A bacterial foraging optimization approach for the index tracking problem

Transcription

1 A bacterial foraging optimization approach for the index tracking problem Hui Qu, Zixu Wang, Sunyu Xu Hui Qu(Corresponding Author) School of Management and Engineering, Nanjing University, , Nanjing, China Phone: ; Fax: Zixu Wang School of Management and Engineering, Nanjing University, , Nanjing, China Sunyu Xu School of Management and Engineering, Nanjing University, , Nanjing, China Abstract Index tracking is a popular passive portfolio management strategy which invests in a subset of stock market index constituents to reproduce the performance of the index. In this paper we apply a heuristic approach based on bacterial foraging optimization (BFO) to the index tracking problem. BFO is a swarm intelligence technique which mimics the foraging behavior of bacteria with three important movements, i.e., chemotaxis, reproduction, and elimination-dispersal. It has strong capability in both global search and local search, and thus has been successfully applied in several problems, but not yet in index tracking. To make our model more realistic, we consider transaction cost when revising the tracking portfolio, and include cardinality and bounding constraints. The BFO approach is compared with a genetic algorithm which is widely adopted as the benchmark in the index tracking problem. Empirical experiments with HangSeng, DAX, FTSE, S&P and Nikkei indexes indicate that, our BFO achieves significantly better overall objective function value under a wide range of tracking error/excess return tradeoffs. Furthermore, rolling window tests show that, our BFO leads to significantly smaller out-of-sample tracking error under a wide range of transaction cost constraints. Keywords bacterial foraging optimization; index tracking; excess return; tracking error; transaction cost

2 1 Introduction Index tracking is a popular passive portfolio management strategy. Fund managers invest in a subset of stock market index constituents to reproduce the performance of the index. Due to its competitive performance over the long run and relatively low cost, index tracking is now widely used, as can be seen from the number of Exchange Trade Funds (ETF) over the world. There have been a lot of works studying the index tracking problem. The majority of them consider the construction of an initial tracking portfolio without further revisions, and thus the transaction cost constraint is not considered. See Beasley et al. (2003) for a complete survey. Beasley et al. (2003) formulate the index tracking problem more realistically, considering transaction cost constraint, cardinality and bounding constraints. The resulting discontinuities and local optima in the search space inhibit the use of standard deterministic optimization tools and call for heuristic methods. They present a genetic algorithm for solving the index tracking problem and make the empirical data sets publicly available. Their data sets have been used in many papers on index tracking (Ruiz-Torrubiano and Suárez 2009; Li et al. 2011; Wang et al. 2012; Guastaroba and Speranza 2012; Li and Bao 2014; Li et al. 2014a, 2014b). Besides, their genetic algorithm has been frequently used as the benchmark to evaluate the tracking performance (Andriosopoulos et al. 2013; Li et al. 2014a; Andriosopoulos and Nomikos 2014). BFO is a swarm intelligence algorithm designed by Passsino (2002). It mimics the foraging behavior of bacteria with three important movements, i.e., chemotaxis, reproduction, and elimination-dispersal. It has strong capability in both global search and local search, and thus has been successfully applied in several problems, mainly in the engineering field. Its application in finance is limited to portfolio optimization problems so far; including portfolio optimization for the efficient frontier (Kao and Cheng 2013), dynamic portfolio optimization (Tan et al. 2014) and portfolio optimization with liquidity risk (Niu et al. 2010). Considering the strong search capability of BFO and the importance of index tracking, we propose a BFO-based algorithm for the index tracking problem in this paper. Empirical results with the public available data sets indicate that, the tracking performance of our BFO significantly outperforms that of the benchmark GA, both in-sample and out-of-sample. The rest of this paper is organized as follows. Section 2 formulates the index tracking problem. Section 3 introduces how BFO is applied to the index tracking problem. Section 4 reports the empirical results. Section 5 summarizes and concludes.

