Index

• 1. INTRODUCTION
• 2. LITERATURE REVIEW
• 3. RESEARCH METHODOLOGIES AND DATA
• 4. EMPIRICAL RESULTS AND ANALYSIS
• 5. CONCLUSIONS
• REFERENCES

Figure 1. The requirements of Zakat Maal.

In Indonesia, the purification process is included in selecting Shariah-compliant stocks. The MUI National Sharia Board issued this rule concerning the categorization of Shariah-compliant stocks in Indeks Saham Syariah Indonesia (ISSI). As shown in Figure 2, the ratio of non-halal revenue to total revenue is equal to, or less than, 10%. 

A Shariah investor must understand that capital market Shariah-compliant stock transactions are long-term investments. The long-term duration is influenced by substantial capital (Abdalloh, 2020). The greater the investment capital, the longer investors will hold the portfolio. If the portfolio generates a profit, the investment is a success. Additionally, investors should be aware that a zakat obligation exists.

2.2. Asset Pricing and Portfolio Modeling

The development of theoretical and applied models with Shariah principles in the economic and financial fields is increasing. Studies focusing on theory means more modeling. Most of the development in the application demonstrates that the traditional model is implemented on assets subject to Shariah principles. Studies on model development in Islamic investments have also been carried out by Hassan, Aliyu, Saiti, & Halim (2020) and Abdullahi (2018). This current study provides related works on developing models for asset valuation or portfolio construction with Shariah principles. The summary models are presented in Table 1.

The adjusted CAPMs on the risk-free rate components (SCAPM-1 to SCAPM-5) are summarized and applied with Indonesian stock market data (Subekti et al., 2020). Compared to the other four models, the model with the sukuk rate is better as a substitute for the risk-free rate in the CAPM under Islamic principles. Sukuk’s replacement of the risk-free rate in the developed CAPM must be revisited because of zakat and purification in stock investment. Considering this reduction reminds investors of their zakat obligation and meticulously calculates a company’s income underlying the assets.

Markowitz’s MV model is the pioneer of modern portfolio theory in portfolio optimization. This MV model is modified by incorporating two Islamic principles: the obligation to issue zakat, and the concept of purification for Shariah shares (Hazny et al., 2012). Shariah-compliant stock portfolio modeling is done by adding the expected return on assets exceeding the zakat rate and using goal programming to accommodate several goals simultaneously (Masri, 2018). Consistent with developing MV by considering reducing zakat and purification, the SCAPM return was used to construct a portfolio with Shariah-compliant stock in Malaysia (Hazny et al., 2017).

Models	Variables	Authors
SCAPM-1	Market returns only	Cyril & Karim (1987)
SCAPM-2	Market and zakat	Ashker (1987)
SCAPM-3	Market and NGDP	Shaikh (2009)
SCAPM-4	Market and inflation rate	Hanif (2011)
SCAPM-5	Market and sukuk rate	Hakim, Hamid, & Mydin Meera (2016)
SCAPM-6	Market, sukuk rate, zakat, and purification	Hazny et al. (2017); Derbali et al. (2017)
MV-modified 1	MV with zakat and purification	Hazny et al. (2012)
MV-modified 2	Multiple objective stochastic programs, additional constraint zakat selection	Masri (2018)
BL with a case study on the Islamic stock market	ARIMA-ARCH, BI Shariah certificate rate, Jakarta Islamic Index	Widodo et al. (2017)
The Augmented BL	Momentum, value, quality investing, Dow Jones Islamic Index	Dewandaru et al. (2015)
BL with SCAPM inflation	SCAPM, inflation rate as a substitute for the risk-free rate	Subekti et al. (2022)

However, the MV model has weaknesses due to using the mean of historical data as a parameter in portfolio calculations. The BL model, which uses the CAPM's expected return as the target return held by most investors, overcomes this problem. The BL model also combines the CAPM with subjective predictions from investors. Many studies confirm the advantages of the BL model by comparing it to other methods (Allaj, 2020; Bessler et al., 2017; Duqi et al., 2014) . This model’s advantages are used in asset allocation. Dewandaru et al. (2015) employed Islamic stocks (such as the Dow Jones Islamic Index) in stock portfolios using an augmented BL model from Cheung (2013). They used momentum, value, and quality investing for the portfolio. Widodo et al. (2017) implemented the BL model using predictions from the ARIMA-ARCH model on stocks listed on the Jakarta Islamic Index utilizing the BI Shariah certificate rate as a substitute for the interest rate in the conventional CAPM.

