Optimal Support for Distressed Subsidiaries -- a Systemic Risk Perspective

Abstract

We consider a network of bank holdings, where every holding has two subsidiaries of different types. A subsidiary can trade with another holding's subsidiary of the same type. Holdings support their subsidiaries up to a certain level when they would otherwise fail to honor their financial obligations. We investigate the spread of contagion in this banking network when the number of bank holdings is large, and find the final number of defaulted subsidiaries under different rules for the holding support. We also consider resilience of this multilayered network to small shocks. Our work sheds light onto the role that holding structures can play in the amplification of financial stress. We find that depending on the capitalization of the network, a holding structure can be beneficial as compared to smaller separated entities. In other instances, it can be harmful and actually increase contagion. We illustrate our results in a numerical case study and also determine the optimal level of holding support from a regulator perspective.

Figures (10)

• Figure 1: Illustration of an exemplary contagion process for a network with three holdings of type $A$ with a support level $x=-1$. The pair of two circles arranged vertically form a holding with the top circle denoting subsidiary $1$ and the bottom circle subsidiary $2$. The edges between type $1$ subsidiaries are drawn in black and the edges between type $2$ subsidiaries are drawn in green. The cascade start (round $0$) is shown in the center upper graph. Subsidiary $1$ of bank $1$ has defaulted because its capital has reached the maximal support level and therefore defaults although the holding capital is still positive. This leads to $\mathcal{S}_{1,0}=\{1_1\}$ and $\mathcal{S}_{2,0}=\emptyset$, which triggers a short cascade evolving in two rounds. In round $1$, shown in the lower left graph, subsidiary $1$ of bank $3$ looses $1$ unit in capital, so its capital then becomes $c_{3,1}^{(1)}=-1$, which causes the entire bank holding $3$ to default and the round ends with ($\mathcal{S}_{1,1}=\{1_1, 3_1 \}$ and $\mathcal{S}_{2,1}=\{ 3_2 \}$). The result of round $2$, which is pictured in the lower right graph, is that subsidiary $2$ of bank $2$ looses one unit of capital, and its terminal capital is $c_{2,1}^{(2)}=2$. The reduced capital of subsidiary $2$ of bank $2$ does not lead to any further defaults ($\mathcal{S}_{1,2}=\{1_1, 3_1 \}$ and $\mathcal{S}_{2,2}=\{ 3_2 \}$).
• Figure 2: In the same setup as Figure \ref{['fig2']}, illustration of a contagion process for a network with three holdings of type $B$ with a support level $x=-1$. The pair of two circles arranged vertically form a holding with the top circle denoting subsidiary $1$ and the bottom circle subsidiary $2$. The edges between type $1$ subsidiaries are drawn in black and the edges between type $2$ subsidiaries are drawn in green. Round $0$ is the same as for type $A$ -- subsidiary $1$ of bank $1$ defaults, $\mathcal{S}_{1,0}=\{1_1\}$ and $\mathcal{S}_{2,0}=\emptyset$. The difference is in round $1$, where while subsidiary $1$ of bank $3$ still looses $1$ unit in capital, and its capital also becomes $c_{3,1}^{(1)}=-1$, this will cause the holding to let it subsidiary to default, while the capital of subsidiary $2$ will remain $c_{3,1}^{(1)}=1.$ Therefore, the cascade of defaults stops.
• Figure 3: (a) Illustration for type $A$ holding. (b) illustration for type $B$ holding. Blue filled area is the default region of subsidiary 1, red filled area is the default region of subsidiary 2. The blue and red dotted lines correspond to the boundaries ${\partial} \mathbf{D}_{A,1}$ and ${\partial} \mathbf{D}_{A,2}$ for type $A$ (respectively ${\partial} \mathbf{D}_{B,1}$ and ${\partial} \mathbf{D}_{B,2}$ for type $B$.)
• Figure 4: Graph with defaults of: subsidiaries of type $1$ (orange squares), subsidiaries of type $2$ (green rhombuses), total defaults (blue circles), as a function of support level $x$. The network parameters are: Holding of type $A$, $r_1=5,~r_2=9,~p_1=14,~p_2=14,~\epsilon_1=\epsilon_2=15\%$.
• Figure 5: Graph with defaults of: subsidiaries of type $1$ (orange squares), subsidiaries of type $2$ (green rhombuses), total defaults (blue circles), as a function of support level $x$. The network parameters are: Holding of type $A$, $r_1=5,~r_2=9,~p_1=14,~p_2=4,~\epsilon_1=\epsilon_2=15\%$.

Theorems & Definitions (16)

• Proposition 2.2
• Theorem 2.3
• Proposition 3.1
• Corollary 3.2
• Corollary 3.3
• Theorem 3.4
• Lemma 3.5
• Example 3.6
• Example 3.7
• proof : Proof of Proposition \ref{['properties:f']}