3 2 Problem Formulation Consider a stock market index consisting of N distinct stocks. We are interested in selecting K (K<N) stocks to make up the best tracking portfolio over the time period [T, T+L], as well as deciding the appropriate quantity of each stock. To achieve this goal, we first use the historical data to construct the best tracking portfolio over the time period [0, T]. Under the assumption that the past guides the future, when this best portfolio is used as the new tracking portfolio during [T, T+L], it will also perform well out-of-sample. Both the tracking error and the excess return are considered when evaluating the tracking portfolio. The transaction cost constraint is also considered to make our problem formulation more realistic. Notations: T : The decision time point. We use the historical data over [0, T] to decide the new tracking portfolio. C : The total value of the current portfolio at time T. I t : The value of the index at time t (t = 0,,T).. V it : The value of one unit of component stock i (i = 1,,N) at time t (t = 0,,T). r t : The single period continuous time return of the tracking portfolio at time t (t = 1,,T). R t : The single period continuous time return of the index at time t (t = 1,,T). γ: The limit on the proportion of C that can be spent on transaction0 1. X i : The number of units of stock i (i = 1,,N) in the current portfolio. x i : The number of units of stock i (i = 1,,N) in the new portfolio. z i : Indicator variable. z i equals 1 when stock i (i = 1,,N) is held in the new portfolio, 0 otherwise. y i : The holding level of stock i (i = 1,,N) in the new portfolio at time T, yi ViTxi C. ε i : The minimum holding level for stock i (i = 1,,N) if it is held (z i =1). δ i : The maximum holding level for stock i (i = 1,,N) if it is held (z i =1). 2.1 Tracking Objective The basic goal of index tracking is to minimize the difference between r t (the tracking portfolio return) and R t (the index return). We consider not only downside risk (index outperforms tracking portfolio) but also upside risk (tracking portfolio outperforms index) when calculating the tracking error, since the latter matters for short sellers. Following

4 Beasley et al. (2003), we use the root mean square error to measure tracking error: T 2 t t, (1) t1 Error r R T N N where the tracking portfolio return rt ln Vitxi Vi, t1xi i1 i1 Rt ln It It. 1, and the index return Some index trackers may want to gain higher investment return as a competitive advantage, at the cost of increasing tracking error and risk. We characterize such desire using excess return: Excess return= r R T T t t. (2) Therefore, the overall objective of our index tracking problem is: where 0 1 return. t1 Minimize Error 1 Excess return, (3) represents the tradeoff between smaller tracking error and higher excess 2.2 Transaction Cost Constraint In reality, investors need to pay commission and stamp tax when selling or buying stocks. The total transaction cost is positively correlated with the trading volume and the transaction cost rate. Therefore, if the number of units of stock i (i = 1,,N) in the tracking portfolio changes from a to b at time t, the transaction cost is calculated as: C a, b, t a b, (4) i where μ is the transaction cost rate. In our problem formulation, we assume transaction costs are paid out of a separate account, and require the total transaction costs to exceed γc. N C i( Xi, xi, T) not i1 2.3 Complete Problem Formulation Following Beasley et al. (2003), we use the holding level y i instead of the number of units x i in the complete problem formulation, since there is no integer constraint for the former:

5 T N N ln Vit yi ViT Vi, t1yi ViT ln ItIt1 T t1 i1 i1 Minimize (5) T N N 1 lnvit yi ViT Vi, t1yi ViT ln ItIt1 T, t1 i1 i1 Subject to: z 0,1, i 1,..., N (6) i N zi K, (7) i1 z y z, i 1,..., N, (8) i i i i i N yi 1, (9) i1 N 0 X i Cyi ViT C. (10) i1 2 3 Index Tracking Using the BFO Algorithm 3.1 The General BFO Algorithm The BFO algorithm is inspired by a kind of bacteria called Escherichia coli (E.coli ) bacteria which live in human intestines. Their whole process of searching for nutrition includes three main activities: chemotaxis, reproduction and elimination-dispersal. In a chemotaxis movement, a bacterium seeks for nutrient-rich regions with two motions, tumbling and swimming. Tumbling means a walk in a random direction. If this random walk leads it to a region with more nutrition, the bacterium continues to move several steps along this direction, i.e., swimming. Otherwise, it stays at the original position. After performing the chemotaxis movements (i.e., performing tumbling and swimming alternatively) for a certain amount of time, all the bacteria are sorted according to their fitness. The worst p percent bacteria are then replaced with the best p percent bacteria. Such reproduction activity can speed up the convergence rate to some extent. Furthermore, after several rounds of reproduction activities, each bacterium in the population is randomly chosen to be eliminated according to a preset probability. Meanwhile, new bacteria are produced at random places to keep the population size constant. Such elimination-dispersal activity can increase the global search ability and avoid getting stuck into local optima.

