# Diversification and Systemic Risk: A Financial Network Perspective

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## Abstract

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## 1. Introduction

Note that Greenspan explicitly entertains the idea that the default of any given financial institution may result in “cascading failures” of other banks via spillover effects in a network of direct credit relationships. In the presence of spillover effects, reducing idiosyncratic risk concentrations may thus be beneficial as it reduces the likelihood that individual banks default in the first place.[In the past year] I, particularly, have been focusing on innovations in the management of risk and some of the implications of those innovations for our global economic and financial system. The development of our paradigms for containing risk has emphasized dispersion of risk to those willing, and presumably able, to bear it. If risk is properly dispersed, shocks to the overall economic systems will be better absorbed and less likely to create cascading failures that could threaten financial stability.

#### Relation to the Literature on Systemic Risk

## 2. Model and Methodology

#### 2.1. The Model

#### 2.1.1. The Financial Network

**Assumption**

**A1.**

- (i)
- All loans in the system are of the same size, normalized to one.
- (ii)
- For every bank k in the network, the ratio ${E}_{k}/{A}_{k}$ (the ratio of equity over total assets) is equal to an exogenously given constant $\gamma <1$.
- (iii)
- For every bank k in the network, the ratio ${A}_{k}^{\mathrm{ib}}/{A}_{k}$ (the ratio of interbank assets to total assets) is equal to an exogenously given constant $\kappa <1$.3

#### 2.1.2. Initial Defaults

#### 2.1.3. Diversification

#### 2.2. Spillover Effects

- Perturb the external assets ${A}_{k}^{\mathrm{ex}}$ of each bank k by the return realization ${r}_{k}$, that is let ${A}_{k}^{\mathrm{ex}}\left(\mathrm{new}\right)={A}_{k}^{\mathrm{ex}}\left(\mathrm{old}\right)(1+{r}_{k})$.
- If any of the banks defaults, propagate the shock to the asset side of its creditors. The new amount of interbank assets satisfies: ${A}_{k}^{\mathrm{ib}}\left(\mathrm{new}\right)={A}_{k}^{\mathrm{ib}}\left(\mathrm{old}\right)-{\sum}_{i}{e}_{ik}{1}_{\left\{i\mathrm{is}\mathrm{in}\mathrm{default}\right\}}$
- If the total value of bank k’s assets falls below its liabilities, that is ${A}_{k}\left(\mathrm{new}\right)<{L}_{k}$, bank k defaults.
- Repeat Steps 2 and 3 until there is no further default.

#### 2.3. The Network-Generation Process

#### 2.3.1. Homogeneous (Erdos–Renyi) Random Graphs

#### 2.3.2. Inhomogeneous (Core–Periphery) Random Graphs

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Financial Network

- Assign the value of interbank assets ${A}_{k}^{\mathrm{ib}}$ and liabilities ${L}_{k}^{\mathrm{ib}}$ of every bank in the network according to Equation (A1).

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1 | For empirical details and real time estimates of the SRISK measure, we refer to the work of the VLAB run by Robert Engle at NYU Stern School of Business, see https://vlab.stern.nyu.edu/. |

2 | Throughout the paper, we use the terms graph and network. When talking about a graph, we are concerned with the structure of financial linkages, whereas, when referring to a network, we mean not just connections themselves but also balance sheet quantities of individual banks. |

3 | This assumption needs to be refined slightly to avoid certain extreme cases where ${L}^{\mathrm{ex}}$ becomes negative; see Appendix A for details. |

4 | Note that we assume that the asset returns of banks are diversified from the outset; potential problems with securitization and credit risk transfer are not the focus of our paper. |

5 | Thanks to this assumption, there is no need for a settlement algorithm in the spirit of Eisenberg and Noe (2001). Assuming non-zero recovery rate on distressed loans would however not change the overall quantitative nature of our results. |

6 | There are just a few countries where regulators possess reasonably good data on the structure of the interbank market, for example Austria, Mexico, Germany or Brazil. |

7 | These parameter values are typical for the Austrian interbank market. |

8 | A regulator with access to real data could apply our approach to an actual interbank network. In that situation, one would use some centrality measure to classify a subset of important banks as hubs that form the core of the network. This immediately yields ${p}_{\mathrm{core}}$, and the other parameters are then easy to estimate. |

9 | In graph theoretic literature, this is known as the average graph degree. |

10 | In a special case where a bank has no connections at all, we assume that it puts a portion $\kappa $ of its total assets into some riskless investment (such as government bonds) and 1-$\kappa $ into risky external assets. We also assume that the size of the balance sheet ${A}_{k}$ is 1. We make this technical assumption to prevent banks from having their balance sheet size equal to zero. Note that this assumption has no effect on the analysis itself since a bank with no connections does not affect the rest of system. |

**Figure 1.**One realization of a random graph for $N=100$ banks: (

**a**) Erdos–Renyi network; and (

**b**) core–periphery network.

**Figure 2.**Probability of a systemic crisis for both network structures on $N=100$ nodes as a function of diversification $\beta $ for particular levels of connectivity C. Bank equity ratio $\gamma =0.35$, integration $\kappa =0.2$ and correlation of project returns $\rho =0.5$ (see Equation (3)) .

**Figure 3.**Probability of a systemic crisis for both network structures on $N=100$ nodes as a function of connectivity C for particular levels of diversification $\beta $. Bank equity ratio $\gamma =0.035$, integration $\kappa =0.2$ and correlation of project returns $\rho =0.5$ (see (3)).

Assets | Liabilities |
---|---|

interbank assets ${A}_{k}^{\mathrm{ib}}$ | interbank liabilities ${L}_{k}^{\mathrm{ib}}$ |

external assets ${A}_{k}^{\mathrm{ex}}$ | external liabilities ${L}_{k}^{\mathrm{ex}}$ |

equity ${E}_{k}$ | |

total assets ${A}_{k}$ | total liabilities ${L}_{k}$ |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Frey, R.; Hledik, J. Diversification and Systemic Risk: A Financial Network Perspective. *Risks* **2018**, *6*, 54.
https://doi.org/10.3390/risks6020054

**AMA Style**

Frey R, Hledik J. Diversification and Systemic Risk: A Financial Network Perspective. *Risks*. 2018; 6(2):54.
https://doi.org/10.3390/risks6020054

**Chicago/Turabian Style**

Frey, Rüdiger, and Juraj Hledik. 2018. "Diversification and Systemic Risk: A Financial Network Perspective" *Risks* 6, no. 2: 54.
https://doi.org/10.3390/risks6020054