Intelligent Stock Portfolio Management Using a Long-Term Fuzzy System

ABSTRACT

The complexity of financial markets is driving researchers to multiply their efforts in order to improve their forecasting methods. This paper inoculates an old trading strategy with fuzzy subjective elements. The aim is to investigate whether the careful synthesis of a few long-term technical indicators, which have a different predictive philosophy, with an appropriately designed stock trading Mamdani fuzzy system, can produce satisfactory returns. More specifically, its purpose is to investigate whether the combination of moving averages, directional movement technical indicators and a fuzzified trading strategy can surpass the performance of buy and hold strategy (B&H). The proposed model has been tested in various (bull and bear) market environments for a period of more than 15 years, using the general index of ASE (Athens Stock Exchange). After taking into consideration transaction costs, it is found that the proposed model can produce better results (higher earnings) than the B&H strategy.

Introduction

Over the previous years, the complexity of financial markets has been well documented by researchers (Lan, Zhang, and Xiong 2011) who also provided empirical evidences that a large number of interrelated economic, political and psychological factors, cause many uncertainties and positively or negatively affect stock values.

Various techniques have been used for time series analysis price forecasting (Chen and Chen 2014), such as the autoregressive moving average model, the factor analysis, the linear multiple discriminant approaches (MDA), and the univariate approaches, while, relatively recently, a variety of artificial intelligence techniques have also been proposed. This is due to the fact that appropriate nonparametric machine-learning approaches may be capable of identifying the dynamic nonlinear relationships that exist between market information and stock values (Chenoweth, Obradovic, and Lee 1996). According to Chang, Fan, and Lin (2011), since stock price data are affected by deterministic and random factors, price forecasting can be successful only with the use of tools and techniques that can overcome the problem of noise and nonlinearity of prices.

Soft computing techniques are nonlinear in nature (Melin et al. 2007) and, therefore, they can capture the nonlinear behavior of stock markets (Atsalakis and Valavanis 2009), emphasizing their usefulness for solving complex problems (Soto 2007). Fuzzy logic is one of the soft computing techniques (Kumar and Ravi 2007) that uses creatively the imprecision and uncertainty of human judgment (Beka Be Nguema et al. 2000). This is actually one of the reasons why it has been used with success in many applications of the real world (Garcia-Crespo et al. 2012).

Technical analysis uses historical prices and volumes in order to forecast future stock prices. The continuous flow of information from the complex “environment” mentioned earlier, and the changing attitudes of investors reacting to this information create opposing forces of supply and demand, which move stock prices in short, medium or long trends. According to technical analysis, all this information is reflected in stock prices. Technical indicators are created using various mathematical formulas and they continually evaluate the strength of the current trend, identifying any change that may occur (Apostolou and Apostolou 2004; Bao and Yang 2008; Cheng, Chen, and Wei 2010). However, the appropriate combination of various technical analysis techniques is a difficult task (Pring 2014) and, as a result, investors usually make decisions using subjective assessments (Dymova, Sevastianov, and Kaczmarek 2012; Kahn 2010). Moreover, technical analysis techniques can provide contradictory results, the evaluation of which demands specific human expertise and appropriate justification (Majhi, Panda, and Sahoo 2009). Thus, the development of novel fuzzy models that can address such issues of human subjectivity in a satisfactory manner is a promising challenge.

Various fuzzy logic techniques, which combine specific methods of technical analysis have been recently tested with rather promising results (Araque et al. 2008; Atsalakis, Dimitrakakis, and Zopounidis 2011; Bekiros and Georgoutsos 2007; Dymova, Sevastianov, and Bartosiewicz 2010; Esfahanipour and Aghamiri 2010; Pereira and Tettamanzi 2008; Sevastianov and Dymova 2009; Svalina et al. 2013). Most of the above researchers use common technical indicators (e.g. moving averages), which are calculated using only the closing daily prices. As a result, important information, such as high and low is missed. However, other technical indicators, like the Directional Movement, exploit these data and it is, therefore, reasonable to assume that the combination of the appropriate technical analysis indicators can improve the predictive power of the models.

This paper proposes an intelligent stock trading fuzzy system, aiming to assist long-term investors in their portfolio management. The purpose of this research is to investigate whether the combination of moving averages, directional movement technical indicators and a fuzzified trading strategy, implemented in an appropriately designed fuzzy system, can surpass the performance of buy and hold strategy (B&H) or the interest earned by the portfolio’s initial capital invested on a safe deposit account. Furthermore, it attempts to the extent the boundaries of previous research in short (Chourmouziadis and Chatzoglou 2016) and medium-term prediction, and to investigate whether even more satisfactory results can be achieved.

