  Financial networks raise a significant computational challenge in identifying
insolvent firms and evaluating their exposure to systemic risk. This task,
known as the clearing problem, is computationally tractable when dealing with
simple debt contracts. However under the presence of certain derivatives called
credit default swaps (CDSes) the clearing problem is $\textsf{FIXP}$-complete.
Existing techniques only show $\textsf{PPAD}$-hardness for finding an
$\epsilon$-solution for the clearing problem with CDSes within an unspecified
small range for $\epsilon$.
  We present significant progress in both facets of the clearing problem: (i)
intractability of approximate solutions; (ii) algorithms and heuristics for
computable solutions. Leveraging $\textsf{Pure-Circuit}$ (FOCS'22), we provide
the first explicit inapproximability bound for the clearing problem involving
CDSes. Our primal contribution is a reduction from $\textsf{Pure-Circuit}$
which establishes that finding approximate solutions is $\textsf{PPAD}$-hard
within a range of roughly 5%.
  To alleviate the complexity of the clearing problem, we identify two
meaningful restrictions of the class of financial networks motivated by
regulations: (i) the presence of a central clearing authority; and (ii) the
restriction to covered CDSes. We provide the following results: (i.) The
$\textsf{PPAD}$-hardness of approximation persists when central clearing
authorities are introduced; (ii.) An optimisation-based method for solving the
clearing problem with central clearing authorities; (iii.) A polynomial-time
algorithm when the two restrictions hold simultaneously.
