An intelligent trading mechanism based on the group trading strategy portfolio to reduce massive loss by the grouping genetic algorithm

Currently, many approaches have been proposed for the TS optimization, and they can be divided into two categories that are the TS optimization without and with SLTPs. For the TS optimization without SLTPs, many approaches have been proposed to solve the TS parameter optimization [14, 26, 34], incorporating TS in stock trading [1, 3, 20, 25, 33, 36], and the TSP optimization [4, 5, 32, 24].

In TS parameter optimization, Fu et al. proposed a genetic-based approach for determining the appropriate parameter settings of the selected technical indicators [14]. It generates TSs in accordance with the seven technical indicators. Then, the parameters of those TSs are encoded into a two-dimension array to represent a possible solution. The profit of a chromosome is used as a fitness function to identify appropriate parameters. Since optimization of TS parameters is time-consuming, Qin et al. presented a MapReduce-based algorithm to speed up the learning processing [34]. The presented approach consists of two MapReduce jobs. The first job is to generate the permutations of parameter combinations. For each combination, the second job is launched to calculate performance metrics of TSs. At last, the best parameter combination is determined according to the performance metrics. Lin et al. proposed a statistical learning method to find the most useful pair from multiple pair assets by combining both diversification and pair trading [29]. The results suggested the strategy can help investments be more diversified and profitable in stock trading.

To incorporate TS in stock trading, Chang et al. considered Markov decision process in GA to formulate TSs for stock markets [1], thus the trading signals are obtained by the Markov decision process. The GA is then employed to search the optimal stock selection strategy and capital allocation. Chien et al. proposed an GA-based approach to build an associative classifier. It can generate trading rules with the given numerical technical indicators [3]. In addition, Wu et al. proposed the adaptive stock trading strategies based on the deep reinforcement learning for trading. It first uses the gated recurrent unit to extract the features for making trading decisions. Two trading strategies with reinforcement learning methods are then presented as gated deep Q-learning trading strategy and gated deterministic policy gradient trading strategy to obtain the state-action table. Results showed that their approach can not only outperform the Turtle trading strategy but also has more stable returns [36]. Part et al. proposed an intelligent financial portfolio trading strategy using the deep Q-learning [33]. In their approach, the deep Q-learning is employed to train the intelligent agent and identify the optimal trading action. Ha et al. proposed an optimal intraday trading algorithm for reducing overall transaction costs when an online portfolio selection method rebalances the portfolio, and the results indicated the algorithm is significant to reduce the transaction costs when the liquidity is limited [20].

In TSP optimization, Chou et al. designed an algorithm to construct a rule-based dynamic trading system for stock [24]. In their approach, the technical indicators are used to generate TSs. An optimal combination of TSs is then obtained using the quantum-inspired Tabu search algorithm. The sliding window is also taken into consideration to avoid over-fitting and to achieve dynamic system. Chen et al. designed a combination genetic algorithm for building investment strategy portfolio [5]. It first uses ten technical indicators that every indicator has the selling and buying signals, and ten stocks to form a thousand security-rule pairs. A possible TSPs is then generated in terms of returns and encoded into a chromosome based on the top ten percent strategies. Return of a portfolio is employed as the evaluation function to score chromosomes. Nuij et al. designed a framework for automatic exploitation of news in stock trading strategies [32]. In that approach, events are first extracted from news. Considering the extracted events and technical indicators, the designed framework is used to find TSs using genetic programming, where the fitness of a trading strategy is evaluated by return calculation according to the given dataset. After several evolutions, they indicated that the news variable is often appeared in the optimized trading strategies, which means that the proposed framework is effective.

For the TS optimization with SLTPs, Kaminski et al. proposed an approach to analyze the stop-loss strategies [21]. Based on the three stop-loss cases, they investigate the results of the stop-loss policy. The cases are the regime-switching models, the mean reversion and momentum, and the random walk hypothesis. The analyzing results showed that the stopping premium which means the marginal impact of stop-loss rules on expected return of a given portfolio is always negative for the random walk hypothesis. However, in other two cases, stop-loss strategies can reach positive stopping premium. Lo et al. presented the closed-form expressions for the impact of stop-loss strategies on security returns that are serially correlated, regime switching, and subject to transaction costs [27]. They describe that tight stop-loss strategies could have worse return because of excessive trading costs when comparing to the BHS. Stop-loss outperformance is also possible for those assets that have high correlation in returns. Using GA, Leu et al. presented an algorithm for obtaining stock portfolio trading strategy with weighted fuzzy time series [25]. In first step, the stock portfolio is obtained using GA. The weighted fuzzy time series is then used to calculate the fitness value. The periodically checking and stop-loss point checking are used to decide trading signals of the stock portfolio.

