Banks are important institutions for maintaining the stability of a nation’s economy. Their main function is to collect funds as deposits and redistribute them in the form of credit, commonly referred to as loans, often interchangeably. In carrying out these operational activities, banks profit from the difference between loan interest and deposit interest, taking risks into consideration. Increases in loans distributed by banks lead to higher profits, but the risk of loss from non-performing loans (NPLs), which are loans that fail to be paid back by borrowers, also increases. A high proportion of NPL reduces the amount of the bank’s assets, meaning that these assets, originally deposits held by the bank’s customers, cannot be withdrawn by them. This creates a solvency risk, where the bank’s assets become less valuable than its liabilities. If customers are unable to withdraw their own money, it can alert customers at other banks, prompting them to withdraw all their funds as a precaution. This phenomenon is called systemic risk, causing people to lose trust in banking, leading to a reluctance to deposit their funds. Consequently, those in need of investment funds may find it challenging to borrow from banks, resulting in slower economic growth.

When banks fail to assess risk properly and experience losses, customers who lose trust tend to withdraw their savings. This can result in liquidity risk, which can lead to banks going bankrupt. This loss of trust can quickly extend to customers of other banks. When too many banks go bankrupt, the economic stability of a nation is severely affected. For example, one of the causes of the 2008 global financial crisis was the distribution of high-risk housing loans in excessive amounts without adequate regulation. This distribution caused an increase in the average portion of non-performing loans (NPLs). When the value of housing loans fell, coupled with a large portion of NPLs, many banks incurred significant losses. In the USA, the bankruptcy of Lehman Brothers triggered a financial crisis and a loss of confidence, which subsequently led to a liquidity crisis. During the 2007–2008 economic crisis, 465 banks were declared bankrupt. When public confidence in the Rupiah currency fell in Indonesia, it led to massive withdrawals and contributed to the 1997–1998 Indonesian Monetary Crisis (1). In this incident, 16 Indonesian banks went bankrupt, triggering massive inflation that caused economic and political instability.

Binding policies are essential to ensure that banking activities operate smoothly within a nation. To establish the right policy, research and data analysis are necessary for modeling the dynamics of banking. One study that has been conducted involves the mathematical modeling of bank balance sheets, known as banking models. These banking models can be expressed using dynamic models that track changes in the values of bank balance sheet components over a specific period. Factors that affect the values of bank balance sheet components are represented as variables in a differential equation.

The use of a system of differential equations allows researchers to determine the growth of observed objects over extended periods and the conditions under which they reach equilibrium. The dynamic model presented in this study, structured as a system of differential equations, is based on the predator–prey model, which has applications in various fields, including banking, as demonstrated by Caravaggio and Sodini (2), Blyuss et al. (3), Fakhry et al. (4), McLean et al. (5), and Saini et al. (6). The development of the dynamic model for creating a banking framework is discussed in Sumarti et al. (7, 8) and Ansori et al. (9–13) as well as in Ansori (14), Atangana and Khan (15), Ananth et al. (16), and Aqsha (17). Previously, the bank balance sheet model consisted solely of deposits and loans. In Sumarti et al. (8), it was expanded to include additional components from the bank’s financial statements. Modifications to the model to incorporate interbank interactions were implemented by Ansori et al. (9, 10), with the introduction of additional operators by Atangana and Khan (15) and Ananth et al. (16). The application of a metaheuristic method to address model optimization problems was initially introduced by Ansori et al. (9). Utilizing the method outlined by Ansori et al. (11), researchers can assess the stability of the bank’s component equilibria and perform sensitivity analyses to predict banking health and extend the stability of the banking system. One advantage of employing a dynamic model is its modular nature, which facilitates the addition or removal of variables. For example, the incorporation of a stochastic component into the dynamic model for representing a bank’s balance sheet is explored by Ansori et al. (12). A similar dynamic model is applied by Ansori et al. (13) to simulate a banking network system and analyze macroprudential policy.

This study discusses a model for modeling the bank’s balance sheet and its implementation. As our new contribution to the existing research, real data from banks in Indonesia and the latest banking regulations in Indonesia that significantly improved the model are used. Using a metaheuristic optimization method, the model’s parameters can be obtained, which are then used to calculate each bank component’s equilibrium and stability. Selected parameter values are then changed to observe the change in equilibrium value and stability. The results from the model can suggest the bank balance’s trend and its sensitivity to some significant factors, which could predict the risks of liquidity and solvency of the bank. The model can be used by the regulator to analyze the impact of proposed regulations related to banks to maintain the stability of the banking and financial industries.

In this section, we show the solutions to the system in Equations 1–12 for appointed banks, the equilibrium points of the system, and the sensitivity analysis using the NPL factor η and the withdrawal factor ω . Figure 1 shows the value of each component in IDR, as stated in the balance sheet. The parameter optimization using SOA results in an MAPE value for each component from each bank, as stated in Table 5. See Appendix A for the obtained parameter values. Simulation for the bank’s components value fitness is indicated with the average MAPE value from each component, with the average MAPE value attributed to each bank below 10%, indicating a good fit for simulation.

