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.

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