# Quantile Risk–Return Trade-Off

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

#### 2.1. Stock Returns

#### 2.2. Risk Measures

## 3. Econometric Method

## 4. Empirical Analysis

#### 4.1. Market Portfolio

#### 4.2. Industry Portfolios

#### 4.3. European Market Portfolios

#### 4.4. Skewness and Kurtosis Effects

#### 4.5. Subsample Analysis

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notes

1 | The US stock returns and risk free interest rate are gratefully available from Kenneth French’s data library. The market portfolio is the value-weight return of all CRSP firms listed on NYSE, AMEX, or NASDAQ, cf. Fama and French (1993). |

2 | The Eurozone stock returns and euro LIBOR rate are available from DataStream. |

3 | The VIX data are available from CBOE’s webpage with VIX historical price data. |

4 | The quantile regression estimation follows Aslanidis and Christiansen (2014). |

## References

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**Figure 1.**Quantile coefficients for the US market portfolio, (

**A**): constant; (

**B**): lagged return; (

**C**): lagged VIX; notes: the figure shows the estimated coefficients from Table 2.

**Figure 2.**Quantile coefficients for US industry portfolios. (

**A**): lagged return; (

**B**): lagged VIX. Notes: The figure shows the estimated coefficients from Table 4.

**Figure 3.**Quantile coefficients for European portfolios. (

**A**) Lagged return; (

**B**) lagged VSTOXX. Notes: The figure shows the estimated coefficients from Table 5.

Mean | Median | Maximum | Minimum | Std. Dev. | Skewness | Kurtosis | |
---|---|---|---|---|---|---|---|

US MKT | 0.03 | 0.07 | 11.35 | −12 | 1.15 | −0.26 | 13.63 |

NoDur | 0.04 | 0.06 | 10.24 | −9.87 | 0.95 | −0.19 | 13.49 |

Durbl | 0.05 | 0.06 | 15.03 | −14.43 | 1.56 | −0.12 | 10.5 |

Manuf | 0.05 | 0.08 | 10.83 | −11.11 | 1.18 | −0.29 | 12.76 |

Enrgy | 0.04 | 0.03 | 19.33 | −19.73 | 1.59 | −0.21 | 17.94 |

HiTec | 0.06 | 0.12 | 16.04 | −13.18 | 1.61 | 0.16 | 9.61 |

Telcm | 0.04 | 0.06 | 14.5 | −9.67 | 1.3 | 0.08 | 12.5 |

Shops | 0.05 | 0.08 | 10.99 | −10.61 | 1.17 | −0.07 | 9.68 |

Hlth | 0.05 | 0.07 | 11.1 | −9.74 | 1.16 | −0.19 | 8.7 |

Utils | 0.04 | 0.07 | 14.43 | −11.61 | 1.07 | 0.13 | 22.09 |

Other | 0.04 | 0.07 | 12.24 | −13.38 | 1.38 | −0.15 | 14.98 |

Belgium | 0 | 0.01 | 8.23 | −14.39 | 1.17 | −0.6 | 12.62 |

Finland | 0 | 0 | 15.33 | −18.26 | 1.77 | −0.42 | 11.77 |

France | 0 | 0.03 | 9.91 | −12.28 | 1.28 | −0.29 | 9.43 |

Germany | 0 | 0.05 | 16.04 | −9.83 | 1.23 | −0.05 | 13.1 |

Ireland | 0 | 0.01 | 9.09 | −13.38 | 1.32 | −0.76 | 11.18 |

Italy | −0.01 | 0.01 | 10.47 | −17.43 | 1.36 | −0.69 | 13.19 |

Luxemburg | −0.01 | 0 | 10.1 | −6.84 | 1.1 | −0.27 | 8.7 |

Netherland | 0 | 0.03 | 9.29 | −10.66 | 1.25 | −0.42 | 9.83 |

Spain | −0.01 | 0.03 | 11.74 | −14.2 | 1.31 | −0.38 | 10.9 |

VIX | 19.42 | 17.38 | 82.69 | 9.14 | 8.12 | 2.22 | 11.37 |

Skewness | −0.33 | −0.34 | 0.36 | −1.13 | 0.16 | −0.02 | 3.87 |

Kurtosis | 4.34 | 4.32 | 5.3 | 4.23 | 0.1 | 2.18 | 13.4 |

VSTOXX | 23.94 | 22.04 | 87.51 | 10.68 | 9.75 | 1.7 | 7.11 |

Quantile | OLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

cons | −0.03 | 0.44 *** | 0.32 *** | 0.19 *** | 0.09 | 0.00 | −0.08 * | −0.16 *** | −0.25 *** | −0.31 *** |