This paper contributes to the literature in a number of ways. First, it investigates whether the use of only a few, very common, carefully selected technical indicators, can be combined with an appropriately designed fuzzy system, in order to create a generic model for forecasting the long-term trading proposals of a portfolio. Contrary to the common practice of using as many as possible, and sometimes with contradictory characteristics technical indicators (Fadlalla and Amani 2014; Patel et al. 2015; Pokropinska and Scherer 2008; Zarandi et al. 2009), this research proposes the use of four only technical indicators, which are all trend following. The selected technical indicators are three moving averages with different parameters and the directional movement. The profitability of moving averages is supported by some researchers, but it is questioned by some others (Brock, Lakonishok, and LeBaron 1992; Levich and Poti 2015; Taylor 2014; Yu et al. 2013), while the directional movement indicator has many alternative interpretations (Metastock Professional 2002). During the research, moving averages’ signals are combined with the signals of the directional movement indicator, in an effort to improve their predictive power, since they are all trend following technical indicators, but with a different constructive philosophy. Further, the predictive power of the model was increased by optimizing the parameters of the above indicators for long-term periods, and from the fact that directional movement, except from the closing price uses the high and low of the daily price as well.

Second, it uses a strategy (proposed by the same authors) in order to transform a rigid trading strategy to a more fuzzy form, enabling thus subjectivity, which is inherent in the real world decisions, to be part of the trading strategy. This is succeeded by inoculating an old trading strategy, with an observation of a researcher and the notion of certainty, as part of the developed system. According to the classic trading strategy, a buy (sell) signal is issued when the moving average is crossing from above (below) the closing price or when the -di is crossing from above (below) the +di of the directional movement indicator. This was extended with an idea of Papoulias (1990), according to which when a stenosis of the space between price and technical indicator appears, the current move of the market is ending, while when the distance between them broadens, the current move continues to be valid. Thus, this notion of the degree of stenosis was fuzzyfied and combined with the classic trading strategy of the cross between price and technical indicator, in order to be implemented through the membership functions and the rules of the fuzzy system, embodying the notion of the degree of certainty and subjectivity.

The proposed model was tested in various market environments (which include bull and bear markets) for a period of more than 15 years, using the general index of ASE (Athens Stock Exchange) in Greece. Two additional tools were created, one to help in correcting errors of historic data and the other to provide information for the daily investments of the portfolio.

Literature Review

Some of the most cited conventional numeric forecasting models are (Wei 2013) ARCH, GARCH, ARMA, and ARIMA.

However, since stock market is a complex and dynamic system with noisy and chaotic data series (Evans, Pappas, and Xhafa 2013; Peters 1994), soft computing systems (Bao and Yang 2008), such as fuzzy logic, neural networks, and neuro-fuzzy systems have been used in an attempt to predict stock market prices. Various models which combine soft computing techniques and technical analysis have also appeared. These include stock price prediction (Chang and Liu 2008), prediction of the direction of the market (Doeksen et al. 2005), prediction of the actual buy/sell signals (Tilakaratne, Mammadov, and Morris 2007), prediction of future prices (Wang and Chan 2009) and stock selection (Quah 2008).

More specifically, some researchers have attempted to develop models for predicting the direction of a market. Vaidehi et al. (2008), for example, proposed a fuzzy system to predict the possibility market prices (for six Indian stocks) to rise or fall with almost 80% accuracy. Similarly, Bekiros (2010) proposed a hybrid neuro-fuzzy system for predicting the direction of the market on the next trading day. The total profits of the system are shown to be superior to a recurrent neural network and a B&H strategy for all indices. Atsalakis and Valavanis (2009) also propose a neuro-fuzzy system, which is composed of an ANFIS controller and a stock market process model to identify the best stock trend prediction.

Further, Chavarnakul and Enke (2008) proposed two generalized regression neural networks (GRNN) for predicting future stock prices with the use of the technical indicators developed from equivolume charting. These networks incorporate the volume-adjusted MA and EMV (Ease of Movement) as predictors, and they outperformed the results of two simple technical indicators, namely the moving average and the volume-adjusted moving average, as well as the B&H strategy. Similarly, Baba and Nomura (2005) proposed an effective decision support system, which uses the artificial neural network in order to predict the intersection of two MA several weeks in advance. Moreover, Chang and Liu (2008) developed a TSK fuzzy system for price predictions using various technical indicators as input. They grouped these indicators in various clusters according to the k-means algorithm. Their predictions were proved to be more accurate than other approaches, such as multiple regression analysis or the Back Propagation Neural network (BPN). Finally, Tan, Quek, and Yow (2008) attempted to identify reversal points through an intelligent Rough Set-based Pseudo Outer-Product (RSPOP) fuzzy neural network and the use of various technical indicators, producing far more accurate predictions (>95%) than that achieved adopting the B&H strategy.

