed sofar. The purpose of this paper is to provide a rigorous statistical analysis of a basic price trendmodel.

The next section introduces the price trend model and describes its properties. The thirdsection describes the estimation o{ model parameters using a Quasi-Maximum Likelihood(QML) method. The QML method has the advantage that it is optimal if the distribution of theerror terms are multivariate normally distributed. The QML method is a standard methodologyin dealing with the state space model and is found to be convenient especially for forecastingpurposes. In the fourth section, an optimal one-step-ahead forecast under the price trendmodel and a new trading strategy based on these optimal forecasts are derived. The new tradingrule is found to bear similarity to a popular trading rule based on moving averages. Themethodology is then applied to forecast the daily returns of the Hong Kong Hang Seng IndexFutures and the results are reported in the fifth seetion in which the performance of the newlydeveloped trading rule is compared with the benchmark buy-and-hold trading rule. The finalsection provides conclusions.

Copyright iC 2000 John Wiley & Sons. Ltd. J. Foreca.Ki. 19. 485-498 (2000)

Forecasting and Trading Strategies 487

A PRICE TREND MODEL

Taylor (1980) proposed various versions of a price trend model. In general, a price trend modelassumes that the Return K,' at trading day / is the sum of two unobservable components: thetrend T^ and the noise i,. i.e.

R^^T^-^-c, t=\.2 n (1)

According to Taylor (1980). tbe trend T, can be written as

7",- ' / , Z, = 0r , when Z,= \ ^ (2)

' " ' t = 2 n/' + '/, when Z, = 0

where 7, is a Bernoulli process with P(Z, = I) = /> and PCZ, - 0) = 1 - /;, n is total number oftrading days, i;, and n, are white noises wbicb are stochastically independent. Under this model.[i + t^ can be interpreted as the new trend value at day t. In this paper, we will discuss theestimation and forecasting methods for this price trend model.

PropertiesIn the following, we present some properties of the price trend model as given in equations (I)and (2). First, it is easily shown that tbe trend T, has mean /i which is then the mean of the returnseries. Second, let Var(//,) = a;, and Var():,) = o;,. Then the autocorrelation function (ACF) p , atlag k is given by

Note that all the autocorrelations are positive and dependent on A and p only. The parameter Acan be interpreted as a measure of the information not reflected instantaneously by the priceswhile tbe parameter p as a measure of the speed at which information is reflected by the prices.Therefore, the quantity 1/(1 - p) measures the expected duration that a trend will remainunchanged.

Comparison with the ARMA(1,1) modelAs mentioned by Taylor (1980), the ACF of tbe price trend model as given in equation (3) has thesame form as that in an ARMA( 1.1) model. Hence, it makes sense to generate forecasts of thefuture returns under the price trend model by using the forecasts under the correspondingARMA(1,1) model. However, since the price trend model is not linear in nature, such forecastsmay not be optimal. In tbe following, we discuss the similarities and differences between a pricetrend model and an ARMA(I,1) model.

' The returti R, is defined ;is \og{P,) - log(P,_|). where P, is the closing price at day /.

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488 J. W. C Kwan et al.

By matching tbe first- and second-order moments under these two models, it is easily shownthat the price trend model has the same ACFs as the following ARMA(1,1) model:

where (I is the root in (0,1) of the equation

and {a,\ is a Gaussian u'hite noise with variance paljO. Therefore, there exists a one-to-onecorrespondence between the class of price trend models and a subclass of ARMA( I.I) models asfar as tbe ACFs are concerned.

However, this does not mean that tbe two models are equivalent. In fact, it can be shown thatthe price trend model and the corresponding ARMA(I,I) model have the same moments up tothe third order only. Their fourth- or higher-order moments are different in general. For example,assuming // ^ 0, we have.

