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This allows us to derive a fric- tionless value of equity that is of repeated use when solv- ing shareholders' optimization problem.
When issuing equity is costless, the value of equity and the optimal default time are given by
Having determined the value of equity when there are no costs of raising funds, we now turn to the solution of the auxiliary problem in which the bank has no access to outside funds.
Lastly, Proposition 3 shows that when outside funds are costly, the value of equity is a concave function of the bank's liquid reserves on the positive real line, so that there are no incentives for shareholders to engage in risk- shifting even if the bank is levered.
The value of equity in a bank subject to a minimum regulatory level of liquid reserves is defined by the opti- mization problem
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