# How to Explain the Cross-Section of Equity Returns through Common Principal Components

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Data and Factors

#### 2.1.1. Data Description

#### 2.1.2. Factors under Study

- Mean of returns for each year and company:$$\overline{x}=\frac{1}{n}\sum _{i=1}^{n}({x}_{i}-\overline{x}).$$
- Standard Deviation (SD) of returns for each year and company:$$s=\sqrt{\frac{1}{n-1}\sum _{i=1}^{n}{({x}_{i}-\overline{x})}^{2}}.$$
- Excess Kurtosis (Kurt) of returns for each year and company:$$Kurt=\frac{\frac{1}{n}{\sum}_{i=1}^{n}{({x}_{i}-\overline{x})}^{4}}{{s}^{4}}-3,$$
- Skewness (Skew) of returns for each year and company:$$Skew=\frac{\frac{1}{n}{\sum}_{i=1}^{n}{({x}_{i}-\overline{x})}^{3}}{{s}^{3}},$$

#### 2.2. Methodology

#### 2.2.1. Common Principal Components

**Proposition**

**1.**

**Lemma**

**1**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3**

**Proof.**

**Preliminary Setup of the Algorithm to Compute the CPC-Model**

`multigroup`, designed to study multigroup data, where the same set of variables are measured on different groups of individuals. Within this package, we specifically use the function

`FCPCA`to perform the CPC calculation.

#### 2.2.2. Portfolio Construction

**Portfolio Setup with the CPC Model**

- Stocks with low CPC1 are included in portfolios 1-b-c-d, while stocks with high CPC1 are included in portfolios 2-b-c-d.
- Stocks with low CPC2 are included in portfolios a-1-c-d, while stocks with high CPC2 are included in portfolios a-2-c-d.
- Stocks with low CPC3 are included in portfolios a-b-1-d, while stocks with high CPC3 are included in portfolios a-b-2-d.
- Stocks with low CPC4 are included in portfolios a-b-c-1, while stocks with high CPC4 are included in portfolios a-b-c-2.

#### 2.2.3. Classical Methodologies

#### 2.2.4. Resampling Techniques: Bootstrap

**Step 1**Estimate benchmark regression models, one for each portfolio. For each portfolio, save $\alpha $, $\beta $, their corresponding t-statistics, residuals, risk factors’ estimates, and GRS statistic.**Step 2**Produce a set of simulation runs equal for each portfolio in order to preserve returns’ cross-correlations.**Step 3**Build a new series of $\alpha $-free portfolio returns by using the simulated time indices.**Step 4**Run the time-series factor model regression on the artificially constructed returns. Calculate $\widehat{\alpha}$, $\widehat{\beta}$ and the corresponding confidence intervals. Next, generate samples of the GRS Statistic, compute different percentiles from the bootstrapped distribution and compare them to the original GRS statistic.In this paper, we improve the methodology proposed in our previous work. We have noticed that, given the construction of the GRS statistic, when we face portfolios with heteroskedasticity, the estimation of the variance through the residual covariance matrix may not be appropriate. Thus, we propose a new statistic, $Q\left(\alpha \right)={\widehat{\alpha}}^{\prime}{\Sigma}^{-1}\widehat{\alpha}$, for which we will calculate the covariance matrix of $\alpha $s, $\Sigma $, through a nested bootstrap. Then, a new step appears:**Step 5**Run the time-series factor model regression on the artificially constructed returns. Calculate $\widehat{\alpha}$, $\widehat{\beta}$ and the corresponding confidence intervals. Finally, generate bootstrap samples of the $Q\left(\alpha \right)$ statistic that makes use of the covariance matrix of the $\alpha $s which is approximated by means of a nested bootstrap and compare the bootstrapped statistics with the original $Q\left(\alpha \right)$ statistic.Concerning cross-sectional regression, we use $\beta $s and average returns from each bootstrapped sample to determine the significance of the risk premia estimates ($\lambda $s). Given that $\beta $s are estimated, the estimates of the $\lambda $s might present substantial bias and we use a reverse bootstrap percentile interval to determine the significance of the factors. To be consistent, we also present this type of bootstrap interval for all the estimated parameters.

