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Factors Determining an Optimal Liquidity Reserve in Islamic Banking . FACTORS DETERMINING AN OPTIMAL LIQUIDITY RESERVE IN ISLAMIC BANKING (Case of Indonesia 2000-2008) Rifki Ismal1 PhD Student Islamic Banking and Finance School of Government and International Affairs Durham University (United Kingdom) Phone : +44 (0) 7900411659 Email: rifki_ismal@yahoo.com Abstract. This paper attempts to model an optimal liquidity reserve for Islamic banking from the idea of the conventional one. It is arranged to find out theoretical variables which construct an optimal liquidity reserve as well as real variables which are applicable for Indonesian case. Such Islamic banking model addresses the role of revenue of operational financing; total marginal financing; return sharing paid by Islamic banks to depositors and; prior position of liquidity reserve which influence bank’s liquidity reserve. Instead of current period, those influential factors affect an optimal level of liquidity reserve from their previous periods with respected to the behavior of the banks’ financing and economic condition. At the end, the paper delivers the importance of those variables to be considered by Islamic banks in order to set up a robust liquidity reserve. Keywords: Liquidity reserve, Financing to Deposit Ratio (FDR), RSD 1 The author address: School of Government and International Affairs, Al Qasimi Building, Elvet Hill Road, Durham University, Durham (DH1 3TU), United Kingdom (UK). 1 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . 1. BACKGROUND Bank functions basically as a financial intermediary between surplus unit of fund and deficit unit of fund. It lends money to business sector for profit motive and for the sake of its stakeholders. However, total amount of financing in bank’s asset side must match with total amount of liquidity in bank’s liability side. Liquidity problem might arise, amongst others, when bank fails to balance supply of fund to entrepreneurs and demand of fund from depositors in terms of amount, maturity, utilization, etc. This is the core cause of liquidity risk problem from internal side of the banking operation2. In this sense, one alternative policy for bank to fulfill routine/regular demand of liquidity from depositors is by having and maintaining an appropriate internal liquidity reserve. It commonly composes of cash reserve and reserve requirement. Cash reserve pursues bank’s discretionary policy to determine its amount while reserve requirement is stipulated by central bank, usually connecting to a certain percentage of total deposit to be deposited as bank’s account in central bank. Hence, total available fund to be given as credit by banks can be simplified as total deposit (D) minus this liquidity reserve (R) or D – R. From such liquidity reserve, banks regularly withdraw money to fill out depositors’ demand of liquidity with the frequency of withdrawal coming from one of three conditions below: (i) In regular demand of liquidity, banks take money from internal liquidity reserve as previously predicted and allocated (ii) In irregular demand of liquidity, besides taking money from liquidity reserve, banks borrow extra fund from money market or selling their short term marketable securities and; (iii) In liquidity run, besides providing liquidity through selling marketable securities above, point (ii), banks also use external liquidity providers such as central bank emergency liquidity, government’s lending facilities, etc (Ismal, 2007a: 15 – 21). This paper attempts to find an optimal internal liquidity reserve which will accomplish depositors’ demand of liquidity and prevent banks from liquidity distress. First of all, model of an optimal liquidity reserve in conventional bank in derived in order to become an initial idea to construct and derive an Islamic banking model. Then, from that conventional model, an Islamic model is built through some adjustments to comply with sharia principles. Then, 2 Externally, bank can only rely on money market to carry out any liquidity shortage but this paper does not take into account this scenario. 2 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . other adjustments are employed to apply the model with the case of Indonesian Islamic banking industry. Finally, the paper will presents factors determining an optimal liquidity reserve in Indonesian Islamic banking industry. 