Mathematical modeling for a new portfolio selection problem in bubble condition, using a new risk measure Alireza Ghahtarani a, Majid Sheikhmohammady a,*, Amir Abbas Najafi b a) Faculty of Industrial and systems Engineering, Tarbiat Modares University, Tehran, Iran b) Faculty of Industrial Engineering, K.N. Toosi University of Technology, Tehran, Iran Abstract A portfolio selection model is developed in this study, using a new risk measure. The proposed risk measure is based on the fundamental value of stocks. For this purpose, a mathematical model is developed and transformed into an integer linear programming. In order to analyze the model's efficiency, the actual data of the Tehran Stock Exchange market are used in 12 scenarios to solve the proposed model. In order to evaluate the scenarios, data mining approaches are employed. Data mining methods which are used in this paper include ANFIS, decision tree, random forest, FDA, and GEP. The best method for scenario evaluation is GEP based on numerical results. Hence, the market values are evaluated by this algorithm. Software packages like MATLAB, GEP xpero tools, and LINGO are used to solve the model. Different trends of market value and fundamental value volatility in the optimum stock portfolio are determined. It is possible to examine the optimum portfolio profitability in different scenarios. By using real- world data, trends are extracted and analyzed. Results show that the developed model can be effectively applied in bubble situations. Keywords: Decision Tree, Financial Bubble, Fundamental Value, Gene Expression Programming, Portfolio Selection Problem, Risk Measure * Correspondence Author TEL: (+98) 21 82884394 Email Addresses: [email protected] (Majid Sheikhmohammady), [email protected] (Alireza Ghahtarani), [email protected] (Amir Abbas Najafi) 1 1. Introduction Portfolio selection problem is one of the most important issues in finance in which investors try to maximize return and minimize risk. Markowitz was the first researcher, who developed a mathematical definition for risk measures[1]. However, his risk measure was not linear and it had some calculation issues. After developing that risk measure, many researchers have tried to develop new risk measures. Konno and Yamazaki [2] presented mean absolute deviation (MAD) as a risk measure. Their risk measure was based on the stocks’ rate of return. It was linear but MAD considered both positive and negative deviation from mean as a risk which was not practically acceptable. Next huge step toward a proper risk measure was taken by Rockafellar and Uryasev [3] who developed conditional value at risk (CVaR). This new risk measure is convex, and it has a linear programming model. CVaR is a downside risk measure. However, it uses historical data based on the assumption that historical trends will happen in the future. Moreover, this risk measure uses the rate of return for its calculation. They also developed conditional drawdown at risk (CDaR) which uses the maximum price of a stock in the investment period as a threshold. However, this risk measure is too conservative. Bernardi et al [4] presented multiple risk measures for multivariate dynamic heavy-tailed models. They studied the evolution of risk interdependence and proposed a new risk measure to capture tail co- movements. This risk measure uses the stock rate of return. Kuzubas et al [5] proposed network centrality measures and systemic risk. They used data from Turkish Interbank market in the financial crisis. They did not consider fundamental values in their analysis. Fu et al [6] developed a convex risk measure based on the generalized lower deviation. This risk measure shows the risk aversion of investors in non-linear equations. They used rate of return of stocks in their risk measure. Yan [6] illustrated deviations and asymptotic behavior of convex and coherent entropic risk measures. Gong and Zhuang [7] presented measuring financial risk and portfolio reversion with time changed. They used stochastic volatility by tempered stable Lévy processes to construct time changed tempered stable Lévy processes. Sorwar and Dowd [8] estimated financial risk measures for options. They used simulation-lattice procedure estimation. However, they did not consider the fundamental factors in their analysis. Many researchers have worked on developing financial risk measures [9-15] But, these researches have gap in considering fundamental factors and bubble condition. Many of these risk measures are based on the difference between expected rate of return at the beginning and that in in the last period of investment. One of the most important criticisms toward these risk measures is regarding the nature of the stock market value that is used for the calculation of risk measures. In fact, the assumption is based on the fact that the stock market value illustrates all events and phenomenon around corporate including good or bad news about the corporate. However, some part of stock value volatility is the result of investor emotion about events. These volatilities are