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ISSN : 1598-7248 (Print)
ISSN : 2234-6473 (Online)
Industrial Engineering & Management Systems Vol.19 No.3 pp.680-693

A New Quantitative Predictability-Based Method of Risk Measurement to Make Profitable Investment Decisions in Financial Markets

Zahra Moeini Najafabadi, Mehdi Bijari*, Mehdi Khashei
Department of Industrial and Systems Engineering, Isfahan University of Technology, Isfahan, Iran.
*Corresponding Author, E-mail:
August 13, 2018 May 3, 2020 July 10, 2020


The ultimate goal of investment in financial markets is to achieve the maximum expected rate of return. Risk is one of the most important factors in making profitable investment decisions. Therefore, in the literature, many efforts have been made to define, model, and determine the effect of it in making financial decisions. The first and the most fundamental attempt to quantitatively model the risk and involve it in making investment decisions is related to Markowitz's studies. The results of these studies, besides many financial applications, have been the basis for other riskbased studies. Such methods have been investors' main instrument in developing profitable investment strategies. However, despite scientific and practical advantages, none of them consider the predictability of under-study systems. While the predictability logically has a non-strict inverse relationship with the risk. In this paper, a new method for measuring risk based on the predictability is presented to eliminate this disadvantage. In the proposed method, Markowitz methods have been developed by entering predictability. Accordingly, the proposed method, besides the advantages of Markowitz methods, enters predictability in risk measurement. In order to evaluate the performance of the proposed method in risk measurement and investment decisions, in comparison to other methods, data from twenty companies listed on the Tehran Stock Exchange were used. The empirical results indicate the efficiency of the proposed method to make financial decisions in comparison with other methods.



    Risk is one of the most important factors in many issues, such as economic, political, social, and technological decision making. Risk measurement plays an important role in modeling economic processes under conditions of uncertainty. In particular, the risk is one of the basic aspects of investment decisions. In other words, risk measurement is essential in making the best investment decisions. In fact, trading in assets whose future is unknown is associated with risk for every investor, so risk measuring is a major concern for the utilization of financial markets, and in the literature, this subject has been repeatedly emphasized. Virlics (2013) believes the risk is an important component of every investment; thus, it is necessary to analyze it as a factor in investment decision making. Byrne (2005) has indicated the central role that risk perceptions play in financial decisions. Peng (2005) has implied that measuring investment risk precisely is critical to investment strategies. Zhai and Bai (2017) and Li et al. (2015) have discussed that in any investment in financial assets, there is a risk background in addition to asset risk in financial markets and the presence of background risk may affect investors' investments.

    Therefore, because of the importance of risk in making investment decisions, many studies have been done into risk measurement and its effect on decision making. In recent studies, Trang and Tho (2017) have explored the effects of perceived risk on investment performance and the intentions of individual investors. Puspitaningtyas (2017) has stated that the estimation of systematic risk is one of the important aspects of the best investment decisions and believes that if the systematic risk can be predicted, it is useful for making investment decisions. Sousa and Sousa (2017) stated that fluctuations in risk are important in explaining consumption and investment behavior and, therefore, business cycle patterns. Buchner (2016) has developed a novel Public Market Equivalent measure to evaluate the risk-adjusted performance of private equity investments.

    Many studies have also examined the impact of risk and return on investment decisions. For example, Li et al. (2015), Li and Mei (2014) have discussed the risks and returns of investment portfolios in the financial market and stock market crashes. Jin (2017) has examined two different methods of calculating risk on investment in China's stock market, pointing out that there is a significant difference between the results of these two methods. Righi and Borensteina (2017) have compared different risk measures regarding the performance of optimal portfolio strategies. Tirea and Negru (2015) discussed that risk handling and evaluation played an important role in optimizing an investment portfolio. Oloko (2017) has used various methods to estimate optimal portfolios to test the different risks of the stock market. Tyminski (2015) has presented the quantitative measures of risk that may assist the investor in making decisions regarding a capital market. Rubio et al. (2017) have proposed, using weighted fuzzy time series, methods to forecast the future performance of returns on portfolios. This fuzzy forecast has made it possible to approximate both the expected return and the risk of the investment. Fu et al. (2017) proposed a new class of risk measures. Based on this new class of risk measures, they have established a realistic portfolio selection model taking market frictions into account.

