# An Intersection–Union Test for the Sharpe Ratio

## Abstract

**:**

## 1. Motivation

- (i)
- I derive closed-form expressions for the standard errors of the test statistics, instead of providing numerical results that have been obtained by bootstrapping, and
- (ii)
- I do this for the case $d\ge 2$ but not (only) for $d=2$.

## 2. The Intersection–Union Test

#### 2.1. Gordin’s Condition

#### 2.2. Asymptotic Properties of Sharpe Ratios

#### 2.3. Empirical Study

## 3. Conclusions

## Conflicts of Interest

## Appendix A. Asymptotic Results

## Appendix B. Correlograms

**Figure A2.**Correlograms with respect to $\left\{{({R}_{t}-{\mu}_{n})}^{2}\right\}$ of the EWP and each G–7 country.

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1. | A different question is whether some asset universe allows the investor to achieve a higher performance compared to another asset universe (Hanke and Penev 2018). |

2. | In order to identify the outperforming strategies, we would have to apply a multiple test. For more details on that topic, see Frahm et al. (2012) as well as Romano and Wolf (2005). |

3. | Any capital income that occurs during Period t is considered part of ${P}_{t}$. |

4. | The total returns have been retrieved from the MSCI webpage (MSCI 2018). |

5. | The data have been obtained from the Federal Reserve Bank of St. Louis (FRED 2018). |

6. | The only exception is Japan, where we can find a relatively large Q-statistic of 31.7637. |

EWP | Canada | France | Germany | Italy | Japan | UK | USA | |
---|---|---|---|---|---|---|---|---|

${\sigma}_{\mathrm{L}n}^{2}/{\sigma}_{n}^{2}$ | 1.4987 | 1.0299 | 1.2036 | 1.1255 | 1.6913 | 2.1828 | 1.2720 | 1.0118 |

${\tau}_{\mathrm{L}n}^{2}/{\tau}_{n}^{2}$ | 2.5962 | 2.7550 | 2.3081 | 2.9514 | 2.3707 | 2.8368 | 2.5027 | 2.6202 |

**Table 2.**Means, standard deviations, and Sharpe ratios for the EWP and the G–7 countries. The standard errors are given in parentheses.

EWP | Canada | France | Germany | Italy | Japan | UK | USA | |
---|---|---|---|---|---|---|---|---|

${\mu}_{n}$ | 0.0053 | 0.0052 | 0.0062 | 0.0060 | 0.0033 | 0.0054 | 0.0052 | 0.0057 |

$\mathbf{SE}\left({\mu}_{n}\right)$ | 0.0023 | 0.0024 | 0.0029 | 0.0028 | 0.0040 | 0.0037 | 0.0020 | 0.0026 |

${\sigma}_{n}$ | 0.0461 | 0.0560 | 0.0640 | 0.0627 | 0.0732 | 0.0599 | 0.0436 | 0.0620 |

$\mathbf{SE}\left({\sigma}_{n}\right)$ | 0.0030 | 0.0040 | 0.0037 | 0.0041 | 0.0038 | 0.0035 | 0.0028 | 0.0077 |

${\eta}_{n}$ | 0.1149 | 0.0923 | 0.0971 | 0.0961 | 0.0449 | 0.0898 | 0.1202 | 0.0927 |

$\mathbf{SE}\left({\eta}_{n}\right)$ | 0.0581 | 0.0462 | 0.0492 | 0.0479 | 0.0537 | 0.0624 | 0.0548 | 0.0508 |

${\mathbf{SE}}_{\mathrm{JK}}\left({\eta}_{n}\right)$ | 0.0419 | 0.0417 | 0.0417 | 0.0417 | 0.0417 | 0.0417 | 0.0418 | 0.0417 |

Canada | France | Germany | Italy | Japan | UK | USA | |
---|---|---|---|---|---|---|---|

$\Delta {\eta}_{n}$ | 0.0226 | 0.0178 | 0.0187 | 0.0700 | 0.0251 | −0.0053 | 0.0222 |

$\mathbf{SE}(\Delta {\eta}_{n})$ | 0.0213 | 0.0317 | 0.0419 | 0.0269 | 0.0374 | 0.0381 | 0.0376 |

t | 1.0635 | 0.5598 | 0.4472 | 2.6054 | 0.6718 | −0.1397 | 0.5891 |

${\mathbf{SE}}_{\mathrm{JK}}(\Delta {\mu}_{n})$ | 0.0291 | 0.0227 | 0.0257 | 0.0299 | 0.0354 | 0.0290 | 0.0274 |

${t}_{\mathrm{JK}}$ | 0.7758 | 0.7821 | 0.7298 | 2.3420 | 0.7083 | −0.1833 | 0.8089 |

© 2018 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Frahm, G.
An Intersection–Union Test for the Sharpe Ratio. *Risks* **2018**, *6*, 40.
https://doi.org/10.3390/risks6020040

**AMA Style**

Frahm G.
An Intersection–Union Test for the Sharpe Ratio. *Risks*. 2018; 6(2):40.
https://doi.org/10.3390/risks6020040

**Chicago/Turabian Style**

Frahm, Gabriel.
2018. "An Intersection–Union Test for the Sharpe Ratio" *Risks* 6, no. 2: 40.
https://doi.org/10.3390/risks6020040