Chapter 3 Clearing Payment Vector
3.1 Basic Definition
Consider a financial system consists of \(n\) financial nodes. For any nodes pair \((i, j)\), there is a nominally non negative liability. The mutual liabilities weave into a network, represented by a \(n\times n\) nominal liability matrix \(\mathbf{L}\equiv \{L_{ij}\}\), where \(L_{ij}\geq 0\) is the nominal amount that node \(i\) owes to node \(j\). Besides, node \(i\) has a operating cash flow, denoted as \(e_i\), from outside of the financial system. Without loss of generality, we assume that every node have a non negative operating cash flow, i.e. \(e_i \geq 0\). The operation cash flow vector is \(\boldsymbol{e}\equiv [e_1\ e_2\ \cdots\ e_n]'\).
We may care about whether node \(i\) can fulfill its’ obligation, which requires the information on total debt of each node. We use \(\bar{p}_i\equiv \sum\limits_{j=1}^n L_{ij}\) to represent the total amount that node \(i\) owes to the remaining part of the system, and use \(\bar{\boldsymbol{p}}\equiv \left[\bar{p}_1\ \bar{p}_2\ \cdots\ \bar{p}_n\right]'\) to represent total obligation vector of the system.
Standardizing the nominal liability matrix \(\mathbf{L}\) by row obtains the relative liabilities matrix \(\mathbf{\Pi}\), which takes the form \[ \mathbf{\Pi} = \begin{bmatrix} 0 & \Pi_{12} & \cdots & \Pi_{1n} \\ \Pi_{21} & 0 & \cdots & \Pi_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ \Pi_{n1} & \Pi_{n2} & \cdots & 0 \end{bmatrix},\ \Pi_{ij} = \begin{cases} L_{ij} / \bar{p}_i & \mbox{if } \bar{p}_i > 0\\ 0 & \mbox{otherwise} \end{cases}. \] The row sum of relative liabilities matrix is 1, i.e. \(\mathbf{\Pi}\boldsymbol{1} = \boldsymbol{1}\). An interconnected financial system consisting of \(n\) firms can be characterized by a tuple \((\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\).
Whether node \(i\) can achieve complete satisfaction hinges on the difference between its’ total cash flow received and payment. Provided that the payment from other nodes to node \(i\) are \(p_j = \bar{p}_j, j\neq i\), then the total payment that node \(i\) receives is \(\sum_{i=1}^n \Pi_{ji} \bar{p}_j + e_i\). If \(\bar{p}_i \leq \sum_{i=1}^n \Pi_{ji} \bar{p}_j + e_i\) holds for all \(i\), and \(\bar{\boldsymbol{p}}\) will be the equilibrium payment vector. However it may be case that some nodes can not fulfill its’ nominal obligation completely so that \(\bar{\boldsymbol{p}}\) can not necessarily clear the system. To see this, suppose \(\bar{p}_i > \sum_{i=1}^n \Pi_{ji} \bar{p}_j + e_i\), then node \(i\) must liquidate all asset to compensate its’ creditors in proportion to their nominal claims. The insolvency of node \(i\) shrinks the \(\bar{p}_i\rightarrow p_i < \bar{p}_i\). The story continues, the liquidation of node \(i\) decreases the payment received by the nodes that hold claim to \(i\), e.g. node \(i+1\), so \(i+1\) may fall into insolvency too, and some elements of payment vector will further shrink.
\[ \begin{bmatrix} \bar{p}_1\\ \bar{p}_2\\ \cdots\\ \bar{p}_{i-1}\\ \bar{p}_i\\ \bar{p}_{i+1}\\ \cdots\\ \bar{p}_n \end{bmatrix} \overset{\bar{p}_i > \sum\limits_{j=1}^n \Pi_{ji} \bar{p}_j + e_i}{\xrightarrow{\hspace{1cm}}} \begin{bmatrix} \bar{p}_1\\ \bar{p}_2\\ \cdots\\ \bar{p}_{i-1}\\ \color{red}{p_i}\\ \bar{p}_{i+1}\\ \cdots\\ \bar{p}_n \end{bmatrix} \overset{\bar{p}_{i+1} > \sum\limits_{j=1}^n \Pi_{j,i+1} \bar{p}_j + e_{i+1}}{\xrightarrow{\hspace{1cm}}} \begin{bmatrix} \bar{p}_1\\ \bar{p}_2\\ \cdots\\ \bar{p}_{i-1}\\ \color{red}{p_i}\\ \color{red}{p_{i+1}}\\ \cdots\\ \bar{p}_n \end{bmatrix} \longrightarrow \vdots \longrightarrow \boldsymbol{p}^*. \]
All stories must have an end. It seems that there exits a payment vector to stop the successive default process. The default process described continues until three criteria are satisfied: (1) limited liability, i.e. the payment of any node can not exceed the total cash flow, (2) priority of debt claim, i.e. stockholders of a node receive no value until the node is able to completely pay off all of its liabilities, (3) proportionality, i.e. if default occurs, all claimant nodes are paid by the defaulting node in proportion to the size of their nominal claim on firm assets. Based on the three criteria, Eisenberg and Noe (2001) declared a key definition: clearing payment vector.