`doParallel`and

`foreach`, designed to do multi-core calculations. The main reason for using the

`foreach`package is that it supports parallel execution, that is, repeated operations can be executed on multiple cores of the computer or on multiple nodes of a cluster, thus reducing the execution time.

## 3. Results

#### 3.1. Time-Series Regressions

#### 3.1.1. Model 1: CAPM

#### 3.1.2. Model 2: Market and Factors CPC1 to CPC4

#### 3.2. Cross-Sectional Regression

#### 3.2.1. Model 1: CAPM

#### 3.2.2. Model 2: Market and Factors CPC1 to CPC4

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

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Sharpe, W.F. Capital asset prices: A theory of market equilibrium under conditions of risk. J. Financ.
**1964**, 19, 425–442. [Google Scholar] - Lintner, J. The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Rev. Econ. Stat.
**1965**, 47, 13–37. [Google Scholar] [CrossRef] - Mossin, J. Equilibrium in a Capital Asset Market. Econometrica
**1966**, 34, 758–783. [Google Scholar] [CrossRef] - Frazzini, A.; Pedersen, L. Betting agains beta. J. Financ. Econ.
**2014**, 111, 1–25. [Google Scholar] [CrossRef] [Green Version] - Miller, M.H.; Scholes, M. Rates of return in relation to risk: A reexamination of some recent findings. In Studies in the Theory of Capital Markets; Jensen, M.C., Ed.; Praeger: New York, NY, USA, 1972; pp. 47–78. [Google Scholar]
- Fama, E.F.; French, K.R. The cross-section of expected returns. J. Financ.
**1992**, 47, 427–465. [Google Scholar] [CrossRef] - Viale, A.M.; Kolari, J.W.; Fraser, D.R. Common risk factors in bank stocks. J. Bank. Financ.
**2009**, 33, 464–472. [Google Scholar] - Ramos, S.; Taamouti, A.; Veiga, H.; Wang, C.W. Do investors price industry risk? Evidence from the cross-section of the oil industry. J. Energy Mark.
**2017**, 10, 79–108. [Google Scholar] [CrossRef] [Green Version] - Elyasiani, E.; Gambarelli, L.; Muzzioli, S. Moment risk premia and the cross-section of stock returns in the European stock market. J. Bank. Financ.
**2020**, 111, 105732. [Google Scholar] [CrossRef] - Lemperiere, Y.; Deremble, C.; Nguyen, T.; Seager, P.; Potters, M.; Bouchaud, J. Risk Premia: Asymmetric Tail Risks and Excess Returns. Quant. Financ.
**2017**, 17, 1–14. [Google Scholar] - Carhart, M.M. On Persistence in Mutual Fund Performance. J. Financ.
**1997**, 52, 57–82. [Google Scholar] [CrossRef] - Misra, A.; Mohapatra, S. Evidence and Sources of Momentum Profits. A Study on Indian Stock Market. Econ. Manag. Financ. Mark.
**2014**, 9, 86–109. [Google Scholar] - Harvey, C.; Liu, Y.; Zhu, H. ... and the cross-section of expected returns. Rev. Financ. Stud.
**2015**, 29, 5–68. [Google Scholar] [CrossRef] [Green Version] - Fama, E.F.; French, K.R. Choosing factors. J. Financ. Econ.
**2018**, 128, 234–252. [Google Scholar] [CrossRef] - Barillas, F.; Shanken, J. Which alpha? Rev. Financ. Stud.
**2017**, 30, 1316–1338. [Google Scholar] [CrossRef] - Fama, E.F.; French, K.R. A five-factor asset pricing model. J. Financ. Econ.
**2015**, 116, 1–22. [Google Scholar] [CrossRef] [Green Version] - Heerden, J.V.; Rensburg, P.V. Common Firm-Specific Characteristics of Extreme Performers on The Johannesburg Securities Exchange. Econ. Manag. Financ. Mark.
**2017**, 12, 25–50. [Google Scholar] - Feng, G.; Giglio, S.; Xiu, D. Taming the Factor Zoo: A Test of New Factors. J. Financ.
**2020**, 75, 1327–1370. [Google Scholar] [CrossRef] - Flury, B.N. Common principal components in k groups. J. Am. Stat. Assoc.
**1984**, 79, 892–898. [Google Scholar] [CrossRef] - Fama, E.F.; French, K.R. Common risk factors in the returns on stocks and bonds. J. Financ. Econ.
**1993**, 33, 3–56. [Google Scholar] [CrossRef] - Fama, E.F.; MacBeth, J.D. Risk, Return and Equilibrium: Empirical Tests. J. Political Econ.
**1973**, 81, 607–636. [Google Scholar] [CrossRef] - Efron, B. Bootstrap Methods: Another Look at the Jackknife. Ann. Stat.
**1972**, 7, 1–26. [Google Scholar] - Cueto, J.M.; Grané, A.; Cascos, I. Models for Expected Returns with Statistical Factors. J. Risk Financ. Manag.
**2020**, 13, 314. [Google Scholar] [CrossRef] - Chou, P.; Zhou, G. Using Bootstrap to Test Portfolio Efficiency. Ann. Econ. Financ.
**2006**, 1, 217–249. [Google Scholar] - Grané, A.; Veiga, H. Accurate minimum capital risk requirements: A comparison of several approaches. J. Bank. Financ.
**2008**, 32, 2482–2492. [Google Scholar] [CrossRef] - Flury, B.N.; Gaustchi, W. An algorithm for simultaneous orthogonal transformation of several positive definite symmetric matrices to nearly diagonal form. SIAM J. Sci. Stat. Comput.
**1986**, 7, 169–184. [Google Scholar] [CrossRef] - Flury, B. Common Principal Components and Related Multivariate Models; John Wiley & Sons: Hoboken, NJ, USA, 1988. [Google Scholar]
- Breiman, L. Bagging predictors. Mach. Learn.
**1996**, 24, 123–140. [Google Scholar] [CrossRef] [Green Version] - Gibbons, M.R.; Ross, S.A.; Shanken, J. A test of the efficiency of a given portfolio. Econometrica
**1989**, 57, 1121–1152. [Google Scholar] [CrossRef] [Green Version] - Fama, E.F. Cross-Section Versus Time-Series Tests of Asset Pricing Models. Fama-Mill. Work. Pap.
**2015**. [Google Scholar] [CrossRef] - Shanken, J. On the estimation of beta-pricing models. Rev. Financ. Stud.
**1992**, 5, 1–33. [Google Scholar] [CrossRef] - Soumaré, I.; Aménounvé, E.J.; Diop, O.; Méité, D.; N’Sougan, Y.D. Applying the CAPM and the Fama-French models to the BRVM stock maket. Appl. Financ. Econ.
**2013**, 23, 275–285. [Google Scholar] [CrossRef] - Blitz, D.; Falkenstein, E.; van Vliet, P. Explanations for the Volatility Effect: An Overview Based on the CAPM Assumptions. J. Portf. Manag.
**2014**, 40, 61–76. [Google Scholar] [CrossRef]

