Dr Abdelkader Derbali
email: derbaliabdelkader@outlook.fr

Dr Tarek Chebbi
email: chebbitarekfsegss@hotmail.fr

Dynamic Equicorrelation between S&P500 Index and S&P GSCI

Keywords: S&P 500, S&P GSCI, Commodities, Equicorrelation, GARCH-DECO JEL Classification: Q43, G15

come

Introduction

As the commodities sector is one of the majority important in the international financial markets in terms of its contribution to world trade, income, speculation and employment, it is not surprising that the time series volatilities in principal agriculture and energy sources, such Wheat, Corn, Soybeans, Cotton, Sugar, Coffee, Cocoa, Crude Oil, Brent Crude Oil, Heating Oil, and Natural Gas, have been examined extensively in terms of finance and economics, thereby leading to significant research in the nexus between stock market and commodities indices [START_REF] Caporin | Are the S&P 500 Index and Crude Oil, Natural Gas and Ethanol Futures Related for Intra-Day Data?[END_REF].

Commodity indices trade on the relationship between supply and demand. Such as, developing countries for example China and India require important investment in commodities such as steel and oil to construct their infrastructures, cotton and metals to produce manufacturing products and food-related commodities to nourish their progressively more middle-class populations. These trends have produced significant demand and higher volatilities of prices for commodities indices. Additionally, the high level of demand has also paying attention of investors, who previously invested in only stocks and bonds markets, to take benefit of commodity markets force and its normal inverse dynamic correlation to the movement of the stock market indices coincided with the 2007-2008 boom in commodity prices [START_REF] Henderson | New Evidence on the Financialization of Commodity Markets[END_REF].

In this context, the returns, volatility and volatility spillovers (namely, the delayed effect of a returns shock in one financial/precious metals/energy/livestock/agricultural asset on the subsequent volatility or co-volatility in another financial/precious metals/energy/livestock/agricultural asset), among alternative stock market indices and commodities indices across diverse markets have been examined using a variety of univariate and multivariate returns, volatility and volatility spillovers specifications, different data frequencies, and alternative data sets.

The period of mainly quick change in stock market index strategies and supplementary financial commodity investment approximately coincided with the 2007-2008 booms in commodity prices, most important to a discuss among both academics and policy makers about whether the flows of conditional dependence between stock market indices and financial commodity indices [START_REF] Irwin | Index funds, financialization, and commodity futures markets[END_REF][START_REF] Bos | A bitter brew? Futures speculation and commodity prices[END_REF][START_REF] Rouwenhorst | Commodity investing[END_REF][START_REF] Cheng | Convective risk flows in commodity futures markets[END_REF].

Then, the time-varying dependency between different asset classes returns, mainly stock market return and commodity indices, can be effectively examined by multivariate Generalized Autoregressive Conditional Heteroskedasticity (MGARCH) models (Tse and Tsui, 2002;[START_REF] Bae | A new approach to measuring financial contagion[END_REF][START_REF] Cho | East Asian financial contagion under DCC-GARCH[END_REF]Xiao and Dhesi, 2010;[START_REF] Égert | Time-varying synchronization of European stock markets[END_REF]. This paper gains its own significance and contribution from the detail that it applies of GARCH-DECO (1,1) for the first time on the examination of the volatility dependency and the time-varying variance of dynamic equicorrelation between S&P500 index and the chosen S&P GSCI commodities indices. The GARCH-DECO specifications, although significant, have become a standard and popular modelling framework for financial time series.

The main findings of estimated GARCH-DECO (1,1) show highly significance of conditional heteroscedasticity parameters can show us to distinguish the nature of the volatility dependency between S&P500 index and the S&P GSCI commodities index. This can make clear the importance of correlation in the process of financialisation of the US stock market index and commodities. Also, the GARCH-DECO (1,1) specification has a good fit of data; it captures all the observed extremely volatility in the conditional dependency of financial time series returns.

The rest of this paper is structured as follow. Section 2 shows an overview of the S&P GSCI composition. In section 3, we present the econometric methodology employed in this paper.

Section 4, indicates data characteristics. In section 5, we report the empirical finding. Finally, section 6 concludes and remarks.

The S&P GSCI composition

The S&P GSCI (formerly Goldman Sachs Commodity Index, GSCI) is a commodity index, which includes 24 different futures, which are traded on commodity exchanges. He was first calculated in 1991 by Goldman Sachs and taken over by Standard & Poor's of 2007.

The S&P GSCI includes 24 different commodities that are weighted according to the value of its annual production with current prices to each other. According to price changes also the weight of the raw material in the index changes. Underlying Down here is the average of the last five years.

As of each commodity index is also available from S&P GSCI Spot a Return Index, an excess return index and a total return index.

By referring to the nearest Future and the regular rolls the second and third variant of the index can be reproduced and are investable. The futures are rolled to an established scheme month. The index adjustment is done annually. This so-called rebalancing is discretely carried out using a procedure by an investment committee.

Consideration is given exclusively futures contracts that have a sufficiently high liquidity, In The S&P GSCI comprises 24 commodities from all sectors (industrial metals, agriculture, energy, livestock, precious metals). The energy sector, especially oil activities, representing the majority of its indices.

which

Econometric Methodology

In this paper, we use the GARCH-DECO models to test the dynamic equicorrelation between the S&P500 index and the sixteen selected S&P GSCI commodities as Energy The Dynamic Conditional Correlation framework is a practical modeling tool, however when the number of test stock market indices become large the estimation can becomes unreliable and even breakdown finally. The Dynamic Equicorrelation Model (DECO Model) group of correlation models is considered to overcome some of these computational difficulties issued by the Dynamic Conditional Correlation models. As such, the documentation for Dynamic Equicorrelation models will be formulated for many stock market return series.

Consider a vector of n different stock market return series    will accede to the same correlation structure as the original two return series.

The Generalized Autoregressive Conditional Heteroskedasticity-Dynamic Equicorrelation (GARCH-DECO) model assumes a specific parametric for the conditional correlation matrix t R . Then, on a given day the GARCH-DECO model supposes that all pairwise correlations are identical. Also, it turns out that despite this seemingly strong restriction, the Dynamic Equicorrelation model can provide consistent estimates of Dynamic Conditional Correlation parameters in large systems. The correlation matrix t R is thus defined as an equicorrelation matrix and evolves as: In order to implement maximum likelihood, we suppose the stacked return series

  1 t t N t N R I J     (5) ,, , , , , 2 ( 1)
i j t t ij i i t j j t q NN qq      (6) , , , , 1 , 1 , , , 1 , ( ) ( )
i j t i j DECO i t j t i j DECO i j t i j qq                (
    ' 1 ' ' 1 , log 2 c DECO DECO t t t t t t t L R R             (8)
As is similar with the Generalized Autoregressive Conditional Heteroskedasticity, the single correlation t  will be stable and mean-reverting so long as 0, 0, 1

DECO DECO DECO DECO         .
The standard restrictions and properties of the univariate Generalized Autoregressive Conditional Heteroskedasticity models that are employed to model each individual return series' volatilities also naturally still hold.

Data

In this paper, the data is collected from daily prices of S&P500 index and the sixteen selected S&P GSCI commodities during the period from January 01, 2003 to December 31, 2015. Then, daily prices are transformed to daily logarithmic returns.

The table 2 reports the descriptive statistics of conditional volatility of S&P 500 and S&P GSCI returns over the period from January 01, 2003 to December 31, 2015. From this table, we can observe that in average the higher conditional volatility is for Heating Oil (49.26%) followed, respectively, by the Sugar (36.25%), the Brent Crude Oil (33.16%) and the Crude Oil (33.11%).

Additionally, we can remark that the higher return is for Heating Oil (2.83%) followed, respectively, by the Feeder Cattle (1.95%), the Natural Gas (1.60%) and the S&P500 index (0.88%).

For the conditional volatility prediction by the GARCH (1,1) model, we can show that the biggest weekly, monthly and yearly prediction is for the energy commodities followed by the Agriculture commodities, the Precious Metals commodities, and the Livestock commodities. According to these figures, we can observe that the conditional volatility prediction of the chosen series returns attained their maximum in the start of 2009. This period is coinciding with the presence of an international financial instability followed by an international liquidity and banking crisis in developing and developed countries.

Table2

Descriptive From this Table, we can show that all parameters are highly significant with a volatility persistence superior than 0.90 (+) for the dynamic equicorrelation between the S&P500 index and the sixteen selected S&P GSCI commodities. Also, the difference between the levels of volatility persistence can be just explained by the reality that commodities cannot be viewed as a homogeneous asset class [START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF][START_REF] Chebbi | The dynamic correlation between energy commodities and Islamic stock market: analysis and forecasting[END_REF][START_REF] Chebbi | A Dynamic Conditional Correlation between Commodities and the Islamic Stock Market[END_REF].

The empirical finding showed in Table 3 balance the existing literature by reviewing the answer of the conditional nexus between stock markets and commodities indices.

Additionally, we can observe that all parameters are statistically significant confirming that the dependence between the S&P500 index and the sixteen selected S&P GSCI commodities are time-varying and extremely volatile.

Besides, there is also a confirmation that the sum of the volatility coefficients (+) is very close to unity, namely for the case between S&P500 index and the sixteen selected S&P GSCI commodities as exposed proving the higher persistence of volatility between the US stock market and commodity markets. There is one possible explanation which finds that such persistence goes along with the financialisation of stock market indices and commodities [START_REF] Creti | On the links between stock and commodity markets' volatility[END_REF][START_REF] Chebbi | The dynamic correlation between energy commodities and Islamic stock market: analysis and forecasting[END_REF][START_REF] Chebbi | A Dynamic Conditional Correlation between Commodities and the Islamic Stock Market[END_REF]. Our empirical results emphasize the importance of using GARCH-DECO (1,1) in modeling the time-varying dynamic equicorrelations.

