Cascades in financial networks are situations in which the failure of onefinancial institution causes acascading failure in another member of the financial network. In an extreme this can cause failure of the whole network in what is known assystemic failure. It can be defined as the discontinuous value loss (e.g. default) of the organization caused by the discontinuous value loss of another organization in the network. There are three conditions required for a cascade, there are; a failure, contagion and interconnection.^{[1]: 3116–3117, 3122}

Diversification and integration in the financial network determine whether and how failures will spread. Using the data on cross-holdings of organizations and on the value of organizations, it is possible to construct the dependency matrix to simulate cascades in the financial network.^{[1]: 3121, 3128}

Elliot, Golub and Jackson (2014) characterize the financial network by diversification and integration. Diversification means to which extentassets of the one organization are spread out among the other members of the network, given the fraction of the assets of the organization cross-held by other organizations is fixed. Integration refers to the fraction of the assets of the organization cross-held by other organizations given the number of the organizations cross-holding is fixed.^{[1]: 3117–3118}

Using random network, the authors show that high integration decreases the percentage offirst failures; and as the network approaches complete integration the percentage of the first failures approaches zero. However, the integration increases the percentage of organizations that fail due to higher interconnection. In addition, up to some threshold, diversification does increase the percentage of discontinuous drops in value. Yet after the threshold level, the diversification decreases the percentage of failures: the authors say the following with respect to diversification: “it gets worse before it gets better”.^{[1]: 3132, 3138}

Intuitively, the higher the threshold value for the discontinuous drop in the organization’s value the higher the percentage of failures is.^{[1]: 3122–3123}

The authors conclude that the financial network is most susceptible to cascades if it has medium diversification and medium integration.^[1]: 3143

Eliot, Golub and Jackson (2014) 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 letteroutside shareholders. Their model starts with the following assumptions:^[1]: 3119

• 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 assetk is${\displaystyle p_k}$
• ${\displaystyle D_{ik}}$ is a share of the assetk that an organizationi holds
• D is then n by m matrix
• ${\displaystyle C_{ij}\ge 0}$ is a fraction of primitive assets of the organizationj held by the organizationi
• ${\displaystyle C_{ii}=0}$
• 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)^[2] and Fedina, Hodder and Trianitis (1994):^[3]

${\displaystyle V_i=\sum _k{D}_{ik}{p}_k+\sum _j{C}_{ij}{V}_j}$

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

The letter implies

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

whereA is the dependence matrix.

The element${\displaystyle A_{ij}}$ represents the fraction ofj's primitive assets thati holds directly and indirectly.

The equity value and the market value equations are extended by introducing threshold value${\displaystyle t_i}$. If the value of the organizationi 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 ofi is below the threshold and 0 if the value ofi 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${\displaystyle j}$ thati incurs ifj fails.

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