6 3.2 Application of BFO in the Index Tracking Problem This study develops a BFO-based algorithm to solve the index tracking problem defined in equations (5-10). In our algorithm, each bacterium represents a portfolio solution and thus is denoted as a linear vector b of size 2K (K is the size of the tracking portfolio). The first K elements, b(1 ik) in the vector indicate the selected stocks, and thus are integers ranging i from 1 to N (N is the total number of distinct stocks). The last K elements, bki(1 i K) indicate the proportion of total capital put into the stocks b(1 i K), i.e., the holding levels i yb i of the corresponding stocks. To satisfy the constraints in equations (8-9), each bki(1 i K) is a real number ranging from levels sum up to 1. b i to 1 (we assume b i =1), and all the holding Fig.1 Flow Chart of the BFO algorithm for index tracking Fig. 1 illustrates the flow chart of our BFO-based algorithm for the index tracking problem. Initially, a population of X artificial bacteria (solution vectors) is randomly generated. We perform an extra repair operation to make sure that the constraints in equations (8-10) are satisfied by each candidate solution 1. The fitness of each bacterium can be evaluated using the objective function in equation (5). Afterwards, our algorithm s optimization process consists of three main operations: chemotaxis, reproduction and elimination-dispersal, whose designs are explained below. Chemotaxis Operation There are two steps in a chemotaxis operation, tumbling and swimming. In the tumbling 1 The repair operation is performed on each newly generated artificial bacterium during the whole BFO process. It is not plotted to make the flow chart clearer.

7 step, for each bacterium b, a random direction vector ΔD of length K is generated and added to the last half of its vector to change the holding levels, which generates a new bacterium. If the objective function value of this new bacterium is smaller, we replace the old bacterium with the new one. The new bacterium then continues with repeated swimming steps, using the vector ΔD to adjust the holding levels in each step. If the objective function value decreases after a swimming step, the new bacterium replaces the current one. If the objection function value does not decrease after some swimming step, or the total number of swimming steps exceeds the preset upper bound, the chemotaxis operation terminates. On the other hand, if a tumbling step doesn t lead to a better objective function value, the chemotaxis operation terminates without further swimming steps, and the old bacterium remains unchanged. Let Δd i be the i th element of ΔD, newb K+i be the new value of b K+i, C tumble be the tumble step size, C swim be the swimming step size, the tumble and swimming step can be represented as: newb b C d (tumble step), (11) K i K i tumble i newb b C d (swimming step ). (12) K i Ki swim i Reproduction Operation After N C rounds of chemotaxis operations as described above, all the bacteria in the population are sorted in ascending order of their objective function values. The last 1/8 bacteria are then replaced with the first 1/8 bacteria, which completes a reproduction operation. Elimination-dispersal Operation For each artificial bacterium b in the population, its each holding level b K+i is set to -1 with the preset elimination-dispersal probability P ed. Since -1 is an illegal holding level, the corresponding stock b i will be replaced with some new stock in the subsequent repair operation. In this way, the old solution vector is eliminated, and the new solution vector is produced. Repair Operation The repair operation is performed whenever a new artificial bacterium is generated, i.e., after the initialization operation, after a tumbling step, after a swimming step, and after an elimination-dispersal operation. Specifically, in the latter three cases, we first identify all the

