# The Optimum Leverage Level of the Banking Sector

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Methodology

- Kelly’s optimal point;
- The maximum return–risk ratio;
- Inflection point.

#### 2.1.1. Kelly Optimal Point

#### 2.1.2. Return–Risk Ratio

#### 2.1.3. Inflection Point

#### 2.2. Assumptions

- The bank issues the same amount of loans every year;
- The bank does not distribute dividends to shareholders;
- The ratio of deposits to equity ($\delta $) is known and it is 6.6. See (A1);
- Refinancing facility is available. When the market interest rates on loans decrease, borrowers have and always take the option to reduce their interest rates by refinancing.

#### 2.3. Problem Formulation

## 3. Estimating the Leverage Level

#### 3.1. Description of Data

- A 5-year treasury constant maturity rate, percent, annual, not seasonally adjusted (1954–2016).
- A 30-year fixed rate mortgage average in the United States, percent, annual, not seasonally adjusted (1971–2016).
- A 15-year fixed rate mortgage average in the United States, percent, annual, not seasonally adjusted (1992–2016).

#### 3.1.1. Projecting 30-Year Maturity Interest Rates from 1954 to 1970

#### 3.1.2. Projecting 15-Year Maturity Interest Rates from 1954 to 1991

#### 3.2. Numerical Simulation

#### Computation Results

#### 3.3. Performance Comparison

- Varying equity and;
- Fixed equity.

#### 3.3.1. Varying Equity

#### 3.3.2. Fixed Equity

#### 3.4. Comparison with Existing Results

## 4. Sensitivity Analysis

#### 4.1. One Fixed Loan Maturity

#### 4.1.1. Kelly’s Point

#### 4.1.2. Approximations

#### 4.1.3. Return–Drawdown Ratio

#### 4.1.4. Inflection Point

#### 4.2. Simulation Study

#### 4.3. Mixed of Loan Maturities in One Model

#### 4.4. Banker’s Strategies

- Change the leverage size by ${\kappa}_{\mu}(\mu )\Delta \mu $ or;
- Change the allocation sizes, $\Delta {\alpha}_{i}$’s.

#### 4.5. Mixed-Model: Performance Test

**Varying Equity**

**Fixing Equity**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

GOP | Growth Optimal Portfolio |

Fed | Federal Reserve Bank |

## Appendix A. Deposit to Equity Ratio

## Appendix B. Calculations

#### Appendix B.1. Total Returns—Single Model

#### Appendix B.2. Derivative of κ

#### Appendix B.3. Derivative of ζ_{Q}

#### Appendix B.4. Derivative of Inflection Point

#### Appendix B.5. Mixed of Loan—Derivatives

#### Appendix B.5.1. Kelly’s Point (GOP)

#### Appendix B.5.2. Return–Drawdown Ratio

#### Appendix B.5.3. Inflection Point

## Appendix C. Raw Data

- Economic Research Division, Federal Reserve Bank of St. Louis GS5: 5-year Treasury Constant Maturity Rate, Percent, Annual, Not Seasonally Adjusted.
- MORTGAGE15US: 15-year Fixed Rate Mortgage Average in the United States, Percent, Annual, Not Seasonally Adjusted.
- MORTGAGE30US: 30-year Fixed Rate Mortgage Average in the United States, Percent, Annual, Not Seasonally Adjusted.