2. CONVENTIONAL MODEL OF AN OPTIMAL LIQUIDITY RESERVE The model is taken from Freixas and Rochet (1999) which is a modification of MontiKlein model. The idea starts from liquidity reserve which is taken from total deposits such that available deposits for credit is formulated as D – R, as written early. Then, the net amount of withdrawal under static framework at the end of the period is the random variable ~ . If the x realization x of ~ is greater than R, the bank has a liquidity shortage and it has to pay a x penalty rp(x-R) which is proportional to the shortage. Assuming deposits are costless and risk neutral, the bank’s expected profit is (Freixas and Rochet, 1999: 228): ( R) r D R rR r Emax 0, ~ R x L p (1) Furthermore, assuming that if the expected cost of the liquidity shortages formula in [.] is a convex function and random variable ~ has a continuous density function f(x), such x shortages are differentiable. Hence, Freixas and Rochet denote C(R) as an expected cost such that: C ( R) rp ( x R) f ( x)dx R (2) (3) (4) C ' ( R) rp R f ( x)dx rp Pr oba~ R x C ' ' ( R) rp f ( R) 0 With that derivation, the maximum profit for bank is achieved when (see appendix A for detail derivation): ' ( R) rL r rp Pr oba~ R = 0 x so the optimal reserve (R*) is found in the following relation: (5) r r Pr oba ~ R* L x rp (6) This implies that the optimal amount of liquidity reserve is the amount for which the marginal opportunity cost of holding reserves equals the expected cost of liquidity shortages or simply stated it equals to the ratio of liquidity premium (rL – r) to the penalty interest rate rp (Freixas and Rochet, 1999: 228-229). Prisman, Slovin and Sushka further improve this model 3 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . by introducing some randomness in the volume of funds collected or distributed by bank. Firstly, they starts from the demand function of loan as L = L(r L) and supply function of deposit as D = D(rD) such that the amount of reserve is simply R = D(rD) - L(rL). With those assumptions, the expected profit function of the bank becomes: or (see appendix B for proof), r L(r ) r D(r ) rR r Emax 0, ~ R x L L D D p (7) (8) rL r LrL r rD D(rD ) rp Emax 0, ~ R x And, we find the maximum expected profit with respected to loan rate is (Freixas and Rochet, 1999: 229-230): rL r L' rL LrL rp Pr oba~ RL' rL = 0 x rL (9) and with an elasticity of the demand for loan: L rL L' rL LrL (10) the optimal reserve (R*) is defined as (see appendix C for proof): 1 rL 1 r L Pr oba ~ R* x rp (11) Equation 11 implies that the optimal reserve is determined by the ratio of the difference between loan interest with respected to its elasticity and the penalty interest rate. 3. SHARIA PRINCIPLES ADJUSTING CONVENTIONAL MODEL From the conventional model of optimal liquidity reserve above, sharia principles purify the model in order to make it applicable for Islamic banking. Such principles with regards to adjusting that conventional idea are: 1. Islamic banks give financing to real sector based on sharia prohibitions of interest (riba), gambling (maysir), fraud (gharar), etc. Thus, Islamic modes of financing take form of either debt based financing; equity based financing or; service financing. And, Islamic banks tend to change the term “loan” into “financing” to distinguish Islamic approach of utilizing deposit with conventional one. 2. Sharia jurisprudence treats any return/loss from financing based on risk/return sharing among parties involved in the business. The item rL in conventional model is then 4 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . modified into rf to represent return of financing based on risk/return sharing between Islamic banks and their business partners. 3. The same case with the relation between Islamic banks and their depositors, rD in conventional model is turned out to be rβ reflecting profit and loss sharing between them. 4. Any return in form of interest is prohibited including return on reserve (r). Sharia does not allow any remuneration for unutilized fund such as liquidity reserve. 4. ISLAMIC OPTIMAL LIQUIDITY RESERVE MODEL FOR INDONESIA Following sharia adjustments into conventional model, the Islamic model is derived taking into account characteristics of the Indonesian Islamic banking industry as in the following: 1. There are two types of financing; Operational Financing (F), mostly Murabahah (61%) and Mudarabah (30%) and (ii) Non Operational Financing (L), dominated by Ijarah. Thus, Islamic banks receive a pre-determined short-term cash inflow (rf and rL) with a minimum probability of bearing looses3. All financing uses saving and time deposit fund4 (79% of total deposit (D)) which is mostly short-term tenor5. Hence, banks know the withdrawal time and behavior of the deposit as well as execution time and behavior of all financing. 2. As part of the liquidity reserve (LR), reserve requirements are stipulated by Bank Indonesia as 5% of the total Rupiah deposit and 3% of total foreign deposit. In addition to that 5% reserve requirement, the latest Bank Indonesia banking regulation number 6/21/2006 article 3 stipulates extra reserve requirements for Islamic banks with FDR below 80% but still have a positive reserve requirement account. In fact, as most of the Islamic banks have FDR beyond 80% FDR (even above 100%), the model does not take into account this extra reserve requirement. 