Research article

Wavelet-based systematic risk estimation: application on GCC stock markets: the Saudi Arabia case

  • Correction on: Quantitative Finance and Economics 7: 117-118.
  • Received: 02 June 2020 Accepted: 01 September 2020 Published: 07 September 2020
  • JEL Codes: G11, C02, C22

  • Systematic risk estimation is widely applied by investors and managers in order to predict risks in the market. One of the most applied measures of risk is the so-called Capital Asset Pricing Model, shortly CAPM. It has been studied empirically focusing on the impact of return interval on the betas. This paper lies in this topic and attempts to estimate the CAPM at different time scales for GCC markets by adapting a wavelet method to examine the relationship between the return of the stock and its systematic risk at different time scales. The main novelty is by applying non-uniform intervals of time. Differently from existing literature, we use random ones. The proposed procedure is acted empirically on a sample corresponding to Saudi Tadawul market as the most important GCC representative market actively traded over the period January 01, 2013 to September 20, 2018, which is characterized by many political, economic and financial movements such as Qatar embargo, Yemen war, NEOM project, 2030 KSA vision and the Arab spring effects. The findings in the present work may be good basis for understanding current and future GCC markets situation and may be thus a basis for investors' decisions in such markets.

    Citation: Anouar Ben Mabrouk. Wavelet-based systematic risk estimation: application on GCC stock markets: the Saudi Arabia case[J]. Quantitative Finance and Economics, 2020, 4(4): 542-595. doi: 10.3934/QFE.2020026

    Related Papers:

    [1] Anouar Ben Mabrouk, Sabrine Arfaoui, Mohamed Essaied Hamrita . Wavelet-based systematic risk estimation for GCC stock markets and impact of the embargo on the Qatar case. Quantitative Finance and Economics, 2023, 7(2): 287-336. doi: 10.3934/QFE.2023015
    [2] Anouar Ben Mabrouk . Wavelet-based systematic risk estim-ation: application on GCC stock markets: the Saudi Arabia case. Quantitative Finance and Economics, 2023, 7(1): 117-118. doi: 10.3934/QFE.2023006
    [3] Raéf Bahrini, Assaf Filfilan . Impact of the novel coronavirus on stock market returns: evidence from GCC countries. Quantitative Finance and Economics, 2020, 4(4): 640-652. doi: 10.3934/QFE.2020029
    [4] Samuel Kwaku Agyei, Ahmed Bossman . Investor sentiment and the interdependence structure of GIIPS stock market returns: A multiscale approach. Quantitative Finance and Economics, 2023, 7(1): 87-116. doi: 10.3934/QFE.2023005
    [5] Fangzhou Huang, Jiao Song, Nick J. Taylor . The impact of business conditions and commodity market on US stock returns: An asset pricing modelling experiment. Quantitative Finance and Economics, 2022, 6(3): 433-458. doi: 10.3934/QFE.2022019
    [6] Ke Liu, Changqing Luo, Zhao Li . Investigating the risk spillover from crude oil market to BRICS stock markets based on Copula-POT-CoVaR models. Quantitative Finance and Economics, 2019, 3(4): 754-771. doi: 10.3934/QFE.2019.4.754
    [7] Ngo Thai Hung . Equity market integration of China and Southeast Asian countries: further evidence from MGARCH-ADCC and wavelet coherence analysis. Quantitative Finance and Economics, 2019, 3(2): 201-220. doi: 10.3934/QFE.2019.2.201
    [8] Ick Jin . Systematic ESG risk and hedge fund. Quantitative Finance and Economics, 2024, 8(2): 387-409. doi: 10.3934/QFE.2024015
    [9] OlaOluwa S. Yaya, Miao Zhang, Han Xi, Fumitaka Furuoka . How do leading stock markets in America and Europe connect to Asian stock markets? Quantile dynamic connectedness. Quantitative Finance and Economics, 2024, 8(3): 502-531. doi: 10.3934/QFE.2024019
    [10] Mohammad Ashraful Ferdous Chowdhury, M. Kabir Hassan, Mohammad Abdullah, Md Mofazzal Hossain . Geopolitical risk transmission dynamics to commodity, stock, and energy markets. Quantitative Finance and Economics, 2025, 9(1): 76-99. doi: 10.3934/QFE.2025003
  • Systematic risk estimation is widely applied by investors and managers in order to predict risks in the market. One of the most applied measures of risk is the so-called Capital Asset Pricing Model, shortly CAPM. It has been studied empirically focusing on the impact of return interval on the betas. This paper lies in this topic and attempts to estimate the CAPM at different time scales for GCC markets by adapting a wavelet method to examine the relationship between the return of the stock and its systematic risk at different time scales. The main novelty is by applying non-uniform intervals of time. Differently from existing literature, we use random ones. The proposed procedure is acted empirically on a sample corresponding to Saudi Tadawul market as the most important GCC representative market actively traded over the period January 01, 2013 to September 20, 2018, which is characterized by many political, economic and financial movements such as Qatar embargo, Yemen war, NEOM project, 2030 KSA vision and the Arab spring effects. The findings in the present work may be good basis for understanding current and future GCC markets situation and may be thus a basis for investors' decisions in such markets.



    In this work, we focus on developing a best mathematical approach for the so-called systematic risk or capital asset pricing model using wavelet theory. Recall that such a theory has been proved to be a powerful tool in the economic/financial field. Financial indicators are subject to volatility and high fluctuations in all markets, which makes their study and their understanding using classical methods un-sufficient. One of the main power characteristic of wavelets is their ability to detect and/or localize fluctuations and volatility.

    Financial markets are the essential component of countries' economies. All industrial countries have at least one purse and most developing countries have created one or have to create. Financial markets have enabled economic agents to reconcile the antagonistic objectives of their clientele. These are objectives of profitability, security and liquidity. To obtain portfolios that combine high levels of profitability with high levels of security and thus being less risky, financial market participants have several financial asset management instruments. Earlier, in the 50th of the last century, Markowitz (1952) has put the basics of the theory of financial asset management and the functioning of the financial markets leading to a rigorous link between risk and return on securities. Next, the studies have been growing up in both theory and empirically. See Sharpe (1964), Lintner (1965a), Lintner (1965b), Mossin (1966) and Black (1972), Black et al. (1972). Henceforth, the CAPM become a central model in financial theory that makes it possible to describe in a simple way the link of profitability of financial assets and their risks. The validity of the CAPM has been by the next investigated by Roll (1977) based on USA market and stating that the omission of some low capitalization securities could affect the CAPM. In addition, all assets such as bonds, gold and real estate should be taken into account.

    More recently, the CAPM has been improved by several empirical studies. In particular, Fama and French (1996) announced the death of systematic risk beta, using annual data of the well known indexes NYSE, AMEX, and NASDAQ returns during the period time 1963 to 1990. They stated that the systematic risk beta may not provide a strong indication for explaining the average change in the market index. Re-considered next by Kothari and Shanken (1998), results of Fama and French (1993) have been proved to be more significant by using monthly data on equity returns rather than annual data. Even though, Kothari and Shanken (1998) noticed that the use of annual returns to estimate the beta may be a cause of measurement problems. This is essentially due to the seasonal effect of returns, the non-synchronization of actions, etc. Based on the annual returns from 1927 to 1990, the authors concluded that the beta is statistically significant and that the regular contribution of size to explain the difference in sample yielded beyond the beta is minimal. For a deep backgrounds on CAPM the readers may refer to Aydogan (1989), Banz (1981), Basu (1977), Black (1972), Black et al. (1972), Breeden (1979), Chae and Yang (2008), Chan and Lakonishok (1993), Fama and MacBeth (1973), Fama and French (2004), Fama and French (2006), Galagedera (2007), Gibbons (1982), Gursoy and Rejepova (2007), Handa et al. (1989), Handa et al. (1993), Ho et al. (2000), Karan and Karadagli (2001), Merton (1973), Perold (2004).

    Some empirical extensions of the CAPM taking into account the time variations of beta and time variations of the risk premium or both of them have been developed. The so-called CAPM Conditional Testing has been established joining an old study on the same subject due to Levhari and Levy (1977) where the use of a shorter time than real time has been proved to provide biased beta estimator. Handa et al. (1993) showed that if the difference in the time interval of the returns is taken into account, for the same action, different estimators can be obtained. Furthermore, Handa et al. (1993) rejected the beta estimator that uses monthly returns and accepted instead the use of annual returns. Cohen et al. (1986) showed that the beta estimator is sensitive to the time intervals used for equity returns. As a result, the problem of the validity of the CAPM arises, and more particularly the validity of the relationship between the systematic risk and the return of securities listed on the financial markets taking into account the time factor in the returns of the shares. This key issue of the CAPM audit, which has already been the subject of extensive research in industrialized countries, is still relevant.

    As mentioned by Marfatia (2017a) and Marfatia (2017b), one of the limitations of conventional approach is the lack of distinction between the time domain and the frequency ones which is a crucial task in both econometric and economic rationale.

    The present study lies in the whole scope of the integration and/or the study and evaluation of risks in stock markets according to time changes. Many studies have been developed to do such task. In Marfatia (2017a) a wavelet-based study has been developed to investigate the impact of risks in international stock markets. The approach proposed in Marfatia (2017a) consists in combining wavelet techniques with time-varying conditional volatility to investigate essentially the co-movement of risks at both the country level and regional level. The authors concluded that co-movement of risks between the US market and European markets are strong mostly at lower frequencies. However, the co-movement of risks between the country and its local region are strong at higher frequencies. Moreover, at high frequencies the relationship between the country in hand and the US or other global markets is also weak. The study concluded also that the spill-over of risks are mostly limited at lower frequencies contrarily to existing studies which claim that especially during the recent financial crisis, the spill-over of risks are largely a global phenomenon.

    In Marfatia (2017b), linkages of the housing market and macroeconomy has been investigated using wavelet approach. Cross-wavelet coherence has shown that the relationship varies significantly relatively to countries, time, frequencies, and the direction of causation. The author claimed that house price movements are related to interest rates at short-run, however per capita income growth is related to the interest rates at the long-run. Moreover, the author studied the role of industrial production and income growth and concluded that the direction of causation between the housing market and macroeconomic variables depends strongly on time-frequency domain.

    Our main aim here is to understand the nature of the relationship linking systematic risk and the return on equities for GCC stock markets by using wavelet theory as a mathematical tool recently introduced in the field of finance and proved its performance compared to classical tools. A new method to forecast systematic risk based on wavelets for missing data and further exploring the relationship between the return of the stock and its systematic risk at different time scales is proposed.

    Wavelets offer efficient algorithms for practical problems where classical techniques have shown limitations. Moreover, they provide an attractive mathematical formalism in the reformulation of several problems and in different scientific fields, especially in time series analysis. Wavelets form a mathematical tool that transforms time domain data into different frequency horizons. They represent the advantage of being localized in both the time and frequency domains. They allow to observe and to analyse data at different time scales, which in turns makes it possible to overcome the inadequacies of the classical analysis of the CAPM. For more backgrounds on wavelets and their applications the authors may refer to Arfaoui et al. (2017), Arfaoui et al. (2020), Arfaoui et al. (2020), Mabrouk et al. (2008), Mabrouk et al. (2008), Mabrouk et al. (2010), Mabrouk et al. (2011), Mabrouk and Zaafrane (2013), Mabrouk et al. (2015), Mahmoud et al. (2016), Percival and Walden (2000), Selcuk (2005), Zemni et al. (2019a), Zemni et al. (2019a).

    From an empirical point of view, we aim to apply the wavelet technique to estimate the systematic risk of an action relatively to the market return. This validity could also be a real foundation for effective financial decisions. Indeed, the effectiveness of a financial decision depends to a large extent on the accuracy of the valuation of the securities as well as the most precise knowledge possible of their subsequent evolutions and their risks. We therefore try, in this work to provide a better comprehension of the GCC financial markets in the face of a modern financial theory such as the CAPM.

    To resume, one aim in the present research is to find a mathematical approach to the GCC market indices by applying wavelet methods. The new idea consists in applying random and/or different time supports allowing the subdivision of the whole support into random parts instead of using the classical sampling (weeks, months, years). We recall in particular, that the Kingdom of Saudi Arabia has established 2030 projects that have a direct impact on the national market as well as the rest of the Arab Gulf and international markets such as and particularly NEOM project which will have a profound impact on the market. This makes it of interest to study such market and understand its complexities.

    The proposed procedure is acted on samples composed of stocks in GCC markets actively traded over a critic period strongly and directly related to the last political changes especially in Arab countries, the GCC embargo against Qatar country. The period may also be considered as a pre-corona stage as we now see that the current pandemic COVID-19 has been spread in quietly all the world. GCC continent is one of the places that are related to all the world because of their geographical position and their strong relation to the worldwide economies as biggest petroleum countries. Moreover, the largest workers community is focusing in these countries. Saudi Arabia is also characterized by the saint cities of Muslims which consequently bring a big number of pilgrims and visitors each year. This may be a strong cause of dispersion of viruses which may be next transferred with pilgrims to other continents. Saudi Arabia has also made a worldwide program known as NEOM international project which is the basic point in the 2030-vision of the kingdom. The sample of study will be based on Tadawul index during the period from January 01, 2013 to September 20, 2018. Notice that this period is also characterized by the direct link to Syrian and Yemen movements and the present Arab war against some parties in Yemen which is somehow leaded by Saudi Arabia. We will discuss the effect of such links on the stability of GCC markets. We may also recall many financial movements such as Aramco subscription and generally the 2030 KSA vision. The findings in the present work may be a good basis for understanding GCC markets situation, behavior and future and thus a basis for investors' decisions in such markets.

    We recall that in the data basis applied, the main problem confronted is the lack of data where the samples present missing values for many cases. Consequently, our work becomes a twofold study. In a first step, we developed a wavelet-based method to reconstruct missing data which by the next leads to complete and adjusted basis on the whole period above. The complete basis is applied next for CAPM and thus the comprehension of the Saudi Tadawul market.

    The rest of the present work is organized as follows. In section 3, a literature review on forecasting systematics risk is briefly presented. Section 4 is devoted to the development of our methodology. We especially re-develop the wavelet analysis of time series briefly to apply it by the next for missing data reconstruction which in turns will be applied for completing the data basis used later. The mathematical formulation of the CAPM is provided by the next. Finally, this section is achieved by the development of the wavelet CAPM. Section 5 is subject of our empirical results and their discussions on the KSA Tadawul stock market. Section 6 is a conclusion. Finally, section 7 is an appendix in which we provided empirical results of the 3-scale law as a supplementary part to section 5 to improve the idea of time changes in stock market prediction.

    We propose in this section to conduct a literature review on the systematic risk forecasting. However, we will not be addressing mathematical formulations in this section and include them instead in the methodology section to provide readers a more comprehensible mathematical methodology.

    The CAPM was firstly discovered by Sharpe (1964) and next Lintner (1965a), Lintner (1965b). It has been applied next for half a century in estimating the capital cost for companies and evaluating the performance of managed portfolios. However, its empirical evaluation has been always affected by the availability or not of the data. The empirical problems of the CAPM might reflect theoretical failures arising from many superficial assumptions.

    The CAPM is based indeed on several assumptions, such as mono-periodicity, the market perfection aspect without taxes nor transaction costs, homogeneity in anticipations, unlimited short selling, loans and borrowing at the risk-free and limitless rate, strictly increasing and strictly concave Von-NeumannMorgenstern utility functions, mean-variance preferences based on restrictions relating to the return or the utility function, investor aversion to risk, competition and market efficiency. However, such assumptions might not be generally accepted simultaneously, which yields limitations for the CAPM. As a consequence, some solutions have been proposed such as the CAPM with taxes (Brennan (1973), Litzenberger and Ramaswamy (1979)), the CAPM with transaction costs (Lévy (1978)), the CAPM in continuous time (Merton (1973)) and the CAPM with non-homogeneous anticipations (Sharpe (1970a), Sharpe (1970c), Sharpe (1970b)).

    The choice of a portfolio is generally made after a financial analysis of a set of more or less independent actions. Robust measurement of the crucial variables in the investment decision should be ensured. Numerous studies have investigated the financial analysis of individual actions allowing to make a good decision. Markowitz (1952) has shown that the investor seeks to optimize his choices by taking into account the expected return on his investments and the risk of his portfolio. His model suggests selecting several stocks based on statistical criteria such as the profitability of a stock in order to obtain optimal portfolios. However, Vasichek et al. (1972) asserted that the profitability of a stock is not sufficient to characterize an investment opportunity. The dispersion of returns around the average return reflects the uncertainty of the investment (the risk). The total risk of a portfolio can always be measured by the variance or the standard deviation of its profitability which constitutes a convenient measure of dispersion. This allows investors to minimize the risk of their actions.

    According to Markowitz, the CAPM is used to solve the problem of the portfolio structure which incorporates the quantified treatment of risk by estimating the demand function of assets. This makes it possible to study the equilibrium of the market. Such a method has a main drawback as it does not take into consideration the time factor, which causes a strong limitation.

    The CAPM is a diagram that analyses the return and risk of an investment. Sharpe (1964) and Lintner (1965b) suggested to estimate the prices of transferable securities allowing supply and demand to be balanced and allowing a general equilibrium of the market. Therefore, the CAPM can determine the prices of equilibrium securities through supply and demand according to a linear relationship between portfolio profitability and total risk. Indeed, the level of risk is obtained by varying the loan or the borrowing which can be made up of all the securities listed on a market. Thus, any combination of an equity portfolio and an investment in the risk-free asset will be indicated by a straight line in the risk-return space. Therefore, any portfolio, by definition, is perfectly diversified, since it involves all stocks in the proposals of their market capitalizations.

    Mathematically, for a market in equilibrium, for any portfolio or asset, the CAPM is expressed by means of a linear relationship between the expected return on the share or portfolio i and the market premium where the linearity coefficient is often denoted βi and is known as systematic risk or beta. This coefficient is mainly used for two purposes. The first involves the ranking of assets and portfolios against systematic risks by practitioners. The second aims to test the mean-variance efficiency. Beta is usually estimated using the standard market model, which is expressed by the linear regression model. It also represents the coefficient of elasticity of the price of the security with respect to the stock index representing the market.

    Sharpe (1964) proved that for each share i relating to a given portfolio the systematic risk βi is expressed by the quotient of the covariance between the rate of return of asset i and the rate of return of the portfolio by the square of market risk. More precisely, the expected return on security i is expressed by the risk-free interest rate increased by a risk premium composed of two elements. The first element is the excess of the expected market rate of return over the return on the risk-free asset, measuring the risk premium that the investor should perceive by agreeing to bear a risk equal to that of the market. The second element βi is the measure of the importance of the risk of security i in relation to the risk of the market. Sharpe (1970b) and Vasichek et al. (1972) concluded firstly that the CAPM is purely normative, subject to empirical validation and secondly that the rate of return of each share in excess of the risk-free interest rate depends only on βi.

    The CAPM gives a coherent answer, as for the evaluation of the expected profitability of an asset according to the risk. This profitability can be used as the discount rate in the valuation of the asset. Indeed, the relationship between the risk and the profitability of a financial asset is linear, provided that this risk is measured by its variance with the market taken as a whole and not its variance or its standard deviation. Each investor has the choice between obtaining a certain but low profitability or taking a risk offset by a higher expected profitability. Markowitz was the first to formalize and quantify the diversification effect according to which a combination of many assets in a portfolio reduces the total risk for a given expected rate of return.

    Sharpe W. F. considered a model which results in the equilibrium relationship between the expected profitability of a financial asset and its risk. This model makes it possible to determine the choice of investment and to compare alternatives to different random gains by using the Esperance-Variance criterion in order to choose the best alternative.

