# Models for Expected Returns with Statistical Factors

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials & Methods

#### 2.1. Data and Portfolio Returns

#### 2.1.1. Data

#### 2.1.2. Statistical Factors

- Coefficient of variation (CV) of prices for each year and company:$$CV\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{{s}_{n}}{\overline{x}},$$
- Skewness (Skew) of prices for each year and company:$$Skew\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{\frac{1}{n}{\sum}_{i=1}^{n}{({x}_{i}-\overline{x})}^{3}}{{s}_{n}^{3}}$$Positive values indicate that the distribution is positively skewed, that is, the right tail is longer than the left one, while the contrary occurs for negative values. When both tails are similar, the skewness is roughly 0.
- Excess Kurtosis (Kurt) of prices for each year and company:$$Kurt\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\frac{\frac{1}{n}{\sum}_{i=1}^{n}{({x}_{i}-\overline{x})}^{4}}{{s}_{n}^{4}}-3$$

#### 2.1.3. Portfolio Returns

- Stocks with low CV will be included in portfolios 1-y-z, while stocks with high CV will be included in portfolios 2-y-z.
- Stocks with low Kurt will be included in portfolios x-1-z, while stocks with high Kurt will be included in portfolios x-2-z.
- Stocks with low Skew (less than the 30-th percentile, P30) will be included in portfolios x-y-1, while stocks with high Skew (greater than P70) will be included in portfolios x-y-3. The rest will be included in portfolios x-y-2.

#### 2.2. Methodology

#### 2.2.1. Classical Methodologies

#### Time-Series Regression (TS)

#### Cross-Sectional Regression (CS) and Fama-MacBeth (FM) Procedure

- First, a Time-Series regression is run to obtain the sensitivity of the expected excess return of each of the portfolios. For each $i=1,\dots ,N$ estimate ${\beta}_{i}$ in the model$${R}_{t}^{i}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{a}_{i}+{\beta}_{i}^{\prime}{f}_{t}+{\u03f5}_{t}^{i},\phantom{\rule{1.em}{0ex}}t=1,2,\dots ,T.$$Note that the intercept here is denoted by a instead of $\alpha $.
- Run a Cross-Sectional regression to get $\lambda $$$E\left({R}^{i}\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{\widehat{\beta}}_{i}^{\prime}\lambda +{\alpha}_{i},\phantom{\rule{1.em}{0ex}}\mathrm{for}\phantom{\rule{4.pt}{0ex}}i=1,\dots ,N,$$

#### 2.2.2. Resampling Techniques: Bootstrap

#### First Step

#### Second Step

#### Third Step

#### Fourth Step

## 3. Results

#### 3.1. Time-Series Regressions

#### 3.1.1. Model 1: CAPM

#### 3.1.2. Model 2: Market and Coefficient of Variation

#### 3.1.3. Model 3: Market and Kurtosis

#### 3.1.4. Model 4: Market and Skewness

#### 3.1.5. Model 5: Market, Coefficient of Variation, and Skewness

#### 3.2. Cross-Sectional Regression

#### 3.3. Comparison between Classical Methodologies and Bootstrap Methods

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Estimates | t-Statistic | Bootstrap CI (2.5%,97.5%) | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Portfolio | Alpha | Market | CV | Skew | Alpha | Market | CV | Skew | Alpha | Market | CV | Skew | Adj. ${\mathit{R}}^{2}$ |

1-1-1 | 0.001 | 0.441 | 0.306 | 0.175 | 0.639 | 9.815 | 3.889 | 1.381 | −0.001,0.005 | 0.348,0.563 | 0.171,0.417 | −0.063,0.485 | 0.702 |

1-1-2 | 0.003 | 0.522 | 0.234 | 0.427 | 2.282 | 15.469 | 3.962 | 4.476 | 0.003,0.009 | 0.455,0.593 | 0.115,0.351 | 0.257,0.608 | 0.842 |

1-1-3 | 0.002 | 0.540 | 0.140 | 0.651 | 1.252 | 10.347 | 1.528 | 4.420 | 0.001,0.009 | 0.425,0.666 | −0.043,0.304 | 0.349,0.991 | 0.692 |

1-2-1 | 0.003 | 0.470 | 0.243 | 0.216 | 1.838 | 12.706 | 3.749 | 2.070 | 0.002,0.008 | 0.387,0.554 | 0.102,0.370 | 0.022,0.428 | 0.779 |