Definition 3.1 (Clearing Payment Vector) A clearing payment vector for the financial system \((\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\) is a vector \(\boldsymbol{p}^* \in [\boldsymbol{0}, \bar{\boldsymbol{p}}]\) that satisfies the following conditions:
- Limited Liability. \(\forall i \in \mathcal{N}\), \(p_i^* \leq \sum\limits_{i=1}^n \Pi_{ji} p_j^* + e_i\),
- Absolute Priority. \(\forall i \in \mathcal{N}\), either obligations are paid in full, \(p_i^* = \bar{p}_i\), or all asset is payed to creditors, that is \(p_i^* = \sum\limits_{j=1}^n \Pi_{ji} p^*_j + e_i\)
Note that clearing payment vector \(\boldsymbol{p}^*\) is the equilibrium payment vector for a certain financial system \((\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\), since it clear all obligations and pin down the states (survive or default) of all nodes.
3.2 Existence and Uniqueness of Clearing Payment Vector
Is the existence of clearing payment vector an illusion? If it indeed exits, is it unique? Or under what conditions it can be unique? To answer the question, we need to further formalize the definition of clearing payment vector.
3.2.1 Existence
By Definition 3.1, the clearing payment vector for the financial system \((\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\) can be rewritten as \[ \boldsymbol{p}^* = \begin{bmatrix} p^*_1 \\ p^*_2 \\ \vdots \\ p^*_n \end{bmatrix} = \begin{bmatrix} \min \left(\sum\limits_{i=1}^n \Pi_{j1} p_j^* + e_1,\ \bar{p}_1\right) \\ \min \left(\sum\limits_{i=1}^n \Pi_{j2} p_j^* + e_2,\ \bar{p}_2\right) \\ \vdots \\ \min \left(\sum\limits_{i=1}^n \Pi_{jn} p_j^* + e_n,\ \bar{p}_n\right) \end{bmatrix} = \left(\mathbf{\Pi}^\intercal \boldsymbol{p}^* + \boldsymbol{e}\right) \wedge \bar{\boldsymbol{p}}, \]
where \(\boldsymbol{x}\wedge \boldsymbol{y} = [\min(x_1, y_1)\ \cdots\ \min (x_n, y_n)]'\). The clearing payment vector is the fixed point of the mapping \(\Phi(\cdot|\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e}):[\mathbf{0}, \bar{\boldsymbol{p}}] \mapsto [\mathbf{0}, \bar{\boldsymbol{p}}]\) ! So we have the following equivalent definition of clearing payment vector.
Definition 3.2 (Another Equivalent Definiton of Clearing Payment Vector) For a financial system \((\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\), a payment vector \(\boldsymbol{p}^*\) is a clearing payment vector if and only if \(\boldsymbol{p}^*\) is a fixed point of the mapping \(\Phi(\cdot|\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e}):[\mathbf{0}, \bar{\boldsymbol{p}}] \mapsto [\mathbf{0}, \bar{\boldsymbol{p}}]\), that is \(\boldsymbol{p}^* = \left(\mathbf{\Pi}^\intercal \boldsymbol{p}^* + \boldsymbol{e}\right) \wedge \bar{\boldsymbol{p}}\).
Eisenberg and Noe (2001) resorted to Tarski’s fixed point theorem to establish the existence of \(\boldsymbol{p}^*\). Tarski’s fixed point theorem requires that the domain of mapping \(\Phi(\cdot)\) to be a complete lattice and the mapping itself to be monotone increasing. First, \([\mathbf{0}, \bar{\boldsymbol{p}}]\) being a complete lattice is obvious. As for \(\Phi(\cdot)\), Eisenberg and Noe (2001) provided some useful propositions.
Proposition 3.1 The mapping \(\Phi(\cdot|\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e}):[\mathbf{0}, \bar{\boldsymbol{p}}] \mapsto [\mathbf{0}, \bar{\boldsymbol{p}}]\) is positive, monotone increasing, concave and nonexpansive.
Theorem 3.1 (Existence of Clearing Payment Vector) For a financial system \((\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\), there exist at least a greatest and smallest clearing payment vector, \(\boldsymbol{p}^{*+}\) and \(\boldsymbol{p}^{*-}\).
Proof. Since \([\mathbf{0}, \bar{\boldsymbol{p}}]\) is a complete lattice, and the mapping \(\Phi(\cdot|\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\) is monotone increasing. By Tarski’s fixed point theorem, there exists a greatest and smallest fixed point in \([\mathbf{0}, \bar{\boldsymbol{p}}]\).
Theorem 3.2 Under all clearing vectors, the value of the equity at each node of the financial system is the same, that is, if \(p_1^*\) and \(p_2^*\) are any two clearing vectors, then \[ \left(\boldsymbol{\Pi}^\intercal \boldsymbol{p}_1^* + \boldsymbol{e} - \bar{\boldsymbol{p}}\right)^+ = \left(\boldsymbol{\Pi}^\intercal \boldsymbol{p}_2^* + \boldsymbol{e} - \bar{\boldsymbol{p}}\right)^+. \] where \(\boldsymbol{x}^+=[\max(x_1, 0)\ \cdots\ \max(x_n, 0)]^\intercal\).