    In the same context, Von Neuman and Morgestern in the 1944-th formally proved that any individual yielding to a few intuitive axioms of rationality aims to maximize the expectation of the utility of their wealth. Indeed, the utility function reflects the preference of each individual, hence each individual is specific, and depends in particular on his initial wealth at the time of the decision and his aversion to risk. The conclusion lies in the fact that the individual faced with alternatives with random consequences aims to maximize the expectation of utility.

    Diversification is also an important factor in estimating the CAPM. It is strongly related to the behaviour of the expectation-variance of the portfolio. The profitability of the portfolio is the weighted average of the expectations. The contribution of each security to the expected profitability of the portfolio is therefore proportional to its expected profitability. In terms of risk, it is possible to measure that of the portfolio by the variance of its profitability.

    According to Markowitz, an efficient portfolio is characterized by a maximum expectation of return or by a minimum variance for a given expectation of return. The efficient frontier is the set of all efficient portfolios. The efficient frontier takes two different forms depending on the absence or presence of a risk-free return asset. The minimum variance frontier determines portfolios of risky assets and shows the lowest risk for a given level of profitability. As for the portfolios with the highest levels of profitability for a given level of risk, they are represented by the efficient frontier corresponding to the upper part of the frontier at minimum variance.

    However, several criticisms have been pointed out for the CAPM (Desmoulins-Lebeault (2003), Magni (2007a), Magni (2007b)). Some are linked to the portfolio and some are linked to market. Roll (1977) introduced an empirical criticism due to the representative portfolio of the market. This portfolio groups together assets held by a set of investments, a set of stocks, bonds and other types of securities that are not in practice negotiable and liquid such as real estate assets of investments and their human capital. This puts into question the results of the econometric tests and the practical utility in terms of management and performance analysis. Roll (1977) claimed that according to CAPM, the ideal portfolio is able to positively manage the different variables. Roll (1977) discussed the problem of the right of the market. It is a conceptual order that challenges the theoretical foundations of investment performance indicators over another period of time. If the market portfolio is efficient, securities are on average to the right of the market. It is therefore sometimes impossible to observe deviations from this line over a long period. In reality, these deviations do not indicate whether a security is over or under-priced.

    The estimated betas are relative for the entire period studied and assumed to be stable relatively to a certain scale. While in reality the variance with the market varies over time and therefore there may be some statistical issues related to errors in the beta estimates of individual securities and their instabilities which are not taken into account.

    Overall, the CAPM formulation showed remarkable empirical robustness in determining excess return, taking systematic risk into account. CAPM is characterized by a special case which forms several types of risk into one defined by systematic risk (non-diversifiable risk or market risk). In this sense, two securities with the same level of risk can lead to different profits. Investors tend to choose the security with the highest profit. This choice will lead to a drop in profit. However, if there are two portfolios A and B of the same risk, where B is the optimal portfolio according to the CAPM, portfolio A generates a higher return than that of B. In such a case, investors who seek new opportunities will be willing to invest in A although it is undervalued by the CAPM, and to sell the optimal portfolio B which has the same systematic risk.

    Besides, other studies have focused on empirical extensions of the CAPM taking into account the time variations of the risk beta or variations in the time of the risk premium or both of them. This area of research is often referred to as the CAPM conditional test. Studies on this topic show the major impact of time intervals on the estimate of beta, understanding of the market and the economic policies in general (See for instance Levhari and Levy (1977), Marfatia (2014), Marfatia (2015), Marfatia (2017a), Marfatia (2017b), Marfatia (2020)). Handa et al. (1993) showed that if the difference in the time interval of returns is taken into account, for the same action, we may obtain different estimators. Handa et al. (1993) rejected the CAPM beta estimator which uses monthly returns, but accepted the one based on annual returns. Cohen et al. (1986) showed that the beta estimator is sensitive to the time intervals used for stock returns.

    Consequently, the problem of the validity of the CAPM according to time variation arises. More precisely, the validity of the relation between the systematic risk and the return of the securities quoted on the financial markets taking into account the time factor in the returns of the equities seems to be necessary. This essential issue of CAPM verification is still relevant today. To our knowledge, few works have looked specifically at this question, in particular by using a method as powerful as that of wavelets. The time-scale division induced by the wavelet transform overcomes the shortcomings of the classical CAPM analysis.

    Our objective here is to test the validity of the CAPM on the Saudi Arabia Tadawul market. We aim firstly to analyze the systematic risk and to point out its limitations taking into account the time factor. The second is to identify the relationship between equity returns and their systematic risks in the Saudi Arabia Tadawul market using the wavelet approach taking already the time factor into account.

    In our knowledge, the application of wavelets for GCC stock markets is still recent and the literature on it needs more developments. However, in other markets such a tool is now developed and has induced more understanding for the markets analyzed such as SP500 (USA), CAC40 (France) and ISE (Istanbul). Gençay et al. (2003) applied wavelets for the stock markets of the US, UK and Germany to estimate the best time scale for measuring systemic risk. The authors concluded that the relationship between risk and return is a multi-scale phenomenon. Fernandez (2006) analyzed the Santiago stock market in Chile using time-scaling methodology. Rhaiem et al. (2007a), Rhaiem et al. (2007b) studied the French CAC40 index as market portfolio and the daily EURIBOR as the risk-free rate. The predictions of the CAPM are claimed to be more relevant in the short term than in the long term, which makes the French market different from those of the US, UK and Germany. Aktan et al. (2009) applied the wavelet multi-scaling method for the Istanbul Stock Exchange during the period from January 2003 to October 2007. It is shown that a positive relationship between risk and returns is most significant at the medium levels, concluding that the effect of market returns on an asset is stronger in this time horizon.

    Before introducing the methodology applied in our work it is necessary to recall some basic facts. Indeed, one of the important factors that may affect the model described here may be the US policy actions and the risk perceptions which significantly impact international stock markets. However, we did not include these factors in the present model as our aim is not to change the CAPM model which already exists but to improve it firstly by using the wavelet time-frequency action. This fact may be considered as a limitation for the present work and opens instead a good idea for an eventual extension. Besides, some recent studies have discussed the role of these factors and their estimations in other models which confirms their importance and the motivation to include them in any eventual extension of the present model. Indeed, in Kishor and Marfatia (2013) time-varying response of foreign stock markets to US monetary policy shocks has been estimated. The authors noticed a significant time-variation in the response of the global equity markets to US monetary policy surprises especially during the crisis periods. The model applied looks like the present one as it aims to estimate a risk-like parameter βi as follows

    Rit=αi+βiΔrut+eit (1)

    where Rit is the abnormal return of country i at the event date t and Δrut is the monetary policy surprise. The coefficient βi reflects the response of the abnormal return of a country i's stock market to unanticipated interest rate increase in the US.

    Now our methodology consists in a first step to apply wavelet decomposition of time series to overcome the problem of missing data confronted when gathering the data applied in our work. Next, as the data is completed the whole basis will be subject of both CAPM and wavelet CAPM processing.

    Wavelet analysis allows the representation of time series into species relative to the time and frequency information known as time-frequency decomposition. It consists in decomposing a series in different frequency components with a scale adapted resolution and thus permits to observe and to analyze data at different scales. Wavelet analysis starts from one source function ψ known as the mother wavelet and next composes dilation-translation copies to get a complete system for finite energy time series. Each wavelet basis element is defined for j,kZ as a copy of ψ at the scale j and the position k by ψj,k(t)=2j/2ψ(2jtk). The quantity 2j corresponds to the frequency of the series while the index k localizes volatility or fluctuations. Let for jZ fixed, Wj=span(ψj,k,k) known as the j-level detail space. A time series X(t) is projected onto Wj yielding a component DXj(t) given by

    DXj(t)=kdj,kψj,k(t) (2)

    The dj,k are the detail coefficients of the series X(t) expressed by means of the ordinary inner product in the functional space L2(R) as

    dj,k=<X,ψj,k>=RX(t)ψj,k(t)dt (3)

    The spaces Wj's form an orthogonal decomposition covering the space of finite energy series L2(R). This means that the series X(t) can be completely reconstructed as a sum of its projections on the detail spaces and that these projections are mutually uncorrelated. In wavelet theory, the mother wavelet yields a second function called father wavelet or scaling function denoted here by φ. (See Daubechies (1992)). Similarly to ψ, the function φ yields dilation-translation copies φj,k(t)=2j/2φ(2jtk) generating subspaces Vj. The sequence (Vj)j is called a multi-resolution analysis (multi-scale analysis) on R and Vj is called the j-level approximation space. It is well known in wavelet theory that VjVj+1, jZ, which means that the approximation of the time series at the level j and j+1 can be viewed from each other and so from any horizon pj+1. In physics-mathematics this is called the zooming rule. It holds also that for all jZ, f(t)Vj iff f(2t)Vj+1, which reflects the fact that, not only the signal f from horizon j can be seen in the horizon j+1 but also his contracted or dilated copies. As for the detail subspaces, the approximation subspaces Vj's satisfy also a completeness relation meaning that no information is lost when considering all approximations and a second property meaning that all the information is lost at finer scales. Finally the Vj's satisfy a shift-invariance property in the sense that f(t)Vj iff f(tk)Vj, j,kZ, which means that the multi-resolution analysis permits to detect the properties of the signal along the whole time support. Combining all the properties above we deduce that the approximation space is decomposed into a low-level approximation part supplemented with a detail one. Under these properties, the following decomposition is proved for jZ,

    X(t)=jDXj(t)=jJDXj(t)+jJ+1DXj(t) (4)

    The component AXJ(t)=jJDXj(t) is called the approximation of X(t) at the level J and it reflects the trend or the global shape of X(t). It also belongs to the space VJ. Thus, using the definition of the VJ's, the component AXJ(t) may be expressed using the basis (φJ,k)k as

    AXJ(t)=kaJ,kφJ,k(t) (5)

    where the aJ,k are the approximation coefficients of the series X(t) expressed by aJ,k=<X,φJ,k>. As a result, we obtain the following relation known as the wavelet decomposition of X(t)

    X(t)=AXJ(t)+jJ+1DXj(t) (6)

    It is composed of one part reflecting the global behavior of the series and a second part reflecting the higher frequency oscillations or the fine scale deviations of the series near its trend. In practice we cannot obviously compute the complete set of coefficients. We thus fix a maximal level of decomposition J and consider the decomposition for any J0<J,

    XJ(t)=AXJ0(t)+J0<jJDXj(t). (7)

    There is no theoretical method for the exact choice of the parameters J0 and J. However, the minimal parameter J0 does not have an important effect on the total decomposition and usually chosen to be 0. But, the choice of J is always critical. One selects J related to the error estimates.

    In finance, economics, management and generally actuarial sciences, compared to classical theories wavelet analysis is still less used although it proved good results and needs to be more developed. Recently the literature starts growing rapidly. See Arfaoui et al. (2017), Arfaoui et al. (2020), Arfaoui et al. (2020), Mabrouk et al. (2008), Mabrouk et al. (2008), Mabrouk et al. (2010), Mabrouk et al. (2011), Mabrouk and Zaafrane (2013), Mabrouk et al. (2015), Conlon et al. (2008), Cifter and Ozun (2007), Cifter and Ozun (2008), DiSario et al. (2008), Fernandez (2006), Gençay et al. (2002), Gençay et al. (2003), Gençay et al. (2005), Mahmoud et al. (2016), In and Kim (2006), In and Kim (2007), In et al. (2008), Percival and Walden (2000), Selcuk (2005), Sharkasi et al. (2006), Xiong et al. (2005), Yamada (2005), Zemni et al. (2019a), Zemni et al. (2019a).

    As mentioned in the introduction, the main problem in studying markets' movements and/or situations is the lack of data and sometimes its uncertainty. This leads researchers to develop prior procedures to complete the data used in the market study. In our situation here this problem is present strongly in the data used. We noticed that the origin of our data results in some missing values in many actions shared in the market index used here and which is the well-known Tadawul index of the Saudi stock market. To overcome this problem and instead of searching left and right for the data and confronting may be with the same problem in other origins, we proposed to act a wavelet method to reconstruct missing values. The data basis is essentially extracted from the web site www.investing.com

    The present section is devoted to present the wavelet-based method to reconstruct such missing data. The method consists in providing a prediction procedure able to predict a short time interval series on an arbitrarily set of backwards and/or forwards (past and/or future, prior and/or post) values. Recall that generally to conduct a best prediction or reconstruction of time series we need usually a long-time interval for training. This fact may not be satisfied in general situations. For some situations such as ownership-structure and diversification variable the samples are usually short. One has one main value on a year. Furthermore, when applying wavelet analysis to approximate and/or to forecast time series, the majority of the existing studies assume the presence of some seasonality, periodicity and/or autoregressive aspect in the series. See for example Mabrouk et al. (2010), Soltani (2002), Soltani et al. (2007) and the references therein.

    In the present paper, we act a simple method already tested in Mabrouk et al. (2010) leading to good prediction. The method is principally characterized by the non-necessity to test it on the detail parts components of the series nor its wavelet coefficients. This is essentially due to the fact that we use few values of series leading to short sub-samples and next act the prediction on the sub-samples. In such parts the dynamic behavior is not important. However, the most positive point in the method is the fact that it necessitates only to compute the values of the source scaling function and the associated wavelet on a suitable grid, the dyadic or the integer grid in the supports of the mother and father wavelets.

    Let X(t),t=1,2,,N, be a time series. The procedure to be applied is a dynamic recursive scheme consisting in applying firstly a partial estimator at short horizons to all the observations (ti,Xi),i=1,2,,N, to yield firstly the predicted value of XN+1. This last is then included as new observation to predict XN+2. We then follow the same steps until reaching the desired horizon. During the procedure, we apply the J-level wavelet decomposition (4). This necessitates to know the values of the mother wavelet ψ on the dyadic grid {2j(N+1)k,k} and the scaling function φ on the integer grid {N+1k,k} in the supports. Assume we have got XJ(N). The next value will be estimated by (J0=0 in (7))

    XJ(N+1)=ka0,kφ(N+1k)+Jj=1kdj,kψ(2j(N+1)k) (8)

    This means that for evaluating the predicted value of XN+1, it suffices to do this for AXJ and the DXj's. This motivates the use of Daubechies compactly supported wavelets which are well evaluated on the integer grid.

    The selection of a portfolio is usually preceded by a financial analysis of a set of stocks. A good analysis provided with a successful quantification of the important variables of the investment decision on the financial markets will be a central factor for investors success. One of the variables is the profitability, defined as the sum of capital gains and dividends reported at the beginning of the period. The expected profitability of a portfolio is equal to the weighted average of the expected returns on the different securities that make it up. A second factor is the risk of an action. Recall that the dispersion of returns near the average profitability reflects the uncertainty or the risk of the investment. The standard deviation and/or the variance are convenient measures of this dispersion. To avoid risk, investors prefer investing with a low variance.

    In its original and/or simple variant, the CAPM is mathematically expressed as

    Ri,t=αi+βiRm,t+ui,t (9)

    where Ri,t is the return rate of an action i at the period time t, Rm,t is the return of the market measured by means of a general index at the same period of time t. The parameter βi is a specific factor to each action i, indicating the relation between the fluctuations of the action i return rate and the fluctuations of the general index of the market, called often the beta coefficient or the systematic risk. The factor ui,t is a random factor representing the hidden fluctuations of Ri,t that are not explained by the market, or generally an error term. Finally, the parameter αi is added to guarantee a null expectation of ui,t. Therefore, CAPM breaks down the total variability of an action into two parts. A first part, due to the influence of the market and which corresponds to the systematic risk and a second due to the specific characteristics of the action and which corresponds to the variations of the specific prices of such action, called sometimes the diversified or the specific risk. The CAPM allows to study the behavior of all operators in a market and to build a theory of equilibrium. The market is a global reference framework defining the conditions that prevail in all the transactions that economic agents carry out. It allows the confrontation of supply and demand. The equilibrium is reached under many hypotheses such as the null transaction cost, the perfect divisibility of assets, free-tax dividends, and capital gains, intervention of both buyers and sellers in the market without influencing prices. We may also consider the possibility that investors can lend or borrow required amount at a pure interest rate, without influencing the level, and the borrowing rate is equal to the loan rate. Some hypothesis are also based on investors behavior such as risk aversion behavior, equality of expectations for operators as to profitability expectations associated with the different assets and their risks, the rationality and safety of operators over a given horizon, to maximize their wealth.

    Many variants of the mathematical CAPM formulation have been developed in the literature. Sharpe (1964) proposed a variant of CAPM based on expected returns stating that

    E(Ri)=rf+σimσ2m(E(Rm)rf) (10)

    which by setting βi=σimσ2m becomes

    E(Ri)=rf+βi(E(Rm)rf) (11)

    It means that the expected return of the action i is estimated by the risk-free interest added with a risk prime. This later is composed of the excess of the expected rate of return of the market minus the remuneration of the risk-free asset multiplied by the factor βi. It thus depends on the risk premium that the investor should receive when agreeing to bear a risk equal to that of the market and the importance of the risk of the action i against the risk of the market. In literature, there are other categories of the CAPM such that CAPM without a risk-free asset and CAPM with transaction costs. See for example Black (1972), Black et al. (1972). Assume that a number K of investors are present. The k-th investor will allocate his wealth Tk in a number Mk of actions in the market. To be in an equilibrium situation, we should have

    Ei=rf+σikσ2k(Ekrf) (12)

    where Ei is the expectation of the return of action i, Ek is the expectation of the return of the optimal portfolio detained by the investor, σ2k is the variance of the optimal portfolio and σik is the covariance between the return of action i and the portfolio. Denoting similarly to previous variants βik=σikσ2k the last equation becomes

    Ei=rf+βik(Ekrf) (13)

    βik is the volatility of the action i relatively to the optimal portfolio. Taking into account all investors wealth, Lévy (1978) proposed the form

    Ei=rf+1TKKk=1Tk(Ekrf) (14)

    where TK=Kk=1Tk. This formula means that the required risk premium of the action i is evaluated as the average of the premiums demanded by the investors, weighted by their respective wealth.

    CAPM may also depend on inflation. Indeed, assuming the existence of an asset with certain nominal return, Friend et al. (1976) proposed that the relation between the risk and the return of the different assets could be written as

    E(Ri)=rf+σiπ+Emrfσmπασ2imσmπ(ασimσiπ) (15)

    where σiπ is the covariance between the rate of return on assets i and the rate of inflation, σmπ is the covariance between the market rate of return and the rate of inflation and α is the ratio between the nominal value of risky assets and that of all assets.

    Finally, taking into account the taxation, the CAPM may be modified to reflect the presence of taxation based on the relationship between the expected return and the risk of the security as

    Ei(1T0)=rf(1T0)+βi(Emrf)(1T0) (16)

    where Ei is the expectation of the yield of action i, T0 is the tax rate, βi is the systematic risk and Em corresponds to the expectation of the market return.

    Different forms of the CAPM may be also found in the literature that take into account other factors such as heterogeneity of anticipations relatively to future performances of actions, etc. See for example Aktan et al. (2009). In the present paper, we assume that the CAPM defines the required return on an investment according to the equation

    E(Ri)=rf+βi(E(Rm)rf) (17)

    where E(Ri) is the asset's expected return, rf is the risk-free rate, E(Rm) is the expected return of the market portfolio, and βi is the measure of risk for asset i evaluated as

    βi=Cov(Ri,Rm)Var(Rm)=σimσ2m (18)

    In empirical finance, βi is estimated usually via the ordinary least square estimate from the linear regression

    Ritrft=αit+βi(Rmtrft)+εit (19)

    where εit is the error term while αit is a constant. Consequently, the risk beta allows to decompose the variance of an asset i as

    σ2i=β2iσ2m+σ2εi (20)

    Hence, the variance σ2i can be decomposed into a first component β2iσ2m corresponding to the firm's systematic risk and a second one σ2εi corresponding to the firm's unsystematic risk.