The main purpose of the grouping problem is to divide instances into a predefined number of groups by considering criteria that not only in groups but also between groups. Give a set of instance INS = {ins₁, ins₂, …, ins_n}, the grouping problem is defined as following:where G_i refers to the group G_i. Since it is time-consuming to obtain a grouping result with the criteria, the genetic algorithms (GA) which is one of evolutionary-based algorithms can be employed to handle it. Note that the merits of GA is that it can handles a variety of optimization problems effectively [16, 17, 19]. Based on the GA, the grouping genetic algorithm (GGA) was designed to solve various grouping problems and indicated that the performance of GGA is better than simple GA [12]13.

In the following, the details of GGA are stated. In encoding schema, the grouping and instance parts are used to present a solution. For example, a chromosome is given as follows:

AAABC: ABC.

In the chromosome, the string "AAABC" is the instance part before the colon, which refer to the five instances that are ins₁, ins₂, ins₃, ins₄ and ins₅. The string "ABC" is the grouping part after the colon, which indicates that instances in instance part can only belong to the three groups either A, B, or C. Therefore, the chromosome indicates that five instances are divided into three groups. The instances ins₁, ins₂, and ins₃ belong to group A. The instance o₄ belongs to group B, and the object o₅ belongs to group C.

As to genetic operators, the GGA has three operators that are crossover, mutation and inversion [13]. For the crossover operator, it switches groups in GGA instead of exchanging genes in GA. For mutation operator, it moves an instance from a group to another group. For the third operator, inversion, it changes the order of the groups in chromosome, and the goal of this operator is to increase the probability of crossover operator to get more diverse chromosomes. Literature also showed that the GGA can be efficiently used to handle stock portfolio optimization problem [6‐8].

The motivation of this paper is to proposed an algorithm that can be utilized to obtain a GTSP and the suitable SLTPs. An example used to describe the motivation is shown in Fig. 1.

From Fig. 1, a GTSP and the SLTPs are then considered in the designed approach to reduce loss and increase profit are derived based on the given set of trading strategies. In this example, the obtained GTSP contains four trading strategy groups. The first group has three trading strategies that are TS₁, TS₃ and TS₈. For other three groups, they have 5, 4 and 3 trading strategies. Hence, totally, 180 trading strategy portfolios can be provided by the GTSP to user. Assume the TSP₁ is suggested, when the user dislikes the trading strategy TS₁, it can be replaced by another trading strategy TS₃ from the same group. In addition, the suggested SLTPs can be utilized to prevent massive loss. To reach the mentioned goal, in other words, many factors should be considered, e.g., trading strategy groups, weight of groups, the SLTPs, and return, among others. To clarify that, the problem definitions are stated as follows.

In the following, six definitions are used for problem definition of the designed model.

While utilizing optimization approach for solving a problem, the design of the chromosome representation or encoding schema always be the first task that should be considered because it seriously influences the final results. In this paper, the aim is to obtain a GTSP and its SLTP. Thus, the SLTP, grouping, weight, and strategy parts are employed to represent a GTSP and its SLTP. The chromosome representation is shown in Fig. 2.

Figure 2 shows that the SLTP part is composed of bit strings to indicate the stop-loss and the take-profit thresholds. They are represented by n and b bits, respectively. In accordance with the SLTP represented in a chromosome, the trading signals, including selling and buying, of each TS can be located. When the return rate is larger than take-profit point (TPP) or less than stop-loss point (SLP), the asset will be sold. On the contrary, when the return rate is between TPP and SLP, the asset will be held. Note that the return rate is the different between selling and buying prices divides by the buying price. The grouping part indicates that the number of TSGs in a GTSP. The TS part reveals what TSs are included in the TSGs. As to the weight part, each c_j is used to indicate the allocated capital ratio of the j-th TSG and c₀ indicates the reserved capital, where a '1′ string is utilized to represent a c_j, and the symbol '0′ is used to separate the two adjacent weight strings. Using the encoding schema, chromosomes can be generated to formed the initial population for the evolution process.