The close dependence between daily movements of commodities and stock markets, which

  Engel (2002) proposes a DCC-GARCH (Dynamic Conditional Correlation-Generalized Autoregressive Conditional Heteroskedasticity) specification to examine the dependence between the assets returns based on their conditional volatilities. Additionally, the Dynamic Conditional Correlation is also developed byTse and Tsui (2002). Subsequently,[START_REF] Silvennoinen | Multivariate GARCH models[END_REF] suggest a Constant Conditional Correlation-GARCH model to investigate the flexibility of univariate Generalized Autoregressive Conditional Heteroskedasticity models and the simplicity of parametric correlation in the model. In large-system conditional covariance modeling, we can show the presence of contrast of Factor ARCH (Autoregressive Conditional Heteroskedasticity) and composite likelihood highlights a fundamental tradeoff.Engel and Kelly (2012) propose a solution to this tradeoff that selectively combines simplifying structural assumptions and composite likelihood adaptability. We use a system in which all pairs of returns have the same correlation on a given day, but this correlation varies over time. The model, called Generalized Autoregressive Conditional Heteroskedasticity-Dynamic Equicorrelation (GARCH-DECO), eliminates the computational and presentational difficulties of high-dimension systems. Also, because equicorrelated matrices have simple analytic inverses and determinants, likelihood computation is dramatically simplified and optimization becomes feasible for vast numbers of financial time series returns and volatilities.Then, the main objective of this paper is to investigate empirically the co-movement of the volatility dependency the S&P500 (Standard & Poor's 500) index and the sixteen selected S&P GSCI commodities as Energy(Crude Oil, Brent Crude Oil, Heating Oil, and Natural Gas), Agriculture (Wheat, Corn, Soybeans, Cotton, Sugar, Coffee, and Cocoa), Precious Metals (Gold and Silver), and Livestock (Live, Cattle, Feeder Cattle, and Lean Hogs) over the period from January 01, 2003 to December 31, 2015. Methodologically, we employ the GARCH-DECO (1,1) to assess the dynamic equicorrelation between S&P500 index and the chosen S&P GSCI commodities indices.

  are denominated in US dollars and listed in a member country of the OECD (Organisation for Economic Co-operation and Development). Moreover, the contracts must be at least 1.0 percent of total revenues of the commodities in the index. The minimum size prevents fragmentation on too many individual items. A raw material is removed when its weight drops below 0.1 percent. The S&P GSCI is considered an indicator of the future development of inflation or the cost of development in the industry. He is at a turnaround on the commodities market, a good leading indicator for the bond market, as raw materials in their tendency towards the bonds generally have a lead time of three to six months. Between the Interest on bonds and commodity prices is also time a close connection. Correlations of the S&P GSCI dollars with the geometric weighted U.S. Dollar Index and the trade-weighted US Trade Weighted Index are recognizable. A falling US dollar is equivalent to inflationary trends and rising trend in commodity prices. This is particularly true for agricultural commodities and the price of oil. The Spot Return Index was launched on January 8, 1991 under the name Goldman Sachs Commodity Index (GSCI). The recalculation was made until January 2, 1970 to a base value of 100 points. Until 1980, the GSCI rose computationally to a record total of 299.71 points. During the recession of the early 1980s lost the index value. In 1986, the low was 144.33 points, up 51.8 percent lower than in 1980. According to an interim high on 6 January 1997 at 231.82 points in commodity index fell during the Asian crisis until December 21, 1998 to a low of 127,94 points. Since the peak in 1997, this represents a decrease of 44.8 percent. In the following years, the index was due to a huge demand for raw materials in mainland China and India strongly. On 11 May 2006 the GSCI overcame the first time the threshold of 500 points. On 3 July 2008 an all time high was marked in the course of trading with 893.85 points. Since bottoming out in 1998 corresponds to an increase of 598.6 percent. On February 02, 2007 Standard & Poor's took over the calculation of the index. To reflect this change, the Goldman Sachs Commodity Index was renamed S&P GSCI. During the international financial crisis, which in the US subprime mortgage crisis had its origin in 2007, the index began to decline. 2008, the financial crisis had an increasing impact on the real economy. Because of a decline in global demand in the commodity markets, it came mainly from the beginning of the fourth quarter of 2008 to strong price declines. On February 19, 2009, the S&P GSCI fell during trading with 305.58 points to its lowest level since 2005. Since the all-time high from July 2008, this represents a decrease of 65.8 percent. This is the biggest fall in the history of the index. The February 19, 2009 marked the end of the descent. From the beginning of 2009, the commodity index was back on the way up. On April 11, 2011 the index rose during trading on 762.22 points. The February 2009, an increase of 149.4 percent. Particularly strong was the increase in prices of agricultural commodities. Especially meat, cereals, sugar and oils and fats prices were up since the middle of 2010. The reasons are several factors are known (rising world population, growing money supply, speculation on agricultural markets, crop losses caused by natural disasters, export restrictions of some countries). Follow the high raw material prices are rising inflation and the outbreak of riots in parts of the world.

(

  Crude Oil, Brent Crude Oil, Heating Oil, and Natural Gas), Agriculture (Wheat, Corn, Soybeans, Cotton, Sugar, Coffee, and Cocoa), Precious Metals (Gold and Silver), and Livestock (Live, Cattle, Feeder Cattle, and Lean Hogs) during the period from January 01, 2003 to December 31, 2015.

  (1,1) specification is obtained after the modeling of the univariate return series as individual Generalized Autoregressive Conditional Heteroskedasticity processes and their standardized residual series as a DECO process. -DECO are estimated by using the system vie quasi maximum likelihood.

Figures 1

 1 Figures 1 to 17 present the evolution of the conditional volatility prediction by GARCH (1,1) specification and the returns of the S&P500 index and the sixteen selected S&P GSCI commodities over the period from January 01, 2003 to December 31, 2015.

Figure 1 .Figure 2 .Figure 3 .Figure 4 .Figure 5 .Figure 6 .Figure 13 .Figure 14 .Figure 15 .Figure 16 .Figure 17 .

 1234561314151617 Figure 1. The conditional volatility and return of S&P500 index over the period January 01, 2003 to December 12, 2015

  have persisted regularly uninterrupted since the outbreak of the financial crisis of 2007, has frayed. The daily movements of the S&P500 and the S&P GSCI commodities index were more or less in tandem for the period from 2008 to 2015. But in recent year from the period of study, the two indexes' correlation as an assessing of how closely markets path one another has fallen to the lowest level since October 2008.

Figure 18 .Figure 19 .Figure 20 .Figure 21 .Figure 22 .Figure 23 .Figure 24 .Figure 25 .Figure 26 .Figure 27 .Figure 28 .Figure 29 .Figure 30 .Figure 31 .Figure 32 .

 181920212223242526272829303132 Figure 18. The Dynamic Equicorrelaton beteewn S&P and Crude Oil indices over the period January 01, 2003 to December 12, 2015

  By using the standard Generalized Autoregressive Conditional Heteroskedasticity model, we can proceed to the estimation of the residuals for each return series after we have fit the univariate Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model.

	Formally, these are defined as:
		, it		, it , it r 	(4)
	1, r  , ,..., t N t r   where all series have been demeaned. Further, define the conditional covariance matrix of all stock market t r return series as ' 1 t t t t r r H     . Then, we can decompose t t t t H D R D  (1) where , () t i t H diag   for (1, 2) i  . , t R denotes the conditional correlation matrix for the two return series. The GARCH-DECO (Generalized Autoregressive Conditional Heteroskedasticity-Dynamic Equicorrelation) model puts specific parametric assumptions on the evolution of t D and t R separately. The conditional variance of each individual return series is modeled as a standard GARCH process: 22 1 , , t i t i t r      (2) 2 2 2 11 t t t         (3) Besides, the volatility residual vector ' 1, t t N t , ,...,   

t H into the following:

it  denotes the conditional volatility of return series i and is the i th diagonal entry of t H . Finally,

Table 3 ,

 3 we present the empirical results from the GARCH-DECO (1,1) model during the period of study from January 01, 2003 to December 31, 2015.

Figure 7. The conditional volatility and return of Corn index over the period January 01, 2003 to December 12, 2015 Figure 8. The conditional volatility and return of Soybeans index over the period January 01, 2003 to December 12, 2015 Figure 9. The conditional volatility and return of Cotton index over the period January 01, 2003 to December 12, 2015 Figure 10. The conditional volatility and return of Sugar index over the period January 01, 2003 to December 12, 2015 Figure 11. The conditional volatility and return of Coffee index over the period January 01, 2003 to December 12, 2015 Figure 12. The conditional volatility and return of Cocoa index over the period January 01, 2003 to December 12, 2015

Conclusion

In this study, we investigate empirically the dynamic equicorrelation between the S&P500 index and the sixteen selected S&P GSCI commodities as Energy (Crude Oil, Brent Crude Oil, Heating Oil, and Natural Gas), Agriculture (Wheat, Corn, Soybeans, Cotton, Sugar, (1,1) to examine the conditional correlation between &P500 index and the sixteen selected S&P GSCI commodities index.

Based on the estimated results of GARCH-DECO (1,1), we conclude the presence of a highly significance of conditional heteroscedasticity parameters can show us to distinguish the nature of the volatility dependency between S&P500 index and the sixteen selected S&P GSCI commodities index. This can make clear the importance of correlation in the process of financialisation of the US index and commodities in the case of our study.

