portfolio can be comprised of at least 5 stocks, which can relatively reduce investment risk. We assume the investor’s market expectations are, respectively, b 0, b 0.3, b 0.5, b 0.8, and b 1. The results are listed in Table 4 .
Table 4 shows that the investmentportfolio only includes a few stocks. Some stocks such as Tellus and ZY Environment have not been selected regardless of the market expectations. The reason is that the expected returns of selected stocks are stable and they have higher expected return range. In Table 4 , we find that the investment ratios of Vanke and Victor onward Textile are 0.2, respectively, because their expected return range is positive. Meanwhile, these two stocks are less risky. Meanwhile, the investment ratio of Advanced Technology is 0.2 as well, since the investment return of the Advanced Technology is more stable than that of other stocks regardless of economic situation. Investment proportion of DaTong Gas is high in bad economic environment. However, when the economy becomes better, the investment proportion gradually declines due to its instable return. Therefore this stock is a good choice for risk-averse investor.
The mean-variance approach originated by Markowitz (1952, 1959) has been a cor- nerstone of asset allocation, investmentanalysis and risk management. In this literature, Merton’s (1969, 1971, 1973) seminal work is considered a benchmark on continuous- time portfolioselection. The single period model is extended by Li and Ng (2000) for multi-period case and developed by Zhou and Li (2000) for continuous-time one, respec- tively. The work of Li and Zhou (2006) reveals the high opportunity of a Markowitz mean-variance strategy hitting the expected return target before the maturity date. Nat- urally, investors also hope to decide when to stop the investment over a given investment horizon so as to maximize their profits. This idea has been further developed to deter- mine the optimal selling time for one stock by Shiryaev et al. (2008), who determined a time point at which investors can sell risky assets as close to the maximum return as possible. This again highlights the efficiency of the mean-variance analysis in the field of investment and portfolioselection. Naturally, an investor also hopes to know the time point to stop the investment over a given investment horizon so as to maximize the profit.
The optimization of investment portfolios is the most important topic in financial decision making, and many relevant models can be found in the literature. According to importance of portfolio optimization in this paper, deals with novel solution approaches to solve new developed portfolio optimization model. Contrary to previous work, the uncertainty of future returns of a given portfolio is modeled using LR- FUZZY numbers while the function of its return are evaluated using possibility theory. We used a novel Lp-metric method to solve the model. The efficacy of the proposed model is tested on criterion problems of portfolio optimization on LINGO provides a framework to optimize objectives when creating the loan portfoliso, in a search for a dynamic markets decision. In addition to, the performance of the proposed efficiently encoded multi-objective portfolio optimization solver is assessed in comparison with two well-known MOEAs, namely NSGAII and ICA. To the best of our knowledge, there is no research that considered NSGAΠ, ICA fuzzy simultaneously. Due to improve the performance of algorithm, the performance of this approach more study is probed by using a dataset of assets from the Iran’s stock market for three years historical data and PRE method. The results are analyzed through novel performance parameters RPD method. Thus, the potential of our comparison led to improve different portfolios in different generations.
AORE is still a hot research area, which focuses on identifying, analyzing, specifying, verifying, and managing the crosscutting concerns at the early stages of software development. Over the last few years, several research efforts have been devoted to developing AORE models that can help in extracting, identifying, and modelling aspects in the early phase of requirements analysis. Some success in this direction has been achieved. But, still managing conflicts among multiple viewpoints or stakeholders is immature and needs more attention with many research issues.