An example is given as follows to show the encoded chromosome used in the designed algorithm. When the bound of the TPP and SLP are respectively set as 15% and − 15%, it indicates the ranges of them are in [0, 15%] and [0, − 15%]. Assume a TPP and SLP are represented by four bits, number of TSs is fifteen and the number of groups is four, the chromosome can be encoded in Fig. 3.

Figure 3 shows that the TPP is 8% and SLP is − 3% according to the bit strings "0100" and "0011". There are four TSGs that are G₁, G₂, G₃ and G₄. The group G₁ has five trading strategies, TS₄, TS₅, TS₆, TS₇ and TS₁₄. Since number of '1′ in c₀ is 1 and total number of '1′ is five in the weight part, it indicates the reserved capital is 20% (= 1/5). In the same way, we can observe that number of '1′ in c₁ to c₄ are 0, 3, 1 and 0. The weights for G₁ to G₄ are 0%, 60%, 20% and 0%. Utilizing the chromosome, 10 (= 2 × 5) TSPs can be generated and suggested to users due to weights of G₁ and G₄ are zero.

To describe how to calculate the return of a TS using the encoded SLTP, trading signals generated by a TS and stock price series in Table 2 with the 8% and -3% as SLTP are employed to explain the process. Note that '1′ and '0′ respectively represent the buying and selling signals.

Table 2 shows that the first buying and selling signals appear on 2/5 and 2/10. Return rate for purchasing the stock is 6% (= (217–203)/203). In this case, the return rate is smaller than 8%, the asset will be held waiting next selling signal. Since the next selling signal appears on 2/12, the return rate is calculated as 11% (= (224.5–203) / 203). In this situation, the asset will be sold because the return rate is larger than the TPP which is 8%.

It is a critical task to obtain a good GTSP and its SLTP by designing a proper fitness function. In other words, many factors should be considered in the design process of a fitness function. To identify a qualified GTSPs and is SLTP, two types of criteria should be considered. The first type is used to evaluate the profitable ability of a GTSP, and the second type is used to evaluate the structure of TSGs. The two types of factors are stated as follows: (1) For the profitable ability of a GTSP, it consists of the two sub-factors that are the return and risk should be maximized and minimized, and are used as the part of evaluation criteria; (2) For the structure of TSGs, the group balance and weight balance are taken into consideration for evaluating the balance degree of TSGs in terms of number of TSs and allocated capital. As a result, the fitness function which is composed of four factors to evaluate a chromosome is given in Formula (1).where PReturn(C_q) and PRisk(C_q) are the return and risk of a GTSP, GB(C_q) and WB(C_q) are the group balance and weight balance of TSGs, and α is a parameter to indicate the impact of group balance. The four criteria are stated as follows. The return of a GTSP in a chromosome is shown in Formula (2).where return(TSP_j) is the return of j-th TSP, and nTSP is the number of TSPs generated from C_q. Note that the higher return of a GTSP is, the better GTSP is obtained. The return(TSP_j) is stated in Formula (3).where avgRRate(TS_i^j) and weight_i are the average return rate of the TS in group G_i of TSP_j and the weight of G_i, and allocatedCap means the allocated capital of the TS for trading. The avgRRate(TS_i ^j) is stated in Formula (4).where frequency_i is the number of transactions during the given trading period, and RRate(TS_ih^j) is defined in Formula (5).where the sellPrice_h and buyPrice_h are selling and buying prices of the h-th transaction using the i-th TS in TSP_j. Note that the buyPrice_h is determined by the trading signal generated using the trading strategy, and the sellPrice_h is determined by the SLTP part in the chromosome. For example, let the bit strings of the SLTP part are "0100" and "0100", the TPP and SLP are 8% and -8%. Then, the TPP and SLP will be used to determine selling price. The PRisk(C_q) is defined in Formula (6).where risk(TSP_j) and nTSP are the risk of a TSP and number of TSPs generated from C_q. The risk(TSP_j) is shown in Formula (7).where MDD(TS_i^j) and K are the maximum draw down (MDD) of the i-th TS of TSP_j and number of groups. It indicates that the risk of a TSP is calculated by the minimum MDD of the TSs in the portfolio. Note that the MDD of a TS is normalized to 0 to 1. The MDD(TS_i^j) is defined in Formula (8).where RRate(TS_ih^j) is given in in Formula (5), and frequency_i is number of transactions using the i-th TS in TSP_j during the given trading period. The group balance of TSGs in a GTSP is given in Formula (9).where |G_i| and N are number of TSs in G_i and the number of the given TSs. The main purpose of group balance is to make the number of TSs in groups as the same as possible. The fourth factor is shown in Formula (10).where |c_i| and TL are the length of the string c_i and the total length of all strings c_i, 0 ≤ i ≤ K. The main purpose of weight balance is utilized for avoiding allocated capital at certain groups. Using the evaluation function, the fitness value of a possible solution can be calculated. According to the selection strategy, the next population will be generated, e.g., the elitist selection, the roulette wheel selection.