    Some studies have also considered the need for new tools to help make investment decisions. For example, Kumari et al. (2017) have explained that various factors affect the return and risk of each asset. Therefore, new tools for use in investment are needed. Teplova et al. (2017) have proposed a new approach to the decisionmaking process, focusing on investment factors, stock selection strategies, and risk and return of each stock. Kang et al. (2017) have provided a method to determine risk and investment based on this method. Dew-Becker et al. (2017) have pointed to the effects of volatilities on investment decisions and posited that investors were looking for a powerful tool to replace variance. González et al. (2016) have introduced a new method to determine stock market beta. They then used the proposed beta in investment strategies. Cejnek and Randl (2016) have examined the risk and return of short-duration equity investments.

    Because of the important role risk plays in investment issues, several approaches have been developed to measure it. In one of the most fundamental efforts to quantify risk, Harry Markowitz (1952) has considered a risk as a concept of variability and variation range and pointed out that risk was related to the dispersion of a random variable. Therefore, Markowitz used the dispersion indices to calculate risk. He considered variance as the most important dispersion index, equivalent to risk. This risk expression is very simple and tangible, and because of its well-known statistical properties, it was extensively used in the 1950s and 1960s (Durlauf and Blume, 2008). Subsequent to the introduction of the Markowitz model (Konno and Yamazaki, 1991;Konno and Koshizuka, 2005), has introduced the L1 risk function based on the expected absolute deviation instead of the L2 risk function (deviation). Following on, Cai et al. (1996, 2000) has introduced the L function as a risk aversion measure because previous studies did not adequately model the concerns of risk-averse investors. In this method, the maximum risk is minimized among all invested assets. Therefore, in this method of risk measurement, the object is to control the highest standard deviation of all assets. An investor who does not want to face high risks can use this risk-averse approach, which measures risk by the maximum risk of its unique assets. After Teo (2001) has provided another risk measurement. In this method, the risk is equivalent to the average of the maximum risk of unique assets over several periods of time. In this method, the data is divided into several periods. In each period, the absolute deviation is calculated. Then, the risk is equivalent to the average of the maximum of these absolute deviations.

    Such methods, despite their unique advantages in risk calculating and improving the quality of investment decisions, do not consider data predictability in the calculation of risk. On the other hand, although these risk measurement methods have certain advantages, all of them are distribution- based. Therefore, they have assumed that the underlying data sets are only random variables without any time series characteristics, such as structures, patterns, and correlation relationships. So, none of them consider the special feature of the time series, i.e., predictability, in risk measurement. In other words, in these methods, there is no difference between a predictable system and an unpredictable system, when the variations are the same. At the same time, it is clear that higher predictability resulted in lower uncertainty and, consequently, the lower risk. Thus, the accuracy of the predictions related to the studied systems should be logically considered in the calculation of risk. For this purpose, this feature is added to the risk measurement models proposed in this paper.

    Accordingly, the ultimate goal of the proposed method is to enter the predictability of the studied systems into risk measurement. Thus, in the proposed method, the Markowitz idea, which determines the distribution of data around their mean, is combined with data prediction. In other words, in the proposed method, the expected rate of return, obtained from the distribution, is affected by the predictions. Therefore, in the first step, an appropriate predictive method is used to predict the data. The residuals obtained from the first step are the values of the studied system, which cannot be predicted. In other words, the residuals are a one-to-one function of the actual values, and predictable patterns have been removed. Accordingly, all of these values will have the same predictability. In other words, contrary to the actual data, which has different predictability, these residuals have almost the same predictability. Therefore, the basic concepts of Markowitz methods can now be applied, which are dispersion, or a function based on dispersion, on the residuals.

    The limitation of the Markowitz risk models, which results from the inaccurate assumption of the same predictability for each data set, can be eliminated. In other words, Markowitz methods, due to considering the same predictability for all data sets, consider the same way to determine the expected rate of return for all of them. However, as noted, data predictability is fundamentally different, so the measured risk based on this inaccurate assumption will be inaccurate. However, in the proposed method, by removing all predictable patterns from the initial data, we can calculate the residuals that have the same predictability. In this way, the basic concepts of Markowitz methods can be applied to these residuals and thus calculate the risk that would overcome the inaccuracies of classic Markowitz methods.