Mkt(−1) | −0.06 ** | 0.07 ** | 0.05 ** | 0.01 | −0.01 | −0.04 ** | −0.05 *** | −0.06 *** | −0.05 *** | −0.07 *** |

VIX(−1) | 0.00 | −0.09 *** | −0.05 *** | −0.03 *** | −0.01 *** | 0.00 * | 0.02 *** | 0.04 *** | 0.06 *** | 0.08 *** |

Pseudo R2 | 0.00 | 0.12 | 0.05 | 0.02 | 0.00 | 0.00 | 0.01 | 0.03 | 0.07 | 0.16 |

Slope equality test | 1067.3 *** |

Quantile | OLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

cons | −0.07 | 0.40 *** | 0.35 *** | 0.21 *** | 0.11 ** | 0.00 | −0.08 * | −0.21 *** | −0.34 *** | −0.39 *** |

VIX(−1) | 0 | −0.09 *** | −0.06 *** | −0.03 *** | −0.01 *** | 0.00 | 0.02 *** | 0.04 *** | 0.06 *** | 0.09 *** |

Pseudo R2 | 0 | 0.13 | 0.06 | 0.02 | 0 | 0.00 | 0.01 | 0.03 | 0.08 | 0.17 |

Slope equality test | 832.1 *** |

Industry | Quantile | 0.1 | 0.5 | 0.9 | Equality Test |
---|---|---|---|---|---|

NoDur | cons | 0.09 * | −0.02 | 0.04 | |

R(−1) | 0.04 | −0.03 ** | −0.05 ** | ||

VIX(−1) | −0.06 *** | 0.00 ** | 0.05 *** | ||

Pseudo R2 | 0.08 | 0.00 | 0.10 | 694.1 ** | |

Durbl | cons | 0.05 | 0.03 | −0.10 | |

R(−1) | 0.07 *** | 0.02 | 0.03 | ||

VIX(−1) | −0.09 *** | 0.00 | 0.10 *** | ||

Pseudo R2 | 0.09 | 0.00 | 0.11 | 630.5 *** | |

Manuf | cons | 0.32 *** | 0.04 | −0.23 *** | |

R(−1) | 0.07 ** | −0.01 | −0.02 | ||

VIX(−1) | −0.08 *** | 0.00 | 0.08 *** | ||

Pseudo R2 | 0.10 | 0.00 | 0.14 | 1005.7 *** | |

Enrgy | cons | −0.11 | −0.04 | 0.16 * | |

R(−1) | 0.04 | −0.03 * | −0.06 ** | ||

VIX(−1) | −0.08 *** | 0.00 | 0.08 *** | ||

Pseudo R2 | 0.06 | 0.00 | 0.08 | 488.0 *** | |

HiTec | cons | 0.11 | 0.06 | −0.23 *** | |

R(−1) | 0.09 *** | 0.01 | −0.02 | ||

VIX(−1) | −0.10 *** | 0.00 | 0.10 *** | ||

Pseudo R2 | 0.09 | 0.00 | 0.12 | 962.7 *** | |

Telcm | cons | 0.28 *** | −0.04 | −0.27 *** | |

R(−1) | 0.04 ** | −0.01 | 0.00 | ||

VIX(−1) | −0.08 *** | 0.01 * | 0.08 *** | ||

Pseudo R2 | 0.11 | 0.00 | 0.13 | 957.1 *** | |

Shops | cons | 0.07 | −0.02 | −0.14 ** | |

R(−1) | 0.09 *** | 0.01 | −0.01 | ||

VIX(−1) | −0.07 *** | 0.01 ** | 0.07 *** | ||

Pseudo R2 | 0.08 | 0.00 | 0.12 | 920.4 *** | |

Hlth | cons | −0.17 ** | −0.02 | 0.17 ** | |

R(−1) | 0.09 *** | −0.02 * | 0.00 | ||

VIX(−1) | −0.06 *** | 0.01 ** | 0.06 *** | ||

Pseudo R2 | 0.05 | 0.00 | 0.08 | 348.4 *** | |

Utils | cons | 0.08 | 0.04 | 0.06 | |