Technical Analysis

In the area of stock market forecasting, technical analysis (TA) is one of the primary analytic approaches used by many investors to make investment decisions and increase their investment returns (Cheng, Chen, and Wei 2010). However, although a series of recent surveys has confirmed the extensive use of TA among investors, practitioners and professional traders (Gehrig and Menkhoff 2006; Menkhoff 2010; Oberlechner 2001), technical analysis’s claims have also found support within the academic community (Neely and Weller 1999; Vanstone and Finnie 2009). As Jasemi, Kimiagari, and Memariani (2011) comment, academics have simply paid more attention to this investment strategy rather recently.

This research uses two types of trend following indicators, (i) Moving Average (MA) and (ii) Directional Movement.

MA is a method for calculating the average value of a stock for a specific time period and it can be characterized as short term (e.g. 15 days), mid-term (e.g. 50 days) and long term (e.g. 200 days). MAs are used to smooth the noise of shorter-term fluctuations in order to more easily identify the significant underlying trends (Appel, 2005). According to Katz and McCormick (2000), the three most important types of MA are the simple MA, exponential MA and weighted MA. The main difference between the various types of MAs is the weight they place on recent data.

A number of in-depth tests were carried out on the various types of MA before the exponential MA was chosen as the most appropriate for the purposes of the present research. This is in line with, Dzikevicius and Saranda’s (2010) claim that exponential MA is more suitable for predicting prices compared to a simple MA. The formula used to calculate the exponential moving average is the one proposed by Wang and Wang (2010).

Milionis and Papanagiotou (2008) examined the predictive ability of MA in the general index of the Athens stock exchange (ASE) and they found that the performance of MA for time lengths 5–100 days surpassed the performance of the B&H strategy. Further, Kuang, Schoder, and Wang (2014) examined the profitability of 10 emerging foreign exchange markets for various technical trading rules (including moving averages rules), and they concluded that nearly all of the apparent success of these rules is simply the result of data mining bias.

The directional movement is also a trend following system which has been developed by Welles Wilder (Wilder 1978), who also suggests the use of five indicators, csi, +di, -di, adx, and adxr with various interpretations. In the present paper, only +di and – di are used. The basic directional movement trading system is to buy when +di rises above the – di and sell when the -di falls below the +di. This rule is followed in the present paper. Chang Chien and Chen (2010) are among the relatively few researchers who have also used this technical indicator.

Collection of Historical Data

Daily data with open, high, low, close and volume were used in this research. However, it is well known that real market data “often suffer from various deficiencies” (Bishop 2004, p.295 raising many doubts about their reliability. Furthermore, fuzzy data are often content-dependent, uncertain, imprecise, vague, inconsistent and ambiguous (Galindo, Urrutia, and Piattini 2006). Therefore, some degree of preparation and pre-processing is required and this procedure must be considered to be an integral part of the modeling process (Garibaldi 2005).

Data pre-processing was consisted of error checking, correction of the historic data and the calculation of various technical indicators. Α methodology to search for and correct errors from data on any market was developed. Then, an appropriate tool was also developed for this purpose. This methodology has helped authors discover and correct four kinds of errors (arithmetic errors, logic errors, date errors and prices out of set limits).

Choosing the Parameters of Technical Indicators

The performance of the various technical indicators depends on the criteria set to create the environment they operate in. In this research, once the environment had been set, extensive back-testing and optimization tasks were carried out in order to select the best parameters for every indicator. The automated system used for back-testing the technical indicators was the Metastock Enhanced System Tester. The criteria (technical strategy) of the testing environment were the following (Chourmouziadis and Chatzoglou 2016): a) each buy or sell order was set at the close of the same day, b) no stops were used, c) the total available cash was used every time a buy signal was given, d) all possessed assets were sold every time a sell signal was given, e) no slippage or margin were used, f) there was no interest received for the cash left in the portfolio.

The index that was used was the ASE (Athens Stock Exchange) general index, while the testing period was from 1987 until 1992, while the period of the research was between 15/11/1996 and 5/6/2012.

Empirical Results, Analysis and Discussion

A new value for each technical indicator is calculated every day which is, then, compared with the daily price of the particular stock or the ASE general index, in order to create the relevant linguistic variable. Linguistic variables (Zadeh 1975) fluctuate between the minimum and the maximum of the historical prices on a scale of 0 to 1. A bell-shaped membership function was chosen because the curve of this function is smooth and nonzero at all points (Dourra and Siy 2002).