The presented study shows that with the application of portfolioselectionmodels, which base only on historical data of securities returns, it is possible to obtain better than average results, when compared to the benchmarks. All the six models, which allowed investment in WIG20 stocks and risk-free rate and do not include transaction costs constraint in the model, obtained much better results when it comes to return, return to risk ratio and skewness in comparison to WIG, Allianz Akcji FIO and equally weighted portfolio of WIG20 stocks. Additionally three of the models – M, MV, MVK, achieved risk to return ratio similar to that of Allianz OFE. Thus, the first research hypothesis is confirmed. As far as the analysis of moments higher than variance is concerned, the results are ambiguous. The inclusion of skewness as an optimization constraint indeed increased the skewness of the portfolio, but at the expense of a lower rate of return. The kurtosis constraint decreased kurtosis of the portfolio in majority of the cases, thus diminishing the probability of extreme events, but it did not work for all the models.
In this paper, based on existing results, decision making about portfolio in- vestment schemes is discussed, ordering method of fuzzy numbers of interval value is shown, corresponding auxiliary models are established and solutions are provided with theories of fuzzy mathematics, optimization theory and numerical calculation, etc. Then it applies software programming to solve the portfolioinvestment situation between investors in savings and four securities according to the established models. The result shows that investors can choose the risk coefficient that they can bear to reach the maximum value of expected returns. The greater the risk coefficient, the greater the income, the smaller the risk coefficient and the smaller the income. Investors can deter- mine their own portfolio strategy according to their own conditions in order to meet their own interests.
3.5. The Framework Decision Support System Based on MAGDM with IVIFSs. As the scale of decision making increases, the procedure solving a MAGDM may be compli- cated. In this case, a decision supporting system (DSS), which is a class of computer-based information system includ- ing knowledge-based systems [39, 40], can be formulated to help experts improve their decision-making level and quality through problem analysis, establishment of models, and simulation of decision-making process in a human- computer interaction way. Figure 1 depicts a framework of DSS designed in this paper for MAGDM with IVIFSs.
Due to their proactive nature, R&D projects are sometimes hard to evaluate. It is often the case that information required for the valuation is actually revealed gradually during the project, and at the beginning of the development opportunity there are no cash ﬂow estimates available that would either justify or invalidate the evaluation of the project. In a sense of information quality, knowledge about the project’s proﬁtability is seldom precise, let alone that sometimes it is not even measurable. However, even in such situa- tions the R&D management has to commit itself to either a positive decision to launch the project, a negative decision to abandon the project, or, which seems most plausible, a decision to wait and see if the information quality improves as time passes. This man- agement position can be described as if the management had some information hidden or in shadow, and it had to make a decision about consuming some resources in order to uncover the information. The decision to use resources for information retrieval leads to the launch of the investment, when the option to start the project is used. On the other hand, the decision to deny resources leads to the abandonment of the underlying invest- ment, when the option to abandon the project is used. Finally, the decision to stand by and wait for new information leads to waiting and deferring the investment opportunity, where both the option to start and the option to abandon are kept alive. In the absence of quantitative value-based statements represented by the cash ﬂows, the R&D manage- ment often relies on qualitative statements made by the technological experts.
Among recent works (2010-2014) we can mention; in  the authors propose an approach for multiobjective heuristic search technique to support a selection of project portfolio in scenarios with a large number of available projects. In  the authors propose an alternative that uses fuzzy logic with a heuristic to choose an optimal portfolio. The same authors present a variation of this alternative in  adding a data mining subsystem. The work presented in  uses an evolutionary algorithm for selecting an optimal portfoliobased on a single objective function. In  the authors propose a tool that identifies a set of portfolios (Pareto Optimal) within a cost range allowing the realization of interactive analysis. In [18, 19], the authors present a tool which implements a heuristic algorithm. The Tab. 2 shows the comparison of related work with our approach.
