An Optimal Investment Strategy and Multiperiod Deposit Insurance Pricing Model for Commercial Banks

Hindawi Journal of Applied Mathematics Volume 2018, Article ID 8942050, 10 pages https://doi.org/10.1155/2018/8942050 Research Article An Optimal Investment Strategy and Multiperiod Deposit Insurance Pricing Model for Commercial Banks Grant E. Muller Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville, Cape Town 7535, South Africa Correspondence should be addressed to Grant E. Muller; Received 3 November 2017; Revised 14 February 2018; Accepted 19 March 2018; Published 2 May 2018 Academic Editor: Lucas Jodar Copyright © 2018 Grant E. Muller. 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 employ the method of stochastic optimal control to derive the optimal investment strategy for maximizing an expected exponential utility of a commercial bank’s capital at some future date 𝑇 > 0. In addition, we derive a multiperiod deposit insurance (DI) pricing model that incorporates the explicit solution of the optimal control problem and an asset value reset rule comparable to the typical practice of insolvency resolution by insuring agencies. By way of numerical simulations, we study the effects of changes in the DI coverage horizon, the risk associated with the asset portfolio of the bank, and the bank’s initial leverage level (deposit-to-asset ratio) on the DI premium while the optimal investment strategy is followed. 1. Introduction Stochastic optimization theory is an extremely important tool to finance as it can be used to solve a vast array of stochastic optimization problems. In banking, specifically, numerous authors have applied the stochastic optimal control technique to solve a variety of optimization problems. Common types of banking optimization problems solved with this approach are optimal asset allocation and optimal capital adequacy management problems. The former entails investing bank funds in assets with the goal of reaching an optimal fund level, while the latter involves optimizing the capital adequacy ratios. The Basel Committee on Banking Supervision introduced the capital adequacy ratios as a measure of the financial strength of banks and other financial institutions. Optimization problems as described above can, for instance, be observed in the papers Fouche et al. [1], Mukuddem-Petersen and Petersen [2, 3], Muller and Witbooi [4], and Chakroun and Abid [5]. Under Basel II, Fouche et al. [1] modelled the capital adequacy ratios and studied an optimal control problem in which they derived an optimal asset allocation strategy for the non-risk-based Leverage Ratio on a given time interval. In particular, Fouche et al. [1] determined the optimal expected terminal utility of the Leverage Ratio and derived the asset allocation strategy that makes it possible to maximize the expected terminal utility of the Leverage Ratio on the given time interval. In their paper [2], Mukuddem-Petersen and Petersen studied a banking problem related to the optimal risk management of banks in a stochastic dynamic setting. The authors of [2] particularly minimized market and capital adequacy risk that, respectively, involves the safety of the securities held and the stability of sources of funds. In this regard, Mukuddem-Petersen and Petersen [2] suggested an optimal portfolio choice and rate of bank capital inflow that will keep the loan level as close as possible to an actuarially determined reference process. This setup leads to a nonlinear stochastic optimal control problem whose solution they determined via the dynamic programming algorithm. In Mukuddem-Petersen and Petersen [3] the authors derived the dynamics of the risk-based Total Capital Ratio or CAR in a stochastic setting. MukuddemPetersen and Petersen [3] further demonstrated how the CAR can be optimized in terms of bank equity allocation and the rate at which additional debt and equity are raised. In their analysis, Mukuddem-Petersen and Petersen [3] employed the dynamic programming algorithm for stochastic optimization. Muller and Witbooi [4] maximized an expected utility of a commercial bank’s asset portfolio/total asset value at a future date by finding the optimal amounts of capital for investment in different assets. Under the optimal investment 2 strategy, the authors of [4] monitored the level of the bank’s CAR numerically. In addition, by controlling the capital of the bank, Muller and Witbooi [4] derived a modified expression for the bank’s asset portfolio under which the CAR will remain fixed at exactly the Basel III prescribed minimum level of 8%. Chakroun and Abid [5] addressed the problem of optimal portfolio choice for banks, for which the bank’s shareholders have a power utility function and the financial market consists of a bank account, securities, and loans. Chakroun and Abid [5] derived the solution to their problem by following the dynamic programming principle. The authors of [5] performed a simulation with their optimal solution with parameters based on the maximum likelihood method. The simulation confirms the practical potential of the results and shows that this model can adequately account for the essential aspects of the bank. van Schalkwyk and Witbooi [6] studied a control problem related to bank liquidity management in a jump diffusion setting. Their paper [6] has close connections with the aforementioned literature in the sense that it established optimal liquidity and a rate of depository consumption that is deemed important during a (random) auditing process of the reserve requirements. The authors of [6] particularly investigate the interplay between a commercial bank and a central bank and what effects the interplay has on the money supply between the two institutions as well as the Liquidity Coverage Ratio (LCR). The LCR was introduced under the Basel III Accord (see [7]) to promote resilience against potential liquidity disruptions over a 30-day horizon (short-term stress scenario) [7]. Their motivation for studying the dynamics of the LCR is to show that, in principle, banks are able to control their liquidity by following an appropriate provisioning strategy. In doing so, the LCR does not fall below an acceptable level. The purpose of deposit insurance (DI) is to protect banks’ depositors against the risks associated with the failure of banks and other depository institutions. DI claims from a pool of funds to which every depository institution regularly contributes and will cover only a fixed maximum amount per bank depositor. The feeling of security that a bank’s depositors have when their deposits are insured reduces the type of fear that caused bank runs in the 1930s. The DIF number of a bank serves as a measure of how a bank’s assets compare to those of problematic banks appearing on the Federal Deposit Insurance Corporation’s (FDIC) quarterl (...truncated)