Capital ideas: optimal capital accumulation strategies for a bank and its regulator

We formulate a continuous-time model of a deposit taking bank, operating subject to capital adequacy regulation, and where the bank's loans are exposed to default risk. The bank maximises its market value of equity by appropriately controlling loan and equity issuance, dividend payments, and endogenous closure. Of interest to regulators, we show how the bank responds to the span of capital adequacy requirements and to changes in default variance. We find a nonmonotonic response of bank lending to increased capital requirements (initially increasing and then decreasing). We also find that the probability of early closure (through either insolvency or endogenous closure) can be minimised for a well-chosen capital adequacy ratio. Moreover, the level of capital requirement that minimises the probability of early closure and maximises lending is fairly robust to changes in the bank's default variance/risk; contributing to the ongoing discussion on optimal bank capital.