Proof. First note that \(\left(\boldsymbol{\Pi}^\intercal \boldsymbol{p}^* + \boldsymbol{e} - \bar{\boldsymbol{p}}\right)^+ = \mathbf{\Pi}^{\intercal} \boldsymbol{p}^* + \boldsymbol{e} - \boldsymbol{p}^*\) since \((\boldsymbol{x} - \boldsymbol{y})^+ = \boldsymbol{x}\vee \boldsymbol{y} - \boldsymbol{y} = \boldsymbol{x} - \boldsymbol{x}\wedge \boldsymbol{y}\). By \(\mathbf{\Pi}\mathbf{1} = \mathbf{1}\), the sum of equity value of all nodes under any clearing payment vector \(\boldsymbol{p}^*\) is \[ \mathbf{1}^\intercal\left(\mathbf{\Pi}^{\intercal} \boldsymbol{p}^* + \boldsymbol{e} - \boldsymbol{p}^*\right) = \mathbf{1}^\intercal \boldsymbol{p}^* + \mathbf{1}^\intercal \boldsymbol{e} - \mathbf{1}^\intercal \boldsymbol{p}^* = \sum_{i=1}^n e_i. \] Consider the smallest clearing payment vector \(\boldsymbol{p}_-^*\) and another clearing payment vector \(\boldsymbol{p}^*\) such that the equity value vectors are different. Since the equity value function \(f(\boldsymbol{p}^*) = \left(\boldsymbol{\Pi}^\intercal \boldsymbol{p}^* + \boldsymbol{e} - \bar{\boldsymbol{p}}\right)^+\) is monotone increasing and \(\boldsymbol{p}^* > \boldsymbol{p}^{*-}\), we have \(\left(\boldsymbol{\Pi}^\intercal \boldsymbol{p}^* + \boldsymbol{e} - \bar{\boldsymbol{p}}\right)^+ > \left(\boldsymbol{\Pi}^\intercal \boldsymbol{p}^{*-} + \boldsymbol{e} - \bar{\boldsymbol{p}}\right)^+ \geq \mathbf{0}\). This contradicts the fact that the sums of equity value of all nodes under different payment vectors are the same.
The intuition behind Theorem 3.2 is that there is no payment arrangement that can increase the equity value of any node. Since the only origin of positive equity value is the operation cash flow \(\boldsymbol{e}>\mathbf{0}\), of which the value is neither destroyed nor created once it enters the financial system. If a payment arrangement endows node \(i\) higher equity value, it must be the case that another node in the system suffers loss. This is analogous to the law of conservation of energy. Tarski’s fixed point theorem ensures the existence of a smallest clearing payment vector and thus of the smallest equity value vector, a larger clearing payment vector increasing the equity value of all nodes at the same time will violate the conservation law.
3.2.2 Regularity and Uniqueness
Definition 3.3 (Surplus Set) A set \(\mathcal{S} \subset \mathcal{N}\) is a surplus set if no node in the set has any obligations to any node outside the set and the set has positive operating cash flows, that is, if \((i,j) \in \mathcal{S} \times \mathcal{S}^c, \Pi_{ij} = 0\) and \(\sum_{i\in \mathcal{S}} e_i >0\).
Lemma 3.1 If \(\boldsymbol{p}^*\) is a clearing vector, then it is not possible for all nodes in a surplus set to have zero equity value.
Proof. Suppose \(\boldsymbol{p}^*\) is a clearing payment vector. If all nodes \(i \in \mathcal{S}\) have zero equity value, the it must be the case that the amount they receive equals that they pay, that is \[ \begin{aligned} p_i^* & = \sum_{j=1}^n \Pi_{ji} p_j^* + e_i \\ & = \sum_{j\in \mathcal{S}^c} \Pi_{ji} p_j^* + \sum_{j\in \mathcal{S}} \Pi_{ji} p_j^* + e_i. \\ \end{aligned} \] Summing the both sides of the equation, we obtain that \[ \begin{aligned} \sum_{i\in \mathcal{S}} p_i^* & = \sum_{i\in \mathcal{S}}\sum_{j\in \mathcal{S}^c} \Pi_{ji} p_j^* + \sum_{i\in \mathcal{S}}\sum_{j\in \mathcal{S}} \Pi_{ji} p_j^* + \sum_{i\in \mathcal{S}} e_i \\ & = \sum_{j\in \mathcal{S}^c} \sum_{i\in\mathcal{S}} \Pi_{ji} p_j^* + \sum_{j\in \mathcal{S}} \left(\sum_{i\in \mathcal{S}} \Pi_{ji} p_j^*\right) + \sum_{i\in \mathcal{S}} e_i \\ & = \sum_{j\in \mathcal{S}^c} \sum_{i\in\mathcal{S}} \Pi_{ji} p_j^* + \sum_{j\in \mathcal{S}} p_j^* + \sum_{i\in \mathcal{S}} e_i, \end{aligned} \\ \Longrightarrow 0 = \sum_{j\in \mathcal{S}^c} \sum_{i\in\mathcal{S}} \Pi_{ji} p_j^* + \sum_{i\in \mathcal{S}} e_i. \] The first term on the RHS of the last equation can not be negative since it is the sum of amount payed to the nodes in \(\mathcal{S}\) by the nodes in \(\mathcal{S}^c\). Therefore, \(\sum_{i\in \mathcal{S}} e_i = -\sum_{j\in \mathcal{S}^c} \sum_{i\in\mathcal{S}} \Pi_{ji} p_j^* \leq 0\), which contradict \(\sum_{i\in \mathcal{S}} e_i > 0\).
