[Paper Review] Pricing of debt and equity in a financial network with comonotonic endowments

Tathagata Banerjee, Zachary Feinstein|arXiv (Cornell University)|Oct 2, 2018

Economic theories and models4 citations

TL;DR

This paper develops analytical pricing formulas for debt and equity in financial networks with comonotonic endowments, leveraging the Eisenberg-Noe framework to derive bounds on default risk. It shows that comonotonicity provides a lower bound for debt prices, while Jensen’s inequality yields an upper bound, offering a tractable alternative to computationally infeasible Monte Carlo methods in large systems with bankruptcy costs.

ABSTRACT

In this paper we present formulas for the valuation of debt and equity of firms in a financial network under comonotonic endowments. We demonstrate that the comonotonic setting provides a lower bound and Jensen's inequality provides an upper bound to the price of debt under Eisenberg-Noe financial networks with bankruptcy costs. Such financial networks encode the interconnection of firms through debt claims. The proposed pricing formulas consider the realized, endogenous, recovery rate on debt claims. Special consideration is given to the CAPM setting in which firms invest in correlated portfolios so as to provide analytical stress testing formulas. We endogenously construct the comonotonic endowment setting from a equity maximizing standpoint with capital transfers. We conclude by, numerically, comparing the network valuation problem with two single firm baseline heuristics which can, respectively, approximate the price of debt and equity.

Motivation & Objective

• To address the computational intractability of network valuation under general random endowments by introducing a comonotonic endowment setting.
• To provide analytical bounds—lower via comonotonicity, upper via Jensen’s inequality—for debt and equity prices in financial networks with bankruptcy costs.
• To endogenously construct comonotonic endowments from an equity-maximizing perspective using capital transfers.
• To compare network-based valuations against single-firm heuristics, demonstrating the necessity of systemic risk pricing.
• To develop a Network Valuation Adjustment (NVA) framework that accounts for contagion effects absent in traditional CVA models.

Proposed method

• Formulates a financial network model using the Eisenberg-Noe clearing mechanism with endogenous recovery rates on debt claims.
• Introduces comonotonic endowments as a tractable proxy for general random endowments, enabling analytical tractability.
• Derives expected equilibrium payments, equity, and wealth under comonotonicity using fixed-point solutions to the clearing equations.
• Applies Jensen’s inequality to the comonotonic setting to derive an upper bound for debt prices in the general random endowment case.
• Uses a lognormal setting for comparative statics to align with Merton (1974) and validate results.
• Employs a two-step heuristic comparison: one approximating debt pricing, another equity pricing, both relative to the full network model.

Experimental results

Research questions

• RQ1 Can comonotonic endowments provide a lower bound for debt prices in a financial network with bankruptcy costs?
• RQ2 Does Jensen’s inequality yield a valid upper bound for debt prices when endowments are not comonotonic?
• RQ3 How do network-based valuations compare to single-firm heuristics in approximating debt and equity prices?
• RQ4 Can comonotonic endowments be endogenously constructed from an equity-maximizing capital transfer mechanism?
• RQ5 To what extent does the network structure affect the pricing of debt and equity beyond individual firm fundamentals?

Key findings

• Comonotonic endowments yield a lower bound for the expected price of debt in financial networks with bankruptcy costs.
• Jensen’s inequality provides an upper bound for the expected debt price under general random endowments, based on the comonotonic approximation.
• The comonotonic endowment setting is endogenously constructed via capital transfers that maximize equity, ensuring consistency with financial incentives.
• Numerical comparisons show that single-firm heuristics systematically underprice or overprice network claims, highlighting the need for systemic risk adjustment.
• The proposed NVA framework avoids the curse of dimensionality inherent in full scenario enumeration ($2^n$) by using comonotonicity to enable analytical solutions.
• Comparative statics under lognormal endowments confirm that network effects significantly alter debt and equity valuations relative to Merton (1974) benchmarks.