In keeping with the main theme of this thesis, we simulate the financial indices ROE and ROA that are two measures of bank profitability. We further discuss optimization with power utility where we show the convergence of the Markov Chain Approximation Method (MCAM) and the impact of varying the model parameters in the form of loan loss severity, and loan loss frequency. We investigate the connections between the banking models and Basel II capital accord as well as the current sub-prime mortgage crises. As way of conclusion, we provide remarks about the main issues discussed in the thesis...