The GARCH-DECO (1,1) specification has a good fit of data; it captures all the observed highly volatility in the conditional dependency of return series.

In addition, the sum of the two parameters (α + β) for the chosen conditional heteroscedasticity specification (GARCH-DECO (1,1) model) is close to 1 which implies the persistence of volatility in the conditional dependency between S&P500 index and S&P GSCI commodities index. There is one possible justification which finds that such persistence goes along with the financialisation of US stock market indices and commodities.

Tse, Y.K. and Tsui, A.K. ( 2002) 'A multivariate generalized autoregressive conditional heteroscedasticity model with time-varying correlations', Journal of Business and Economic Statistics, Vol. 20, No. 3, pp.351-362. Xiao, L. and Dhesi, G. (2010) 'Volatility spillover and time-varying conditional correlation between the European and US stock markets ',Global Economy and Finance Journal,Vol. 3,No. 2,


Dr Abdelkader Derbali
email: derbaliabdelkader@outlook.fr

Abderrazek El Khaldi
email: abderrazek.elkhaldi@yahoo.fr

Dr Fathi Jouini
email: fathi_75@hotmail.fr

Abderrazek El Khaldi

Dr Abderazzek El Khaldi

Shariah-compliant capital asset pricing model: New mathematical modeling

Keywords: CAPM, shariah, Islamic finance, zakat, purification

HAL is

Introduction

The link between ethical finance and financial performance is difficult to establish. In terms of risk-adjusted returns, most studies do not find significant differences between ethical and conventional investments [START_REF] Bauer | The ethical mutual fund performance debate: New evidence from Canada[END_REF]. The financial literature reports that performance evaluation depends on several variables, including benchmark selection, performance metrics, and number of ethical filters used, investment style, geographic factors, and economic Investment horizon.

The foundation for modem investment theories were studier for the first time by [START_REF] Markowitz | Portfolio selection[END_REF], who concluded that a portfolio is mean-variance efficient if it has the highest expected (mean) return p R for a given variance (risk) 2 p  , or, similarly, if it has the negligible variance for a given expected return [1]. Building on Markowitz's earlier work, Sharpe (1964) develop the CAPM (Capital Asset Pricing Model) for valorizing risky assets; [START_REF] Lintner | The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets[END_REF], [START_REF] Mossin | Equilibrium in a Capital Asset Market[END_REF], [START_REF] Fama | The cross-section of expected stock returns[END_REF] and [START_REF] Black | The Capital Asset Pricing Model: Some empirical tests[END_REF], all made some contributions by extending the model. [START_REF] Roll | A critique of the asset pricing theory's tests Part I: On past and potential testability of the theory[END_REF] investigates the test of two-parameter asset pricing theory. According to the mathematical equivalence among the individual return/'beta' linearity nexus and the market portfolio's mean-variance efficiency, any valid test presupposes complete knowledge of the true market portfolio's composition. This implies, inter alia, that every individual asset must be included in a correct test. Errors of inference inducible by incomplete tests are examined and some ambiguities in published tests are explained.

Despite these criticisms, [START_REF] Markowitz | Portfolio selection[END_REF] model is at the root of scientific developments in finance, and has paved the way for further work on performance evaluation, Birth to several models. Thus, one-factor or multifactorial models have emerged with the work of [START_REF] Sharpe | A simplified model for portfolio analysis[END_REF]Sharpe ( , 1964)), [START_REF] Lintner | The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets[END_REF], [START_REF] Treynor | How to rate management of investment funds[END_REF] and other models, including those of Ross (1976) Fama andFrench (1992, 1993) and Carhart (1997).

The progression of Islamic finance leads to the fundamental question whether or not the practice of modern investment theories and analyses-Markowitz's Mean Variance Analysis and Capital Asset Pricing Model (CAPM), are in accord to shariah and could be employed in pricing Islamic financial assets [2]. [START_REF] Usmani | An introduction to Islamic finance[END_REF] urges that the Islamic financial institutions should liberate themselves from this practice.

A few researchers have agreed that Capital Asset Pricing Model can be applied in Islamic finance [START_REF] Selim | An Islamic capital asset pricing model[END_REF][START_REF] Hakim | Capital Asset Pricing Model and pricing of Islamic financial instruments[END_REF]. One of the variables in CAPM requires the presence of a risk-free interest rate which known as risk-free return. However, a risk-free interest rate is not shariah compliant as Islam prohibits the payment or acceptance of interest charge. So theoretically there is no equivalent of risk-free interest rate in Islamic markets. [START_REF] El-Ashker | The Islamic Business Enterprise[END_REF] establishes a theoretical framework of the traditional CAPM in pricing assets by changing the risk-free return by a zakat rate on pricing assets of 2.5% [3]. Then, it is a minimum rate of return from assets where investors and traders would accept for investments and speculations to cover zakat and to expose on their risks. [START_REF] Hanif | Risk and return under Shari'a framework: An attempt to develop Shari'a[END_REF] suggests the using of the traditional Capital Asset Pricing Model with a minor modification by eliminating the risk-free return rate with inflation rate. [START_REF] Shaikh | Proposal for a new economic framework based on Islamic principles[END_REF] proposes to employ the traditional Capital Asset Pricing Model in pricing treasury bonds by changing the risk-free return with NGDP (Nominal Gross Domestic Product) growth rate. But, their assumptions on risk-free return are theoretical and conjectural which is unlikely to be simulated in the financial market.

In the light of shariah compliance, the main objective of our paper is to explore key fundamentals of the Capital Asset Pricing Model (CAPM) and the set of assumptions fundamental the model in terms of its constancy to the principles of Islamic finance (shariah compliance) [4]. On the basis of CAPM, our study then shows a mathematical modeling of pricing Islamic financial assets. This particular modeling shariah-compliant CAPM is a change of the traditional CAPM that integrates principles of Islamic finance and incorporates other Islamic variables such as prohibition of short selling, purification and zakat. To do so, we utilize a data of 10 shariah-compliant public listed companies in Bursa Malaysia.

The empirical findings show that the proposed Islamic Capital Asset Pricing Model (CAPM) is suitable and pertinent in investigating the link among risk and return in the Islamic stock market. Also, our study contributes to existing body of knowledge by presenting an algorithm and mathematical derivation of the shariah-compliant Capital Asset Pricing Model (CAPM) which has been lacking in the literature of Islamic finance.

The rest of this paper is organized as follow. Section 2 presents an overview of the The assumptions of the traditional CAPM from shariah perspective. Section 3 shows econometric Modeling of the Islamic Capital Asset Pricing Model. Section 4, reports data characteristics utilized in our paper and discuss the empirical results. Then, section 5 concluding remarks. In section 6, we present the policy implications. Finally, section 7 reports the suggestions for future research.

The assumptions of the traditional CAPM from shariah perspective

The CAPM is a demand-side model that is based on maximizing the investor's utility function and assumes the market equilibrium. This model is relying on a set of assumptions introduced by [START_REF] Markowitz | Portfolio selection[END_REF] with a number of other additional assumptions set introduced by Sharpe (1964) and [START_REF] Lintner | The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets[END_REF].

Based on [START_REF] Elton | Modern Portfolio Theory and Investment Analysis[END_REF], the assumptions are as follows: There is no transaction cost; the assets are considerably divisible; the nonexistence of personal income tax; an investor cannot influence the price of a stock by his buying and selling actions; investors' decisions are solely in terms of expected values and variance of returns; unlimited short sales are allowed;

unlimited lending and borrowing at the riskless rate; and homogeneity of expectations and all assets are marketable.

In this context, we present an overview of the assumptions under which the Capital Asset Pricing Model was derived, in terms of Islamic finance (shariah compliance). Some of these assumptions may seem unlikely (unrealistic) but such simplification of reality make the Capital Asset Pricing Model more good from a mathematical perspective.

2.1.No transaction cost

The Capital Asset Pricing Model assumes 'no transaction cost' to decrease complexity of the model [START_REF] Elton | Modern Portfolio Theory and Investment Analysis[END_REF]. The 'no transaction cost' assumption is not opposing with any principles of Islamic finance and shariah. But, it is levelheaded to point out that the insertion of transaction cost causes serious errors when numerous transactions are performed as in permanent-time models [START_REF] Steinbach | Markowitz revisited: Mean-variance models in financial portfolio analysis[END_REF]. In the same alignment of shariah, these errors could possibly lead to a violation of the principles of Islamic finance where the results of the model may be ambiguous, or unfortunately give a speculative (maysir) and extreme uncertainties (gharar) finding. Therefore, in this study, we follow the CAPM assumption of 'no transaction cost' in the proposed model.

2.2.The absence of personal income tax

Under Capital Asset Pricing Model it is assumed that there is no personal income tax. This assumption is not differing with any principles of Islamic finance and shariah as there is no concept of income tax in Islam. Conversely, Islam has a concept of zakat a religious tax for all Muslim who meet the essential criteria of assets, which is only levied on excess wealth and not on income. Consequently, it is decisive to include the provision of zakat in the proposed modelling that we shall discuss further in part 2.3.

2.3.Divisibility and marketability of asset

The Capital Asset Pricing Model (CAPM) assumes that the assets are infinitely divisible and marketable. In practice, it is not always possible to buy only single unit of an asset and investors are not continuously in the position of ideal liquidity. But, this assumption is meant for oversimplification of the model and compatible with the principles of Islamic finance and shariah. Some researchers find that by relaxing these assumptions lead to better empirical findings. In this study, we follow these assumptions in our proposed modelling.