    To estimate the CAPM with wavelets the first step is to pass by the variance and the covariance of the statistics (time) series and introduce the analogues for the components due to the wavelet decomposition. Therefore, the wavelet analysis consists here also in splitting the variance of the series into sub-variances relative to scales or the levels j which will be called the variance of the series at the scale j (Percival and Walden (2000)). Let υ2X be the variance of X(t) and υ2X(j) be the variance of the projection at the scale or the level j, we have

    υ2X=j=1υ2x(j) (21)

    This permits to focus on the sub-variances of the components at a level j instead of considering the whole series. This in turns facilities the analysis of the fluctuations and the dynamics of the series. Denote L the length of the wavelet support, Nj=[N/2j] the number of wavelet coefficients at the level j and Lj=[2j(L2)(2j1)] the number of boundary wavelet coefficients at the level j. The variance at the level j is estimated as

    ˆυ2X(j)=12j(NjLj)Nj1k=Lj1d2j,k (22)

    Similarly, we have an analogue formulation for the covariance at the level j for a couple of series (X,Y) as

    ˆυ2XY(j)=12j(NjLj)Nj1k=Lj1dXj,kdYj,k (23)

    To apply wavelets for evaluating the CAPM we firstly decompose returns into their wavelet decompositions. This leads to component-wise returns according to the levels. These new time series will be considered as the returns to be applied. See for example Gençay et al. (2003). The risk beta will be estimated by the wavelet risk beta at the level or the scale j as

    ˆβi(j)=ˆυRiRm(j)ˆυ2Rm(j) (24)

    where ˆυRiRm(j) is the wavelet covariance of the component i of the portfolio with the market at the scale j. ˆυ2Rm(j) is the wavelet variance of the market at the scale j.

    Next, in order to illustrate the explanatory power of market returns on the determination of portfolio returns, we estimate the coefficient R2i(j) for each time scale used as follows

    R2i(j)=βi(j)2σ2Rm(j)σ2Ri(j) (25)

    One of the main facts in analyzing stock markets is the uncertainty phenomenon related to many factors such as data collection and recording, policy makers, etc. In Marfatia (2014) the impact of monetary policy on the real economy has been investigated based on policy makers and market participants. A novel approach has been developed for evidence supporting the Lucas island model predictions. Based on the estimation of the time varying response of the S & P stock returns to monetary policy surprises derived from the high frequency Federal funds futures market, the study concluded that at higher level of uncertainty, the impact of FOMC policy surprise on the time varying S & P returns decreases. Besides, the volatility in the short-term bond market has been shown to offer highest explanatory power in explaining the impact of uncertainty on the effectiveness of Fed's policy surprises. This study interconnects with the present one in the common point of taking into consideration of the uncertain character of financial data. However, in the present work we considered these uncertain values to be part of missing data and thus a post reconstruction step based on wavelet theory has been conducted to fill the gaps of missing and uncertain data. The study of the affects of such missing and/or uncertain data on the risk may be a good eventual future direction. Moreover, taking into consideration the US policy is of great interest in the model as US policy is strongly affecting the one in GCC countries especially KSA and vice-versa. We may recall here the last phenomenon of floating the oil market which affected immediately the US oil market and thus the US policy makers. We have noticed enormous politicians from both republic and democratic parties that strongly criticizing and threatening to change the way of collaboration with KSA despite their strategic relations.

    In Marfatia (2015) the role of financial stress and risks in the monetary policy's time-varying impact on the US bond markets has been studied. The author investigated precisely the link between the treasury and monetary policy shocks and the impact of the financial market volatility. Besides the author concluded that a time-varying coefficient is more adequate to express the impact of the monetary policy on the market and to express adequately the uncertainty in economic data and its role in estimating interest rates as well as the level of financial risks. These facts join in some sense the present work as we suggest here also that time variation is an essential character in estimating the coefficient of correlation between the whole market index and the firms and did not assume that such coefficient remains constant over time. However, our present model did not include neither the US monetary policy nor the GCC monetary policy as explicit factors, despite their importance and their hidden role and hidden affect on the market studied. These factors, which constitute some limitations of the present model, may be interesting extending studies in the future for both the theoretical mathematical model and the decision makers in the market as well as policy makers.

    The main object of our work is to examine the effect of time scales on the systematics risk of the stock components of the market. We test the following hypotheses:

    ● Hypothesis 0: The risk beta is stable according to time scales.

    ● Hypothesis 1: There is a linear dependence between action's returns and their systematic risks.

    This paper empirically tests a wavelet methodology of beta estimation proposed previously on daily data for GCC markets collected on the period of January 01, 2013 to September 20, 2018 resulting of a sample size N=1427 due to the well-known Tadawul index for Saudi Arabia stock market. We focused on a portfolio composed of 148 actions as listed in Table 1 with corresponding sectors. Table 1 provides the components of Tadawul market applied in our study and their classification according to the global classification standards.

    Table 1.  Tadawul companies and sectors.
    Sector Company Name Abbreviation
    Materials Alujain Corporation ACAlujain
    Arabian Cement Co ACCAC
    Al Jouf Cement Company AJCC
    Advanced Petrochemical Company APC1
    Arabian Pipes Company APC2
    Basic Chemical Industries Co BCIC
    Eastern Province Cement Co EPCC
    Filing & Packing Materials Co FPMC
    Hail Cement Company HCC
    Methanol Chemicals Company MCC
    Najran Cement Company NCC
    Nama Chemicals Co NCC2
    National Gypsum Company NGC
    National Industrialization Co NIC
    National Metal Manufacturing Co NMMC
    National Petrochemical Company NPC
    Rabigh Refining & Petrochemical Co RRPC
    Saudi Arabia Fertilizers Co SAFC
    Sahara SAHARA
    Saudi Arabian Mining Company SAMC
    Saudi Basic Industries Corp SBIC
    Saudi Cement Company SCC2
    Saudi Chemical Company SCC4
    Saudi Industrial Investment Group SIIG
    Saudi Int Petrochemical Co SIPC
    Saudi Kayan Petrochemical Company SKPC
    Southern Province Cement Co SPCC
    Saudi Paper Manufacturing Co SPMC
    Saudi Steel Pipe Company SSPC
    Takween Advanced Industries TAI
    Tabuk Cement Co TCC
    The National Co for Glass Industry TNCGI
    The Qassim Cement Co TQCC
    United Wire Factories Company UWFC
    Yanbu Cement Co YCC
    Yanbu National Petrochemical Co YNPC
    Yamamah Saudi Cement Co YSCC
    Zamil Industrial Investment Co ZIIK
    Energy Aldrees Petroleum & Transport Svcs APTS
    Saudi Arabia Refineries Co SARC
    Capital Goods Al-Ahsa Development Co AADC
    Abdullah A. M. Al-Khodari Sons Co AKSC
    Al-Babatain Power & Telecom Co ABPTC
    Astra Industrial Group AIG
    Middle East Specialized Cables Co MESCC
    Saudi Arabian Amiantit Co SAAC
    Saudi Cable Company SCC
    Saudi Ceramic Co SCC3
    Saudi Industrial Export Co SIEC
    Saudi vitrified clay pipes co SVCPC
    Commercial & Professional Svc The National Shipping Co TNSC
    Allianz SE VNA O. N ASVON
    Saudi Printing & Packaging Company SPPC
    Transportation Components Saudi Industrial Services Co SISC
    Saudi Public Transport Co SPTC
    Saudi Transport & Investment Co STIC
    United Int Transportation Company UITC
    Consumer Al-Abdullatif Co ACAbdullatif
    Al Sorayai Trading & Industrial Group ASTIG
    Saudi Industrial Development Co SIDC
    Tourism Enterprise Co TEC
    Al Khaleej Training & Education Co ATEC
    Dur Hospitality DH
    Herfy Food Services Co HFSC
    Retailing United Electronics Company UEC
    Al Hassan Ghazi Ibrahim Shaker AKGIS
    Jarir Marketing Co JMC
    Fawaz Abdulaziz AlHokair Company FAAC
    Saudi Automotive Services Co SASC
    Foods Almarai Company ACAlmarai
    Al-Jouf Agriculture Development Co AJADC
    Ash-Sharqiyah Development Company ASDC
    Food Products Co FPC
    Halwani Bros HB
    Jazan Development Co JDC
    National Agriculture Development Co NADC
    Qassim Agriculture Co QAC
    Saudia Dairy and Foodstuff Co SDFC
    Saudi Fisheries Co SFC
    Savola Group SG
    Tabuk Agriculture Development Co TADC
    Anaam International Holding Group AIHG
    National Agriculture Marketing Co NAMC
    Abdullah Al Othaim Markets AAOMC
    Media and Entertainment Saudi Research and Marketing Group SRMG
    Tihama Advertising & Public Relations TAPR
    Health & Life Science Mouwasat Medical Services Company MMSC
    Saudi Pharmaceutical Appliances SPA
    Banks Alawwal Bank ABALAwal
    Alinma Bank ABAINma
    Arab National Bank ANB
    Al Rajhi Bank ARB
    Bank AL Bilad BA
    Bank Al Jazira BA2
    Banque Saudi Fransi BSF2
    Riyad Bank RB
    Samba Financial Group SFG
    The Saudi British Bank TSBB
    The Saudi Investment Bank TSIB
    Diversified Financials Al-Baha Investment and Development Company ABIDC
    Al Ahli Takaful Company ATC
    Aseer Trading Tourism & Manufacturing ATTM
    Falcom Financial Services FFS
    Kingdom Holding Company KHC
    Saudi Advanced Industries Co SAIC
    Solidarity Saudi Takaful Co SSTC
    SABB Takaful ST
    Insurance Al-Ahlia Insurance AAIC
    Al Alamiya Cooperative Insurance AACI
    Arabia Cooperative Insurance ACI
    Amana Cooperative Insurance ACIC
    Allied Cooperative Insurance Group ACIG
    Arabia Insurance Cooperative Co AICC
    Al-Rajhi Cooperative Insurance ARCI
    AS-AXA SA ASAXASA
    Al Sagr Co-operative Insurance Co ASCIC
    Arabian Shield Coop Insurance Co ASCIC2
    Bupa Arabia for Coop. Insurance BACI
    Buruj Cooperative Insutance Co BCIC2
    Gulf General Cooperative Insurance GGCI
    Gulf Union Cooperative Insurance GUCI
    Malath Cooperative Insurance Co MCIC
    Rade Union Cooperative Insurance RUCI
    Saudi Arabian Cooperative Insurance SACI
    Salama Cooperative Insurance Co SCIC
    Saudi Enaya Cooperative Insurance SECI
    Saudi Indian Company Insurance SICI
    Saudi RE Cooperative Reinsurance SRCR
    Saudi United Cooperative Insurance SUCI
    The Company for Coop. Insurance TCCI
    The Mediterranean & Gulf Insurance Co TMGIC
    United Cooperative Assurance Co UCAC
    Wataniya Insurance Company WIC
    Telecommunication Services Etihad Atheeb Telecommunication EAT
    Etihad Etisalat Co EEC
    Mobile Telecommunications Company MTC
    Saudi Telecom ST2
    Utilities National Gas & Industrialization Co NGIC
    Saudi Electricity Company SEC
    REITs Taiba Holding Co THC
    Real Estate Mgmt & Dev't. Arriyadh Development Co ADC
    Dar Alarkan Real Estate Development DARED
    Emaar The Economic City ETEC
    Jabal Omar Development Company JODC
    Knowledge Economic City KEC
    Makkah Construction & Development Co MCDC
    Red Sea Housing Services Company RSHSC
    Saudi Real Estate Co SREC

     | Show Table
    DownLoad: CSV

    In fact, Tadawul stock market is already classified into industrial sectors since 2008. Next, with the continuous developments in KSA economy, new companies as well as industries have emerged into the market. Consequently, necessary changes have been conducted in the classification of companies listed on Tadawul to reflect the emergence of the new elements. However, Tadawul's previous sector classification was not based on global classification standards which induces some limitations in the classification.

    In the new Tadawul market structure there are 20 sectors presented as follows: Energy, Materials, Capital Goods, Commercial & Professional Svc, Transportation, Consumer Durables & Apparel, Consumer Services, Media and Entertainment, Retailing, Food & Staples Retailing, Food & Beverages, Health Care Equipment & Svc, Pharma Biotech & Life Science, Banks, Diversified Financials, Insurance, Telecommunication Services, Utilities, REITs, Real Estate Mgmt & Dev't. Table 1 provides the re-organization of the Tadawul stock market components according to the global classification standards reminiscent of a bit modification where we merged some sectors into one according to their closeness. The Consumer Durables & Apparel Components is merged with Consumer Services to constitute the new sector Consumer. The sectors Foods & Bevarage and Food & Staples Retailing are merged to compose the new sector Foods. Finally we merged the sector Health Care Equipment & Svc with the sector Pharma Biotech & Life Science into the new sector Health and Life Science. Consequently we obtain 17 sectors as shown in Table 1, which resembles to the classical segmentation of Saudi market.

    The choice of this market is motivated by the fact that Tadawul is the best representative index of the Saudi market supervised by the Capital Market Authority. It is also considered as the largest capital market in the Middle East and North Africa. Recently at the end of 2019, Tadawul has taken place in the first 10 largest stock markets in the world. It lists more than 150 publicly traded companies. We applied in our study a number of 148 ones because of the non-availability of sufficient data and or the very weak effect of some companies in the applied period of study. For example, a first test has yielded for ABIDC action a zero beta for both scale laws of the periods of time applied.

    Recall as mentioned in the introduction that when collecting the data about financial GCC markets such as KSA Tadawul applied here, the main problem confronted is the lack of data where the samples present for many cases missing values. Consequently, we started by acting a wavelet method to reconstruct missing data which by the next leads to complete and adjusted basis on the period of study. On the total basis used we noticed 231 missing values dispersed on the whole market and on different time dates (daily) on 14 actions in the market. The complete basis is applied next for CAPM and thus the comprehension of the market. The missing values are distributed as in Table 2. We applied as in Mabrouk et al. (2010) a training set of 5 prior values for different time dates (daily) on 14 actions in the market.

    Table 2.  Missing data.
    Stock Number of missing values
    AADC 12
    ACAbdullatif 18
    AJCC 21
    AKGIS 6
    ASAXASA 23
    ATEC 21
    DH 17
    FFS 19
    JDC 20
    MCDC 21
    QAC 19
    SAHARA 15
    YSCC 10
    ZIIK 8
    Total 231

     | Show Table
    DownLoad: CSV

    In fact several wavelet functions have been tested to provide a best filtering. The tests results in the well-known Daubechies wavelet Db6. The authors may refer to Daubechies (1992) where a mathematical method has been developed to compute numerically the values of the mother/father wavelet on dyadic points in the compact supports. The essential idea there was based on the fact that the vector of values of the wavelet at these points is an eigenvector for a special matrix associated also to dyadic eigenvalues. In fact, in Mabrouk et al. (2010) we applied a newer version of Daubechies wavelets developed differently and applied also in Mabrouk et al. (2010) based on connection coefficients of wavelets and leading to the fact that the matrices obtained in Daubechies (1992) may be transformed into circulant and hollow ones. This permits a great gain in machine memory, algorithms speed and error estimates.

    Next as our basis of data has been completed to obtain 100% trading days we are able to develop the CAPM analysis. We propose to study the relationship between excess return on each individual stock and the time scales of market portfolio using the usual OLS estimator for βi issued from the regression (19). The daily return of each stock is calculated as the log-price difference

    Rit=logPi,tlogPi,t1 (26)

    where Pi,t is the price of asset i at day t. The market return Rmt is taken as the log-difference of the index due to the action i as

    Rmt=logCtlogCt1 (27)

    where Ct is the index value at day t.

    Table 3 below shows the descriptive statistics of the market. The statistics corresponds precisely to the return excess for each company relatively to the risk-free. Table 3 presents descriptive statistics of excess return on the stocks in the sample and on the proxy for the market portfolio the Tadawul. The median value of all assets in the present sample is approximately zero except for the one of Tadawul being equal to 0.662.103. The flatness and distortion features of all stocks' returns are different from each other. Moreover, the Jarque-Bera test leads to JB=1 which rejected the null hypothesis at the 5% significance level, and 0 otherwise.

