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Tractable bank capital structure: optimal control under Basel III constraints

Erhan Bayraktar, Etienne Chevalier, Vathana Ly Vath, Yuqiong Wang

Abstract

Banks must optimize risky investments, dividend payouts, and capital structure under tight Basel III solvency and liquidity constraints, while costly equity issuance serves as a distress-recovery tool. We formulate this as a stochastic control problem that reduces the high-dimensional balance-sheet dynamics to a tractable one-dimensional process in the leverage ratio, with state-dependent investment limits. The resulting policy is simple and interpretable: pay dividends at an upper reflection barrier and, when needed, recapitalize only at the distress boundary, jumping to a unique target level. We characterize these thresholds analytically and show their sensitivity to regulatory parameters. From a regulatory viewpoint, we solve an outer optimization problem that maps the efficient frontier between shareholder value and survival probability (via Monte Carlo), with and without leverage caps. Results highlight that tightening solvency requirements often yields the best safety-profitability trade-off.

Tractable bank capital structure: optimal control under Basel III constraints

Abstract

Banks must optimize risky investments, dividend payouts, and capital structure under tight Basel III solvency and liquidity constraints, while costly equity issuance serves as a distress-recovery tool. We formulate this as a stochastic control problem that reduces the high-dimensional balance-sheet dynamics to a tractable one-dimensional process in the leverage ratio, with state-dependent investment limits. The resulting policy is simple and interpretable: pay dividends at an upper reflection barrier and, when needed, recapitalize only at the distress boundary, jumping to a unique target level. We characterize these thresholds analytically and show their sensitivity to regulatory parameters. From a regulatory viewpoint, we solve an outer optimization problem that maps the efficient frontier between shareholder value and survival probability (via Monte Carlo), with and without leverage caps. Results highlight that tightening solvency requirements often yields the best safety-profitability trade-off.
Paper Structure (14 sections, 9 theorems, 132 equations, 6 figures, 7 tables)

This paper contains 14 sections, 9 theorems, 132 equations, 6 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Balance-sheet dynamics, regulatory constraints, and the bank's optimization problem
  3. Characterization of the optimal strategies
  4. Analytical properties of the value function
  5. Lower and upper bounds for the value function
  6. Viscosity characterization of the value function
  7. Optimal dividend and capital issuance strategies
  8. Sensitivity analysis and safety-profitability frontiers under regulation
  9. Value function and optimal strategies
  10. Sensitivity to regulatory parameters
  11. Regulatory parameter optimization
  12. Case without a leverage restriction
  13. Case with the leverage restriction $\pi\leq 1$
  14. Appendix

Key Result

Proposition 2.1

Let $\alpha:=((\tau_n)_{n\in\mathbb{N}^*},({\xi}_n)_{n\in\mathbb{N}^*},{Z},\pi)$ where $(\tau_n)_{n\in\mathbb{N}^*}$ is an increasing sequence of stopping times, $({\xi}_n)_{n\in\mathbb{N}^*}$ a sequence of positive $\mathcal{F}_{\tau_n^-}$-measurable random variables, and $Z$ an increasing process. where $\mu(\pi)=(1-\pi)r+\pi\mu$. Define the stopping time $T^\alpha=\inf\{t\ge 0:\ Y^\alpha_t< 1\}

Figures (6)

  • Figure 1: The value function of $y$ and $(x,l)$.
  • Figure 2: Trajectory from $y_0 =1.2$ of $50$ years
  • Figure 2: Pareto frontier without restriction
  • Figure 3: Total capital and dividend of $50$ years
  • Figure 4: Optimization without restriction
  • ...and 1 more figures

Theorems & Definitions (18)

  • Proposition 2.1: Reduction to the leverage ratio
  • Proposition 3.1
  • Proposition 3.2
  • Corollary 3.1
  • Proposition 3.3: Viscosity characterization of the value function
  • Proposition 3.4: Concavity of the value function
  • Corollary 3.2
  • Theorem 3.1
  • proof
  • proof
  • ...and 8 more