2.4.Investors ' decisions are solely in terms of expected return and variance (risk)

All expected returns, variance and covariance should be identified so that investors can make decisions only on the basis of expected return and variance. Based on [START_REF] Rosly | Critical issues on Islamic banking and financial markets[END_REF], this is, in fact, in line with the Islamic principles of al-ghunm bil ghurm (there is no return without risk)

and al-kharaj bil damam (profit is accompanied with responsibility). In fact, the shariah maxim of al-ghorm bil ghnom is invoke to invite investors to participate in ventures involving both risk and return such as al-bay (trading), al-ijarah (hire, lease or rent), salam (forward contract), mudarabah (silent partnership, where one of the parties provides capital and the other contributes expertise to conduct a exacting business, and musyarakah (joint venture).

2.5.Short sales are allowed

The assumption of short sales makes Capital Asset Pricing Model simplified for its mathematical derivation. This assumption means that assets may be held in an arbitrary amount-positive and negative. [START_REF] Dusuki | Fiqh issues in short selling as implemented in the Islamic Capital Market in Malaysia[END_REF] study the issue of small sale, which is selling an asset the investor does not hold, and it obviously violates the principle of Islamic finance, bay'ma'dum (selling what the seller does not own). Any revenue from this activity is deemed riba and hence not shariah compliant. The constituent of speculation (maysir) involved in small sales further suggests that short selling is intolerable in Islamic finance. So, in this proposed modeling, we assume short sales are not allowed.

2.6.Unlimited lending and borrowing at a risk-free rate

This assumption is very significant in developing the Capital Asset Pricing Model since this assumption lead to a piecewise linear relationship between expected return and beta for efficient portfolios. From the perspective of Islamic finance, the prohibition of interest leads to the nonexistence of risk-free assets, and the elimination of risk-free risk. But, there are cases that this assumption can be utilized in agreement to shariah, for instance, the employ of Islamic financial assets that could resemble risk-free assets such as sukuk (Islamic bonds). [START_REF] Omar | Research Paper: Islamic pricing benchmark[END_REF] develop the utilizing of a 3-month Islamic treasury bills as a proxy for the risk-free assets.

2.7.Homogeneity of expectation

The Capital Asset Pricing Model assumes all investors have homogeneous expectations about expected returns, variances (risks) of assets and correlation coefficients among assets. In the existence of homogeneous expectations, investors hold the identical optimal portfolio. In Islamic finance, investors are encouraged to trade in shariah-compliant investments to produce return (or profit) in the usually accepted manner (al-ribh al-ma'ruf) and involving risk [START_REF] Islahi | Economic Concepts of Ibn Taimiyah[END_REF].

Hence, it is reasonable to expect investors to assert homogeneous expectations in the market.

Therefore, each asset will have a market price and it should be deemed as the price when investors have homogeneous expectation. Moreover, shariah allows tagging the price according to the market price [START_REF] Omar | Research Paper: Islamic pricing benchmark[END_REF]; and this will lead to mutual consent, and ultimately homogeneity of expectations.

2.8.An investor cannot affect the price of a stock by his buying and selling actions

The homogeneity of anticipation implies that investors are price takers (no single investor can influence the price of a stock by buying and selling actions). In Islamic finance, investors cannot influence the price through their transactions. Thus, the homogeneous expectations assumption in the traditional Capital Asset Pricing Model (CAPM) is preserved and investors will constantly select a most advantageous portfolio.

Therefore, based on the above considerations, we concluded that most of the assumptions in the traditional CAPM and Markowitz's Mean-Variance Analysis are not contradictory to any principles of Islamic finance and therefore compliance to shariah. In this study, we propose a mathematical modelling of pricing Islamic financial assets using the traditional CAPM as our basis, with a small number of modifications, which is, integrating zakat, purification of return and exclusion of short sales.

Modelling the Islamic Capital Asset Pricing Model

In this section, we outline the assumptions underlying an Islamic capital asset pricing model. viii. An investor cannot influence the price of a stock by his buying and selling actions.

3.1.Mean-Variance under Islamic framework

Markowitz (1952) develops a relationship among expected return and risk in a portfolio context. Let i X indicates the weight of asset i in the portfolio 1, 2,..., iN  of a set of N shariah-compliant assets which is through applying shariah screening procedure from the asset universe 1, 2,..., IM  with iI  . According to [START_REF] Derigs | Review and analysis of current Shariah-compliant equity screening practices[END_REF], applying qualitative screens to the entire universe of globally accessible assets to limit on the set of admissible assets is always a first step in portfolio optimization, and results in a abridged shariah-compliant assets [5]. This screening process, but, may have some advantageous consequences as it reduces excessive risk [START_REF] Basov | On Sharia'a compliance, positive assoratative matching, and return to investment Banking[END_REF].

Given that all available shariah-compliant assets must be allocated, the N portfolio weight must add up to 1, so:

1 1 N i i X    (1)
The fundamental principle of the mean-variance is to utilize the expected return of a portfolio to represent the investment return and its variance as the investment risk. Basically, portfolio selection is the study of risk and return. This is reliable with the ethical principle of "no risk, no gain" in Islamic finance [START_REF] Chapra | Innovation and authenticity in Islamic finance[END_REF], as stipulated in the shariah maxim al-ghunm bil ghurm (there is no return without risk).

In our study, it is useful to point out that Islamic ethically accountable investors do not face any adverse consequence from the shariah screening procedure on the asset universe; they can expect as much return for a given risk as an investor would gain from a conventional assets [START_REF] Hassan | Impact of ethical screening on investment performance: The case of the Dow Jones Islamic Index[END_REF]. This is reliable with a recent study by [START_REF] Abbes | Risk and return of Islamic and conventional indices[END_REF], who investigates the risk and the return characteristics of the Islamic stock markets employing a large international data of 35 countries. The results find that the shariah-compliant stock markets do not perform significantly differently, nor do they appear less risky than their conventional counterparts. Broadly speaking, the paper infers that investors can pursue investing in conventionality with shariah without sacrificing return or incur extra risk. Therefore, we utilize the same approach in deriving the expected return p R , and variance of the portfolio 2 p  which are given by: 1 ()

N p i i i R X R    (2) 2 2 2 1 1 1 N N N p i i i j i j ij i i j X X X            (3)
We represent the variance of the expected return of asset i by of and the correlation coefficient between the expected returns on assets i and j by 2 i  .

An investor can decrease their exposure to individual asset risk by investment a diversified portfolio of assets. Due to the ethical fundamentals of Islamic finance, for example in terms of the shariah screening procedure, it is a common faith that Islamic portfolios suffer a diversification difficulty in comparison to conventional portfolios. In opposing, advocates of Islamic finance disagree that the universe of shariah-compliant assets appears adequate for diversification to be achieved [6]. Now, we are integrating zakat and purification of return in equations ( 2) and (3). In Islamic finance, zakat and purification would naturally be connected and prejudiced by ethical principles and shariah constraints forced on investment [7].

In deriving the mathematical modelling, zakat and purification are treated as two independent entities although zakat is a subset of purification. Let z denotes the zakat rate and i  denotes the individual purification rate of asset i. Therefore, the expected return and variance of individual asset i are given by:

      1 1 1 1 i i i i E z R z R          (4)        22 2 1 1 1 1 i i i i Var z R z          
(5)

After integrating zakat and purification of return, we derive the equations for the expected return ' p R , and variance of the portfolio '2 p  which are given by:

   ' 1 11 N p i i i i R X z R         (6)          2 2 2 '2 2 2 1 1 1 1 1 1 1 1 N N N p i i i i j i j i j ij i i j X z X X z                     (7)

3.2.Deriving the Islamic CAPM

Let us consider the problem of constructing a portfolio with the objective function of achieving maximum expected return, agreed this constraint of maximum capital to be invested. To obtain the objective function, we employ equations ( 6), ( 7) and the sukuk profit rate, as we have seen above in the assumption of unlimited lending and borrowing at the sukuk profit rate. Let s R denotes the sukuk profit rate of asset i:

  1 N s i s i R X R    (8)
The objective function is given by:

  ' ' 1 ps p R z R     (9) Subject to the constraints 1 1 N i i X   
and 0 i X  for all i to avoid small sales which are not allowable in Islamic finance.

Therefore, from (9), we can write:

              1 1 2 2 2 2 22 1 1 1 1 1 1 1 1 1 1 1 N i i i s i N N N i i i i j i j i j ij i i j X z R z R X z X X z                                 (10)
To maximize the objective function, we get the derivative with admiration to weight of each asset, and set it equal to zero.