The output of the developed fuzzy system is a number ranging from 0 to 1 which expresses the percentage of the portfolio that should be invested on a daily basis in the ASE. Every day, complementary trades should be carried out in order to ensure that the invested proportion of the portfolio is exactly the same as the suggested percentage by the system. A working prototype was developed in order to calculate all these changes, along with the relevant daily transaction costs, which are subtracted from the respective gains or added to the losses.

Table 1 shows that for the first day (15/11/1996), the general index of the ASE was 890,39 units and the output of the fuzzy system was 0,074339303, suggesting an investment of 7,4339% (743,39€) of the initial amount of the portfolio (assumed to be 10.000€) in the ASE. The next day, the output of the fuzzy system was 0,084423054, indicating that a bigger part (8,4423% or 844,23€) of the portfolio should be invested. Of course, the shares bought the previous day would already worth more since the general index of the ASE was higher (914,82). The assumptions made for the portfolio management are the following:

• The initial capital is 10.000€.

• The portfolio would be invested solely in the general index of ASE in the Greek stock market.

• The value of the portfolio is calculated on a daily basis, taking into consideration the closing price of the general index.

• All transactions are made ATC, i.e. “at the close” of the same day.

• The amount of the portfolio which should be invested on a daily basis depends on the output of the fuzzy system.

• For every transaction made, there are two measurements, with and without subtracting the transaction costs. The transaction costs adopted for this study are those which were valid in the ASE on 31/12/2012.

• No interest is calculated for the remaining cash in the portfolio, nor are they invested in any “safe” asset, although this is particularly common (Brock, Lakonishok, and LeBaron 1992) since it strongly increases the portfolio’s performance. The reason for choosing not to invest the portfolio’s remaining cash is that the goal of this research is to determine the performance of the proposed model and not try to maximize it with any of the available methods.

Table 1. An extract of the first and last few days of the portfolio management.

Date	GI of ASE (close)	Output of model – PoIP	PCV	TAiS	NoPS	NoAS	VoNS	TrCo	VoNSTC	Cash	Value of portfolio
Initial prices			10000,00							10000,00
15/11/1996	890,39	0,074339303	10000,00	743,39	0,835	0,835	743,39	3,07	746,46	9253,54	9996,93
18/11/1996	914,82	0,084423054	10017,33	845,69	0,924	0,090	81,90	0,34	82,24	9171,30	10016,99
19/11/1996	917,1	0,084423054	10019,10	845,84	0,922	−0,002	−1,96	0,01	−1,95	9173,25	10019,09
20/11/1996	913,1	0,084423054	10015,40	845,53	0,926	0,004	3,38	0,01	3,39	9169,85	10015,39
21/11/1996	900,65	0,074650692	10003,86	746,79	0,829	−0,097	−87,21	0,53	−86,67	9256,53	10003,32
22/11/1996	906,02	0,07433932	10007,77	743,97	0,821	−0,008	−7,28	0,04	−7,23	9263,76	10007,73
25/11/1996	909,03	0,074650692	10010,20	747,27	0,822	0,001	0,83	0,00	0,83	9262,93	10010,20
26/11/1996	911,96	0,074650692	10012,61	747,45	0,820	−0,002	−2,23	0,01	−2,22	9265,15	10012,59
… .	… .	… .	… .	… .	… .	… .	… .	… .	… .	… .	… .
24/5/2012	502,52	0,074339303	50219,01	3733,25	7,429	0,313	157,22	0,65	157,87	46485,12	50218,36
25/5/2012	485,18	0,074339303	50089,54	3723,62	7,675	0,246	119,20	0,49	119,69	46365,43	50089,05
28/5/2012	518,49	0,074339303	50344,70	3742,59	7,218	−0,456	−236,68	1,45	−235,23	46600,66	50343,25
29/5/2012	528,14	0,074339303	50412,90	3747,66	7,096	−0,122	−64,59	0,40	−64,19	46664,85	50412,51
30/5/2012	511,29	0,074339303	50292,94	3738,74	7,312	0,216	110,65	0,46	111,11	46553,74	50292,48
31/5/2012	525,45	0,074339303	50396,03	3746,41	7,130	−0,182	−95,88	0,59	−95,29	46649,03	50395,44
1/6/2012	501,9	0,074339303	50227,53	3733,88	7,439	0,310	155,38	0,64	156,02	46493,01	50226,89
5/6/2012	476,36	0,074339303	50036,88	3719,71	7,809	0,369	175,83	0,73	176,56	46316,45	50036,16

Close of the General Index of ASE (close)

Output of the model – PoIP: Output of the proposed model on a scale of 0 to 1 which is equal to the Percentage of Portfolio Invested.