## References

- Barron, Andrew R., and Thomas M. Cover. 1988. A Bound on the Financial Value of Information. IEEE Transactions of Information Theory 34: 1097–988. [Google Scholar] [CrossRef]
- Bell, Robert M., and Thomas M. Cover. 1980. Competitive Optimality of Logarithmic Investment. Mathematics of Operations Research 5: 161–166. [Google Scholar] [CrossRef]
- Berg, Tobias, and Jasmin Gider. 2017. What Explains the Difference in Leverage between Banks and Non-Banks? Journal of Financial and Quantitative Analysis 52: 2677–702. [Google Scholar] [CrossRef]
- Birge, John R., and Pedro Júdice. 2012. Long-term Bank Balance Sheet Management: Estimation and Simulation of Risk-Factors. Journal of Banking and Finance 37: 4711–20. [Google Scholar] [CrossRef]
- Browne, Sid. 2000. Risk-Constrained Dynamic Active Portfolio Management. Management Science 46: 1188–99. [Google Scholar] [CrossRef]
- Chekhlov, Alexei, Stanislav Uryasev, and Michael Zabarankin. 1993. Drawdown Measure in Portfolio Optimization. International Journal of Theoretical and Applied Finance 8: 13–58. [Google Scholar] [CrossRef]
- Chopra, Vijaya K., and William T. Ziemba. 1993. The Effect of Errors in Mean, Variance and Covariance Estimates on Optimal Portfolio Choice. Journal of Portfolio Management 19: 6–11. [Google Scholar] [CrossRef]
- Cover, Thomas M. 1991. Universal Portfolio. Mathematical Finance 1: 1–29. [Google Scholar] [CrossRef]
- Davis, Mark, and Sébastien Lleo. 2010. Fractional Kelly Strategies for Benchmarked Asset Management. Kelly Capital Growth Investment Criterion 10: 387–409. [Google Scholar]
- Gornall, Will, and Ilya A. Strebulaev. 2013. Financing as a Supply Chain: The Capital Structure of Banks and Borrowers. The National Bureau of Economic Research 43: 127–48. [Google Scholar]
- Grossman, Sanford J., and Zhongquan Zhou. 1993. Optimal investment strategies for controlling drawdowns. Mathematical Finance 3: 241–76. [Google Scholar] [CrossRef]
- Hahm, Joon-ho, Hyun Song Shin, and Kwanho Shin. 2013. Non-Core Bank Liabilities and Financial Vulnerability. Journal of Money, Credit and Banking 45: 3–36. [Google Scholar] [CrossRef]
- Hakansson, Nils H. 1970. Optimal Investment and Consumption Strategies Under Risk for a Class of Utility Functions. Econometrica 38: 587–607. [Google Scholar] [CrossRef]
- Harding, John P., Xiaozhong Liang, and Stephen L. Ross. 2013. Bank Capital Requirements, Capital Structure and Regulation. Journal of Financial Services Research 43: 127–48. [Google Scholar] [CrossRef]
- Kelly, John. L. 1956. A New Interpretation of Information Rate. Bell System Technical Journal 35: 917–26. [Google Scholar] [CrossRef]
- Latane, Henry. A. 1959. Criteria for Choice among Risky Ventures. Journal of Political Economy 67: 144–55. [Google Scholar] [CrossRef]
- MacLean, Leonard C., William T. Ziemba, and George Blazenko. 1992. Growth versus Security in Dynamic Investment Analysis. Management Science 38: 1562–85. [Google Scholar] [CrossRef]
- MacLean, Leonard C., William. T. Ziemba, and Yuming Li. 2005. Time to Wealth Goals in Capital Accumulation. Quantitative Finance 5: 343–55. [Google Scholar] [CrossRef]
- MacLean, Leonard C., Edward O. Thorp, and William T. Ziemba. 2010. Good and Bad Properties of Kelly Criterion. In Kelly Capital Growth Investment Criterion. Hackensack: World Scientific, pp. 32–58. [Google Scholar]
- MacLean, Leonard C., Edward. O. Thorp, and William T. Ziemba. 2011. Introduction to the Relationship of Kelly Optimization to Asset Allocation. In Kelly Capital Growth Investment Criterion. Hackensack: World Scientific, pp. 303–6. [Google Scholar]
- MacLean, Leonard C., Edward O. Thorp, and William. T. Zeimba. 2012. The Kelly Capital Growth Investment Criterion. Singapore: World Scientific, vol. 3. [Google Scholar]
- Maier-Paape, Stanislaus, and Qiji Zhu. 2018. A General Framework for Portfolio Theory—Part II: Drawdown risk measures. Risks 6: 76. [Google Scholar] [CrossRef]
- Markowitz, Harry M. 1976. Investment for the Long Run: New Evidence for an Old Rule. Journal of Finance 31: 273–86. [Google Scholar] [CrossRef]
- Ordentlich, Erik, and Thomas M. Cover. 1998. The Cost of Achieving the Best Portfolio in Hindsight. Mathematics of Operations Research 23: 960–82. [Google Scholar] [CrossRef]
- Phelps, Edmund S. 1962. The Accumulation of Risky Capital: A Sequential Utility Analysis. Econometrica 30: 729–43. [Google Scholar] [CrossRef]
- Samuelson, Paul A. 1979. Why We Should Not Make Mean Log of Wealth Big Though Years to Act are Long. Journal of Banking and Finance 3: 305–7. [Google Scholar] [CrossRef]
- Shannon, Claude E. 1948. A Mathematical Theory of Communication. Bell System Technical Journal 27: 379–423. [Google Scholar] [CrossRef]
- Thorp, Edward O. 1971. Portfolio Choice and Kelly Criterion. In Proceedings of the Business and Economics Section of the American Statistical Association, Fort Collins, CO, USA, August 23–26; pp. 215–224. [Google Scholar]
- Thorp, Edward O. 2011. Understanding the Kelly Criterion. Kelly Capital Growth Investment Criterion 3: 511–25. [Google Scholar]
- Vince, Ralph, and Qiji Jim Zhu. 2015. Optimal Betting Size for the Game of Blackjack. Risk Journals: Portfolio Management 4: 53–75. [Google Scholar] [CrossRef]
- Ziemba, William T., and Donald B. Hausch. 1986. Betting at the Racetrack. San Luis Obispo: Dr. Z. Investments. [Google Scholar]