Trichilli et al. (2020) analyzed the MV and Bayesian models on portfolios with Islamic stocks, using investor sentiment as information for the Bayesian approach. The study results state the advantages of MV’s diversification with conventional stocks over Islamic stocks when the market is depressed and bullish. Diversification is better in Islamic markets than traditional markets when using the Bayesian method. In a calm market, a portfolio with an Islamic index is more profitable when using MV. But the opposite is true when using Bayesian modeling. Hooi & Parsva (2012) suggested that the Shariah market would attract more Muslim investors. They examined the Islamic indices in Malaysia during and after the global financial crisis in 2008 to show that the risk was high but tolerable. Various approaches to obtaining BL returns exist, such as Theil’s mixed estimation, Bayesian, and sampling theory (Walters, 2014). Black & Litterman (1992) used Theil’s mixed estimation approach and provided Bayes’ theorem. However, the explanation was not provided in detail, so the BL development emerged through the description of the method to obtain the formula for the expected BL return. Kolm & Ritter (2017) explained the description of the Bayesian method in the Black–Litterman model. The process toward the expected BL return starts from the return equilibrium (identical to the CAPM). But in Black and Litterman’s original article, they did not state that the CAPM condition was equilibrium. Investor predictions are vital in preparing a portfolio related to assets.

Combining the two returns results in a new target expectation. Several open problems have led to other studies developing the BL model, from its elaboration to its application (Subekti et al., 2021a). For example, Nocera (2016) proposed a modification of the BL model using the three-stage dividend discount model (DDM) instead of the CAPM. Krishnan & Mains (2005) developed the BL model using a multifactor CAPM. Subekti et al. (2022) used the SCAPM as a new starting point to adjust the BL strategy because it better suits portfolios containing Shariah-compliant stocks. The SCAPM uses the inflation rate instead of the interest rate in the CAPM for the BL model. Their result is that the BL model with the SCAPM outperformed BL model with the CAPM.

So, these studies motivate a new development of the BL model by utilizing a unique SCAPM for the Shariah portfolio. We develop a modifying BL model with a new SCAPM for Shariah-compliant stocks regarding zakat and purification.

3. RESEARCH METHODOLOGIES AND DATA

3.1. Research Methodologies

The Black–Litterman model combines the CAPM return with investor predictions. We established the portfolio using two strategies to demonstrate the proposed model and offer a new Black–Litterman model with zakat, and purification reductions as a model for Shariah-compliant stock portfolios. This section briefly describes the stages in the BL-CAPM and the proposed BL-SZP model. The experimental results with Shariah-compliant stock data were taken from the Indonesian stock market, specifically the Jakarta Islamic Index (JII) 70.

3.2. Research Model

The development of the Black–Litterman model in the 90s provided critical solutions because the MV model is sensitive. The range of the historical means used in its calculation indicates the classical MV model’s problem (Black & Litterman, 1992). Suppose a portfolio has n assets and assume that . The return on assets is normally distributed with a vector of expected return, , and a covariance matrix, Σ. The expected return, and portfolio variances are calculated in Equations 1 and 2, respectively, as follows:

Where n is the number of assets in a portfolio and is the weight allocation for asset i.

The MV problem can be expressed in matrix form, such as Equation 3 that presents the objective function, and Equation 4 as the constraint.

The new expected return from the Black–Litterman model derives from a combination of prior estimations from the CAPM  and the views vector (Q) is expressed in Equation 5 where the covariance values are  and , respectively, and P is a pick matrix related to the Q (Subekti et al., 2022).

Therefore, Equation 8 presents the weight of asset in the Black–Litterman model.