PCV: Portfolio’s Current Value (cash + amount of yesterday’s stocks * today’s close)

TAiS: The Total Amount in Stocks (yesterday’s value of portfolio * percentage of invested portfolio, i.e. [cash + yesterday’s amount of stocks * today’s price]) * today’s output

NoPS: Number of Portfolio Stocks that should be possessed by the portfolio in order to ensure that is invested the correct amount be invested (according to today’s closing price)

NoAS: Number of Additional Shares that should be Bought+/Sold-

VoNS: Value (Buy+/Sell-) of these New Stocks

TrCo: Transaction Costs

VoNSTC: VoNS + Transaction Costs

Cash remaining in the portfolio

Value of the portfolio: (New cash + amount invested in stocks). This is the same as PCV butit is calculated differently, i.e. by adding today’s cash in today’s value of the stocks possessed.

For the process of the fuzzy inference, the Mamdani method was chosen, because it is widely accepted for capturing expert knowledge (Negnevitsky 2005), a very important aspect for the present study since fuzzy inference rules were based on the expert knowledge of a technical analyst.

Long-Term Fuzzy System

The performance of the fuzzy system has been explored using the following four long-term technical indicators: the exponential moving average of 105 days, the exponential moving average of 145 days, the Directional Movement of 73 days and, finally, the “classic” simple moving average of 200 days (Table 2). According to technical analysis, for the first, second and fourth indicator, a buy signal is given when their value becomes less than the daily close, while for the third indicator, a buy signal is given when the -di crosses underneath (becomes less than) the +di. Therefore, a buy signal is issued when the values CL-MA(105), CL-MA(145), +di(73)-(-di(73)) and CL-MA(200) become positive, while a sell signal is issued when their value becomes negative. These values are calculated daily and they constitute the inputs of the fuzzy system.

Table 2. Inputs of the long-term fuzzy system.

Long-term technical indicators	Adjusted parameters	Linguistic variables
An exponential ΜΑ	105 days	CL-MA(105)
An exponential ΜΑ	145 days	CL-MA(145)
Directional Movement	73 days	+di(73)-(-di(73))
The “classic” simple MA	200 days	CL-MA(200)

According to Papoulias (1990), the stenosis (narrowing) of the space between the MA line and stock prices is an indication that the market’s movement will shortly end. Extending this observation, an improved technical trading strategy was explored: when the positive (negative) values of CL-MA(105), CL-MA(145), +di(73)-(-di(73)) and CL-MA(200) become smaller and smaller, there is an indication that progressively the degree of certainty of the present order for buying (selling) decreases. However, at the same time, this increases the possibility that these values will reach zero or even turn negative (positive) and, consequently, the opposite order for selling (buying) will be issued. Similarly, when the space between the MA (or any of the above technical indicators) and stock prices increases, the positive (negative) values of CL-MA(105), CL-MA(145), +di(73)-(-di(73)) and CL-MA(200) become larger and larger, indicating that the degree of certainty of the present order of buy (sell) increases.

The curves of the two membership functions (e.g. MA105overCL and MA105underCL) are plotted in a specific way. When one of them (e.g. MA105underCL) has very high values on the $x$ axis, it expresses the almost absolute certainty that the current buy order is correct. When MA105overCL has very low prices on the $x$ axis, it expresses the almost absolute certainty that the current sell order is correct. These two curves are plotted in the opposite manner. In a specific moment $t$, when both curves have a specific value on the $x$ axis, one of them expresses the almost certainty that the buy order is correct, while the other expresses the minimum certainty that the sell order is correct. When both membership functions are in the area of zero (i.e. when MA approaches the price of the ASE general index), they denote uncertainty about the strength of the order they represent and that it is entirely possible that the opposite order will be given.

In this case, the $x$ axis ranges from the minimum value of the linguistic variable (which for the more than 15-year testing period is −1427,8 for CL-MA(105)) to the maximum (which for CL-MA(105) is 1835,6). Thus, it becomes apparent that the minimum and maximum values are not symmetrical to zero. The two membership functions intersect at the value 0 on the horizontal $x$ axis and each one of them is bell-shaped. One of them is MA105underCL, which represents the case where MA is under the close price (so a buy signal is issued), while the other is MA105overCL, which represents the case where MA is over the close price (so a sell signal is issued).

The area around zero is where each curve moves from absolute certainty to absolute uncertainty. The slope of the curve in this area determines the width of the values between which this transfer occurs. The steeper the slope, the smaller is the range of values on the $x$ axis within which this transfer occurs and the smaller is the elasticity of the curve. Similarly, the more gentle the slope, the more progressive is the transfer from one stage to another and wider is the range of values on the $x$ axis between absolute certainty and absolute uncertainty.