R(−1) | 0.06 * | −0.03 * | −0.05 *** | ||

VIX(−1) | −0.06 *** | 0.00 | 0.05 *** | ||

Pseudo R2 | 0.08 | 0.00 | 0.09 | 598.3 *** | |

Other | cons | 0.54 *** | 0.04 | −0.46 *** | |

R(−1) | 0.03 | 0.00 | −0.03 | ||

VIX(−1) | −0.10 *** | 0.00 | 0.10 *** | ||

Pseudo R2 | 0.13 | 0.00 | 0.16 | 874.9 *** |

Country | Quantile | 0.1 | 0.5 | 0.9 | Equality Test |
---|---|---|---|---|---|

Belgium | cons | 0.05 | 0.02 | −0.13 * | |

R(−1) | 0.14 *** | 0.02 | 0.03 | ||

VSTOXX(−1) | −0.06 *** | 0.00 | 0.06 *** | ||

Pseudo R2 | 0.10 | 0.00 | 0.10 | 642.5 *** | |

Finland | cons | 0.22 | −0.01 | −0.37 *** | |

R(−1) | 0.08 ** | 0.01 | −0.03 | ||

VSTOXX(−1) | −0.09 *** | 0.00 | 0.09 *** | ||

Pseudo R2 | 0.09 | 0.00 | 0.12 | 821.7 *** | |

France | cons | 0.33 *** | 0.02 | −0.28 *** | |

R(−1) | 0.09 *** | −0.01 | −0.03 | ||

VSTOXX(−1) | −0.07 *** | 0.00 | 0.07 *** | ||

Pseudo R2 | 0.11 | 0.00 | 0.13 | 813.5 *** | |

Germany | cons | 0.13 | 0.04 | −0.24 ** | |

R(−1) | 0.17 *** | 0.01 | 0.00 | ||

VSTOXX(−1) | −0.07 *** | 0.00 | 0.07 *** | ||

Pseudo R2 | 0.10 | 0.00 | 0.12 | 675.9 *** | |

Ireland | cons | −0.13 | 0.01 | 0.20 * | |

R(−1) | 0.11 *** | 0.03 ** | −0.01 | ||

VSTOXX(−1) | −0.05 *** | 0.00 | 0.05 *** | ||

Pseudo R2 | 0.06 | 0.00 | 0.06 | 254.5 *** | |

Italy | cons | −0.05 | 0.02 | −0.13 ** | |

R(−1) | 0.13 *** | −0.02 | −0.10 *** | ||

VSTOXX(−1) | −0.06 *** | 0.00 | 0.07 *** | ||

Pseudo R2 | 0.07 | 0.00 | 0.11 | 616.9 *** | |

Luxemburg cons | −0.46 *** | 0.05 * | 0.59 *** | ||

R(−1) | 0.02 | −0.02 * | −0.07 ** | ||

VSTOXX(−1) | −0.03 *** | 0.00 * | 0.02 *** | ||

Pseudo R2 | 0.03 | 0.00 | 0.02 | 105.4 *** | |

Netherland cons | 0.32 *** | 0.04 | −0.37 *** | ||

R(−1) | 0.12 *** | 0.04 ** | −0.04 | ||

VSTOXX(−1) | −0.07 *** | 0.00 | 0.07 *** | ||

Pseudo R2 | 0.12 | 0.00 | 0.14 | 754.8 *** | |

Spain cons | −0.13 | −0.01 | −0.19 ** | ||

R(−1) | 0.10 *** | 0.01 | −0.01 | ||

VSTOXX(−1) | −0.06 *** | 0.00 | 0.07 *** | ||

Pseudo R2 | 0.08 | 0.00 | 0.10 | 399.4 *** |

Quantile | OLS | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
---|---|---|---|---|---|---|---|---|---|---|

cons | 1.01 | 4.17 * | 2.02 | 0.78 | 0.73 | 0.92 | 0.22 | −0.87 | 0.13 | 1.14 |

Mkt(−1) | −0.06 * | 0.07 ** | 0.05 * | 0.01 | −0.01 | −0.04 ** | −0.05 *** | −0.06 *** | −0.05 *** | −0.06 ** |