Definition 3.4 (Risk Orbit) For each node \(i \in \mathcal{N}\), define the risk orbit of node \(i\), denoted by \(o(i)\), as follows: \[ o(i) = \{j \in \mathcal{N}: \mbox{there exists a directed path from } i \mbox{ to } j \}. \]
Lemma 3.2 Suppose that \(\boldsymbol{p}^*\) is a clearing vector for \((\boldsymbol{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\). Let \(o(i)\) be a risk orbit that satisfies \(\sum_{j\in o(i)} e_j > 0\). Then, under \(p^*\), at least one node of i has positive equity value, that is \[ \exists j \in o(i), \left(\boldsymbol{\Pi}^T \boldsymbol{p}^* - \boldsymbol{e}\right)_j - \bar{p}_j > 0. \]
Proof. A risk orbit with the condition \(\sum_{j\in o(i)} e_j >0\) is also a surplus set. Thus by Lemma 3.1, \(o(i)\) must contain a node with positive equity value.
Definition 3.5 (Regularity) A financial system is regular if every risk orbit, \(o(i)\), is a surplus set.
Theorem 3.3 (Uniqueness of Clearing Payment Vector) If the financial system is regular, the greatest and least clearing vectors are the same, i.e., \(p_+^*\) = \(p_-^*\), implying that the clearing vector is unique.
Proof. We prove this theorem by justifying \(\boldsymbol{p}^{*+} = \boldsymbol{p}^{*-}\).
Denote the equity value of node \(i\) under \(\boldsymbol{p}^{*+}\) and \(\boldsymbol{p}^{*-}\) as \(E_i^+\) and \(E_i^-\) respectively. By Theorem 3.2, the equity value vectors under \(\boldsymbol{p}^{*+}\) and \(\boldsymbol{p}^{*-}\) are the same, \(E_i^+ = E_i^-\). For the nodes that have positive equity value, it must be the case that they just pay their nominal obligation, that is \(p_i^{*+} = p_i^{*-} = \bar{p}_i\), \(p_i^{*+} > p_i^{*-}\) only occurs for the nodes with zero equity value.
By Definition 3.4 and Lemma 3.2, for a zero equity node \(i\), the risk orbit \(o(i)\) must contain such a path \[ i=\underbrace{i_0 \rightarrow i_1 \rightarrow \cdots \rightarrow i_{l-1}}_{\mbox{zero equity}} \rightarrow \underbrace{i_l}_{{\mathrm{positive}\\ \mathrm{equity}}} \] where \(i_k \in o(i), k = 0, 1,\cdots, l\), \(l \geq 1\) and \(i_0, i_1, \cdots, i_{l-1}\) have zero equity value and \(i_l\) has positive equity value.
Now we would like to show that \(E_l^+ > E_l^-\), which results in contradiction. For any zero equity value node \(i_k\), the payment equals their cash inflows under both \(\boldsymbol{p}^{*+}\) and \(\boldsymbol{p}^{*-}\), that is \[ p_{i_k}^{*+} = \sum_{j=1}^n \Pi_{ji_k} p_j^{*+} + e_{i_k},\ p_{i_k}^{*-} = \sum_{j=1}^n \Pi_{ji_k} p_j^{*-} + e_{i_k}. \] Suppose \(p_{i_{k-1}}^{*+} > p_{i_{k-1}}^{*-}\) holds, then the difference between \(p_{i_k}^+\) and \(p_{i_k}^-\) is \[ \begin{aligned} p_{i_k}^{*+} - p_{i_k}^{*-} & = \sum_{j=1}^n \Pi_{ji_k} \left(p_j^{*+} - p_j^{*-}\right) \\ & = \sum_{j\neq i_{k-1}} \Pi_{ji_k} \left(p_j^{*+} - p_j^{*-}\right) + \Pi_{i_{k-1}i_k} \left(p_{i_{k-1}}^{*+} - p_{i_{k-1}}^{*-}\right) \\ & > 0. \end{aligned} \] We have presumed that \(p_{i_0}^{*+} > p_{i_0}^{*+}\), then by mathematical induction, we can include that \(p_{i_k}^{*+} > p_{i_k}^{*+}\) for all \(k \leq l-1\).
Now let’s see node \(i_l\), of which the equity value is positive. The difference between \(E_k^+\) and \(E_k^-\) is \[ \begin{aligned} E_{i_l}^+ - E_{i_l}^- & = \sum_{j=1}^n \Pi_{ji_l} \left(p_j^{*+} - p_j^{*-}\right)- \left(p_{i_l}^+ - p_{i_l}^-\right) \\ & = \sum_{j\neq i_{l-1}} \Pi_{ji_l} \left(p_j^{*+} - p_j^{*-}\right) + \Pi_{i_{l-1}i_l} \left(p_{i_{l-1}}^{*+} - p_{i_{l-1}}^{*-}\right) \\ & > 0. \end{aligned} \] We reach the contradiction between \(E_{i_l}^+ > E_{i_l}^-\) and \(E_{i_l}^+ = E_{i_l}^-\).
