We consider capital adequacy management and an asset allocation problem for a bank under a risk process modeled by a jump-diffusion process. In our framework, the bank optimizes the amount for investment in the risky assets, a stock and loan portfolio using the constant absolute risk aversion (CARA) utility function. We use separate jump processes to model expected losses, covered by the loan loss provision, and unexpected losses. The bank's liability dynamics are governed by a Cramér–Lundberg model. Using the martingale approach, we derive closed-form optimal allocations for stocks and loans, which we subsequently use to model the Capital Adequacy Ratio (CAR) under Basel III/IV.

We calibrate jump-diffusion parameters for the equity process using S&P 500 daily data from 2015 to 2024, and estimate loan dynamics from the Federal Reserve's TOTLL aggregate (Loans and Leases in Bank Credit, All Commercial Banks) using the method of moments. The parameter $\beta$, which drives the systematic drift of the CAR process $\Gamma$ in the model, is estimated directly from the FDIC Quarterly Banking Profile data. Risk process parameters are benchmarked from the existing literature on stochastic bank management.

We perform 10,000 simulations for $\Gamma$ under three values of the risk aversion parameter $\delta \in \{0.5, 1.0, 2.0\}$ and find that the breach probability $P(\Gamma < 8\%)$ equals 1 under the estimated asset cost factor $\beta = 0.0275$. Setting $\beta = 0$ produces zero breach events across 10,000 simulated trajectories and a terminal mean CAR of approximately 11.95%, essentially unchanged from the initial CAR of 12%. This establishes that CAR erosion is driven entirely by the asset cost factor. Our analysis also shows that higher risk aversion slightly increases the terminal mean CAR but does not change the breach probability.

Sensitivity analysis confirms the robustness of our conclusions. When loan volatility $\sigma_2$ increases from the system-level estimate toward the individual-bank range documented in the literature, CAR compresses further. If we vary the risk-free rate $r$, the core results remain unchanged for $r \leq 3\%$; at high rates the optimal loan allocation turns negative, which is consistent with the observed shift of U.S. banks toward Treasury holdings in 2022-2023.

In further analysis, we will derive risk-neutral probabilities of default (PD) for individual institutions from CDS spreads and use high-frequency equity return data to estimate realized asset return correlations. We will also consider institution-level PDs and the associated loss-given-default (LGD) parameters from CDS market data for the loan appreciation rate $\mu_L$ and for the jump size $\gamma_2$ governing expected losses in the risk process, comparing the analysis under the aggregate TOTLL proxy with institution-level credit risk measures. Our framework will thereby connect micro-level solvency dynamics, captured through the jump-diffusion CAR model, to macro-prudential stress conditions, providing an empirical tool for assessing bank resilience under the Basel IV output floor constraint.

Keywords: capital adequacy management, martingale approach, asset allocation, systemic risk, credit default swap.

This is joint work with Dr. Luis Enrique Garcia-Perez (Universidad de las Américas Puebla, Mexico).