# Optimal Risk Budgeting under a Finite Investment Horizon

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## Abstract

**:**

## 1. Introduction

## 2. An Example

## 3. The General Model

## 4. Return/Risk Paths

**Definition**

**1.**

- 1.
- f has piecewise continuous second order derivatives.
- 2.
- $f\left(a\right)=0$ and $f\left(b\right)=\kappa $.
- 3.
- $t\mapsto {l}_{X}\left(f\left(t\right)\right)$ is an increasing function on $[a,b]$.
- 4.
- There is a risk measure m on the leverage space such that $t\mapsto m\left(f\right(t\left)\right)$ is an increasing function on $[a,b]$.
- 5.
- $\nabla {l}_{X}\left(f\left(t\right)\right){f}^{\prime \prime}\left(t\right)=0$ for all $t\in [a,b]$ where ${f}^{\prime \prime}\left(t\right)$ exists.

**Theorem**

**1.**

**Proof.**

## 5. Determining the Manifold of Inflection Points Using Sylvester’s Criterion

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

## 6. Determining the Manifold of Return/Risk Maximum Points

**Definition**

**2.**

**Lemma**

**1.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 7. Applications

**Example**

**1.**

**Example**

**2.**

## 8. Conclusions and Further Research

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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1 |

**Figure 1.**Return/risk ratios as slopes with the top line at the tangent, the middle line at the growth optimal, and the bottom line at the inflection point.

Coin 1 | Coin 2 | Probability |
---|---|---|

2 | 1 | 0.3 |

2 | −1 | 0.2 |

−1 | 1 | 0.3 |

−1 | −1 | 0.2 |

$\mathit{\nu}$ | $\mathit{\zeta}$ | $\mathit{\kappa}$ | |
---|---|---|---|

Path 1 | (0.1885, 0.077) | (0.2175, 0.133) | (0.243, 0.18) |

Path 2 | (0.148, 0.1385) | (0.1935, 0.18) | (0.243, 0.18) |

Company A | Company B | Probability |
---|---|---|

−5000 | −9200 | 0.12 |

−5000 | 5000 | 0.48 |

15,300 | −9200 | 0.068 |

15,300 | 5000 | 0.32 |

Company A | Company B | Probability |
---|---|---|

−5000 | −9200 | 0.2 |

−5000 | 5000 | 0.36 |

15,300 | −9200 | 0.06 |

15,300 | 5000 | 0.38 |

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**MDPI and ACS Style**

López de Prado, M.; Vince, R.; Zhu, Q.J. Optimal Risk Budgeting under a Finite Investment Horizon. *Risks* **2019**, *7*, 86.
https://doi.org/10.3390/risks7030086

**AMA Style**

López de Prado M, Vince R, Zhu QJ. Optimal Risk Budgeting under a Finite Investment Horizon. *Risks*. 2019; 7(3):86.
https://doi.org/10.3390/risks7030086

**Chicago/Turabian Style**

López de Prado, Marcos, Ralph Vince, and Qiji Jim Zhu. 2019. "Optimal Risk Budgeting under a Finite Investment Horizon" *Risks* 7, no. 3: 86.
https://doi.org/10.3390/risks7030086