3.3 Clearing Payment Vector Searching Algorithm
3.3.1 Fictitious Default Algorithm
How to find \(\boldsymbol{p}^*\) ? Obviously it is difficult to determine an analytic solution to the equation \(\boldsymbol{p} = \Phi(\boldsymbol{p}|\mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e})\). Eisenberg and Noe (2001) proposed a iteration algorithm named “fictitious default algorithm” to find clearing payment vector, as shown in below definition.
Definition 3.6 (Fictitious Default Algorithm) For the financial system \(()\), the fictitious default algorithm consists of following three steps:
- Initialize \(p^{(0)} = \bar{\boldsymbol{p}}\), the set of default nodes \(D\left(\boldsymbol{p}^{(0)}\right)=\left\{i\in \mathcal{N}: \left(\mathbf{\Pi}^\intercal \boldsymbol{p}^{(0)} + \boldsymbol{e}\right)_i < \bar{p}_i\right\}\) and the iteration index \(s=1\);
- For the iteration index \(s\), let \(\mathbf{\Lambda}^{(s-1)}\) be the diagonal matrix whose the \(i\)th diagonal equals 1 if node \(i\) default under \(\boldsymbol{p}^{(s-1)}\) and equals 0 otherwise. Solve the unique fixed point of the following function \[ FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p})=\mathbf{\Lambda}^{(s-1)}\left\{\mathbf{\Pi}^\intercal \left[\mathbf{\Lambda}^{(s-1)}\boldsymbol{p} + \left(\mathbf{I} - \mathbf{\Lambda}^{(s-1)}\right) \bar{\boldsymbol{p}} \right]+\boldsymbol{e}\right\} + \left(\mathbf{I} - \mathbf{\Lambda}^{(s-1)}\right) \bar{\boldsymbol{p}}. \]
- Let \(\boldsymbol{p}^{(s)}=FF_{\boldsymbol{p}^{(s)}}\left(\boldsymbol{p}^{(s)}\right)\), i.e. the fixed point of \(FF_{\boldsymbol{p}^{(s)}}\). Update the set of default nodes \(D\left(\boldsymbol{p}^{(s)}\right) = D\left(\boldsymbol{p}^{(s-1)}\right) \cup \left\{i\in \mathcal{N}: \left(\mathbf{\Pi}^\intercal \boldsymbol{p}^{(s)} + \boldsymbol{e}\right)_i < \bar{p}_i\right\}\). If \(D\left(\boldsymbol{p}^{s}\right) = D\left(\boldsymbol{p}^{s-1}\right)\), break the iteration and \(\boldsymbol{p}^* = \boldsymbol{p}^{(s)}\). Else return to the step 2.
The idea behind the algorithm is simple: we begin with the nominal obligation vector, supposing that all nodes pay in full. If no node default under \(\bar{\boldsymbol{p}}\), then the algorithm stops. However It possibly the case that some nodes default even though other nodes fulfill their obligations. If so, then we find the fixed point assuming that only the first round of default occurs. The algorithm terminates if the new payment vector no longer causes the second round default, and continues otherwise. there are \(n\) nodes in the system, so the iteration repeats at most \(n\) times.
It is firstly worth noting that the fictitious default algorithm works only when the function in step 2, \(FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p})\), indeed have unique fixed point, or is well defined as Eisenberg and Noe (2001) said. A necessary condition for this is that \(\mathbf{\Lambda}(\boldsymbol{p}^{(s)})\neq \mathbf{I}\), i.e. there must be some nodes being non default. To see this more clearly, suppose that the all nodes default under \(\boldsymbol{p}^{(s-1)}\), then the function in step 2 is \(FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p}) = \mathbf{\Pi}^\intercal \boldsymbol{p} + \boldsymbol{e}\), to find the fixed point, we need to solve \[ \mathbf{\Pi}^\intercal \boldsymbol{p} + \boldsymbol{e} = \boldsymbol{p} \iff (\mathbf{I} - \mathbf{\Pi}^\intercal) \boldsymbol{p} = \boldsymbol{e}. \] Unfortunately, the equation does not has since \(\left|\mathbf{I} - \mathbf{\Pi}^\intercal\right|=0\) (note that 1 is one eigenvalue of \(\mathbf{\Pi}\)). Thus it is necessary to ensure that not all nodes default at the same time. To achieve this, Eisenberg and Noe (2001) introduced a new concept named “supersolution”, that is \(\boldsymbol{p}\) is