any clearing solution must be in the domain so that the relative liabilities and exposures coincide. This additionally provides the invertibility of this matrix using standard input-output results as discussed in [@EN01; @feinstein2017sensitivity].

We wish to finish up our discussion of the discrete-time Eisenberg-Noe framework by considering some extensions involving loans.

The theoretical framework presented in this paper can be easily extended to incorporate the concepts of loans until some (deterministic) insolvency condition is hit. In particular, we will consider loans made from a central bank or lender of last resort who we will assume are part of the societal node $0$. From this perspective we consider three cases that a firm might be in:

-   [**solvent and liquid**]{} in which case the firm has positive equity and pays off its obligations in full;

-   [**solvent and distressed**]{} in which case the firm has negative equity, but receives an overnight loan (with interest rate set at the risk-free rate for simplicity) to cover all obligations due on that date; and

-   [**insolvent**]{} in which the firm will not receive any loans and is sent to a bankruptcy court.

The determination whether a firm is solvent can be done with an 