# Modeling Financial System with Interbank Flows, Borrowing, and Investing

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## Abstract

**:**

## 1. Introduction

**Remark**

**1.**

**Remark**

**2.**

#### 1.1. Existing Models and Contributions

- First, we extend the model in (Carmona et al. 2013) to include the central bank. This affects the response of the private banks who now make decisions considering the monetary policy of the central bank. Our model allows for the central bank to play a more active role in stabilizing the banking system. This important feature was missing in the work of (Carmona et al. 2013) where the central bank was considered merely as a clearing house.
- Second, we reformulate the problem discussed in (Carmona et al. 2013) as a principal–agent problem where both, the principal (central bank) and the agent (private banks) maximize their respective utility. We assume that the private banks are risk neutral and central bank as risk averse. This allows us to get closed form solutions for the optimal policy of both the players.
- Third, we generalize the flow rates in our model to be different for each pair of private banks. This generalization converts the representative agent model discussed in (Carmona et al. 2013) into a heterogeneous agent model. This heterogeneity demonstrates a unique feature of the banking system where banks with higher interbank flows are less likely to default than their counterparts with lower interbank flows.
- Finally, as we show in Section 2, the volatility of the wealth process of the private banks is controlled, in contrast to (Carmona et al. 2013) where the volatility is constant for each bank.

#### 1.2. Organization of the Paper

#### 1.3. Notation

## 2. Description of the Model

#### 2.1. Formal Description

**1.a**) All ${W}_{1},\dots ,{W}_{N}$ are independent. Then, the matrix A is diagonal:

**1.b**) All ${W}_{1},\dots ,{W}_{N}$ are the same: ${W}_{1}={W}_{2}=\dots ={W}_{N}$. This means that all banks, in fact, use the same portfolio, and they are perfectly correlated. Then, it makes sense to let ${\mu}_{1}=\dots ={\mu}_{N}$ and ${\sigma}_{1}=\dots ={\sigma}_{N}$.

**1.c**) An intermediate case: for some i.i.d. Brownian motions ${\tilde{W}}_{i},\phantom{\rule{0.166667em}{0ex}}i=0,\dots ,N$, and some coefficients ${\rho}_{0},{\tilde{\rho}}_{0}$ with ${\rho}_{0}^{2}+{\tilde{\rho}}_{0}^{2}=1$ we have:

#### 2.2. Main System of Driving Stochastic Equations

**Remark**

**3.**

**2.a**) All ${c}_{ij}(t)\equiv 0$. Then, there are no cash flows between banks.

**2.b**) All ${c}_{ij}(t)\equiv c(t)>0$. For a constant c, this is the model from (Carmona et al. 2013).

**2.c**) Let G be a graph on vertices $\{1,\dots ,N\}$. Fix a $c(t)>0$ for all $t>0$. Let

#### 2.3. Interpretation

## 3. Optimal Behavior of Private Banks

#### 3.1. Statement of the Problem

#### 3.2. Solution of the Problem

**Theorem**

**1.**

**Remark**

**4.**

**Remark**

**5.**

**Proof.**

#### 3.3. The Dynamics of Banks under Their Optimal Investment Choices

#### 3.4. Systemic Risk

## 4. Optimal Central Bank Policy

**Theorem**

**2.**

**Remark**

**6.**

**Proof.**

**3.a**) Assume ${S}_{1}=\dots ={S}_{N}$: all investments are the same. Then, we have:

**3.b**) Independent portfolio process: ${a}_{ij}={\sigma}_{i}^{2}{\delta}_{ij}$, where ${\delta}_{ij}$ is as defined in Equation (24). Then,

**3.c**) Correlated portfolio process with same growth rates $\mu ={\mu}_{i}$ and volatilities ${\sigma}^{2}={\sigma}_{i}^{2}$. Assume the driving Brownian motions of these portfolio process are correlated as in Equation (10). After calculation, we get:

**Remark**

**7.**

**Remark**

**8.**

## 5. Long-Term Stability

**Theorem**

**3.**

**Proof.**

**Lemma**

**1.**

**Proof.**

## 6. Concluding Remarks

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CARA | Constant Relative Risk Aversion |

NSF | National Science Foundation |

SDE | Stochastic Differential Equation |

## Appendix A

**Lemma**

**A1.**

**Lemma**

**A2.**

**Lemma**

**A3.**

**Proof.**

**Lemma**

**A4.**

**Proof.**

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**Figure 1.**We use the following parameters for the simulations: $N=30$ bank, time horizon $T=1$, no correlation ${\rho}_{0}=0$, no interbank flows ${c}_{i,j}=0$, 1000 time steps and ${\mu}_{i},{\sigma}_{i},\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,N$ i.i.d. uniform $[0.1,0.2]$.

**Figure 2.**Evolution of the logarithmic capital ${Y}_{i}(t)$ of banks, $i=1,\dots ,N$. We use the following parameters: interest rate $r=0$, $N=30$ banks, time horizon $T=1$, no correlation: ${\rho}_{0}=0$, interbank flows ${c}_{i,j}$ are as in Equation (38), 1000 time steps, and ${\mu}_{i},{\sigma}_{i},\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,N$ are i.i.d. uniform on $[0.1,0.2]$.

**Figure 3.**Evolution of ${Y}_{i}(t)$, which represents the log capital of the ith bank. We use the following parameters: interest rate r = 8%, $N=30$ banks, time horizon $T=1$, correlation coefficient ${\rho}_{0}=0.5$, interbank flows ${c}_{i,j}$ as in Equation (38), 1000 time steps and ${\mu}_{i},{\sigma}_{i},\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,N$ i.i.d. uniform $[0.1,0.2]$.

**Figure 4.**Number of banks in default, whose log capital ${Y}_{i}(t)$ at some time $t\in [0,T]$ goes below $D=-1$. We use the following parameters: $N=100$ banks, 1000 simulations, no correlation: ${a}_{ij}={\sigma}_{i}^{2}{\delta}_{ij}$, no interbank flows: ${c}_{ij}=0$ for $i,j=1,\dots ,N$, 100 time steps, and ${\mu}_{i},{\sigma}_{i},\phantom{\rule{0.166667em}{0ex}}i=1,\dots ,N$ are i.i.d. uniform on $[0.1,0.2]$.

**Figure 5.**Empirical CDF of $\mathfrak{D}$, the number of banks in default, with $N=100$ banks, 1000 simulations, ${\mu}_{i}={\sigma}_{i}=0.1$ for $i=1,\dots ,N$.

**Figure 6.**Histogram of the number of banks defaulting. We use the following parameters: $N=100$ banks, 1000 simulations, ${\mu}_{i}={\sigma}_{i}=0.1$ for $i=1,\dots ,N$, and no interbank flows: ${c}_{ij}=0$ for $i,j=1,\dots ,N$.

**Figure 7.**Empirical estimates of the probabilities of large and small defaults: $\mathbf{P}(\mathcal{D}>60)$ and $\mathbf{P}(\mathcal{D}<5)$, respectively, as a function of correlation between portfolio process ${\rho}_{0}$ at different interest rates. We use the following parameters: $N=100$ banks, 5000 simulations, ${\mu}_{i}={\sigma}_{i}=0.1$ for $i=1,\dots ,N$, and interbank flow rates ${c}_{ij}=0$ for $i,j=1,\dots ,N$.

**Figure 8.**Empirical CDF of the number $\mathcal{D}$ of banks in default. We use the following parameters: $N=100$ banks, 1000 simulations, ${\mu}_{i}={\sigma}_{i}=0.1$ for $i=1,\dots ,N$, and constant interbank flows ${c}_{ij}=a$ for $i,j=1,\dots ,N$, where $a\in \{0,0.5,1\}$.

**Figure 9.**Optimal interest rate with $N=30$ uncorrelated portfolio process: ${\rho}_{0}=0$, with ${\mu}_{i}={\sigma}_{i}=0.1$ for $i=1,\dots ,N$.

**Figure 10.**Optimal interest rate with $N=30$ uncorrelated assets. with mean and standard deviation ${\mu}_{i},{\sigma}_{i},i=1,\dots ,N$ i.i.d uniform on $[0.1,0.2]$.

**Figure 11.**Optimal interest rate with $N=30$ portfolio process, with correlation ${\rho}_{0}=0.8$ and ${\mu}_{i},{\sigma}_{i},i=1,\dots ,N$ i.i.d uniform on $[0.1,0.2]$.

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**MDPI and ACS Style**

Maheshwari, A.; Sarantsev, A. Modeling Financial System with Interbank Flows, Borrowing, and Investing. *Risks* **2018**, *6*, 131.
https://doi.org/10.3390/risks6040131

**AMA Style**

Maheshwari A, Sarantsev A. Modeling Financial System with Interbank Flows, Borrowing, and Investing. *Risks*. 2018; 6(4):131.
https://doi.org/10.3390/risks6040131

**Chicago/Turabian Style**

Maheshwari, Aditya, and Andrey Sarantsev. 2018. "Modeling Financial System with Interbank Flows, Borrowing, and Investing" *Risks* 6, no. 4: 131.
https://doi.org/10.3390/risks6040131