mostly short term. In fact, after a while, stock price back to the former price channel. Nevertheless, traditional risk measures have no procedure to confront these volatilities. The stock market crash in financial and economic crisis has shown that traditional risk measure cannot evaluate risk properly, especially in emotional and crisis situations. This circumstance leads to a bubble condition. Some stocks’ market prices are greater than the fundamental values, which cause bubble phenomena. In recent years, some researchers focused on bubble analysis in the financial market. Domeij and Ellingsen [16] proposed a quantitative analysis of rational bubbles and public debt policy. They used bubble analysis in securities in public sectors. Barberis et al [17] proposed an 2 article entitled “Extrapolation and bubbles”. They developed a model for extrapolative of bubbles. Lee et al [18] proposed an asset pricing with financial bubble risk. They provided an empirical investigation of risk factors on bubble pricing. Bosi et al [19] proposed an article entitled “financial bubbles and capital accumulation in altruistic economies”. They studied the global dynamics of capital stocks and assets’ values. Maynard and Ren [20] assessed the power of long-horizon predictive tests in a model that contains financial asset bubbles. Michaelides et al [21] developed a non-linearities in financial bubbles. They used theory and Bayesian evidence from S&P500. The authors attempted to find and date non-linear bubble episodes, which are captured by using a neural network. Miao and Wang [22] surveyed banking bubbles and financial crises. Wigniolle [23] considered optimism, and pessimism in financial bubbles. They showed that it is possible to extend the scope of the existence of rational bubbles when uncertainty is introduced associated with rank-dependent expected utility. Harvey et al [24] developed tests for explosive financial bubbles in the presence of non-stationary volatility. Kunieda and Shibata [25] presented a tractable model in which asset bubbles can exist in spite of infinitely lived agents. Their results show that a policy of purchasing the asset avoids financial crises. Many researchers have focused on bubble analysis [26-30]. The above-mentioned researches are among the newest of researches in this context. However, there are still some important gaps in the literature. Researchers who developed risk measure have not considered a fundamental factor and bubble analysis in their models. On the other hand, Researchers who worked on bubble analysis have not developed risk measures. Consequently, there is a gap in this research area. Developing a new risk measure which takes into account bubble conditions can fill this gap. To remove the weakness of traditional risk measures we need the development of a new paradigm in risk measures. Many people in financial market invest in financial assets based on the fundamental value of stocks. The attitude of these investors is that if the market value of a stock is less than the fundamental value then the probability of increasing market value to fundamental value is high. In this situation, investors try to buy stocks. On the other hand, if the market value of a stock is higher than the fundamental value then the probability of decreasing market value to fundamental value is high. In this situation, investors often sell stocks. Accordingly, investors in this strategy have profit and loss zones witch is illustrated in figure 1: Please Insert Figure 1 The aim of this paper is to present a formulation for portfolio selection problem based on a new definition of risk measure. Risk is measured, in this paper, based on the difference between fundamental value and the market value of stocks. The structure of this study is as follows; in the second part, the innovation and research method is presented. The third part, problem modeling and the method of calculating the stock fundamental value are presented. The forth market value prediction based on different scenarios is presented. The fifth part contains the numerical results. Finally, the conclusion of the study is presented in the last part. 2. Innovation and Research Method The most important innovation in this research is the conceptual and mathematical development of the portfolio selection model. In fact, in this research, a new concept is introduced as a risk, and then this new concept is transformed into a mathematical model using mathematical foundations. The concept which is developed in this research is generally different from financial 3 risk measures in the literature of the financial field. On the other hand, the proposed risk measure is more consistent with the existing strategies of investment in the stock markets. To develop a risk measure and thus to develop a new portfolio selection model, this paper uses the concept of market value and fundamental value of stocks. In the real world, many investors use the strategy of estimating the fundamental value and comparing it with the actual stock value to choose their optimum portfolio.