called a supersolution to \(\Phi\left(\boldsymbol{p}|\boldsymbol{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e}\right)\) if \(\Phi(\boldsymbol{p}) \leq \boldsymbol{p}\). Under any supersolution, it must be the case that some nodes pay in full, or equivalently, some nodes have positive equity value because \[ \boldsymbol{E} = \mathbf{\Pi}^\intercal \boldsymbol{p} + \boldsymbol{e} - \Phi(\boldsymbol{p}) \geq \mathbf{\Pi}^\intercal \boldsymbol{p} + \boldsymbol{e} - \boldsymbol{p} \\ \Longrightarrow \mathbf{1}^\intercal \boldsymbol{E} \geq \mathbf{1}^\intercal \left( \mathbf{\Pi}^\intercal \boldsymbol{p} + \boldsymbol{e} - \boldsymbol{p}\right) = \sum_{i=1}^n e_i > 0. \] Moreover, supersolution \(\boldsymbol{p}^{(s)}\) is sufficient for the existence of the fixed point to \(FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p})\). To justify the statement, we need to dissect the \(FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p})\). Without loss of generality, we assume that the fist \(k\) nodes default under \(\boldsymbol{p}^{(s)}\) and partition the payment vector \(\boldsymbol{p}\) into \(\boldsymbol{p}= \left[\boldsymbol{p}_{1:k}^\intercal\quad \boldsymbol{p}_{(k+1):n}^{\intercal}\right]^\intercal\) where \(\boldsymbol{p}_{i:j}\) represents the subvector of \(\boldsymbol{p}\) starting from the \(i\)th element and end with the \(j\)th element, then the equation \(FF_{\boldsymbol{p}^{(k)}}(\boldsymbol{p})=\boldsymbol{p}\) can be written as \[ FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p}) = \begin{bmatrix} \mathbf{\Pi}_{11}^\intercal & \mathbf{\Pi}_{21}^\intercal \\ \boldsymbol{0} & \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{p}_{1:k} \\ \boldsymbol{p}_{(k+1):n} \end{bmatrix} + \begin{bmatrix} \boldsymbol{e}_{1:k} \\ \boldsymbol{0} \end{bmatrix} + \begin{bmatrix} \boldsymbol{0} \\ \bar{\boldsymbol{p}}_{(k+1):n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{p}_{1:k} \\ \boldsymbol{p}_{(k+1):n} \end{bmatrix}. \] Obviously, the \((k+1)\)th to \(n\)th element of the fixed point \(\boldsymbol{p}^{(s+1)}\) requires that should be the nominal obligation, that is \(\boldsymbol{p}_{(k+1):n}^{(s+1)}=\bar{\boldsymbol{p}}_{(k+1):n}\). The remaining unsolved part is \[ \mathbf{\Pi}_{11}^\intercal \boldsymbol{p}_{1:k} +\mathbf{\Pi}_{21}^\intercal \boldsymbol{p}_{(k+1):n} + \boldsymbol{e}_{1:k} = \boldsymbol{p}_{1:k}, \\ \iff \left(\mathbf{I} - \mathbf{\Pi}_{11}^\intercal\right) \boldsymbol{p}_{1:k} = \mathbf{\Pi}_{21}^\intercal \boldsymbol{p}_{(k+1):n} + \boldsymbol{e}_{1:k}. \] By Definition 3.3, the default set \(\{i\in \mathcal{N}: 1 \leq i \leq k\}\) is not a surplus set, so \(\mathbf{\Pi}_{11}\) has a row sum less than \(\mathbf{1}\), which implies that 1 is not an eigenvalue of \(\mathbf{\Pi}_{11}\) and thus 0 is not an eigenvalue of \(\mathbf{I} - \mathbf{\Pi}_{11}\). Therefore, the remaining part of equation has an unique solution.
These facts described below the Definition 3.6 reminds us that we should guarantee that every payment vector \(\boldsymbol{q}^{(s)}\) is a supersolution to \(\Phi(\boldsymbol{q})\) during every iteration. In addition, the step 3 of Definition 3.6 shows that the algorithm produces a decreasing sequence \(\{\boldsymbol{q}^{(s)}\}\). The two things are formally proved in below Theorem.
Theorem 3.4 The fictitious default algorithm in 3.6 produces a well-defined sequence of vectors, \(\boldsymbol{p}^{(s)}\). This sequence decreases to the clearing vector in at most \(n\) iterations.
Proof. We complete the proof by mathematical induction. Suppose \(\boldsymbol{p}^{(s)}\) is a supersolution to \(\Phi(\cdot)\), which obviously holds for \(s=0\). Then we prove that \(\boldsymbol{p}^{(s+1)} \leq \boldsymbol{q}^{(s)}\) and \(\Phi\left(\boldsymbol{p}^{(s+1)}\right) \leq \boldsymbol{p}^{(s+1)}\).
