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Incentive-Aware Models of Financial Networks

Akhil Jalan, Deepayan Chakrabarti, Purnamrita Sarkar

TL;DR

We study an endogenous, belief-driven framework for weighted financial networks in which contract sizes $W_{ij}$ arise from heterogeneous agents maximizing mean-variance utilities. The main result is that a unique stable network exists almost surely, with edge weights depending on all agents’ beliefs; the network can be reached via iterative pairwise negotiations and responds to changes in beliefs, with regulatory interventions having potentially far-reaching ripple effects. The model yields insights for regulators (limited ability to pinpoint belief changes) and firms (outlier detection within communities and limitations in interpreting risk-versus-return shifts). Practically, the framework highlights how seemingly minor information can trigger large network reconfigurations and how belief updates propagate endogenously through the contract network. The appendix provides rigorous proofs and experiments showing convergence, identifiability limits, and empirical validations on macro-financial datasets.

Abstract

Financial networks help firms manage risk but also enable financial shocks to spread. Despite their importance, existing models of financial networks have several limitations. Prior works often consider a static network with a simple structure (e.g., a ring) or a model that assumes conditional independence between edges. We propose a new model where the network emerges from interactions between heterogeneous utility-maximizing firms. Edges correspond to contract agreements between pairs of firms, with the contract size being the edge weight. We show that, almost always, there is a unique "stable network." All edge weights in this stable network depend on all firms' beliefs. Furthermore, firms can find the stable network via iterative pairwise negotiations. When beliefs change, the stable network changes. We show that under realistic settings, a regulator cannot pin down the changed beliefs that caused the network changes. Also, each firm can use its view of the network to inform its beliefs. For instance, it can detect outlier firms whose beliefs deviate from their peers. But it cannot identify the deviant belief: increased risk-seeking is indistinguishable from increased expected profits. Seemingly minor news may settle the dilemma, triggering significant changes in the network.

Incentive-Aware Models of Financial Networks

TL;DR

We study an endogenous, belief-driven framework for weighted financial networks in which contract sizes arise from heterogeneous agents maximizing mean-variance utilities. The main result is that a unique stable network exists almost surely, with edge weights depending on all agents’ beliefs; the network can be reached via iterative pairwise negotiations and responds to changes in beliefs, with regulatory interventions having potentially far-reaching ripple effects. The model yields insights for regulators (limited ability to pinpoint belief changes) and firms (outlier detection within communities and limitations in interpreting risk-versus-return shifts). Practically, the framework highlights how seemingly minor information can trigger large network reconfigurations and how belief updates propagate endogenously through the contract network. The appendix provides rigorous proofs and experiments showing convergence, identifiability limits, and empirical validations on macro-financial datasets.

Abstract

Financial networks help firms manage risk but also enable financial shocks to spread. Despite their importance, existing models of financial networks have several limitations. Prior works often consider a static network with a simple structure (e.g., a ring) or a model that assumes conditional independence between edges. We propose a new model where the network emerges from interactions between heterogeneous utility-maximizing firms. Edges correspond to contract agreements between pairs of firms, with the contract size being the edge weight. We show that, almost always, there is a unique "stable network." All edge weights in this stable network depend on all firms' beliefs. Furthermore, firms can find the stable network via iterative pairwise negotiations. When beliefs change, the stable network changes. We show that under realistic settings, a regulator cannot pin down the changed beliefs that caused the network changes. Also, each firm can use its view of the network to inform its beliefs. For instance, it can detect outlier firms whose beliefs deviate from their peers. But it cannot identify the deviant belief: increased risk-seeking is indistinguishable from increased expected profits. Seemingly minor news may settle the dilemma, triggering significant changes in the network.
Paper Structure (37 sections, 33 theorems, 60 equations, 6 figures, 1 algorithm)

This paper contains 37 sections, 33 theorems, 60 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. The Proposed Model
  4. Characterizing Stable Points
  5. Finding the Stable Point via Pairwise Negotiations
  6. Pairwise Negotiations under Random Covariances
  7. Inferring Beliefs from the Network Structure
  8. Non-identifiability of beliefs.
  9. Insights for Regulators
  10. Effect of Friction in Contract Formation
  11. Effect of Changes in Firms' Beliefs
  12. Insights for Firms
  13. Detecting Outlier Firms
  14. Risk-Aversion versus Expected Returns
  15. Conclusions
  16. ...and 22 more sections

Key Result

Theorem 1

Define $n\times n$ matrices $A$, $B_{(i,j)}$, and $C_{(i,j)}$ as follows: Let $Z_F$ be the $|F|\times |F|$ matrix whose rows are the ordered sets $\{\textrm{uvec}(C_{(i,j)})_F \mid (i,j)\in F\}$. A stable point exists under $\{\Psi_i\}$ if and only if $\textrm{uvec}(A-A^T)_F$ lies in the column space of $Z_F$.

Figures (6)

  • Figure 1: Example of a stable point for a borrower (Firm 1) and a lender (Firm 2): (a) When the borrower cannot pay the lender an additional payment, the firms may be unable to agree to a contract, even if trading improves their utilities. (b) By allowing for contract-specific payments, both firms can agree on a contract size. In effect, the borrower (Firm 2) shares its utility with the lender (Firm 1) to achieve agreement. (c) The stable network is shown.
  • Figure 2: Pairwise negotiations for the setting of Example \ref{['example:pairwise']}: The contracts matrix $W_t$ and payments matrix $P_t$ after $t = 0, 1, 2$ steps of Algorithm \ref{['alg:pairwise']} ($\eta=0.5$) converge quickly to the stable point $(W, P) = (W_\infty, P_\infty)$. Cells corresponding to forbidden edges are empty.
  • Figure 3: Source Detection Problem in a noisy scaled equi-correlation model of $\Sigma$: We rank the entries of $W$ by the magnitude of change induced by a change in one entry of $M$ ($M_{ij}$). Plot (a) shows the fraction of times $W_{ij}$ is most-changed entry of $W$. Plot (b) shows the fraction of times $W_{ij}$ is among the top-$10$ most changed entries of $W$. The success rate goes to zero as $\alpha$ and $\epsilon$ increase.
  • Figure 4: Source Detection Problem on real-world data: The success rate scales monotonically with the number of samples used to construct the data-driven covariance matrix $\hat{\Sigma}$.
  • Figure 5: The eigenvalues of estimated covariance matrices are skewed, and the degree of skew depends on the number of samples $m$. As $m$ decreases, so does the smallest eigenvalue $\lambda_n$ and the ratio $\lambda_n/\lambda_{n-1}$.
  • ...and 1 more figures

Theorems & Definitions (56)

  • Example 1: Loan contract
  • Example 2: Interest rate swap contract
  • Example 3: Insurance Contract
  • Definition 1: Network Setting
  • Definition 2: Feasibility
  • Definition 3: Stable point
  • Example 4
  • Theorem 1: Existence and Uniqueness of Stable Point
  • Theorem 2: Stable points cannot be dominated
  • Definition 4: Agent Action
  • ...and 46 more