Combining Argumentation and Hybrid Evolutionary Systems in a Portfolio Construction Application Nikolaos Spanoudakis1 and Konstantina Pendaraki2 and Grigorios Beligiannis2 Abstract. In this paper we present an application for the accurately predicting the evolution of stock values in the Greek construction of mutual fund portfolios. It is based on a market (its application on economic data is presented in [2]). combination of Intelligent Methods, namely an argumentation Moreover, there is a lot of work on hybrid evolutionary algorithms based decision making framework and a forecasting algorithm and their application on many difficult problems has shown very combining Genetic Algorithms (GA), MultiModel Partitioning promising results [4]. The problem of predicting the behavior of (MMP) theory and Extended Kalman Filters (EKF). The the financial market is an open problem and many solutions have argumentation framework is employed in order to develop mutual been proposed. However, there isn't any known algorithm able to funds performance models and to select a small set of mutual identify effectively all kinds of behaviors. Also, many traditional funds, which will compose the final portfolio. The forecasting methods have been applied to the same problem and the results algorithm is employed in order to forecast the market status obtained were not very satisfactory. There are two main (inflating or deflating) for the next investment period. The difficulties in this problem, firstly the search space is huge and, knowledge engineering approach and application development secondly, it comprises of many local optima. steps are also discussed.12 In this contribution, we present the whole application resulting from the combination of argumentation with hybrid evolutionary systems along with the respective results. 1 INTRODUCTION The rest of the paper is organized as follows: Section two presents an overview of the concepts and application domain Portfolio management [8] is concerned with constructing a knowledge. Section three outlines the main features of the portfolio of securities (e.g., stock, bonds, mutual funds [13], etc.) proposed argumentation based decision-making framework and that maximizes the investor’s utility. In a previous study [14], we the developed argumentation theory. The forecasting hybrid constructed mutual fund (MF) portfolios using an argumentation evolutionary system is presented in section four, followed by based decision making framework. We developed rules that section five, which presents the developed application and characterize the market and different investor types policies using discusses the obtained empirical results. Finally, section six evaluation criteria of fund performance and risk. We also defined summarizes the main findings of this research. strategies for resolving conflicts over these rules. Furthermore, the developed application can be used for a set of different investment policy scenarios and supports the investor/portfolio manager in 2 DOMAIN KNOWLEDGE composing efficient MF portfolios that meet his investment preferences. The traditional portfolio theories ([8], [11], [12]) This section describes the criteria (or variables) used for were based on unidimensional approaches that did not fit to the creating portfolios and the knowledge on how to use these criteria multidimensional nature of risk ([3]), and they did not capture the in order to construct a portfolio. complexity presented in the data set. In [14], this troublesome The data used in this study is provided from the Association of situation was resolved by the high level of adaptability in the Greek Institutional Investors and consists of daily data of domestic decisions of the portfolio manager or investor when his equity mutual funds (MFs) over the period January 2000 to environment is changing and the characteristics of the funds are December 2005. multidimensional that was demonstrated by the use of The proposed framework is based on five fundamental argumentation. variables. The return of the funds is the actual value of return of Our study showed that when taking into account the market an investment defined by the difference between the nominal context, the results were better if we could forecast the status of return and the rate of inflation. This variable is based on the net the market of the following investment period. In order to achieve price of a fund. At this point, it is very important to mention that this goal we employed a hybrid system that combines Genetic transaction costs such as management commission are included in Algorithms (GA), MultiModel Partitioning (MMP) theory and the the net price. Frond-end commission and redemption commission Extended Kalman Filter (EKF). A general description of this fluctuate depending on the MF class and in most cases are very algorithm and its application in linear and non-linear data is low. The standard deviation is used to measure the variability of discussed in [2], while the specific version used in this the fund’s daily returns, thus representing the total risk of the contribution is presented in [1], where its successful application to fund. The beta coefficient (β) is a measure of fund’s risk in non-linear data is also presented. This algorithm captured our relation to the capital risk. The Sharpe index [13] is a useful attention because it had been successfully used in the past for measure of performance and is used to measure the expected return of a fund per unit of risk, defined by the standard deviation. 