**Figure 5.**Autocorrelation charts for errors in TS regression for 5-factor model for the whole period.

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

2009 | 1230 | 3 | 34.70 | 455.92 | 0.05 | 7.61 | 18,043.85 |

2010 | 1230 | 12 | 33.76 | 363.74 | 0.03 | 8.15 | 15,513.50 |

2011 | 1230 | 12 | 29.23 | 176.98 | 0.03 | 8.57 | 6515.83 |

2012 | 1230 | 12 | 26.55 | 132.78 | 0.02 | 7.82 | 4486.71 |

2013 | 1230 | 12 | 28.28 | 116.58 | 0.03 | 9.01 | 3430.00 |

2014 | 1230 | 12 | 31.64 | 127.10 | 0.01 | 10.95 | 3800.00 |

2015 | 1230 | 12 | 35.24 | 138.22 | 0.01 | 12.16 | 4000.00 |

2016 | 1230 | 12 | 37.46 | 166.02 | 0.01 | 12.29 | 5999.00 |

2017 | 1230 | 12 | 45.42 | 194.01 | 0.01 | 14.95 | 5999.99 |

2018 | 1230 | 12 | 47.43 | 211.38 | 0.01 | 14.70 | 6600.00 |

2019 | 1230 | 10 | 46.17 | 241.41 | 0.01 | 13.34 | 9200.00 |

Market | Momentum | Mean | SD | Kurt | Skew | Market Cap | P/B | ROA | ROE | Total Assets | |
---|---|---|---|---|---|---|---|---|---|---|---|