This section describes the three genetic operators used in the GTSP-SLTP algorithm. The first operator is crossover which is executed on only the SLTP and weight parts. The one-point crossover and two-point crossover operators are applied on the SLTP and the weight parts to generate new offspring, respectively. Because applying crossover on the weight part may disrupt the number of “0” and “1” in a chromosome, the suitable arrangement should be done to correct them.

As to mutation operators, they are executed on the SLTP, TS and weight parts. To perform mutation on the SLTP part, one gene is randomly chosen for mutation. If the gene value is 0, it will be changed to 1; otherwise, it will be changed to 0. To perform mutation on the TS part, it will select and move a TS from a TSG to another TSG. For mutation on the weight part, a “0″ and a”1″ genes will be selected for exchanging. The third operator, the inversion, is performed on the grouping part. Due to the aim of this operator is to increase the diversity of a chromosome when executing crossover operator, it only exchanges the order of two random selected TSGs.

Based on the mentioned definitions above, in this paper, we propose an optimization algorithm, namely GTSP-SLTP algorithm, to solve the GTSP-SLTPO problem. The flowchart of the GTSP-SLTP algorithm is illustrated in Fig. 4.

Figure 4 shows that in accordance with the selected technical indicators and the stock price series, the initial population is first generated. Every chromosome means a potential GTSP and its SLTPs. The four parts of a chromosome are generated as follows. In the SLTP part, two randomly generated bit strings are used to represent the stop-loss threshold and take-profit threshold. Then, K groups are initialized for the grouping part. The TSs in groups are generated using m candidate TSs that are generated by the candidate TS generation procedure which will be described in Fig. 5. The weight part is represented by a randomly generated bit string. In the chromosome evaluation, four factors that are the portfolio return, the risk of portfolio, the group balance and the weight balance of groups are employed to calculate the fitness value of a chromosome. The four factors can be calculated by Formulas (2), (6), (9) and (10). Three genetic operators, including crossover, mutation and inversion, are executed on the population to generate new chromosomes. Finally, the obtained GTSP with its SLTPs is provided to investors. Below, Fig. 5 shows how the candidate TSs generation procedure work.

In the Fig. 5, it shows the processed m TSs are formed based on the given the stock prices series, technical indicators, and the SLTP part in a chromosome. The process consists of four phases: (1) It first forms candidate TSs by the selected technical indicators; (2) Then, according to the given stock price series and candidate TSs, the selling and buying signals are identified. Using the SLTP part, the generated trading signals will be relocated. Take the take-profit point 5% as an example. Although the TS generates a selling signal on 2014/2/19, the stock will still be held because the cumulative return is 2% which is smaller than the threshold. Thus, that selling signal is removed. Since the next selling signal will be generated on 2014/03/11 and its cumulative return is 8%, the stock will be sold and the new selling signal is added; (3) After relocating the trading signals, the ranking functions that are the average return, trading frequency and maximum draw down (MDD) are used to calculate scores of the candidate TSs. When using the trading strategy for trading, the MDD can be used to evaluate its risk degree. Given a set of transactions with returns, the MDD means the one causes the highest loss. Hence, if the MDD value is larger than 0, it indicates that the used trading strategy is better than that smaller than 0. Hence, in the candidate TSs generation procedure, the MDD is employed as a ranking function to determine the m trading strategies; (4) Finally, using the preferred ranking function selection strategy, m TSs are selected to form the trading strategy part.

To state the proposed approach clearly in this section, the pseudo code of the GTSO-SLTP algorithm is given in Fig. 6.

