Journal of Applied Mathematics

Maximizing Banking Profit on a Random Time Interval

J. Mukuddem-Petersen, M. A. Petersen, I. M. Schoeman, and B. A. Tau

Full-text: Open access


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 bank's 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 bank's 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.

Article information

J. Appl. Math., Volume 2007 (2007), Article ID 29343, 22 pages.

First available in Project Euclid: 5 July 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Mukuddem-Petersen, J.; Petersen, M. A.; Schoeman, I. M.; Tau, B. A. Maximizing Banking Profit on a Random Time Interval. J. Appl. Math. 2007 (2007), Article ID 29343, 22 pages. doi:10.1155/2007/29343.

Export citation


  • Basel Committee on Banking Supervision, ``The New Basel Capital Accord,'' 2001, Bank for International Settlements,
  • Basel Committee on Banking Supervision, ``International Convergence of Capital Measurement and Capital Standards; A Revised Framework,'' June 2004, Bank for International Settlements,
  • F. Modigliani and M. H. Miller, ``The cost of capital, corporation finance and the theory of investment,'' The American Economic Review, vol. 48, no. 3, pp. 261--297, 1958.
  • A. N. Berger, R. J. Herring, and G. P. Szegö, ``The role of capital in financial institutions,'' Journal of Banking & Finance, vol. 19, no. 3-4, pp. 393--430, 1995.
  • D. W. Diamond and R. G. Rajan, ``A theory of bank capital,'' The Journal of Finance, vol. 55, no. 6, pp. 2431--2465, 2000.
  • H. E. Leland, ``Corporate debt value, bond covenants, and optimal capital structure,'' The Journal of Finance, vol. 49, no. 4, pp. 1213--1252, 1994.
  • J.-C. Rochet, ``Capital requirements and the behaviour of commercial banks,'' European Economic Review, vol. 36, no. 5, pp. 1137--1170, 1992.
  • T. Dangl and J. Zechner, ``Credit risk and dynamic capital structure choice,'' Journal of Financial Intermediation, vol. 13, no. 2, pp. 183--204, 2004.
  • J.-P. Decamps, J.-C. Rochet, and B. Roger, ``The three pillars of Basel II: optimizing the mix,'' Journal of Financial Intermediation, vol. 13, no. 2, pp. 132--155, 2004.
  • M. A. Petersen and J. Mukuddem-Petersen, ``Stochastic behaviour of risk-weighted bank assets under the Basel II capital accord,'' Applied Financial Economics Letters, vol. 1, no. 3, pp. 133--138, 2005.
  • R. Repullo, ``Capital requirements, market power, and risk-taking in banking,'' Journal of Financial Intermediation, vol. 13, no. 2, pp. 156--182, 2004.
  • J.-C. Rochet, ``Rebalancing the three pillars of Basel II,'' Economic Policy Review, vol. 10, no. 2, pp. 7--21, 2004.
  • D. Hancock, A. J. Laing, and J. A. Wilcox, ``Bank capital shocks: dynamic effects on securities, loans, and capital,'' Journal of Banking & Finance, vol. 19, no. 3-4, pp. 661--677, 1995.
  • S. Altug and P. Labadie, Dynamic Choice and Asset Markets, Academic Press, San Diego, Calif, USA, 1994.
  • R. Bliss and G. Kaufman, ``Bank procyclicality, credit crunches, and asymmetric monetary policy effects: a unifying model,'' Working Paper 2002-18, Federal Reserve Bank of Chicago, Chicago, Ill, USA, 2002.
  • E. Catarineu-Rabell, P. Jackson, and D. P. Tsomocos, ``Procyclicality and the new Basel Accord-banks' choice of loan rating system,'' Working Paper 181, Bank of England, London, UK, 2003.