1-2-2 | 0.002 | 0.558 | 0.161 | 0.416 | 1.080 | 14.183 | 2.341 | 3.743 | 0.000,0.006 | 0.460,0.654 | 0.020,0.277 | 0.187,0.643 | 0.797 |

1-2-3 | 0.003 | 0.442 | 0.140 | 0.730 | 1.853 | 11.680 | 2.113 | 6.832 | 0.002,0.008 | 0.365,0.526 | −0.015,0.255 | 0.522,0.959 | 0.769 |

2-1-1 | 0.003 | 0.484 | 1.267 | −0.196 | 1.913 | 10.782 | 16.143 | −1.551 | 0.003,0.010 | 0.387,0.575 | 1.125,1.403 | −0.442,0.055 | 0.894 |

2-1-2 | 0.003 | 0.539 | 1.175 | 0.187 | 2.002 | 14.727 | 18.352 | 1.813 | 0.003,0.008 | 0.465,0.607 | 1.022,1.303 | −0.022,0.402 | 0.930 |

2-1-3 | 0.005 | 0.404 | 1.276 | 0.840 | 2.412 | 7.795 | 14.082 | 5.749 | 0.005,0.013 | 0.308,0.514 | 1.094,1.445 | 0.528,1.176 | 0.873 |

2-2-1 | 0.001 | 0.427 | 1.242 | −0.298 | 0.613 | 10.021 | 16.643 | −2.476 | −0.001,0.005 | 0.339,0.523 | 1.083,1.394 | −0.479,−0.065 | 0.891 |

2-2-2 | 0.001 | 0.344 | 1.299 | 0.446 | 0.561 | 5.058 | 10.897 | 2.320 | −0.002,0.008 | 0.196,0.498 | 1.067,1.487 | 0.141,0.844 | 0.772 |

2-2-3 | −0.001 | 0.489 | 1.034 | 1.242 | −0.598 | 11.200 | 13.550 | 10.084 | −0.005,0.001 | 0.391,0.595 | 0.900,1.159 | 1.009,1.510 | 0.906 |

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Year | Companies | Months | Mean | SD | Min | Median | Max |
---|---|---|---|---|---|---|---|

2009 | 1631 | 12 | 69.55 | 1619.75 | 0.01 | 5.59 | 92,316.88 |

2010 | 1643 | 12 | 47.39 | 765.65 | 0.01 | 6.48 | 42,907.84 |

2011 | 1672 | 12 | 36.20 | 516.56 | 0.00 | 6.50 | 26,316.80 |

2012 | 1706 | 12 | 23.77 | 120.80 | 0.00 | 5.48 | 4203.77 |

2013 | 1729 | 12 | 24.98 | 109.47 | 0.00 | 5.98 | 3400.01 |

2014 | 1737 | 12 | 28.37 | 123.28 | 0.00 | 7.20 | 3789.00 |

2015 | 1789 | 12 | 30.98 | 139.74 | 0.00 | 7.25 | 3999.00 |

2016 | 1833 | 12 | 32.39 | 164.73 | 0.01 | 7.20 | 6000.00 |

2017 | 1864 | 12 | 38.99 | 195.06 | 0.00 | 8.74 | 6149.00 |

2018 | 1878 | 2 | 41.34 | 204.65 | 0.00 | 9.08 | 6000.00 |

Market | CV | Kurt | Skew | |
---|---|---|---|---|

Market | 1.0000 | 0.5970 | −0.4210 | 0.1770 |

CV | 0.5970 | 1.0000 | −0.5450 | 0.2930 |

Kurt | −0.4210 | −0.5450 | 1.0000 | 0.1650 |

Skew | 0.1770 | 0.2930 | 0.1650 | 1.0000 |

Portfolio | CV | Kurt | Skew | |
---|---|---|---|---|

(1) | 1-1-1 | Low | Low | Low |

(2) | 1-1-2 | Low | Low | Medium |

(3) | 1-1-3 | Low | Low | High |

(4) | 1-2-1 | Low | High | Low |

(5) | 1-2-2 | Low | High | Medium |

(6) | 1-2-3 | Low | High | High |

(7) | 2-1-1 | High | Low | Low |

(8) | 2-1-2 | High | Low | Medium |

(9) | 2-1-3 | High | Low | High |

(10) | 2-2-1 | High | High | Low |

(11) | 2-2-2 | High | High | Medium |

(12) | 2-2-3 | High | High | High |

Estimates | t-Statistic | Bootstrap CI (2.5%,97.5%) | |||||
---|---|---|---|---|---|---|---|