Differentiation of the Slope of the Curve

Every day, the minimum and maximum of each linguistic variable (e.g. MA105overCL) are calculated in order to define the $x$ axis range. Therefore, if a unique value of b was used for all linguistic variables (i.e. the input variables of the fuzzy system), the slope of the curve would be different for every curve. To avoid this and to have curves with similar slopes, the value of b should be altered according to the minimum and maximum boundaries of the input variable (x). Hence, the value of b would be different for every input variable, but the slope would have the same characteristics in order to ensure that the relative $x$ axis boundaries, where the differentiation of the degree of certainty for the specific order takes place, will remain the same. Moreover, since the minimum and maximum values of each input variable on the $x$ axis are not symmetrical to zero, it should be noted that in order to have analogically the same slope of the curve in the negative and positive values, a different slope of the curve on the left and right sides of zero is necessary.

Taking into consideration all these characteristics, research for a wide range of b values needs to be conducted. However, in order to do this, the fuzzy system has to be completed by creating fuzzy rules. Initially, all fuzzy rules created have the same maximum weight, which is equal to one. Thereafter, for each b value, the output of the long-term fuzzy system is firstly calculated, followed by the calculation of the return of the portfolio and, finally, the rate of growth of this return.

It must be stressed, though, that, there is a potential weakness with these fuzzy rules, emerging from the fact that the profitability of the initial four technical indicators (from which the four input variables of the fuzzy system were created) is very uneven. Therefore, new fuzzy rules had to be created, where each rule was strengthened or weakened, according to the performance of the relevant technical indicator. After the completion of the new fuzzy system, the new fuzzy rules for each b value, the output of the long-term fuzzy system were again calculated. The return of the portfolio and the rate of growth for each return were also calculated. The combined results of all these tests are shown in Table 3.

Table 3. Returns of the long-term fuzzy system when the b parameter varies.

The influence of b in the returns of long-term fuzzy system (%)
	Without transaction costs		With transaction costs
b	Rules with weights equal to 1	Rules with varying weights(from 0 to 1)	Rules with weights equal to 1	Rules with varying weights (from 0 to 1)
10/1	609,47	620,97	401,02	406,68
7/1	609,22	620,75	400,85	406,57
5/1	608,69	620,20	400,41	406,20
4/1	608,06	619,47	399,95	405,77
3/1	606,64	617,78	398,96	404,73
2/1	602,26	612,81	395,62	400,36
1/1	583,62	595,53	381,79	384,90
1/2	548,93	569,17	356,31	364,03
1/3	517,87	546,11	334,41	347,77
1/4	489,44	522,62	315,84	331,99
1/5	464,24	500,67	300,25	318,05
1/10	380,54	427,41	251,86	275,74
1/20	316,44	362,68	214,92	238,27
1/30	289,59	333,48	197,51	220,51
1/40	272,20	314,74	184,64	208,01
1/50	255,64	295,29	172,01	191,44

The performance of the general index of ASE for the same period was −46,50%.

The results in Table 3 imply that in both cases, i.e. with or without transaction costs, the best value of b equals to 2/1 of the relevant width on the $x$ axis of each input variable. This value is chosen because after this point, the increasing rhythm of the return changes dramatically and it continuously decreases. This value of b will be used in the next stages of the current analysis since it is the value which achieves high returns of the fuzzy system, while simultaneously maintains the highest possible elasticity (so that the degree of certainty expressed by the slope of the curve does not change very abruptly).

As it is shown in Table 4, in total 16 fuzzy rules were created. The weights of the rules were given in a way to enhance those rules which contain the most profitable technical indicators giving a signal (Buy or Sell) in the same direction of the outcome of the rule.

Table 4. The rules which have been created for the long-term system.

Rule	CL-MA(105)	CL-MA(145)	DM(73)	CL-MA(200)	Output	Weight
1	S	S	S	S	All Out	1
2	B	S	S	S	Low	0,1
3	S	B	S	S	Low	0,4
4	S	S	B	S	Low	0,7
5	S	S	S	B	Low	1
6	B	B	S	S	Medium	0,1
7	B	S	B	S	Medium	0,5
8	B	S	S	B	Medium	1
9	S	B	B	S	Medium	1
10	S	B	S	B	Medium	0,5
11	S	S	B	B	Medium	0,1
12	S	B	B	B	High	0,1
13	B	S	B	B	High	0,4
14	B	B	S	B	High	0,7
15	B	B	B	S	High	1
16	B	B	B	B	All In	1

B = Buy, S = Sell

Low = Buy or Sell Small % of the portfolio, Medium = Buy or Sell Medium % of the portfolio, High = Buy or Sell Large % of the portfolio.

