# TPLVM: Portfolio Construction by Student’s t-Process Latent Variable Model

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## Abstract

**:**

## 1. Introduction

## 2. Short Review of Gaussian Process

#### 2.1. Gaussian Process

#### 2.2. Gaussian Process Latent Variable Model

## 3. Proposed Model: Student’s t-Process Latent Variable Model

#### 3.1. Introduction of the Student’s t-Process

#### 3.2. Student’s t-Process Latent Variable Model

#### 3.3. Variational Inference

## 4. Problem Formulation in Finance

#### 4.1. Factor Model

#### 4.2. Portfolio Theory

## 5. Experiment

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

GARCH | Generalized AutoRegressive Conditional Heteroscedasticity |

DCC | Dynamic Conditional Correlation |

GPLVM | Gaussian Process Latent Variable Model |

TPLVM | Student’s t-Process Latent Variable Model |

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US | Canada | UK | France | Germany | Spain | Italy | Netherlands | |
---|---|---|---|---|---|---|---|---|

Mean [%] | 6.00 | 5.41 | 2.39 | 4.08 | 6.87 | 3.20 | 1.35 | 2.96 |

Std. [%] | 14.93 | 14.92 | 13.62 | 18.12 | 21.13 | 20.66 | 21.71 | 19.13 |

R/R | 0.40 | 0.36 | 0.18 | 0.23 | 0.33 | 0.15 | 0.06 | 0.15 |

Skew | −0.66 | −0.92 | −0.55 | −0.38 | −0.50 | −0.17 | 0.03 | −0.74 |

Kurtosis | 5.23 | 7.36 | 4.53 | 4.52 | 6.12 | 4.96 | 4.80 | 5.88 |

Sweden | Switzerland | Japan | HongKong | Australia | Korea | Norway | Singapore | |

Mean [%] | 6.32 | 2.80 | 3.35 | 7.27 | 4.70 | 12.98 | 10.72 | 5.05 |

Std. [%] | 19.51 | 14.68 | 19.24 | 23.46 | 12.40 | 28.80 | 21.49 | 21.71 |

R/R | 0.32 | 0.19 | 0.17 | 0.31 | 0.38 | 0.45 | 0.50 | 0.23 |

Skew | −0.19 | −0.73 | −0.54 | 0.28 | −0.69 | 1.39 | −0.93 | −0.26 |

Kurtosis | 5.29 | 6.11 | 4.75 | 5.78 | 4.54 | 11.63 | 6.84 | 6.81 |

${\mathbf{Port}}_{\mathit{G}}$ | ${\mathbf{Port}}_{\mathit{t}}$ | Difference | |
---|---|---|---|

Anterior half (Jun 2008–Jun 2013) | |||

Return | −4.89% | −2.63% | 2.25% |

Risk | 19.57% | 18.33% | −1.24% |

R/R | −0.25 | −0.14 | 0.11 |

Posterior half (Jul 2013–Jun 2019) | |||

Return | 6.08% | 6.30% | 0.22% |

Risk | 11.16% | 10.56% | −0.60% |

R/R | 0.54 | 0.60 | 0.05 |

Whole period (Jun 2008–Jun 2019) | |||

Return | 0.64% | 1.87% | 1.23% |

Risk | 15.92% | 14.93% | −0.99% |

R/R | 0.04 | 0.12 | 0.09 |

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**MDPI and ACS Style**

Uchiyama, Y.; Nakagawa, K.
TPLVM: Portfolio Construction by Student’s *t*-Process Latent Variable Model. *Mathematics* **2020**, *8*, 449.
https://doi.org/10.3390/math8030449

**AMA Style**

Uchiyama Y, Nakagawa K.
TPLVM: Portfolio Construction by Student’s *t*-Process Latent Variable Model. *Mathematics*. 2020; 8(3):449.
https://doi.org/10.3390/math8030449

**Chicago/Turabian Style**

Uchiyama, Yusuke, and Kei Nakagawa.
2020. "TPLVM: Portfolio Construction by Student’s *t*-Process Latent Variable Model" *Mathematics* 8, no. 3: 449.
https://doi.org/10.3390/math8030449