Portfolio | Alpha | Market | Alpha | Market | Alpha | Market | Adj. ${\mathit{R}}^{2}$ |

1-1-1 | −0.0005 | 0.558 | −0.282 | 14.287 | −0.005,0.003 | 0.467,0.668 | 0.651 |

1-1-2 | 0.0010 | 0.627 | 0.661 | 19.470 | −0.001,0.005 | 0.555,0.699 | 0.776 |

1-1-3 | 0.0003 | 0.623 | 0.163 | 13.442 | −0.004,0.006 | 0.492,0.744 | 0.622 |

1-2-1 | 0.0011 | 0.567 | 0.749 | 17.456 | −0.001,0.006 | 0.482,0.647 | 0.736 |

1-2-2 | 0.0000 | 0.637 | −0.015 | 18.314 | −0.004,0.003 | 0.534,0.738 | 0.754 |

1-2-3 | 0.0003 | 0.530 | 0.192 | 14.014 | −0.003,0.004 | 0.437,0.627 | 0.642 |

2-1-1 | −0.0011 | 0.919 | −0.358 | 13.760 | −0.008,0.004 | 0.765,1.075 | 0.633 |

2-1-2 | −0.0022 | 0.962 | −0.758 | 15.497 | −0.010,0.002 | 0.819,1.109 | 0.687 |

2-1-3 | −0.0022 | 0.896 | −0.603 | 11.332 | −0.012,0.003 | 0.729,1.080 | 0.539 |

2-2-1 | −0.0030 | 0.848 | −1.014 | 13.132 | −0.012,0.000 | 0.692,1.016 | 0.611 |

2-2-2 | −0.0046 | 0.824 | −1.201 | 9.916 | −0.017,−0.002 | 0.649,1.009 | 0.472 |

2-2-3 | −0.0080 | 0.918 | −2.355 | 12.478 | −0.023,−0.009 | 0.731,1.112 | 0.587 |

Estimates | t-Statistic | Bootstrap CI (2.5%,97.5%) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Portfolio | Alpha | Market | CV | Alpha | Market | CV | Alpha | Market | CV | Adj. ${\mathit{R}}^{2}$ |

1-1-1 | 0.0008 | 0.441 | 0.332 | 0.447 | 9.776 | 4.322 | -0.002,0.005 | 0.344,0.562 | 0.209,0.430 | 0.700 |

1-1-2 | 0.0021 | 0.523 | 0.297 | 1.537 | 14.263 | 4.765 | 0.001,0.007 | 0.441,0.607 | 0.178,0.407 | 0.814 |

1-1-3 | 0.0012 | 0.541 | 0.235 | 0.587 | 9.562 | 2.449 | −0.002,0.007 | 0.393,0.689 | 0.054,0.405 | 0.639 |

1-2-1 | 0.0022 | 0.471 | 0.275 | 1.539 | 12.520 | 4.300 | 0.001,0.007 | 0.386,0.561 | 0.140,0.390 | 0.773 |

1-2-2 | 0.0008 | 0.559 | 0.222 | 0.527 | 13.401 | 3.139 | −0.002,0.005 | 0.453,0.669 | 0.087,0.326 | 0.773 |

1-2-3 | 0.0013 | 0.443 | 0.247 | 0.753 | 9.791 | 3.218 | −0.001,0.006 | 0.346,0.552 | 0.061,0.395 | 0.670 |

2-1-1 | 0.0036 | 0.484 | 1.238 | 2.139 | 10.709 | 16.133 | 0.004,0.011 | 0.387,0.574 | 1.097,1.374 | 0.892 |

2-1-2 | 0.0024 | 0.539 | 1.202 | 1.746 | 14.576 | 19.131 | 0.002,0.008 | 0.462,0.608 | 1.065,1.317 | 0.929 |

2-1-3 | 0.0031 | 0.404 | 1.400 | 1.419 | 6.850 | 13.948 | 0.002,0.011 | 0.286,0.551 | 1.197,1.570 | 0.835 |