Market | 1.000 | 0.042 | 0.025 | −0.002 | −0.001 | 0.010 | −0.006 | −0.005 | −0.016 | −0.005 | 0.001 |

Momentum | 0.042 | 1.000 | 0.959 | −0.066 | −0.062 | 0.231 | 0.008 | 0.049 | 0.049 | 0.199 | 0.022 |

Mean | 0.025 | 0.959 | 1.000 | −0.070 | −0.062 | 0.243 | 0.009 | 0.054 | 0.059 | 0.207 | 0.023 |

SD | −0.002 | −0.066 | −0.070 | 1.000 | 0.200 | 0.036 | −0.117 | 0.010 | −0.020 | −0.115 | −0.009 |

Kurt | −0.001 | −0.062 | −0.062 | 0.200 | 1.000 | 0.068 | −0.059 | 0.003 | −0.011 | −0.029 | −0.009 |

Skew | 0.010 | 0.231 | 0.243 | 0.036 | 0.068 | 1.000 | −0.088 | 0.008 | −0.014 | 0.012 | −0.003 |

Market Cap | −0.006 | 0.008 | 0.009 | −0.117 | −0.059 | −0.088 | 1.000 | 0.001 | 0.081 | 0.047 | 0.010 |

P/B | −0.005 | 0.049 | 0.054 | 0.010 | 0.003 | 0.008 | 0.001 | 1.000 | 0.015 | 0.065 | 0.008 |

ROA | −0.016 | 0.049 | 0.059 | −0.020 | −0.011 | −0.014 | 0.081 | 0.015 | 1.000 | 0.187 | 0.197 |

ROE | −0.005 | 0.199 | 0.207 | −0.115 | −0.029 | 0.012 | 0.047 | 0.065 | 0.187 | 1.000 | 0.051 |

Total Assets | 0.001 | 0.022 | 0.023 | −0.009 | −0.009 | −0.003 | 0.010 | 0.008 | 0.197 | 0.051 | 1.000 |

Dim 1 | Dim 1 (Bagging) | Dim 2 | Dim 2 (Bagging) | Dim 3 | Dim 3 (Bagging) | Dim 4 | Dim 4 (Bagging) | |
---|---|---|---|---|---|---|---|---|

Momentum | 70.92 | 70.77 | −0.06 | 2.15 | −4.92 | −5.61 | 2.12 | 2.32 |

Mean | 70.12 | 69.82 | 0.60 | 2.76 | −4.08 | −4.62 | 1.93 | 2.14 |

SD | −3.31 | −3.78 | 4.66 | 4.60 | −7.11 | −6.04 | 99.58 | 99.52 |

Kurt | −4.40 | −7.29 | 72.27 | 72.81 | −68.46 | −67.39 | −8.42 | −7.81 |

Skew | 4.30 | 3.02 | 68.95 | 68.02 | 72.26 | 72.95 | 2.08 | 1.27 |

Market Cap | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | −0.01 |

P/B | 0.00 | 0.01 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

ROA | 0.25 | 0.32 | −0.05 | −0.11 | −0.08 | −0.04 | −0.65 | −0.85 |

ROE | 2.23 | 2.33 | −0.04 | 0.08 | −0.27 | −0.21 | −0.69 | −1.06 |

Total Assets | 0.02 | 0.01 | −0.03 | 0.00 | −0.04 | −0.06 | −0.05 | −0.10 |

Portfolio Number | Portfolio Description | Portfolio Composition | |||
---|---|---|---|---|---|

CPC1 | CPC2 | CPC3 | CPC4 | ||

1 | 1-1-1-1 | Low | Low | Low | Low |

2 | 1-1-1-2 | Low | Low | Low | High |

3 | 1-1-2-1 | Low | Low | High | Low |

4 | 1-1-2-2 | Low | Low | High | High |

5 | 1-2-1-1 | Low | High | Low | Low |

6 | 1-2-1-2 | Low | High | Low | High |

7 | 1-2-2-1 | Low | High | High | Low |

8 | 1-2-2-2 | Low | High | High | High |

9 | 2-1-1-1 | High | Low | Low | Low |

10 | 2-1-1-2 | High | Low | Low | High |

11 | 2-1-2-1 | High | Low | High | Low |

12 | 2-1-2-2 | High | Low | High | High |

13 | 2-2-1-1 | High | High | Low | Low |

14 | 2-2-1-2 | High | High | Low | High |

15 | 2-2-2-1 | High | High | High | Low |

16 | 2-2-2-2 | High | High | High | High |

Portfolio | Portfolio | Portfolio Composition (Mean Values) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Number | Description | Market Cap | Total Assets | PB | ROA | ROE | MOM | Mean | sd | Kurt | Skew |