    The rest of this paper is organized as follows: In section II, the proposed method of risk measurement will be introduced and formulated. In section III, the characteristics and method of collecting data on the shares of companies that have more trading volume in the Tehran Stock Exchange are briefly described. In section IV, the proposed model is implemented in order to make investment decisions. In Section V, the results of the proposed method are compared with other methods. Section VI explains the conclusions.


    In this paper, four common approaches to risk measurement, including Markowitz, Konno, Cai, and Teo, are the basis for presenting the proposed method to measure risk. In order to present the proposed method, based on the Markowitz model, the basic Markowitz method is considered. According to Markowitz, the risk of each random variable corresponds to the variance or standard deviation of that random variable. Equation (1) expresses this (Markowitz, 1952).

    R i s k M a r k o w i t z ( x ) = V a r ( x ) = E ( x E ( x ) ) 2

    In calculating the risk of studied systems, by ignoring the predictability of data, this method considers the variance as risk measurement: while the proposed method of risk measurement first attempts to remove the predictable patterns from the studied systems. Therefore, expected returns are determined by one of the common methods in forecasting. Then, the risk of residuals, according to equation (2), is calculated as the risk of the studied system.

    R i s k ( x ) = 1 n t = 1 n ( x t x ^ t ) 2

    where, x ^ represents the expected return from the prediction that can be calculated using any common method. In this paper, in order to eliminate the effect of prediction methods, three common methods have been used to predict. These three methods include autoregressive, autoregressive moving average, and artificial neural network. Equations (3), (4), and (5) respectively show these methods.

    R i s k ( x ) = 1 n t = 1 n ( x t x ^ t ) 2

    x ^ t = i = 1 d ( α i x t i + β i U t i )

    x ^ t = f ( x t 1 , x t 2 , , x t d )

    Here, by combining equations (1), (2), and (3), the first proposed method of risk measurement, called autoregressive- Markowitz, is presented in accordance with equation (6).

    R i s k ( x ) = 1 n t = 1 n ( x t 2 + α 2 x t 1 2 2 α x t x t 1 ) = E ( x 2 ) + α 2 E ( x 2 ) 2 α E 2 ( x ) = R i s k M a r k o w i t z ( x ) + α 2 E ( x 2 ) + ( 1 2 α ) E 2 ( x )

    Equation (6) shows the difference between the proposed method of autoregressive (1)-Markowitz and the Markowitz method. Similarly, if the predictive method used in the determination of x ^ demonstrates higher rankings in the autoregressive model, the difference between the classic risk measurement and the proposed risk measurement will be increased.

    In the same way, the proposed method of autoregressive moving average-Markowitz is achieved by combining equations (1), (2), and (4) according to (7).

    R i s k ( x ) = 1 n t = 1 n ( x t α x t 1 β U t 1 ) 2 = R i s k M a r k o w i t z ( x ) + α 2 E ( x 2 ) + β 2 E ( U 2 ) + ( 1 2 α ) E 2 ( x )

    Equation (8) also presents the proposed method of the artificial neural network-Markowitz, which is derived from equations (2), (3), and (5).

    R i s k ( x ) = 1 n t = 1 n ( x t f ( x t 1 , x t 2 , , x t d ) ) 2

    Other combined methods are presented in the following way, considering the classic methods of Konno, Cai, and Teo and using the three predictive methods. To introduce the proposed methods based on the Konno risk measurement approach, first consider the Konno risk measurement method for a unique stock, according to equation (9) (Konno and Yamazaki, 1991).

    R i s k K o n n o ( x ) = 1 n t = 1 n | x t E ( x ) |

    By entering any equations (3), (4) and (5) into equation (9), in fact, predictability is added to the base model by using an autoregressive, autoregressive moving average and artificial neural network methods. Thus, equations (10), (11), and (12) represent the risk of each unique asset by the proposed methods, including autoregressive- Konno, autoregressive moving average-Konno, and artificial neural network-Konno.

    R i s k ( x ) = 1 n t = 1 n | x t α x t 1 |

    R i s k ( x ) = 1 n t = 1 n | x t α x t 1 β U t 1 |

    R i s k ( x ) = 1 n t = 1 n | x t f ( x t 1 , x t 2 , , x t d ) |

    Equation (13) shows how to calculate the risk of each unique asset by using the Cai method (Cai et al., 1996).

