Systemic Risk & Clearing

Setup

A network of $n$ financial institutions. Bank $i$ has:

• $e_i \ge 0$: exogenous assets (non-interbank cash and tradeable assets).
• $L_{ij} \ge 0$: nominal obligation owed by $i$ to $j$ ($L_{ii} = 0$).
• $\bar{p}_i = \sum_j L_{ij}$: total liabilities of $i$ to other banks.
• $\pi_{ij} = L_{ij} / \bar{p}_i$ (if $\bar{p}_i > 0$): share of $i$’s debt owed to $j$.

Clearing-vector definition

A clearing payment vector $p^* \in [0, \bar{p}]$ satisfies: every bank pays pro-rata with what it has (own assets plus payments received from others), up to its total liabilities:

Existence, uniqueness, computation

Eisenberg & Noe proved the operator $\Phi(p)_i = \min\{\bar{p}_i, e_i + \sum_j \pi_{ji} p_j\}$ is monotone non-decreasing and maps $[0, \bar{p}]$ into itself, so by Tarski’s fixed-point theorem there is a greatest clearing vector. The algorithm fictitious default computes it in $O(n^3)$:

  p ← p-bar
  repeat:
      solve linear system assuming default set D, clear to obtain p
      D' ← {i : p_i < p-bar_i}
      if D' == D: return p
      else: D ← D'
            

LP characterisation

Equivalently, the clearing vector solves the LP:

Contagion metric

The number of defaults $|\{i : p_i^* < \bar{p}_i\}|$ is the primary contagion statistic. Shortfall $\sum_i (\bar{p}_i - p_i^*)$ captures aggregate loss. Glasserman-Young (2015) show that amplification from network linkages is often modest relative to direct losses, which pushed the regulatory literature toward fire-sale and liquidity-based contagion channels.