1 | 1-1-1-1 | 4922.44 | 8816.40 | 1.94 | 4.57 | 8.47 | −0.086 | −0.008 | 0.057 | 2.935 | −0.555 |

2 | 1-1-1-2 | 2072.12 | 6275.54 | 1.64 | 1.80 | −0.28 | −0.233 | −0.021 | 0.116 | 2.865 | −0.580 |

3 | 1-1-2-1 | 4063.73 | 6734.97 | 1.88 | 4.48 | 7.93 | −0.094 | −0.009 | 0.055 | 2.161 | 0.025 |

4 | 1-1-2-2 | 1839.13 | 5504.61 | 1.56 | 1.84 | −4.27 | −0.233 | −0.021 | 0.108 | 2.079 | 0.000 |

5 | 1-2-1-1 | 3263.27 | 6024.39 | 1.84 | 4.22 | 7.74 | −0.075 | −0.007 | 0.059 | 4.308 | −0.189 |

6 | 1-2-1-2 | 1044.30 | 3209.33 | 1.46 | 1.37 | −14.23 | −0.241 | −0.022 | 0.146 | 4.263 | −0.227 |

7 | 1-2-2-1 | 2795.40 | 4915.16 | 1.75 | 4.05 | 5.53 | −0.080 | −0.007 | 0.055 | 2.961 | 0.575 |

8 | 1-2-2-2 | 1025.77 | 3036.45 | 1.62 | 1.41 | −9.23 | −0.208 | −0.019 | 0.118 | 2.853 | 0.558 |

9 | 2-1-1-1 | 6655.51 | 8231.53 | 2.65 | 7.33 | 14.65 | 0.186 | 0.017 | 0.058 | 2.729 | −0.494 |

10 | 2-1-1-2 | 2654.52 | 4986.17 | 2.50 | 5.76 | 11.55 | 0.306 | 0.028 | 0.106 | 2.614 | −0.440 |

11 | 2-1-2-1 | 6033.27 | 7107.41 | 2.52 | 7.26 | 14.15 | 0.172 | 0.016 | 0.055 | 2.089 | 0.023 |

12 | 2-1-2-2 | 2570.95 | 4606.20 | 2.42 | 5.76 | 10.76 | 0.284 | 0.026 | 0.099 | 1.999 | 0.068 |

13 | 2-2-1-1 | 2946.72 | 3876.48 | 2.33 | 6.85 | 12.89 | 0.169 | 0.016 | 0.060 | 4.068 | 0.222 |

14 | 2-2-1-2 | 1218.32 | 2307.18 | 2.29 | 5.23 | 10.51 | 0.369 | 0.034 | 0.140 | 4.121 | 0.502 |

15 | 2-2-2-1 | 3016.33 | 3701.31 | 2.32 | 7.13 | 13.15 | 0.172 | 0.016 | 0.056 | 2.943 | 0.600 |

16 | 2-2-2-2 | 1031.39 | 2519.19 | 2.15 | 6.31 | 8.45 | 0.333 | 0.031 | 0.111 | 2.877 | 0.672 |

Statistics (p-Value) | Number of Portfolios | |||||
---|---|---|---|---|---|---|

GRS | GRS Boot. | $\mathit{Q}\left(\mathbf{\alpha}\right)$ | Non-Normal | Heteros. | Autocorr. | |

Whole period | $2.024$ ($1.93\%$) | $2.024$ ($17.4\%$) | $42.57$ ($29.5\%$) | 6 | 4 | 5 |

Months 1–80 | $1.924$ ($3.43\%$) | $1.924$ ($8.2\%$) | $38.62$ ($27.5\%$) | 3 | 5 | 4 |

