# Nash Bargaining Over Margin Loans to Kelly Gamblers

## Abstract

**:**

## 1. Introduction

#### Contribution

## 2. Definitions and Notation

**asymptotic growth rate**is denoted

## 3. Nash Bargaining

**Example**

**1.**

**Example**

**2.**

**Proposition**

**1.**

**Theorem**

**1.**

**Example**

**3.**

**Example**

**4.**

#### Utility Possibility Frontier

## 4. Monopoly Threat Point

**Example**

**5.**

## 5. Several Risk Assets

#### 5.1. Monopoly Disagreement Point

#### 5.2. General Solution for CRRA Utility

## 6. Summary and Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | This was on 12 March 2019. The rate has dropped since then, to 3.62% as of 13 August 2019. |

2 | The principals are assumed to re-negotiate the margin loan contract after every differential tick $dt$ of the market clock. However, the Nash bargaining solution never actually changes, on account of the fact that the Nash product is directly proportional to the client’s wealth ${V}_{t}\left(b\right)$. |

**Figure 1.**Kelly gambler’s instantaneous demand for margin loans per dollar of account equity, $\nu :=0.09,\sigma :=0.15,\mu :=\nu +{\sigma}^{2}/2,r:=0.03$.

**Figure 2.**Gains from cooperation under Nash bargaining, $\nu :=0.09,\sigma :=0.15,\mu :=\nu +{\sigma}^{2}/2,r:=0.03$.

Tier | Interest Rate | $log(1+\mathbf{Interest}\phantom{\rule{4.pt}{0ex}}\mathbf{Rate})$ | Kelly Bet, b |
---|---|---|---|

0–100,000 | 3.9% | 3.83% | 2.8 |

100,000.01–1,000,000 | 3.4% | 3.34% | 3.01 |

1,000,000.01–3,000,000 | 2.9% | 2.86% | 3.23 |

3,000,000.01–200,000,000 | 2.7% | 2.66% | 3.32 |

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**MDPI and ACS Style**

Garivaltis, A.
Nash Bargaining Over Margin Loans to Kelly Gamblers. *Risks* **2019**, *7*, 93.
https://doi.org/10.3390/risks7030093

**AMA Style**

Garivaltis A.
Nash Bargaining Over Margin Loans to Kelly Gamblers. *Risks*. 2019; 7(3):93.
https://doi.org/10.3390/risks7030093

**Chicago/Turabian Style**

Garivaltis, Alex.
2019. "Nash Bargaining Over Margin Loans to Kelly Gamblers" *Risks* 7, no. 3: 93.
https://doi.org/10.3390/risks7030093