Credit line takedown and endogenous bank capital

This paper investigates the implications of the uncertain timing and usage of loan commitments for the optimal level of bank capital. We use trended Brownian motion to proxy the stochastic takedown of credit lines. Relying on "time to first passage" mathematics, we derive a probability density function for the time to depletion of the bank credit line as well as the likelihood for the time to exhausting the sources of liquidity that fund the loan takedown. Armed with these analytical results, we solve for the optimal level of bank capital within a simultaneous equation framework in order to capture the interrelationships of the endogenous variables. The optimality conditions produce a system of integral differential equations which refuse to yield reduced form solutions and provide no immediate intuition. Therefore, the maximizing values of the bank's decision variables were simulated over a host of realistic scenarios. We document the comparative static behavior of the bank's decision variables when equity is unencumbered by capital requirements and, also, examine the impact of the same parametric changes on bank behavior when equity is a fixed proportion of lending. Further simulations produce the expected time to liquidity depletion under different capital requirement schemes.