This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points and . Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF.
Citation: Mahmoud Abul-Ez, Mohra Zayed, Ali Youssef. Further study on the conformable fractional Gauss hypergeometric function[J]. AIMS Mathematics, 2021, 6(9): 10130-10163. doi: 10.3934/math.2021588
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Abstract
This article presents an exhaustive study on the conformable fractional Gauss hypergeometric function (CFGHF). We start by solving the conformable fractional Gauss hypergeometric differential equation (CFGHDE) about the fractional regular singular points and . Next, various generating functions of the CFGHF are established. We also develop some differential forms for the CFGHF. Subsequently, differential operators and contiguous relations are reported. Furthermore, we introduce the conformable fractional integral representation and the fractional Laplace transform of CFGHF. As an application, and after making a suitable change of the independent variable, we provide general solutions of some known conformable fractional differential equations, which could be written by means of the CFGHF.
1.
Introduction
Since the 1960s, the world has shown a multiplicity of financial crises: the first generation crisis in the 1960s and 1970s that affected Latin America. Then, the crisis of the European Monetary System in the years 1992–1993 which characterizes the second generation of crisis. Then, the models of foreign exchange crisis of the third generation which are after the Asian crisis of 1997, and more recently the "subprime" crisis which arose in the United States considered to be the purest of them. It upsets the gaze of agents towards the principles of conventional finance and, on the other hand, improves confidence in Islamic finance, as long as it demonstrates that it is escaping from this crisis. As such, the UN commission of experts, meeting in New York under the chairmanship of Joseph Stiglitz (2009), suggests the importance of reforming the financial and monetary sector to get rid of financial crises. Among the guidelines of this commission, a new door was opened to draw inspiration from the principles of Islamic finance.
From there arises a definite interest in the establishment of banking governance mechanisms to remedy this problem of banking crisis. It is within this framework that we have begun to observe in recent years the emergence, all over the world, of what we call codes or guides of "good practices" of governance. Generally produced by institutional investors, multinationals, associations of directors, stock exchanges or national and international organizations, these guides most often translate recommendations in matters of corporate governance.
Indeed, considerable attention has been paid in academic books and in the business press to the impact of governance mechanisms, such as the ownership structure, the composition of the board of directors and executive compensation, on the bank performance (Macey and O'Hara, 2003; Levine, 2004; Adams and Mehran, 2005; Caprio et al., 2007; Dahya et al., 2008) without taking into account the ethical aspect that governs Islamic finance in general and Islamic banks in particular.
The objective of our work is part of this perspective. It focuses on the awareness of Islamic banking firms of the ethical dimension and the identification and quantification of the relationship between governance mechanisms and the performance of these banks, to fill the void in Islamic banking literature.
We focus on the case of Malaysia because, despite its small size it has rapidly became a global leader in Islamic Finance, playing host to the largest concentration of financial institution in the Asian countries.
Our study is organized as follows. The first section presents the theoretical framework and research hypotheses. The second section discusses the methodological aspects. The analysis and discussion of our results are presented in the third section. The conclusion summarizes the main results, points out the limits and the openings of this research.
2.
Theoretical framework
In the banking literature, several studies have studied the relationship between governance mechanisms and performance (Adams and Mehran, 2003), however, at the level of Islamic banks, there is almost an absence of work that studies this relationship, except for a few conceptual works (Shahul et al., 2006; Ghayad, 2008; Siagh and Hafsi, 2004).
We study the relationship between governance mechanism variables and the Islamic banks performance with reference to agency theory.
2.1. The relation between the board of directors characteristics and the Islamic banks performance
In literature, to study the relation between the board of directors and the Islamic Banks performance, we should study the impact of the three characteristics of the board of directors. It's the size of the board, the independence of their members, and finally the duality management.
2.1.1. The impact of the size of the board of directors
Several works have studied the effect of the size of the board of directors on banking performance.
Indeed, contrary to the Jensen explanations (1993), Adams and Mehran(2003, 2005) were studied the impact of the size of the Board on the banking performance. They found a positive relationship between the two variables, namely, when performance are measured by Tobin's Q. However, Zulkafli and Samad (2007) observed no significant relationship between performance ratios such as the Return on Assets (ROA)1, Tobin's Q, and the size of the board of directors.
1 This is the ratio of earnings before interest and taxes to total assets.
In the same context, Staikouras et al. (2007) examined the impact of the size of the board of directors on banking performance. Their analysis shows that there is a negative and significant impact of the size of the board of directors on banks' performance, measured in terms of the Return on Assets, the Return on Equity (ROE)2 and Tobin's Q.
2 This is the ratio between net income and equity.
In Malaysia, Adnan et al. (2011) analyzed the relationship between governance mechanism variables and the banking performance during the period 1996 to 2005. The authors showed that the performance increases with a smaller board of directors.