Months 30–108 | $2.736$ ($0.24\%$) | $2.736$ ($17.8\%$) | $61.35$ ($30.6\%$) | 3 | 1 | 8 |

Estimates | t-Statistics | Bootstrap CI $(2.5\%,97.5\%)$ | |||||
---|---|---|---|---|---|---|---|

Portfolio | $\mathbf{\alpha}$ | Market | $\mathbf{\alpha}$ | Market | $\mathbf{\alpha}$ | Market | ${\mathit{R}}^{\mathbf{2}}$ |

1 | −0.001 | 0.685 | −0.546 | 10.856 | −0.007, 0.002 | 0.533, 0.845 | 0.526 |

2 | −0.003 | 0.954 | −0.695 | 9.109 | −0.013, 0.002 | 0.746, 1.161 | 0.439 |

3 | −0.004 | 0.543 | −1.068 | 4.982 | −0.016, −0.002 | 0.324, 0.736 | 0.190 |

4 | −0.005 | 0.946 | −1.235 | 8.853 | −0.018, −0.002 | 0.722, 1.172 | 0.425 |

5 | 0.000 | 0.485 | −0.041 | 7.141 | −0.006, 0.004 | 0.317, 0.649 | 0.325 |

6 | −0.003 | 0.823 | −0.680 | 7.380 | −0.014, 0.002 | 0.604, 1.024 | 0.339 |

7 | −0.002 | 0.542 | −0.905 | 7.948 | −0.010, 0.001 | 0.389, 0.716 | 0.373 |

8 | −0.004 | 0.831 | −1.032 | 7.702 | −0.016, −0.001 | 0.577, 1.078 | 0.359 |

9 | 0.004 | 0.632 | 1.655 | 9.304 | 0.003, 0.012 | 0.432, 0.840 | 0.450 |

10 | 0.005 | 0.865 | 1.398 | 8.898 | 0.003, 0.016 | 0.622, 1.097 | 0.428 |

11 | 0.004 | 0.619 | 1.647 | 8.967 | 0.003, 0.013 | 0.398, 0.827 | 0.431 |

12 | 0.003 | 0.852 | 0.960 | 8.613 | 0.000, 0.014 | 0.602, 1.112 | 0.412 |

13 | 0.004 | 0.532 | 1.936 | 8.128 | 0.004, 0.013 | 0.362, 0.725 | 0.384 |

14 | 0.000 | 0.657 | 0.128 | 6.932 | −0.006, 0.008 | 0.472, 0.847 | 0.312 |

15 | 0.006 | 0.454 | 2.300 | 6.467 | 0.007, 0.015 | 0.264, 0.628 | 0.283 |

16 | 0.005 | 0.729 | 1.459 | 7.672 | 0.003, 0.016 | 0.500, 0.991 | 0.357 |

Statistics (p-Value) | Number of Portfolios | |||||
---|---|---|---|---|---|---|

GRS | GRS Boot. | $\mathit{Q}\left(\mathbf{\alpha}\right)$ | Non-Normal | Heteros. | Autocorr. | |

Whole period | $2.242$ ($0.90\%$) | $2.242$ ($17.1\%$) | $56.59$ ($22.6\%$) | 10 | 1 | 12 |

Months 1–80 | $1.806$ ($5.19\%$) | $1.806$ ($22\%$) | $44.92$ ($32.8\%$) | 1 | 1 | 7 |

Months 30–108 | $2.625$ ($0.38\%$) | $2.625$ ($13\%$) | $56.19$ ($27.8\%$) | 8 | 1 | 7 |

Estimates | t-Statistics | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Portfolio | $\mathbf{\alpha}$ | Market | CPC1 | CPC2 | CPC3 | CPC4 | $\mathbf{\alpha}$ | Market | CPC1 | CPC2 | CPC3 | CPC4 | Adj-${\mathit{R}}^{\mathbf{2}}$ |

1 | 0.001 | 0.436 | −0.292 | −0.381 | 0.626 | 0.687 | 0.731 | 6.674 | −2.891 | −1.503 | 3.029 | 6.199 | 0.688 |

2 | 0.002 | 0.286 | −0.315 | −1.601 | 0.497 | 1.607 | 0.693 | 3.819 | −2.713 | −5.497 | 2.092 | 12.621 | 0.825 |