1 The Treynor index [15] is similar to the Sharpe index except that Technical University of Crete, Greece, email: nikos@science.tuc.gr 2 University of Ioannina, Greece, email: {dpendara, gbeligia}@cc.uoi.gr 59 performance is measured as the risk premium per unit of communication. A single agent may use argumentation techniques systematic (beta coefficient) and not of total risk. to perform its individual reasoning because it needs to make On the basis of the argumentation framework for the selection decisions under complex preferences policies, in a highly dynamic of a small set of MF, which will compose the final multi- environment (see e.g. [6]). This is the case used in this research. portfolios, the examined funds are clustered in three groups for In the following paragraphs we describe the theoretical framework each criterion for each year. For example, we have funds with that we adopted: high, medium and low performance (return), the same for the Definition 1. A theory is a pair (T, P) whose sentences are other criteria. formulae in the background monotonic logic (L, ⊢ ) of the form The aforementioned performance and risk variables visualize L←L1,…,Ln, where L, L1, …, Ln are positive or negative ground the characteristics of the capital market (bull or bear) and the type literals. For rules in P the head L refers to an (irreflexive) higher of the investor according to his investment policy (aggressive or priority relation, i.e. L has the general form L = h_p(rule1, rule2). moderate). Further information is represented through variables The derivability relation, ⊢ , of the background logic is given by that describe the general conditions of the market and the investor the simple inference rule of modus ponens. policy (selection of portfolios with high performance per unit of An argument for a literal L in a theory (T, P) is any subset, T, risk). of this theory that derives L, T ⊢ L, under the background logic. A The general conditions of the market are characterized through part of the theory T0 ⊂ T, is the background theory that is the development of funds which have high performance levels considered as a non defeasible part (the indisputable facts). (high return). Regarding the market context, in a bull market, An argument attacks (or is a counter argument to) another funds are selected if they have high systematic or total risk. On the when they derive a contrary conclusion. These are conflicting other hand, in a bear market, we select funds with low systematic arguments. A conflicting argument (from T) is admissible if it and total risk. An aggressive investor is placing his capital upon counter-attacks all the arguments that attack it. It counter-attacks funds with high performance and high systematic risk. an argument if it takes along priority arguments (from P) and Accordingly, a moderate investor selects funds with high performance and low or medium systematic risk. Some types of makes itself at least as strong as the counter-argument (we omit investors select portfolios with high performance per unit of risk. the relevant definitions from [6] due to limited space). Such portfolios are characterized by high Sharpe ratio and high Definition 2. An agent’s argumentative policy theory is a Treynor ratio. theory T = ((T, T0), PR, PC) where T contains the argument rules in the form of definite Horn logic rules, PR contains priority rules which are also definite Horn rules with head h_p(r1, r2) s.t. r1, r2 3 ARGUMENTATION-BASED DECISION ∈ T and all rules in PC are also priority rules with head h_p(R1, MAKING R2) s.t. R1, R2 ∈ PR ∪ PC. T0 contains auxiliary rules of the agent’s background knowledge. In this section we firstly present the argumentation framework that Thus, in defining the decision maker’s theory we specify three we used and then we describe the domain knowledge modeling levels. The first level (T) defines the (background theory) rules based on the argumentation framework. that refer directly to the subject domain, called the Object-level Decision Rules. In the second level we have the rules that define priorities over the first level rules for each role that the agent can 3.1 The Argumentation Framework assume or context that he can be in (including a default context). Autonomous agents, be they artificial or human, need to make Finally, the third level rules define priorities over the rules of the decisions under complex preference policies that take into account previous level (which context is more important) but also over the different factors. In general, these policies have a dynamic nature rules of this level in order to define specific contexts, where and are influenced by the particular state of the environment in priorities change again. which the agent finds himself. The agent's decision process needs to be able to synthesize together different aspects of his preference policy and to adapt to new input from the current environment. 3.2 The Decision Maker’s Argumentation Such agents are the mutual fund managers. Theory In order to address requirements like the above, Kakas and Using the presented argumentation framework, we transformed Moraitis ([6]) proposed an argumentation based framework to the criteria for all MFs and experts knowledge (§2) to background support an agent's self deliberation process for drawing theory (facts) and rules of the first and second level. Then, we conclusions under a given policy. defined the strategies (or specific contexts) in the third level rules. Argumentation can be abstractly defined as the principled The goal of the knowledge base is to select some MFs in order interaction of different, potentially conflicting arguments, for the to construct our portfolio. Therefore our rules have as their head sake of arriving at a consistent conclusion (see e.g. [10]). The the predicate selectFund/1 and its negation. We write rules nature of the “conclusion” can be anything, ranging from a supporting it or its negation and use argumentation for resolving proposition to believe, to a goal to try to achieve, to a value to try conflicts. We introduce the hasInvestPolicy/2, preference/1 and to promote. Perhaps the most crucial aspect of argumentation is market/1 predicates for defining the different contexts and roles. the