Optimal Capital Management in Banking

With the drafting of new banking regulation via the Basel II capital accord, bank regulatory capital and its adequacy has become the subject of much debate. In our contribution, we strive to construct a stochastic dynamic model to describe the evolution of bank capital that incorporates capital gains and losses. In our paper, such gains and losses are represented by loan loss reserves and the unexpected loan losses, respectively. In this regard, we recognize that bank capital consists of Tier 1 capital (mainly equity which is modeled via an exponential Lévy process), Tier 2 and Tier 3 capital (collectively known as supplementary capital). The latter two types of capital mainly consist of (shortand long-term) subordinate debt held by debtholders and loan losses reserves. Furthermore, we setup an optimal capital management problem which maximizes the expectation of bank capital under a risk constraint on the Capital-at-Risk (CaR), where CaR is defined in terms of Valueat-Risk (VaR). In particular, we seek an optimal bank capital management strategy in a mean-CaR paradigm. Historical evidence from Organization for Economic Corporation and Development (OCED) countries assists us in establishing the relationship between output gap (as the proxy of business cycle) and capital adequacy ratios (CARs). Finally, we provide a brief analysis of some of the optimal capital management issues and suggestions for topics of possible future direction.