# Alpha Beta Risk and Stock Returns—A Decomposition Analysis of Idiosyncratic Volatility with Conditional Models

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## Abstract

**:**

## 1. Introduction

## 2. Overview

## 3. Alpha Beta Risk

## 4. Methodology and Data

## 5. Empirical Results

## 6. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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1 | MKT is the excess return on the market minus the one-month Treasury bill rate. SMB (Small Minus Big) is the average return on the three small portfolios minus the average return on the three big portfolios; HML (High Minus Low) is the average return on the two value portfolios minus the average return on the two growth portfolios. |

Mean | S.D. | 5% | 25% | Median | 75% | 95% | N | |
---|---|---|---|---|---|---|---|---|

Vol_Alpha | 3.89 | 1.65 | 2.07 | 2.74 | 3.48 | 4.64 | 6.99 | 2,334,870 |

Vol_Bmkt | 8.07 | 4.44 | 2.93 | 4.87 | 7.14 | 10.13 | 16.75 | 2,335,918 |

Vol_Bsmb | 12.11 | 5.75 | 4.96 | 8.13 | 11.08 | 14.95 | 22.28 | 2,335,918 |

Vol_Bhml | 14.74 | 7.32 | 6.31 | 9.98 | 13.06 | 17.83 | 27.67 | 2,335,918 |

Vol Alpha Quintiles | Size Quintiles | All Stocks | ||||
---|---|---|---|---|---|---|

1-Small | 2 | 3 | 4 | 5-Large | ||

1-Low | 1.02 | 1.15 | 1.22 | 1.21 | 1.07 | 1.13 |

2 | 1.16 | 1.19 | 1.25 | 1.23 | 1.04 | 1.21 |

3 | 1.01 | 1.09 | 1.19 | 1.25 | 1.08 | 1.20 |

4 | 0.99 | 1.06 | 1.16 | 1.12 | 0.98 | 1.13 |

5-High | 0.51 | 0.28 | 0.57 | 0.80 | 0.83 | 0.80 |

High-Low | −0.51 *** | −0.87 *** | −0.65 *** | −0.42 *** | −0.24 | −0.32 ** |

(−4.19) | (−7.41) | (−5.39) | (−3.32) | (−1.68) | (−2.20) | |

CAPM Alpha | −0.63 *** | −1.03 *** | −0.80 *** | −0.59 *** | −0.43 *** | −0.49 *** |

(−5.46) | (−9.93) | (−7.34) | (−5.48) | (−3.43) | (−3.85) | |

FF3 Alpha | −0.65 *** | −1.03 *** | −0.77 *** | −0.52 *** | −0.36 *** | −0.51 *** |

(−5.92) | (−11.14) | (−8.08) | (−5.79) | (−3.26) | (−5.46) |

Vol Bmkt Quintiles | Size Quintiles | All Stocks | ||||
---|---|---|---|---|---|---|

1-Small | 2 | 3 | 4 | 5-Large | ||

1-Low | 0.93 | 1.16 | 1.22 | 1.21 | 1.04 | 1.06 |

2 | 1.17 | 1.22 | 1.23 | 1.21 | 1.05 | 1.22 |

3 | 1.04 | 1.15 | 1.20 | 1.28 | 1.06 | 1.23 |

4 | 1.00 | 0.94 | 1.09 | 1.17 | 1.03 | 1.13 |

5-High | 0.54 | 0.31 | 0.61 | 0.78 | 0.75 | 0.75 |

High-Low | −0.39 *** | −0.85 *** | −0.61 *** | −0.43 ** | −0.29 ** | −0.29 * |

(−3.35) | (−7.04) | (−5.30) | (−3.52) | (−2.07) | (−1.96) | |

CAPM Alpha | −0.51 *** | −1.01 *** | −0.76 *** | −0.61 *** | −0.48 *** | −0.44 *** |

(−4.72) | (−9.96) | (−7.25) | (−5.71) | (−3.98) | (−3.56) | |

FF3 Alpha | −0.58 *** | −1.03 *** | −0.73 *** | −0.52 *** | −0.41 *** | −0.48 *** |