interaction between arguments. This means that argumentation For example, John, an aggressive investor is expressed with the can give us means for allowing an agent to reconcile conflicting predicate hasInvestPolicy(john, aggressive). information within itself, for reconciling its informational state The knowledge base facts are the performance and risk with new perceptions from the environment, and for reconciling variables values for each MF, the thresholds for each group of conflicting information between multiple agents through 60 values for each year and the above mentioned predicates The problem with the above rules is that the facts market(bear) characterizing the investor and the market. The following rules are or (exclusive) market(bull) could not be safely determined for the an example of the object-level rules (level 1 rules of the next investment period. In the application version presented in framework - T): [14] it was just assumed to remain the same as at the time of the investment. This strategy, however produced quite poor results for r1(Fund): selectFund(Fund) ← highR(Fund) this context if it should change in the next period. r2(Fund): ¬selectFund(Fund) ← highB(Fund) 4 FORECASTING THE STATUS OF THE The highR predicate denotes the classification of the MF as a FINANCIAL MARKET high return fund and the highB predicate denotes the classification of the MF as a high risk fund. Thus, the r1 rule states that a high One of the most prominent issues in the field of signal processing performance fund should be selected, while the r2 rule states that a is the adaptive filtering problem, with unknown time-invariant or high risk fund should not be selected. Such rules are created for time-varying parameters. Selecting the correct order and the three groups of our performance and risk criteria. estimating the parameters of a system model is a fundamental Then, in the second level we assign priorities over the object issue in linear and nonlinear prediction and system identification. level rules. The PR are the default context rules or level 2 rules. The problem of fitting an AutoRegressive Moving Aaverage These rules are added by experts and express their preferences in model with eXogenous input (ARMAX) or a Nonlinear the form of priorities between the object level rules that should AutoRegressive Moving Aaverage model with eXogenous input take place within defined contexts and roles. For example, the (NARMAX) to a given time series has attracted much attention level 1 rules with signatures r1 and r2 are conflicting. In the because it arises in a large variety of applications, such as time default context the first one has priority, while the bear market series prediction in economic and biomedical data, adaptive context reverses this priority: control, speech analysis and synthesis, neural networks, radar and sonar, fuzzy systems, and wavelets [5]. R1: h_p(r1(Fund),r2(Fund)) ← true The forecasting algorithm used in this contribution is a generic applied evolutionary hybrid technique, which combines the R2: h_p(r2(Fund),r1(Fund)) ← market(bear) effectiveness of adaptive multimodel partitioning filters and GAs’ robustness [1]. This method has been first presented in [7]. Rule R1 defines the priorities set for the default context, i.e. an Specifically, the a posteriori probability that a specific model, of a investor selects a fund that has high return on investment (RoI) bank of the conditional models, is the true model, can be used as even if it has high risk. Rule R2 defines the default context for the fitness function for the GA. In this way, the algorithm identifies bear market context (within which, the fund selection process is the true model even in the case where it is not included in the cautious and does not select a high RoI fund if it has high risk). filters’ bank. It is clear that the filter’s performance is Finally, in PC (level 3 rules) the decision maker defines his considerably improved through the evolution of the population of strategy and policy for integrating the different roles and contexts the filters’ bank, since the algorithm can search the whole rules. When combining the Aggressive investor role and bear parameter space. The proposed hybrid evolutionary algorithm can market context, for example, the final portfolio is their union be applied to linear and nonlinear data; is not restricted to the except that the aggressive investor now would accept to select Gaussian case; does not require any knowledge of the model high and medium risk MFs (instead of only high). The decision switching law; is practically implementable, computationally maker’s strategy sets preference rules between the rules of the efficient and applicable to online/adaptive operation; and exhibits previous level but also between rules at this level. Relating to the very satisfactory performance as indicated by simulation level 2 priorities, the bear market context’s priority of not buying experiments [2]. The structure of the hybrid evolutionary system a high risk MF, even if it has a high return, is set at higher priority used is depicted in Figure 1. than that of the general context. Then, the specific context of an The representation used for the genomes of the population of aggressive investor in a bear market defines that the bear market the GA is the following. We use a mapping that transforms a fixed context preference is inverted. See the relevant priority rules: dimensional internal representation to variable dimensional problem instances. Each genome consists of a vector x of real C1: h_p(R2, R1) ← true values xi∈ ℜ , i = 1, ..., k, and a bit string b of binary digits bi∈{0,1}, i = 1, ..., k. Real values are summed up as long as the C2: h_p(R1, R2) ← hasInvestPolicy(Investor, aggressive). corresponding bits are equal. Obviously, k is an upper bound for the dimension of the resulting parameter vector. We use the first C3: h_p(C2, C1) ← true k/3 real values for the autoreggressive part, the second k/3 real values for the moving average part, and the last k/3 real values for Thus, an aggressive investor in a bear market context would the exogenous input part. An example of this mapping is continue selecting high risk funds. In the latter case, the argument presented in Figure 2. For a more detailed description of this r1 takes along the priority arguments R1, C2 and C3 and becomes mapping refer to [2]. stronger (is the only admissible one) than the conflicting r2 At first, an initial population of m genomes is created at argument that can only take along the R2 and C1 priority random (each genome consists of a vector of real values and a bit arguments. Thus, the selectFund(Fund) predicate is true and the string). As stated before, each vector of real values represents a fund is inserted in the portfolio. possible value of the NARMAX model order and its parameters. For each such population we apply an MMAF with EKFs and 61 have as result the model-conditional probability density function (one) which is the maximum value it is able to have as a (pdf) of each candidate model. This pdf is the fitness of each probability For a more detailed description of this hybrid candidate model, namely the fitness of each genome of the evolutionary system refer to [2]. population (Figure 3). Figure 3: The fitness of each candidate model is the model conditional pdf (m is the number of the extended Kalman filters in the multimodel adaptive filter) In this contribution we apply a slightly different approach compared to the one presented in [2]. In [2], at the algorithm’s step where the value of the estimation (output) x of each filter is calculated, the past values of x that are used in order to estimate the next value of x are always taken from the estimation file (the file of all past values of x that have been estimated by the algorithm till this point). All these values are used in each generation in order to estimate the next value of the estimation (output) vector x. The method presented in this contribution uses a different approach in order to estimate x. At the algorithm’s step where the value of x for each filter is calculated, the past values of x that are used in order to estimate the next value of x are smaller than the total length of the time series that has been estimated till this point. The length of past values used in each generation in order to estimate the next value of x equals to n/2, where n is the total length of the time series to be estimated. Every new value of Figure 1: The structure of the hybrid evolutionary system used x, estimated by the algorithm, is added to this time series of length for forecasting n/2 and the oldest one is removed in order this time series to sustain a length of n/2. The value of n/2 was not selected arbitrarily. We have conducted exhaustive experiments using many different values. The value of n/2, that has been finally selected, was the most effective one, that is, the one that resulted in the best prediction results. Thus, the hybrid evolutionary system presented in Figure 1 is used in order to forecast the behavior of the financial market in relation to its current status. The market is characterized as bull market if it is forecasted to rise in the next semester, or as bear market if it is forecasted to fall. We used the return values of the Greek market index for each semester starting from year 1985 to Figure 2: Mapping from a fixed dimensional internal the years of our sample data (2000 to 2005). The algorithm representation to a variable length NARMAX parameter performed very well considering that it could forecast the next vector. The resulting order is n(p, q, r) = (4, 3, 2). semester market behavior with a success rate of 85.17% (12 out of The reproduction operator we decided to use is the classic 14 right predictions). biased roulette wheel selection according to the fitness function value of each possible model order [9]. As far as crossover is concerned, we use the one-point crossover operator for the binary 5 THE PORTFOLIO CONSTRUCTION strings and the uniform crossover operator for the real values [9]. APPLICATION Finally, we use the flip mutation operator for the binary strings and the Gaussian mutation operator for the real values [9]. Every In this section we firstly present the system architecture, i.e. the new generation of possible solutions iterates the same process as combination method for the argumentation decision making sub- the old ones and all this process may be repeated as many system and the hybrid forecasting sub-system that resulted in a generations as we desire or till the fitness function has value 1 62 coherent application. Then we present the results of this to different investment choices and leads to the selection of combination. different number and combinations of MFs. 5.1 System Architecture The portfolio generation application is a Java program creating a human-machine interface and managing its modules, namely the decision making module, which is a prolog rule base (executed in SWI-prolog1) using the Gorgias2 framework, and the forecasting module, which is a Matlab3 implementation of the forecasting hybrid system (see Figure 4). The application connects to the SWI-Prolog module using the provided Java interface (JPL) that allows for inserting facts to an existing rule-base and running it for reaching goals. The goals can be captured and returned to the Java program. The application connects to Matlab by executing it in a system shell. The matlab program writes the results of the algorithm to a MySQL4 database Figure 5: A screenshot for portfolio generation for a scenario using SQL (Structured Query Language). The application first of a moderate investor in a bull market context executes the forecasting module, then updates the database, using JDBC (Java DataBase