]_,) = }[P4 - Op^alu\ + (J(l - 20p + (f)^]] (5)

for the ARMA(l.l) model but

for the corresponding basic trend model, where (T^ ^ (1 - (^)"/(l - pfal.Asan illustration, forthe typical parameter values A ^ 0.03 and p = 0.95 stated in Taylor (1986, p. 142), we calculatethe fourth moments implied by tbe price trend and the corresponding ARMA(I,1) models.According to the quadratic equation in (4), the moving average parameter 0 is equal to 0.9267.Substituting 0 in equations (5) and (6), we get the values ofE{R^R^_,) for both the priee trend andARMA(1,1) models as 7.6734(7* and ^.OIIICT^ respectively. As we can see, the two fourthmoments are quite similar and their magnitude is comparable to the squared variance(Tfl = 4.63 lOrr,,. For more detailed description about their similarities and differences, see Kwan(1994). Becau.se of the difference between a price trend model and an ARMA(I,1) model, a newstatistical method which is tailor-made for the price trend model is required for forecastingpurposes.

FSTIMATION

As mentioned in the previous section, there are three parameters to be estimated. They arethe probability/7 of maintaining the same trend, the signal variance afj and the noise variance

Copyright f 2000 John Wiley & Sons, Ltd. j . Foreca.st. 19, 485-498 (2000)


(JI. Taylor (1980) used tbe GMM estimation in which parameter estimates are obtainedby minimising

where k^ and kj are pre-specified constants and r^ is the /th-order sample autocorrelation. In thispaper, we propose a QML procedure for the parameter estimation In the price trend model.

Let 7^ , ^ (/?, R,)' and Z^, - (Z^ Z,)' for \ ^s ^ t ^ n. The likelihood function orthe joint density of /i/s can be written as a mixture form:

possible 2 , „

Since there are n — 1 elements in the vector Z^,, and each element has two possible values, thesummation in equation (7) involves 2""' terms. Hence, it is not computationally feasible toevaluate /(7î „) through (7) when n is large. This explains why the maximum likelihood approachwas not used for parameter estimation in the literature.

In what follows, we provide an alternative way of computing/^ÎTC,,,), the likelihood of /?/s,given that c, and ?/, are Gaussian, The subscript A' is used to emphasize that the joint density/• ,(7 i „) is calculated under the Gaussian assumption of R^. Define g^, as the eonditional normal

^,î_, - 1), i.e.

where 1 is a column vector with all the elements equal to 1 and L is a covariance matrix ofdimension t - s + \ with variance fr," + o;^ and constant correlation A. Also, let /, = max{.v:.9—1 / and Z^ ^ Oj, the latest time from I to / in which there is a change in the trendcomponent. Using the law of total probability, we have

Copyright ij 2000 John Wiley & Sons, Ltd. J. Forecast. 19, 485-49S (2000)


The second term in equation (8) arises because

/ = ,v for .V = 2, . . .

(A, /?^_|) and ( .. ./?,) are independent

and there is no change in the trend after ,v — 1

Given the initial value of/y(^ | ,) ^ ^| | . f^{R^ ,) for / = 2 n can be computed recursivelyvia equation (8). By maximising the quasi-likelihood function/\,(/?i ,) with respect to theunknown parameters. QML estimates are obtained. As compared with the GMM approach, theQML estimation procedure has an advantage of producing the best linear forecasts of futurereturns as by-products. This property is referred to as tbe minimum mean square linear estimatiorin Harvey (1993). These forecasts are crueial elements for the construction of a trading rule whichwill be discussed fully in the next section.