(−5.80) | (−11.35) | (−7.67) | (−5.50) | (−3.80) | (−5.32) |

Vol Bsmb Quintiles | Size Quintiles | All Stocks | ||||
---|---|---|---|---|---|---|

1-Small | 2 | 3 | 4 | 5-Large | ||

1-Low | 0.84 | 1.00 | 1.22 | 1.23 | 1.08 | 1.13 |

2 | 1.08 | 1.19 | 1.23 | 1.26 | 1.05 | 1.21 |

3 | 1.08 | 1.17 | 1.19 | 1.20 | 1.02 | 1.20 |

4 | 0.99 | 0.99 | 1.15 | 1.18 | 1.03 | 1.17 |

5-High | 0.60 | 0.38 | 0.60 | 0.76 | 0.77 | 0.89 |

High-Low | −0.24 * | −0.63 *** | −0.62 *** | −0.47 *** | −0.31 ** | −0.23 |

(−2.07) | (−5.30) | (−5.25) | (−3.92) | (−2.17) | (−1.59) | |

CAPM Alpha | −0.35 *** | −0.78 *** | −0.77 *** | −0.64 *** | −0.48 *** | −0.38 *** |

(−3.25) | (−7.47) | (−7.26) | (−5.96) | (−3.88) | (−3.07) | |

FF3 Alpha | −0.40 *** | −0.79 *** | −0.74 *** | −0.56 *** | −0.41 *** | −0.41 *** |

(−3.88) | (−8.31) | (−7.86) | (−6.09) | (−3.65) | (−4.54) |

Vol Bhml Quintiles | Size Quintiles | All Stocks | ||||
---|---|---|---|---|---|---|

1-Small | 2 | 3 | 4 | 5-Large | ||

1-Low | 1.02 | 1.25 | 1.26 | 1.21 | 1.05 | 1.20 |

2 | 1.11 | 1.13 | 1.19 | 1.27 | 1.06 | 1.21 |

3 | 1.04 | 1.04 | 1.26 | 1.22 | 1.02 | 1.20 |

4 | 0.96 | 0.93 | 1.02 | 1.10 | 1.06 | 1.14 |

5-High | 0.55 | 0.41 | 0.64 | 0.76 | 0.74 | 0.88 |

High-Low | −0.48 *** | −0.84 *** | −0.62 *** | −0.46 *** | −0.31 ** | −0.30 ** |

(−4.16) | (−7.01) | (−5.29) | (−3.61) | (−2.17) | (−2.07) | |

CAPM Alpha | −0.60 *** | −1.00 *** | −0.77 *** | −0.64 *** | −0.50 *** | −0.46 *** |

(−5.76) | (−9.83) | (−7.32) | (−5.90) | (−4.10) | (−3.71) | |

FF3 Alpha | −0.63 *** | −1.00 *** | −0.74 *** | −0.56 *** | −0.43 *** | −0.49 *** |

(−6.24) | (−11.13) | (−8.34) | (−5.96) | (−3.85) | (−5.38) |

Vol_Alpha | Panel A | Panel B | Panel C | ||||||
---|---|---|---|---|---|---|---|---|---|

Low Book-to-Market | Medium Book-to-Market | High Book-to-Market | |||||||

Small Cap | Mid Cap | Large Cap | Small Cap | Mid Cap | Large Cap | Small Cap | Mid Cap | Large Cap | |

Low | 0.66 | 0.94 | 1.05 | 1.14 | 1.26 | 1.07 | 1.37 | 1.38 | 1.20 |

Medium | 0.50 | 0.96 | 1.00 | 1.12 | 1.25 | 1.11 | 1.44 | 1.49 | 1.24 |

High | −0.09 | 0.37 | 0.81 | 0.61 | 0.98 | 1.02 | 0.88 | 1.08 | 1.19 |

High-Low | −0.75 *** | −0.57 *** | −0.24 *** | −0.53 *** | −0.28 *** | −0.05 | −0.49 *** | −0.31 *** | −0.01 |