3. However, if bank’s reserve requirements falls below the standard, the central bank charges 125% of the daily Islamic money market indicative return for RR which falls below 5% but is still positive. An additional 150% of that amount is further charged for RR which lies below 5% and if the balance is negative (Bank Indonesia banking regulation number 6/21/2006 article 14). What is called penalty (rp) in the following Islamic model captures both charges (125% and an extra 150%). 3 4 Repullo (2005:48) categorizes it as a safe and perfectly liquid asset. Wadiah demand deposit (21% of total deposit) is assumed idle (excluded in bank’s financing). 5 56.96% of total deposit is short term. It also implies consumers consumption behavior. 5 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . 4. Bank Indonesia does not pay any remuneration of the reserve requirement held by Islamic banks (Bank Indonesia banking regulation number 6/21/2006 article 7). Therefore, the Islamic model eliminates r of R as in conventional model above. 5. Islamic banks adopt revenue sharing concept (rβ) rather than profit and loss sharing in their deposit scheme. Technically, operational return is shared with depositors but non operational return is not shared. Thus, depositors do not have to compensate any looses instead they get a continuous return on their account. As such, expected profit of Indonesian Islamic banking allowing for penalty cost of less reserve requirement is much simpler than the conventional one and derived in the following: rf F rf 1 r rL LrL rp Emax 0, ~ R x The maximum expected profit is then: 1 r F rf rp Pr oba~ R F ' rf = 0 x rf (12) (13) and with an elasticity of the demand for financing of: f rf F ' rf F rf (14) the optimal reserve (R*) is defined as (see appendix D for proof): 1 r rf Pr oba ~ R* x rp f (15) Model (15) above implies that an optimal liquidity reserve is determined by the bank’s return over penalty rate ratio and the rate of return of financing over financing elasticity ratio. Thus, some variables which have to be considered for econometric modeling are rate of return of financing, elasticity of financing, penalty rate of an incompliance amount of reserve requirement, the bank’s return from operational financing, realization of the return sharing paid by Islamic banks to depositors and total Islamic bank financing. 5. ECONOMETRICS ANALYSIS Based on an optimal model in equation (15) above, a dynamic model (Autoregressive Distributed Lag – ARDL model) is employed to analyze the exact factors determining an optimal liquidity reserve for Islamic banking in Indonesia. ARDL is used to the fact that 6 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . dependent variable is often influenced by both lag of independent variable(s) and lag of itself (Studentmund, 2005: 173-175). The econometrics analysis starts from (1) Defining variable and model specification; (2) Constructing the models including testing them to fulfill the requirement of classical normal error term in order to be a robust Gauss-Markov model (BLUE) and; (3) Interpreting the result of the models. 5. 1. Definition of Variables and Model Specification All time series data are using Bank Indonesia (BI) monthly data, from December 2000 into September 2008. And, referring to a couple of deterministic variables in equation (15), the model will find out the most significant and influential variables through some statistical tests and position of the variables in the actual condition. Finally, the model finds total financing (PF); bank’s return from operational financing (OP) and; return sharing paid by Islamic banks to depositors (RSD) as variables that drive an optimal liquidity reserve (LR). Detailed description of those variables is in the following: 1. Total financing (PF). This variable is part of elasticity of financing which influence the optimal liquidity reserve to be managed. 2. Bank’s return from operational financing (OP). This variable is the end result of the financing return owned by Islamic banks after return sharing with depositors. 3. Return sharing paid by Islamic banks to depositors (RSD). It approaches (rβ) which determine total amount of liquidity sharing between banks and depositors. 