    Table 3.  Descriptive statistics of excess returns.
    Stocks Mean Maximum Minimum SD. Skewness Kurtosis
    Tadawul 0.0001 0.0855 −0.0755 0.0111 −0.6594 13.1546
    AADC −0.0003 0.0976 −0.1042 0.0214 −0.4020 7.8649
    AAIC −0.0009 0.6868 −0.7257 0.0410 −0.5965 126.3768
    AKSC −0.0010 0.1211 −0.1079 0.0246 −0.2688 8.1129
    AAOMC 0.0009 0.1088 −0.1056 0.0174 0.1619 8.2193
    ABALAwal −0.0004 0.3614 −0.7054 0.0283 −11.0296 310.8295
    ABAINma 0.0003 0.0950 −0.1077 0.0168 −0.0143 10.4981
    ABIDC −0.0002 0.0953 −0.1102 0.0172 0.4497 19.5665
    ABPTC −0.0002 0.0953 −0.1126 0.0218 −0.6165 8.6596
    AACI −0.0004 0.1166 −0.5887 0.0365 −2.9527 51.2243
    ACAbdullatif −0.0005 0.0917 −0.1089 0.0188 −0.9383 9.3029
    ACAlmarai 0.0003 0.0948 −0.2262 0.0164 −1.6522 33.7767
    ACAlujain −0.0000 0.0970 −0.1102 0.0249 −0.1402 6.5993
    ACCAC −0.0002 0.0949 −0.0827 0.0161 −0.0995 8.0130
    ACI −0.0010 0.1060 −0.6712 0.0346 −5.0857 101.9765
    ACIC −0.0002 0.8330 −0.1121 0.0460 4.1390 78.3266
    ACIG −0.0001 0.1007 −0.1181 0.0315 −0.3735 5.9002
    ADC 0.0001 0.1010 −0.1141 0.0175 −0.1528 11.3075
    AICC −0.0004 0.4090 −0.2500 0.0322 1.0842 25.2380
    AIG −0.0006 0.0744 −0.1141 0.0196 −0.7648 8.1017
    AIHG −0.0009 0.1010 −0.1056 0.0248 −0.5832 7.6882
    AJADC −0.0000 0.0932 −0.1041 0.0185 −0.2863 8.7256
    AJCC −0.0003 0.1028 −0.1069 0.0167 −0.6405 13.4413
    AKGIS −0.0013 0.1218 −0.6005 0.0256 −9.1548 215.2429
    ANB 0.0000 0.0741 −0.0914 0.0142 −0.0283 7.0484
    APC1 0.0005 0.0947 −0.0794 0.0174 0.2549 8.2650
    APC2 −0.0005 0.0950 −0.1136 0.0229 −0.3841 8.1570
    APTS −0.0000 0.0992 −0.2109 0.0200 −0.9530 15.0490
    ARB 0.0000 0.0916 −0.0694 0.0132 0.1622 9.5371
    ARCI 0.0006 0.0950 −0.1044 0.0252 −0.1468 6.4127
    ASAXASA −0.0002 0.1092 −0.5073 0.0314 −3.0477 51.8985
    ASCIC −0.0001 0.0987 −0.2144 0.0278 −0.6780 9.3565
    ASCIC2 0.0003 0.0984 −0.1153 0.0298 −0.0923 6.2951
    ASDC −0.0001 0.1152 −0.1104 0.0282 −0.1505 6.4707
    ASTIG −0.0004 0.4990 −0.1088 0.0255 4.9823 107.8562
    ASVON −0.0008 0.1001 −0.1097 0.0297 −0.2984 6.5046
    ATC −0.0004 0.0950 −0.1040 0.0231 −0.2889 7.2039
    ATEC −0.0002 0.0983 −0.1363 0.0235 −0.2669 7.3756
    ATTM −0.0003 0.0953 −0.1132 0.0205 −0.7734 8.7852
    BA 0.0003 0.0984 −0.0943 0.0189 0.1910 8.3323
    BA2 −0.0000 0.0945 −0.1067 0.0186 −0.1549 7.6460
    BACI 0.0009 0.1159 −0.6777 0.0310 −7.1585 164.7862
    BCIC −0.0002 0.0953 −0.1286 0.0213 −0.9222 9.6679
    BCIC2 −0.0002 0.0984 −0.4638 0.0318 −2.3196 36.2178
    BSF2 0.0001 0.0835 −0.1040 0.0168 0.0432 6.6258
    DARED −0.0001 0.1021 −0.1086 0.0257 −0.1485 6.9724
    DH −0.0003 0.1094 −0.1131 0.0199 −0.2447 9.6004
    EAT −0.0008 0.9017 −0.1158 0.0351 11.9202 307.3399
    EEC −0.0010 0.0953 −0.1214 0.0200 −0.7863 11.2386
    EPCC −0.0005 0.0906 −0.1001 0.0137 −0.7080 14.8314
    ETEC 0.0002 0.1041 −0.1082 0.0239 0.0271 7.2838
    FAAC −0.0004 0.0984 −0.6727 0.0297 −8.0928 186.2733
    FFS 0.0003 0.0925 −0.1027 0.0136 −0.4737 14.0483
    FPC −0.0003 0.1007 −0.1054 0.0266 −0.3694 7.2488
    FPMC −0.0002 0.1078 −0.1105 0.0228 −0.6224 7.8977
    GGCI −0.0005 0.0975 −0.1085 0.0285 −0.3174 6.3612
    GUCI −0.0001 0.4143 −0.1087 0.0295 1.5882 32.0246
    HB 0.0002 0.1220 −0.1033 0.0205 0.1787 7.4574
    HCC −0.0005 0.0963 −0.1102 0.0160 −0.4340 10.2036
    HFSC 0.0001 0.0925 −0.1049 0.0175 −0.3081 7.8099
    JDC −0.0000 0.0968 −0.1126 0.0221 −0.9225 9.1503
    JMC 0.0003 0.0945 −0.0975 0.0147 0.0394 10.2247
    JODC 0.0008 0.0988 −0.1138 0.0200 0.1267 8.7210
    KEC −0.0001 0.1004 −0.1138 0.0241 −0.2723 7.7153
    KHC −0.0005 0.1021 −0.1101 0.0197 0.0212 11.1638
    MCC −0.0004 0.0927 −0.1037 0.0195 −0.5431 9.3482
    MCDC 0.0005 0.0953 −0.1262 0.0188 0.5072 11.2096
    MCIC −0.0001 0.9192 −0.4867 0.0403 6.9176 206.2428
    MESCC −0.0005 0.1082 −0.1084 0.0244 −0.2775 7.8676
    MMSC 0.0009 0.0949 −0.0996 0.0182 0.1273 6.3078
    MTC −0.0007 0.6035 −0.1069 0.0276 7.2330 164.4192
    NADC 0.0000 0.0978 −0.2533 0.0221 −1.2536 19.7337
    NAMC 0.0000 0.0990 −0.1124 0.0270 −0.3427 8.1218
    NCC −0.0006 0.0950 −0.0996 0.0178 −0.1085 8.1755
    NCC2 −0.0001 1.7154 −0.1201 0.0521 24.9600 824.2936
    NGC −0.0006 0.1004 −0.1063 0.0210 −0.2816 9.3964
    NGIC 0.0002 0.0930 −0.1035 0.0147 0.3508 13.0524
    NIC −0.0005 0.0997 −0.1082 0.0195 −0.2293 8.5197
    NMMC −0.0002 0.1023 −0.1085 0.0224 −0.6541 8.0132
    NPC −0.0001 0.0990 −0.1054 0.0220 −0.1101 7.9040
    QAC −0.0004 0.5358 −0.1086 0.0273 4.8470 109.9920
    RB −0.0004 0.0970 −0.6845 0.0216 −22.1634 706.8519
    RRPC 0.0000 0.0971 −0.1066 0.0231 0.0671 7.8090
    RSHSC −0.0004 0.0977 −0.3537 0.0244 −2.4811 37.1443
    RUCI −0.0001 0.0970 −0.1094 0.0250 −0.5178 7.6858
    SAAC −0.0007 0.1032 −0.1071 0.0183 −0.4872 11.1093
    SACI −0.0006 0.1036 −0.6155 0.0330 −4.5685 88.1574
    SAFC −0.0007 0.0942 −0.2485 0.0160 −4.5416 73.6872
    SAHARA 0.0001 0.0950 −0.1077 0.0180 −0.2328 8.4039
    SAIC −0.0002 0.0956 −0.1059 0.0223 −0.4760 7.7089
    SAMC 0.0004 0.1037 −0.1057 0.0205 −0.0872 8.5748
    SARC −0.0005 0.0989 −0.1087 0.0236 −0.4868 7.9098
    SASC −0.0001 0.1086 −0.2045 0.0239 −1.0248 11.6036
    SBIC 0.0001 0.0958 −0.1048 0.0156 0.0344 11.9477
    SCC −0.0001 0.0980 −0.1128 0.0183 −0.3730 10.0518
    SCC2 −0.0001 0.6317 −0.1022 0.0275 8.4612 198.6055
    SCC3 −0.0004 0.0868 −0.1058 0.0150 −0.8061 13.2266
    SCC4 −0.0009 0.0936 −0.2701 0.0187 −2.2048 36.5825
    SCIC −0.0002 0.1021 −0.5216 0.0331 −2.8175 46.8998
    SDFC 0.0005 0.0948 −0.1025 0.0181 0.1999 6.5451
    SECI −0.0005 0.7122 −0.1132 0.0354 5.5794 119.1722
    SEC 0.0003 0.0941 −0.1058 0.0157 0.2316 12.4987
    SFC −0.0001 0.9771 −0.1107 0.0365 13.3332 361.5026
    SFG −0.0005 0.0976 −0.7988 0.0266 −18.9221 571.6588
    SG 0.0012 1.5817 −0.1038 0.0463 27.8451 951.0885
    SICI −0.0007 0.5166 −0.5062 0.0437 0.0462 29.0768
    SIDC −0.0005 0.0970 −0.1091 0.0234 −0.7096 8.0909
    SIEC 0.0009 2.2792 −0.1200 0.0669 27.6153 942.0233
    SIIG −0.0001 0.1020 −0.1139 0.0212 −0.1086 6.6162
    SIPC 0.0000 0.0915 −0.1030 0.0196 −0.3179 6.8247
    SISC −0.0002 0.0930 −0.1129 0.0194 −0.7895 8.7647
    SKPC −0.0002 0.1047 −0.1141 0.0219 0.0579 8.5149
    SPA 0.0000 0.0926 −0.1038 0.0173 −0.2850 7.9182
    SPCC −0.0005 0.1044 −0.0993 0.0152 0.2098 11.5496
    SPMC −0.0006 0.6261 −0.6241 0.0380 2.5707 147.2114
    SPPC −0.0004 0.1014 −0.1360 0.0289 0.1974 7.5492
    SPTC 0.0000 0.1018 −0.1030 0.0222 −0.5027 8.8293
    SRCR −0.0003 0.1766 −0.1103 0.0222 0.0748 10.2966
    SREC −0.0001 0.0953 −0.1077 0.0211 −0.3305 8.2718
    SRMG 0.0007 0.1058 −0.1091 0.0295 0.5594 7.1668
    SSPC −0.0003 0.1014 −0.1109 0.0199 −0.2914 7.8505
    SSTC −0.0005 0.0946 −0.1051 0.0270 −0.3907 6.9996
    ST 0.0004 0.0975 −0.0804 0.0147 0.2952 10.6360
    STIC 0.0002 0.0953 −0.2765 0.0275 −0.8529 13.4305
    ST2 −0.0003 0.0986 −0.1054 0.0266 −0.3334 7.1394
    SUCI 0.0004 0.1017 −0.2443 0.0291 −0.4557 9.1944
    SVCPC −0.0003 0.1004 −0.1241 0.0186 0.0852 9.3941
    TADC −0.0003 0.0939 −0.1054 0.0235 −0.5464 7.5586
    TAI −0.0011 0.1121 −0.3038 0.0254 −1.6922 20.8887
    TAPR −0.0003 0.6959 −0.1077 0.0372 4.6034 89.7945
    TCC −0.0003 0.0957 −0.1060 0.0165 0.1651 13.5109
    TCCI 0.0003 0.0987 −0.3808 0.0263 −2.7115 40.6446
    TEC −0.0001 0.1054 −0.1181 0.0272 −0.1333 7.1379
    THC 0.0003 0.0979 −0.1050 0.0177 0.1665 11.2939
    TMGIC −0.0002 0.9370 −0.2231 0.0397 8.8970 219.8138
    TNCGI −0.0003 0.0950 −0.1132 0.0184 −0.6498 9.6728
    TNSC 0.0004 0.1280 −0.1258 0.0208 −0.3959 8.7984
    TQCC −0.0004 0.0656 −0.1018 0.0114 −0.7792 12.6196
    TSBB −0.0001 0.0901 −0.4053 0.0194 −6.3189 137.0194
    TSIB −0.0001 0.0822 −0.1619 0.0145 −0.6161 17.7199
    UCAC −0.0007 0.1007 −0.6010 0.0299 −5.8647 118.2236
    UEC −0.0000 0.0982 −0.1156 0.0220 −0.0306 8.2912
    UITC 0.0005 0.0960 −0.1019 0.0207 −0.0130 7.0139
    UWFC −0.0007 0.1010 −0.2848 0.0218 −1.9298 27.3173
    WIC −0.0007 0.0984 −0.4847 0.0312 −2.4851 45.1754
    YCC −0.0003 0.0950 −0.0969 0.0158 0.1400 10.5054
    YNPC 0.0002 0.1014 −0.1030 0.0195 0.0737 7.4320
    YSCC −0.0007 0.0899 −0.1045 0.0136 −0.6175 12.2179
    ZIIK −0.0000 0.0922 −0.1051 0.0183 −0.5785 7.9267

     | Show Table
    DownLoad: CSV

    Notice from Table 3 that as in the majority of studies of financial markets, return excess of actions relatively to the risk-free as well as the return excess of the market relatively to its risk-free have always low skewness and high kurtosis. Our analysis consists of projecting Equation (19) relatively to time scales to test the effect of time on the systematic risk beta. This will be conducted by splitting the market returns into crystals or horizons relative to different time scales instead of using the classical periods such as weeks, moths, years. The coefficients of the linear regressions will be estimated by the usual OLS of the returns (Ritrf)j on the one of the market (Rmtrf)j for each level j. This leads to a j-level mathematical formulation as

    (Rirf)j=αji+βji(Rmrf)j+εji=αji+αiDjm+εji (28)

    In the empirical study a set of 6 levels j=1,2,3,4,5,6 will be applied. We will proceed differently to classical methods by testing random periods of time instead of classical ones using weeks, months and years. The idea is an extension of our papers Rhaiem et al. (2007a), Rhaiem et al. (2007b) and Aktan et al. (2009). The correspondence scale and dynamic days applied here is resumed in Table 4 where we considered the first two prime numbers as scale laws. For the two scale laws we stopped at the higher-level J=6 as the next level J=7 corresponds to 128–256 dynamic days and 192–384 dynamic days for the 2-scale and the 3-scale laws respectively and this corresponds to approximately one year.

    Table 4.  Time scales.
    Scale law Number of dynamic days
    2-scale law J=1 2–4 dynamic days
    J=2 4–8 dynamic days
    J=3 8–16 dynamic days
    J=4 16–32 dynamic days
    J=5 32–64 dynamic days
    J=6 64–132 dynamic days
    3-scale law J=1 3–6 dynamic days
    J=2 6–12 dynamic days
    J=3 12–24 dynamic days
    J=4 24–48 dynamic days
    J=5 48–96 dynamic days
    J=6 96–192 dynamic days

     | Show Table
    DownLoad: CSV

    We propose in this section to discuss the validity of the main hypotheses raised above about the time-scale stability of the risk beta and the linear dependence between action's returns and their systematic risks.

    Table 5 shows the different regressions of the return excess of actions relatively to the one of the market at different scales j=1,2,3,4,5,6 by the OLS estimates methods. It represents the estimations of the Betas of each stock component at the scales j=1,2,3,4,5,6 relatively to the 2-scale law.

    Table 5.  Estimations of the return excess of actions on the market for 2-scale law.
    STOCKS Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6
    AADC 2.0454 1.2592 4.7203 −0.1344 0.2696 −0.1017
    AAIC −0.8136 −0.5988 0.7101 3.6282 2.7476 −0.2771
    AKSC 0.2910 −0.6088 0.2336 3.6697 0.3298 −0.4860
    AAOMC −2.0076 −1.6669 2.4864 −0.6651 0.5414 −0.0907
    ABALAwal 2.2558 1.2156 0.7113 2.1137 0.2595 0.0006
    ABAINma 0.2077 −0.9598 −0.1373 2.9759 0.3678 −0.1966
    ABIDC 0 0 0 0 0 0
    ABPTC 2.0344 −3.1069 0.8582 0.2941 0.2484 0.0735
    AACI −4.9193 −6.9013 −0.3877 0.8579 2.9001 0.4560
    ACAbdullatif 1.3031 1.8143 0.0771 1.3672 −1.1050 0.6233
    ACAlmaraI 0.1346 0.4174 0.3182 1.6072 −0.4814 −0.0170
    ACAlujain −2.2558 −5.5011 0.2792 −5.3093 1.9286 0.4987
    ACCAC 1.5871 −1.0429 0.4471 3.2303 −0.2090 −0.1456
    ACI −1.1459 −0.6471 1.8266 −3.3696 0.6706 −0.4927
    ACIC 1.3312 −0.0810 0.6685 −3.1870 −4.6848 0.6405
    ACIG 1.3704 −1.8459 −0.9784 0.2556 3.8544 −0.5202
    ADC −1.6338 −1.2348 −0.3794 −1.2973 0.8679 −0.2050
    AICC 1.7936 −0.8655 −0.7359 1.4150 0.8197 0.2657
    AIG 0.1469 −0.7044 −0.7536 1.0553 −0.6053 −0.1658
    AIHG 3.9209 3.6844 −0.4308 3.9178 2.3019 −0.6122
    AJADC 0.1802 −0.6197 0.7412 0.8990 −0.3237 0.0060
    AJCC 0.5851 0.2148 0.2926 1.1417 0.5260 −0.2301
    AKGIS −0.3550 −3.7764 0.2050 0.8631 0.2353 0.0068
    ANB 0.1510 −1.1475 1.5114 −0.0872 −0.6937 −0.1717
    APC1 1.2912 0.5540 2.0165 1.2257 −0.2257 −0.1791
    APC2 0.6019 −0.2898 0.5744 3.9534 0.8820 −0.3703
    APTS −0.3484 0.5055 0.0923 2.0748 0.2792 −0.1150
    ARB 1.7859 0.5454 −0.0968 1.9171 −0.1537 −0.0129
    ARCI −0.1832 0.4277 0.2845 0.2703 −0.4891 −0.1190
    ASAXASA −1.5885 −1.8101 −0.9836 2.6300 −1.2146 −0.2040
    ASCIC −0.9028 0.5524 −0.1869 −1.3193 −2.8156 −0.3546
    ASCIC2 −1.1030 −0.7915 2.0132 1.0996 0.7185 −0.2994
    ASDC 2.1719 0.8584 1.0935 3.1043 −0.3542 −0.7126
    ASTIG 0.0798 −0.6475 2.5086 1.6947 0.0239 −0.5799
    ASVON −0.9754 −10.8147 −1.4902 0.4733 −1.0793 0.1823
    ATC 0.2324 −0.3313 −0.7571 1.0422 −1.3387 −0.0540
    ATEC −0.5190 −1.0538 0.2905 3.3250 −0.2892 −0.4030
    ATTM 1.5430 −0.0771 −0.1022 4.6420 −0.2408 −0.3051
    BA 0.2042 −0.3765 −1.6786 0.8267 0.0514 −0.0353
    BA2 0.6170 0.0255 −0.0960 3.1852 1.1053 0.0912
    BACI −1.7953 6.4710 −1.9693 0.6657 −1.2850 −0.9257
    BCIC 1.6692 1.3176 0.0553 5.2654 −1.6853 −0.0222
    BCIC2 −0.3402 −0.8038 3.1566 3.1345 0.9370 −0.3572
    BSF2 0.9532 0.8619 0.0967 0.1826 −0.2565 0.0949
    DARED 0.0819 −1.6201 −1.3135 5.0073 −0.0942 −0.4284
    DH −0.1791 −0.1964 −0.0991 0.5983 0.8619 −0.3453
    EAT 3.5175 1.8236 −0.1227 7.9564 −0.1084 −0.1180
    EEC 0.7504 1.3335 −1.3436 −2.8282 −0.5041 0.2862
    EPCC −1.7460 −1.8496 −0.8482 −1.0594 0.6062 0.0826
    ETEC −0.1419 −1.1883 −0.9012 4.7261 −0.1504 −0.2432
    FAAC 3.4837 2.7514 0.5599 1.5045 1.1037 0.1344
    FFS −1.3546 −2.0219 −0.7033 0.0681 0.2875 −0.2461
    FPC 0.0702 0.1569 0.0989 2.6568 −0.2760 −0.3351
    FPMC 1.0234 −0.0070 0.4789 2.1010 −0.5076 −1.1162
    GGCI −1.6632 0.5200 0.6573 3.6334 1.2126 −0.3616
    GUCI 3.0394 2.2841 4.2901 1.6468 −0.2954 0.2031
    HB −0.8458 −1.5923 3.7476 0.9129 0.5309 −0.1082
    HCC 1.0593 0.3255 −0.1368 1.0735 −0.2680 −0.4327
    HFSC −0.3633 −0.9863 −0.8563 −0.2454 −1.5168 0.2498
    JDC 0.0592 −0.1913 0.3649 3.5112 −1.2266 −0.4731
    JMC 0.4594 0.6729 0.3395 0.2675 0.4964 −0.1651
    JODC 1.4972 0.9974 −0.1874 2.5024 −0.6180 −0.2581
    KEC 0.4610 5.0965 3.4134 0.7002 0.4510 −0.5444
    KHC −3.0494 −3.7322 −2.0498 0.7790 −0.4837 0.0575
    MCC −0.9612 −0.6497 −0.4138 1.7948 −0.3047 −0.2888
    MCDC 0.2186 0.6632 −0.2014 0.4837 0.6552 −0.1054
    MCIC 3.1832 1.1088 2.0118 0.5240 −1.0701 0.1967
    MESCC −2.3770 −2.8424 −2.5796 −1.7246 0.3690 −0.2658
    MMSC −0.1815 0.8594 0.1113 −1.6686 −0.1210 −0.0233
    MTC −3.4109 −1.9792 −0.5610 1.2879 1.7197 −0.1438
    NADC −0.6231 −0.3392 0.9297 0.8857 0.2977 −0.1760
    NAMC −1.9429 −0.7066 0.0368 2.8776 1.3819 −0.3872
    NCC 1.1487 0.8169 0.1470 0.3079 −0.3792 −0.5390
    NCC2 9.1468 1.2602 −5.2381 −10.6298 0.5239 −0.1011
    NGC 2.1524 1.1171 0.0046 2.2828 −0.6344 −0.3704
    NGIC 1.7953 1.5640 0.5471 0.1670 0.1474 −0.0615
    NIC 1.4052 0.8410 −0.5249 1.6480 −0.6017 −0.2061
    NMMC 2.6260 0.1592 0.0231 3.3927 −0.2717 −0.5809
    NPC 1.1125 0.7752 −0.4892 2.8176 0.0757 −0.3156
    QAC 1.2273 −0.1425 −0.2098 4.4579 0.1034 −0.1594
    RB 1.1363 0.8378 0.1758 1.3161 0.2652 −0.0258
    RRPC 0.6483 −0.1850 0.4136 0.3193 −0.0602 0.0378
    RSHSC −0.3698 −0.7114 0.4449 3.1914 0.2107 −0.1487
    RUCI −0.2733 −1.0438 −0.0889 1.4671 1.2418 −0.3125
    SAAC 1.0423 2.1967 0.7674 0.2750 −0.4898 −0.0547
    SACI −1.0528 0.1017 0.2823 4.2541 −2.4115 −0.0486
    SAFC 1.1512 0.6889 0.5044 1.1865 0.1779 −0.0260
    SAHARA −0.0427 −0.0528 0.0045 0.7286 −0.3619 −0.2953
    SAIC −1.4498 0.7550 −2.3830 3.1324 −0.1545 −0.9078
    SAMC 3.4219 2.5712 0.2028 0.6863 1.4786 −0.2563
    SARC 1.0320 0.1089 −0.2897 0.3245 −0.1400 −0.9298
    SASC −0.5931 0.0990 −0.8007 5.8086 −1.0631 −0.6298
    SBIC 0.9065 0.4762 −0.4144 1.8419 −0.4755 −0.1427
    SCC 0.6076 0.4292 0.5561 0.4767 0.6291 −0.0205
    SCC2 −14.4916 −1.9066 −4.0903 1.1632 0.1599 −0.5467
    SCC3 2.7830 2.1234 0.5116 1.1399 0.8658 −0.0871
    SCC4 −0.6128 0.7771 0.1343 2.2970 1.0891 −0.2841
    SCIC −0.2775 0.4804 −0.4168 2.0828 1.4853 −0.0950
    SDFC −1.4744 0.1362 0.6110 −0.0223 −1.0306 0.0109
    SECI −2.1329 −9.3723 −0.5578 −4.1358 0.5618 −0.1817
    SEC −0.1276 −0.1430 −0.0693 0.9053 0.2668 −0.0806
    SFC 0.5515 0.0295 0.3074 2.8764 0.1732 −0.5060
    SFG 2.3863 1.0691 −0.4912 1.0404 −0.1379 0.1002
    SG 0.0942 −1.1869 0.2493 1.0254 0.1091 0.0541
    SICI 3.3619 0.3848 −0.6766 3.9122 0.3650 −0.1877
    SIDC −1.3162 −0.4311 −1.0642 3.2360 −0.0909 −0.4698
    SIEC 2.7579 0.7454 4.4298 6.5821 1.9027 −0.3040
    SIIG 0.7388 0.4496 0.3281 1.3524 0.1179 −0.2102
    SIPC 0.0949 1.1080 0.2261 −0.9483 −0.5055 −0.0899
    SISC −1.0568 0.5250 −0.3734 4.1423 0.4105 −0.3802
    SKPC 0.6351 0.2604 −0.3634 1.3533 −0.1275 −0.3112
    SPA 1.6959 0.4021 −0.1265 2.3174 0.5124 −0.1977
    SPCC 0.1245 0.4730 −1.0012 1.2827 −0.5697 −0.0033
    SPMC −0.0953 0.1594 0.2309 −0.5832 1.4201 0.0667
    SPPC 3.0029 2.0906 1.7543 1.0952 −0.2096 −0.5743
    SPTC 0.1662 1.4353 −0.0313 4.3419 0.4840 −0.0461
    SRCR 2.4947 0.9071 5.0771 −0.8209 −0.0912 −0.4913
    SREC −0.3585 0.0772 −0.4963 2.4197 0.1505 −0.1294
    SRMG 4.7696 −0.7324 2.4767 2.4454 −0.5290 0.0696
    SSPC 1.9187 1.6177 0.2899 1.2718 0.0358 −0.0169
    SSTC 0.4567 1.4386 −0.4690 1.2473 0.7117 1.6682
    ST 0.4615 0.7855 0.1862 0.1500 −0.8559 −0.0837
    STIC −0.2576 −0.7365 1.6165 5.3658 −1.0183 −0.0929
    ST2 −0.0839 −0.9605 −0.2524 2.7731 −2.3866 −0.2827
    SUCI −1.4984 0.3416 −0.2582 8.6798 1.2857 −0.4694
    SVCPC −0.9153 −0.8431 −0.9134 0.9804 −0.1793 0.1746
    TADC 1.3985 0.2504 0.7664 0.3929 −0.5896 −0.2982
    TAI 0.8440 0.5883 0.4978 3.9641 0.5515 −0.6518
    TAPR 0.1290 2.4773 −0.9535 −0.4211 −1.9360 0.3635
    TCC 1.0496 1.4458 −0.7647 −1.0836 0.2322 −0.2075
    TCCI 0.9107 −0.1210 0.8547 0.5367 1.4837 −0.2641
    TEC −0.3004 −0.2483 0.1187 5.7231 0.6804 −0.8181
    THC −1.2204 −0.6896 0.1092 2.2283 −0.0053 −0.0980
    TMGIC −1.4057 −2.6089 −0.7206 −0.1546 0.4604 −0.1210
    TNCGI 0.7529 −1.1795 1.2706 2.3802 −0.6989 −0.1479
    TNSC 2.5977 2.3481 0.1663 2.2420 −0.9197 −0.4980
    TQCC 2.4470 1.7056 0.0552 0.4815 −0.4503 −0.1157
    TSBB 2.2765 1.7355 0.1684 0.5770 0.4966 −0.1079
    TSIB −0.4280 −0.8878 0.7389 −0.3927 0.4265 −0.2391
    UCAC 1.9711 0.3970 −1.8811 −0.7071 0.6535 0.1020
    UEC −0.6223 −0.3789 0.3593 0.6947 1.2473 0.3361
    UITC 0.6165 −0.1991 0.4392 1.6566 −0.7038 −0.2740
    UWFC −0.1181 −1.8048 0.3520 4.1857 0.6254 −0.2395
    WIC −2.2706 −3.5767 −0.0353 −0.6691 0.4868 0.1453
    YCC 0.5451 0.5711 1.0556 1.5989 −0.0201 0.0016
    YNPC 1.7776 0.5990 0.3653 2.2086 0.6763 −0.4501
    YSCC 0.3955 0.4720 0.2649 1.2843 0.5708 −0.1295
    ZIIK 0.4253 1.6005 −0.2111 −0.5111 −0.7499 −0.0979
    Mean 0.3514 −0.0826 0.1707 1.4273 0.0707 −0.1693