Figure 6 shows the process of the GTSP-SLTP algorithm to obtain a GTSP and its SLTP using the GGA in accordance with the given stock price series sp and the technical indicators nTech. From lines 2 to 9, the initial population is first generated. From lines 11 to 18, based on the designed fitness function, every chromosome is evaluated by return and risk of portfolio, and group and weight balances. From lines 19 to 23, the genetic operators such as the crossover, mutation, inversion and selection operators, are utilized to generate new chromosomes. Finally, while reaching the predefined number of iterations, the chromosome with the highest fitness value will be produced as the optimized GTSP and the SLTP at line 25. The pseudo code of the TS procedure is given in Fig. 7.

Figure 7 indicates how the m TSs to be generated based on the selected technical indicators, stock price series and SLTP part of a chromosome. Firstly, the combinations of the selected technical indicators are generated to from the candidate TSs at line 2. Then, according to the candidate TSs and SLTPs, the trading signals can be identified from the given stock price series at line 3. Then, the average return, trading frequency and maximum draw down are used as the ranking functions to score TSs at lines 4 to 9. At last, the m TSs are selected based on the scores at lines 10 to 11.

To illustrate the GTSP-SLTP algorithm, an example is provided in this section with eight steps.

Sub-step 1.1 Eight bits are generated to represent SLTP part for each chromosome randomly. For instance, let the SLTP part of a chromosome C₁ is generated as "11,110,111". The first four bits mean the value of TPP and the followed four bits represent the value of SLP. Assume that the bounds of the TPP and SLP are 15% and -15%, according to the string "11,110,111", it means that the values of TPP and SLP are 15% (= 0.15 / 15 × 15) and -7% (= 0.15 / 15 × 7), respectively.

Sub-step 1.2 For each chromosome, the TS procedure is employed to generate 15 TSs according to its SLTP part, the stock price series and technical indicators. In this example, assume that the generated fifteen TSs of C₁ and their related data are given in Table 3.

In Table 3, the first attribute is the average return rate which is calculated using Formula (4). During the given trading period, the second attribute (MDD) represents the maximum loss of a TS and it can be calculated by the Formula (8).

Sub-step 1.3 Because the number of groups is three, the fifteen TSs are randomly divided into three groups to generate the grouping and strategy parts. For instance, the grouping and strategy parts of C₁ could be [G₁: {6, 8, 9, 12}, G₂: {4, 11}, G₃: {0, 1, 2, 3, 5, 7, 10, 13, 14}].

Sub-step 1.4 Assume that number of bits used to represent the weight part is 100, a weight part of C₁ could be generated as [1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1]. The length is 103 (= 100 + 3) since the number of groups is 3. In other words, it indicates that weights of reserved capital and three groups are 0.13, 0.12, 0.72, 0.03. Note that for the sake of being concise, we will directly use real numbers transformed from the bit string in the following steps. Combining the four parts, the chromosome C₁ is represented as C₁: "11,110,111", [{6, 8, 9, 12}, {4, 11}, {0, 1, 2, 3, 5, 7, 10, 13, 14}], 0.13, 0.12, 0.72, 0.03.

Sub-step 1.5 Repeating the sub-steps 1.1 to 1.5 and since the population size is set as 10, the initial population are generated and shown as follows:

Sub-step 2.1 The TSPs are first generated. Take chromosome C₁ as an example. In accordance with the grouping part [G₁: {6, 8, 9, 12}, G₂: {4, 11}, G₃: {0, 1, 2, 3, 5, 7, 10, 13, 14}], it generates 72 (= 4 × 2 × 9) possible TSPs. All of them are collected in a set tspSet = {tsp₁, tsp₂, …, tsp₇₂} = {{6, 4, 0}, {6, 4, 1}, {6, 4, 2}, …, {12, 11, 14}}.

Sub-step 2.2.1 Return of every TSP in the set tspSet is calculated. Take tsp₁: {6, 4, 0} of the chromosome C₁ as an example. In accordance with the weight part: [0.13, 0.12, 0.72, 0.03], let the investment capital is 100,000, the return of tsp₁ is -3237 (= [-0.052 × (100,000 × 0.12) + -0.048 × (100,000 × 0.72) + 0.281 × (100,000 × 0.03)]). In the same way, returns of remaining TSPs can be calculated.

Sub-step 2.2.2 After the previous subsetp, average return of TSPs is set as the return of a chromosome by Formula (2). The results of the ten chromosomes are given in Table 4.

Sub-step 2.3.1 The MDD of every TS is normalized firstly and the results are shown in Table 5.