4. Liquidity reserve (LR) which composes of reserve requirement and cash reserve. The position of liquidity reserve is decided by Islamic banks upon taking into account the performance of financing; pattern of liquidity demanded by depositors; penalty rate (rp) if reserve requirement falls below the stated level by central bank; opportunity cost of holding this non profitable account. Next, a complete optimal liquidity reserve model is written afterwards followed by a list of variables and their historical statistics plotted in Table 1. ΔLRt = c - β1Δ(OPt-3) + β2Δ(PFt-2) - β3(RSD) + β2Δ(LRt-12) + e (16) Table 1 suggests that total financing faces an increasing trend as shown by the high value of mean and median. Fortunately, because of the domination of debt based financing, such an increasing trend benefits the depositors in the form of growing profit from operational 7 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . financing (OP) and higher percentage of revenue sharing over deposit (RSD). These numbers demonstrate good financial management. Table 1. Statistical Summary Variable Revenue from Operational Financing (OP)* Total Financing (PF)* Return Sharing to Depositors (RSD)** Liquidity Reserve (LR)* * In million Rp; ** In percentage per year Mean 825,576 13,673,828 4.029 653,115 Median Std Deviation 496,401 774,681 11,350,808 10,595,069 3.565 2.213 573,813 475,261 Nevertheless, in the current business and economic situation it is very challenging for Islamic banks predominantly to arrange a robust level of liquidity reserve in line with their financing activities and to provide well-demanded liquidity to depositors. LR in Table 1 shows a high median and standard deviation meaning that despite successful financing management as shown above, business activities is getting riskier than before and depositors’ liquidity behavior is getting more difficult to predict and anticipate. 5. 2. Construction of Models A. Stationary Test Before modeling, a unit root test is conducted to check the stationary of every variable. The basic idea of stationary can be explained by taking a simple AR (Autoregressive) (1) process: Yt a0 a1Yt 1 t (17) where Yt-1 is a lag independent variable which might contain a constant and trend; a is a constant and; ε is assumed to be a white noise (Enders, 1995: 70). If |a1|≥1, Yt is a non stationary series meaning it has a trend, it does not have constant mean and; the variance is time variant. So, the hypothesis of stationary can be evaluated by testing whether absolute value of a1 is strictly less than one. Two common tests used in this stage are Augmented Dickey-Fuller (ADF) and Phillip and Perron (PP). ADF re-estimates (24) by subtracting Yt-1 (Lutkepohl and Kratzig, 2004:54): Yt Yt 1 a j Yt j t j 1 p 1 (18) where α = -a, null and alternative hypothesis of H0: α = 0 and H1: α < 0; with tα< α/(se(α)). The basic idea of ADF is to correct high order serial correlation by adding lagged difference 8 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . terms in the right hand side of the equation. Meanwhile, Phillips and Perron (PP) use nonparametric statistical methods to take care of the serial correlation in the error terms without adding lagged difference terms (Gujarati, 2004: 818). ADF and PP t-statistics of each variable attached in models are depicted in the following Table 2: Table 2. Stationary Test Variable Name OP PF RSD LR Augmented Dickey-Fuller Phillip and Perron Level 1st Difference Level 1st Difference -2.461 -10.542*** -2.388 -11.992*** 6.064 -6.682*** 6.355 -6.734*** -4.0511*** -4.0511*** 5.85 0.059 5.87 -11.276*** Note: *** refers to stastical significance of 1% Table 2 reveals that LR is stationary in order 1 and is followed by other variables (OP, PF and R). Nonetheless, return sharing paid by Islamic banks to depositors (RSD) is already stationary in level or I(0). Thus, the model will integrate all variables in their 1st order of integration except RSD to build a robust liquidity reserve models. B. Correlation and Causality Test To asses the strength of the linear relation between dependent and independent variables and its causality direction, the correlation coefficient test and granger causality test are used. The correlation coefficient formula is: r1, 2 [( X X )( X X )] (X X ) (X X ) 1i 1 2i 2 2 1i 1 2i 2 2 (19) with r value ranges between -1 ≤ r ≤ 1. Basically it detects the correlation of two variables without explaining causality or direction of the correlation (if it exists). Then, if two variables have a perfect positive linear correlation, r = 1; if they have a perfect negative linear correlation, r = -1; and if there is no linear correlation, r = 0. The granger causality specifically detects how much the current dependent variable (Yt) can be explained by its past value (Yt-n) and lag value of independent variables (Xt-n). Mathematically, the granger causality function is (Gujarati, 2004: 697): Yt i X t i jYt j u1t i 1 j 1 n n and X t i X t i jYt j u2t i 1 j 1 n n (20) 9 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . Yt is said to be granger caused by Xt if the latter explains the former as well as lag of the former and vice versa. The assesment