# The Distribution of Cross Sectional Momentum Returns When Underlying Asset Returns Are Student’s t Distributed

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Notation and Preliminaries

#### 2.1. Notation

#### 2.2. Multivariate Normal Distributions

**Theorem**

**1.**

**Theorem**

**2.**

**Corollary**

**1.**

**Proof.**

#### 2.3. Multivariate Student’s t Distribution

- each component, ${z}_{i}$, of $\mathit{z}$ is normalized by independent ${Y}_{i}=\sqrt{{g}_{i}/{\nu}_{i}}$ with same or differing ${\nu}_{i}$,
- ${Y}_{i}$ could be jointly dependent,
- single common Y.

- idiosyncratic shocks,
- common factor shocks,
- economy-wide or market-wide shock.

**Theorem**

**3.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**2.**

**Proof.**

#### 2.4. Unified Skew t Family of Distributions

## 3. Cross-Sectional Momentum Returns with Student’s $\mathit{t}$ Distributed Asset Returns

**Definition**

**1.**

**Assumption**

**1.**

**Theorem**

**5.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Theorem**

**6.**

**Proof.**

## 4. Special Case of Two Assets

**Lemma**

**1.**

**Proof.**

**Theorem**

**7.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Theorem**

**8.**

**Proof.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Theorem 4

## Appendix B. Proof of Theorem 5

## Appendix C. Proof of Theorem 6

## Appendix D. Proof of Lemma 1

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1. | |

2. | The normalization factor is, in fact, the probability of the event $\mathbf{0}\prec {\mathit{X}}_{1}$, where ${\mathit{X}}_{1}$ is as defined in (20). |

3. | Rewritten in the notation of this paper. |

4. |

**Figure 1.**Expected CSM return, ${\mu}_{{1}_{\pm},t+1}$, as a function of $\nu $ and ${\varrho}_{t,t+1}$.

**Figure 2.**Standard deviation, ${\sigma}_{{1}_{\pm},t+1}$, of CSM return as a function of $\nu $ and ${\varrho}_{t,t+1}$.

**Figure 3.**Skewness, ${\gamma}_{{1}_{\pm},t+1}$, of CSM return as a function of $\nu $ and ${\varrho}_{t,t+1}$.

**Figure 4.**Excess kurtosis, ${\kappa}_{{1}_{\pm},t+1}$, of CSM return as a function of $\nu $ and ${\varrho}_{t,t+1}$.

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**MDPI and ACS Style**

Kwon, O.K.; Satchell, S.
The Distribution of Cross Sectional Momentum Returns When Underlying Asset Returns Are Student’s *t* Distributed. *J. Risk Financial Manag.* **2020**, *13*, 27.
https://doi.org/10.3390/jrfm13020027

**AMA Style**

Kwon OK, Satchell S.
The Distribution of Cross Sectional Momentum Returns When Underlying Asset Returns Are Student’s *t* Distributed. *Journal of Risk and Financial Management*. 2020; 13(2):27.
https://doi.org/10.3390/jrfm13020027

**Chicago/Turabian Style**

Kwon, Oh Kang, and Stephen Satchell.
2020. "The Distribution of Cross Sectional Momentum Returns When Underlying Asset Returns Are Student’s *t* Distributed" *Journal of Risk and Financial Management* 13, no. 2: 27.
https://doi.org/10.3390/jrfm13020027