3.3. Shariah-Compliant Asset Pricing Model with Zakat and Purification

Various research types dealing with investing according to Islamic principles have emerged. Some countries focus on Shariah financial products. For example, a sukuk appears as an alternative asset to replace the conventional bond and the Shariah capital market. Sukuk and bonds differ in the traditional context (Tiby, El, & Grais, 2015). The CAPM adjustment is continued by involving the sukuk’s return rate rather than the interest rate (Hakim et al., 2016). This method is used because, in Islamic principles, no investment is profitable without risk. The CAPM’s development in Shariah principles continued, including zakat reduction and purification in calculating investment returns (Derbali et al., 2017; Hazny et al., 2017). The proposed model uses the yield level of sukuk assets instead of the risk-free level. Therefore, in this Shariah adjustment model, one must consider the obligation of zakat and the existence of purification from the beginning of the expected return calculation.

Two investment reductions are zakat and purification. Zakat is a tax that becomes an obligation and applies to Muslims if the requirements have been met. Zakat purifies Muslims’ wealth. But investment activities also require the consideration of the existence of prohibited activities. So, purification is a critical step in receiving profits. The amount of zakat to be paid is 2.5% in this study. Purification can be calculated as the percentage of total income that is non-halal. In Indonesia, the asset to be selected in the Indeks Saham Syariah Indonesia (ISSI) or the Islamic Composite Index must be included in the Indonesia Stock Exchange and follow the process illustrated in Figure 2. The term purification in this process is defined separately from zakat. The purification process can be seen in Figure 2. The ratio of non-halal revenue to total revenue is less than or equal to 10%. The MUI National Shariah Board issued this rule concerning the categorization of Shariah-compliant stocks. According to Hazny et al. (2012), this purification is denoted by δi (Equation 9):

So, to model the determination of the CAPM under Shariah principles, zakat reduction and purification components are assumed for Islamic investors. Starting from the CAPM (with the sukuk as a substitute for the risk-free rate), it is developed further by considering zakat reduction and purification (Derbali et al., 2017; Hazny et al., 2017).

The following Lemma 3.1 re-formulate a new return that adjusted for the reduction of zakat and purification. To distinguish the notation of the conventional portfolio and Shariah, the new expected return of the Shariah portfolio is written as . Two reductions exist here, namely the reduction of zakat and the reduction of purification. Each expected return and variance of individual Shariah-compliant stock (i) is reformulated, with adjustments to zakat (z) and purification (δ).

Lemma 3.1

Equation 18 is an ideal condition defined to derive the market purification (Equation 19). Therefore, it is assumed that  is the observed market price of risk on the capital market line (Derbali et al., 2017; Hazny et al., 2017).

3.4. A New Black–Litterman Model with a SCAPM Incorporating Zakat and Purification

We focus, in this study, on incorporating zakat and purification in the Black–Litterman model for Shariah-compliant stock portfolios. The new procedure in the BL model, modified for Shariah-compliant stock portfolios, is as follows:

Step 1: The SCAPM return is utilized as a benchmark for asset returns in the market. According to Lai & Xing (2008) and Ruppert & Matteson (2015), the CAPM can be associated with a regression model. Thus, it is assumed that the SCAPM is viewed from the regression analysis. The SCAPM in regression form can be written as follows:

Step 2: The Black–Litterman model is identical to blending the investor prediction, called a view return. The return prediction expressed quantitatively in this model is not the value of numbers taken arbitrarily. According to Islamic principles, buying and selling are allowed. Naturally, the experts predict the future through their experience in market transactions. In Islamic principles, gharar means uncertainty, obscurity, or ambiguity. Consider that choosing portfolio assets is a rational investment process by understanding fundamental analysis. We follow the rule that the expected return will be recalculated considering the zakat and purification reduction, like the step-by-step SCAPM process.

An additional assumption, such as short sales activity, is prohibited in Shariah modeling. So, we add a limitation to the BL model with Shariah principles. We suppose that BL-SZP is limited for absolute views prediction. In this case, relative views have a short sales effect on optimum allocation results.

The following step is the combination process of two sources. For that reason, the SCAPM used in this modified Black–Litterman model can be written as Equation 22. The investor's prediction must also consider the reduction of zakat and purification as per Equation 23.

Step 3: Theorem 3.2. (the BL model with the SCAPM-ZP for Shariah portfolios). If the prior is the SCAPM-ZP in Equation 22 and investor views in Equation 23, the posterior distribution of the expected excess return in a new Black–Litterman model with the SCAPM is:

Step 4: The same procedure is employed with the MV method to find the optimal weight of the Black–Litterman portfolio. We need to add a limitation in the optimization model since short selling is against Shariah principles.