(I) \(\left\{\boldsymbol{q}^{(s)}\right\}\) is decreasing. Note that \(\mathbf{\Lambda}^{(s)} \boldsymbol{p}^{(s)} + \left(\mathbf{I} - \mathbf{\Lambda}^{(s)}\right)\bar{\boldsymbol{p}} = \boldsymbol{p}^{(s)}\), so we have \[ \begin{aligned} FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p}^{(s)}) & = \mathbf{\Lambda}^{(s)}\left\{\mathbf{\Pi}^\intercal \left[\mathbf{\Lambda}^{(s)}\boldsymbol{p} + \left(\mathbf{I} - \mathbf{\Lambda}^{(s)}\right) \bar{\boldsymbol{p}} \right]+\boldsymbol{e}\right\} + \left(\mathbf{I} - \mathbf{\Lambda}^{(s)}\right) \bar{\boldsymbol{p}} \\ & = \mathbf{\Lambda}^{(s)} \left(\mathbf{\Pi}^\intercal \boldsymbol{p}^{(s)} + \boldsymbol{e} \right) + \left(\mathbf{I} - \mathbf{\Lambda}^{(s)}\right) \bar{\boldsymbol{p}} \\ & = \Phi\left(\boldsymbol{p}^{(s)} \Big| \mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e}\right). \end{aligned} \] Since \(\Phi\left(\boldsymbol{p}^{(s)} \Big| \mathbf{\Pi}, \bar{\boldsymbol{p}}, \boldsymbol{e}\right) \leq \boldsymbol{p}^{(s)}\), we have \(FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p}^{(s)}) \leq \boldsymbol{p}^{(s)}\). By definition, \(\boldsymbol{p}^{(s+1)}\) is the unique fixed point of \(FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p})\), i.e. \(FF_{\boldsymbol{p}^{(s)}}(\boldsymbol{p}^{(s+1)}) = \boldsymbol{p}^{(s+1)}\). To compare \(\boldsymbol{p}^{(s)}\) and \(\boldsymbol{p}^{(s+1)}\), suppose the first to the \(k\)th element under \(\boldsymbol{p}^{(s)}\) are default, we can write the inequality and the equality as \[ \begin{aligned} & FF_{\boldsymbol{p}^{(s)}}\left(\boldsymbol{p}^{(s)}\right) = \begin{bmatrix} \mathbf{\Pi}_{11}^\intercal & \mathbf{\Pi}_{21}^\intercal \\ \boldsymbol{0} & \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{p}_{1:k}^{(s)} \\ \boldsymbol{p}_{(k+1):n}^{(s)} \end{bmatrix} + \begin{bmatrix} \boldsymbol{e}_{1:k} \\ \boldsymbol{0} \end{bmatrix} + \begin{bmatrix} \boldsymbol{0} \\ \bar{\boldsymbol{p}}_{(k+1):n} \end{bmatrix} \leq \begin{bmatrix} \boldsymbol{p}_{1:k}^{(s)} \\ \boldsymbol{p}_{(k+1):n}^{(s)} \end{bmatrix}, \\ & FF_{\boldsymbol{p}^{(s)}}\left(\boldsymbol{p}^{(s+1)}\right) = \begin{bmatrix} \mathbf{\Pi}_{11}^\intercal & \mathbf{\Pi}_{21}^\intercal \\ \boldsymbol{0} & \boldsymbol{0} \end{bmatrix} \begin{bmatrix} \boldsymbol{p}_{1:k}^{(s+1)} \\ \boldsymbol{p}_{(k+1):n}^{(s+1)} \end{bmatrix} + \begin{bmatrix} \boldsymbol{e}_{1:k} \\ \boldsymbol{0} \end{bmatrix} + \begin{bmatrix} \boldsymbol{0} \\ \bar{\boldsymbol{p}}_{(k+1):n} \end{bmatrix} = \begin{bmatrix} \boldsymbol{p}_{1:k}^{(s+1)} \\ \boldsymbol{p}_{(k+1):n}^{(s+1)} \end{bmatrix}. \end{aligned} \] For the nondefalut nodes, the inequality and equality hold only when \(\boldsymbol{p}_{(k+1):n}^{(s+1)}=\boldsymbol{p}_{(k+1):n}^{(s)} = \bar{\boldsymbol{p}}_{(k+1):n}\) obviously. For the default nodes, \[ \begin{aligned} & (\mathbf{I} - \mathbf{\Pi}_{11}^\intercal) \boldsymbol{p}_{1:k}^{(s)} \geq \mathbf{\Pi}_{21}^\intercal \bar{\boldsymbol{p}}_{(k+1):n} + \boldsymbol{e}_{1:k}, \\ & \boldsymbol{p}_{1:k}^{(s+1)} = \left(\mathbf{I} - \mathbf{\Pi}_{11}^\intercal\right)^{-1} \left(\mathbf{\Pi}_{21}^\intercal \bar{\boldsymbol{p}}_{(k+1):n} + \boldsymbol{e}_{1:k}\right). \end{aligned} \] Note again that the invertibility of \(\mathbf{I} - \mathbf{\Pi}_{11}\) is assured by the fact that \(\boldsymbol{p}^{(s)}\) is a supersolution to \(\Phi(\cdot)\). Since \((\mathbf{I} - \mathbf{\Pi}_{11})^{-1} = \sum_{t=0}^\infty \mathbf{\Pi}_{11}^t \geq 0\), i.e. all elements in \((\mathbf{I} - \mathbf{\Pi}_{11})^{-1}\) is nonnegative, premultiplying both sides of the inequality by \((\mathbf{I} - \mathbf{\Pi}_{11})^{-1}\) will not reverse the direction of inequality. Therefore we have \[ \boldsymbol{p}_{1:k}^{(s)} \geq (\mathbf{I} - \mathbf{\Pi}_{11}^\intercal)^{-1} \left( \mathbf{\Pi}_{21}^\intercal \bar{\boldsymbol{p}}_{(k+1):n} + \boldsymbol{e}_{1:k}\right) = \boldsymbol{p}_{1:k}^{(s+1)}. \] (II) \(\boldsymbol{q}^{(s)}\) is a supersolution to \(\Phi(\cdot)\). Based on the result in (I), the default set node must be no smaller under \(\boldsymbol{p}^{(s+1)}\) than under \(\boldsymbol{p}^{(s)}\), that is \(D\left(\boldsymbol{p}^{(s+1)}\right) \geq D\left(\boldsymbol{p}^{(s)}\right)\). We divide into two cases for discussion.