Let,

       1 1 1 1 1 N i i i s i F X X z R z R           (11) Then,       1 11 k i s k dF X z R R dX        (12) Let,            2 2 2 22 2 1 1 1 1 1 1 1 1 N N N i i i i j i j i j ij i i j F X X z X X z                    (13) Then,            2 2 2 2 2 1, 2 1 1 2 1 1 1 N k k k j k j k j kj j j k k dF X X z X z dX                 (14) Thus,                              1 3 2 2 2 2 22 11 2 2 2 2 1 2 2 2 22 11 1 1 1 1 1 1 1 1 1 * 2 2 1 1 1 1 1 1 1 1 1 N i i i s i i N N N i i i i j i k i j ij i i j i NN k i i j k j k j kj i j k N N N i i i i j i i j i d X z R z R dX X z X X z X z X z X z X X z                                                                            1 2 1* 11 i k i j ij k k s z R R              (15)
Multiplying the equation above by:

         1 2 2 2 2 22 11 1 1 1 1 1 N N N i i i i j i k i j ij i i j i X z X X z                      (16)
And equating 0 i d dX   and the rearranging give the following results:

                           1 2 2 2 22 11 2 2 2 2 1 11 1 1 1 * 1 1 1 1 1 1 1 1 1 1 k k s N i i i s i N N N i i i i j i k i j ij i i j i NN k i i j k j k j kj i j k z R R X z R z R X z X X z X z X z                                                  (17)
This gives,

             2 2 2 2 1, 11 1 1 1 1 1 k k s N k k k j k j k j kj i j k z R R X z X z                           (18)
If there are homogeneous expectations, then all investors must select the same optimal portfolio. If all investors select the same portfolio, then in equilibrium, that portfolio must be a portfolio that has all assets held in the same percentage they represent of the market. Therefore, the right-hand side of the equation is given follow:

     2 1 1 1 k M k M kM z          (19)
This gives the following expression:

         2 1 1 1 1 1 k k s k M k M kM z R R z                 ( 20
)
Since this must hold for all assets (all possible values of k), it must hold for all portfolios of assets including market portfolio. Then,

     2 2 1 11 M M s MM RR z          (21)
Substituting this value for  in equation ( 20) and rearranging, then, we obtain:

              2 2 2 1 1 1 1 1 1 11 M M s k k s k M k M kM MM RR z R R z z                       (22) Then,         1 11 1 k k k s k M M s M R R R R                (23) Thus,         1 1 1 11 k k s k M M s kM R R R R                      (24) Therefore,     1 11 s k s k M kM R R R R            (25)
Finally, we can note that the term M  is an optimal purification factor or the market purification rate. The equation for market purification rate is discussed in the next part.

3.3.The optimal market purification rate

In this part, we assess the optimal market purification rate M  for the market portfolio. This rate is necessary in the equilibrium model of the Islamic CAPM that we have developed above. We use the objective function and the observed market price of risk (in the capital market line) from the efficient frontier to get the purification rate for market portfolio. From equation ( 9), the objective function is expressed as follow:

  ' ' 1 Ps P R z R     (26)
Where, the expected return ' P R from equation ( 6) is obtained by:

   ' 1 11 N P i i i i R X z R         (27) This is subject to the constraints 10 1 1 i i X   
and 0 i X  for all i to avoid short sales.

Therefore, the objective function for market portfolio is given by the following expression:

     ' 1 1 1 M M s M z R z R         (28)
This gives the equation of market purification rate as follow:

    ' 1 1 1 Ms M M zR zR       (29)
Where, '  indicates the observed market price of risk on the capital market line.

rate of return on the market portfolio at time t, iM  is the market beta of asset i and it  is the corresponding random disturbance term in the regression equation at time t. The i  denotes the intercept term in the time series regression. The CAPM implies that the alpha is zero.

Monthly returns of each asset are regressed on the monthly surplus market returns to find betas for each asset. Using these betas, the expected rate of returns and the standard deviation of each asset are assessing. Table 1 presents the summary statistics for the annualized returns of each asset.

The R², or coefficient of determination, reflects the percentage of a fund's movements that can be explained by the movements of its benchmark index. A R² of 100 indicates that all changes in the fund's price can be explained by those of its index. On the other hand, a weak R² indicates that a very small part of the price movements of a fund can be explained by the evolution of its benchmark index. A R² of 35 means that only 35% of the fund's price movements are explained by changes in the index. Such a fund provides interesting diversification opportunities because 65% of its risk can be eliminated through diversification.

In our case, the various coefficients R² are all greater than 80%. Subsequently, it can be concluded that all changes in the Islamic fund price can be explained by those of their index by applying the traditional CAPM. Finding the efficient frontier requires solving the following problem: find the portfolio weights 1 2 10 , ,..., X X X that minimize the variance of the portfolio 2 p  as given in equation ( 3), subject to the constraints

  10 1 P i i i R X R    and 10 1 1 i i X   
allowing for small sales and with no short sales by adding a constraint 0 i X  . Then, Figure 1 indicates the efficient frontiers for the 10 assets used in our paper. The analysis continues using the proposed Islamic CAPM. From equation ( 25) that has been derived previous, the proposed Islamic CAPM in the regression form is given by the following equation:

1

St it st i iM Mt it Mt R R R R             (32)
Where, the equation of market purification rate is obtained by the following expression:

    '1 1 1 Ms M M zR zR       (33)
Where, it R indicates the return of asset i at time t, st R is the sukuk profit rate, Mt R is the rate of return on the market portfolio, iM  is the market beta of asset i and it  is the corresponding random disturbance term in the regression equation.

Under the Islamic framework, it is essential to subtract zakat, as well as to cleanse the income by deducting purification from the expected rate of returns of each asset. In this alignment, we utilize a zakat rate, z of 2.5% [8]. For purification, after removing assets from prohibited business activities that are beyond the scope of a company's primary business activities; the remaining assets are augmented by a purification rate. We follow a technique by [START_REF] Rosly | Critical issues on Islamic banking and financial markets[END_REF] where the purification rate can be intended by dividing total prohibited income by total income. The data employed for this analysis is obtained from the individual company's annual report.

Table 2 summarizes the statistics for the annualized returns of each asset using the proposed Islamic CAPM.

In our case, the various coefficients R² are all greater than 90%. Subsequently, it can be concluded that all changes in the Islamic fund price can be explained by those of their index by applying the Islamic CAPM. To produce the Markowitz mean-variance efficient frontier, we impose a constraint to avoid small sales which are not allowable in Islamic finance by using the constraints 1 1

N i i X    and 0 i X  for all i.
Then, Figure 2 presents the efficient frontiers for the 10 assets by utilizing the proposed Islamic CAPM. 3 presents the expected rate return and standard deviation of the minimum variance portfolio. For the traditional CAPM, the range of portfolio returns is among 14.04% and 36.00%, and the range of portfolio risk is 7.96% to 29.41%. For the shariah-compliant CAPM, the range of portfolio returns is among 13.44% and 34.32%, and the range of portfolio risk is among 7.69% to 28.13%. Figure 3 presents the dissimilarity among the efficient frontiers for the traditional and the proposed shariah-compliant CAPM.   in the Islamic model. At an inferior risk, the efficient frontier for the shariah-compliant CAPM outperforms the efficient frontier for the traditional CAPM and consequently underperforms at an elevated risk. The constraint of small sales causes the efficient frontiers to have limited higher and lesser bound.

Conclusion

The

Policy implications

In our case, the new Islamic CAPM can help investors to choose and to select their portfolios composed by Islamic assets.

Suggestions for future research

For the possible future works, we can modify our sample. Also, we can use the analysis by the cross section methodologies.

Acknowledgement

I would like to thank the editor and anonymous reviewers for their supportive comments and suggestions.

Notes:

[1]: A portfolio * w is said to be mean-variance efficient if there exists no portfolio w with * pp RR  and 2 *2 pp   except itself. That is, you cannot find a portfolio that has a higher return and lower risk than those for an efficient portfolio.

[2]: The term shariah comes from the Arabic language means Islamic principles based on the Quran and the traditions of the Prophet (known as the hadith and sunnah).

[3]: Zakat is a religious tax for all Muslims who meet the necessary criteria of wealth, at a rate of 2.5% of total assets each year, for the benefit of the poor in the Muslim community.

[4]: There are 7 key principles of Islamic finance [START_REF] Lewis | Islamic Banking[END_REF]Iqbal, 1997): (i) the prohibition of riba, (ii) the basis of halal activities, (iii) the prohibition of speculative behavior (maysir) and extreme uncertainties (gharar), (iv) the obligation of zakat payment, (v) shariah board as the ruling body of Islamic finance, (vi) profit risk sharing and (vii) sanctity of contracts.

[5]: Qualitative screens are sector screens through which companies operating within specific business areas that are non-permissible under shariah are excluded. Shariah clearly defines a number of aspects which are not permissible for Muslims, such as the consumption of alcohol and pork, and thus compliant companies are not allowed to participate in business earning primarily or even partially from such activities [START_REF] Derigs | Review and analysis of current Shariah-compliant equity screening practices[END_REF].

[6]: For detail on potential diversification benefits on Islamic financial assets see [START_REF] Kok | Performance of Shariah-compliant indices in London and NY stock markets and their potential for diversification[END_REF], Abd Majid and Kassim (2010) and [START_REF] Kamil | Is there a diversification! 'cost' o shari'ah compliance? Empirical evidence from Malaysian equities. 4th Islamic Banking and Finance Conference[END_REF].

[7]: Zakat and purification are two different concepts. The literal meaning of zakat is to cleanse or purification, but in reality zakat operates as a religious tax, which is levied on excess wealth that a person is accumulating and has had in their possession for over a year.

But the term 'purification' intended for discussion in this paper is the act of deducting the non-halal tainted income which is not permissible in shariah, from the total return on investment [START_REF] Noor | A Shari'ah compliance review on investment linked takaful in Malaysia[END_REF]. Due to the complexity and the generally non-Islamic nature of the current capital market, the return obtained from both portfolios and individual assets has to be cleansed through post-investment purification practices [START_REF] Elgari | Purification of Islamic Equity Funds: Methodology and Shariah Foundation[END_REF]. It is a process in which the proportion of non-shariah compliant income is identified and donated [START_REF] Derigs | Review and analysis of current Shariah-compliant equity screening practices[END_REF]. The main purification practices are dividend purification and non-halal income purification. Non-halal income includes non-operating income from interest bearing investment and income from any prohibited business activities [START_REF] Delorenzo | Shariah supervision of Islamic mutual funds[END_REF]. The purification is calculating by dividing non-halal income by the total income of asset ia purification factor of asset i, i  .