    R i s k C a i ( x ) = M a x 1 t n | x t E ( x ) |

    In order to provide each of the proposed methods based on Cai risk measurement, including autoregressive- Cai, autoregressive moving average-Cai, and artificial neural network-Cai, it is necessary that each of the equations (3), (4), and (5) are added to equation (13). Thus, the proposed method based on Cai risk measurement, according to equations (14), (15), and (16), is presented.

    R i s k ( x ) = M a x 1 t n | x t α x t 1 |

    R i s k ( x ) = Max 1 t n | x t α x t 1 β U t 1 |

    R i s k ( x ) = Max 1 t n | x t f ( x t 1 , x t 2 , , x t d ) |

    Finally, in equation (17), the risk of each unique asset is calculated by the Teo method. In this method, the data are divided into P periods of time, then in each period, the maximum deviation from the expected rate of return is obtained and, finally, the average of these maximums shows the risk (Teo and Yang, 2001).

    R i s k T e o ( x ) = 1 P p = 1 P M a x n ( p 1 ) / P t n p / P | x t E ( x ) |

    In this method, as in the previous methods, to achieve proposed models of risk measurement based on the Teo approach, which is called autoregressive-Teo, autoregressive moving average-Teo, and artificial neural network-Teo, it is necessary that each of the equations (3), (4), and (


    ) enter into equation (17). By entering these equations into equation (17), the proposed model is introduced according to equations (18), (19) and (20).

    R i s k ( x ) = 1 P p = 1 P Max n ( p 1 ) / P t n p / P | x t α x t 1 |

    R i s k ( x ) = 1 P p = 1 P Max n ( p 1 ) / P t n p / P | x t α x t 1 β U t 1 |

    R i s k ( x ) = 1 P p = 1 P Max n ( p 1 ) / P t n p / P | x t f ( x t 1 , x t 2 , , x t d ) |

    The basic methods, which are the basis for introducing the proposed methods, are the fundamental methods in risk measurement. The literature is not clear on which of these methods is preferred. In the following section, the performance of each of the basic methods and proposed methods are evaluated by using it to make investment decisions.


    To evaluate the proposed method and to test it versus classic methods, data from the Tehran Stock Exchange is used. Among the stocks present at the main hall and the first market of the Tehran Stock Exchange, stocks are selected that had a high financial transaction in the 5-year period between 2012 and 2017. The returns of these stocks are extracted and then refined on a weekly basis. These data included 259 weeks, of which 233 weeks are used for training and modeling, and the remaining 26 weeks are used to test the performance of the methods. Table 1 introduces these stocks (Tehran Stock Exchang,

    First, each prediction method is modeled based on training data. Then, the risk of each data set was calculated using four classic methods of risk measurement, including Markowitz, Konno, Cai, and Teo. Finally, the risk was calculated using the proposed model. Because the measured risks are not comparable, the performance of each of these methods on investment decision making is considered as the basis of comparison. In other words: first, the investment strategy (buying stock) or the elimination of investment (selling stock) in each stock was adopted based on the measured risk. Then the performance of investment decisions is compared. The main idea in this comparison is based on the expectation that a method that more accurately measures the risk is associated with higher investment performance as well. Therefore, a method that has a higher investment performance can calculate the most favorable risk: a method that has led to more favorable investment decisions is better able to measure risk.


    This section describes how the proposed method is used to measure the risk of each stock, and how the investment decision based on this measured risk was made at nine different risk levels, which were arranged from 0.1 to 0.9 at intervals of 0.1. This investment approach is implemented for all training and testing data, and ultimately the average investment performance was selected as the investment rate of return. In order to determine the different effects of predictive methods on risk measurement and investment decisions, three different prediction methods were selected, including autoregressive, autoregressive moving average, and artificial neural network. In other words, the goal was to evaluate the effectiveness of the proposed method of risk measurement, independent of the predictive method. Therefore, assuming that the transaction cost is ignored, the decision on how to invest in each stock and in each of the 26 tests is summarised in eight steps as follows.

    • 1. The expected return on each stock is determined.

    • 2. The risk of each stock is measured using the method corresponding to the method of deter-mining the expected return.

    • 3. Measured risks are normalized in such a way that in each risk measurement approach and for each asset, the measured risks over consecutive weeks are normalized in the range of [0, 1].

