setting the clearing wealths process $V: {\mathbb{T}}\to {\mathbb{R}}^{n+1}$ can be found by the following algorithm. Initialize $t = -1$ and $V(-1) \geq 0$ as a given. Repeat until $t = \max{\mathbb{T}}$:

1.  Increment $t = t+1$.

2.  \[alg:v\] Initialize $k = 0$, $V^0 = V(t-1) + c(t)$, and $D^0 = \emptyset$. Repeat until convergence:

    1.  Increment $k = k+1$;

    2.  Denote the set of insolvent banks by $D^k := \left\{i \in \{1,2,...,n\} \; | \; V_i^{k-1} < 0\right\}$.

    3.  If $D^k = D^{k-1}$ then terminate and set $V(t) = V^{k-1}$.

    4.  Define the matrix $\Lambda^k \in \{0,1\}^{n \times n}$ so that $\Lambda_{ij}^k = \begin{cases}1 &\text{if } i = j \in D^k \\ 0 &\text{else}\end{cases}$.

    5.  \[alg:vk\] Define $V^k = (I - \Pi(t,V_{t-1})^{\top}\Lambda^k)^{-1}\left(V(t-1) + c(t) + A(t-1,V_{t-1})^{\top}V(t-1)^-\right)$.

\[rem:loans\] Note that in the construction of $V^k$ in step of the fictitious default algorithm we utilize the relative liabilities $\Pi(t,V_{t-1})$ in the matrix inverse rather than the relative exposures $A(t,(V_{t-1},V^k))$. This has the added benefit that this definition of $V^k$ is *not* a fixed point problem, which it would be if the relative exposures matrix at time $t$ were considered. This change is possible since, as discussed in the proof of Theorem \[thm:discrete\], 