[8]: For details see [START_REF] Al-Qaradawi | Fiqh az zakat: A comparative study: The rules, regulations and philosophy of zakat in the light of the Quran and Sunna[END_REF].

Figure 1 :

 1 Figure 1: The efficient frontier for the ten assets with and without short sales using the traditional CAPM

Figure 2 :

 2 Figure 2: The efficient frontier for the 10 assets using the proposed Islamic CAPM

Figure 3 :

 3 Figure 3: The difference between the traditional and Islamic efficient frontiers

Table 1 :

 1 Summary statistics for the annualized returns of each asset i by utilizing the traditional CAPM

	Asset, i		Expected rate of return i R	Standard Deviation i 	R²
	Hup Seng Industries 35.96372%	0.085050	0.8329
	IJM	Plantations	14.49241%	0.076787	0.8930
	IOI Berhad Petronas Berhad Sime Darby Berhad 7.70236% Corporation 14.52682% Gas 11.69837% Berhad Nestle (Malaysia) 12.50002%	0.078124 0.041404 0.055833 0.036194	0.8002 0.8289 0.9321 0.9102
	Pos Berhad Genting Plantations Malaysia Berhad Tasek Corporation Berhad Berhad DRB-Hicom Berhad 6.49814% 16.34169% 22.17836% 10.30622%	0.080119 0.088649 0.075129 0.110899	0.8374 0.8510 0.8321 0.9062

Table 2 :

 2 Summary statistics for the annualized returns of each asset i using the proposed

	Islamic CAPM
	Asset, i		Expected rate of	Standard	Deviation	R²
			return i R		i
	Hup Seng Industries 34.32018%	0.081163	0.9235
	IJM	Plantations	13.96025%	0.073968	0.9834
	IOI Berhad Petronas Berhad Sime Darby Berhad 7.32271% Corporation 13.92876% Gas 11.14223% Berhad Nestle (Malaysia) 12.00471%	0.074908 0.039436 0.053080 0.034760	0.9738 0.9943 0.9939 0.9543
	Pos Berhad Genting Plantations Malaysia Berhad Tasek Corporation Berhad Berhad DRB-Hicom Berhad 6.27348% 15.73173% 20.96376% 9.84816%	0.077128 0.083795 0.071790 0.107065	0.9948 0.9994 0.9930 0.9905

Table 3 :

 3 The minimum-variance portfolio Analysis

Expected Rate of Return Standard Deviation

	Traditional CAPM	14.04%	0.07967
	Islamic CAPM	13.44%	0.07690

  quick growth in Islamic finance urges us to not only appear for alternatives to traditional asset pricing model, but to develop its own model to assess risk and the relation among expected return and risk of sharia-compliant assets. This study finds that the traditional CAPM is a pertinent model in Islamic finance. Most of the assumptions underlying the Markowitz's Mean-Variance Portfolio Theory are not contradictory to shariah principles. In the development of the mathematical modelling of pricing Islamic financial assets utilizing the traditional Capital Asset Pricing Model, we let some of the assumptions remain for simplification of the model.

	The shariah-compliant Capital Asset Pricing Model that we developed with a few changing's
	of the traditional Capital Asset Pricing Model is integrating zakat, purification of return and
	exclusion of short sales. As a proof-of-concept, we use the data of 10 shariah-compliant
	public listed companies in Bursa Malaysia. The results show that the proposed Islamic Capital
	Asset Pricing Model (CAPM) is suitable and pertinent in investigating the link among risk
	and return in the Islamic stock market.
	Our paper contributes to existing body of knowledge by presenting an algorithm and
	mathematical derivation of the shariah-compliant Capital Asset Pricing Model (CAPM)
	which has been lacking in the literature of Islamic finance. This study proposes a novel
	approach in pricing Islamic financial assets in accordance to shariah, advocated by
	contemporary investment theories of Markowitz's Mean-Variance Analysis and Capital Asset
	Pricing Model (CAPM).

Empirical Results

Our empirical analysis is intended to serve as a proof-of-concept for applying the Islamic CAPM. We compare empirically the mean-variance framework of Markowitz and the Islamic CAPM based on the Markowitz mean-variance criterion as derived in the section 3 of our study.

We investigate empirically monthly rates of return from 10 shariah-compliant public listed companies in Bursa Malaysia and they are utilized to perform the analysis in this section.

They are Hup Seng Industries, IJM Plantations Berhad, IOI Corporation Berhad, Petronas Gas Berhad, Sime Darby Berhad, Nestle (Malaysia) Berhad, Pos Malaysia Berhad, Genting Plantations Berhad, Tasek Corporation Berhad and DRB-Hicom Berhad.

The sample period is from January 01, 2003 to December 31, 2015. To serve as a benchmark, the returns on the Kuala Lumpur Composite Index (KLCI) are employed as a proxy for market portfolio returns, the return from a 3-month Malaysia Treasury Bills as a proxy for the risk-free rate in the traditional Capital Asset Pricing Model (CAPM) analysis, and the sukuk profit rate is proxied by 3-month Malaysia Islamic Treasury Bills for the analysis of the proposed Islamic Capital Asset Pricing Model.

We express the return of asset i at time t by the following equation:

Where, t P indicates the adjusted close price at time t and 1 t P  indicates the asset price of time t-1.

To illustrate the employ of the proposed Islamic Capital Asset Pricing Model in our paper, we compare the analysis of the traditional Capital Asset Pricing Model and the Islamic Capital Asset Pricing Model.

Let us revisit the traditional Capital Asset Pricing Model (CAPM) regression model is obtained as follow:

Where, it R indicates the return of asset i at time t, ft R is the risk-free rate at time t, Mt R is the


Dr Abdelkader Derbali
email: derbaliabdelkader@outlook.fr

Dr Lamia Jamel
email: lajamel@yahoo.fr

Determinants of performance of Tunisia insurance companies: Case of life insurance

Keywords: Performance, life insurance, insurance companies, ROA, panel data

Business performance has attracted researchers' attention in the previous literature of corporate finance. However, in the framework of the life insurance sector, they have given a number of attentions. In our paper, we investigate empirically the effect of firm-specific characteristics (size, leverage, tangibility, risk, growth, liquidity and age) on the profitability of eight Tunisian insurance companies during the period of study from 2005 to 2015. The empirical results show that the variables size, age and growth are the most important determinants of the performance of Tunisian insurance companies as measured by Return on Asset ratio. Then, leverage, tangibility, liquidity and risk don't affect the performance of Tunisian insurance companies.

Introduction

Insurance companies make known significance for businesses and because individuals compensate losses and put them in positions where they were before they occur. In addition, insurers give economic, financial and social benefits for firms such as, loss prevention and diminution of anxiety.

However, it is true that investments are important for economic development and savings are the primary source of investments. Insurance company's products are the major instruments for the mobilization of savings of the families, particularly from middle and lower group households. Several research works have identified and analyzed the nexus among the role of insurance sector for stimulating the economic development by mitigating sudden and devastating occurrences [START_REF] Horng | Does Insurance Demand or Financial Development Promote Economic Growth? Evidence from Taiwan[END_REF][START_REF] Lee | The Link between Life Insurance Activities and Economic Growth: Some New Evidences[END_REF][START_REF] Yinusa | Insurance Development and Economic Growth in Nigeria 1986-2010[END_REF][START_REF] Taiwo | Relationship between Insurance and Economic Growth in Sub-Saharan African: A Panel Data Analysis[END_REF].

Insurance activities can be distinct as a service that provides a benefit upon the occurrence of a risk. Delivery, usually financial, may be for an individual, association or business in exchange for a perceived contributions or premiums. Thus, insurance activities are economic sector which includes the design, production and marketing of their services.

Given the quick expansion of financial markets, banks and insurance companies are in front of intense competition. Traditional performance measures come into sight to be insufficient to assemble the needs of strategic development of financial institutions [START_REF] Zhang | Study on Balanced Scorecard of Commercial Bank in Performance Management System[END_REF].

The performance is an association between the functional efficiency and strategic effectiveness. Thus, functional performance is to improve the products, services, production processes and marketing and human resources management. Then, the strategic performance is ahead of the competition by positioning itself on a growing market [START_REF] Chandler | Organization and performance of enterprises[END_REF].

The problem of measuring the performance of insurance companies has been well developed in the literature of financial theory. Thus, the profitability of insurance companies can be influenced by external factors and internal factors [START_REF] Demerguç-Kunt | Determinants of trading bank interest margins and Profitability: Some international evidence[END_REF]. In our study, we will examine the impact of specific features at the enterprise level on the performance of insurance companies in the case of life insurance. In this alignment, it begs the question: what are the main determinants of profitability of life insurance companies in Tunisia?

To answer this question, we examine empirically the impact of firm-specific characteristics (size, leverage, tangibility, risk, growth, liquidity and age) on the profitability of eight Tunisian insurance companies during the period of study from 2005 to 2015. The empirical results show that the variables size, age and growth are the most important determinants of the performance of Tunisian insurance companies as measured by Return on Asset ratio. Then, leverage, tangibility, liquidity and risk don't influence the performance of Tunisian insurance companies.

The rest of the paper is organized as follow; in Section 2, we expose a review of related literature. In Section 3, we present the methodology and we illustrate the data employed for the empirical investigation. Section 4 finds the empirical findings and a discussion of the important results. Concluding remarks are showed in section 5. Section 6 presents the policy implications. Finally, in section 7, we found the suggestions for future researchs.