VIX(−1) | 0.00 | −0.08 *** | −0.05 *** | −0.03 *** | −0.01 *** | 0.00 * | 0.02 *** | 0.03 *** | 0.05 *** | 0.08 *** |

Skewness(−1) | −0.17 | 0.12 | 0.07 | −0.02 | −0.08 | −0.23 * | −0.19 | −0.22 | −0.42 * | −0.59 *** |

Kurtosis(−1) | −0.25 | −0.86 | −0.39 | −0.14 | −0.16 | −0.23 | −0.08 | 0.15 | −0.11 | −0.37 |

Pseudo R2 | 0.00 | 0.12 | 0.06 | 0.02 | 0.00 | 0.00 | 0.01 | 0.03 | 0.08 | 0.16 |

Slope equality test | 988.0 *** |

Industry | Quantile | 0.1 | 0.5 | 0.9 | Equality Test |
---|---|---|---|---|---|

NoDur | cons | 2.84 | −0.38 | 0.63 | |

R(−1) | 0.05 * | −0.03 ** | −0.05 *** | ||

VIX(−1) | −0.06 *** | 0.00 ** | 0.05 *** | ||

Skewness(−1) | 0.01 | 0.03 | −0.29 | ||

Kurtosis(−1) | −0.64 | 0.09 | −0.15 | ||

Pseudo R2 | 0.08 | 0.00 | 0.11 | 920.1 *** | |

Durbl | cons | −2.19 | −2.99 | 1.98 | |

R(−1) | 0.07 ** | 0.02 | 0.02 | ||

VIX(−1) | −0.09 *** | 0.00 | 0.10 *** | ||

Skewness(−1) | 0.76 * | 0.16 | −0.38 | ||

Kurtosis(−1) | 0.58 | 0.71 | −0.51 | ||

Pseudo R2 | 0.09 | 0.00 | 0.11 | 1045.6 *** | |

Manuf | cons | 4.96 * | 1.42 | 0.00 | |

R(−1) | 0.07 *** | −0.01 | −0.03 | ||

VIX(−1) | −0.08 *** | 0.00 | 0.08 *** | ||

Skewness(−1) | −0.17 | −0.29 | −0.36 * | ||

Kurtosis(−1) | −1.09 | −0.34 | −0.07 | ||

Pseudo R2 | 0.11 | 0.00 | 0.14 | 1093.9 *** | |

Enrgy | cons | 4.91 | 2.51 | 0.20 | |

R(−1) | 0.05 * | −0.03 ** | −0.05 ** | ||

VIX(−1) | −0.08 *** | 0.00 | 0.08 *** | ||

Skewness(−1) | −0.29 | −0.35 | −0.32 | ||

Kurtosis(−1) | −1.19 | −0.62 | −0.04 | ||

Pseudo R2 | 0.06 | 0.00 | 0.08 | 491.1 *** | |

HiTec | cons | 5.92 | −1.63 | 0.81 | |

R(−1) | 0.09 *** | 0.01 | −0.02 | ||

VIX(−1) | −0.09 *** | 0.00 | 0.10 *** | ||

Skewness(−1) | 0.06 | 0.16 | −0.85 *** | ||

Kurtosis(−1) | −1.36 | 0.40 | −0.29 | ||

Pseudo R2 | 0.09 | 0.00 | 0.12 | 1099.3 *** | |

Telcm | cons | 4.80 * | −0.67 | 1.36 | |

R(−1) | 0.05 ** | −0.01 | 0.00 | ||

VIX(−1) | −0.08 *** | 0.01 * | 0.08 *** | ||

Skewness(−1) | −0.16 | 0.05 | −0.52 ** | ||

Kurtosis(−1) | −1.06 | 0.15 | −0.41 | ||

Pseudo R2 | 0.11 | 0.00 | 0.13 | 849.1 *** | |

Shops | cons | 2.87 | −0.01 | 0.98 | |

R(−1) | 0.10 *** | 0.00 | −0.01 | ||

VIX(−1) | −0.07 *** | 0.01 ** | 0.07 *** | ||

Skewness(−1) | 0.10 | −0.05 | −0.75 *** | ||

Kurtosis(−1) | −0.65 | 0.00 | −0.32 | ||