3 | 0.000 | 0.290 | −0.503 | −1.257 | 0.719 | 0.288 | −0.078 | 2.189 | −2.453 | −2.444 | 1.712 | 1.280 | 0.263 |

4 | 0.001 | 0.316 | −0.547 | −1.001 | 1.195 | 1.734 | 0.384 | 4.574 | −5.127 | −3.738 | 5.471 | 14.813 | 0.853 |

5 | 0.003 | 0.208 | −0.311 | −0.546 | 0.073 | 0.643 | 1.314 | 2.883 | −2.791 | −1.954 | 0.318 | 5.260 | 0.533 |

6 | 0.003 | 0.386 | −0.592 | 0.358 | −0.170 | 1.440 | 0.972 | 4.113 | −4.072 | 0.981 | −0.570 | 9.030 | 0.712 |

7 | 0.000 | 0.325 | −0.303 | −0.218 | 0.976 | 0.682 | 0.174 | 4.473 | −2.690 | −0.771 | 4.234 | 5.525 | 0.562 |

8 | −0.001 | 0.306 | −0.271 | −0.598 | 1.428 | 1.654 | −0.212 | 3.303 | −1.896 | −1.664 | 4.872 | 10.531 | 0.710 |

9 | 0.002 | 0.345 | 0.406 | −0.879 | 0.625 | 0.832 | 1.129 | 5.021 | 3.821 | −3.296 | 2.873 | 7.138 | 0.654 |

10 | 0.003 | 0.312 | 0.561 | −1.874 | 0.442 | 1.368 | 1.225 | 3.784 | 4.404 | −5.861 | 1.695 | 9.784 | 0.747 |

11 | 0.002 | 0.378 | 0.376 | −1.018 | 1.040 | 0.659 | 1.084 | 5.254 | 3.376 | −3.649 | 4.568 | 5.396 | 0.620 |

12 | 0.000 | 0.362 | 0.709 | −0.871 | 0.948 | 1.670 | 0.195 | 4.622 | 5.854 | −2.867 | 3.825 | 12.567 | 0.774 |

13 | 0.002 | 0.404 | 0.475 | −0.025 | 0.478 | 0.640 | 0.885 | 5.457 | 4.150 | −0.085 | 2.039 | 5.094 | 0.517 |

14 | −0.001 | 0.259 | 0.370 | −0.668 | 0.576 | 1.280 | −0.220 | 2.679 | 2.469 | −1.779 | 1.880 | 7.792 | 0.558 |

15 | 0.003 | 0.279 | 0.470 | −0.302 | 0.414 | 0.679 | 1.416 | 3.489 | 3.802 | −0.973 | 1.636 | 5.002 | 0.430 |

16 | 0.003 | 0.401 | 0.471 | 0.264 | 1.637 | 1.585 | 1.283 | 5.005 | 3.800 | 0.847 | 6.447 | 11.649 | 0.719 |

Bootstrap CI (2.5%,97.5%) | ||||||
---|---|---|---|---|---|---|

Portfolio | $\mathbf{\alpha}$ | Market | CPC1 | CPC2 | CPC3 | CPC4 |

1 | −0.840, 0.435 | 0.299, 0.605 | −0.528, −0.044 | −0.966, 0.179 | 0.105, 1.153 | 0.376, 0.945 |

2 | −0.851, 0.452 | 0.145, 0.435 | −0.582, −0.085 | −2.288, −1.054 | −0.129, 1.333 | 1.188, 1.888 |

3 | −1.288, 0.666 | 0.038, 0.538 | −0.869, −0.165 | −2.966, −0.060 | −0.236, 1.505 | −0.453, 0.758 |

4 | −1.410, 0.684 | 0.157, 0.496 | −0.762, −0.315 | −1.611, −0.462 | 0.680, 1.619 | 1.403, 1.978 |

5 | −0.343, 0.473 | 0.065, 0.363 | −0.567, −0.071 | −1.242, −0.086 | −0.461, 0.659 | 0.279, 0.876 |

6 | −0.161, 0.797 | 0.147, 0.628 | −0.930, −0.242 | −0.916, 1.272 | −0.900, 0.427 | 0.881, 1.785 |