3.5. Data

The portfolio sample is constructed based on the Jakarta Islamic Index (JII) 70 list issued in June 2019. We can choose Shariah-compliant stock from the ISSI that contains 423 stocks. We selected JII 70, which has 70 liquid Shariah-compliant stocks. We collected the historical monthly returns for ten years and chose complete data from June 2009 to 2019. Due to the availability, only 42 stocks from 70 have comprehensive data in the sample period. We found 12 stocks with normal distribution (see Table 2), but we limit the portfolio to only six assets. The six Shariah-compliant stocks are selected from different sectors in this portfolio, and the chosen assets are ADRO, ASII, CTRA, LSIP, SMGR, and UNTR (see Table 3 for details).

No.	Stock	Company	Sector	JB test	ADF test
1.	AALI	Astra Agro Lestari Tbk	Agriculture	6.0918 (0.0475)	-5.3028 (0.0100)
2.	ADHI	Adhi Karya (Persero) Tbk	Property & Construction	3.2618 (0.1958)	-4.9976 (0.0100)
3.	ADRO	Adaro Energy Tbk	Mining	5.6487 (0.0593)	-4.5960 (0.0100)
4.	ASII	Astra International Tbk	Misc. Industries	3.5778 (0.1671)	-5.8971 (0.0100)
5.	CTRA	Ciputra Development Tbk	Property & Construction	5.2161 (0.0736)	-6.1205 (0.0100)
6.	INTP	Indocement Tunggal Prakarsa Tbk	Chemical Industry	0.5657 (0.7536)	-5.2411 (0.0100)
7.	LPKR	Lippo Karawaci Tbk	Property & Construction	2.8888 (0.2359)	-5.5120 (0.0100)
8.	LSIP	PP London Sumatra Indonesia Tbk	Agriculture	3.7944 (0.1500)	-5.1649 (0.0100)
9.	PTBA	Tambang Batubara Bukit Asam Tbk	Mining	1.9128 (0.3843)	-4.0032 (0.0100)
10.	SMGR	Semen Indonesia (Persero) Tbk	Chemical Industry	1.3184 (0.5173)	-5.3073 (0.0100)
11.	UNTR	United Tractors Tbk	Trade, Service, & Investment	1.1167 (0.5721)	-5.5476 (0.0100)
12.	WIKA	Wijaya Karya (Persero) Tbk	Property & Construction	2.6504 (0.2657)	-4.7821 (0.0100)

Note: The p-values are in parentheses.

The Jarque–Bera test and stationarity test results can be seen in Table 2. The augmented Dickey–Fuller (ADF) value in the fifth column in Table 2 indicates that all assets are stationary.

Stock	Sector	Min.	Median	Mean	Max.	Std. Dev
ADRO	Mining	-0.2620	0.0057	0.0082	0.4117	0.1213
ASII	Miscellaneous industries	-0.1492	0.0189	0.0146	0.2800	0.0798
LSIP	Agriculture	-0.3410	0.0000	0.0071	0.3508	0.1204
CTRA	Property and construction	-0.2631	0.0205	0.0190	0.4117	0.1285
SMGR	Chemical industry	-0.2140	0.0151	0.0115	0.2540	0.0835
UNTR	Trade, service, and investment	-0.2203	0.0173	0.0124	0.2663	0.0886

Table 3 reports the descriptive statistics for six assets. All assets have positive means and medians of around 1%, the standard deviations range from 7% to 12%, while ADRO, LSIP, and CTRA have greater volatility than the others. The market return is represented by the Jakarta Composite Index, which has a mean return of 1.15% and a standard deviation of 4.53%. The sukuk rate is calculated for monthly returns as 0.56%, and the risk-free rate is 0.51%, on average.

