Modelling and Simulating Dynamics Efficiency of Rural-Community Banks (RCBs) in Ghana

The present work seeks to develop a model for measuring efficiency of RCBs in Ghana by means of financial key performance indicators pairing macroeconomic indicators. A stochastic differential equation model for predicting the efficiency of RCBs in the future is developed and simulated using gaussian jumps to evaluate the models’ performance in unpredicted situations with four distinct phases of efficiency. Unique solution Exit multiple 4-dimensional stochastic differential equations and Macroeconomic model is proven to be the best-fitting model for the data with the lowest information criterion.


Introduction
Banks serve as financial intermediaries that accept deposits from customers and use them in the form of loan facilities to deficit spending units in the economy [1] In Ghana, the business of banking remains one of the most lucrative industries despite increasing competition and in the attempt to remain competitive; banks are exposed to several factors which can affect their profitability [2]. One of the key players in the Ghanaian banking industry is the Rural and Community Banks (RCBs) to fast track the development of rural areas of Ghana. They are regulated by the Bank of Ghana and thereby form part of the regulated financial sector in Ghana.
investigations to identify the factors that predict efficiency. A common observation made from these existing studies is that accounting-based financial ratios approach is consistently used to measure and rank RCB's performance. The accounting ratios approach to performance measurement focuses only on either the input or output side of the financial intermediation process. It does not consider the input-output combinations in the intermediation process. Therefore, it is not able to fully identify inefficiencies and changes in efficiency. This research seeks to develop a model for measuring efficiency of RCBs in Ghana by means of financial key performance indicators pairing macroeconomic indicators, stochastic differential equation model formulated was simulated with jumps incorporation to predict and evaluate the performance of various models.

Methodology and Model Development
Developing a prediction methodology for rural bank financial position in Ghana is the primary goal of this research. According to the literature, there are three methods for determining a bank's efficiency: financial ratios; parametric techniques; and non-parametric approaches. Using financial ratios to evaluate a decision-making unit's efficiency has the drawback of creating a false sense of priority for various sorts of input and output. Simulated Data from R was used for model simulation, with the results divided into four categories: profitability, liquidity, solvency, and macroeconomic indicators, to meet the study's goals. For this investigation, repeated responses over time are evaluated and assessed using Stochastic Differential Equations algorithm.

Model Formulation
The Efficiency equation can be derived using conditional stationary SDE's. Let Thus, the state's parameter can be defined as:

Model Assumptions
1) The efficiency process is Multivariate discrete-time stochastic process with a countable state space Markov process.
2) Efficiency consists of 4 time-dependent subpopulations Open Journal of Modelling and Simulation 3) The transition of

Model Process
From the states Equation (1), the derivative of Equation (1) can be written as.
Using the flow chart and model assumptions above, ODE can be formulated below.

Incorporating White Noise (Brownian Motion,
) into ODE Due to Randomness in the Efficiency Rate We Obtain

Simplifying the Equation (5) of the System into SDE Form by
Multiplying dt

The Stochastic Differential Equations Can Be Written in
Matrix Form As This Equation (7) is the matrix form of SDE's using Brownian motion for efficiency of RCB's.

Profitability Stochastic Differential Equation
With the solution (10), we can characterize the qualitative behavior of the process at t → ∞ : We observe that profitability is always positive, assuming the initial 0 j P is positive. Since

Liquidity Stochastic Differential Equation Model
Let; We observe that Liquidity is continuously positive, assuming the initial 0

Solvency Stochastic Differential Equation Model
( ) ( ) ( ) ( ) We observe that Solvency is each time positive, supposing the initial 0 We observe that Macroeconomic is constantly positive, assuming the initial

Model Simulation
In Figure 1, show simulated profitability SDE model performance of RCB's over the period to assess the trend of company's ability to earn profits from its sales or operations, balance sheet assets, or shareholders' equity. The series however exhibited cyclic movements which are cycles of rising and falling from initial time point zero (0) to one (1), data values that do not repeat at regular intervals. Such oscillatory movements of time series often have the duration of more than a Business Cycle. Cyclic variations of prosperity, recession, depression, and recovery are due to a combination of two or smore economic forces and their interactions. A company's ability to meet its short-term financial obligations can be assessed by looking at the simulated performance of RCB's liquidity model over time, as shown in Figure 2. However, simulated values tend to rise over time, but the magnitude of the seasonal change remains the same, indicating an additive seasonal pattern.
As seen in Figure 3, simulated solvency SDE model show how the banking industry has maintained its resiliency and robustness with trend pattern. This indicates that the industry's ability to withstand losses has increased because of the reforms and the recapitalization.
A phenomenon known as the business cycle occurs in Figure 4, when long-term trends in macroeconomic growth are superimposed on the levels and rates of change of macroeconomic SDE model. These include the levels and rates of change of major macroeconomic variables, which is expected to go through occasional fluctuations grow on expansions and recessions. The Great Depression of the 1930s served as the impetus for the development of most of the modern macroeconomic theory, and the financial crisis that occurred in 2008 serves as an obvious recent example. According to [17] Ghana's inflation show a decreasing pattern and there is non-seasonal and seasonal pattern in the series. [18] highlighting those changes in money supply, changes in Government of Ghana Treasury bill rates as well as changes in exchange rate as determinants of inflation in the short run. Open Journal of Modelling and Simulation

Model Evaluation
The performance of the models is evaluated from the likelihood-ratio tests and j j t where d t Z is a Gaussian process with

Discussion
In To confirm the best-fitting stochastic differential equation models, an information criterion is applied to each model, and the model with the lowest information criterion is the best. Typically, the criteria try to minimize the expected dissimilarity between the chosen model and the true model. In Table 1, the AIC estimates the relative amount of information lost by all the four SDE models, the less information a model loses, the higher the quality of that model. However Macroeconomic model prove to minimize information loss with AIC value (372.3814), BIC (377.5918) and logLik (−184.1907) which offer a better fit. In Figure 1, the series however exhibited cyclic movements which are cycles of rising and falling from initial time point zero (0) to one (1) for profitability SDE model, as shown in Figure 2. However, simulated values tend to rise over time, but the magnitude of the seasonal change remains the same, indicating an additive seasonal pattern liquidity SDE model. Whiles Figure 3 indicated trend pattern that the industry's ability to withstand losses has increased because of the reforms and the recapitalization. However, a phenomenon known as the business cycle occurs in Figure 4, when long-term trends in macroeconomic growth are covered on the levels and rates of change of macroeconomic SDE model.

Conclusions
Non-financial metrics like loan coverage, productivity, service quality, and management quality have not been included in the current analysis. Based on profitability, liquidity, solvency, and microeconomic variables, the bank's efficiency has been modelled, simulated, and rated. The report focused on these areas because they are of special relevance to investors, prospective investors, employees, management, and the Central Bank. In Ghana, a stochastic differential equation model has been created to measure the efficiency of RCBs.
The model's performance was evaluated using simulated data, and the results demonstrate that it may be utilized to make predictions. Over time, this study advises that these institutions apply real data to simulated stochastic differential models to improve their efficiency. Findings from this study could be useful to develop country rural banking institutions and policymakers, as well as scholars studying banking efficiency.