Banks must hold capital — equity and retained earnings — as a buffer against losses. Basel III regulations specify minimum capital ratios: the ratio of a bank's capital to its risk-weighted assets must exceed prescribed thresholds. The regulation is a constraint on the bank's optimization problem: maximize shareholder value subject to maintaining the capital ratio above the regulatory floor.
The paper formulates this as an optimal control problem with state constraints. The state is the bank's capital ratio; the controls are dividend payments, asset allocation, and new equity issuance; the constraint is the Basel III floor. The problem is tractable — meaning the optimal policy can be characterized analytically — because the constraint is a simple inequality on a single state variable.
The optimal policy has three regimes. Well-capitalized: the bank pays dividends and invests normally. Near the constraint: the bank reduces dividends and shifts to safer assets, building a buffer. At the constraint: the bank issues equity (the most expensive funding source) to avoid violation. The transitions between regimes are smooth, not abrupt — the bank anticipates the constraint before hitting it and adjusts gradually.
The structural insight: the capital constraint changes the bank's behavior even when it's not binding. A bank with capital well above the minimum still modifies its dividend and investment policies to reduce the probability of approaching the constraint in the future. The constraint's shadow — its influence on decisions made far from the boundary — extends further than the constraint itself. The regulatory floor is felt through the entire capital distribution, not just at the floor.