This reasoning allows us to formulate the following hypothesis:
H1: There is a negative relationship between the size of the board of directors and Islamic banks performance.
2.1.2. The impact of the independent directors
Works examining the relation between the independence of the Board and the performance of banks are not abundant and found conflicting results.
On the other hand, the results of several works found a positive association between the percentage of outside directors and bank performance (Skully, 2002; Sierra et al., 2006). Thus, the following hypothesis:
H2: The independence of the board affects positively the Islamic banks performance.
2.1.3. The impact of the dual management
To study the relation between duality and banking performance, we found conflicting results in literature.
Indeed, some authors found an insignificant relationship between duality and the banking performance measured by the EVA3 (Fogelberg and Griffith, 2000). Besides, many others found a negative relation between the two variables (Pi and Timme, 1993). In contrast, others noted a positive correlation between duality and banking performance, whether accounting or market (Louizi, 2007). From these intersection earlier studies, we make the following hypothesis:
3 Economic Value Added.
H3: There is no relation between duality and Islamic banks performance.
2.2. The impact of the Sharia committee characteristics on the Islamic bank performance
As in the case of the board of directors, the effectiveness of the Sharia committee in its mission of monitoring depends on various characteristics. It's the size of the SBS, the competence of their members, and finally, the visibility of supervisors.
2.2.1. The impact on the size of the Sharia Supervisory Board
To our knowledge, the impact of the size of the Sharia committee on the Islamic bank performance wasn't studied by any works; that's why we will base ourselves on the literature of conventional finance (Jensen and Meckling, 1976) to examine this relationship. So, the following hypothesis:
H4: There is a negative relationship between the size of SSB and Islamic banks performance.
2.2.2. The impact of the competence of the Sharia Supervisory Board
Some works assume that the supervisors having financial and/or accounting skill in the Sharia committee can influence the performance of Islamic banks (Khan, 2004; Ghayad, 2008). Thus, the following hypothesis may be formulated:
H5: There is a positive relationship between the supervisors having financial and/or accounting skill and Islamic banks performance.
2.2.3. The impact of the visibility of the SSB
Many authors believe that when a supervisor of the SSB is solicited by many banks at the same time, can increase the performance of Islamic banks due to the access to several operations, transactions and more knowledge in business of different banks. The following hypothesis can be deduced:
H6: There is a positive relationship between the percentage of supervisors of the SSB who are members in other religious board of other Islamic bank and Islamic banks performance.
3.
The methodology
3.1. Sample and data collection
To meet our research objective, our sample is based on 17 Malaysian Islamic Banks.These are: Affin Islamic Bank, Al Rajhi Banking & Investment, Alliance Islamic Bank, AmIslamic Bank, Asian Finance Bank, Bank Islam Malaysia, Bank Muamalat Malaysia, CIMB Islamic Bank, Islamic Bank EONCAP, HongLeong Islamic Bank, HSBC Amanah Malaysia, Kuwait Finance House Malaysia Maybank Islamic, RHB Islamic Bank, Standard Chartered Saadiq Bhd, Public Islamic Bank and OCBC Al-Amin Bank.
Based on this sample, we collect financial and accounting information from financial statements and annual activity reports of Islamic banks over a period of seven years from 2005 to 2011. Data on the governance system are collected from activity reports issued by banks in the sample on their website.
In the next section, we present dependant and independent variables used to estimate the impact of governance mechanisms on the ethical performance of Islamic banks.
3.2. Definitions and measurement of variables
3.2.1. The endogenous variables: the ethical performance
To comply with the principles of Sharia et al. (2006) use several indicators to measure the ethical performance, namely: the profit sharing ratio, the zakat performance ratio, equitable distribution ratio, ratio between welfare director-employee, Islamic investment versus non-Islamic investment ratio, AAOIFI ratio and the Islamic income versus non Islamic income ratio. In our case, and given the availability of data, we will focus on the following variables to estimate the ethical performance:
♦The Profit Sharing Ratio (PSR): The PSR variable is calculated by the ratio between investment from the musharaka and mudaraba contract on overall investment.
♦The Zakat Performance Ratio (ZPR): It is the ratio of Zakat over net assets. Authors assume that Zakat, is one of the objectives of Islamic Accounting.
♦The Islamic Income versus Non-Islamic Income Ratio (IIVNIR): The IIVNIR variable is calculated by the ratio between the Islamic income and the overall income of the Islamic bank.
3.2.2. The exogenous variables
The exogenous variables are divided into two groups. These are measures of variables related to the Sharia committee and the Board of directors.