     | Show Table
    DownLoad: CSV

    Table 5 reflects a quite positive relationship between multi-scale return and systematic risk coefficients. Moreover, the beta changes according to the scale. The looking at the individual results indicated that this relationship becomes quietly stronger at the fourth and fifth scales (16–32 and 32–64 days): The effect of the market return on an individual asset's return will be greater at these scales than the others ones. This means that Tadawul is more efficient at scales 4 and 5. Therefore, the CAPM is a multi-scale phenomenon, and quite longer periods are more relevant in explaining the relationship between stock return and its beta. However, this relevance returns to be perturbed for long periods which means the non-resistance of the market according to time indicator. These facts lead us to think again about other factors that may be included in the model to understand more the market movement. Factors such as US policy actions, local economic policy may improve the studies on such market.

    Table 5 shows that the linear dependence is always justified even-though being negative and very weak in many cases and on all the 6 scales. This means that some crisis is always present in the market explained by an opposite variation of the actions and the market. The table shows also that no law may be expected simultaneously for all the contribution of the Dim of all actions relatively to the increasing of time scale.

    To explain more such a contribution, we computed the determination coefficient R2 in Table 6. It represents the estimations of the determination coefficient R2 relative to the Betas of each stock component estimated in Table 5 at the scales j=1,2,3,4,5,6 relatively to the 2-scale law. Notice from Table 6 that the coefficient R2 is decreasing as the time scale increases with some perturbed cases where no monotony is conserved along all time horizons. This means among the negative movement of the market already observed that the major influencing parts of the market portfolio on the actions returns is localized in high frequencies. This concluded that at low horizons the market is going down although the linearity is strong at high levels (5 and 6). Economically speaking this a bad information for small companies and/or short investments.

    Table 6.  The determination coefficient R2 relative to Table 5.
    R2 for each scale
    STOCKS Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6
    AADC 0.5513 0.1359 0.6325 0.0002 0.0033 0.0017
    AAIC 0.1450 0.0281 0.0120 0.0563 0.0375 0.0034
    AKSC 0.3626 0.0286 0.0026 0.3691 0.0031 0.0392
    AAOMC 0.8296 0.1393 0.3241 0.0180 0.0075 0.0038
    ABALAwal 0.9866 0.2454 0.0666 0.1087 0.0069 0.0000
    ABAINma 0.2077 0.9598 0.1373 2.9759 0.3678 0.1966
    ABIDC NaN NaN NaN NaN NaN NaN
    ABPTC 0.5584 0.2241 0.3029 0.0035 0.0065 0.0005
    AACI 0.9971 0.5625 0.0016 0.0066 0.0374 0.0095
    ACAbdullatif 0.6384 0.3413 0.0008 0.0812 0.0748 0.0350
    ACAlmaraI 0.0150 0.0519 0.0118 0.1490 0.0065 0.0001
    ACAlujain 0.7835 0.5221 0.0074 0.1169 0.0562 0.0271
    ACCAC 0.1562 0.0155 0.0114 0.2168 0.0014 0.0122
    ACI 0.3076 0.5430 0.0597 0.0385 0.0033 0.0047
    ACIC 0.3465 0.0007 0.0034 0.0256 0.0517 0.0038
    ACIG 0.4663 0.0483 0.0118 0.0003 0.0473 0.0089
    ADC 0.7977 0.1493 0.0056 0.0196 0.0527 0.0173
    AICC 0.1696 0.0508 0.0137 0.0152 0.0076 0.0045
    AIG 0.0051 0.1327 0.0430 0.0911 0.0580 0.0054
    AIHG 0.6664 0.8132 0.0083 0.1172 0.1047 0.0187
    AJADC 0.0220 0.1731 0.0668 0.0179 0.0123 0.0000
    AJCC 0.0522 0.0037 0.0041 0.0825 0.0458 0.0292
    AKGIS 0.3382 0.6773 0.0033 0.0260 0.0068 0.0000
    ANB 0.0201 0.5334 0.3266 0.0008 0.0216 0.0177
    APC1 0.2548 0.0774 0.1005 0.0187 0.0114 0.0117
    APC2 0.1998 0.0040 0.0072 0.1661 0.0431 0.0143
    APTS 0.1252 0.0125 0.0011 0.2254 0.0128 0.0038
    ARB 0.3688 0.0939 0.0014 0.0917 0.0030 0.0001
    ARCI 0.0168 0.0087 0.0017 0.0004 0.0047 0.0015
    ASAXASA 0.7668 0.1040 0.0275 0.0314 0.0146 0.0021
    ASCIC 0.1493 0.0121 0.0007 0.0118 0.0859 0.0085
    ASCIC2 0.2443 0.0782 0.1228 0.0135 0.0024 0.0023
    ASDC 0.2244 0.0378 0.0218 0.1038 0.0031 0.0578
    ASTIG 0.2014 0.1077 0.4786 0.1102 0.0001 0.0208
    ASVON 0.5520 0.6574 0.1278 0.0004 0.0112 0.0007
    ATC 0.0162 0.0075 0.0142 0.0081 0.0475 0.0003
    ATEC 0.6875 0.1231 0.0041 0.1438 0.0038 0.0301
    ATTM 0.2088 0.0007 0.0003 0.2669 0.0031 0.0227
    BA 0.0152 0.0326 0.2408 0.0096 0.0001 0.0005
    BA2 0.0468 0.0000 0.0004 0.0741 0.0283 0.0014
    BACI 0.9822 0.2175 0.0504 0.0053 0.0405 0.1130
    BCIC 0.6013 0.2346 0.0001 0.1289 0.0764 0.0001
    BCIC2 0.0943 0.0259 0.0887 0.0702 0.0056 0.0082
    BSF2 0.5051 0.1925 0.0041 0.0031 0.0077 0.0042
    DARED 0.0064 0.0844 0.0304 0.1732 0.0001 0.0321
    DH 0.0251 0.0121 0.0042 0.0226 0.0406 0.0681
    EAT 0.3689 0.0500 0.0006 0.1937 0.0002 0.0014
    EEC 0.6922 0.4296 0.0290 0.0477 0.0317 0.0512
    EPCC 0.6784 0.2600 0.0692 0.1606 0.0575 0.0092
    ETEC 0.0047 0.0365 0.0191 0.1689 0.0005 0.0158
    FAAC 0.9998 0.3874 0.0314 0.0878 0.0612 0.0060
    FFS 0.5238 0.4407 0.1663 0.0002 0.0171 0.0080
    FPC 0.0181 0.0210 0.0003 0.0537 0.0024 0.0177
    FPMC 0.2145 0.0000 0.0052 0.0200 0.0074 0.1441
    GGCI 0.8051 0.0050 0.0098 0.0761 0.0132 0.0034
    GUCI 0.9193 0.6496 0.3417 0.0154 0.0005 0.0026
    HB 0.6265 0.3599 0.5291 0.0107 0.0056 0.0026
    HCC 0.0958 0.0322 0.0006 0.0266 0.0038 0.0462
    HFSC 0.0323 0.1624 0.0494 0.0062 0.0778 0.0184
    JDC 0.0019 0.0032 0.0024 0.1340 0.0410 0.0338
    JMC 0.0951 0.1530 0.0504 0.0087 0.0792 0.0350
    JODC 0.5510 0.1758 0.0009 0.1263 0.0180 0.0165
    KEC 0.6406 0.2771 0.3365 0.0093 0.0174 0.0332
    KHC 0.9947 0.2511 0.6098 0.0295 0.0079 0.0004
    MCC 0.7362 0.0374 0.0039 0.0509 0.0062 0.0202
    MCDC 0.9850 0.2134 0.0208 0.0576 0.0279 0.0197
    MCIC 0.6028 0.2023 0.1153 0.0012 0.0134 0.0070
    MESCC 0.4411 0.9385 0.1787 0.0273 0.0030 0.0065
    MMSC 0.0146 0.1928 0.0042 0.1552 0.0007 0.0001
    MTC 0.4446 0.2618 0.0046 0.0098 0.0306 0.0029
    NADC 0.3605 0.0585 0.1084 0.0223 0.0047 0.0106
    NAMC 0.9296 0.0180 0.0000 0.1074 0.0408 0.0152
    NCC 0.0442 0.1038 0.0019 0.0026 0.0095 0.0667
    NCC2 0.9996 0.0044 0.3526 0.2712 0.0052 0.0015
    NGC 0.3648 0.1052 0.0000 0.1620 0.0090 0.0400
    NGIC 0.3314 0.6376 0.0132 0.0025 0.0017 0.0022
    NIC 0.3651 0.2964 0.0154 0.1665 0.0448 0.0157
    NMMC 0.9193 0.0013 0.0000 0.0888 0.0023 0.0736
    NPC 0.5774 0.2832 0.0173 0.1008 0.0003 0.0128
    QAC 0.1159 0.0008 0.0005 0.1427 0.0003 0.0020
    RB 0.2963 0.5422 0.0229 0.0588 0.0062 0.0024
    RRPC 0.8476 0.0153 0.0284 0.0083 0.0003 0.0002
    RSHSC 0.0595 0.0982 0.0080 0.1678 0.0019 0.0089
    RUCI 0.1524 0.0873 0.0002 0.0260 0.0266 0.0146
    SAAC 0.4508 0.5554 0.2000 0.0031 0.0148 0.0026
    SACI 0.2828 0.0031 0.0013 0.1097 0.0307 0.0001
    SAFC 0.8476 0.2866 0.0772 0.2063 0.0013 0.0005
    SAHARA 0.0005 0.0007 0.0000 0.0283 0.0107 0.0396
    SAIC 0.9068 0.0147 0.0875 0.1527 0.0005 0.0942
    SAMC 0.6533 0.5972 0.0056 0.0262 0.0857 0.0190
    SARC 0.9850 0.0025 0.0022 0.0010 0.0003 0.0548
    SASC 0.9570 0.0004 0.0177 0.4427 0.0500 0.0714
    SBIC 0.5176 0.0946 0.0223 0.0879 0.0185 0.0111
    SCC 0.6040 0.1211 0.0512 0.0118 0.0566 0.0002
    SCC2 0.9896 0.0610 0.3424 0.0179 0.0027 0.0447
    SCC3 0.4674 0.5461 0.0502 0.1523 0.0972 0.0078
    SCC4 0.9570 0.3738 0.0022 0.3199 0.0783 0.0326
    SCIC 0.0564 0.0085 0.0045 0.0249 0.0158 0.0007
    SDFC 0.9850 0.0011 0.0465 0.0000 0.0666 0.0000
    SECI 0.0975 0.6284 0.0018 0.0223 0.0013 0.0027
    SEC 0.1524 0.0113 0.0028 0.3588 0.0774 0.0095
    SFC 0.0259 0.0002 0.0020 0.1340 0.0007 0.0301
    SFG 0.8971 0.1041 0.0214 0.0376 0.0014 0.0039
    SG 0.1524 0.6770 0.0066 0.0652 0.0006 0.0009
    SICI 0.9912 0.0022 0.0091 0.0741 0.0014 0.0022
    SIDC 0.8273 0.0084 0.0207 0.0905 0.0001 0.0343
    SIEC 0.9820 0.0345 0.5092 0.1512 0.0257 0.0061
    SIIG 0.9850 0.0268 0.0079 0.0850 0.0015 0.0166
    SIPC 0.0150 0.8035 0.0151 0.0958 0.0781 0.0040
    SISC 0.6464 0.0238 0.0032 0.1814 0.0105 0.0396
    SKPC 0.3652 0.0863 0.0065 0.1093 0.0006 0.0463
    SPA 0.4194 0.6744 0.0029 0.1834 0.0108 0.0150
    SPCC 0.0053 0.0451 0.0657 0.2596 0.0458 0.0000
    SPMC 0.0038 0.0086 0.0447 0.0354 0.0220 0.0005
    SPPC 0.9989 0.1971 0.2367 0.0292 0.0013 0.0184
    SPTC 0.3653 0.2404 0.0000 0.2722 0.0095 0.0003
    SRCR 0.7167 0.1152 0.5949 0.0079 0.0004 0.0273
    SREC 0.0255 0.0030 0.0229 0.0995 0.0029 0.0071
    SRMG 0.9915 0.0864 0.1634 0.0293 0.0110 0.0013
    SSPC 0.9849 0.8655 0.0077 0.1149 0.0000 0.0001
    SSTC 0.0935 0.8642 0.0045 0.0164 0.0045 0.1762
    ST 0.3597 0.4065 0.0098 0.0028 0.0349 0.0029
    STIC 0.3626 0.2447 0.0870 0.1124 0.0104 0.0008
    ST2 0.0040 0.0174 0.0013 0.0362 0.0550 0.0050
    SUCI 0.9995 0.0053 0.0012 0.2316 0.0085 0.0117
    SVCPC 0.8539 0.1033 0.1238 0.0525 0.0075 0.0038
    TADC 0.5053 0.0126 0.0136 0.0021 0.0102 0.0071
    TAI 0.3679 0.0196 0.0045 0.1628 0.0119 0.0337
    TAPR 0.1524 0.2582 0.2177 0.0095 0.0441 0.0145
    TCC 0.9851 0.2749 0.1572 0.1094 0.0052 0.0313
    TCCI 0.9574 0.0068 0.1052 0.0098 0.0672 0.0251
    TEC 0.0253 0.0044 0.0006 0.2845 0.0111 0.0694
    THC 0.5221 0.0451 0.0011 0.1061 0.0000 0.0021
    TMGIC 0.8498 0.4677 0.0494 0.0010 0.0048 0.0040
    TNCGI 0.2239 0.1142 0.0503 0.3044 0.0348 0.0067
    TNSC 0.8762 0.6472 0.0018 0.0412 0.0419 0.0557
    TQCC 0.5566 0.4555 0.0009 0.0309 0.0266 0.0194
    TSBB 0.9958 0.2533 0.0045 0.0109 0.0180 0.0043
    TSIB 0.4233 0.1382 0.1285 0.0229 0.0218 0.0261
    UCAC 0.6874 0.0102 0.0670 0.0047 0.0067 0.0016
    UEC 0.0893 0.1027 0.0171 0.0208 0.1551 0.0590
    UITC 0.1772 0.0096 0.1136 0.0707 0.0563 0.0137
    UWFC 0.0047 0.3556 0.0074 0.2828 0.0376 0.0207
    WIC 0.9397 0.1827 0.0000 0.0025 0.0019 0.0010
    YCC 0.1972 0.2168 0.0772 0.1497 0.0001 0.0000
    YNPC 0.4871 0.0711 0.0056 0.1923 0.0290 0.0611
    YSCC 0.1287 0.0558 0.0147 0.0563 0.0865 0.0200
    ZIIK 0.6492 0.2813 0.0057 0.0327 0.0809 0.0036

     | Show Table
    DownLoad: CSV

    To test more the link between the market return and the actions we plotted for the 2-scale law of time the recomposed crystal of the excess return on the stock versus the corresponding crystal on the market portfolio in Figure 1. Figure 1 plots daily stock returns versus corresponding stock beta at different time scales. The inspection of the figure confirms the relation between the average betas of stocks and average returns at every scale which enforces our earlier conclusions. Furthermore, as the scale increase from low (scale 1) to high (scale 6), the relation between the beta and the return becomes more and more clear. This evidence supports the proposition that the major part of the market's influence on individual asset prices is at higher horizons.