FORECASTING AND TRADING STRATEGY

Forecasting of future returnsIn this section, we discuss the optimal prediction of future returns under tbe price trend model.The optimal one-step-ahead forecast of /?,,_j_| under square loss is simply the conditionalexpectation E(R,,^^ l^i.,,) Indeed, the forecast can be expressed as a smoothed estimate of thetrend at time // as follows

(9)

Therefore, the optimal forecast is the product of/?, the probability of no change in trend, and thesmoothed estimate of T ,, i.e. E(7',J7S| J . In order to obtain the one-step-ahead forecast, itsuffices to calculate E(T^^ \ 7^,,). However, without any distribution assumption for >/ and ^ in theprice trend model, the eonditional expectation cannot be evaluated. In this paper, we propose toevaluate £"( 7",, | "R^ „) under the Gaussian assumption and denote tbe condition expectation underGaussian assumption by ^^(r,,!??.!,,). Tbe evaluation of £^..(r,,7?.| ,) can be carried out by furtherconditioning on the random variable I^^. Using conditioning argument, we have

t=\

,J (10)

Copyright €} 2000 John Wiley & Sons, Ltd. J. Forecast. 19, 485-498 (2000)


The terms in the right-hand side of equation (10) can further be evaluated as follows:

t> 1 (11)

and

(12)

Fora proof of equation (12), see the Appendix. Note that E^.{T^^ \ 'U.^J is simply the expected valueof T,, given 7^, „ under the Gaussian assumption and pE^{T^^\ 7?., „) will then be used to forecastR,,^\. In light of expressions (II) and (12), £,,y(7'^|7S|^) can be computed easily as all termsappearing in (11) and (12) are by-products of tbe QML procedure. Although EîR^^ ^ ^ \ TC, „) is notthe optimal estimator of ,,_|_j without tbe Gaussian assumption, it is the best linear estimator ofy?,, i based on observations up to time n. In other words, among all the estimators formulatedas linear combinations of observations, it is the one which minimises tbe mean squarederror. Therefore, E,J{R^|_^_^ \ ^Z^ „) provides a natural alternative to E{R„^^ \ TZ^J as an estimator ofi?,,î, especially when we do not want to impose distribution assumptions on /;, and tj^ inforecasting R,,_^] •

Trading strategyUsing the best linear one-step-ahead forecast of returns, a natural trading rule can be derived.First, we choose a parameter oO and generate a buy (sell) signal according to whether£^.(/i,,î|K, ,,)>(• (£'.\,(V?,,_ i|TC, „) < - (')• In other words, if the forecasted return of a certainsecurity exceeds c. we buy the security and keep holding it until a sell signal is obtained. On theother hand, if tbe expected return is smaller than - r while we are holding the securities, we coverthe long position and go short immediately. The short position is maintained until another buysignal emerges. Notice that the larger is the critical value e. the smaller is the number of switchesfrom long to short or from short to long. Thus, the choice of r depends on the transaction cost.According to Lam and Lam (1999), a good choice of e is roughly equal to two tmies thetransaction cost.

In particular, let us consider the trivial case: c = 0, in which the buy and sell signals aregenerated according to whether the expected return is positive or negative. In order to have aclearer description of the trading rule, we express the expected trend in equation (10) as follows:

(13)t=\

Copyright < 2000 John Wiley & Sons. L(d. J. Foreca.st. 19. 485-498 (2000)


where

Since equation (13) can be simplified as

a buy signal is generated if

n

'08 . > Jl''MPt-\ (14)

;=1

where

Since

the right-hand side of equation (14) is a weighted average of the past log prices. The trading rulederived here is based on the comparison between the current log price and a weighted average oi'log prices. It is similar to the well-known Moving Average (MA) rule in which a simple average isused instead of a weighted one. This result is interesting in that the optimal trading rule derivedunder the price trend model is in fact a modification of a very popular trading rule which is wellreceived by market participants. For similar results, see Acar and Satchell (1998).

In general, the parameter c can be different from zero. The larger tbe value of r, the smaller is thenumber of transactions involved and vice versa. Therefore, the parameter c can be used to controlthe transaction cost. The suitable choice of c is essential in implementing the new trading strategy.

ILLUSTRATION

In this section, we illustrate how the methodology developed above works by performingsimulation trading of the Hang Seng Index Futures contact.

The Hang Seng Index Futures (HSIF) contract was first introduced by the Hong Kong FuturesExchange in May 1986. Soon after its inception, HSIF quickly became the most actively tradedfutures contracts in Hong Kong. Figure I shows the daily futures price series and Table I gives abrief specification of HSIF contracts. The most recent historical data of this futures product canbe downloaded from the website of Hong Kong Futures Exchange http://www.hkfe.com.