proposed by Oh, Kim, and Min . Fernandez  proposed control model that includes ecological and economic uncertainty for managing both types of natural resources. Topaloglou et al  worked on international portfolio management using dynamic stochastic programming model to determine capital allocations to international markets and the selection of assets within each market. Talebnia and Fathi  compared the Markowitz model with the value at risk for creating the optimal stock portfolio in Tehran Stock Exchange from 2001 to 2008. In a study conducted by Nikoomaram and Hemmati  the benchmark of intellectual capital was measured by using six accounting model, and the portfolios network matrix were selected based on the market value of companies from listed companies in Tehran Stock Exchange during 5 years from 2006 to 2010. PortfolioSelection and Risk Management: An Introduction, Empirical Demonstration and R-Application for Stock Portfolios by Angela Hei-Yan Leung  this paper serves as an introduction to PortfolioSelection and Risk Management theories founded upon Harry Markowitz's Value-at-Risk calculation methods are described. Masoud Mansoury et.al  proposed an enhanced decision support system for portfolio management using financial indicators using a hybrid approach that offers the best suggestions in buying and selling stocks. Amit D.Narote et.al  has proposed a mixed portfolio theory model using genetic algorithm and vector quantization to find the effective data related to the stock market behaviour.
The AHP, first introduced by , is a MCDM method for solving MCDM problems by setting their priorities. This method uses precise numbers in the rating of alternatives. The AHP uses objective mathematics to process the subjective and personal preferences of an individual or a group in decision-making . The AHP works on a premise that decision making of complex problems can be handled by structuring it into a simple and comprehensible hierarchical structure. Solution of the AHP hierarchical structure is obtained by synthesizing local and global preference weights to obtain the overall priority . The classical MCDM methods such as AHP cannot handle problems with imprecise information effectively. One of the tools which has been used for transmission of uncertainties in decision-making problems during recent decades is the type-1 fuzzy sets (FSs) introduced by . To date, many researchers have extended the AHP based on fuzzy sets (fuzzy AHP methods). The most important and earliest fuzzy AHP methods are summarized below. Van Laarhoven and Pedrycz  proposed the first study that applied the fuzzy logic principle to AHP. Buckley  initiated trapezoidal fuzzy numbers to express the decision maker’s evaluation on alternatives with respect to each criterion while Van Laarhoven and Pedrcyz  have used triangular fuzzy numbers. Chang  introduced a new approach for handling fuzzy AHP with the use of triangular fuzzy numbers for pair-wise comparison scale of fuzzy AHP, and the use of the extent analysis method for the synthetic extent values of the pair-wise comparisons. Cheng  proposed a new algorithm for evaluating naval tactical missile systems by the fuzzy analytical hierarchy process based on grade value of membership function. Deng  presented a fuzzyapproach for tackling qualitative multi-criteria analysis problems in a simple and straightforward manner. Others research works are
Portfolioinvestment is quoted securities investment, a narrow sense of investment. It refers to the behavior that an enterprise or individual buys negotiable profits. Securities such as stocks and bonds with accumulated money to earn profits (Yin, 2018). Portfolioselection problem (PSP) is a well- know problem in the field of economics, which aims to allocate the capital to a pre- given set of securities and mean-while obtain the maximum return (Fang et al., 2017). In the real- world applications, there exists two types of uncertainties in the decision- making process one is randomness; the other is fuzziness. In general, if enough sample data are available, we can use the statistic methods to estimate the probability distribution of the involved uncertain parameters, and the probability theory can be used as an effective tool to deal with them. On the other hand when there are not enough sample data or even no sample data, a common method is to treat the parameters as fuzzy variables by using professional judgments or expert experience. With these concerns, two classes of methods can be adopted in the literature to investigate the PSP, i.e., random optimization and fuzzy optimization, in order to maximize the total return and decrease the risks in the uncertain environment (Feng et al., 2017; Markowitz, 1952, and 1959). Huang and Ying, 2013 considered the portfolio adjusting problem. A mean- variance- based PSP in a complete market with unbounded random coefficients is investigated by Shen et al., 2014. Tanaka et al., 2000 extended the probability into fuzzy probability for the Markowitz's model. Hassuike et al. 2009 proposed several models for PS problems, particularly scenario model including the ambiguous factors. A multi- period PSP with market random uncertainty of asset prices is introduced by He and Qu, 2014. Zhang and Chen, 2016 present a mean- variance PSP with regime switching under the constraint of short- selling being prohibited. Shi et al., 2015 proposed three multi- period behavioral portfolioselectionmodels under cumulative prospect theory. Lv et al., 2016 explored a continuous- time mean- variance PSP with random market parameters and a random time horizon in an incomplete market. According to Dowd, 2002, in conditions of risky investments, a strategy that can be done to reduce the magnitude of the risk of investment is to build a portfolio. A single index model used for the assessment of the stock price introduced by Azizah et al., 2008.