4. EMPIRICAL RESULTS AND ANALYSIS

4.1. Results

Step 1: CAPM and SCAPM Calculation

We present the calculation of both CAPM and SCAPM as a starting return point for BL. The results of beta CAPM and beta SCAPM can be seen in Tables 4 and 5. The details of the significance test for alpha and beta can be seen in Columns 4 and 6 in Table 4. Toward SCAPM, we calculate the delta purification for each asset based on the annual financial report. Worldwide, one condition for a share to be categorized into Shariah-compliant stock groups in Indonesia is that the purification level is less than 10%. Accounting experts assist in calculating individual purification of assets based on annual financial statements. The expected return of SCAPM in Table 5 resembles that of CAPM in Table 4. SCAPM’s value is slightly below CAPM’s expected return. This result is consistent with Hazny. et al. (2017) finding that zakat and purification reduce CAPM's expected return.

No.	Stock	Alpha	Pval(Alpha)	Beta	Pval(Beta)	CAPM
1	ADRO	-0.0024	0.8160	0.8594	3.38E-04	0.0053
2	ASII	0.0009	0.8487	1.3536	9.49E-25	0.0083
3	LSIP	-0.0020	0.8491	0.6261	9.38E-03	0.0038
4	CTRA	0.0044	0.6627	1.4976	5.32E-10	0.0092
5	SMGR	-0.0009	0.8813	1.1413	4.30E-14	0.0070
6	UNTR	0.0010	0.8776	0.9677	8.49E-09	0.0059

Step 2: The input of a return prediction reflects the investor's prediction based on the asset observation. One of the best ways to understand a company's condition is to use a fundamental analysis. The investment decision-making process is based on a fundamental analysis of macroeconomic conditions, the entire industry, and related companies’ performance. Risks in stock investments can be anticipated under the “buy what you understand and understand what you buy” principle. Investors' return predictions can be determined from several considerations simultaneously, such as technical and fundamental analyses. The moving average (MA) method is often a simple consideration, but it can help to see the movement of stocks in the future. Based on financial statements, investors can also provide advice on which assets will perform well in the future. We chose to use MA 10 and discuss it with practitioners.

No.	Stock			P-value (Alpha)	Beta SCAPM	P-value (Beta)
1	ADRO	0.0065	0.0056	0.8013	0.8606	3.25E-04	0.0052
2	ASII	0.0077	0.0056	0.8751	1.3558	7.22E-25	0.0082
3	LSIP	0.0154	0.0057	0.8306	0.6293	8.94E-03	0.0090
4	CTRA	0.0328	0.0058	0.6823	1.4968	5.14E-10	0.0038
5	SMGR	0.0059	0.0056	0.8614	1.1404	4.17E-14	0.0069
6	UNTR	0.0090	0.0056	0.9003	0.9693	7.72E-09	0.0058

These results were discussed with investors as practitioners of Shariah divisions in Indonesia. Each asset’s purification was determined from stocks’ financial statements in the portfolio. Then, three assets were selected, namely, ADRO, LSIP, and SMGR, for prediction with statistical tools.

ADRO’s reported operating income increased 11% annually to US$846 million in the first quarter of 2019. This achievement was consistent with the annual production growth and a positive return for the next period. LSIP is a company engaged in agribusiness, such as palm oil plantations. The government policy on expanding palm oil land has been issued.

Palm oil is a raw material that is increasingly in demand. Analysts predict that it affects the increase of LSIP profits. SMGR’s performance (PT Semen Indonesia (Persero) Tbk) until May 2019 was still not positive, but throughout the year, Semen Indonesia Group’s sales volume increased. The consolidated sales volume of Semen Indonesia Group increased by 17.05% to 15.16 million tons from the previous year’s volume of 12.95 million tons.

The predicted statistical methods and collaboration discussions were used with investment managers for academic purposes. The moving averages (MA) for ADRO, LSIP, and SMGR are 2.8%, 7.4%, and 2.1%, respectively. Formally, these predictions are stated in the Black–Litterman procedure as follows:

• View 1: Stock ADRO is predicted to yield an expected return of 2.8%.
• View 2: Stock LSIP is predicted to generate an expected return of 7.4%.
• View 3: Stock SMGR is predicted to yield an expected return of 2.1%.

Step 3: The expected returns of the BL model with CAPM and SCAPM are calculated using Equation 5 and Equation 27, respectively. The results are presented in Table 6, and the results of both expectations appear almost the same. The return of BL-SZP expectations is only slightly below that of BL-CAPM. The zakat reduction and purification influence this difference.