Copyright V 2000 John Wiley & Sons. Ltd. J. Forecast. 19. 485-498 (2000)

Eoreeasting and Trading Strategies 493

18000

teooo

14000

12000

10000

8000

6000

4000

2000

1986 1987 1988 1989 1990 1991 1992 1993 1994 1995

Figure I. The daily futures price series

1996 1997 1998

DataThedatausedin this paper include daily HSIF over the period from May 1986 to DecemberFigure 1 gives a plot of the time series of futures prices. Daily HSIF return series are generatedwitb the return at trading day / defined as the difference between the logarithm of the two closingprices on the same nearest contract. A price trend model is fitted to the return series in 9 overlappedsubperiods: 86-89. 87-90, . . . , 94-97. The fitted model is used to forecast the future trend of the

Table I. Specification of Hang Seng Index Ftitures contracts (as of January 1999)

Contract multiplierPrice determinationContract monthTrading hoursExpiration dayFinal settlement daySettlement method

HK$50 pointOpen outcrySpot month, the next month, the next two calendar quarter month9:45 a.m. 12:30 p.m. and 2:30 p.m.-4:l5 p.m.-'The business day preceding the last business day of the monthThe tirst business day after the last trading dayCash settlement

'On the last trading day, the second trading session is 2:30 p.m. to 4:00 p.m.

Copyright '^Q 2000 John Wiley & Sons, Ltd. J. Forecast. 19, 485-498 (2000)


Table II. Descriptive statistics for the return series

86-89 87-90 88-9! 89-92 90- 93 91-94 92-95 93-96 94-97

Mean 0.007% -0.010% 0.046% 0.060% 0.145% 0.100% 0.085% 0.084% -0.019%Std. dev. 2.77% 2.69% 1.72% 1.74% 1.38% 1.65% 1.72% 1.68% 2.06%Skewness -11.43 -11.41 -5.44 -5.26 -0.16 -0.25 -0.10 -0.19 0.43Kurtosis'' 226.98 232.86 85.82 S0.82 4.64 2.94 2.05 2.72 18.83

•' Excess kuriosi:i.

Table III. Parameter estimates of the price trend model

86-89 87 90 88 91 89-92 90-93 91-94 92-95 9.^96 94 97

AP(1 - / ? ) - '

0.840.92

12.50

0.830.92

12.50

0.750,743.85

0.750.733,70

0.700.281.39

0.780.191,23

0.730.121.14

0,770,081,09

0.700.261.35

returns in the subsequent year. In other words, we obtain the respective trend forecasts for theyears 90. 91, . . . and 98 after the model fitting. Table II gives some descriptive statistics for thereturns series.

Model fittingUsing the methodology as deseribed above, we estimate tbe unknown parameters of tbe pricetrend model using the QML approach. Table III reports the estimate of .,•! and/j. Among tbe ninesubperiods. the estimates of/J are fairly large and do not vary too much, ranging from 0.70 to0.84. This indicates that the signal variation (IT^) is much larger than the noise variation (aj:). Inaddition, it is observed that the estimates of/) drop over time, from 0.92 in the subperiod 86 89 to0.26 in the subperiod 94-97. This implies that futures prices have been becoming more likely tohave changes in trend since its inception. As a result, the quantity (1 - / ? ) " ' which reflects theexpected duration for a trend to remain unchanged, also drops over time. The price trend isexpected to change from biweekly (12.5 days) in the subperiod 86-89 to daily (1.35 days) in thesubperiod 94-97. Notice that the values of .4 and p are relatively large in the first two subperiods.This may be explained by the fact that the worldwide 1987 crash wbicb is included in these twosubperiods contributes significantly to the estimates of t7,-j and /;. Finally, diagnostic checking ofthe residtials demonstrates that the price trend model is adequate in all cases.