    • 4. Investment decisions are made in relation to each stock on nine different risk levels based on normalized risk and expected returns.

    • 5. As the ultimate goal of measuring risk and de-termining the expected return is using these indi-cators in decision making, each of the expected returns will determine the buying or selling strat-egy. Then, the measured risk will complete the process, and a final decision will be made. Fig-ure 1 shows how investment decisions are made based on measured risk and expected return.

    • 6. The performance of each investment decision is determined as follows: if the correct decision is made, the value of the actual return with the positive sign is considered as the decision per-formance. But, where the decision making does not lead to the correct decision, the actual return with the negative sign is recorded as the decision performance.

    • 7. The average of these performances at different risk levels indicates the performance of invest-ment decisions at each stage of the proposed method tested.

    • 8. The average of method performance in all 26 tests indicates the overall performance of the proposed method on each stock.

    These steps have been done for each of the listed stocks, using the various methods presented in calculat-ing the expected returns and risks. Table 2 illustrates these steps from the implementation of the proposed method of autoregressive moving average -Markowitz on the Karafrin Bank stock.

    The average measured risk for each of the studied stocks, measured using any of the classic methods and the proposed methods, is presented in Table 3. Each cell in Table 3 represents the average of risk incurred for each of the stocks during the 26 tests. The detail of ac-cessing the contents of each cell consists of three prima-ry columns of Table 2. For example, the total risk ob-tained from Table 2 is shown in the first row of the third column of Table 3.

    Results of the measured risk of the studied stocks, using classic methods and proposed methods, show that in the Markowitz risk measurement approach by im-proving the prediction method from autoregressive to the artificial neural network the risk has reduced, also, the measured risk based on the proposed methods has been lower than Konno, Cai, and Teo methods. Figure 2 shows the difference between the risk measured based on the proposed methods versus the Markowitz-based methods.

    According to Figure 2, using the proposed method did not have the same effect on the measured risks. This difference resulted from the different predictability of the data sets. But in general, the proposed method has had more effect on the Markowitz model than the other studied methods. It should be noted, despite, that there is no strict reduction in the measured risk, but there is no significant difference between the measured risks using different proposed methods simultaneously improvement the prediction rate. The statistical tests show that there has been a significant reduction in the measured risk using the proposed methods in this study in comparison to the classical approaches.

    Also, normalizing the measured risks shows that, among these approaches, the Cai approach has the most risk aversion and can only be used for trading in situations where a high level of risk is accepted. In contrast, the Markowitz and the Konno models display the least risk aversion, and they are able to be used to determine whether to buy or sell at lower risk levels. Finally, Table 4 shows the average of the performance of decision making based on the 16 risk measurement methods described for each stock. Each cell in this Table represents the weekly investment rate of return, based on each proposed method and the decision-making strategy. For example, the total performance obtained from Table 2 is shown in the first row of the third column of Table 4.

    Considering that evaluating each risk measurement method relies on the performance of investment decisions, the results show that using the proposed method for measuring risk and investment decision making was associated with more favorable investment performance than using classic methods. In particular, the method based on the Markowitz approach, using the expected returns determined by the artificial neural network, resulted in the best performance. The second most successful performance was achieved using the method based on the Konno approach and the artificial neural network. Therefore, considering data predictability, more risky methods that have the ability to influence decision making at lower risk levels led to more favorable results. In general, in all approaches, methods that used the artificial neural network to determine the expected returns resulted in the best outcomes. None of the basic methods earned a positive return on investment overall at 26 weeks and over 20 stocks. However, among the basic methods, the Cai risk measurement had the least loss. This result is due to the risk aversion aspect of the Cai method. Due to this feature, this method often results in investment decisions at high-risk levels and so faces less chance of loss.


    In general, the most important effect of risk measurement is on investment decisions. Logically, the method that has been shown to be more desirable in decision making had the highest accuracy in determining the risk and return on investment. Accordingly, Table 5 shows comparisons of the performance of different risk measurement methods. Table 5 shows how much their annual rate of return would be if investors made their investment decisions based on each of the risk measurement methods discussed in this paper.

    Figure 3 summarizes the effect of the proposed methods in making investment decisions based on different risk measurement approaches.

    Then by multiplying the weekly rate of return of each stock listed in Table 4 by 52 weeks of the year, each cell in Table 5 is calculated. So table 5 shows the annual rate of return of the studied stocks based on different risk measurement methods.

