Mathematical models for optimal management of bank capital, reserves and liquidity

Abstract

The aim of this study is to construct and propose continuous-time mathematical models for optimal management of bank capital, reserves and liquidity. This aim emanates from the global financial crisis of 2007 − 2009. In this regard and as a first task, our objective is to determine an optimal investment strategy for a commercial bank subject to capital requirements as prescribed by the Basel III Accord. In particular, the objective of the aforementioned problem is to maximize the expected return on the bank capital portfolio and minimize the variance of the terminal wealth. We apply classical tools from stochastic analysis to achieve the optimal strategy of a benchmark portfolio selection problem which minimizes the expected quadratic distance of the terminal risk capital reserves from a predefined benchmark. Secondly, the Basel Committee on Banking Supervision (BCBS) introduced strategies to protect banks from running out of liquidity. These measures included an increase of the minimum reserves that the bank ought to hold, in response to the global financial crisis. We propose a model to minimize risk for a bank by finding an appropriate mix of diversification, balanced against return on the portfolio. Thirdly and finally, in response to the financial crises, the Basel Committee on Banking Supervision (BCBS) designed a set of precautionary measures (known as Basel III) for liquidity imposed on banks and one of its purposes is to protect the economy from deteriorating. Recently, bank regulators wanted banks to depend on sources such as core deposits and long-term funding from small businesses and less on short-term wholesale funding.