Profit evaluationUsing price trend parameters to be estimated in a three-year period, a new trading strategy canbe constructed based on the forecasts from the price trend model in the subsequent one-yearperiod. With a proper choice of critical value c. buy-and-sell signals can be generated. We applythe new trading rule to the Hang Seng Index Futures assuming a transaction cost to range from0.4% to 0.5Vo. As discussed in the previous section, we let e vary from 0.8 to 1.0 and buy-and-sell trading decisions are simulated in the subsequent year after the estimation. To compute theprofit of trading HSIF. we follow the trading performance measurement method proposed byTaylor (1994). The main idea behind Taylor's method is to assume that investors can use a

Copyright r. 2000 John Wiley & Sons, Ltd. J. Foreca.st. 19. 485 498 (2000)

Forecasting and Trctding Strategies 495

Table IV, Profit eviiluation using the proposed trading rule and the buy-and-hold strategy

90 91 92Tesling period

93 94 95 96 97

Net proiit 0.1033 0.2749No. of (11) (5)

transactions

Net profit 0.0209 0.3710

Net profit 0.1370 0.1875No. of (9) (5)

transactions

Net profit 0.0204 0.3705

Net profit 0.1325 0.2325No. of (9) (3)

transactions

Net profit 0.0199 0.3700

Proposed trading rule with c = 0.80.0529 0.1416 b.OI 10 0.0779 0.0000 -0.2434 0.4624

(18) (23) (7) (2) (0) (2) ( 17 )

Buv-and-hold trading strategy0.2352 0.7765 -0.3875 0.2145 0.2792

Propo.sed trading rule with c = (1.00.1104 0.0556 0.0436 0.2313 0.0000

(14) (23) (5) (1) (0)

Buv-and-hold trading strategy0.2347 0!7760 -0.3880 0.2140 0.2787

Propo.sed trading rule with c = 1.00.1034 0.0599 -0.0497 0.0000 0.0000

(14) (21) (4) (0) (0)

Btiv-anct-hold trading strategy0.2342 0.7755 -0.3885 0.2135" 0.2782

0.2316 -0.0616

-0.0184(I)

0.2828(14)

-0.2321 -0.0621

-0.0189(I)

0.3453(6)

-0.2326 -0.0626

security which is regarded as free from default risk such as US Treasury bills as margin. Bydoing so, the investment used to finance futures trades will be covered by the margin securityand hence the futures contract could be treated as a primary security. Therefore, the return ofthe trading strategy can be determined very easily. We assume that an investor starts with onedollar for each transaction in the testing periods and remains fully investigated throughout tbewhole period. The gross profit of the investment is the profit derived from trading the marketindex in lieu of any transaction eost. The net profit is the profit after deducting the transactioncost which is assumed to range from 0.4% to 0.5% of the futures price for a one-waytransaction. As a benchmark for comparison, we apply also the naive buy-and-hold strategy.Table IV reports the net profit using both trading strategies for these selected values of c. Thenumber of transactions refers to tbe number of buy and sell decisions made in a one-yearperiod. In Table IV, when tbe trading rule's yearly net profit exceeds the buy-and-hold netprofit, the trading rule's net profit is given in bold. Otherwise, tbe buy-and-hold net profit isgiven in bold. Among 27 comparistms, 12 arc in favour of the price-trend trading rule. Table IValso shows that tbe new trading rule performs poorer ihan the naive buy-and-hold strategy inthe periods 91-93 and 95-96. This result is not surprising because the market is bullish in thesetwo periods (see Figure 1) and it is hard for a trading rule other than buy-and-bold strategy tomatch. However since the market becomes up and down in the years 90. 94. 97 and 98. tbetrading rule which takes into account the trend structure embodied in the returns could producea larger profit than the commonly used buy-and-hold rule. When net profit is averaged out overall years and all transaction costs, the net return for the buy-and-hold rule is 13.5% and thetrading strategy based on the price trend model derived a net return of 10.1%, which is quiteclose to that obtained by tbe buy-and-hold rule.