8 illegal b K+i (<0), and replace each corresponding stock b i with some stock not included in the current solution vector, as well as set b K+i to a random real number between 0 and 1. Then we perform a normalization step and a validation step which are common in all these four cases. In the normalization step, the K holding levels are first normalized to satisfy the constraint in equation (9), and then we set bk i bki( 1 b i ) b i to make sure that the constraint in equation (8) is satisfied. While in the validation step, each bacterium is tested using equation (10) to decide whether it can be kept in the population. 4 Empirical Results In the section, we apply our heuristic to real markets and compare its computational results to those of the benchmark GA introduced in Beasley et al. (2003). 4.1 Data Description We consider the tracking of five different capital market indexes: Hang Seng (Hong Kong), DAX 100 (Germany), FTSE 100 (UK), S&P 100 (USA) and Nikkei 225 (Japan). The data are downloaded from (We thank Beasley for making the data publicly available.), which are weekly prices from March 1992 to September Therefore, we have 291 prices for each index/stock, and the length of the corresponding return series is 290. The numbers of stocks (N) for the above indexes are 31, 85, 89, 98 and 225, respectively. Unless otherwise stated, we set the minimum holding level ε i = 0.01, the maximum holding level δ i = 1 and the transaction cost rate μ= 0.01 in the following empirical experiments. 4.2 In-sample Tracking Performance We first consider an investor who just enters the market. He does not have an initial portfolio and needs to create one using heuristic algorithms. Here we set T = 290 and compare the in-sample tracking performance of the benchmark GA and our BFO during the period [0, 290]. There is no transaction cost constraint at this stage. Different values of the error-return tradeoff parameter λ are considered (λ = 1, 0.8, 0.6) to reflect the utility of heterogeneous investors; as λ decreases, the investor puts more emphasis on excess return. Furthermore, the tracking portfolio size K is assumed to be 5 and 10 respectively, so as to evaluate the

9 robustness of our performance comparison results. Table 1 In-sample tracking performance of GA and BFO in terms of overall objective ( Error 1 Excess return ) Index Hang Seng DAX FTSE S&P Nikkei Number of stocks Objective function value Tracking portfolio size Tracking portfolio size K=5 K=10 BFO GA BFO GA E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-04 For each of the five indexes, we run the heuristic algorithms five times with the same set of parameters and the best solutions (minimal objective function values) are reported in table 1. The corresponding results obtained with GA and BFO are compared, and the smaller ones are in bold. When the tracking portfolio contains 5 stocks, our BFO outperforms the benchmark GA 10 out of 15 times. When the tracking portfolio contains 10 stocks, our BFO outperforms the benchmark GA 13 out of 15 times. Besides, most of the outperformance of the benchmark GA happens with the tradeoff parameter λ = 1; while the GA never outperforms our BFO when λ = 0.6. Such empirical results indicate that, our BFO has better in-sample tracking performance than the benchmark GA in general, and its advantage is more obvious when the investor puts more weight on excess return in the overall objective function. We also test the null hypothesis that the overall objective function value of BFO is not smaller than that of GA using the data in table 1. The null hypothesis is significantly rejected

10 with the p-value of when K equals 5, and significantly rejected with the p-value of when K equals 10, which indicates that the outperformance of our BFO is significant. 4.3 Out-of-sample Tracking Performance In reality, investors periodically revise their tracking portfolios with the change of the market. Therefore, we now consider an investor with an initial tracking portfolio who uses heuristic algorithms to revise the portfolio periodically. Naturally, the transaction cost constraint in equation (10) is considered at this stage. Specifically, assume the initial portfolio of size K = 10 is generated with the benchmark GA or our BFO using the historical data in [0,150] with no transaction cost constraint. The error-return tradeoff parameter λ is set to 1, which means the investor only cares about tracking error. Afterwards, the investor runs the same heuristic algorithm as used at the initial stage to revise his portfolio every 20 weeks, using the historical data in [0, w], w = 1,2,3,4,5,6, respectively. Different levels of transaction cost constraint (γ = 0, , 0.005, , 0.01) are considered at this revision stage. The corresponding out-of-sample tracking performance in [150+20w, w], w = 0,1,2,3,4,5,6, is summarized in Table 2. Table 2 Out-of-sample tracking performance of GA and BFO in terms of tracking error Index Number of stocks Hang Seng 31 Average tracking error Transaction GA BFO cost limit γ In-sample Out-of-sample In-sample Out-of-sample E E E E E E E E E E E E E E E E E E E E-04 DAX E E E E E E E E E E E E E E E E E E E E-03 FTSE E E E E E E E E E E E E-03