Sub-step 2.3.2 The risk of each TSP tsp_j in the set tspSet is calculated. Take tsp₁: {6, 4, 0} as an example. The risk of tsp₁ is 0.148 (= min(0.148, 0.155, 1)) by Formula (7). After risk values of other TSPs are calculated, the risk of a chromosome can be set according to Formula (6). The results of all chromosomes are given in Table 6.

Sub-step 2.4 Based on the grouping part, the group balance of every chromosome is evaluated. Since the grouping part of C₁ is [G₁: {6, 8, 9, 12}, G₂: {4, 11}, G₃: {0, 1, 2, 3, 5, 7, 10, 13, 14}], the GB(C₁) is calculated as 0.860. In the same way, the group balance scores of the ten chromosomes are given in Table 7.

Sub-step 2.5 Based on the weight part of a chromosome, the weight balance score is calculated. Take C₁ as an example. Since the weight part of C₁ is [0.13, 0.12, 0.72, 0.03] using Formula (10), the weight balance of chromosome C₁ is 0.742. The results of ten chromosomes are given in Table 8.

Sub-step 2.6 Let α is 2, the fitness value of C₁ can be set at -359.744 (= − 5161.653 × 0.127 × 0.860² × 0.742) according to Formula (1). The fitness values of the ten chromosomes are shown in Table 9.

Step 4 The crossover operator is executed to generate new chromosomes. Take chromosomes C₆ and C₉ as an example. Because the SLTP parts of C₆ and C₉ are "01011101" and "01100100", assume that the cutoff point is 3, after crossover, the SLTP parts of the new chromosomes are "01000100" and "01111101". For crossover on the weight part, take chromosomes C₃ and C₇ as an example, their weight parts are shown as follows:

[1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1]

and

[0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1].

Assume that the cutoff points are respectively set as 3 and 74, the newly formed weight parts after transforming to real numbers are C₃′: [0.42, 0.31, 0.21, 0.06] and C₇′: [0.0, 0.42, 0.32, 0.26].

Step 5 The mutation operator is executed on the population to generate new chromosomes. Take chromosome C₃ as an example. For mutation on the SLTP part, let the mutation point is three, the SLTP part of C₃ changes from “11110111” to "11010111". For mutation on the TS part, assume the TS₄ in G₁ is selected and moved to G₂, the TS part of C₃ changes from [G₁: {4, 9, 12}, G₂: {0, 1, 3, 7, 11}, G₃: {2, 5, 6, 8, 10, 13, 14}] to [G₁: {9, 12}, G₂: {0, 1, 3, 4, 7, 11}, G₃: {2, 5, 6, 8, 10, 13, 14}]. For mutation on the weight part, assume two genes, the 5-th and 44-th genes, are picked to exchange, the weight part of C₃ changes from [0.43, 0.3, 0.21, 0.06] to [0.04, 0.69, 0.21, 0.06].

Step 6 The inversion operator is performed. Take chromosome C₅ as an example. Because the grouping part of C₅ is [G₁: {2, 3, 11, 12, 13, 14}, G₂: {7, 8}, G₃: {0, 1, 4, 5, 6, 9, 10}], let G₂ and G₃ are selected for exchanging, it changes to [G₁: {2, 3, 11, 12, 13, 14}, G₂: {0, 1, 4, 5, 6, 9, 10}, G₃: {7, 8}].

Step 8 The chromosome with the highest fitness value is prodcued as the optimized GTSP and its SLTP. In this example, after 100 iterations, the final best chromosome C_best is ["10000100", G₁: {1, 3, 8, 9, 10, 12, 14}, G₂: {2, 4, 11, 13}, G₃: {0, 5, 6, 7}, 0.16, 0.15, 0.16, 0.53]. The C_best indicates that the fifteen TSs are divided into three groups. The TPP is 8% and the SLP is − 4%. Group G₁ contains TS₁, TS₃, TS₈, TS₉, TS₁₀, TS₁₂ and TS₁₄. Group G₂ composes of TS₂, TS₄, TS₁₁ and TS₁₃. Group G₃ has TS₀, TS₅, TS₆, and TS₇. The weight part represents that allocated capital for groups. Based on the obtained GTSP, there are 112 TSPs (= 7 × 4 × 4) can be provided to users making investment plans.