    Figure 1.  Excess market return (horizontal axis) versus excess return of the action (vertical axis) for different time scales with the 2-scale law.

    Figure 1 shows some linearity between the individual stock and the market portfolio where on the horizontal axis the excess of the market return is assigned versus the excess of the stock return assigned to the vertical axis. We notice that the linear relationship is particularly strong at the high scales especially scale 5 and 6. Medium and low scales show some perturbation in the movement of the market which confirms our conclusion previously about the market being non encouraging for small companies and short investments.

    Next, we acted in a similar way the second 3-scale law on the market in order to study and understand more the movement of the market and its influence on actions and vice-versa. We obtained the results of Table a. For reasons of readability and to help readers focusing we provided the results of the 3-scale law in the Appendix later. Table a represents the estimations of the Betas of each stock component at the scales j=1,2,3,4,5,6 relatively to the 3-scale law.

    Table a confirms the linear dependence noticed previously and confirms the perturbations in the market with very weak dependence in many cases and on all the 6 scales. Table a shows also that no law may be expected simultaneously for all the contribution of the Djm of all actions relatively to the increasing of time scale. To explain more such a contribution, we computed as previously the determination coefficient R2 in Table b below. It represents the estimations of the determination coefficient R2 relative to the Betas of each stock component estimated in Table a at the scales j=1,2,3,4,5,6 relatively to the 3-scale law. In fact, the changing in the time scale permits to quantify more the link between the return excess for each action and the time dimension of the market portfolio.

    Notice here also from Table b that the coefficient R2 is decreasing as the time scale increases with some perturbed cases. This explains here also the negative movement of the market already observed and that the major influencing parts of the market portfolio on the actions returns is localized in high frequencies. This confirms that at low horizons the market is going down but no law is conserved for all companies and all levels.

    As for the case of the 2-scale law of time intervals we proposed here to test the linearity between the market return and the actions. This is illustrated by Figure A which corresponds to the 3-scale law and which is provided also in Appendix. Figure A shows some linearity between the individual stock and the market portfolio where on the horizontal axis the excess of the market return is assigned versus the excess of the stock return assigned to the vertical axis. We notice that the linear relationship is particularly strong at the high scales especially scale 5 and 6. Medium and low scales show some perturbation in the movement of the market.

    However, we noticed from both figures (Figure 1 and Figure A) that the market is somehow perturbed and a high-level J=6 is not somehow compatible strongly in level 5. The same remark may be also noticed for the medium-high level J=4. This means that the application of random periods even-though it confirms the nature and the structure of the market did not permit to discover the hidden aspects that induced the perturbation observed from some short periods.

    To resume, the results make possible to quantify the correlation between the return of the stock and its beta at different time scales. The empirical results show according to Figure 1 and Figure A that the relationship between the return of a stock and its beta shows an important ambiguity although being linear. Recall that the CAPM implies that the excess return from assets (in excess of the risk-free asset return) should be proportional to the market premium (market return in excess of the risk–free asset return). Tables 6 and b show some positivity and significance for the relationship between the endogeneous and exogeneous variables at all scales although being very weak in some cases. When looking at individual excess return, the mean contribution of Djm is not clear enough to decide except quietly and sometimes at higher scales J4. This is always explained by the weak determination coefficient R2. This implies that it is unclear to conclude about the major part of the market portfolio's influence on individual stocks; is it at the medium or higher frequencies.

    This allows us to adapt the wavelet methods to discover more aspects in the Saudi Tadawul market. It is well known that wavelets are nowadays the most powerful tools in detecting hidden phenomena in financial markets. To show the volatility and the hidden dynamics of the market index Tadawul we applied firstly a wavelet filtering that consists in decomposing the time series associated into wavelet decomposition at the level 6. The result is illustrated by Figure 2.

    Figure 2.  Wavelet decomposition of Tadawul return series at level 6.

    The wavelet approach in estimating the systematic risk of an asset in the CAPM is based on a wavelet multi-scaling by decomposing the corresponding financial time series due to the stock return and the market one on a scale-by-scale wavelet basis. At each scale, the wavelet variance of the market return and the wavelet covariance between the market return and the stock return is computed to obtain a scale-by-scale estimate of stock's beta. The empirical results are gathered in Table 7.

    Table 7.  Wavelet Beta for each level.
    Beta for wavelet levels J=1 to J=6.
    STOCKS A6 D1 D2 D3 D4 D5 D6
    AADC 0.0456 0.0542 0.0179 0.1145 0.2421 0.0009 0.0273
    AAIC 0.0147 0.0081 0.1691 0.1342 0.0624 0.7116 0.1077
    AKSC 0.0146 0.0049 0.0910 0.1801 0.0312 0.4310 0.7640
    AAOMC 0.0022 0.0936 0.0098 0.0616 0.2043 0.3210 0.0341
    ABALAwal 0.0673 0.0284 0.1195 0.0003 0.1863 0.2360 0.8566
    ABAINma 0.0415 0.0424 0.0475 0.1100 0.1312 0.2777 0.0323
    ABIDC 0.0242 0.0569 0.0617 0.0920 0.0203 0.0006 0.1124
    ABPTC 0.0266 0.0268 0.0001 0.0281 0.1581 0.1581 0.2604
    AACI 0.0281 0.0679 0.1736 0.3365 0.2494 1.0131 0.1229
    ACAbdullatif 0.0411 0.0297 0.0013 0.0275 0.0980 0.2472 0.2536
    ACAlmaraI 0.0461 0.0893 0.1196 0.0724 0.3399 0.1337 0.3615
    ACAlujain 0.0968 0.0039 0.1484 0.3781 0.1976 0.1978 0.0943
    ACCAC 0.0686 0.0669 0.0045 0.0525 0.2210 0.1984 0.0702
    ACI 0.0697 0.1585 0.0762 0.0461 0.2419 0.0863 0.6547
    ACIC 0.0375 0.0327 0.1805 0.3293 0.2982 0.7254 0.3424
    ACIG 0.0507 0.0400 0.0705 0.1302 0.3809 0.3830 0.2661
    ADC 0.0190 0.0311 0.0623 0.1224 0.0979 0.0768 0.0702
    AICC 0.0112 0.0615 0.0712 0.0048 0.1443 0.3192 0.1213
    AIG 0.0676 0.0908 0.0320 0.1739 0.0568 0.1343 0.1885
    AIHG 0.0580 0.0233 0.2209 0.0852 0.0832 0.0607 0.3612
    AJADC 0.0060 0.0003 0.0215 0.0347 0.0678 0.2266 0.2843
    AJCC 0.0060 0.0033 0.0218 0.1552 0.1513 0.0814 0.2852
    AKGIS 0.0546 0.0453 0.2659 0.2000 0.1226 0.1847 0.1372
    ANB 0.0268 0.0506 0.0283 0.0674 0.0760 0.0548 0.2062
    APC1 0.0374 0.0536 0.0135 0.1091 0.2764 0.2207 0.7767
    APC2 0.0123 0.0329 0.0178 0.0574 0.1145 0.1385 0.1497
    APTS 0.0470 0.0911 0.0024 0.0020 0.0018 0.1974 0.0723
    ARB 0.0223 0.0224 0.0373 0.0208 0.1418 0.2026 0.0611
    ARCI 0.1525 0.2613 0.1309 0.2157 0.1349 0.0898 0.2411
    ASAXASA 0.0972 0.1594 0.0708 0.0277 0.0418 0.5093 0.1081
    ASCIC 0.0530 0.0495 0.2044 0.1794 0.0412 0.1838 0.2804
    ASCIC2 0.1415 0.2969 0.1227 0.1461 0.0200 0.4611 0.1952
    ASDC 0.0770 0.1209 0.1090 0.3411 0.1059 0.1814 0.0541
    ASTIG 0.0199 0.0211 0.1032 0.1846 0.0056 0.1017 0.2861
    ASVON 0.0351 0.0789 0.0799 0.2179 0.0809 0.2327 0.3148
    ATC 0.1226 0.1647 0.1447 0.0564 0.0239 0.1422 0.0088
    ATEC 0.0136 0.0351 0.0492 0.0251 0.0541 0.0419 0.3493
    ATTM 0.0115 0.0402 0.1117 0.0373 0.0256 0.0809 0.1285
    BA 0.0241 0.0742 0.0410 0.1302 0.3286 0.0013 0.2471
    BA2 0.0245 0.0005 −0.0289 0.0601 0.1783 0.1511 0.0892
    BACI −0.0338 −0.1625 −0.1457 0.0285 0.5634 0.3169 0.0614
    BCIC −0.0612 −0.0732 −0.1072 0.0494 0.0063 0.0639 −0.2716
    BCIC2 0.0598 −0.0354 0.0842 0.3015 0.0639 0.6607 0.2274
    BSF2 0.0299 −0.0036 0.0418 −0.0750 0.3108 0.0878 −0.0789
    DARED 0.0332 0.0541 −0.1733 −0.0101 0.3716 0.3324 −0.0017
    DH −0.0157 −0.0057 −0.0923 −0.0019 0.2259 −0.1083 −0.0520
    EAT −0.0633 −0.1789 −0.1411 0.3081 0.0860 0.5561 0.5497
    EEC 0.0900 −0.0058 0.1621 0.0578 0.1009 0.2936 0.9460
    EPCC 0.0027 −0.0005 0.0537 −0.0043 −0.0976 0.2280 −0.3686
    ETEC −0.0780 −0.0424 −0.1387 −0.2074 0.2058 −0.2062 −0.0935
    FAAC −0.0364 −0.0923 0.0080 −0.1250 0.0767 −0.0097 0.2983
    FFS −0.0376 −0.0093 −0.1266 −0.0069 −0.0209 0.0271 −0.0250
    FPC −0.0072 0.0486 −0.2269 0.2946 0.0863 −0.1109 −0.1828
    FPMC −0.1131 −0.1256 −0.1212 −0.0607 −0.0738 −0.1013 −0.5218
    GGCI −0.0141 −0.0012 −0.1053 0.1598 −0.1022 0.2126 0.1017
    GUCI 0.0346 −0.0494 0.1068 0.1781 0.0482 0.3336 0.6681
    HB −0.0031 −0.0287 0.0521 0.0272 0.1787 −0.3348 −0.6415
    HCC −0.0184 −0.0067 −0.0145 −0.0269 0.0276 −0.1004 −0.1397
    HFSC −0.0289 0.0374 −0.1611 −0.0272 −0.0048 0.1469 −0.1942
    JDC −0.0413 −0.0328 −0.1512 −0.0011 0.1987 0.0130 0.0080
    JMC −0.0396 −0.0267 −0.1830 −0.0113 0.1340 0.1954 0.1677
    JODC −0.0396 0.0117 −0.0927 −0.1693 −0.0684 0.2397 0.3081
    KEC 0.0228 0.0221 0.0901 −0.7417 0.0984 0.5667 1.4308
    KHC −0.0315 0.0111 0.0801 −0.4839 −0.2192 −0.2052 1.1686
    MCC −0.0131 0.0437 −0.1318 −0.0062 0.0266 −0.0479 0.2546
    MCDC 0.0192 0.0079 −0.0850 0.2193 0.1774 −0.1346 0.0068
    MCIC −0.0856 −0.0649 0.0001 0.0876 −0.1942 −0.5787 0.9745
    MESCC 0.0624 −0.0036 0.1909 0.0119 0.1185 0.1184 −0.3774
    MMSC −0.0165 0.0246 −0.0466 −0.0205 −0.0512 −0.2765 0.1979
    MTC 0.0621 0.0193 −0.0586 0.1656 0.2843 0.5829 −0.0508
    NADC −0.0055 0.0106 −0.0288 −0.1748 0.3539 −0.2887 0.3566
    NAMC 0.0176 −0.0139 0.0025 0.0920 0.1666 0.2011 −0.5989
    NCC −0.0208 −0.0568 −0.0378 0.0056 0.1093 0.1240 0.1683
    NCC2 −0.0486 −0.1032 0.3798 0.0441 −0.5495 −0.6516 1.4675
    NGC −0.0328 −0.0027 −0.1242 −0.0748 0.1352 0.1267 0.1416
    NGIC −0.0325 −0.0249 −0.1076 −0.0713 0.0575 0.2202 −0.0242
    NIC 0.0056 −0.0294 −0.0704 0.1314 0.0778 0.1146 0.0940
    NMMC −0.0580 0.0075 −0.2024 −0.0421 0.0416 0.0608 −0.3702
    NPC −0.0802 −0.0287 −0.1793 0.0420 0.0791 0.1705 0.2557
    QAC 0.0070 0.1227 −0.1421 −0.1661 0.2926 0.0481 −0.0265
    RB 0.0417 0.0822 0.1209 −0.1099 −0.0373 −0.0026 −0.3406
    RRPC −0.0057 0.0226 −0.0950 −0.1592 0.0971 0.2582 1.4673
    RSHSC −0.1031 −0.1218 −0.1347 −0.0627 −0.1571 0.2579 0.7422
    RUCI −0.0746 −0.1569 −0.1086 0.1516 0.0528 −0.0863 0.6127
    SAAC −0.0211 −0.0015 0.0017 −0.6231 0.0361 0.4772 1.3337
    SACI 0.0465 −0.0492 0.3521 0.0392 0.0144 −0.0173 −0.7094
    SAFC −0.0392 −0.0753 −0.0291 0.0116 0.0703 −0.0545 −0.0540
    SAHARA −0.0001 0.0100 −0.0833 −0.0273 0.0647 −0.0915 0.7992
    SAIC −0.0435 −0.0287 −0.1793 0.0420 0.0791 0.1705 0.2557
    SAMC −0.0554 −0.0288 −0.0690 −0.0728 −0.0460 −0.1104 0.1703
    SARC −0.0556 −0.0074 −0.2230 0.1207 −0.0827 0.3700 −0.7227
    SASC −0.1013 −0.2558 0.1039 −0.0344 0.1160 −0.4093 0.1326
    SBIC −0.0063 −0.0007 −0.0411 −0.0692 0.1383 −0.1662 0.4092
    SCC −0.0235 0.0223 −0.1505 0.0182 −0.0065 0.3886 −0.0727
    SCC2 0.0233 −0.0196 0.0826 −0.0464 0.0946 0.2569 −0.6639
    SCC3 0.0264 0.0216 0.0130 0.0811 0.0034 0.0011 0.1613
    SCC4 0.0137 −0.0368 −0.1199 0.0677 0.3075 0.4753 0.2284
    SCIC −0.0645 −0.1311 −0.2697 0.2887 0.1641 0.2769 −0.0454
    SDFC −0.0181 0.0428 −0.2401 0.1097 0.1630 −0.1407 −0.4301
    SECI 0.0403 0.0403 0.1164 0.0695 −0.3204 0.4207 0.7303
    SEC 0.0524 0.0577 −0.0225 0.1559 −0.0170 0.1915 0.4767
    SFC 0.0708 0.1561 0.0207 0.1582 −0.1650 0.3554 0.3235
    SFG 0.0239 0.0406 0.0093 −0.0620 0.1675 −0.4243 0.7986
    SG 0.0682 −0.0119 0.0034 −0.1310 0.5997 0.4311 0.3788
    SICI 0.0808 0.1858 −0.1974 0.5537 −0.2980 0.3184 −0.4706
    SIDC 0.0005 0.0041 0.0166 −0.0081 0.0289 0.0964 0.2958
    SIEC 0.1569 0.1548 0.3015 0.1853 0.3910 −0.3421 −0.6848
    SIIG −0.0193 −0.0115 −0.0666 −0.2546 0.1769 0.0918 1.0752
    SIPC 0.1220 0.0570 0.0461 −0.5714 0.6840 0.7317 1.5726
    SISC −0.0284 0.0242 −0.1223 −0.0651 0.0169 0.1226 0.1384
    SKPC −0.0116 −0.0139 −0.0799 −0.1090 0.2074 0.1536 0.1100
    SPA −0.0043 0.0342 −0.1003 0.0008 0.0688 0.1896 −0.3133
    SPCC 0.0039 0.0659 −0.1328 0.0018 0.0911 −0.0435 −0.1378
    SPMC 0.0466 0.0674 0.1489 −0.1305 −0.5855 0.0198 2.1554
    SPPC −0.1349 −0.0717 −0.3389 −0.0615 −0.0492 −0.1068 0.2281
    SPTC 0.0015 −0.0258 0.0063 −0.1330 0.2367 0.2255 0.5886
    SRCR 0.0795 −0.0163 0.0922 0.1830 0.2519 0.6456 0.6302
    SREC −0.0435 −0.0113 −0.1283 0.0082 0.0159 0.1174 −0.2488
    SRMG −0.0822 −0.0163 −0.2175 −0.0583 −0.0043 −0.0628 0.0482
    SSPC −0.0021 −0.0076 −0.1187 0.0045 0.1936 0.3004 0.0021
    SSTC 0.1072 0.1209 −0.0049 0.1628 0.2037 0.1308 0.0637
    ST −0.0142 −0.0234 −0.0529 −0.0817 0.1553 −0.1020 0.2956
    STIC −0.0300 −0.0376 0.0693 0.0188 0.0581 −0.1068 −0.4726
    ST2 −0.0877 −0.1114 −0.1397 0.1198 −0.1517 0.3265 −0.1733
    SUCI −0.1081 −0.1897 −0.0812 −0.0051 0.0272 −0.1135 0.0635
    SVCPC −0.0933 −0.0127 −0.1554 −0.0354 −0.7366 0.3252 0.7706
    TADC 0.0218 0.0159 0.0245 0.0616 0.0529 −0.0515 0.0022
    TAI −0.0085 0.0656 −0.0769 −0.1610 0.1383 0.0659 0.9873
    TAPR 0.0865 −0.0391 0.2627 0.0648 −0.0169 0.1629 −0.1250
    TCC 0.0662 0.0291 0.1601 −0.0269 0.0820 0.4433 0.0911
    TCCI 0.0048 −0.0362 −0.0630 −0.0027 0.3080 0.7444 −0.4341
    TEC 0.0098 0.0413 −0.1368 0.1357 0.0449 0.0228 −0.3267
    THC −0.0064 −0.0257 −0.0740 0.1256 0.0902 −0.0468 −0.1454
    TMGIC 0.0893 −0.0840 0.3027 −0.0005 0.1539 1.4404 0.1353
    TNCGI 0.0131 0.0284 −0.0248 −0.0269 0.1447 0.1147 0.1534
    TNSC −0.0507 −0.1198 −0.0972 0.0384 0.2365 0.3383 0.0835
    TQCC −0.0131 −0.0264 −0.0498 0.0454 −0.0007 0.1612 0.0034
    TSBB 0.0299 0.0060 −0.0291 0.0442 0.2327 −0.1622 0.3065
    TSIB 0.0455 0.1177 −0.0475 −0.0747 0.0499 −0.0638 0.2830
    UCAC −0.1449 −0.2060 −0.0364 −0.1649 −0.2076 −0.0638 0.9574
    UEC 0.0038 −0.0059 −0.1177 0.0701 0.1616 0.3211 0.3109
    UITC −0.0424 −0.0231 −0.1561 −0.0926 0.0089 0.3628 0.2038
    UWFC 0.0001 0.0216 −0.0804 0.1801 −0.0675 −0.2087 0.0294
    WIC −0.1450 −0.2493 −0.0908 −0.0261 −0.0027 0.3320 −0.0153
    YCC −0.0224 0.0166 −0.0026 −0.0658 −0.1775 0.2109 −0.3489
    YNPC −0.0060 −0.0205 −0.0110 −0.1010 0.2394 −0.5251 0.7140
    YSCC 0.0184 0.0849 −0.0399 0.0296 −0.1814 0.0618 0.0721
    ZIIK −0.0277 0.0439 −0.1857 −0.0914 0.1403 0.1091 0.1613
    Mean −0.0111 −0.0178 −0.0450 −0.0036 0.0649 0.1208 0.1682

     | Show Table
    DownLoad: CSV

    It represents the estimations of the Betas of each stock component at the wavelet levels J=1,2,3,4,5,6. In the first column we provided also the estimation of the Beta relative to the wavelet approximation at the level J=6 to get a global idea on the systematic risk beta. Recall that in wavelet theory the approximation component AJ provides a global description of the series estimated. This column is no longer provided in all the previous studies.