Literature review

In general, the majority of studies on bank performance have been established on samples of U.S. banks and some parts of some European banks such as; [START_REF] Angbazo | Commercial bank net interest margins, default risk, interest-rate risk, and off-balance sheet banking[END_REF][START_REF] Berger | The relationship between capital and earnings in banking[END_REF], [START_REF] Neely | Why does bank performance vary across states?[END_REF]. For others banking systems of countries, the emphasis on a few studies in Colombia [START_REF] Barajas | Interest spreads in banking in Colombia from 1974 to 1996[END_REF], Brazil [START_REF] Afanasieff | The determinants of bank interest spread in Brazil[END_REF], in Malaysia [START_REF] Guru | Determinants of Profitability Commercial bank in Malaysia[END_REF] and in Tunisia [START_REF] Ben Naceur | The determinants of the Tunisian deposit banks' performance[END_REF].

Determinants of performance have been widely studied in the literature of finance company in recent decades. In 1995, Berger conducts a survey on a sample of U.S. banks to determine the impact on the capital return on equity. He used in his study the ratio ROE (Return on Equity) to measure the performance of banks. He concluded that this ratio reveals the existence of a positive impact on the capital return on equity. The impact of the characteristics of banks in the United States on the net interest margin is examined. The results showed that the net interest margins of banks have a positive relationship with leverage, the opportunity cost, the risk of default and management efficiency [START_REF] Ahmed | Determinants of Performance: A case of life Insurance Sector of Pakistan[END_REF][START_REF] Angbazo | Commercial bank net interest margins, default risk, interest-rate risk, and off-balance sheet banking[END_REF]. [START_REF] Prakash | Validation of multipleitem scale for measuring service quality in life insurance business: structural equation modelling approach[END_REF] validate the multiple-item scale for measuring service quality in a way that it predicts customer satisfaction and patronage intentions in life insurance business in India. They utilize a structural equation modelling approach going after the model development strategy, which takes account of proposing a basic structural model framework and subsequently it improves the framework through modifications of competing structural models for arriving at the best-fit structural model. Their study has recognized that a satisfied life insurance customer is not necessarily a loyal one as the best-fit structural model does not include patronage intention. Based on the significant relationships observed in the best-fit structural model, content, welcome and alignment are identified as the main factors of service quality in life insurance business; and service quality has not been found to be leading to customer satisfaction significantly. This paper suffers from methodological limitations associated with convenience sampling and anonymous survey-based research. [START_REF] Wahjudi | The relationship between organisational culture and firm performance: an empirical study on Indonesian manufacturing firms[END_REF] examine the impact of organisational culture on firm performance among Indonesia manufacturing firms. The empirical results from 152 organisations confirm that organisational culture has significant effect on firm performance. Only individualism and uncertainty avoidance significantly impact firm performance. Their implication is that, even in a country with many subcultures, firms' top management still need to pay attention to the impact of national culture at organisational level. In addition, since Indonesia has a very low score on individualism, managers need to create a conducive and rewarding environment for individuals to contribute.

George (2016) compares the productive efficiency and service quality of major foreign banks operating in India with that of the new private banks and also to study the linkages between efficiency, service quality and financial performance. [START_REF] George | Productive efficiency, service quality and profitability: a comparative analysis of foreign and private banks in India[END_REF] uses the data envelopment analysis (DEA) methodology for benchmarking of productive efficiency while SERVQUAL framework was used for service quality analysis. The linkages between productive efficiency, service quality and financial performance of a selected subset of both foreign and private banks were investigated. The empirical results of the analysis have clearly indicated that though both efficiency and service quality are extremely important they need not necessarily result in superior profitability of a bank. George (2016) contributes to the emerging need for a comprehensive framework for measuring efficiency, service quality and financial performance as well as their inter linkages for the Indian banks of different ownership structure.

Using data collected from 200 bank customers in two major cities in Klang Valley, Malaysia, Tan et al. (2016) investigate whether the five dimensions of SERVQUAL have an impact on customer satisfaction in Malaysian banking industry. They find out that the expectations of Malaysian banking customers are higher than perceptions in terms of service quality and tangible dimension has the largest influence on customer satisfaction. Their empirical findings provide several implications for bank management to improve upon their customer service quality in order to benefit from customer satisfaction which will lead to greater competitive advantage and profitability to the institutions concerned. [START_REF] Sweis | Total quality management practices and organisational performance in Jordanian courier services[END_REF] examine the relationship between TQM practices and the organisational performance in the Jordanian courier service firms, and to determine which of the studied practices control the performance wheel. The methodology used is based on a cross sectional study carried out using a questionnaire survey. The survey is distributed to a sample of departments in a leading courier service firm in Jordan. Their empirical results

show that TQM practices correlate positively with organisational performance, with a significant positive value for continuous improvement practice. Consequently, managers should consider TQM practices as a way to enhance performance, allocate resources and develop an environment that supports continuous improvement ideas.

This study aims to propose a conceptual framework to investigate the impact of organisational culture on quality practices and performance. Competing values framework has been used to capture the underlying value orientation of organisational culture and used the same to check its impact on performance. This study integrated both TQM and Six Sigma quality practices and defined a new set of infrastructure and core quality practices. In case of infrastructure practices, five practices have been identified which are top management support, customer relationship, supplier relationship, workforce management and Six Sigma role structure. Also, core practices resulted into five practices viz. quality information, product/service design, process management, Six Sigma focus on metrics and Six Sigma structured methodology. Further, a pilot study has been conducted in 40 medium scale manufacturing organisations to test the reliability of the constructs developed.

Kumar and Kumar (2016) present a real life example of improvement gained through implementation of VSM enabler of lean manufacturing. Their purpose is to discuss the measurable impact of lean manufacturing implementation on operational performance with the use of value stream mapping enabler of lean manufacturing.

In the study of Hosseini and Saravi-Moghadam (2016) In order to study the impact of bank characteristics, financial structure and macroeconomic indicators, net interest and performance of Tunisian banks for the period concerned margins.

It should be remembered that the different characteristics of banks account for a substantial part of the variation in net interest margin and the performance of the bank. The net interest margin and high performance tend to be associated with banks that hold a relatively large amount of capital.

Other determinants of bank performance on the net interest margin on bank loans have a positive and significant impact. Thus, the size has negative coefficient significance on net interest margins. This finding may simply reflect inefficiencies in the balance.

In addition, macro-economic indicators such as inflation and growth rates have no impact on net interest margins and bank profitability.

Finally, the financial structure has an impact on bank performance and the net interest margin. In fact, the concentration is less beneficial to commercial banks Tunisian competition.

In addition, the stock market development has a positive effect on the performance of banks. This reflects the complementarities between bank performance and stock market growth, these complementarities is probably a factor of economic growth. Furthermore, the development of disintermediation of the Tunisian financial system supports the performance of the banking sector in particular and economic management.

The determinants of performance of Malaysian banks over the period of 10 years (1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993)(1994)(1995) are studied. At this point, they chose two features at the micro and macro level. The results showed that inflation and effective expenditure management have a negative impact on profitability, as the interest rate has a positive impact on profitability [START_REF] Guru | Determinants of Profitability Commercial bank in Malaysia[END_REF].

Other authors developed a study of the determinants of profitability financial institutions.

The results indicated that the credit risk and intensity loans are proportional inversely to profitability. Thus, financial institutions and large sizes that have a level of significant expenses tend to a high profitability ratio [START_REF] Sufian | Determinants of bank efficiency falling on unstable macroeconomic environment: Empirical evidence from Malaysia[END_REF].

But, in other studies we noted that liquidity, capital and investment are important determinants of bank profitability [START_REF] Hakim | Profitability and Risk Management in Banking: A Comparative Analysis of Egypt and Lebanon[END_REF].

In addition, the growth of the money supply has a negligible effect on profitability, while GDP and the capitalization of assets on the stock market have a negative relationship with ROA (Return on Asset). Then, the profitability is positively affected by the size, sales growth and investment. On the other hand, current assets and leverage are negatively correlated with profitability. Several studies have been conducted to measure the performance of insurance companies. For example the functional status of insurers has no impact on the profitability to be provided by public coverage, but have a significant impact on the profitability of insurance companies. However, the size, the investment and the liquidity are the main determinants of the financial viability of insurance companies.

For the role of consolidation and deregulation in the insurance sector in Spain, we focuses on estimating the effect of EU directives on insurance (1994) and the policy of the Spanish Government in 1980 on changes in the market structure of the Spanish insurance. Demerguç-Kunt and Huizinga (1999) also determine that the consolidation has eliminated inefficient and poorly performing companies in the market. In the Spanish insurance sector, the role of organizational form is analyzed by comparing the stock and mutual insurers.

Then, other authors studied the impact of characteristics for insurance companies in Pakistan on the performance thereof. They selected a sample of five insurance companies for a period of seven years (2001)(2002)(2003)(2004)(2005)(2006)(2007). In their study, the performance is expressed by the ratio of economic profitability (ROA) based on seven variables. They made an estimate by the method of OLS (Ordinary Least Square) to justify the most important determinants of the performance of insurance companies. These authors concluded that performance is positively influenced by the level of risk and the size of the firm. But this performance was negatively impacted by the debt ratio (leverage) [START_REF] Sufian | Determinants of bank efficiency falling on unstable macroeconomic environment: Empirical evidence from Malaysia[END_REF].

In addition, insurance companies must diversify their investment techniques and use an effective hedge for them to create adequate financial income. Since diversification minimizes the risk level. Our sample consists of eight life insurance companies, In order to measure their performance over a period of 11 years (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015). This choice is justified by the availability of data. Therefore, the financial data were collected from the annual financial statements (balance sheet, income statement and statement of cash flows) of the life insurance companies that have been published by them.