**Figure 1.**The 5-year treasury constant maturity rate (X) vs. the 30-year fixed rate mortgage average (Y) in the United States.

**Figure 5.**Returns from $\kappa $-red, $\zeta $-blue, and $\nu $-green vs. year in x-axis for fixed equity.

**Figure 8.**Simulation of variation of $\kappa ,\zeta ,\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\nu $.

**Figure 9.**Variation of $\kappa \phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\nu $.

**Figure 11.**Return on changing leverage levels $\kappa $-red, $\zeta $-blue, and $\nu $-green vs. year in x-axis.

**Figure 12.**Return on changing leverage levels $\kappa $-red, $\zeta $-blue, and $\nu $-green vs. year in x-axis.

Horizon | GOP ($\mathit{\kappa}$) | Ret/Drawdown (${\mathit{\zeta}}_{\mathit{Q}}$) | Inflection (${\mathit{\nu}}_{\mathit{Q}}$) |
---|---|---|---|

50 years | 13.2 | 10.2 | 8 |

30 years | 13.2 | 8.8 | 6.3 |

20 years | 13.2 | 7.8 | 5.6 |

Horizon | GOP ($\mathit{\kappa}$) | Ret/Drawdown (${\mathit{\zeta}}_{\mathit{Q}}$) | Inflection (${\mathit{\nu}}_{\mathit{Q}}$) |
---|---|---|---|

50 years | 16.7 | 14.0 | 10.6 |

30 years | 16.7 | 12.0 | 8.7 |

20 years | 16.7 | 10.9 | 7.8 |

Loans | Third-Moment |
---|---|

30-years | $2.577232\times {10}^{-6}$ |

15-years | $1.659951\times {10}^{-6}$ |

$\Delta {\mathit{\alpha}}_{1}$ | Kelly (GOP) | Return-Drawdown Ratio | Inflection Point |
---|---|---|---|

50-years | 0.04379085 | 0.56454794 | 0.02352097 |

30-years | 0.04379085 | 1.16005581 | 0.03888651 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Dewasurendra, S.; Judice, P.; Zhu, Q. The Optimum Leverage Level of the Banking Sector. *Risks* **2019**, *7*, 51.
https://doi.org/10.3390/risks7020051

**AMA Style**

Dewasurendra S, Judice P, Zhu Q. The Optimum Leverage Level of the Banking Sector. *Risks*. 2019; 7(2):51.
https://doi.org/10.3390/risks7020051

**Chicago/Turabian Style**

Dewasurendra, Sagara, Pedro Judice, and Qiji Zhu. 2019. "The Optimum Leverage Level of the Banking Sector" *Risks* 7, no. 2: 51.
https://doi.org/10.3390/risks7020051