11 E E E E E E E E-03 S&P E E E E E E E E E E E E E E E E E E E E-03 Nikkei E E E E E E E E E E E E E E E E E E E E-03 Each "Average tracking error" in table 2 is the arithmetic average of the in-sample/out-of-sample tracking errors in 7 windows. Specifically, the in-sample results are calculated using the tracking errors in [0,150], [0,170], [0,190], [0,210], [0,230], [0,250] and [0, 270]; while the out-of-sample results are calculated using the tracking errors in [150,170], [170,190], [190,210], [210,230], [230,250], [250,270] and [270, 290]. For each of the five indexes, we run the heuristic algorithms five times with the same set of parameters in each window and take the best solution. The corresponding out-of-sample results obtained with GA and BFO are compared, and the smaller ones are in bold. We can see that our BFO outperforms the benchmark GA 22 out of 25 times, which indicates the competitiveness of our BFO in out-of-sample tracking. Furthermore, we test the null hypothesis that "the average out-of-sample tracking error of BFO is not smaller than that of GA" using the data in table 2. This null hypothesis is significantly rejected with the p-value of , which confirms that our BFO is significantly superior to the benchmark GA out-of-sample. It is worth mentioning that, when the transaction cost limit γ equals 0, the two heuristic algorithms cannot revise the initial portfolio. Thus what we get out-of-sample are the average tracking errors of the fixed initial tracking portfolios. These errors are relatively larger than their respective comparatives, indicating that periodically revising the tracking portfolio is meaningful. Furthermore, we calculate the pearson correlation between the in-sample and out-of-sample tracking errors based on the data in table 2, which are and for GA and BFO, respectively. Such high correlations between the in-sample and out-of-sample

12 tracking errors indicate that our basic assumption of the past guiding the future is reasonable. 5 Summary and Conclusion In this paper, we formulate an index tracking problem with transaction cost constraint, cardinality and bounding constraints. To solve this problem, we proposed a BFO-based heuristic algorithm which is composed of three main operations, i.e., chemotaxis, reproduction, and elimination-dispersal. Empirical results with five different capital market indexes show that, the tracking performance of our BFO algorithm significantly outperforms that of the benchmark GA, both in-sample and out-of-sample. Possible extensions of our work include applying the BFO-based algorithm to track different indexes, using genetic algorithm to optimize the parameters of our BFO algorithm, as well as evaluating the performance of the BFO algorithm with different index tracking problem formulations. We leave these for future work. Acknowledgments This work was supported by the National Natural Science Foundation of China ( ), the Natural Science Foundation of Jiangsu Province (BK ), and the Specialized Research Fund for the Doctoral Program of Higher Education of China ( ). References Andriosopoulos, K., Doumpos, M., Papapostolou, N. C., & Pouliasis, P. K. (2013). Portfolio optimization and index tracking for the shipping stock and freight markets using evolutionary algorithms. Transportation Research Part E: Logistics and Transportation Review, 52, Andriosopoulos, K., & Nomikos, N. (2014). Performance replication of the Spot Energy Index with optimal equity portfolio selection: Evidence from the UK, US and Brazilian markets. European Journal of Operational Research, 234(2), Beasley, J. E., Meade, N., & Chang, T. J. (2003). An evolutionary heuristic for the index tracking problem. European Journal of Operational Research,148(3), Guastaroba, G., & Speranza, M. G. (2012). Kernel search: an application to the index tracking problem. European Journal of Operational Research, 217(1), Kao, Y., & Cheng, H. (2013). Bacterial foraging optimization approach to portfolio optimization. Computational Economics, 42, 4, Li, Q., & Bao, L. (2014). Enhanced index tracking with multiple time-scale analysis. Economic Modelling, 39, Li, Q., Bao, L., & Zhang, Q. L. (2014a). Multi-scale tracking dynamics and optimal index replication. Applied

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