  The DebtRank algorithm has been increasingly investigated as a method to
estimate the impact of shocks in financial networks, as it overcomes the
limitations of the traditional default-cascade approaches. Here we formulate a
dynamical "microscopic" theory of instability for financial networks by
iterating balance sheet identities of individual banks and by assuming a simple
rule for the transfer of shocks from borrowers to lenders. By doing so, we
generalise the DebtRank formulation, both providing an interpretation of the
effective dynamics in terms of basic accounting principles and preventing the
underestimation of losses on certain network topologies. Depending on the
structure of the interbank leverage matrix the dynamics is either stable, in
which case the asymptotic state can be computed analytically, or unstable,
meaning that at least one bank will default. We apply this framework to a
dataset of the top listed European banks in the period 2008 - 2013. We find
that network effects can generate an amplification of exogenous shocks of a
factor ranging between three (in normal periods) and six (during the crisis)
when we stress the system with a 0.5% shock on external (i.e. non-interbank)
assets for all banks.