Figure 1 and Table 5 show the following:

From the system, we identified more than one equilibrium point. Here, we only consider the equilibrium point to be positive and stable, as shown in Table 6. In the sensitivity analysis, the results are categorized as stable equilibrium or unstable equilibrium. A stable equilibrium indicates that the bank’s component value is almost certain to reach the equilibrium value. An unstable equilibrium has a lower probability of reaching the equilibrium value, or the solution could blow up to infinity or fall to minus infinity, impacting the bank’s ability to operate normally. Banks that have at least one component with a stable negative equilibrium value have a higher likelihood of going bankrupt than those that have at least one component with an unstable negative equilibrium value.

The results of the sensitivity analysis of the NPL value to equilibrium points in Figure 2 show that the obtained equilibrium points are all in blue color. It means that they are all stable. The behavior of the equilibrium points for each component is described in the following paragraphs.

When the NPL value rises, the loan equilibrium depicted in the center graph of all banks in Figure 2 declines rapidly. When the loan equilibrium reaches zero, the pace of decline slows. Long-term high non-performing loan (NPL) values make it more likely for banks to lose their loan balances.

The left and right graphs of all banks in Figure 2 show very similar behaviors for the deposit and equity equilibrium points, respectively. For low NPL levels, they are declining, but then they start to rise again. According to Equation 8, the growth rate of the deposit α D is influenced by the equity value, and it appears that the behavior of the deposit equilibrium mimics that of the equity equilibrium.

The equity equilibrium decreases until a certain η then increases, which is the reflection of the behavior of the equity value. According to Equation 1, the equity growth rate depends on the difference between the portion of profit value and the portion of loan value. For the low value of NPL, the increase in profit cannot withstand the increase of the loan, so the equity decreases. For a large value of NPL, even though the profits decrease, the value of the loan is much smaller, so the equity will increase.

The equity equilibrium for high NPL value is negative for Bank Victoria, suggesting that if the manager does nothing to control the non-performing loan, there may be an insolvency condition. The situation is different with BPD East Java, where the negative value is only for certain values of NPL.

As the NPL value rises, so do the equity equilibriums of Bank Mega and BPD East Java. The rising NPL does not appear to be an issue. The behavior of the original related solution in Figure 2 is consistent with the original equilibrium points of the deposit for BPD East Java and Bank Mega in Table 6, where they are almost identical or larger than those of the loan. It indicates that deposit growth is outpacing loan growth. Strong deposit growth influences the rise in liquid assets, which ultimately leads to a profit and an increase in quantity. It could be the cause of the NPL growth’s seeming lack of issues for both banks.

However, in this sensitivity analysis, the values of other parameters remain constant, as shown in Appendices A, B. Their values cannot remain constant with these outcomes, particularly for the parameters of deposit and equity growth, if the situation is changing. There may be a psychological impact of putting the banks in serious danger of going bankrupt, which causes customers and bank owners to lose faith in the bank’s management in the handling of their funds.

Now, we discuss the results of the sensitivity analysis for the withdrawal parameter ω . With the exception of BRI, Figure 3 illustrates that if the bank’s withdrawal parameter is more than 50%, even than 40% for BPD East Java, the bank will become bankrupt since the equilibrium points of equity, loans, and deposits are zero. Because loan value is reliant on reserves and liquid assets, it will go to zero as deposit value does. Zero loan value results in zero equity value since the bank’s profit is linked to loans. However, the unstable status, highlighted in red in Figure 3, for some equilibrium points means the values could just not be permanent. They can increase to positive values or decrease to negative values for equity values.

The deposit equilibrium in Figure 3d for BRI decreases but does not approach zero if the withdrawal parameter value increases. The loan equilibrium remains constant at its carrying capacity K L for all values of the withdrawal parameter ω . It appears that the strong growth of loans has already separated from the growth of deposits. Conversely, for small values of ω up to around 40%, deposit growth has little effect on the equity equilibrium; after that, a high value of ω causes it to deteriorate.

Based on the data used, the conditions of each bank determine the sensitivity analysis of parameter ω . It is not necessarily the case that banks in a higher KBMI category have a stable equilibrium than those in a lower KBMI. Furthermore, when banks from different KBMI groups are analyzed, no specific pattern has emerged from the movement of equity equilibrium. Therefore, more investigation is required to ascertain how KBMI groups affect a bank’s ability to withstand changes in the ω parameter.

Using data from Indonesian banks, the process of optimizing and simulating the value of bank balance sheet components produces good simulation results, with the average MAPE below 10% for each bank observed. Based on the simulation results, the following conclusions can be drawn:

Based on the model’s performance, several conclusions can be drawn:

Based on the study, we could propose several recommendations for the regulator:

For further research, we can consider modeling operational costs and securities that fluctuate with respect to time or as a function of time. This approach requires more data on these variables and aims to approximate the parameters of the function quite accurately. Another suggestion is to modify the existing model so that the deposit is more sensitive to the bank’s condition when the loan and equity values are zero or negative. This adjustment could reflect the immediate cause of customer dissatisfaction with the bank’s condition, which can be contagious in real life.