by $L(t)\vec{1}$ and the new incoming interbank obligations are given by $L(t)^{\top}\vec{1}$. We can also consider $c_i(t)$ to be the *net cash flow* for firm $i$ at time $t$. Further, we introduce the functional matrix $A: {\mathbb{T}}\times {\mathbb{R}}^{n+1} \to [0,1]^{(n+1) \times (n+1)}$ to be the relative exposure matrix. That is, $a_{ij}(t,V_t)V_i(t)^-$ provides the (negative) impact that firm $i$’s losses have on firm $j$’s wealth at time $t \in {\mathbb{T}}$. This is in contrast to $\Pi$, the relative liabilities, in that it endogenously imposes the limited exposures concept. In this work the two notions will generally coincide, but for mathematical simplicity we introduce this relative exposure matrix. For the equivalence we seek, we define the relative exposures so that $$L(t)^{\top}\vec{1} + A(t-1,V_{t-1})^{\top}V(t-1)^- - A(t,V_t)^{\top}V(t)^- = \Pi(t,V_{t-1})^{\top}[\bar p(t,V_{t-1}) - V(t)^-]^+$$ for any $V(t) \in {\mathbb{R}}^{n+1}$. This formulation is such that if the positive part were removed from the right hand side, the relative exposures $A$ would be defined exactly as the relative liabilities $\Pi$ by construction. In particular, we will define the relative exposures element-wise and pointwise so as to encompass the limited exposures as in . If $\bar p_i(t,V_{t-1}) > 0$ then we can simplify this further as $a_{ij}(t,V_t) = \frac{L_{ij}(t) 