M. A. Petersen

Bank management via stochastic optimal control

This paper examines a problem related to the optimal risk management of banks in a stochastic dynamic setting. In particular, we minimize market and capital adequacy risk that involves the safety of the securities held and the stability of sources of funds, respectively. In this regard, we suggest 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 set-up leads to a nonlinear stochastic optimal control problem whose solution may be determined by means of the dynamic programming algorithm. The above analysis is reliant on the construction of continuous-time stochastic models for bank behaviour upon which a spread method for loan capitalization is imposed.

Dynamic modelling of bank profits

A topical issue in financial economics is the development of a stochastic dynamic model for bank behaviour. Under the assumption that the loan market is imperfectly competitive, we investigate the evolution of banking items such as loans, provisions for loan losses and deposit withdrawals, Treasuries and deposits and their relationship with profit. A motivation for studying this type of problem is the need to generalize the more traditional discrete-time models that are being used in the majority of studies that analyse banks and their operational idiosyncracies. An important outcome of our research is an explicit model for bank profit based solely on the stochastic dynamics of bank assets (loans, Treasuries and reserves) and liabilities (deposits). By way of conclusion, we provide a brief discussion of some of the economic aspects of the dynamic bank modelling undertaken in the main body of the article.

Stochastic behaviour of risk-weighted bank assets under the Basel II capital accord

The primary objective of this paper is to add to the growing debate about the impact of the Basel II Capital Accord (see Base II, June 2004) on the functioning of internationally active banks. A technical contribution is made to this discussion by constructing a stochastic continuous-time model for the dynamics of the total risk-weighted assets (RWAs) of such a bank. RWAs exhibit random behaviour because they partly depend on the uncertain rates of return of bank investments. Also, such assets are weighted by considering the main risks that banks have to bare, viz., credit, operational and market risk. Here the credit risk capital requirement is viewed from the perspective of the internal ratings-based (IRB) approach, operational risk capital is considered within the framework of the standardized approach and a Value-at-Risk (VaR) model describes the capital charge for market risk. Total RWAs (TRWAs) are used to calculate the capital adequacy ratio that is regarded as the most important component of bank supervision and risk management. In particular, according to Basel II, the capital adequacy of banks is determined by the ratio of eligible regulatory capital to TRWAs.

Inner-outer factorization for nonlinear noninvertible systems

This paper considers inner-outer factorization of asymptotically stable nonlinear state space systems in continuous time that are noninvertible. Our approach will be via a nonlinear analogue of spectral factorization which concentrates on first finding the outer factor instead of the inner factor. An application of the main result to control of nonminimum phase nonlinear systems is indicated.

Maximizing Banking Profit on a Random Time Interval

We study the stochastic dynamics of banking items such as assets, capital, liabilities and profit. A consideration of these items leads to the formulation of a maximization problem that involves endogenous variables such as depository consumption, the value of the banks investment in loans, and provisions for loan losses as control variates. A solution to the aforementioned problem enables us to maximize the expected utility of discounted depository consumption over a random time interval, [ t , τ ] , and profit at terminal time τ . Here, the term depository consumption refers to the consumption of the banks profits by the taking and holding of deposits. In particular, we determine an analytic solution for the associated Hamilton-Jacobi-Bellman (HJB) equation in the case where the utility functions are either of power, logarithmic, or exponential type. Furthermore, we analyze certain aspects of the banking model and optimization against the regulatory backdrop offered by the latest banking regulation in the form of the Basel II capital accord. In keeping with the main theme of our contribution, we simulate the financial indices return on equity and return on assets that are two measures of bank profitability.

Optimal allocation between bank loans and treasuries with regret

The main categories of assets held by banks are loans, Treasuries (bonds issued by the national Treasury), reserves and intangible assets. In our contribution, we investigate the investment of bank funds in loans and Treasuries with the aim of generating an optimal final fund level. Our results take behavioral aspects such as risk and regret into account. More specifically, we apply a branch of optimization theory that enables us to consider a regret attribute alongside a risk component as an integral part of the utility function. In this case, regret-aversion corresponds to the convexity of the regret function and the bank’s preference is assumed to be representable by optimization subject to the utility. In addition, we provide a comparison between risk- and regret-averse banks in terms of optimal asset allocation between loans and Treasuries. A feature of our contribution is that these and other optimization issues are analyzed briefly and, where possible, represented graphically. Furthermore, we comment on the claim that an investment away from loans towards Treasuries is responsible for credit crunches in the banking industry.

Bank Valuation and Its Connections with the Subprime Mortgage Crisis and Basel II Capital Accord

The ongoing subprime mortgage crisis (SMC) and implementation of Basel II Capital Accord regulation have resulted in issues related to bank valuation and profitability becoming more topical. Profit is a major indicator of financial crises for households, companies, and financial institutions. An SMC-related example of this is the U.S. bank, Wachovia Corp., which reported major losses in the first quarter of 2007 and eventually was bought by Citigroup in September 2008. A first objective of this paper is to value a bank subject to Basel II based on premiums for market, credit, and operational risk. In this case, we investigate the discrete-time dynamics of banking assets, capital, and profit when loan losses and macroeconomic conditions are explicitly considered. These models enable us to formulate an optimal bank valuation problem subject to cash flow, loan demand, financing, and balance sheet constraints. The main achievement of this paper is bank value maximization via optimal choices of loan rate and supply which leads to maximal deposits, provisions for deposit withdrawals, and bank profitability. The aforementioned loan rates and capital provide connections with the SMC. Finally, OECD data confirms that loan loss provisioning and profitability are strongly correlated with the business cycle.

A note on the subprime mortgage crisis: dynamic modelling of bank leverage profit under loan securitization

In this brief research article, we consider the financial modelling of the process of mortgage loan securitization that has been a root cause of the ongoing Subprime Mortgage Crisis (SMC). In particular, we suggest a Lévy process-driven model of bank leverage profit that arises from the securitization of a pool of subprime mortgage loans. To achieve this, we develop stochastic models for mortgage loans, mortgage loan losses, credit ratings and mortgage loan guarantees in a subprime context. These models incorporate some of the most important issues related to the SMC and its causes. Finally, we provide a brief analysis of the models developed earlier in our contribution and its relationship with the SMC.

Minimizing Banking Risk in a Lévy Process Setting

The primary functions of a bank are to obtain funds through deposits from external sources and to use the said funds to issue loans. Moreover, risk management practices related to the withdrawal of these bank deposits have always been of considerable interest. In this spirit, we construct Levy process-driven models of banking reserves in order to address the problem of hedging deposit withdrawals from such institutions by means of reserves. Here reserves are related to outstanding debt and acts as a proxy for the assets held by the bank. The aforementioned modeling enables us to formulate a stochastic optimal control problem related to the minimization of reserve, depository, and intrinsic risk that are associated with the reserve process, the net cash flows from depository activity, and cumulative costs of the banks provisioning strategy, respectively. A discussion of the main risk management issues arising from the optimization problem mentioned earlier forms an integral part of our paper. This includes the presentation of a numerical example involving a simulation of the provisions made for deposit withdrawals via treasuries and reserves.

Subprime mortgage funding and liquidity risk

In this article, we use actuarial methods to solve a nonlinear stochastic optimal liquidity risk management problem for subprime originators with deposit inflow rates and marketable securities allocation as controls. The main objective is to minimize liquidity risk in the form of funding and credit crunch risk in an incomplete market. In order to accomplish this, we construct a stochastic model that incorporates originator mortgage and deposit reference processes. Finally, numerical examples that illustrate the main modeling and optimization features of the article are provided.