2-2-1 | 0.0016 | 0.427 | 1.198 | 0.949 | 9.783 | 16.144 | 0.000,0.007 | 0.345,0.518 | 1.062,1.341 | 0.886 |

2-2-2 | 0.0006 | 0.345 | 1.364 | 0.232 | 4.963 | 11.550 | −0.004,0.006 | 0.196,0.514 | 1.115,1.564 | 0.763 |

2-2-3 | −0.0033 | 0.490 | 1.217 | −1.463 | 8.056 | 11.780 | −0.012,−0.002 | 0.334,0.666 | 1.007,1.409 | 0.818 |

Estimates | t-Statistic | Bootstrap CI (2.5%,97.5%) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Portfolio | Alpha | Market | Kurt | Alpha | Market | Kurt | Alpha | Market | Kurt | Adj. ${\mathit{R}}^{2}$ |

1-1-1 | −0.0007 | 0.541 | −0.168 | −0.396 | 12.563 | −0.923 | −0.005,0.002 | 0.436,0.674 | -0.563,0.211 | 0.650 |

1-1-2 | 0.0008 | 0.614 | −0.131 | 0.545 | 17.275 | −0.871 | −0.002,0.005 | 0.537,0.696 | −0.486,0.191 | 0.776 |

1-1-3 | 0.0003 | 0.623 | −0.004 | 0.158 | 12.130 | −0.018 | −0.004,0.006 | 0.482,0.767 | −0.505,0.487 | 0.619 |

1-2-1 | 0.0014 | 0.588 | 0.213 | 0.925 | 16.507 | 1.421 | 0.000,0.006 | 0.506,0.681 | −0.152,0.564 | 0.738 |

1-2-2 | 0.0005 | 0.682 | 0.452 | 0.348 | 18.387 | 2.892 | −0.002,0.005 | 0.595,0.780 | 0.033,0.857 | 0.770 |

1-2-3 | 0.0008 | 0.563 | 0.337 | 0.437 | 13.692 | 1.942 | −0.002,0.005 | 0.478,0.667 | −0.106,0.757 | 0.651 |

2-1-1 | −0.0026 | 0.796 | −1.232 | −0.921 | 11.644 | −4.279 | −0.011,0.001 | 0.655,0.952 | −2.078,−0.541 | 0.684 |

2-1-2 | −0.0037 | 0.841 | −1.210 | −1.392 | 13.373 | −4.567 | −0.013,−0.002 | 0.700,1.001 | −1.977,−0.571 | 0.736 |

2-1-3 | −0.0037 | 0.779 | −1.179 | −1.048 | 9.344 | −3.357 | −0.015,0.000 | 0.620,0.983 | −2.103,−0.372 | 0.579 |

2-2-1 | −0.0038 | 0.783 | −0.656 | −1.304 | 11.192 | −2.224 | −0.014,−0.002 | 0.654,0.933 | −1.506,0.028 | 0.625 |

2-2-2 | −0.0045 | 0.832 | 0.075 | −1.162 | 9.037 | 0.194 | −0.017,−0.001 | 0.647,1.045 | −0.951,0.936 | 0.467 |

2-2-3 | −0.0079 | 0.923 | 0.057 | −2.305 | 11.337 | 0.166 | −0.023,−0.009 | 0.751,1.127 | −0.998,0.956 | 0.583 |

Estimates | t-Statistic | Bootstrap CI (2.5%,97.5%) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Portfolio | Alpha | Market | Skew | Alpha | Market | Skew | Alpha | Market | Skew | Adj. ${\mathit{R}}^{2}$ |

1-1-1 | 0.0002 | 0.543 | 0.292 | 0.116 | 13.924 | 2.229 | −0.003,0.004 | 0.461,0.645 | 0.070,0.602 | 0.663 |

1-1-2 | 0.0022 | 0.600 | 0.516 | 1.661 | 20.448 | 5.228 | 0.002,0.007 | 0.546,0.657 | 0.362,0.690 | 0.820 |

1-1-3 | 0.0021 | 0.586 | 0.704 | 1.052 | 13.703 | 4.893 | 0.000,0.008 | 0.490,0.676 | 0.439,1.010 | 0.689 |

