Without Failure Costs

Eliot, Golub and Jackson (2013) provide an empirical method how to model cascades in financial networks. They assume that organizations in the network can cross hold assets of other organizations in the network. Also, they assume that players outside of the network can hold assets of the organizations in the network. They call the letter outside shareholders. Their model starts with the following assumptions (all notations are borrowed from Elliot, Golub and Jackson (2013)):

• There are n organizations that form a set N=[1,...,n]
• There are m "primitive" assets (e.g. factors of production)
• Market price of an asset k is ${\displaystyle p_{k}}$
• ${\displaystyle D_{ik}}$ is a share of the asset k that an organization i holds
• D is then n by m matrix
• ${\displaystyle C_{ij}\geq \ 0}$ is a fraction of primitive assets of the organization j held by the organization i
• ${\displaystyle C_{ii}=0}$
• C is a n by n matrix with zeros as diagonal elements
• ${\displaystyle F_{ii}=1-\sum _{j}C_{ji}}$
• F is a n by n matrix whose diagonal element is :${\displaystyle F_{ii}}$

The authors find the equity value of an organization using the works by Brioschi, Buzzachi and Colombo (1989)^[5] and Fedina, Hodder and Trianitis (1994):^[6]

${\displaystyle V_{i}=\sum _{k}D_{ik}p_{k}+\sum _{j}C_{ij}V_{j}}$

The equity value is defined as the value of primitive assets and the value of claims on the primitive assets in other organizations in the network.

The counterpart of the equation above in terms of matrix algebra is given by

${\displaystyle V=Dp+CV}$

The letter implies

${\displaystyle V=(I-C)^{-1}Dp}$

The market value is defined by

${\displaystyle v_{i}=\sum _{k}D_{ik}p_{k}+\sum _{j}C_{ij}-\sum _{j}C_{ji}V_{i}}$

Market value of i is the equity value of i less the claims of other organizations in the network on i.

The letter implies

${\displaystyle v=FV=F(I-C)^{-1}Dp=ADp}$

where A is the dependence matrix.

The element ${\displaystyle A_{ij}}$ represents the fraction of j's primitive assets that i holds directly and indirectly.

With Failure Costs

The equity value and the market value equations are extended by introducing threshold value ${\displaystyle t_{i}}$. If the value of the organization i goes below this value, then a discontinuous drop in value happens and the organization fails. The cap on failure costs is ${\displaystyle k_{i}}$.

Further, let ${\displaystyle I}$ be an indicator function that is equal to 1 if the value of i is below the threshold and 0 if the value of i is above the threshold.

Then the equity value becomes

${\displaystyle V_{i}=\sum _{k}D_{ik}p_{k}+\sum _{j}C_{ij}V_{j}-k_{i}I_{i}}$

Using matrix algebra, the expression above is equivalent to

${\displaystyle V=(I-C)^{-1}(Dp-b(v))}$

where ${\displaystyle b(v)}$ is a vector whose element ${\displaystyle b_{i}=k_{i}I_{i}}$.

The market value including failure costs is given then by

${\displaystyle v=F(I-C)^{-1}(Dp-b(v))=A(Dp-b(v))}$

The element ${\displaystyle A_{ij}}$ represents the fraction of failure costs of ${\displaystyle j}$ that i incurs if j fails.