The Performance of the Proposed Long-Term System in the ASE General Index

Markets change character over time (Kirkpatrick and Dahlquist 2011) and this is why a long study period is needed in order to extract useful and valid results. The time period for the performance tests was from 15/11/1996 to 5/6/2012, which includes all the changes made in the institutional framework of the ASE (1995), the entry of Greece into the European Exchange Rates Mechanism II (1998), the final entrance of the country into the Economic Monetary Union (2001), the characterisation of the ASE as a developed market and the Olympic Games of 2004 (Kenourgios and Samitas 2008; Liroudi et al. 2004). It also includes the Asian crisis (1997), the world financial crisis (2008) and the Greek fiscal crisis (2010). Overall, it includes three major bull markets and three major bear markets. On 5^th June 2012, the ASE general index made the lowest low in the past 19 years.

The ASE general index during the 15-year period chosen had a negative return of −46,50% and this is also the exact return for the B&H strategy. On the other hand, the return of the proposed long-term system was 612,81% without taking into consideration the transaction costs, and 400,36% when transaction costs were considered (Table 5). Moreover, if the initial amount of 10.000€ was deposited in a safe account for the same period, taking into consideration the interest rates of every period, the total interest gained would have been 45,62%. It is obvious, therefore, that the return of the proposed system is quite positive, especially when this performance is compared with such a big loss of the ASE General Index, over the tested period. It should also be underlined that no interest gains were calculated for the part of the portfolio which is maintained in cash, which would have increased significantly the overall value of the portfolio.

Table 5. The return of the proposed system compared with the interest gained from a safe deposit account and the B&H strategy.

	Comparison of performances
	Interest on deposits in a safe account	B&H Strategy	Proposed long-term fuzzy system
ASE general index: 15/11/1996 till 5/6/2012	10.000€ → 14.562,40€	890,39 → 476,36 units	10.000€ → 71.280,71€ (without transaction costs)
			10.000€ → 50.036,16€ (with transaction costs)
Return	45,62%	−46,50%	612,81% (without transaction costs)
			400,36% (with transaction costs)

During the examined period (15/11/1996 to 5/6/2012), there were periods of bull and bear markets. In order to further explore the performance of the system during these distinctive periods, they were examined separately (Table 6).

It is found that for the first bull market (15/11/1996 to 17/9/1999), the performance for the B&H strategy was 613,74%, while the return of the proposed system was 437,89% if transaction costs were not subtracted, and 411,37% if they were subtracted (Table 7). For the second bull market (01/04/2003 to 31/10/2007), the performance for the B&H strategy was 261,75%, while the return of the proposed system was 108,98% (without transaction costs) and 97,83% (with transaction costs). Finally, for the third and smallest bull period (between 09/03/2009 and 14/10/2009), the performance for the B&H strategy was 97,15%, while the return of the proposed system was 39,54% without taking into consideration transaction costs and 38,25% with transaction costs considered.

Table 6. The major bull and bear markets of ASE between 15/11/1996 and 5/6/2012.

Bull Markets	Bear Markets
15/11/1996 till 17/9/1999	20/9/1999 till 31/3/2003
01/4/2003 till 31/10/2007	1/11/2007 till 9/3/2009
09/03/2009 till 14/10/2009	15/10/2009 till 05/6/2012

Table 7. The return of the proposed system during the three ASE bull market periods.

	Performance
Bull Markets, ASE	B&H Strategy (BHS)	Proposed long-term fuzzy system (LTFS)	LTFS – BHS	(LTFSb-LTFSa)/months
15/11/1996–17/9/1999 (approx, 34 months)	613,74%	437,89% (a)	−175,85	−26,52%/34 = -0,78%
		411,37% (b)	−202,37
01/4/2003–31/10/2007 (approx. 54 months)	261,75%	108,98% (a)	−152,77	−11,15%/54 = -0,76%
		97,83% (b)	−163,92
09/03/2009–14/10/2009 (approx. 7 months)	97,15%	39,54% (a)	−57,61	−1,29%/7 = -0,18%
		38,25% (b)	−58,90

a = without considering transaction cost, b = with transaction cost being considered

The results in Table 7 also show that transaction costs have little effect on the results achieved by the proposed long-term fuzzy system and that they unevenly affect the return of the system. Thus, while for the first bull period the return of the system with transaction costs was roughly 6% lower than the return of the proposed system without transaction costs (411,37% compared with 437,89%), for the second bull market period it was 10% lower (97,83% compared with 108,98%), while for the third bull market it was roughly 3% lower than the return without transaction costs (38,25% compared with 39,54%). Probably, this is related to the number of transactions in each period, which is an indication of the frequency of the buy or sell signals produced by the proposed system (sensitivity of the long-term indices used). Since the bull market periods have different lengths (34, 54 and 7 months, respectively), transaction costs reduce the monthly performance of the system only by 0,78%, 0,76% and 0,18% (during the first, second and third bull market periods, respectively).