(−7.19) | (−7.14) | (−2.76) | (−4.74) | (−3.06) | (−0.68) | (−4.57) | (−3.07) | (−0.08) | |

CAPM Alpha | −0.84 *** | −0.65 *** | −0.34 *** | −0.64 *** | −0.40 *** | −0.14 ** | −0.59 *** | −0.42 *** | −0.13 * |

(−8.33) | (−8.37) | (−4.19) | (−6.19) | (−4.87) | (−2.13) | (−5.82) | (−4.80) | (−1.82) | |

FF3 Alpha | −0.86 *** | −0.63 *** | −0.31 *** | −0.65 *** | −0.42 *** | −0.12 * | −0.64 *** | −0.47 *** | −0.15 ** |

(−9.10) | (−8.57) | (−4.21) | (−6.44) | (−5.65) | (−1.90) | (−6.53) | (−6.03) | (−2.35) |

Model 1 | Model 2 | Model 3 | Model 4 | Model 5 | |
---|---|---|---|---|---|

Beta | 0.55** | 0.56** | 0.57** | 0.57** | 0.56** |

(2.32) | (2.35) | (2.39) | (2.40) | (2.34) | |

Ln(ME) | −0.19 *** | −0.19 *** | −0.18 *** | −0.20 *** | −0.16 *** |

(−4.74) | (−4.92) | (−4.73) | (−5.52) | (−4.28) | |

Ln(B/M) | 0.16 *** | 0.16 *** | 0.16 *** | 0.15 *** | 0.16 *** |

(3.07) | (3.06) | (3.12) | (3.02) | (3.05) | |

Ret(−2, −7) | 0.86*** | 0.86*** | 0.89*** | 0.86*** | 0.86*** |

(5.16) | (5.01) | (5.08) | (5.18) | (5.12) | |

Ln(TURN) | −0.25*** | −0.24*** | −0.26*** | −0.22*** | -0.24*** |

(−5.01) | (−4.92) | (−5.13) | (−4.76) | (−4.93) | |

Ln(CVTURN) | −0.38 *** | −0.37 *** | −0.38 *** | −0.36 *** | −0.38 *** |

(−6.43) | (−6.44) | (−6.56) | (−6.29) | (−6.46) | |

VOL_Alpha | −0.04 *** | −0.04 *** | −0.03 *** | −0.03 *** | −0.04 *** |

(−4.62) | (−5.90) | (−4.08) | (−4.93) | (−5.78) | |

VOL_Bmkt | 0.005 | 0.007 | 0.005 | 0.004 | |

(1.06) | (1.55) | (1.38) | (0.98) | ||

VOL_Bsmb | 0.002 | 0.003 | 0.003 | 0.002 | |

(0.66) | (1.06) | (1.04) | (0.68) | ||

VOL_Bhml | −0.007 *** | −0.007 ** | −0.005 * | −0.007 ** | |

(−2.66) | (−2.56) | (−1.99) | (−2.49) | ||

Alpha | −0.02 *** | ||||

(−8.41) | |||||

Idio_VOL | −0.03*** | ||||

(−4.09) | |||||

Z (TERM) | −3.53 | ||||

(−1.29) | |||||

VOL_Z | 1.35 | ||||

(0.02) | |||||

Adj. R2 | 0.10 | 0.11 | 0.12 | 0.12 | 0.11 |

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**MDPI and ACS Style**

Fu, C.
Alpha Beta Risk and Stock Returns—A Decomposition Analysis of Idiosyncratic Volatility with Conditional Models. *Risks* **2018**, *6*, 124.
https://doi.org/10.3390/risks6040124

**AMA Style**

Fu C.
Alpha Beta Risk and Stock Returns—A Decomposition Analysis of Idiosyncratic Volatility with Conditional Models. *Risks*. 2018; 6(4):124.
https://doi.org/10.3390/risks6040124

**Chicago/Turabian Style**

Fu, Chengbo.
2018. "Alpha Beta Risk and Stock Returns—A Decomposition Analysis of Idiosyncratic Volatility with Conditional Models" *Risks* 6, no. 4: 124.
https://doi.org/10.3390/risks6040124