Connectivity interface) technology, with In Table 1 the reader can inspect the average return on the investor profile (selected roles) and, finally, queries the investment (RoI) for the six years for all different contexts. The decision making module setting as goal the funds to select for reader should notice that the table contains two RoI columns, the participation in the final portfolio. Thus, after the execution of the first (“Previous RoI”) depicts the results before changing the forecasting module the predicate market/1 is determined as bull or system as they appeared in [14]. The second presents the results of bear and inserted as a fact in the rule base before the decision upgrading the application by combining it with the hybrid making process is launched. The reader can see in Figure 5 a evolutionary forecasting sub-system and by fixing the selected screenshot of the integrated system. funds participation to the final portfolio. The latter modification is out of the scope of this paper but the reader can clearly see that it has greatly influenced the performance of all scenarios. Table 1, however, shows the added value of this contribution as the market context has become the most profitable in the “New RoI” column (8.17% RoI), while in the “Previous RoI” column it was one of the worst cases (3.72% RoI). Consequently the specific contexts containing the market context have better results. Table 1: Average RoI for six years. The New RoI column shows the gains after the evolutionary hybrid forecasting system’s integration Previous Figure 4: System Architecture Context type Context New RoI (%) RoI (%) simple general 3.53 6.86 role aggressive 2.65 7.38 5.2 System Evaluation role moderate 4.02 6.09 context market 3.72 8.17 For evaluating our results we defined scenarios for all years for role high performance 4.98 7.16 which we had available data (2000-2005) and for all combinations specific context aggressive – market 3.56 7.92 of contexts. That resulted to the two investor types combined with specific context moderate – market 4.72 6.08 the market status, plus the two investor types combined with the specific context aggressive - high p. 4.88 7.46 high performance option, plus the market status combined with specific context moderate - high p. 4.98 7.16 the high performance option, all together five different scenarios specific context Market - high perf. 4.59 7.23 run for six years each. Each one of the examined scenarios refers ASE-GI RASE 6.75 1 SWI-Prolog offers a comprehensive Free Software Prolog environment, Moreover, Table 1 also shows the added value of our approach http://www.swi-prolog.org as the reader can compare our results with the return on 2 Gorgias is an open source general argumentation framework that investment (RASE) of the General Index of the Athens Stock combines the ideas of preference reasoning and abduction, Exchange (ASE-GI). According to the results of this table, the http://www.cs.ucy.ac.cy/~nkd/gorgias/ average return of the constructed portfolios for all contexts, except 3 MATLAB® is a high-level language and interactive environment for two, achieves higher return than the market index. The two cases performing computationally intensive tasks, http://www.mathworks. where the constructed portfolios did not beat the market index are com/products/matlab 4 the moderate simple context and moderate-market specific MySQL is an open source database, http://www.mysql.com 63 context. This is, maybe, due to the fact that in these two contexts The developed application allows a decision maker (fund we have an investor who wishes to earn more without taking into manager) to construct multi-portfolios of MFs under different, account any amount of risk in relation to the variability which possibly conflicting contexts. Moreover, for medium to long term characterizes the conditions of the market during the examined investments, the returns on investment of the constructed period. This fact makes it very difficult to implement investment portfolios are better than those of the General Index of the Athens strategies that can help a fund manager outperform a passive Stock Exchange, while the best results are those that involve the investment policy. forecasting of the financial market. Furthermore, we notice that in some specific contexts the Our future work will be to develop a new rule base for the results are more satisfying than the results obtained by simple problem of determining when to construct a new portfolio for a contexts, while in others there is little or no difference. This specific investor. We will also make the application web-based so means that by using effective strategies in the third preference that it can get on-line financial data available from the internet for rules layer the decision maker can optimize the combined computing the decision variables and for allowing the investors to contexts. Specifically, the aggressive-high performance specific insert their profiles by filling on-line forms. Finally, we will context provides better results than both the simple contexts continue evaluating our application as new data become available aggressive and high performance (the ones that it combines) and for years after 2005. Our aim is to be able to guarantee a better the general context. The moderate-high performance specific RoI than that of the ASE. context’s returns on investment are equal to the higher simple context’s returns (high performance) while the aggressive-market specific context returns are closer to the higher simple context’s 7 REFERENCES returns (market). [1] A. V. Adamopoulos, Anninos, P. A., Likothanassis, S .D., Finally, in Figure 6, we present the RoI of all contexts Beligiannis, G. N., Skarlas, L. V., Demiris E. N. and Papadopoulos, separately for each year. This view is also useful, as it shows that P., 2002. 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