# 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 |

## References

- Carhart, Mark M. 1997. On Persistence in Mutual Fund Performance. The Journal of Finance 52: 57–82. [Google Scholar] [CrossRef]
- Chang, Bo Young, Peter Christoffersen, and Kris Jacobs. 2013. Market skewness risk and the cross section of stock returns. Journal of Financial Economics 107: 46–68. [Google Scholar] [CrossRef]
- Conrad, Jennifer, Robert F. Dittmar, and Eric Ghysels. 2013. Ex Ante Skewness and Expected Stock Returns. The Journal of Finance 68: 85–124. [Google Scholar] [CrossRef][Green Version]
- Efron, Bradley. 1972. Bootstrap Methods: Another Look at the Jackknife. The Annals of Statistics 7: 1–26. [Google Scholar] [CrossRef]
- Elyasiani, Elyas, Luca Gambarelli, and Silvia Muzzioli. 2020. Moment risk premia and the cross-section of stock returns in the European stock market. Journal of Banking & Finance 111: 105732. [Google Scholar]
- Fama, Eugene F., and Kenneth R. French. 1992. The cross-section of expected returns. The Journal of Finance 47: 427–65. [Google Scholar] [CrossRef]
- Fama, Eugene F., and Kenneth R. French. 1993. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33: 3–56. [Google Scholar] [CrossRef]
- Fama, Eugene F., and Kenneth R. French. 2010. Luck versus skill in the cross section of mutual fund returns. The Journal of Finance 75: 1915–47. [Google Scholar] [CrossRef]
- Fama, Eugene F., and James D. MacBeth. 1973. Risk, Return and Equilibrium: Empirical Tests. Journal of Political Economy 81: 607–36. [Google Scholar] [CrossRef]
- Feng, Guanhao, Stefano Giglio, and Dacheng Xiu. 2020. Taming the Factor Zoo: A Test of New Factors. The Journal of Finance 75: 1327–70. [Google Scholar] [CrossRef]
- Gibbons, Michael R., Stephen A. Ross, and Jay Shanken. 1989. A test of the efficiency of a given portfolio. Econometrica 57: 1121–52. [Google Scholar] [CrossRef][Green Version]
- Grané, Aurea, and Helena Veiga. 2008. Accurate minimum capital risk requirements: A comparison of several approaches. Journal of Banking and Finance 32: 2482–92. [Google Scholar] [CrossRef]
- Harvey, Campbell, and Akhtar Siddique. 2000. Conditional skewness in asset pricing tests. The Journal of Finance 55: 1263–95. [Google Scholar] [CrossRef]
- Jensen, Michael C. 1968. The performance of mutual funds in the period 1945–1964. The Journal of Finance 23: 389–416. [Google Scholar] [CrossRef]
- Kosowski, Robert, Allan Timmermann, Russ Wermers, and Hal White. 2006. Can mutual fund “stars” really pick stocks? New evidence from a bootstrap analysis. The Journal of Finance 61: 2551–95. [Google Scholar] [CrossRef]
- Kristjanpoller, Werner, and Carolina Liberona. 2010. Comparación de modelos de predicción de retornos accionarios en el Mercado Accionario Chileno: CAPM, Fama y French y Reward Beta. EconoQuantum 7: 121–40. [Google Scholar] [CrossRef]
- Kumar, Alok. 2009. Who gambles in the stock market? The Journal of Finance 64: 1889–933. [Google Scholar] [CrossRef]
- Lintner, John. 1965. The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics 47: 13–37. [Google Scholar] [CrossRef]
- Markowitz, Harry. 1952. Portfolio selection. The Journal of Finance 7: 77–91. [Google Scholar]
- Miller, Merton H., and Myron Scholes. 1972. Rates of return in relation to risk: A reexamination of some recent findings. In Studies in the Theory of Capital Markets. Edited by Michael C. Jensen. New York: Praeger, pp. 47–78. [Google Scholar]
- Morelli, David. 2010. European capital market integration: An empirical study based on a European asset pricing model. Journal of International Financial Markets, Institutions and Money 20: 363–75. [Google Scholar] [CrossRef]
- Mossin, Jan. 1966. Equilibrium in a Capital Asset Market. Econometrica 34: 758–83. [Google Scholar] [CrossRef]
- Ramos, Sofia, Abderrahim Taamouti, Helena Veiga, and Chih-Wei Wang. 2017. Do investors price industry risk? Evidence from the cross-section of the oil industry. Journal of Energy Markets 10: 79–108. [Google Scholar] [CrossRef][Green Version]
- Shanken, Jay. 1992. On the estimation of beta-pricing models. The Review of Financial Studies 5: 1–33. [Google Scholar] [CrossRef]
- Sharpe, William F. 1964. Capital asset prices: A theory of market equilibrium under conditions of risk. The Journal of Finance 19: 425–42. [Google Scholar]
- Sørensen, Lars Qvigstad. 2009. Testing Mutual Fund Performance at the Oslo Stock Exchange. Available online: https://ssrn.com/abstract=1488745 (accessed on 19 October 2018).
- Soumaré, Issouf, Edo Jossi Aménounvé, Ousmane Diop, Dramane Méité, and Yao Djifa N’Sougan. 2013. Applying the CAPM and the Fama-French models to the BRVM stock maket. Applied Financial Economics 23: 275–85. [Google Scholar] [CrossRef]
- Viale, Ariel M., James W. Kolari, and Donald R. Fraser. 2009. Common risk factors in bank stocks. Journal of Banking and Finance 33: 464–72. [Google Scholar] [CrossRef][Green Version]

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 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. 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/).

## Share and Cite

**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