7 | −1.271, 0.413 | 0.190, 0.483 | −0.518, −0.074 | −0.894, 0.365 | 0.365, 1.622 | 0.363, 0.894 |

8 | −1.770, 0.425 | 0.119, 0.465 | −0.526, −0.006 | −1.532, 0.035 | 0.732, 1.973 | 1.197, 1.938 |

9 | −0.892, 0.008 | 0.209, 0.494 | 0.149, 0.623 | −1.446, −0.413 | 0.091, 1.232 | 0.483, 1.067 |

10 | −0.912, 0.201 | 0.137, 0.490 | 0.170, 0.879 | −2.840, −1.137 | −0.292, 1.313 | 0.809, 1.694 |

11 | −1.329, 0.008 | 0.213, 0.554 | 0.052, 0.680 | −1.854, −0.372 | 0.445, 1.737 | 0.212, 0.968 |

12 | −1.255, 0.005 | 0.177, 0.566 | 0.453, 0.980 | −1.680, −0.145 | 0.332, 1.513 | 1.316, 1.931 |

13 | −0.759, 0.008 | 0.216, 0.596 | 0.221, 0.720 | −0.708, 0.548 | −0.082, 1.022 | 0.274, 0.907 |

14 | −1.154, 0.646 | 0.114, 0.424 | −0.166, 0.753 | −2.052, 0.300 | −0.491, 2.086 | 0.581, 1.698 |

15 | −0.821, 0.010 | 0.087, 0.458 | 0.260, 0.706 | −1.298, 0.412 | −0.343, 0.966 | 0.351, 0.910 |

16 | −2.012, 0.010 | 0.265, 0.589 | 0.210, 0.751 | −0.252, 0.906 | 0.876, 2.468 | 1.254, 1.916 |

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

Estimate | FM | CS | $\mathbf{95}\%$CI | |

Market | 0.0108 | 2.054 | 1.5447 | 0.0091, 0.0254 |

CPC1 | 0.0071 | 9.625 | 3.5836 | 0.0017, 0.0181 |

CPC2 | −0.0001 | −0.085 | −0.0502 | −0.0126, 0.0038 |

CPC3 | −0.0012 | −2.241 | −1.1303 | −0.0150, 0.0014 |

CPC4 | 0.0002 | 0.445 | 0.1139 | −0.0120, 0.0043 |

Number of Portfolios Where | ||||||||
---|---|---|---|---|---|---|---|---|

Model | Factors | ${\mathbf{\beta}}_{\mathit{m}}=\mathbf{0}$ | ${\mathbf{\beta}}_{\mathit{CPC}\mathbf{1}}=\mathbf{0}$ | ${\mathbf{\beta}}_{\mathit{CPC}\mathbf{2}}=\mathbf{0}$ | ${\mathbf{\beta}}_{\mathit{CPC}\mathbf{3}}=\mathbf{0}$ | ${\mathbf{\beta}}_{\mathit{CPC}\mathbf{4}}=\mathbf{0}$ | GRSp-Value | Adj-${\mathit{R}}^{\mathbf{2}}$ |

Model 1 | Market | 0 | - | - | - | - | 0.019 | 0.190 to 0.526 |

Model 1b | Market | 0 | - | - | - | - | 0.174 | |

Model 2 | Market & CPCs | 0 | 1 | 9 | 6 | 1 | 0.009 | 0.263 to 0.853 |

Model 2b | Market & CPCs | 0 | 1 | 8 | 8 | 1 | 0.171 |

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## Share and Cite

**MDPI and ACS Style**

Cueto, J.M.; Grané, A.; Cascos, I.
How to Explain the Cross-Section of Equity Returns through Common Principal Components. *Mathematics* **2021**, *9*, 1011.
https://doi.org/10.3390/math9091011

**AMA Style**

Cueto JM, Grané A, Cascos I.
How to Explain the Cross-Section of Equity Returns through Common Principal Components. *Mathematics*. 2021; 9(9):1011.
https://doi.org/10.3390/math9091011

**Chicago/Turabian Style**

Cueto, José Manuel, Aurea Grané, and Ignacio Cascos.
2021. "How to Explain the Cross-Section of Equity Returns through Common Principal Components" *Mathematics* 9, no. 9: 1011.
https://doi.org/10.3390/math9091011