Pseudo R2 | 0.08 | 0.00 | 0.13 | 1095.0 *** | |

Hlth | cons | 1.88 | −0.01 | −3.06 | |

R(−1) | 0.10 *** | −0.03 ** | 0.00 | ||

VIX(−1) | −0.05 *** | 0.01 ** | 0.06 *** | ||

Skewness(−1) | 0.14 | −0.10 | −0.03 | ||

Kurtosis(−1) | −0.47 | −0.01 | 0.74 | ||

Pseudo R2 | 0.06 | 0.00 | 0.09 | 543.1 *** | |

Utils | cons | 1.82 | 1.78 | 5.30 *** | |

R(−1) | 0.07 ** | −0.03 ** | −0.06 ** | ||

VIX(−1) | −0.06 *** | 0.00 | 0.05 *** | ||

Skewness(−1) | 0.02 | −0.32 ** | −0.66 *** | ||

Kurtosis(−1) | −0.41 | −0.43 | −1.26 *** | ||

Pseudo R2 | 0.08 | 0.00 | 0.09 | 648.1 *** | |

Other | cons | 5.24 * | −0.85 | −0.77 | |

R(−1) | 0.05 | 0.00 | −0.03 | ||

VIX(−1) | −0.09 *** | 0.00 | 0.10 *** | ||

Skewness(−1) | −0.26 | 0.02 | −0.21 | ||

Kurtosis(−1) | −1.12 * | 0.21 | 0.06 | ||

Pseudo R2 | 0.13 | 0.00 | 0.16 | 1173.4 *** |

Period | Quantile | 0.1 *** | 0.5 | 0.9 | Equality Test |
---|---|---|---|---|---|

1990–2006 | cons | 0.29 *** | 0.01 | −0.20 ** | |

Mkt(−1) | 0.14 *** | 0.02 | 0.01 | ||

VIX(−1) | −0.07 *** | 0.00 | 0.07 *** | ||

Pseudo R2 | 0.10 | 0.00 | 0.10 | 380.8 *** | |

1990–2006 | cons | 0.30 *** | 0.01 | −0.19 ** | |

VIX(−1) | −0.07 *** | 0.00 | 0.07 *** | ||

Pseudo R2 | 0.10 | 0.00 | 0.10 | 468.0 *** | |

2007–2020 | cons | 0.51 *** | −0.03 | −0.40 *** | |

Mkt(−1) | 0.01 | −0.09 *** | −0.14 *** | ||

VIX(−1) | −0.10 *** | 0.01** | 0.09 *** | ||

Pseudo R2 | 0.14 | 0.01 | 0.22 | 524.5 *** | |

2007–2020 | cons | 0.53 *** | −0.04 | −0.50 *** | |

VIX(−1) | −0.10 *** | 0.01 * | 0.09 *** | ||

Pseudo R2 | 0.14 | 0.00 | 0.21 | 553.7 *** |

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## Share and Cite

**MDPI and ACS Style**

Aslanidis, N.; Christiansen, C.; Savva, C.S.
Quantile Risk–Return Trade-Off. *J. Risk Financial Manag.* **2021**, *14*, 249.
https://doi.org/10.3390/jrfm14060249

**AMA Style**

Aslanidis N, Christiansen C, Savva CS.
Quantile Risk–Return Trade-Off. *Journal of Risk and Financial Management*. 2021; 14(6):249.
https://doi.org/10.3390/jrfm14060249

**Chicago/Turabian Style**

Aslanidis, Nektarios, Charlotte Christiansen, and Christos S. Savva.
2021. "Quantile Risk–Return Trade-Off" *Journal of Risk and Financial Management* 14, no. 6: 249.
https://doi.org/10.3390/jrfm14060249