of both tests to all variables is listed in the following Tables 3 and 4 below. Table 3. Coefficient of Correlation Table 4. Granger Causality Test Variable Name LR Coeff of Correlation OP PF RSD 0.738 0.9981 0.1829 Null Hypothesis OP does not Granger Cause D(LR) PF does not Granger Cause D(LR) RSD does not Granger Cause D(LR) F-Stat 23.0317 35.6202 9.1028 P-value Conclusion 6.7E - 6 Not Accepted 5.4E - 8 Not Accepted 0.0033 Not Accepted Coefficient correlation shows that LR has more than a 50% indication of a perfect positive linear correlation with both OP and RSD and a greater indication with PF (Table 3). LR and OP denote coefficient of correlation of 0.74, LR and RSD records 0.18, while LR and PF depict the strongest among others with 0.99. This is evidence that these four variables have strong correlation compared to other variables considered in equation (15) and the value of LR associates with the value of OP, RSD and PF. Investigation with granger causality strengthens the idea further. As shown in Table 4, granger causality indicates that LR is explained by its independent variables (OP, PF and RSD) with the strongest influence from PF followed by OP and RSD respectively. This finding means that the decision to locate LR as dependent variable to be explained by OP, PF and RSD as independent variable and lag of LR is very reasonable and statistically significant. It is because causality relationship goes from OP, PF and RSD to LR and not vice versa. The following section describes the result of the regression between LR and its independent variables as well as explaining the exact period of every independent variable which dominantly explains LR. C. Regression Result The estimated model of liquidity reserve is depicted in Table 5 in the following. The regression has fit the requirement of classical normal error term such as autocorrelation test, heteroskedasticity test, multicolinearity test including Ramsey Reset test for correctly specified equation. The coefficient of individual variable and the whole independent variable has also been robust with Gauss-Markov requirement of Best (minimum variance), Linier, Unbiased Estimator (BLUE). 10 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . Particularly, the model expresses that all of the following mutually determine marginal liquidity reserve in the current period (ΔLRt): (i) lag 3 periods of marginal profit from operational financing (ΔOPt-3); (ii) lag 2 periods of marginal investment in all financing (ΔPFt-2) and; (iii) lag 9 periods of percentage revenue sharing over deposit (RSDt-9) as depicted in figure 5. Table 5. Estimated Model of Liquidity Reserve Dependent Variable: D(LR) Independent Variable Coefficient t-statistic Constant 13182.39 2.5588 D(OP(-3)) -0.0146 -3.0008 D(PF(-2)) 0.0100 2.0032 RSD(-9) -0.2212 -2.2014 D(LR(-12)) 0.8568 8.0721 Diagnostic Analysis Value P-value R-squared 0.5242 Residual Sum of Square 2.95E+10 Akaike Info Criterion 22.7330 F-Statistics 21.9357 0.0000 Jarque Bera 0.7315 0.6936 LM Test 2.6846 0.1057 ARCH LM test 2.2985 0.1182 Ramsey RESET 1.4110 0.2388 Therefore, there is no variable influencing liquidity reserve in its current level. Rather, the most influential period is the prior (lag) period of every variable. Further, the model finds a positive direction between the current marginal liquidity reserve with the last 2 periods of marginal investment in all financing and the previous year’s marginal liquidity reserve. Meanwhile, there is a negative direction occurs between the current marginal liquidity reserve with the last 3 periods of marginal profit from operational financing and the last 9 periods of percentage revenue sharing over deposit In fact, it tells that the decision of an optimal liquidity reserve must go in the same direction (positive) with total investment in all financing and prior (annual) record of the liquidity reserve. When Islamic banks extended higher investment in the last two periods than the previous one, they have to prepare extra liquidity reserves to anticipate financing failure, unexpected liquidity withdrawal from depositors, etc. The same is true with the last 12 periods of liquidity reserve, if it was high meaning they have to locate extra liquidity lately. 11 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . One main reason is because of yearly demand of business activities as indicated by annual pattern of currency in circulation. However, Islamic banks have to hold extra liquidity reserve in the opposite direction with profit from operational financing. This is because if Islamic banks face any business loss leading to lower profit from operational financing and percentage revenue sharing over deposits, they have to prepare some additional liquidity reserves to anticipate liquidity withdrawal from return oriented (rational) depositors. Such depositors have two accounts (Islamic bank’s account and conventional bank’s account) and will switch their deposit to the one who pays higher return (Ismal, 2009:5-7). 