As far as bear markets are concerned, Table 8 shows that in the first bear market the ASE general index had a negative return of −76,83% which equals to the performance of the B&H strategy, while the return of the proposed system was −32,56% in the absence of transaction costs and −37,54% when transaction costs were considered. During the second bear market, the B&H strategy return was again negative (−72,09%), while the proposed system had a return of −15,36% without transaction costs and −17,70% when transaction costs were considered. Finally, the B&H strategy for the third and largest ASE bear market again produced negative results (−83,54%), while the proposed system had a negative performance of −31,57% without transaction costs and −35,48% with transaction costs considered. Transaction costs reduce the monthly performance of the system by 0,12%, 0,15% and 0,12% (during the first, second and third bear market periods, respectively). The main conclusion that can be drawn by looking at these results is that the proposed system outperforms by far the B&H strategy in all three ASE bear market periods, even when transaction costs are considered.

Table 8. The return of the proposed system during the three ASE bear markets periods.

	Performance
Bear Markets, ASE	B&Η Strategy (BHS)	Proposed long-term fuzzy system (LTFS)	LTFS – BHS	(LTFSb-LTFSa)/months
20/9/1999–31/3/2003 (approx. 42 months)	−76,83%	−32,56% (a)	+44,27	−5,01%/42 = -0,12%
		−37,54% (b)	+39,26
1/11/2007–9/3/2009 (approx. 16 months)	−72,09%	−15,36% (a)	+56,73	−2,34%/16 = -0,15%
		−17,70% (b)	+54,39
15/10/2009–5/6/2012 (approx. 32 months)	−83,54%	−31,57% (a)	+51,97	−3,91%/32 = -0,12%
		−35,48% (b)	+48,06

a = without considering transaction cost, b = with transaction cost being considered

In general, the returns of the B&H strategy are higher than the returns of the proposed system in bull markets, while the opposite is observed during bear markets. This conclusion is valid even when transaction costs are considered. Transaction costs are higher during the three bull market periods (0,78%, 0,76% and 0,18% of the monthly cost, respectively) than during the three bear market periods (0,12%, 0,15% and 0,12% of the monthly cost, respectively). These results indicate that, with the exception of the third bull market period, transaction costs can be considered to be much higher, namely 5–7 times, in bull market periods compared with bear market periods. This finding can be attributed to the frequency of the transactions (signals), which is usually higher during bull market periods.

Conclusions

The proposed long-term fuzzy system was tested for the ASE general index for a long period (over 15 years, between 15/11/1996 and 5/6/2012) producing a 612,81% return. After taking into consideration transaction costs, the total return was 400,36%, which is exceptionally higher than the −46.50% of the B&H strategy in the ASE general index for the same 15-year period and the 45,62% interest gain, which might have been earned during the same period, if the total amount of the initial portfolio (cash) were deposited in a safe account. It should be noted here that no interest gains were added to the amount of the portfolio remained in cash, which would have increased considerably the final performance.

The system has achieved this performance mostly by avoiding big losses during bear markets since during bull markets the gains made were significantly lower than those under the B&H strategy. Therefore, it can be considered as a conservative system, which manages to achieve very satisfactory returns, even though the testing period was over 15 years, a rather long period. The noticeable high performance of the proposed system when compared with the buy and hold strategy suggests that it can be used as a valuable investment tool in the real world of stock trading.

The significance of the paper is two-fold. First, it examines whether the use of a few only, carefully selected common technical indicators, combined with a well-structured fuzzy system, can create a generic model for long-term forecasting investment proposals for portfolio management. Contrary to the usual practice of using as many as possible various technical indicators, the model was designed from the beginning to use a set of only four, very common, but with different constructive philosophy, trend following indicators. Although the research conducted in a preliminary stage has shown that if only the three indicators with the highest initial performances (i.e. not the classic simple moving average of 200 days) had been used, the returns would have been much higher. However, it was decided to develop the model as it was initially designed, in order to investigate the possibility of creating such a generic robust model. Second, it implements the inoculation of an old and rigid trading strategy of technical analysis, with the fuzzy subjective elements inherent in the real world of trading. This is achieved through the transformation of the classic trading strategy with appropriately designed membership functions and fuzzy rules of the fuzzy system.

Further research could focus on the identification of the right combination of the optimum number of the appropriate indicators. Also, the construction of three complementary fuzzy systems with different time frames, e.g. short, medium and long-term and the combination of them in a platform of fuzzy systems, which would constitute a new fuzzy system, could be the target of future research. Finally, according to the investor’s profile, it should be interesting to place different weights to the fuzzy systems, which will act as inputs of the platform.