There are several models available for portfolio manag- ers to help select projects, prioritize and weight alterna- tives. The Balanced scorecard, developed by Kaplan and Norton in 1992 is now applicable in non-profit organiza- tion project management , the public sector  and municipalities . When a balanced set of indicators are produced and managed over time, municipalities must use the information in order to manage the selection of projects. Rough set theory developed by [4-6] is regarded as a mathematical tool for imperfect data analysis and may be applied in engineering, banking, medicine . We propose the use of Rough set theory modified by  and called Dominance-based Rough Set Approach (DRSA) in portfolio management, more specifically in the man- agement of a portfolio of projects for a group of munici- palities.
After computing the MODWT crystals (details and smooths) for every stock market return, and from the decomposed series (d1, . . . , d8, s8) we classify the short, medium and long term series as follows: Short term=d1+d2+d3; Medium term=d4+d5+d6; Long term=d7+d8+s8. This choice of time-horizon decomposition is used to classify three types of investors or traders, such as short, medium and long term ones, i.e. to analyze the behavior of investors among different time-horizons. Here the highest frequency component Short term, (d1+d2+d3) represents the short-term variations due to shocks occuring at a time scale of 2 to 16 days, it provides daily and weekly spillovers, the next component Medium term, (d4+d5+d6) represents the mid-term variations at time scale of 32 to 128 days, it defines the monthly and quarterly spillovers, and the third component Long term, (d7+d8+s8) represents the long-term variations of 256 days and more, it provides the annual spillovers. The main advantage of this classification is to decompose the risk and the volatility spillovers into three investment horizons. Therefore, we focus in three sub-spillovers. All market participants, such as regulators, traders and investors, who trade in stock markets (in our study, U.K., U.S. and Japan stock markets) make decisions over different time scales. In fact, due to the different decision-making time scales among investors, the time-varying volatilities and correlations of stock market indices will vary over the different time scales associated with those horizons (investment strategies).
Most studies have considered more than one objective in portfolioselection. Huang (2006), for example, developed a bi-objective portfolioselection model to maximize in- vestors ’ returns and the likelihood of achieving a specified return level. Abdelaziz et al. (2007), meanwhile, developed a multiobjective deterministic portfolio-selection model for the Tunisian stock market. Mathematical models have been integrated with other techniques, such as fuzzy logic. Carlsson et al. (2007), for example, proposed a fuzzy mixed-integer programming approach to select R&D portfolios. Li et al. (2010), mean- while, developed a skewness concept for fuzzy variables in portfolioselection. MCDM techniques have also been investigated in recent portfolioselection research. Jeng and Huang (2015), for example, developed a systematic MCDM approach and applied decision-making trial and evaluation laboratory (DEMATEL), analytic network process (ANP), and the modified Delphi method (MDM) to portfolioselection. Adopting add- itional criteria, Mehlawat (2016) applied risk, wealth, liquidity, number of assets, and transaction cost to portfolio assessment. Meanwhile, according to Huang and Di (2016), uncertain portfolioselection can be conducted in the presence of background risk, background assets, and security returns based on expert assessment rather than historical data. Mashayekhi and Omrani (2016) developed a multiobjective mathemat- ical model that integrated the Markowitz MV model with data envelopment analysis (DEA) cross-efficiency considering risk, efficiency, and returns. Recently, Nystrup et al. (2018) developed multiperiod forecasting for the mean and covariance of financial returns from a time-varying portfolioselection model.