Copyright '(J. 2000 Johu Wiley & Sons. Lid. Forecast. 19, 485-498 (2000)


In comparing the proposed trading rule with the buy-and-boid rule, note that in a buy-and-hold setting, an investor will not leave the market and will bear the market risk throughout theyears. However, according to the proposed strategy, there may be an initial period in whichthe investor will take a neutral position and hence will not bear any market risk. In this sense, theproposed strategy is less risky than the buy-and-hold strategy. This compensates somehow theslightly lower return of the proposed trading strategy.

CONCLUSIONS

The focus of this paper is the price trend proposed by Taylor (1980). According to the model.financial returns can be written as the sum of two components, the trend and the noise. The trendof the return changes only when new information is available. Switching of the trend occurs witha fixed probability. The contributions of this paper are twofold. First, recursive methods incomputing the quasi-likelihood function are derived so that tbe method of QML estimation isfeasible. As a by-product of the likelihood evaluation, forecasts of future trend can be obtained.The forecasts are very useful in developing a new trading rule. In other words, the secondcontribution of this paper is the introduction of a new trading strategy by capturing the trendmovements of the returns. In the special case of no transaction cost which corresponds to c = 0,the proposed trading rule is similar to the commonly used moving average rule by replacingsimple averages by weighted averages.

The method is illustrated by fitting the Hang Seng Index Futures since its inception in 1986.Forecasts of the future returns' trend based on the fitted price trend model are generated.Buy-and-sell signals are produced using the forecasts of the trend. When comparing with thenaive buy-and-hold rule, the proposed trading rule with parameter c chosen to match the round-trip transaction cost could earn more net profit than the commonly used buy-and-hold strategy inan up-and-down market.

APPENDIX: PROOF OF EQUATION (12)

When /„ = /, r,, and (R^ R^) are independent oi {R^ R,_^). Therefore.

=UT,,\n,_,, i,,^t)

^f.iT^jr,, I,, ^ t)f,(Tji,, = t)

oc exp xexp

exp - -2^ 2 f j ' \

Copyright 'C' 2000 John Wiley & Sons. Ltd, J. Forecast. 19, 4S5-498 (2000)


where

k— -^ — ^IR.+ ... +R)dr\(\

G] + (« + I - ,)(T2

The result follows as T^^\TZ^,^, f,, ^ t -^ N{/., n-) under the Gaussian assumption of

R E F E R E N C E S

.Acar, E. and Satchell. S. Advanced Trading Rules. Oxford: Butterworth-Heinetnann. 1998.Alexander. S. S, 'Price movements in speculative markets: Trends or random walks'. Industrial Management

Review. 5 (1964). 22 46.Bachelier. L. Theory of speculation', Annaies Seientificpies dc I'Fcole Normale Superictire, Paris: Giiiitheier-

Villars. (1910). Ser. 3. No. 1018.Brock, W. A., Lakonishok, J. and LeBaron. B. "Simple technical trading rules and the stochastic properties

of stock returns". Journal of Finance. 41 (1992). 1731-1764.Cootner. P. H. The Random Character of Stock Market Prices. Cambridge. MA: MIT Press. 1964.Fama. E. F. and French. K. R. 'Permanent and temporary components of stock prices". Journal of Poiitical

Economy, ^k {\m). 246-273.Farrell. C. H. and Olszewski. E. A. 'Assessing inetliciency in the S&P 500 ttitures market". Journal of

Forecasting, 12 (1993), 395 420.Fortune, P. "Stock market efficiency; an autopsy'. New England Economic Review Federal Reserve Bank of

Boston. (!991). 17-40,Gencay, R. 'Non-linear prediction of security returns with moving averaae rules'. Jounut! <>/ Forecasting:. 15

(1996). 165-174.Harvey. A. C. Time Series Models. 90. New York: Harvester, Wheatsheaf. !993.Jensen. M. C. 'Some anomalous evidence regarding market etlicieney, an editorial introduction". Journal of

Financial Economics. 6 (1978), 95 101.Kendall, M. G, The analysis of economic lime series. Part I: Prices". Jotimal of the Roval Statistical

Society. %, Pan I (1953). 11 -33.Kwan. W. C. Trend models for price movements in financial markets, MPhil thesis. Department of Statistics.