The three datasets with different stock trends used for experimental evaluations including uptrend, sideway trend and downtrend are described in this section. The time period from 2011/01 to 2016/12 of the stock prices were collected. The stock price series of them are illustrated in Figs. 8, 9 and 10.

In Fig. 8, we can observe that the stock prices are between 50 and 200. When using the buy and hold trading strategy (BHS), the returns are calculated as 29%, 8%, 32%, 1% and 27% for 2012, 2013, 2014, 2015 and 2016. The stock price of the sideway trend dataset in Fig. 9 is between 100 and 400. Again, when using BHS, the returns of the years from 2012 to 2016 are 32%, 25%, 10% and -21%. The downtrend stock price series showed in Fig. 10, the highest and lowest values are around 1500 and 100. The returns can be calculated using the BHS as − 40%, − 53%, − 1%, − 45% and − 2% for 2012, 2013, 2014, 2015 and 2016. To generate candidate TSs, the used ten technical indicators are selected based on the literatures [5]14, and are shown in Table 11.

In accordance with the parameter setting used in [22], the generated trading rules with appropriate parameters for finding trading signals, including selling and buying signals, are given in Table 12.

Table 12 lists twenty rules, and ten of them are selling rules and others are buying rules. Then, a TS can be formed by selecting a buying rule and a selling rule. Take S₁ and B₁ as an example. The generated TS is “Buying signal: MA5 cross over MA20; Selling signal: MA5 cross down MA20”. In the same way, totally 100 TSs can be generated. Based on the candidate TSs generation procedure, top-m TSs will be then selected for obtaining a GTSP and its SLTP according to the used ranking function.

Experiments were made on the three kinds of trends to show the effectiveness of the GTSP-SLTP algorithm. The three datasets are stock prices series with uptrend, sideway trend, and downtrend. The results for the three datasets are shown and discussed in Sects. 6.2.1, 6.2.2 and 6.2.3. To form needed TSs for generating initial population, two different ranking function selection strategies are adopted in the following experiments. The first selection strategy uses only average return rate as ranking function to select top-15 TSs out of 100 candidate TSs, and named TOP15R. The second selection strategy uses the three ranking functions to select the 15 TSs, and named TOP5R5F5D. In other words, it used the average return rate as the first ranking function to select top-5 TSs, the trading frequent as the second ranking function to select another top-5 TSs, and maximum draw down as the third ranking function to select last top-5 TSs. Then, the TSs generated using the TOP15R and TOP5R5F5D are utilized to find the GTSPs and its SLTPs based on the given training and testing datasets.

To show the merits of the GTSP-SLTP algorithm, we applied it on the group stock portfolio (GSP) which is a type of stock portfolio that can be generated by the algorithm presented in [6]. Since a GSP is composed of stock groups, many stock portfolios can be generated. This case study is conducted on a real financial dataset with 31 companies that are collected from 2010/01 to 2012/12. We first used 2 years dataset 2010–2011 as the training period to generate ten GSPs, and used one-year dataset 2012 as the testing period for the comparison. The GTSP-SLTP algorithm with TOP15R and TOP5R5F5D has then been employed to obtain suitable GTSP and SLTP for each company. The comparison results of the generated GSPs with BHS and the GTSP-SLTP algorithm with TOP15R and TOP5R5F5D in terms of MiR are shown in Table 19.

Table 19 shows that the average MiR value of the GSP with BHS and with GTSP-SLTP with TOP15R and TOP5R5F5D are 26.98%, 12.28% and 11.82% that indicate the GSP with BHS is better than the GSP with the GTSP-SLTP. However, we also observe that the MiR values of the GSP with the GTSP-SLTP are always positive which means that using the GTSP-SLTP algorithm has a higher ability to avoid risk than using the BHS. To verify whether the GSP with the GTSP-SLTP algorithm can reach more stable returns clearly, comparison results of the GSP with BHS and with GTSP-SLTP in terms of variance of returns are shown in Table 20.

From Table 20, we can easily see that the average values of variance of returns of GSP with BHS and with the GTSP-SLTP using TOP15R and TOP5R5F5D are 3.63%, 0.061% and 0.57%, respectively. These results reveal that the GSP with the GTSP-SLTP can actually provide more robust returns than that with BHS. In other words, by using the proposed algorithm, the advantage is that the appropriate trading signals generated by various trading strategies can be suggested for reaching a better trading performance. Through this case study, we can conclude that the GTSP-SLTP algorithm can provide a more safety way for users making investment plans.