sheet, all firms are assumed to begin with some amount of external assets $x_i \geq 0$ for all firms $i \in {\mathcal{N}}_0$. The resultant *clearing payments*, under a no priority of payments assumption, satisfy the fixed point problem in payments $p \in [0,\bar p]$ $$\label{eq:EN-p}
p = \bar p \wedge \left(x + \Pi^{\top}p\right).$$ That is, each bank pays the minimum of what it owes ($\bar p_i$) and what it has ($x_i + \sum_{j \in {\mathcal{N}}} \pi_{ji} p_j$). The resultant vector of *wealths* for all firms is given by $$\label{eq:equity}
V = x + \Pi^{\top}p - \bar p.$$ Noting that payments can be written as a simple function of the wealths ($p = \bar p - V^-$), we provide the following proposition. We refer also to [@veraart2017distress; @barucca2016valuation; @banerjee2017insurance] for similar notions of utilizing clearing wealth instead of clearing payments.

\[prop:EN-e\] A vector $p \in [0,\bar p]$ is a clearing payments in the Eisenberg-Noe setting if and only if $p = [\bar p - V^-]^+$ for some $V \in {\mathbb{R}}^{n+1}$ satisfying the following fixed point problem $$\label{eq:EN-e}
V = x + \Pi^{\top}[\bar p - V^-]^+ - \bar p.$$ Vice versa, a vector $V \in {\mathbb{R}}^{n+1}$ is a clearing wealths (i.e., satisfying 