No.	Stock
1	ADRO	0.0243	0.0236
2	ASII	0.0173	0.0168
3	LSIP	0.0402	0.0387
4	CTRA	0.0128	0.0124
5	SMGR	0.0162	0.0158
6	UNTR	0.0157	0.0153

Step 4: Portfolio optimization using BL utilizes the mean-variance method by updating its target expectations. As mentioned, short selling is prohibited. We can consider using the quadprog package in the R program. The following weight is positive for each asset in the portfolio. Using BL-SZP, the weight allocation obtained from the BL-SZP in Figure 3 resembles that obtained from the BL-CAPM. But the BL-SZP process is modeled with Shariah principles. The result of asset allocation does not contain negative weighting, so the constraints have been met without short selling. The allocation weights for LSIP, SMGR, and ADRO are corrected due to the views.

Figure 3 shows the distribution of MV, BL-CAPM, and BL-SZP. Compared to the MV results without short selling, MV and BL weighting is dominated by SMGR. There is a correction for the proportion of UNTR and LSIP with the BL model. The difference in weight correction of BL and BL-SZP is quite small. However, this can cause differences in the allocation of capital to certain assets when the investment capital is large.

4.2. Portfolio Performance

We also demonstrate the portfolio results with the MV method as a consideration. Table 7 presents the comparison of the portfolio's expected return, risk, and Sharpe ratio.

Portfolio	Expected Return	Risk	SR
MV	0.0121	0.0041	2.9683
MV-ZP	0.0117	0.0038	3.0746
BL-CAPM	0.0211	0.0043	3.7175
BL-SZP	0.0204	0.0040	3.8281

A portfolio using Black–Litterman analysis generates a higher Sharpe ratio than the MV model. This result is consistent with Bessler et al. (2017) , who found that the BL model outperformed the MV model. The Shariah version of the BL model compared with the conventional model showed that the Sharpe ratio of the BL model with the integration of zakat and purification outperformed the BL-CAPM. Incorporating purification and zakat in the model reduces each asset's modified expected return and variance. Then the Sharpe ratio of the portfolio increases. This finding shows that despite changing the BL benchmark, it produces a better portfolio in diversifying assets and portfolio performance. The expected return and portfolio risks have changed from the traditional to the Shariah version with zakat reduction and purification. Thus, it embeds a new Shariah-compliant strategy.

5. CONCLUSIONS

Recently, confidence has increased in the Islamic economic and financial system nearly worldwide, consistent with the development of relevant research and the investment world of Shariah finance. But the literature regarding Shariah modeling for portfolios remains insufficient. The Shariah system’s presence in the investment world is expected to accommodate Muslim investors. Some studies explain the advantages of the Black–Litterman model for portfolio optimization. But most portfolio construction for Shariah-compliant stock with the Black–Litterman model still uses the CAPM as an initial benchmark. This research is part of the development of Shariah compliance for investment. The idea of Shariah principles seems to be considered unfavorable. Many new terms are not well understood, and one such term is riba (usury). Usury is an exaggeration or addition prohibited in Islam, such as bank interest. The other related terms are zakat from income and purification in stock markets with Shariah principles.

This study proposes integrating the Black–Litterman model with Shariah principles by considering the zakat and purification concepts. The procedure and its application contribute to Islamic finance literature. Zakat can also be interpreted as a form of purification. The term purification also exists in business but is distinguished from the obligation of zakat. The term purification is identical to the cleansing process of returns from prohibited activities, such as interest rates. The SCAPM involving sukuk, zakat, and purification inspires portfolio construction with the BL process and Shariah framework. The new BL model is expected to produce results resembling the BL-CAPM but starts with a process that integrates Islamic principles.

The result shows a slight difference in allocation between the BL and BL-SZP models due to the reduction of zakat and purification. This research is expected to provide new discourse in the actualization of Islamic finance and business by adding illustrations of the zakat calculation and applying purification in the model. In future research, more sample experiments can supplement this model’s development and are expected to strengthen the model offered to obtain an optimal portfolio and follow Shariah principles.

Funding: This study received no specific financial support.

Competing Interests: The authors declare that they have no competing interests.

Authors’ Contributions: All authors contributed equally to the conception and design of the study.

Acknowledgement: The authors thank LPDP Indonesia for the financial support under the Doctoral program.

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