x(t) + \Pi(t,V_{t-1})^{\top}\left[\bar p(t,V_{t-1}) - V(t)^-\right]^+ - \bar p(t,V_{t-1}).$$ That is, all firms have a clearing wealth that is the summation of their positive equity at the prior time, the new incoming external cash flow, and the payments made by all other firms minus the total obligations of the firm (including the prior unpaid liabilities). In this way we can construct the wealths of firms forward in time. This can be considered a discrete-time extension of .

We now wish to consider a reformulation of . To accomplish this, we consider a process of cash flows $c$ and functional relative exposures $A$. These we define by $$\begin{aligned}
\nonumber c(t) &:= x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1}\\
\label{eq:discrete-A} a_{ij}(t,V_t) &:= \begin{cases} \pi_{ij}(t,V_{t-1}) &\text{if } \bar p_i(t,V_{t-1}) \geq V_i(t)^-\\ \frac{L_{ij}(t) + \pi_{ij}(t-1,V_{t-2})V_i(t-1)^-}{V_i(t)^-} &\text{if } \bar p_i(t,V_{t-1}) < V_i(t)^-\end{cases} \quad \forall i,j \in {\mathcal{N}}_0.\end{aligned}$$ In the above, $\vec{1} := (1,1,\ldots,1)^{\top}\in {\mathbb{R}}^n$ is the vector of ones. Here we consider $c(t) = x(t) + L(t)^{\top}\vec{1} - L(t)\vec{1} \in {\mathbb{R}}^{n+1}$ to be the vector of book capital levels at time $t$, i.e., the new wealth of each firm assuming all other firms pay in full. We wish to note that the new total liabilities are given 