# The Properties of Alpha Risk Parity Portfolios

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Framework

#### 2.1. Literature Review

#### 2.1.1. Risk Parity

#### 2.1.2. Using Entropy and Information Geometry for Portfolio Allocation

#### 2.2. Notations and Assumptions

#### 2.3. Equal Risk Contributions and Risk Budgeting

#### 2.3.1. The Direct Approach

#### 2.3.2. The Variational Approach

#### 2.4. A Generalized Approach to Risk Budgeting

#### 2.4.1. Using a Generalized Logarithm

#### 2.4.2. Introducing Alpha Risk Parity

**Definition**

**1.**

#### 2.5. Properties of Alpha Risk Parity

#### 2.5.1. A Simple Case

**Proposition**

**1.**

#### 2.5.2. Independence from the Constraint

**Proposition**

**2.**

**Proposition**

**3.**

**Proposition**

**4.**

- For any $\alpha <-1$, the alpha risk parity portfolios are found by normalizing the solution to optimization problem:$$min\left(\right)open="\{"\; close="\}">R\left(y\right)|\phantom{\rule{4.pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}i,\phantom{\rule{4.pt}{0ex}}{y}_{i}\ge 0,\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{4.pt}{0ex}}\sum _{i}{b}_{i}^{\frac{1-\alpha}{2}}{y}_{i}^{\frac{1+\alpha}{2}}\le 1$$
- For $\alpha =-1$, the alpha risk parity portfolio is the risk-budgeting portfolio. It is found by normalizing the solution to:$$min\left(\right)open="\{"\; close="\}">R\left(y\right)|\phantom{\rule{4.pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}i,\phantom{\rule{4.pt}{0ex}}{y}_{i}\ge 0,\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{4.pt}{0ex}}\sum _{i=1}^{n}{b}_{i}log\left(\right)open="("\; close=")">\frac{{y}_{i}}{{b}_{i}}.$$
- For $-1<\alpha \le 1$, alpha risk parity portfolios are found by normalizing solutions to:$$min\left(\right)open="\{"\; close="\}">R\left(y\right)|\phantom{\rule{4.pt}{0ex}}for\phantom{\rule{4.pt}{0ex}}all\phantom{\rule{4.pt}{0ex}}i,\phantom{\rule{4.pt}{0ex}}{y}_{i}\ge 0,\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{4.pt}{0ex}}\sum _{i}{b}_{i}^{\frac{1-\alpha}{2}}{y}_{i}^{\frac{1+\alpha}{2}}\ge 1$$

**Proposition**

**5.**

#### 2.5.3. A Risk-Budgeting Equation for Alpha Risk Parity

**Proposition**

**6.**

- If $\alpha \to -\infty $, Equation (20) converges to $\tilde{y}\propto {b}_{i}$. Budget weighting is a limit case of alpha risk parity.

#### 2.6. Utility Maximization under Divergence Constraint

#### 2.6.1. Risk Budgeting Portfolios and Maximum Utility

#### 2.6.2. Beyond Kullback–Leibler

**Proposition**

**7.**

#### 2.6.3. Alpha Risk Parity and Alpha Divergences

#### 2.7. What Happens if Some Risk Budgets Are Equal to Zero?

## 3. Materials and Methods

**Definition**

**2.**

## 4. Results

#### 4.1. Allocation within the Reference Universe

#### 4.1.1. Diversity and Duplication Sensitivity

#### 4.1.2. Risk and Returns

- Volatility decreases continuously with $\alpha $. The minimum-variance portfolio has a minimum out-of-sample volatility.
- After normalization by volatility, the maximum drawdown is worst for equal weights and fairly stable across the range of $\alpha $ considered here.
- Tail risk—as measured by CVaR—is a higher multiple of volatility when $\alpha $ becomes too close to 1. There is some risk in extreme portfolio concentration.
- Equal weights and minimum variance offer suboptimal Sharpe ratios.
- The equal-weighted portfolio does not have the lowest turnover. Turnover is minimal for equal risk contributions and increases gradually when alpha risk parity becomes close to minimum variance.