  • R. Chami and T. F. Cosimano, ``Monetary policy with a touch of Basel,'' Working Paper 01/151, International Monetary Fund, Washington, DC, USA, 2001.
  • D. Tsomocos, ``Equilibrium analysis, banking, contagion and financial fragility,'' Working Paper 175, Bank of England, London, UK, 2003.
  • S. van den Heuwel, ``The bank capital channel of monetary policy,'' 2001, Mimeo, University of Pennsylvania.
  • S. van den Heuwel, ``Does bank capital matter for monetary transmission?'' Economic Policy Review, vol. 8, no. 1, pp. 259--265, 2002.
  • D. Hackbarth, J. Miao, and E. Morellec, ``Capital structure, credit risk, and macroeconomic conditions,'' Journal of Financial Economics, vol. 82, no. 3, pp. 519--550, 2006.
  • R. A. Korajczyk and A. Levy, ``Capital structure choice: macroeconomic conditions and financial constraints,'' Journal of Financial Economics, vol. 68, no. 1, pp. 75--109, 2003.
  • J. A. Bikker and P. A. J. Metzemakers, ``Bank provisioning behaviour and procyclicality,'' Journal of International Financial Markets, Institutions and Money, vol. 15, no. 2, pp. 141--157, 2005.
  • C. Borio, C. Furfine, and P. Lowe, ``Procyclicality of the financial system and financial stability: issues and policy options,'' Working Paper 1, Bank for International Settlements, Basel, Switzerland, 2001,
  • M. Cavallo and G. Majnoni, ``Do banks provision for bad loans in good times? empirical evidence and policy implications,'' in Ratings, Rating Agencies and the Global Financial System, R. Levich, G. Majnoni, and C. Reinhart, Eds., Kluwer Academics Publishers, Boston, Mass, USA, 2002.
  • J. Mukuddem-Petersen and M. A. Petersen, ``Bank management via stochastic optimal control,'' Automatica, vol. 42, no. 8, pp. 1395--1406, 2006.
  • J. Mukuddem-Petersen, M. A. Petersen, I. M. Schoeman, and B. A. Tau, ``A profit maximization problem for depository financial institutions,'' in Proceedings of the International Conference on Modelling and Optimization of Structures, Processes and Systems (ICMOSPS '07), Durban, South Africa, January 2007.
  • J. Mukuddem-Petersen, M. A. Petersen, I. M. Schoeman, and B. A. Tau, ``An application of stochastic optimization theory to institutional finance,'' Applied Mathematical Sciences, vol. 1, no. 28, pp. 1359--1385, 2007.
  • L. Laeven and G. Majnoni, ``Loan loss provisioning and economic slowdowns: too much, too late?'' Journal of Financial Intermediation, vol. 12, no. 2, pp. 178--197, 2003.
  • T. Björk, Arbitrage Theory in Continuous Time, Oxford University Press, New York, NY, USA, 1998.
  • The Federal Deposit Insurance Corporation (FDIC), March 2007,
  • R. C. Merton, Continuous-Time Finance, Blackwell Publishers, Cambridge, Mass, USA, 1992.
  • J. C. Cox and C.-F. Huang, ``A variational problem arising in financial economics,'' Journal of Mathematical Economics, vol. 20, no. 5, pp. 465--487, 1991.
  • I. Karatzas, J. P. Lehoczky, and S. E. Shreve, ``Optimal portfolio and consumption decisions for a ``small investor'' on a finite horizon,'' SIAM Journal on Control and Optimization, vol. 25, no. 6, pp. 1557--1586, 1987.
  • C. H. Fouche, J. Mukuddem-Petersen, and M. A. Petersen, ``Continuous-time stochastic modelling of capital adequacy ratios for banks,'' Applied Stochastic Models in Business and Industry, vol. 22, no. 1, pp. 41--71, 2006.
  • P. Jackson and W. Perraudin, ``Regulatory implications of credit risk modelling,'' Journal of Banking & Finance, vol. 24, no. 1-2, pp. 1--14, 2000.