The Sharia Supervisory Board (The Sharia committee)
To investigate the effectiveness of the control of this mechanism, we will identify several criteria. It is mainly the size of the SSB, Accounting and finance ability of SSB, and finally the visibility of SSB.
♦The Size of the Sharia committee (SSSB): We calculate this variable by the total number of the member of the Sharia committee.
♦The Competence of the Sharia Supervisory Board (CSSB): It is the number of the supervisor having financial and accounting skill in the Sharia committee.
♦The Visibility of the Sharia Supervisory Board (VSSB): It is the number of mandates cumulated in other Islamic banks by the members of the Sharia committee.
The Board of Directors
In the literature, several variables have been used to measure the impact of the board of directors on banking performance.
♦The Board of Directors Size (SBD): We measure this variable by the total number of the member of the board of directors. This measure has also been used by Pathan et al. (2007).
♦The Independence of the Board of directors (INDEP): This variable is calculated by the percentage of independent directors on the board of directors. This measure was also used by Fogelberg and Griffith (2000); Adam and Mehran (2003) and Belkhir (2005).
♦The Duality of the Board of directors (DUAL): The duality variable is measured by a dichotomous variable that takes the value 1 when the two positions are occupied by the same person and 0 in the case otherwise. This measure has been used by several previous researches, we cite as an example Pi and Timme (1993).
3.2.3. The control variables
In addition to the explanatory variables, the control variables used in this study are:
♦The Bank Size (SB): We measure this variable by the the natural logarithm of the total assets. Several studies have used this measurement, such Mak and Ong (1999), Godard (2001) and Fernandez and Arrondo (2005).
♦The Bank Age (AB): In our study, the bank age is measured by the natural logarithm of the number of years since the creation of the bank.
♦The Bank Leverage (LB): We calculate this variable by the following ratio: Total debt to total assets. This standard measurement has been used by several studies such Knoeber and Agrawal (1996), Mak and Ong (1999) and Fernandez and Arrondo (2002).
Table 1a gives the meanings and measures of different variables. (See Supplementary)
Before making the statistical estimation of our theoretical model, we will start with descriptive statistics of the dependent and independent variables.
4.1. Descriptive statistics
The Table 1 shown that the average size of the sharia committee is four Malaysian banks over the period 2005 to 2011 and between 3 and 7 supervisors. The percentage of members of the Sharia committee with qualifications in accounting and/or finance is low (on average equal to 0.001889). So on average, the number of supervisors qualified and competent in accounting and/or finance is largely less than the number of qualified supervisors and not proficient in accounting and/or finance. These results indicate the rarity of dual-qualified specialists, both conventional and Sharia finance. This may be a good sign of skills, but it is also a bad sign as supervisors will not have sufficient time to fulfill their responsibilities properly.
Moreover, the size of the director's board of Malaysian Islamic banks varies between 4 and 11 with a mean of 7. This result is consistent with the disciplinary vision board advocated by Jensen (1993). Indeed, among the conditions for the director's Board to be an effective control mechanism, Jensen (1993) cites a director's board relatively small (consisting of seven to eight members) so that it is not dominated by the executive. However, this medium is lower than that found in the literature for conventional banks. We can therefore explain this average (7), relatively modest by the presence of a second board, the Sharia committee. The existence of independent directors varies between 1/4 and 1 with an average of 1/2. This small proportion shows that the notion of independence of director's board members does not matter in Islamic banks. Besides, we note that the average of the dual roles of CEO and chairman is almost zero (0.154639) and varies between 0 and 1. So, the duality of management is absent in Islamic banks of Malaysia during the study period. This separation of functions is also recommended by the Malaysian Code of Governance (2001).
The average size of Islamic banks is small and does not exceed during the study period 9.075117 RM (1RM Dollar = 0.7814). Otherwise, the majority of these banks are newly created and their average age is low and does not exceed 2 years (1.92368 years). In fact, the bank is more old more experience in the financial field is wider and the accumulations of skills allow bank executives and staff to better select profitable investment projects. Thus, the LB ratio of these banks is high and it is equal to 87.5034% on average during the seven years of study. This intense debt is due to the important role of Islamic Banks to finance economic agents who need funding and they make heavy use of various financing techniques, including the musharaka and mudaraba. This confirms the specificity of Islamic banks governance.
What about the ethical performance? The average ratio of profit sharing (PSR) is equal to 16.23633 times the average performance ratio of Zakat (ZPR) is equal to 17.7776% and the average ratio of income vs. non Islamic Islamic income (IIVNIR) is 44, 4042%. This means that the financing techniques used by most Islamic banks are musharaka and mudaraba as the PSR ratio is well above 1. The importance of the value of this ratio can be explained by confusion, sometimes, for some banks between different modes of Islamic financing used. Thus, the Zakat fund, which is fed by the alms collected from shareholders, depositors and employees, represents on average 17.77% of total net assets. According to this ratio, we can infer that Islamic banks play an important role in achieving social justice and equity. In addition, these banks seem to achieve their goals in terms of ethical investment. Indeed, the Islamic transactional income is on average equal to 44.4% of total revenue and revenue transactions as non Islamic (haram) are equal to 55.6%.