If \(D\left(\boldsymbol{p}^{(s+1)}\right)= D\left(\boldsymbol{p}^{(s)}\right)\), then \(\mathbf{\Lambda}^{(s+1)} = \mathbf{\Lambda}^{(s)}\), so we have \[ \Phi\left(\boldsymbol{p}^{(s+1)}\right) = FF_{\boldsymbol{p}^{(s+1)}}\left(\boldsymbol{p}^{(s+1)}\right) = FF_{\boldsymbol{p}^{(s)}}\left(\boldsymbol{p}^{(s+1)}\right) = \boldsymbol{p}^{(s+1)}. \]
If \(D\left(\boldsymbol{p}^{(s+1)}\right) < D\left(\boldsymbol{p}^{(s)}\right)\), then the nodes can be devided into three groups: the first group contains \(l\) nodes that default under \(\boldsymbol{p}^{(s+1)}\) but do not under \(\boldsymbol{p}^{(s)}\), another group contains \(m\) nodes that are default under both \(\boldsymbol{p}^{(s+1)}\) and \(\boldsymbol{p}^{(s)}\), and the last group contains the remaining nodes which keep solvent under both \(\boldsymbol{p}^{(s+1)}\) and \(\boldsymbol{p}^{(s)}\). Then it must be the case that \[ \mathbf{\Lambda}^{(s)} = \begin{bmatrix} \mathbf{0}_l & \cdot & \cdot \\ \cdot & \mathbf{I}_m & \cdot \\ \cdot & \cdot & \mathbf{0}_{n-l-m} \end{bmatrix}, \mathbf{\Pi} = \begin{bmatrix} \mathbf{\Pi}_{11} & \mathbf{\Pi}_{12} & \mathbf{\Pi}_{13} \\ \mathbf{\Pi}_{21} & \mathbf{\Pi}_{22} & \mathbf{\Pi}_{23} \\ \mathbf{\Pi}_{31} & \mathbf{\Pi}_{32} & \mathbf{\Pi}_{33} \end{bmatrix},\ \boldsymbol{p}^{(s+1)} = \begin{bmatrix} \bar{\boldsymbol{p}}_{1:l} \\ \boldsymbol{p}_{(l+1):(l+m)}^{(s+1)} \\ \bar{\boldsymbol{p}}_{(l+m+1):n} \end{bmatrix}. \] Using the block matrix representation again, we discuss the three situations as following: \[ \begin{aligned} & \Phi\left(\boldsymbol{p}^{(s+1)}\right)_{1:l} = \mathbf{\Pi}_{11}^\intercal \bar{\boldsymbol{p}}_{1:l} + \mathbf{\Pi}_{21}^\intercal \boldsymbol{p}_{(l+1):(l+m)}^{(s+1)} + \mathbf{\Pi}_{31}^\intercal \bar{\boldsymbol{p}}_{(l+m+1):n} + \boldsymbol{e}_{1:l} < \bar{\boldsymbol{p}}_{1:l},\\ & \begin{aligned} \Phi\left(\boldsymbol{p}^{(s+1)}\right)_{(l+1):(l+m)} & = \mathbf{\Pi}_{12}^\intercal \bar{\boldsymbol{p}}_{1:l} + \mathbf{\Pi}_{22}^\intercal \boldsymbol{p}_{(l+1):(l+1)}^{(s+1)} + \mathbf{\Pi}_{32}^\intercal \bar{\boldsymbol{p}}_{(l+m+1):n} +\boldsymbol{e}_{(l+1):(l+m)} \\ & = FF_{\boldsymbol{p}^{(s)}}\left(\boldsymbol{p}^{(s+1)}\right)_{(l+1):(l+m)} = \boldsymbol{p}_{(l+1):(l+m)}^{(s+1)}, \end{aligned}\\ & \Phi\left(\boldsymbol{p}^{(s+1)}\right)_{(l+m+1):n} = \mathbf{\Pi}_{13}^\intercal \bar{\boldsymbol{p}}_{1:l} + \mathbf{\Pi}_{23}^\intercal \boldsymbol{p}_{(l+1):(l+1)}^{(s+1)} + \mathbf{\Pi}_{33}^\intercal \bar{\boldsymbol{p}}_{(l+m+1):n} +\boldsymbol{e}_{(l+m+1):n} = \bar{\boldsymbol{p}}_{(l+m+1):n}. \end{aligned} \] Combining these results, we can conclude that \(\Phi\left(\boldsymbol{p}^{(s+1)}\right) \leq \boldsymbol{p}^{(s+1)}\).