The University of Hong Kong. Hong Kong. 1994.Lam. K. and Lam. K. C. "Foreeasting for the generation of trading signals in financial markets".

Forthcoming in Journal of Forecasting. (1999).LeRoy. S. F. 'Capital market etiiciency: an update". Ecotwmic Review, Federal Reserve Bank of San

Francisco. 2 (1990|. 29-40.Lo. A. W. and MacKinlay. A. C. "Stock market prices do not follow random walks: Evidence from a simple

specification test'. Review of Financial Stitdies. I (I9HIS). 41-66.Neftei. N. 'Naive trading rules in financial markets and Wiener-Kolmogorov prediction theory; a study of

lechnical analysis'. Journal of Busine.ss. 64 (1991), 549 571.Poterba, J. M. and Summers, L. H. "Mean reversion in stock returns: evidence and implications". JourtutI of

Financial Economics, 11 {\9m. 27-59,Summers. L. "Does the stock market rationally reflect fundametUal values''" Journal of Finance. 41 (19S6).

591-601.Taylor, S. J. "Conjectured models for trends in financial prices, tests and forecasts". Journal of the Rovaf

Statistical Society Series A. 143 (19X0), 338-362.Taylor. S. J. Modellinti Financial Time Series. Chiehester: Wiley. 1986.

Copyright ' 2000 John Wiley & Sons. Ltd. J. Forecast. 19. 485-498 (2000)


Taylor, S. J. "Rewards available to currency futures speculators: compensation for risk or evidence ofinefficient pricing?" Ecotwmic Record.. 68(Supplement) (1992), 105-116.

Taylor. S. J. "Trading futures using a channel rule: a sttidy of the predictive power of technical analysis withcurrency examples'. Journal of Futures Markets. 14 (1994), 215-235.

Authors' biographies:Josephine VV. C. Kwan received her M. Phil, in Statistics from The University of Hong Kong. Her researchinterests include modeling of price movement in securities markets and study of stock market efficiency.

K. Lam is a Chair Professor and Head of the Department of Finance and Decision Sciences at the HongKong Baptist University. He received his PhD from the University of Wisconsin al Madison. He is thedirector of the Business SchooKs focused research area in financial derivatives and investment strategy whichcoincided with his research direction.

Mike K. P. So is an Assistant Professor of the Department of Information and Systems Management at theHong Kong University of Science and Technology. He received his PhD in Statistics from The University ofHong Kong in 1996. His research interests include nonlinear time series analysis and its applications,modeling of financial time series and market volatility sttidy.

Philip L. H. Yu is an Assistant Professor o'^ the Deparlinent of Statistics and Actuarial Science at TheUniversity oi Hong Kong. He received his PhD in Stalislics from The University of Hong Kong. Hisresearch interests include stattslical models for financial data and its applications to stock market volatilitystudy, and analysis o{ discrete choice and ranking data and its applications in market research.

Authors' addres.scs:Josephine W. C. Kwan. Department of Statistics and Actuarial Science, The University of Hong Kong,Pokfulam Road. Hong Kong.

K. Lam, Department of Finance and Decision Sciences. Hong Kong Baptist University. Kowloon Tong.Hong Kong.

Mike K. P. So. Department of Information and Systems Management. The Hong Kong University ofScience and Technology. Clear Water Bay. Kowloon. Hong Kong.

Philip L. H. Yu. Department of Statistics and Actuarial Science. The University of Hong Kong. PokfulamRoad. Hong Kong.

Copyright r 2000 John Wiley & Sons. Ltd. J. Forecast. 19, 485- 498 (2000)

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