The importance of non-Islamic income earned by the Malaysian banks, according to their annual reports, the incentive to invest in human capital to meet their ongoing demand expert, doubling proficient in Sharia and accounting and/or finance.
4.2. Examination of multicollinearity
Before proceeding to test our basic model, it is necessary to make sure of the adequacy of the variables used. First, the explanatory variables must be independent. From Table 2, all Pearson correlation coefficients between the independent variables are less than 0.6, so the problem of multicollinearity is absent.
Our explanatory model of ethical performance is as follows:
(1)
where;
EP: the dependant variable measuring the ethical performance of bank I at time t, measured by PSR, ZPR and IIVNIR; with i = 1………..N, t = 1…………T; SSSBit, CSSBit, VSSBit, SBDit, INDEPit, DUALit, SBit, ABit, LBit: are the explanatory variables; , , , , , , , , : Are the constant coefficients;
: The intercept of the regression;
: The error term.
The first step is to make sure that the panel structure is well-founded for our model. The Table 3 corresponds to the homogeneity-heterogeneity tests Hsiao (1986).
From this table, we can see that the coefficients of our theoretical model are homogeneous, but the constants are heterogeneous. We specify then our theoretical model by a panel with individual effects and we will use the within and GLS methods to estimate the ethical performance of Islamic banks from the three endogenous variables, namely: the PSR, ZPR and IIVNIR.
4.4. Estimate of the ethical performance
To estimate the ethical performance in the three cases, we will use the Within and GLS techniques. Table 4 summarizes the results obtained.
Or these results are expected but statistically insignificant, which pushes us to make use of the Hausman test (1978) to choose between the two techniques.
We will use the Hausman statistic (1978) to specify the individual effects for our models. The results are reported in the following table.
According to Table 6, our results show that the relation of the three variables of corporate governance mechanisms on ethical performance (SBD, INDEP and DUAL) are not statistically significant; thus the three first hypotheses are unconfirmed.
On the other hand, the results indicate that the three Islamic corporate governance mechanisms (SSSB, CSSB and VSSB) have significant impacts on ethical performance. Indeed, the impact of the size of the Sharia committee on the ethical performance expressed by (PSR) is negative. So, hypothesis 4 is confirmed.
In addition, the impact of the competence and visibility of Sharia committee members on ethical performance are respectively positive and negative, which confirms our hypothesis 5 and 6.The signs of Islamic governance mechanisms show that when the size of the Sharia committee increases, the competence and the visibility of their members affect positively and negatively, respectively, the ethical performance of the Islamic banks.
We deduce from our results that the the number of Sharia committee members are still low, despite their know-how in finance and/or accounting.
5.
Conclusion
To sum up, in our paper we tried to examine empirically the relation between Islamic governance system and ethical performance in Malaysia. We investigated this relationship using a sample of 17 Islamic banks during the period 2005–2011.
Our findings are summarized as follows: first, the relationship between ethical performance and conventional governance variables is non-significant. Second, the ethical performance (when measured by the PSR and ZPR) is explained by Islamic governance variables and control variables (the bank age).
These findings support clearly the important role played by Islamic banks to achieve socio-economic objectives (Shahul et al., 2006). Indeed, the Zakat fund, which is fed by the alms collected from shareholders, depositors and employees, represents on average 17.77% of total net assets. According to this ratio, we can infer that Islamic banks play an important role in achieving social justice and equity.
However, the effectiveness of the Sharia committee, as governance structure, is not clear since it doesn't seem to impact the ethical performance when explained by the ratio IIVNIR. In this context Ghayad (2008) stated that: "the members of Sharia board were a serious handicap for the directors of the Islamic banks. Directors and members of Sharia board did not speak the same language. The members of the Sharia board were not very specialized in the fields other than Sharia".
The contributions of our work are huge, but some limits can be reported: first, the reduced size of our sample. Indeed, if the size increases the resutats of the estimate can be improved. Secondly, the relation between the Sharia committee characteristics and the ethical performance of Islamic banks remains ambiguous. Then the use of others variables, such as the presence of the woman in the Sharia committee, the number of meetings of the members of the Sharia committee...... etc. can remove this ambiguity. Thus, more evidence is needed in order to confirm or reject the efficiency of the Sharia committee as a value creator.
Conflict of interest
The authors declare no conflict of interest.
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