3. Cross-entropy minimization models
In this section, we consider the Kapur cross-entropy minimization model under fuzzy environment. Suppose there is a priori fuzzyinvestment return η for an investor, and his objective is to minimize the divergence of the fuzzyinvestment return from η and allocate the investment with considerations such that (i) the return is above a given level; and (ii) the risk remains below a given level. Generally speaking, we use the cross-entropy to measure the degree of divergence and use the expected value to measure the return. The main problem is how to measure the risk. There are three most popular ways, that is, variance, semivariance and chance of bad outcome. If investment return ξ 1 x 1 + ξ 2 x 2 + · · · + ξ n x n is symmetry, we
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As far as minimum risk level where efficient frontier is vertical is concerned, the risk levels of GM and HM are bounded by that AM curve and GDM. Given that GDM are highest in all four means, its variance-covariance is greatest at lower level of k values. The variances of HM are even greater because of negative mean returns. However, five of ten covariances in HM are negative and thus the portfolio risk could be reduced from proper diversification. The high values of the GDM returns translate into higher variances and covariances when com- pared to that of AM and GM. It is little wonder that the efficient frontier has the largest risk in the vertical segment or v(GDM) > v(HM) > v(GM) > v(AM). As k value increases to 30% the risk become the greatest for HM that has 5 largest va- riances of all the four means. The rank of the portfolio risk become v(HM) > v(GM) > v(AM) > v(GDM). Note that v(GDM) is still at its vertical segment at
is, if it is possible to choose among models so that the performance of one of them is always better than the worst case performance.
In trying to answer the above question the follow- ing procedure is proposed. At each time step, from iteration 1 to 25, the performance of each one of the models is registered. Then, an ideal ex-post portfolio from the Markowitz model is considered, which allows one to measure how the stock markets really moved, and take the ﬁrst ﬁve worst results. That is, one considers 20% of the times that the market performed worst. Afterwards, one checks what the performance of the other models was and this set is ordered for the previously detected worst cases. To the extent that a particular model behaves better than the others in these extreme conditions, one can say that it should be, ceteris paribus, chosen by risk adverse investors.
In this study, we employ three multivariate GARCH models, such as the CCC model of Bollerslev (1990), the DCC model of Engle (2002) and the cDCC model of Aielli (2008). These models impose a useful structure on the many possible model parameters. However, parameters of the model can easily be estimated and the model can be evaluated and used in straightforward way. Our empirical methodology follows a two-step approach. The first step applies these dynamic conditional correlation models to model conditional volatility in the returns of three developed stock markets (U.K., U.S. and Japan stock markets), in order to examine the evidence of time-varying conditional variances and correlations between stock markets. Moreover, in order to show the asymmetric effects of positive and negative return shocks on conditional volatility the EGARCH and GJR-GARCH models of Nelson (1991) and Glosten et al. (1993), respectively, were employed for modeling univariate conditional volatility. In the second step, we re-examine the dynamic conditional correlation analysis among the three major developed stock markets through a novel approach, wavelet analysis. This technique is a very promising tool as it is possible to capture the time and frequency varying features of co-movement within an unified framework which represents a refinement to previous approaches. This wavelet-basedanalysis takes account the distinction between the short and long-term investor. From a portfolio diversification view, there exist a kind of investors whose are more interested in the co-movement of stock returns at higher frequencies (lower scales), that is, short-term fluctuations, and also, there exist a kind of investors whose focuse on the relationship at lower frequencies (higher scales), that is, long-term fluctuations. The study of the co-movement of stock market returns, i.e. dynamics of variances and correlations, across scales is crucial for risk assessment of a portfolio. In terms of risk management, a higher co-movement (higher covariances) among assets of a given portfolio implies lower gains. According to investors or traders, evaluating the co-movement of assets is a