Research methodology

3.2.Model

In this paper, we use the model who studied the impact of specific variables of a firm (size, leverage, tangibility, risk, premium growth, liquidity and age) on the performance of companies Insurance in Pakistan. We employ the ratio ROA (Return on Asset) as a measure of performance that will be regressed in terms of the explanatory variables.

In this context, the model used in our study is represented as follows:

              0 1 2 3 4 5 6 7          it it it it it it it it it PR β β LG β TA β SZ β LQ β AG β RC β GR ε Where:
 𝐏𝐑 𝐢𝐭 = Performance (ROA) = (earnings before interest and taxes divided by total assets).  𝐋𝐆 𝐢𝐭 = Leverage (Leverage) = (total debt / total assets).  𝐓𝐀 𝐢𝐭 = Tangibility (Tangibility) = (capital / total assets).  𝐒𝐙 𝐢𝐭 = Size (Size) = Ln (premiums).  𝐋𝐐 𝐢𝐭 = Liquidity (Liquidity) = (Current Assets / Current Liabilities).  𝐀𝐆 𝐢𝐭 = Age (Age) (The difference between the current year and the year of establishment of the company).  𝐑𝐂 𝐢𝐭 =Risk (Risk) = (Standard deviation of the ratio of total applications premiums).  𝐆𝐑 𝐢𝐭 = (Growth) = (Percent change in premiums).  𝛆 𝐢𝐭 = The error term.

Empirical results

4.1.Descriptive statistics

In Table 1, we present a descriptive analysis of the different variables associated with insurance companies' life in Tunisia. In fact, in this study, we consider the performance as a dependent variable expressed in terms of specific characteristics of insurance companies such as Tunisian, leverage, size, growth, tangibility, liquidity, risk and age (independent variables). The analysis in Table 1 indicates that the minimum value of the debt ratio (Leverage) is (0.030), while its maximum value is (0.436). Based on the value of the median, we notice that 50% of Tunisian life insurance companies have an average value (0.1428), which shows us the importance of debt in the activity of these firms.

Thus, we notice that 50% of life insurance companies have a ratio of Tunisian tangibility (Tangibility) of (0.716) (the median value) so the assets have a large enough value in the composition of the assets of those thereof. During the study period we found that half of the life insurance companies have an average value of the variable size (Size) of (17.94) which is expressed by the natural logarithm of the premiums received by them it from the insured. In addition, we noticed that 50% of the population in our sample age (Age) 33 years over the study period.

In this context, it fits the importance of age in the business of life insurance companies as may affect the market share and the performance thereof. Indeed, among the life insurance companies in our sample, those with a growth rate (Growth) premiums received is equal to 0.0766. Therefore, changes in premiums may largely influence the profitability of life insurance companies as premiums are the main resources for this type of service business. In addition, the results show no coefficient exceed the tolerance limit (0.7), which does not cause problems during the regression of performance (ROA). The results for these tests are given in Table 2.

4.2.Analysis of the results of the estimation

After interpreting the results from the table of descriptive and statistical correlation table, we present the results of the estimation of our model that measures the performance of life insurance companies.

First, we present a regression on panel data (double dimension: individual and temporal).

In this case, there is a problem in the estimation. That is to say, we will choose between the through the age of the life insurance companies and their presence on the Tunisian insurance market.

The positive coefficient on the variable Growth (premium growth) indicates positive relationship between growth and yield. Thus, the variable Growth has a positive coefficient (0.284792). This variable is statistically significant positive (2.64) at 1%. Thereafter, the increase in premiums received positively affects the level of performance of insurance companies in Tunisia. In this case, premiums may be an important variable and admits a direct impact on the profitability of insurers.

However, other variables have no influence on the levels of performance of life insurance companies. In this context, the performance of a company, regardless of their type of activity depends on several internal and external factors which are connected with the activity performed by each type of firm.

Conclusion

This study examines the impact of the characteristics of life insurance Tunisian firms on the level of performance the insurance sector in Tunisia during a period of 11 years (2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015).

Then, we investigate empirically the effect of firm-specific characteristics (size, leverage, tangibility, risk, growth, liquidity and age) on the profitability of eight Tunisian insurance companies during the period of study from 2005 to 2015.

The empirical results of the estimation of a regression model on panel data show that three variables, Size (size), age (age) and Growth (growth) are the most important determinants of the performance of the sector of insurance in Tunisia during the period going from 2005 until 2012.

Thus, the two variables Age and Growth have a positive impact on performance while the Size variable has a negative impact on the level of performance. Then, the other variables (Leverage, Tangibility and Liquidity Risk) are insignificant in relation to the performance of life insurance Tunisian firms.

Policy implications

All the aspects concluded about the empirical analysis are to be taken care for the healthy growth of the insurance industry in Tunisia. If the insurance industry wants to remain cost competitive, both life insurers and non-life insurers have to improve their distribution techniques and develop products to change customer behavior.

Suggestions for future research

For the possible future works, we can add an analysis for the case of non life insurance. Also, we can compare between the two insurance sectors; life and non life.

  , to evaluate of brand equity dimension, the measuring tools suggested by different researcher, regarding local circumstances and Persian bank specifications, is used. The present research is a descriptive study by using a sample of 266 actual customers from seven branches in north provinces Saman bank of Iran. The findings in this paper support the five-dimensional model of customer-based brand equity in bank industry. The present study contributes to the understanding of customer-based brand equity measurement by examining the dimensionality of this construct. The paper shows that bank managers and executives should try to influence; perceived quality, brand loyalty, brand image, brand value and brand awareness in their organisations and design their service delivery process by considering relations between customer-based brand equity components.In 1997, Ben Naceur and Goaied (2001) study the determinants of the profitability of commercial banks. They found that profitability is positively related to changes in the gross domestic product (GDP) per capita[START_REF] Neely | Why does bank performance vary across states?[END_REF]. In fact, other study relates to the measurement of the performance of Tunisian commercial banks. The empirical data used in this work were extracted from the database of the CBT (Central Bank of Tunisia). The sample included the largest commercial banks in Tunisia (10 banks) during the period 1980-2000.

  Currently, there are nineteen insurance companies in Tunisia. According to the annual report(2015) published by the Tunisian Federation of insurance companies, we can show the increase generated by the insurance companies in Tunisia income. This report has enabled us to draw the following conclusions:  The premiums of the insurance sector totaled 1 026.223 MD in 2015 against 961.921 MD in 2014, an increase of 6.68%.  The paid claims registered an increase of 18.69% from 505.709 MD in 2014 to 600.250 MD in 2015.  Management fees totaled an amount of 181.415 MD in 2015 against 167.351 MD in 2014, an increase of 8.40%.  Technical reserves recorded an increase of 12.35% in 2015 from 1 820.791 MD and 2014 for 2 045.696 MD in 2015.  Investment amounts entered in balance sheet assets totaled 243.374 MD in 2015 against 205.227 MD in 2014, a growth rate of 11.88%.  The technical result for the year deteriorated in 2015, it recorded a surplus of 125.529 MD against 132.061 MD in 2014.  The consolidated balance sheets were cleared in 2009 to a profit of 129.448 MD against a profit of 117.083 MD in 2008.

Table 1 .

 1 Descriptive Statistics

	Stats	N	Average	Max	Min	Standard	Median
						deviation
	Leverage	88	0.1687585	0.4365028	0.0305468	0.1033324	0.1428552
	Tangibility	88	0.6951185	0.90702	0.4352791	0.1279787	0.7162244
	Size	88	17.57351	19.44382	14.04266	1.457735	17.94376
	Liquidity	88	4.965466	40.99003	0.2161963	8.514908	2.397961
	Age	88	33,125	60	13	12.45955	33
	Risk	88	10.41076	35	0.2435627	9.588427	8.432577

Table 2 .

 2 Pearson Correlation

		Leverage	Tangibility Size	Liquidity Age	Risk	Growth
	Leverage	1
	Tangibility -0.0483	1
	Size	-0.5256	0.1183	1
	Liquidity	-0.2338	0.1914	-0.0109	1
	Age	-0.2659	0.3293	0.3414	0.0466	1
	Risk	-0.2036	0.4023	0.1058	0.4541	0.1150	1
	Growth	-0.1237	-0.0634	0.2042	-0.3703	0.0097	-0.1539	1

Acknowledgement

We would like to thank the editor and anonymous reviewers for their supportive comments and suggestions.

estimate of the fixed effects or random effects by. Our choice is justified by the probability of the Hausman test which must be compared to a value of 10%. Based on the results shown in Table 3 we note that although the probability of Hausman test is equal to (0.0002). This probability is less than 10%, so we chose the fixed effects model.

In fact, the likelihood is equal to Fisher (0.0273) is less than 5%. So the estimated model is globally significant.

The value of R² is (0.6213) indicating that the performance of insurance companies in Tunisia depends almost 62.13% of independent variables namely, leverage, size, growth, tangibility, age, risk and liquidity.

Therefore, the performance of insurers is defined mainly by these seven variables insurers during the study period. Table 3 indicates that the variable size is inversely proportional to the performance of Tunisian life insurance companies.

The results show that the coefficient on Size (size) is negative (-0.2777893). Thus, this variable is statistically significant negative (-2.17) the 5% threshold. For this, life insurance Tunisian small size companies are more efficient than those of larger sizes.

The coefficient on age is positive (0.0565387). Thus, this variable has a significant and positive depending on the value of the Student's t which is equal to (3.40) at the 1% impact.

Therefore, the age of an insurance company may affect its performance. This is explained