# The Optimum Leverage Level of the Banking Sector

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## 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.

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**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