#### 4.1.3. Alpha Risk Parity and Shrinkage

#### 4.2. Allocation Outside the Reference Universe

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

RC | Risk contribution |

KL | Kullback–Leibler |

MDD | Maximum drawdown |

CVaR | Conditional value at risk |

## Appendix A. Minimum-Risk Portfolios

## Appendix B. The $q$-Logarithm and Related Functions

- The $q$-logarithm satisfies the following quasimorphism property.$$\mathrm{For}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}\lambda ,x>0,\phantom{\rule{1.em}{0ex}}{log}_{q}\left(\lambda x\right)={\lambda}^{1-q}{log}_{q}\left(x\right)+{log}_{\alpha}\left(\lambda \right).$$
- The inverse of the $q$-logarithm is the $q$-exponential. For $q=1$ and $\alpha =-1$, this inverse is the classical exponential function. For other values of the entropic index and using the alpha parametrization,$${exp}_{q}\left(y\right)={\left(\right)}^{1}\frac{2}{1+\alpha}$$Of course, ${exp}_{q}\left(0\right)=1$. Regarding the domain of definition, if $\alpha <-1$, ${exp}_{q}$ is defined for any $y<-\frac{2}{1+\alpha}$. This upper bound is positive. If $-1\le \alpha $, ${exp}_{q}$ is defined for any $y$. For any nonnegative number $y$ from the domain of definition, ${exp}_{q}\left(y\right)\ge 1$.
- The quasimorphism of the $q$-logarithm implies a similar property for the ${F}_{\alpha}$ log-barrier defined in Equation (11).$$\mathrm{For}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}\lambda >0,\phantom{\rule{0.277778em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}\mathrm{all}\phantom{\rule{4.pt}{0ex}}y\in {\mathbb{R}}_{+}^{n},\phantom{\rule{1.em}{0ex}}{F}_{\alpha}\left(\right)open="("\; close=")">b,\lambda y-{log}_{q}\left(\lambda \right).$$

## Appendix C. Using the Negative $q$-Logarithm as a Barrier

## Appendix D. Simple Formulas for Alpha Risk parity

- In the case of volatility-based equal risk contribution portfolios with constant correlation, Lagrangian (5) can be symmetrized by substracting $\mu {\sum}_{i}{b}_{i}log\left(\right)open="("\; close=")">{\sigma}_{i}$ and using the morphism property of the logarithm. Considering squared volatility, the problem is solved by minimizing $F({\sigma}_{1}{y}_{1},\dots ,{\sigma}_{n}{y}_{n})$ with:$$F({z}_{1},\dots ,{z}_{n})=\sum _{i}{z}_{i}^{2}+\rho \phantom{\rule{0.277778em}{0ex}}\sum _{i\ne j}{z}_{i}{z}_{j}-\mu \sum _{i}{b}_{i}log\left(\right)open="("\; close=")">{z}_{i}$$A symmetric and convex function achieves its minimum on the diagonal. Therefore, ${\sigma}_{i}{\tilde{y}}_{i}$ is constant across assets.
- If $\alpha \ne -1$, volatility-based alpha risk parity portfolios with uncorrelated assets are determined by minimizing the following Lagrangian criterion:$$\mathcal{L}(y,\mu )=\sum _{i}{y}_{i}^{2}{\sigma}_{i}^{2}-\mu \phantom{\rule{0.222222em}{0ex}}\frac{2}{1+\alpha}\sum _{i}{b}_{i}^{\frac{1-\alpha}{2}}{y}_{i}^{\frac{1+\alpha}{2}}.$$
- If the correlation is constant and different from 0, Lagrangian (A6) is symmetric if volatility is constant across assets. In this case, the equal-weighted portfolio solves the alpha risk parity problem. If volatility varies across assets and $\alpha \ne -1$, Lagrangian (A6) is not symmetric. The portfolio must be determined using numerical optimization.

## Appendix E. Independence from the Constraints

**Proposition**

**A1.**

**Proof**

**of**

**Proposition A1.**

**Proof**

**of**

**Proposition 2.**

## Appendix F. The Risk-Budgeting Equation

**Proof**

**of**

**Proposition 6.**

## Appendix G. Why Alpha Risk Parity Is a Maximum-Utility Problem

**Proof**

**of**

**Proposition 7.**

## Appendix H. Using Divergence Functions

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**Figure 2.**Duplication sensitivity and diversity. Simulation based on data from 7 August 2019 to 10 August 2022. Source: BNP Paribas, Bloomberg.

**Figure 3.**Alpha risk parity portfolios. The portfolios are rescaled in such a way that their volatility equals that of the equal risk contributions portfolio. Source: BNP Paribas, Bloomberg.