    This Table shows that using classic methods in making investment decisions in relation to the studied stocks not only leads to no profits but also leads to losses. Meanwhile, making investment decisions based on the proposed method showed a significant increase in profits overusing the basic methods. Also, according to Table 5, if decision-makers make their investment decisions based on the artificial neural network-Markowitz or the artificial neural network-Konno, in most of the studied stocks, they will earn high annual profits. Generally, the use of these two methods results in the highest increase in profits than using the basic methods. In general, as shown in Table 5, the Markowitz approach, because of its higher risk-taking, has a higher potential for profit or loss. Also, the risk aversion of the Cai approach is well illustrated in this Table. In fact, using the Cai approach involves trying to avoid losses ahead of making a profit, so only small profits can be earned.

    Finally, regardless of the predictive methods used in the proposed methods for measuring the risk, the removal of predictable patterns from historical data has led to improved decision making. Also, as can be seen, if predictable patterns in historical data are eliminated by a powerful method such as an artificial neural network, what remains is most closely related to the assumption that the predictability of data is the same, which is considered by classic methods. Therefore, the performance of decision making based on the use of the artificial neural network is more desirable. Finally, Table 6 shows that each of the methods discussed in this paper has shown the percentage of profitable earnings earned.

    According to Table 6, although the proposed methods succeeded in increasing profits significantly, in general, all of these methods have, at best, realized less than 36 percent of their potential profits. It suggests that the prediction methods used to remove the predictable patterns of the studied systems have not been able to eliminate all the patterns, and despite the significant advances made by the proposed method, the path to improving the methods is still open. Finally, in order to more describe the effect of the proposed methods in investment decisions, the results of the investment decisions in the three stock exchanges, including the Tehran Stock Exchange, S&P, and DAX, are compared. Table 7 shows the results of investment decisions at different risk levels based on the Markowitz approach in each of these markets.

    According to Table 7, previous results in relation to the proposed risk measurement methods are confirmed. So using the ANN-Markowitz risk measurement method has improved investment decisions. Figure 4 summarizes the impact of prediction- based risk measurement methods, including AR-Markowitz and ANN-Markowitz on the efficient frontier and performance of investment decisions.


    Risk is an important aspect of decision making in many activities. Therefore, different methods of risk measurement have been presented in the literature. Most of these methods, considering the same predictability for different systems, ignore predictable patterns in each system. For this reason, the risk measured in classical methods is not sufficiently accurate. The existence of this inaccuracy in the measurement of risk becomes more important when making the investment decision based on this risk measurement. In this situation, classical risk measurement methods will be associated with the loss of many profit opportunities. But, in today’s competitive world, it is not wise to ignore the slightest available information to achieve more profit. Hence, the basis of this study was to provide a method for measuring risk using predictability in risk measurement. In fact, the proposed method attempts to show better performance in decision making by identifying extractable patterns of data.

    Therefore, the proposed methods of risk measurement, by using predictive methods in determining the expected returns, first remove the extractable patterns of data. Then, the risk of the residual is measured when the predictability of the data has been somewhat reduced. Therefore, the proposed method, as opposed to classical methods, will lead to more accurate risk measurement. The performance of each method is measured in the context of investment decision making to test the proposed method against the previously used methods. For this purpose, the percentage of return on investment decisions based on each proposed method and the classic methods was calculated for 20 selected stocks on the Tehran Stock Exchange during 26 periods. The results show that the proposed method is more effective in studied stocks. Finally, it should be noted that the implemented models in this paper were crisp. Therefore, the development of risk measurement models to fuzzy prediction based risk measurement models will lead to more accurate risk measurement. So, using uncertain time series prediction methods in the prediction-based risk measurement models is suggested in future studies.



    Investment decision-making diagram.


    The difference between the risk measured based on different methods.


    The effect of the proposed methods on investment decisions.


    The impact of prediction- based risk measurement methods on the efficient frontier and performance of investment decisions.


    The under-study stocks

    The implementation of autoregressive moving average Markowitz method on the Karafrin bank stock

    The average of measured risk for the studied stocks

    Average of the performance of decision making

    The annual rate of return of each studied stock based on risk measurement methods

    Percentage of profits identified by each method

    The results of investment decisions based on the Markowitz approach in three stock exchanges


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