    Notice from Table 7 that more dependence between the market and actions is detected. For example, the ABIDC which has resulted in a zero beta for previous classical methods even by changing the time scale law has yielded by using the wavelet processing a non-zero beta although being negative and small (in absolute value) for all wavelet levels. This means that in such type of markets one should apply always macroscopic and microscopic tools able to detect the hidden facts.

    Next to confirm the linear dependence and the efficiency of the wavelet tool we established as previously a computation of the R2 coefficient of determination in Table 8. It represents the estimations of the determination coefficient R2 relative to the Betas of each stock component estimated in Table 7 at the wavelet levels J=1,2,3,4,5,6. It shows a coherence with results on wavelet estimation of beta.

    Table 8.  Wavelet R2 relative to Table 7 for each level.
    R2
    STOCKS D1 D2 D3 D4 D5 D6
    AADC 0.0007 0.0001 0.0036 0.0229 0.0000 0.0002
    AAIC 0.0000 0.0021 0.0014 0.0005 0.0402 0.0006
    AKSC 0.0000 0.0014 0.0056 0.0005 0.0254 0.0626
    AAOMC 0.0033 0.0000 0.0017 0.0252 0.0264 0.0004
    ABALAwal 0.0001 0.0027 0.0000 0.0108 0.0118 0.1438
    ABAINma 0.0007 0.0009 0.0054 0.0167 0.0524 0.0008
    ABIDC 0.0013 0.0012 0.0070 0.0002 0.0000 0.0028
    ABPTC 0.0002 0.0000 0.0003 0.0092 0.0064 0.0108
    AACI 0.0004 0.0030 0.0074 0.0102 0.0881 0.0010
    ACAbdullatif 0.0003 0.0000 0.0003 0.0058 0.0285 0.0175
    ACAlmaraI 0.0030 0.0068 0.0030 0.0755 0.0075 0.0985
    ACAlujain 0.0000 0.0044 0.0308 0.0112 0.0099 0.0012
    ACCAC 0.0020 0.0000 0.0013 0.0299 0.0130 0.0034
    ACI 0.0025 0.0005 0.0002 0.0165 0.0008 0.0318
    ACIC 0.0001 0.0017 0.0060 0.0066 0.0308 0.0027
    ACIG 0.0002 0.0006 0.0016 0.0198 0.0139 0.0042
    ADC 0.0004 0.0012 0.0072 0.0066 0.0024 0.0029
    AICC 0.0004 0.0005 0.0000 0.0071 0.0114 0.0021
    AIG 0.0025 0.0003 0.0083 0.0020 0.0066 0.0105
    AIHG 0.0001 0.0080 0.0016 0.0027 0.0005 0.0204
    AJADC 0.0000 0.0001 0.0005 0.0031 0.0126 0.0165
    AJCC 0.0000 0.0002 0.0076 0.0212 0.0030 0.0324
    AKGIS 0.0004 0.0114 0.0068 0.0045 0.0054 0.0035
    ANB 0.0015 0.0005 0.0026 0.0051 0.0022 0.0186
    APC1 0.0012 0.0001 0.0048 0.0436 0.0265 0.2068
    APC2 0.0002 0.0001 0.0007 0.0061 0.0041 0.0029
    APTS 0.0025 0.0000 0.0000 0.0000 0.0101 0.0019
    ARB 0.0003 0.0009 0.0003 0.0216 0.0304 0.0040
    ARCI 0.0122 0.0027 0.0097 0.0093 0.0024 0.0169
    ASAXASA 0.0031 0.0006 0.0001 0.0005 0.0278 0.0013
    ASCIC 0.0004 0.0058 0.0056 0.0005 0.0041 0.0092
    ASCIC2 0.0128 0.0016 0.0035 0.0001 0.0190 0.0045
    ASDC 0.0024 0.0016 0.0200 0.0023 0.0033 0.0003
    ASTIG 0.0001 0.0019 0.0064 0.0000 0.0018 0.0084
    ASVON 0.0010 0.0008 0.0063 0.0017 0.0028 0.0066
    ATC 0.0058 0.0040 0.0008 0.0003 0.0054 0.0000
    ATEC 0.0003 0.0005 0.0001 0.0009 0.0006 0.0260
    ATTM 0.0005 0.0035 0.0003 0.0004 0.0018 0.0057
    BA 0.0019 0.0005 0.0051 0.0565 0.0000 0.0419
    BA2 0.0000 0.0003 0.0011 0.0212 0.0074 0.0025
    BACI 0.0029 0.0027 0.0001 0.0666 0.0194 0.0008
    BCIC 0.0013 0.0033 0.0006 0.0000 0.0009 0.0317
    BCIC2 0.0002 0.0007 0.0095 0.0008 0.0554 0.0076
    BSF2 0.0000 0.0008 0.0023 0.0631 0.0041 0.0040
    DARED 0.0005 0.0054 0.0000 0.0485 0.0161 0.0000
    DH 0.0000 0.0022 0.0000 0.0432 0.0047 0.0012
    EAT 0.0026 0.0021 0.0111 0.0010 0.0426 0.0344
    EEC 0.0000 0.0074 0.0012 0.0029 0.0151 0.1693
    EPCC 0.0000 0.0019 0.0000 0.0093 0.0216 0.0693
    ETEC 0.0003 0.0042 0.0094 0.0155 0.0087 0.0022
    FAAC 0.0011 0.0000 0.0023 0.0015 0.0000 0.0103
    FFS 0.0001 0.0094 0.0000 0.0006 0.0004 0.0003
    FPC 0.0004 0.0075 0.0149 0.0022 0.0020 0.0051
    FPMC 0.0041 0.0025 0.0009 0.0024 0.0028 0.0498
    GGCI 0.0000 0.0014 0.0034 0.0030 0.0073 0.0011
    GUCI 0.0003 0.0014 0.0044 0.0005 0.0112 0.0464
    HB 0.0002 0.0008 0.0002 0.0146 0.0273 0.1334
    HCC 0.0000 0.0001 0.0003 0.0007 0.0036 0.0078
    HFSC 0.0006 0.0087 0.0002 0.0000 0.0090 0.0157
    JDC 0.0003 0.0048 0.0000 0.0194 0.0001 0.0000
    JMC 0.0004 0.0170 0.0001 0.0141 0.0153 0.0203
    JODC 0.0000 0.0027 0.0082 0.0027 0.0172 0.0279
    KEC 0.0001 0.0018 0.1057 0.0046 0.0618 0.4610
    KHC 0.0000 0.0017 0.0561 0.0328 0.0150 0.2636
    MCC 0.0006 0.0048 0.0000 0.0005 0.0007 0.0197
    MCDC 0.0000 0.0021 0.0138 0.0142 0.0061 0.0000
    MCIC 0.0003 0.0000 0.0006 0.0043 0.0254 0.1106
    MESCC 0.0000 0.0061 0.0000 0.0049 0.0024 0.0214
    MMSC 0.0002 0.0008 0.0001 0.0016 0.0509 0.0260
    MTC 0.0001 0.0005 0.0062 0.0230 0.0439 0.0002
    NADC 0.0000 0.0002 0.0073 0.0558 0.0217 0.0140
    NAMC 0.0000 0.0000 0.0017 0.0078 0.0037 0.0337
    NCC 0.0012 0.0005 0.0000 0.0069 0.0047 0.0089
    NCC2 0.0004 0.0063 0.0001 0.0225 0.0127 0.1011
    NGC 0.0000 0.0041 0.0016 0.0109 0.0051 0.0067
    NGIC 0.0003 0.0074 0.0026 0.0028 0.0231 0.0002
    NIC 0.0003 0.0014 0.0055 0.0032 0.0051 0.0031
    NMMC 0.0000 0.0077 0.0004 0.0008 0.0009 0.0259
    NPC 0.0002 0.0069 0.0004 0.0025 0.0069 0.0143
    QAC 0.0025 0.0029 0.0046 0.0229 0.0003 0.0001
    RB 0.0016 0.0038 0.0033 0.0006 0.0000 0.0307
    RRPC 0.0001 0.0017 0.0062 0.0030 0.0165 0.3417
    RSHSC 0.0028 0.0032 0.0009 0.0108 0.0156 0.1135
    RUCI 0.0048 0.0021 0.0045 0.0011 0.0011 0.0349
    SAAC 0.0000 0.0000 0.1209 0.0007 0.0950 0.3936
    SACI 0.0002 0.0133 0.0002 0.0000 0.0000 0.0333
    SAFC 0.0023 0.0004 0.0001 0.0043 0.0017 0.0015
    SAHARA 0.0000 0.0023 0.0003 0.0042 0.0035 0.1846
    SAIC 0.0002 0.0069 0.0004 0.0025 0.0069 0.0143
    SAMC 0.0002 0.0013 0.0016 0.0012 0.0047 0.0142
    SARC 0.0000 0.0092 0.0029 0.0031 0.0224 0.1287
    SASC 0.0136 0.0021 0.0003 0.0059 0.0261 0.0023
    SBIC 0.0000 0.0008 0.0025 0.0169 0.0111 0.0952
    SCC 0.0002 0.0082 0.0001 0.0000 0.0578 0.0024
    SCC2 0.0001 0.0011 0.0004 0.0026 0.0117 0.0850
    SCC3 0.0003 0.0001 0.0031 0.0000 0.0000 0.0148
    SCC4 0.0005 0.0050 0.0015 0.0549 0.0698 0.0149
    SCIC 0.0019 0.0076 0.0087 0.0052 0.0067 0.0002
    SDFC 0.0006 0.0196 0.0044 0.0226 0.0072 0.0832
    SECI 0.0002 0.0013 0.0004 0.0185 0.0155 0.0382
    SEC 0.0015 0.0003 0.0119 0.0002 0.0156 0.1007
    SFC 0.0021 0.0000 0.0021 0.0054 0.0139 0.0117
    SFG 0.0003 0.0000 0.0006 0.0073 0.0377 0.0955
    SG 0.0000 0.0000 0.0011 0.0335 0.0113 0.0090
    SICI 0.0023 0.0025 0.0170 0.0080 0.0043 0.0071
    SIDC 0.0000 0.0001 0.0000 0.0004 0.0018 0.0138
    SIEC 0.0006 0.0025 0.0010 0.0081 0.0030 0.0107
    SIIG 0.0000 0.0011 0.0174 0.0107 0.0022 0.2091
    SIPC 0.0010 0.0007 0.1038 0.1837 0.1835 0.3832
    SISC 0.0002 0.0047 0.0011 0.0002 0.0050 0.0045
    SKPC 0.0000 0.0016 0.0030 0.0234 0.0073 0.0031
    SPA 0.0005 0.0031 0.0000 0.0037 0.0134 0.0411
    SPCC 0.0020 0.0101 0.0000 0.0068 0.0006 0.0132
    SPMC 0.0004 0.0012 0.0028 0.0766 0.0001 0.3092
    SPPC 0.0008 0.0134 0.0005 0.0007 0.0011 0.0070
    SPTC 0.0002 0.0000 0.0035 0.0260 0.0112 0.0756
    SRCR 0.0001 0.0022 0.0104 0.0225 0.0672 0.0786
    SREC 0.0000 0.0041 0.0000 0.0002 0.0020 0.0144
    SRMG 0.0000 0.0071 0.0004 0.0000 0.0003 0.0002
    SSPC 0.0000 0.0036 0.0000 0.0283 0.0258 0.0000
    SSTC 0.0024 0.0000 0.0038 0.0107 0.0028 0.0006
    ST 0.0003 0.0012 0.0038 0.0237 0.0070 0.0852
    STIC 0.0002 0.0008 0.0000 0.0011 0.0016 0.0268
    ST2 0.0022 0.0026 0.0023 0.0091 0.0288 0.0039
    SUCI 0.0051 0.0008 0.0000 0.0002 0.0017 0.0008
    SVCPC 0.0000 0.0085 0.0006 0.3510 0.0345 0.2701
    TADC 0.0001 0.0001 0.0008 0.0011 0.0004 0.0000
    TAI 0.0008 0.0011 0.0042 0.0068 0.0004 0.0964
    TAPR 0.0002 0.0068 0.0003 0.0000 0.0008 0.0008
    TCC 0.0003 0.0103 0.0004 0.0047 0.0940 0.0061
    TCCI 0.0002 0.0006 0.0000 0.0324 0.0895 0.0249
    TEC 0.0003 0.0024 0.0033 0.0006 0.0001 0.0155
    THC 0.0002 0.0020 0.0055 0.0065 0.0008 0.0138
    TMGIC 0.0005 0.0063 0.0000 0.0033 0.0930 0.0009
    TNCGI 0.0003 0.0002 0.0002 0.0173 0.0051 0.0055
    TNSC 0.0038 0.0022 0.0005 0.0281 0.0389 0.0020
    TQCC 0.0006 0.0021 0.0020 0.0000 0.0144 0.0000
    TSBB 0.0000 0.0003 0.0006 0.0196 0.0108 0.0433
    TSIB 0.0080 0.0011 0.0031 0.0023 0.0027 0.0439
    UCAC 0.0057 0.0002 0.0032 0.0101 0.0006 0.0683
    UEC 0.0000 0.0035 0.0010 0.0120 0.0275 0.0203
    UITC 0.0001 0.0054 0.0029 0.0000 0.0325 0.0137
    UWFC 0.0001 0.0014 0.0072 0.0021 0.0111 0.0002
    WIC 0.0080 0.0008 0.0001 0.0000 0.0138 0.0000
    YCC 0.0001 0.0000 0.0020 0.0238 0.0149 0.0649
    YNPC 0.0001 0.0000 0.0042 0.0232 0.0913 0.1429
    YSCC 0.0048 0.0010 0.0006 0.0266 0.0015 0.0046
    ZIIK 0.0007 0.0115 0.0029 0.0150 0.0049 0.0112

     | Show Table
    DownLoad: CSV

    We notice that the market is going to be efficient at high scales and the dependence grow up in fact as the level increases. This let investors to conclude that investing in such a market may need to come over the microscopic and macroscopic scales of the market and study may be different panels in it to conclude in what sector investments will be of good returns.

    In the present case, it appears that food sectors seems to be the most important. This is somehow natural as KSA is an importing country more than a self-producing in this sector. Otherwise, sectors such as energy, transportation and security and safety are somehow encouraging. This is also a good task as KSA is planning to lunch the international project NEOM which will be from the financial and economic points of view an international market where different sectors will intersect and interchange.

    Besides, some sectors present a small and sometimes negative relationship between the growth in the market and the one of the company and thus should be corrected to firstly pass to positive influence of the company in the whole market and increase the return. Some steps should be taken into consideration such as the Yemen war at the south and Syrian one where a great part of KSA money is lost.

    The finings in Tables 7 and 8 are confirmed in Figure 3 hereafter which illustrates the overage of wavelet excess return of the actions against the wavelet excess return of the market at the different wavelet levels j=1,2,3,4,5,6.

    Figure 3.  Wavelet excess market return (horizontal axis) versus wavelet excess return of the action (vertical axis) at levels j = 1, 2, 3, 4, 5, 6.

    According to the classification into sectors we notice from Table 9 that by grouping the components of the market according to the sectors this allows us to carry out a global view of the market. Yet the disruption is very clear in the market in hand. Most sectors show a positive relationship in the medium level J=4 according to the 2-scale law except for a few cases such as health. However, this sector itself is based mainly on imported equipment and a workforce which suffers from qualification except a great part of foreign one. By applying the wavelet model which generally allows to go further into the microscopic state of the market, we notice that the positive relationship between market and sectors becomes more and more important for large scales, J=4 and mostly J=5 and J=6. These conclusions join previous ones which require to introduce other factors such as those due to Marfatia H.A. and may be political and geopolitical factors to better understand this market.