6. INTERPRETATIONS OF MODELS The model delivers some important messages with respected to factors determining an optimal position of liquidity reserve. First of all, liquidity reserve mainly depends on return sharing paid by Islamic banks to all depositors. Because of the potential of displaced commercial risk, once return sharing is paid lower than before, Islamic banks might need to prepare extra liquidity reserves. Secondly is the previous position of liquidity reserve. Islamic banks submit to this factor because it contains regular demand of liquidity by business sector. The third message relates to prior investment in operational financing. If it moves up, liquidity reserve is also up anticipating either regular or irregular liquidity withdrawal. One reason is that expansion of operational financing is caused by an increasing trend of deposits which requires Islamic banks to reserve higher positions of liquidity than before (Ismal, 2008a: 2). The forth message concerns profit from operational financing. Dependency of current liquidity reserve position on this factor mimics the dependency of liquidity reserve on return sharing paid by Islamic banks to depositors. Once profit from operational financing falls, it will suggest that Islamic banks need to prepare extra liquidity since payment of return sharing to depositors will be falling as well. 7. CLOSING REMARKS Referring to the model of conventional bank’s optimal liquidity reserve, the same model for Islamic banking is constructed with some sharia adjustments and modification. Upon considering some characteristics and practices of Islamic banks in Indonesia, an optimal liquidity reserve model is modeled. It discloses some variables determining an optimal 12 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . position of liquidity reserve to be taken care by Islamic banking regulators and Islamic bankers. References Freixas, Xavier and Rochet, Jean-Charles (1999). Microeconomics of Banking. The MIT Press, 3rd Printing, London, England, 1999. Ismal, Rifki (2007a). “Sharia Issues in Liquidity Risk Management”. Academic paper to be presented in The 3rd International Conference on Islamic Banking and Finance: Risk Management, Regulation and Supervision, Islamic Research and Training Institute (IRTI) – Islamic Development Bank (IDB), previously scheduled in Karachi, Pakistan, October 27-28 2008, the new schedule will be in 2009. Ismal, Rifki (2007b). “Industrial Analysis of Liquidity Risk Management in Islamic Banking”. Unpublished academic paper, Durham University, England, October, 2007. Ismal, Rifki (2008a). “Islamic Banks Portfolio Risk Measurement”. Unpublished academic paper, Durham University, England, December, 2008. Ismal, Rifki (2008b). “Factors Determining Islamic Bank’s Asset Liability Balancing”. Unpublished academic paper, Durham University, England, December, 2008. Ismal, Rifki (2009). “Managing Liquidity Risk in Islamic Banking Industry”. Unpublished academic paper, Durham University, England, April, 2009. Bank Indonesia (2006). Banking regulation number 6/21/2006. Bank Indonesia, Jakarta, Indonesia, 2006. Enders, Walter (1995). Applied Econometric Time Series. John Wiley & Son, 1st Edition, Canada, 1995, p 70. Lutkepohl, Helmut and Kratzig, Markus (2004). Applied Time Series Econometrics. Cambridge University Press, 2004, p 54. Gujarati, Damodar (2004). Basic Econometrics. The Mc Graw- Hill Companies, 4th Edition, USA, 2004. Studenmund A.H (2005). Using Econometrics: A Pratical Guide. Addison-Wesley Series in Economics, 5th Edition, Person Higher Education Inc, United Kingdom, 2005. 13 Factors Determining an Optimal Liquidity Reserve in Islamic Banking . Appendix A ( R) rL D R rR rp Emax 0, ~ R x ( R) rL D rL R rR rp Emax 0, ~ R x ' ( R) rL r rp Pr oba~ R = 0 x so that, such that, finally, (proven) r r Pr oba ~ R* L x rp Appendix B or, rL L(rL ) rD D(rD ) rR rp Emax 0, ~ R x rL L(rL ) rD D(rD ) rDrD LrL rp Emax 0, ~ R x rL L(rL ) rD D(rD ) rDrD rLrL rp Emax 0, ~ R x so that, finally it becomes, (proven) rL r LrL r rD D(rD ) rp Emax 0, ~ R x Appendix C rL r L' rL LrL rp Pr oba~ RL' rL = 0 x rL so that, then, rL r L' rL and knowing L L LrL rL rL L' rL rL' rL LrL rp Pr oba~ RL' rL = 0 x LrL rL rp Pr oba~ R r x L' rL by multiplying with 1 rL 1 r L We comes up with the optimal reserve (R*) as Pr oba ~ R* x rp (proven) Appendix D 1 r F rf rp Pr oba~ R F ' rf = 0 x rf then, 1 r F Pr oba~ R x rp F ' by multiplying with rf rf and knowing f rf F ' F 1 r rf x We comes up with the optimal reserve (R*) as Pr oba ~ R* rp f (proven) 14

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