**Figure 4.**Allocation into equities and bonds for the three regions represented in the investment universe. Allocation based on risk metrics measured over 1994–2022. Source: BNP Paribas, Bloomberg.

**Figure 5.**Allocation into US equities and bonds using alpha risk parity and applying covariance shrinkage to selected portfolios. 1994–2022. Source: BNP Paribas, Bloomberg.

Market | Index | Bloomberg Ticker |
---|---|---|

US equities | BNP Paribas US Equity Futures | BNPIFUS |

EU equities | BNP Paribas Eurozone Equity Futures | BNPIFEU |

Japan equities | BNP Paribas Japan Equity Futures | BNPIFJP |

10y Bonds US | BNP Paribas Bond Futures US Tsy 10Y | BNPIFU10 |

10y Bonds DE | BNP Paribas Bond Futures Germany 10Y | BPBFE10 |

10y Bonds JP | BNP Paribas Bond Futures Japan JGB 10Y | BPBFJ10 |

Gold | S&P GSCI Gold Index CME | SPGSGCP |

Commodities | Bloomberg Commodity ex-Agriculture and Livestock Capped | BBUXALC |

Alpha | $-\mathbf{\infty}$ | −5 | −1 | −0.5 | −0.2 | 0 | 0.2 | 0.5 | 1 |
---|---|---|---|---|---|---|---|---|---|

Portfolio | Equal weights | ERC | Hellinger | Min variance | |||||

Volatility | 8.1% | 5.1% | 3.7% | 3.4% | 3.2% | 3.1% | 3.0% | 2.8% | 2.7% |

Sharpe | 0.532 | 0.742 | 0.853 | 0.867 | 0.871 | 0.870 | 0.864 | 0.843 | 0.754 |

MDD/vol | −3.37 | −2.78 | −2.70 | −2.74 | −2.79 | −2.83 | −2.87 | −2.92 | −2.80 |

CVaR/vol | 4.02 | 4.07 | 4.00 | 3.95 | 3.96 | 3.97 | 4.01 | 4.06 | 4.14 |

TurnOver | 2.7% | 2.9% | 2.6% | 2.7% | 2.7% | 2.8% | 2.9% | 3.2% | 4.0% |

Alpha | −3000 | −5 | −1 | −0.5 | −0.2 | 0 | 0.2 | 0.5 | 1 |
---|---|---|---|---|---|---|---|---|---|

US bonds | 16.5% | 15.3% | 13.4% | 13.0% | 12.5% | 12.1% | 11.4% | 10.5% | 5.0% |

DE Bonds | 12.7% | 12.6% | 12.7% | 12.3% | 12.3% | 12.1% | 12.1% | 11.4% | 10.4% |

JP bonds | 13.8% | 13.2% | 12.6% | 12.5% | 12.2% | 12.2% | 11.9% | 11.9% | 11.9% |

US eq. | 11.4% | 11.7% | 12.3% | 12.7% | 13.0% | 13.3% | 13.8% | 15.0% | 24.9% |

EU eq. | 11.4% | 10.8% | 9.8% | 9.4% | 9.0% | 8.7% | 8.3% | 7.2% | 0.0% |

JP eq. | 11.4% | 11.2% | 10.9% | 10.8% | 10.7% | 10.6% | 10.6% | 10.5% | 10.6% |

Gold | 11.4% | 13.4% | 16.2% | 17.3% | 18.2% | 19.0% | 20.0% | 22.0% | 29.5% |

Commodities | 11.4% | 11.8% | 12.1% | 12.1% | 12.1% | 12.0% | 11.9% | 11.4% | 7.7% |

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**MDPI and ACS Style**

Gava, J.; Turc, J.
The Properties of Alpha Risk Parity Portfolios. *Entropy* **2022**, *24*, 1631.
https://doi.org/10.3390/e24111631

**AMA Style**

Gava J, Turc J.
The Properties of Alpha Risk Parity Portfolios. *Entropy*. 2022; 24(11):1631.
https://doi.org/10.3390/e24111631

**Chicago/Turabian Style**

Gava, Jérôme, and Julien Turc.
2022. "The Properties of Alpha Risk Parity Portfolios" *Entropy* 24, no. 11: 1631.
https://doi.org/10.3390/e24111631