An Optimal Portfolio and Capital Management Strategy for Basel III Compliant Commercial Banks

Hindawi Publishing Corporation Journal of Applied Mathematics Volume 2014, Article ID 723873, 11 pages http://dx.doi.org/10.1155/2014/723873 Research Article An Optimal Portfolio and Capital Management Strategy for Basel III Compliant Commercial Banks Grant E. Muller and Peter J. Witbooi University of the Western Cape, Private Bag X17, Bellville 7535, South Africa Correspondence should be addressed to Peter J. Witbooi; Received 3 October 2013; Accepted 5 January 2014; Published 19 February 2014 Academic Editor: Francesco Pellicano Copyright © 2014 G. E. Muller and P. J. Witbooi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We model a Basel III compliant commercial bank that operates in a financial market consisting of a treasury security, a marketable security, and a loan and we regard the interest rate in the market as being stochastic. We find the investment strategy that maximizes an expected utility of the bank’s asset portfolio at a future date. This entails obtaining formulas for the optimal amounts of bank capital invested in different assets. Based on the optimal investment strategy, we derive a model for the Capital Adequacy Ratio (CAR), which the Basel Committee on Banking Supervision (BCBS) introduced as a measure against banks’ susceptibility to failure. Furthermore, we consider the optimal investment strategy subject to a constant CAR at the minimum prescribed level. We derive a formula for the bank’s asset portfolio at constant (minimum) CAR value and present numerical simulations on different scenarios. Under the optimal investment strategy, the CAR is above the minimum prescribed level. The value of the asset portfolio is improved if the CAR is at its (constant) minimum value. 1. Introduction Successful bank management can be achieved by addressing four operational concerns. Firstly, the bank should be able to finance its obligations to depositors. This aspect of bank management is called liquidity management. It involves the bank acquiring sufficient liquid assets to meet the demands from deposit withdrawals and depositor payments. Secondly, banks must engage in liability management. This aspect of bank management entails the sourcing of funds at an acceptable cost. Thirdly, banks are required to invest in assets that have a reasonably low level of risk associated with them. This process is referred to as asset management. It aims to encourage the bank to invest in assets that are not likely to be defaulted on and to adopt investment strategies that are sufficiently diverse. The fourth and final operational concern is capital adequacy management. Capital adequacy management involves the decision about the amount of capital the bank should hold and how it should be accessed. From a shareholder’s perspective, utilizing more capital will increase asset earnings and will lead to higher returns on equity. From the regulator’s perspective, banks should increase their buffer capital to ensure the safety and soundness in the case where earnings may end up below an expected level. In this paper, we address problems associated with asset and capital adequacy management. The Basel Committee on Banking Supervision (BCBS) regulates and supervises the international banking industry by imposing minimal capital requirements and other measures. The 1998 Basel Capital Accord, also known as the Basel I Accord, aimed to assess the bank’s capital in relation to its credit risk, or the risk of a loss occurring if a party does not fulfill its obligations. The Basel I Accord launched the trend toward increasing risk modeling research. However, its oversimplified calculations and classifications have simultaneously called for its disappearance. This paved the way for the Basel II Capital Accord and further agreements as the symbol of the continuous refinement of risk and capital. The 2004 (revised) framework of the Basel II Capital Accord (see [1]) laid down regulations seeking to provide incentives for greater awareness of differences in risk through more risk-sensitive minimum capital requirements based on numerical formulas. The Total Capital Ratio or Capital Adequacy Ratio (CAR) (see for instance [2–6]) measures the amount of the bank’s capital relative to its amount of credit exposures. Internationally, a standard has been adopted 2 Journal of Applied Mathematics that requires banks to adhere to minimum levels of capital requirements. By complying with minimum capital levels, banks are guaranteed the ability to absorb reasonable levels of losses before becoming insolvent. Thus, CARs ensure the safety and stability of the banking system. Under the Basel II Accord, banks were required to maintain a CAR, that is, a ratio of total bank capital to total risk-weighted assets (TRWAs), with a minimum value of 8%. In response to the 2007-2008 financial crisis, the Basel Committee on Banking Supervision (BCBS) introduced a comprehensive set of reform measures known as the Basel III Accord. The Basel III Accord is aimed at improving the regulation, supervision, and risk management within the banking sector and is part of the continuous effort made by the BCBS to enhance the banking regulatory framework. The Basel III Accord builds on the Basel I and II documents and seeks to improve the banking sector’s ability to deal with financial and economic stress, improve risk management, and strengthen the banks’ transparency. The focus of the Basel III Accord is to foster greater resilience at the individual bank level in order to reduce the risk of system wide shocks. In this regard, the Basel III Accord contains changes in the following areas: (i) augmentation in the level and quality of capital; (ii) introduction of liquidity standards; (iii) modifications in provisioning norms; (iv) introduction of a leverage ratio. For a detailed discussion on the aforementioned enhancements in the Basel III Accord over the Basel II Accord, see the paper [7] of Jayadev or the book [8] of Petersen and MukuddemPetersen. In our paper, we are concerned with the level of bank capital. Thus we study the Total Capital Ratio or CAR which, if denoted by Λ, is defined as Λ= 𝐶 . 𝑎rw (1) In the expression for the CAR above, 𝐶 denotes the total bank capital and 𝑎rw the TRWAs of the bank. Under Basel III rules, the minimum prescribed value of the bank’s CAR remains unchanged at 8%. However, banks are now required to hold a Capital Conservation Buffer (CCB) of 2.5% and Countercyclical Buffer Capital (CBC) in the range of 0– 2.5%. The CCB ensures that banks can absorb losses without breaching the minimum capital requirement and are able to carry on business even in a downturn without deleveraging. The CBC is a preemptive measure that requires banks to build up capital gradually as imbalances in the credit market develop. It may be in the r (...truncated)