1-2-1 | 0.0019 | 0.551 | 0.309 | 1.276 | 17.242 | 2.877 | 0.001,0.007 | 0.467,0.626 | 0.114,0.533 | 0.753 |

1-2-2 | 0.0011 | 0.612 | 0.478 | 0.760 | 18.687 | 4.335 | −0.001,0.005 | 0.513,0.700 | 0.269,0.693 | 0.789 |

1-2-3 | 0.0023 | 0.489 | 0.784 | 1.561 | 15.590 | 7.430 | 0.002,0.007 | 0.418,0.559 | 0.578,0.996 | 0.762 |

2-1-1 | −0.0004 | 0.904 | 0.289 | −0.126 | 13.357 | 1.268 | −0.007,0.005 | 0.746,1.065 | −0.178,0.778 | 0.635 |

2-1-2 | −0.0006 | 0.928 | 0.637 | −0.216 | 15.306 | 3.121 | −0.006,0.005 | 0.786,1.079 | 0.231,1.066 | 0.710 |

2-1-3 | 0.0011 | 0.827 | 1.329 | 0.323 | 11.617 | 5.549 | −0.004,0.008 | 0.665,1.003 | 0.847,1.878 | 0.639 |

2-2-1 | −0.0026 | 0.839 | 0.177 | −0.852 | 12.762 | 0.801 | −0.011,0.001 | 0.677,1.013 | −0.344,0.723 | 0.610 |

2-2-2 | −0.0023 | 0.775 | 0.943 | −0.617 | 9.635 | 3.484 | −0.012,0.003 | 0.594,0.964 | 0.322,1.651 | 0.521 |

2-2-3 | −0.0040 | 0.832 | 1.638 | −1.471 | 14.217 | 8.322 | −0.013,−0.002 | 0.684,0.978 | 1.244,2.090 | 0.747 |

t-Statistic | ||||
---|---|---|---|---|

Estimate | FM | CS | Bootstrap Basic 95% CI | |

Market | 0.0102 | 2.0582 | 1.4853 | 0.0039, 0.0245 |

CV | −0.0015 | −0.5816 | −0.4020 | −0.0070, 0.0039 |

Skew | −0.0026 | −1.8070 | −1.2878 | −0.0060, −0.0006 |

Number of Portfolios Where: | GRS | ||||||
---|---|---|---|---|---|---|---|

Model | Factors | ${\mathit{\beta}}_{\mathit{m}}=0$ | ${\mathit{\beta}}_{\mathbf{cv}}=0$ | ${\mathit{\beta}}_{\mathit{k}}=0$ | ${\mathit{\beta}}_{\mathit{s}}=0$ | p-Value | Adj. ${\mathit{R}}^{2}$ |

Model 1 | M | 0 | - | - | - | 0.038 | 0.776–0.472 |

Model 1b | M | 0 | - | - | - | 0.054 | |

Model 2 | M, CV | 0 | 0 | - | - | 0.038 | 0.929-0.639 |

Model 2b | M, CV | 0 | 0 | - | - | 0.133 | |

Model 3 | M, K | 0 | - | 7 | - | 0.065 | 0.776–0.467 |

Model 3b | M, K | 0 | - | 8 | - | 0.086 | |

Model 4 | M, S | 0 | - | - | 2 | 0.051 | 0.820–0.521 |

Model 4b | M, S | 0 | - | - | 2 | 0.078 | |

Model 5 | M, CV, S | 0 | 1 | - | 3 | 0.063 | 0.930–0.692 |

Model 5b | M, CV, S | 0 | 2 | - | 3 | 0.097 |

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**MDPI and ACS Style**

Cueto, J.M.; Grané, A.; Cascos, I.
Models for Expected Returns with Statistical Factors. *J. Risk Financial Manag.* **2020**, *13*, 314.
https://doi.org/10.3390/jrfm13120314

**AMA Style**

Cueto JM, Grané A, Cascos I.
Models for Expected Returns with Statistical Factors. *Journal of Risk and Financial Management*. 2020; 13(12):314.
https://doi.org/10.3390/jrfm13120314

**Chicago/Turabian Style**

Cueto, José Manuel, Aurea Grané, and Ignacio Cascos.
2020. "Models for Expected Returns with Statistical Factors" *Journal of Risk and Financial Management* 13, no. 12: 314.
https://doi.org/10.3390/jrfm13120314