    Table 9.  The mean Beta for each sector.
    2-scale law Beta Wavelet Beta
    Sectors Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6 Scale 1 Scale 2 Scale 3 Scale 4 Scale 5 Scale 6
    Energy 0.3418 0.3072 0.0987 1.1997 0.0696 0.5224 0.0493 0.1127 0.0614 0.0422 0.0863 0.3252
    Materials 0.6043 0.2231 0.1048 1.0446 0.0677 0.2219 0.0017 0.0470 0.0443 0.0598 0.0605 0.2630
    Cap. Go. 1.1992 0.1656 1.1567 1.5932 0.3301 0.1080 0.0166 0.0060 0.0630 0.0294 0.1691 0.1994
    CoPr SvC 1.5417 2.1253 0.1435 1.2702 0.7362 0.2967 0.0901 0.1720 0.0649 0.0355 0.1547 0.2088
    Transport 0.1329 0.2562 0.4128 3.8766 0.2069 0.1983 0.0156 0.0507 0.0680 0.0801 0.1510 0.1145
    Consumer 0.1850 0.2499 0.1393 2.2427 0.2051 0.2490 0.0173 0.0750 0.0480 0.0492 0.0784 0.0874
    Media Ent. 2.4493 0.8724 0.7616 1.0121 1.2325 0.2165 0.0277 0.0226 0.0032 0.0106 0.0500 0.0384
    Retailing 0.4745 0.1264 0.1326 1.8277 0.4039 0.0635 0.0671 0.0909 0.0601 0.1222 0.0564 0.2093
    Foods 0.1943 0.0608 0.7407 1.8958 0.0772 0.2536 0.0150 0.0778 0.0434 0.1819 0.0106 0.0292
    Health 0.7572 0.6308 0.0076 0.3244 0.1957 0.1105 0.0294 0.0735 0.0099 0.0088 0.0435 0.0577
    Banks 1.0496 0.2654 0.0821 1.2413 0.1573 0.0457 0.0115 0.0148 0.0459 0.1605 0.0281 0.1804
    Div. Fin. 0.3950 0.3979 0.7848 1.3826 0.2593 0.0161 0.0139 0.0598 0.0457 0.0224 0.0051 0.1886
    Insurance 0.1504 0.6573 0.4598 1.0127 0.2895 0.1567 0.0823 0.0187 0.0917 0.0005 0.3518 0.1177
    TelecoM 0.1933 0.0543 0.5699 2.2973 0.3199 0.0646 0.0692 0.0443 0.1628 0.0799 0.4398 0.3179
    Utilities 0.8338 0.7105 0.2389 0.5362 0.2071 0.0711 0.0164 0.0650 0.0423 0.0203 0.2059 0.2263
    REITs 1.2204 0.6896 0.1092 2.2283 0.0053 0.0980 0.0257 0.0740 0.1256 0.0902 0.0468 0.1454
    R Estate 0.0307 0.2600 0.0474 2.2167 0.1841 0.2578 0.0061 0.0906 0.1052 0.0927 0.1371 0.2768

     | Show Table
    DownLoad: CSV

    In this paper a wavelet study of the largest GCC market Saudi Tadawul has been developed in order to understand such market on a critic period which post follows many important movements infusing directly and strongly on the market. Saudi Tadawul is chosen based on its value as the most representative market for GCC continent and thus its influence and relation to worldwide markets.

    In this paper we proposed essentially to test the impact of the time scale on the estimation of the systematic risk Beta in the presence of missing data by using essentially wavelet tools. We essentially focused on the stability of the systematic risk beta according to time scales and to the eventual linear linkage between the assets returns and their systematic risks.

    At a first step the empirical results conducted showed that the relationship between the return of a stock and its beta becomes stronger quietly at the medium-higher, but the test of the linearity between the return and its betas showed that there is an important ambiguity. This led us to think about wavelets as more adoptable and relevant methods to enlighten the ambiguity. Wavelets are suitable adopted mathematical tools that have been world-wise developed until the 80's. They have been shown successful applications in physics, mathematics, finance, statistics, etc.

    The application of wavelets has proved that the relationship between the return of a stock and its beta is more robust at higher scales 4, 5 and 6. This evidence means that Saudi stock market is more efficient in the high scales. Therefore, the predictions of the CAPM are more relevant at the higher-horizons in a multi-scale framework as compared to other horizons.

    Nevertheless the market presents sometimes some perturbations which remain clear even when splitting it into sectors. We think that such perturbations are due to other factors that should be included for any future study to understand more the movement and the situation of the market. Factors such as US policy, local political factors as well as geopolitical ones may be good extensions.

    The study shows that such a market may be encouraging for investments on long time horizons reminiscent of some prudence that should be taken into account for short investments and small companies. The study may be a good example for both investors and researchers to understand other GCC markets that will be future extending studies of the present case.

    The author would like to thank the editor and the anonymous reviewers for their valuable comments which improved the paper considerably. The present work is subscribed under the research project S-1439-0180, University of Tabuk, KSA. The author would like to thank the deanship of scientific research at the University of Tabuk, KSA for considering this project.

    The author declares here that no conflicts of interest in this paper.



    [1] Aktan B, Ozturk M, Rhaeim N, et al. (2009) Wavelet-Based Systematic Risk Estimation An Application on Istanbul Stock Exchange. Int Res J Financ Econ 23: 34-45.
    [2] Arfaoui S, Rezgui I, Mabrouk AB (2017) Wavelet Analysis On The Sphere, Spheroidal Wavelets, Degryuter, Degruyter, 2017, ISBN 978-3-11-048188-4.
    [3] Arfaoui S, Mabrouk AB, Cattani C (2020) New type of Gegenbauer-Hermite monogenic polynomials and associated Clifford wavelets. J Math Imaging Vision 62: 73-97.
    [4] Arfaoui S, Mabrouk AB, Cattani C (2020) New type of Gegenbauer-Jacobi-Hermite monogenic polynomials and associated continuous Clifford wavelet transform. Acta Applicandea Math. https://doi.org/10.1007/s10440-020-00322-0.
    [5] Aydogan K (1989) Portfolio Theory, Capital Market Board of Turkey Research Report: Ankara.
    [6] Banz RW (1981) The relationship between return and market value of common stock. J Financ Econ 9: 3-18.
    [7] Basu S (1977) The relationship between earnings' yield, market value and return for NYSE common stocks-further evidence. J Financ 32: 663-681.
    [8] Mabrouk AB, Mohamed MLB, Omrani K (2008) Numerical solutions for PDEs modeling binary alloy-solidification dynamics, In: Proceedings of 2007 International Symposium on Nonlinear Dynamics, J Phys, conference series 96 (2008) 012067.
    [9] Mabrouk AB, Kortass H, Ammou SB (2008) Wavelet Estimators for Long Memory in Stock Markets. Int J Theor Appl Financ 12: 297-317.
    [10] Mabrouk AB, Kahloul I, Hallara SE (2010) Wavelet-Based Prediction for Governance, Diversification and Value Creation Variables. Int Res J Financ Econ 60: 15-28.
    [11] Mabrouk AB, Abdallah NB, Hamrita ME (2011) A wavelet method coupled with quasi self similar stochastic processes for time series approximation. Int J Wavelets Multiresolution Inf Process 9: 685-711.
    [12] Mabrouk AB, Zaafrane O (2013) Wavelet Fuzzy Hybrid Model For Physico Financial Signals. J Appl Stat 40: 1453-1463.
    [13] Mabrouk AB, Rabbouch B, Saadaoui F (2015) A wavelet based methodology for predicting transmembrane segments, In: Poster Session, The International Conference of Engineering Sciences for Biology and Medecine, 1-3 May 2015, Monastir, Tunisie.
    [14] Black F, Jensen MC, Scholes M(1972) The capital asset pricing model: Some empirical tests, in Jensen MC (ed), Studies in the theory of Capital, New York: Praeger, 1-54.
    [15] Black F (1972) Capital Market Equilibrium with Restricted Borrowing. J Bus 45: 444-455.
    [16] Breeden D (1979) An Intertemporal Asset Pricing Model with Stochastic Consumption and Investment Opportunities. J Financ Econ 73: 265-296.
    [17] Brennan MJ (1973) Taxes, market valuation and corporate financial policy. Natl tax J 23: 417-427.
    [18] Chae J, Yang C (2008) Which idiosyncratic factors can explain the pricing errors from asset pricing models in the Korean stock market? Asia-Pasific J Financ Stud 37: 297-342.
    [19] Chan LK, Lakonishok J (1993) Are the reports of beta's death premature? J Portf Manage 19: 51-62.
    [20] Cifter A, Ozun A (2007) Multiscale systematic risk: An application on ISE 30, MPRA Paper 2484, University Library of Munich: Germany.
    [21] Cifter A, Ozun A (2008) A signal processing model for time series analysis: The effect of international F/X markets on domestic currencies using wavelet networks. Int Rev Electr Eng 3: 580-591.
    [22] Cohen K, Hawawin G, Mayer S, et al. (1986) The Microstructure of Securities Markets, Prentice-Hall: Sydney.
    [23] Conlon T, Crane M, Ruskin HJ (2008) Wavelet multiscale analysis for hedge funds: Scaling and strategies. Phys A 387: 5197-5204.
    [24] Daubechies I (1992) Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, Philadelphia.
    [25] Desmoulins-Lebeault F (2003) Distribution of Returns and the CAPM Empirical Problems. Post-Print halshs-00165099, HAL.
    [26] DiSario R, Saroglu H, McCarthy J, et al. (2008) Long memory in the volatility of an emerging equity market: The case of Turkey. Int Mark Inst Money 18: 305-312.
    [27] Fama EF (1970) Efficient capital markets: A review of theory and empirical work. J Financ 25: 383-417.
    [28] Fama E, French K (1992) The Cross-section of Expected Stock Returns. J Financ 47: 427-465.
    [29] Fama E, French K (1993) Common risk factors in returns on stocks and bonds. J Financ Econ 33: 3-56.
    [30] Fama E, French KR (1996) The CAPM is Wanted, Dead or Alive. J Financ 51: 1947-1958.
    [31] Fama E, French KR (2004) The Capital Asset Pricing Model: Theory and Evidence. J Econo Perspect 18: 25-46.
    [32] Fama E, French KR (2006) The Value Premium and the CAPM. J Financ 61: 2163-2185.
    [33] Fama E, MacBeth J (1973) Risk, return and equilibrium: Empirical tests. J Polit Econ 81: 607-636.
    [34] Fernandez V (2006) The CAPM and value at risk at different time-scales. Int Rev Financ Anal 15: 203-219.
    [35] Friend L, Landskroner Y, Losq E (1976) The demand for risky assets and uncertain inflation. J Financ 31: 1287-1297.
    [36] Galagedera DUA (2007) A review of capital asset pricing models. Managerial Financ 33: 821-832.
    [37] Gençay R, Selçuk F, Whitcher B (2002) An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press, San Diego.
    [38] Gençay R, Whitcher B, Selçuk F (2003) Systematic Risk and Time Scales. Quant Financ 3: 108-116.
    [39] Gençay R, Whitcher B, Selçuk F (2005) Multiscale systematic risk. J Inte Money Financ 24: 55-70.
    [40] Gibbons MR (1982) Multivariate tests of financial models: A new approach. J Financ Econ 10: 3-27.
    [41] Gursoy CT, Rejepova G (2007) Test of capital asset pricing model in Turkey. J Dogus Univ 8: 47-58.
    [42] Handa P, Kothari SP, Wasley C (1989) The relation between the return interval and beta: Implications for size-effect. J Financ Econ 23: 79-100.
    [43] Handa P, Kothari SP, Wasley C (1993) Sensitivity of multivariate tests of the CAPM to the return measurement interval. J Financ 48: 1543-1551.
    [44] Ho YW, Strange R, Piesse J (2000) CAPM anomalies and the pricing of equity: Evidence from the Hong Kong market. Appl Econ 32: 1629-1636.
    [45] Hubbard BB (1998) The world according to wavelets: The story of a mathematical technique in the making, 2e, Ak Peters Ltd., MA.
    [46] Mahmoud IMM, Ben Mabrouk A, Hashim MHA (2016) Wavelet multifractal models for transmembrane proteins' series. Int J Wavelets Multires Inf Process 14: 36.
    [47] In F, Kim S (2006) The hedge ratio and the empirical relationship between the stock and futures markets: A new approach using wavelet analysis. J Bus 79: 799-820.
    [48] In F, Kim S (2007) A note on the relationship between Fama-French risk factors and innovations of ICAPM state variables. Financ Res Lett 4: 165-171.
    [49] In F, Kim S, Marisetty V, et al. (2008) Analysing the performance of managed funds using the wavelet multiscaling method. Rev Quant Financ Accounting 31: 55-70.
    [50] Karan MB, Karadagli E (2001) Risk return and market equilibrium in Istanbul stock exchange: The test of the capital asset pricing model. J Econ Administrative Sci 19: 165-177.
    [51] Kishor NK, Marfatia HA (2013) The time-varying response of foreign stock markets to US monetary policy surprises: Evidence from the Federal funds futures market. J Int Financ Mark Inst Money 24: 1-24.
    [52] Kothari S, Shanken J (1998) On defense of beta, J. Stern, and D. Chew, Jr. (Eds.), The Revolution in Corporate Finance, 3e, Blackwell Publishers Inc., 52-57.
    [53] Levhari D, Levy H (1977) The Capital Asset Pricing Model and the Investment Horizon. Rev Econ Stat 59: 92-104.
    [54] Lévy H (1978) Equilibrium in an imperfect market: a constraint on the number of securities in the portfolio. Am Econ Rev 68: 643-658.
    [55] Lintner J (1965a) Security Prices and Maximal Gaines from Diversification. J Financ 20: 587-615.
    [56] Lintner J (1965b) The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Rev Econ Stat 47: 13-37.
    [57] Litzenberger RH, Ramaswamy K (1979) The effect of personal taxes and dividends on capital asset prices: Theory and empirical evidence. J Financ Econ 7: 163-195.
    [58] Magni CA (2007a) Project selection and equivalent CAPM-based investment criteria. Appl Financ Econ Lett 3: 165-168.
    [59] Magni CA (2007b) Project valuation and investment decisions: CAPM versus arbitrage. Appl Financ Econ Lett 3: 137-140.
    [60] Marfatia HA (2014) Impact of uncertainty on high frequency response of the US stock markets to the Fed's policy surprises. Q Rev Econ Financ 54: 382-392.
    [61] Marfatia HA (2015) Monetary policy's time-varying impact on the US bond markets: Role of financial stress and risks. North Am J Econ Financ 34: 103-123.
    [62] Marfatia HA (2017a) A fresh look at integration of risks in the international stock markets: A wavelet approach. Rev Financ Econ 34: 33-49.
    [63] Marfatia HA (2017b) Wavelet Linkages of Global Housing Markets and macroeconomy. Available at SSRN 3169424.
    [64] Marfatia HA (2020) Investors' Risk Perceptions in the US and Global Stock Market Integration. Res Int Bus Financ 52: 101169.
    [65] Markowitz H (1952) Portfolio Selection. J Financ 7: 77-91.
    [66] Merton RC (1973) An Intertermporal Capital Asset Pricing Model. Econometrica: J Econometric Society 41: 867-887.
    [67] Mossin I (1966) Equilibrium in a Capital Asset Market. Econometrica: J Econometric Society 34: 768-783.
    [68] Perold A (2004) The Capital Asset Pricing Model. J Econ Perspect 18: 3-24.
    [69] Percival DB, Walden AT (2000) Wavelet methods for time series analysis, Camridge University Press, NY.
    [70] Rhaiem R, Ammou SB, Mabrouk AB (2007a) Estimation of the systematic risk at different time scales: Application to French stock market. Int J Appl Econ Financ 1: 79-87.
    [71] Rhaiem R, Ammou SB, Mabrouk AB (2007b) Wavelet estimation of systematic risk at different time scales, Application to French stock markets. Int J Appl Econ Financ 1: 113-119.
    [72] Roll R (1977) A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory. J Financ Econ 4: 129-176.
    [73] Selcuk F (2005) Wavelets: A new analysis method (in Turkish). Bilkent J 3: 12-14.
    [74] Sharkasi A, Crane M, Ruskin HJ, et al.(2006) The reaction of stock markets to crashes and events: A comparison study between emerging and mature markets using wavelet transforms. Phys A 368: 511-521.
    [75] Sharpe WF (1964) Capital asset prices: A theory of market equilibrium under conditions of risk. J Financ 19: 425-442.
    [76] Sharpe WF (1970a) Computer-Assisted Economics. J Financ Quant Anal, 353-366.
    [77] Sharpe WF (1970b) Stock market price behavior. A discussion. J Financ 25: 418-420.
    [78] Sharpe WF (1970c) Portfolio theory and capital markets, McGraw-Hill College.
    [79] Soltani S (2002) On the use of the wavelet decomposition for time series prediction. Neurocomput 48: 267-277.
    [80] Soltani S, Modarres R, Eslamian SS (2007) The use of time series modeling for the determination of rainfall climates of Iran. Int J Climatol 27: 819-829.
    [81] Vasichek AA, McQuown JA (1972) Le modèle de marché efficace. Analyse financière, 15, 1973, traduit de "The effecient market model". Financ Anal J.
    [82] Xiong X, Zhang X, Zhang W, et al. (2005) Wavelet-based beta estimation of China stock market, In: Proceedings of 4th International Conference on Machine Learning and Cybernetic, Guangzhou. IEEE: 0-7803-9091-1.
    [83] Yamada H (2005) Wavelet-based beta estimation and Japanese industrial stock prices. Appl Econ Lett 12: 85-88.
    [84] Zemni M, Jallouli M, Mabrouk AB, et al. (2019a) Explicit Haar-Schauder multiwavelet filters and algorithms. Part II: Relative entropy-based estimation for optimal modeling of biomedical signals. Int J Wavelets Multiresolution Inf Process 17: 1950038.
    [85] Zemni M, Jallouli M, Mabrouk AB, et al. (2019b) ECG Signal Processing with Haar-Schauder Multiwavelet, In: Proceedings of the 9th International Conference on Information Systems and Technologies—Icist 2019.
  • QFE-04-04-026-s001.pdf
  • This article has been cited by:

    1. Lean Yu, Lihang Yu, Kaitao Yu, A high-dimensionality-trait-driven learning paradigm for high dimensional credit classification, 2021, 7, 2199-4730, 10.1186/s40854-021-00249-x
    2. Mounir Sarraj, Anouar Ben Mabrouk, The Systematic Risk at the Crisis—A Multifractal Non-Uniform Wavelet Systematic Risk Estimation, 2021, 5, 2504-3110, 135, 10.3390/fractalfract5040135
    3. Tahani S. Alotaibi, Luciana Dalla Valle, Matthew J. Craven, The Worst Case GARCH-Copula CVaR Approach for Portfolio Optimisation: Evidence from Financial Markets, 2022, 15, 1911-8074, 482, 10.3390/jrfm15100482
    4. Juan Meng, Sisi Hu, Bin Mo, Dynamic tail dependence on China's carbon market and EU carbon market, 2021, 1, 2769-2140, 393, 10.3934/DSFE.2021021
    5. Majed S. Balalaa, Anouar Ben Mabrouk, Habiba Abdessalem, A Wavelet-Based Method for the Impact of Social Media on the Economic Situation: The Saudi Arabia 2030-Vision Case, 2021, 9, 2227-7390, 1117, 10.3390/math9101117
    6. Sabrine Arfaoui, Maryam G. Alshehri, Anouar Ben Mabrouk, A Quantum Wavelet Uncertainty Principle, 2021, 6, 2504-3110, 8, 10.3390/fractalfract6010008
    7. Xu Zhang, Zhijing Ding, Ning Cai, Multiscale Systemic Risk and Its Spillover Effects in the Cryptocurrency Market, 2021, 2021, 1099-0526, 1, 10.1155/2021/5581843
    8. Zhenghui Li, Zimei Huang, Pierre Failler, Dynamic Correlation between Crude Oil Price and Investor Sentiment in China: Heterogeneous and Asymmetric Effect, 2022, 15, 1996-1073, 687, 10.3390/en15030687
    9. Ying Chen, Xuehong Zhu, Jinyu Chen, Spillovers and hedging effectiveness of non-ferrous metals and sub-sectoral clean energy stocks in time and frequency domain, 2022, 111, 01409883, 106070, 10.1016/j.eneco.2022.106070
    10. Anouar Ben Mabrouk, Sabrine Arfaoui, Mohamed Essaied Hamrita, Wavelet-based systematic risk estimation for GCC stock markets and impact of the embargo on the Qatar case, 2023, 7, 2573-0134, 287, 10.3934/QFE.2023015
    11. Sabrine Arfaoui, Nidhal Ben Abdallah, Robustness and sensitivity of some wavelet multifractal models in fractal data modelling, 2023, 0266-4720, 10.1111/exsy.13268
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(4616) PDF downloads(149) Cited by(11)

Figures and Tables

Figures(3)  /  Tables(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog