# The Role of Entropy in Estimating Financial Network Default Impact

## Abstract

**:**

## 1. Introduction

**L**

**L**and their corresponding shares of the grand totals are ${L}_{i}={l}_{i}/{\displaystyle \sum _{k}{l}_{k}}$ and ${A}_{i}={a}_{i}/{\displaystyle \sum _{k}{a}_{k}}$.

The merchant or trader who relied on credit lived constantly on the edge. The still relatively primitive state of communication, travel, and production meant that he could not be sure when he would receive the next shipment or the next payment on which his ability to pay his own creditors depended. His goal was to “synchronize the payments being made to him as a creditor with those he had to make as a debtor”, and this he could never do with complete assurance. As all merchants and traders who depended on credit existed in this state of financial instability, the insolvency of one person who owed significant debts could lead to the failure of many others.

**L**. This raises the issue of whether or not this estimator of unknown cells systematically biases estimation of the distributional impact index. That issue is investigated in Section 4, using simulations that failed to uncover a systematic bias. Section 5 provides a statistical rationale for this. Section 6 concludes.

## 2. The Proportional Payment Rule and the Entropic Index of Distributional Impact

**L**, but also on both the magnitudes and the distribution of the collateral, complicating our goal of understanding the relationships between the estimation of

**L**and the distributional impact of the default resolution process. That understanding is enhanced by assuming situations in which default is not a rare event, as it will be when assumed collateral is high enough. Readers interested in estimates for a particular financial network can easily modify the analysis herein to incorporate that network’s distribution of assignable collateral. So the proportional payment rule requires that the vector $\mathsf{\theta}$ satisfy the linear inequalities ${\theta}_{i}{\displaystyle \sum _{j\ne i}{L}_{ij}}-{\displaystyle \sum _{j\ne i}{\theta}_{j}}{L}_{ji}\le 0;\text{\hspace{0.17em}}i=1,\dots ,N$.

**L**.

## 3. The Entropy of the Liabilities Matrix

**L**. One first normalizes it by dividing each of its cells by the grand total of all cells, i.e., define ${P}_{ij}={L}_{ij}/{\displaystyle \sum _{i}{\displaystyle \sum _{j}{L}_{ij}}}$, and compute the Shannon entropy of the normalized matrix $H=-{\displaystyle \sum _{i}{\displaystyle \sum _{j}{P}_{ij}}}\mathrm{log}{P}_{ij}$. By adopting the convention $0\mathrm{log}0=0$, cells containing zeros, e.g., those along the diagonal (no agent owes anything to itself), contribute nothing to entropy. Hence we calculate $H=-{\displaystyle \sum _{ij;i\ne j}{P}_{ij}}\mathrm{log}{P}_{ij}=2.10$ using the data in (1).

**L**are unknown, but that its N row sums l

_{i}(total liabilities of each agent i) and column sums a

_{j}(total assets of each agent j) are known. This situation is faced by researchers with access to financial reports that list total liabilities and assets of agents without breaking out the bilateral specifics. A researcher could estimate values for the unknown cells by maximizing this entropy subject to the constraints that row and column sums have their observed values. That is

**L**are observed, we need only subtract them from their respective row and column totals, and then drop the corresponding probabilities from the estimation problem (6).

## 4. Will Entropic Estimation of L Bias Estimation of the Distributional Impact?

**L,**the lower the impact might be, but suppose instead that the more evenly-spread liabilities are larger than what the other agents can absorb without also defaulting. This suggests that the estimation procedure might lead to overestimates of the distributional impact.

## 5. Why Doesn’t Mutual Information Estimation Systematically Bias Estimates of Distributional Impact?

## 6. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## References

- Kadens, E. The last bankrupt hanged: Balancing incentives in the development of bankruptcy law. Duke Law J.
**2010**, 59, 1229–1319. [Google Scholar] - Elimam, A.; Girgis, M.; Kotab, S. The use of linear programming in disentangling the bankruptcies of Al-Manakh stock market crash. Oper. Res.
**1996**, 44, 665–676. [Google Scholar] [CrossRef] - Eisenberg, L.; Noe, T. Systemic risk in financial systems. Manag. Sci.
**2001**, 47, 236–249. [Google Scholar] [CrossRef] - Demange, G. Contagion in Financial Networks: A Threat Index. Manag. Sci.
**2016**, 64, 955–970. [Google Scholar] [CrossRef] - Cover, T.; Thomas, J. Elements of Information Theory; John Wiley: New York, NY, USA, 1991. [Google Scholar]
- Shore, J.E.; Johnson, R.W. Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy. IEEE Trans. Inf. Theory
**1980**, 26, 26–37. [Google Scholar] [CrossRef] - Golan, A.; Judge, G.; Robinson, S. Recovering information from incomplete or partial multisectoral economic data. Rev. Econ. Stat.
**1994**, 76, 541–549. [Google Scholar] [CrossRef] - Upper, C.; Worms, A. Estimating bilateral exposures in the German Interbank market: Is there a danger of contagion? Eur. Econ. Rev.
**2004**, 48, 827–849. [Google Scholar] [CrossRef] - Croux, C.; Dehon, C. Influence functions of the Spearman and Kendall correlation measures. Stat. Methods Appl.
**2010**, 19, 497–515. [Google Scholar] [CrossRef] - Chen, Y. Modeling maximum entropy distributions for financial returns by moment combination and selection. J. Financ. Econ.
**2015**, 13, 414–455. [Google Scholar] [CrossRef] - Hansen, L.; Jagannathan, R. Assessing specification errors in stochastic discount factor models. J. Financ.
**1997**, 52, 557–590. [Google Scholar] [CrossRef] - Ghosh, A.; Juillard, C.; Taylor, A. What is the Consumption-CAPM missing? An information-theoretic framework for the analysis of asset pricing models. Rev. Financ. Stud.
**2018**, 30, 442–504. [Google Scholar] [CrossRef] - Golan, A. Foundations of Info-Metrics: Modeling, Inference, and Imperfect Information; Oxford University Press: Oxford, UK, 2017. [Google Scholar]

**Figure 1.**

**L**matrices paired with minimum mutual information estimates: results of permutations of example (1).

**Figure 2.**

**L**matrices paired with minimum mutual information estimates: results of bootstrapping example (1).

**Figure 3.**Kendall’s τ is negatively related to the distributional impact index I: results of permutations of example (1).

**Figure 4.**Kendall’s τ is negatively related to the distributional impact index I: results of bootstrapping example (1).

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**MDPI and ACS Style**

Stutzer, M. The Role of Entropy in Estimating Financial Network Default Impact. *Entropy* **2018**, *20*, 369.
https://doi.org/10.3390/e20050369

**AMA Style**

Stutzer M. The Role of Entropy in Estimating Financial Network Default Impact. *Entropy*. 2018; 20(5):369.
https://doi.org/10.3390/e20050369

**Chicago/Turabian Style**

Stutzer, Michael. 2018. "The Role of Entropy in Estimating Financial Network Default Impact" *Entropy* 20, no. 5: 369.
https://doi.org/10.3390/e20050369