# Goodness-of-Fit versus Significance: A CAPM Selection with Dynamic Betas Applied to the Brazilian Stock Market

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## Abstract

**:**

## 1. Introduction

## 2. Conditional CAPM Learning Model

#### 2.1. The Adrian and Franzoni Model (2009)

#### 2.2. Empirical A&F exercise

- −
- Term Spread: Difference between the 10 years treasury rate of the USA and the 3 months one. This variable refers to the risk premium between sovereign bonds of long-term and short-term.
- −
- Value Spread: Difference between the average returns between companies with high BE/ME (Book Value/Market Value) and low BE/ME. This factor is HML (high minus low) from Fama and French (1993);
- −
- Value-Weighted Market Portfolio: weighted market return variable as Campbell and Vuolteenaho (2004);
- −
- CAY (Consumption; Asset Holdings and Labor Income relationship): Variable created by Lettau and Ludvigson (2001) that captures the innovations for the cointegrating relationship between Consumption, Asset Holdings and Labor Income relationship. This is the variable that underlies market expectations in microeconomics, adding consumption between the market conditioners.

- −
- RMSE (Root Mean Squared pricing errors): mean square error of the returns of assets;
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- CPE (Composite Pricing Error). Defined as ${\widehat{\alpha}}^{\prime}{\widehat{\Omega}}^{-1}\widehat{\alpha}$, where $\widehat{\alpha}$ is error vector with an N assets dimension whose models were estimated and $\widehat{\Omega}$ is the diagonal matrix of the returns variances returns estimated by the model.

#### 2.3. CAPM Model with Beta as Random Walk

## 3. Models Estimation and Their Results

#### 3.1. Data

**Credit Spread in Brazil**—Due to a lack of long-term bonds in Brazil before 2000, when were issued several government bonds series LTN, NTN-F and NTN-B, in this work the country’s credit spread used was the SWAP360 (swap rate—DI—360 days—period average -% p.y.) available in IPEADATA. The correlation of the used SWAP360 with the fixed 1-year rates for the actual LTN bonds also available in IPEADATA from May 2000 presents a correlation of 99.85%. So, this was the component of the model that represents the Brazilian credit spread.

**Variables related to stock prices**—Share Price over Book-Value (P/BV): ratio between the share price and its book value. This parameter is used by the market to observe how far the price of the shares is from its book value. Price over Earnings (P/E): ratio between the share price and the earnings per share in a one-year period. The parameter is used to observe if the share has good returns comparing to its price traded on the stock exchange.

**Proxy for Consumption**—A difference between this study and that of Mazzeu et al. (2013) is the inclusion of the above variables and a proxy for consumption. In the A&F model, it was used the variable related to consumption, called CAY (Consumption; Asset Holdings and Labor Income relationship), in the state-space model together with other macroeconomic variables. However, given that in Brazil there is not an analogous variable available, it was decided to use an alternative variable: the total electricity consumption (obtained in IPEADATA). As the results show, it was significant in the models of several of the analyzed shares.

#### 3.2. Parameters Estimation

#### 3.3. Learning CAPM Estimation Errors

## 4. VaR Calculation Using the Model Results

#### 4.1. VaR Calculation

#### 4.2. Backtesting Results

## 5. Final Remarks

## Author Contributions

## Conflicts of Interest

## Appendix A

Stocks | Fi | p-Value | Bi | p-Value | ${\mathit{\sigma}}_{\mathit{\eta}}$ | p-Value | ${\mathit{\sigma}}_{\mathit{\mu}}$ | p-Value |
---|---|---|---|---|---|---|---|---|

PETR4 | 0.097 | 0.886 | 0.967 | (0.000) | 0.004 | 0.000 | 0.077 | 0.009 |

ABEV3 | 0.221 | 0.641 | 0.385 | (0.000) | 0.005 | 0.000 | 0.157 | 0.026 |

BBDC4 | −0.675 | 0.000 | 0.928 | (0.000) | 0.003 | 0.000 | 0.154 | 0.004 |

BBAS3 | −0.077 | 0.883 | 1.092 | (0.000) | 0.005 | 0.000 | 0.149 | 0.058 |

BRKM5 | 0.889 | 0.000 | 0.825 | (0.007) | 0.012 | 0.000 | 0.093 | 0.011 |

ELET6 | 0.941 | 0.000 | 0.872 | (0.001) | 0.009 | 0.000 | 0.022 | 0.024 |

CMIG4 | −0.283 | 0.536 | 0.731 | (0.000) | 0.006 | 0.000 | 0.000 | 1.000 |

CSNA3 | 1.000 | 0.000 | −20.832 | (0.982) | 0.007 | 0.000 | 0.000 | 1.000 |

GGBR4 | −0.181 | 0.821 | 1.207 | (0.000) | 0.006 | 0.000 | 0.000 | 1.000 |

ITSA4 | 0.020 | 0.962 | 0.934 | (0.000) | 0.003 | 0.000 | 0.069 | 0.007 |

KLBN4 | −0.792 | 0.006 | 0.704 | (0.000) | 0.006 | 0.000 | 0.052 | 0.111 |

OIBR4 | 0.965 | 0.000 | 0.662 | (0.001) | 0.007 | 0.000 | 0.000 | 1.000 |

CRUZ3 | −0.851 | 0.000 | 0.487 | (0.000) | 0.005 | 0.000 | 0.035 | 0.008 |

USIM5 | −0.080 | 0.884 | 1.343 | (0.000) | 0.009 | 0.000 | 0.000 | 1.000 |

VALE5 | 0.922 | 0.000 | 0.786 | (0.000) | 0.004 | 0.000 | 0.026 | 0.000 |

VIVT4 | −0.253 | 0.269 | 0.415 | (0.000) | 0.004 | 0.000 | 0.318 | 0.002 |

EMBR3 | 0.205 | 0.310 | 0.686 | (0.000) | 0.007 | 0.000 | 0.751 | 0.510 |

Stocks | Fi | p-Value | Bi | p-Value | ${\mathit{\sigma}}_{\mathit{\eta}}$ | p-Value |
---|---|---|---|---|---|---|

PETR4 | 0.750 | 0.000 | 0.754 | (0.021) | 0.004 | 0.000 |

ABEV3 | 0.073 | 0.905 | 0.598 | (0.113) | 0.005 | 0.000 |

BBDC4 | −0.385 | 0.032 | 0.887 | (0.019) | 0.003 | 0.000 |

BBAS3 | −0.193 | 0.614 | 1.349 | (0.000) | 0.005 | 0.000 |

BRKM5 | 0.914 | 0.000 | 1.644 | (0.142) | 0.012 | 0.000 |

ELET6 | 0.886 | 0.000 | 0.791 | (0.000) | 0.009 | 0.000 |

CMIG4 | 0.936 | 0.000 | 0.656 | (0.000) | 0.005 | 0.000 |

CSNA3 | 0.958 | 0.000 | 0.754 | (0.105) | 0.007 | 0.000 |

GGBR4 | −0.176 | 0.858 | 1.063 | (0.000) | 0.006 | 0.000 |

ITSA4 | −0.910 | 0.000 | 1.201 | (0.000) | 0.003 | 0.000 |

KLBN4 | −0.775 | 0.010 | 0.656 | (0.059) | 0.006 | 0.000 |

OIBR4 | 0.030 | 0.972 | 0.926 | (0.003) | 0.007 | 0.000 |

CRUZ3 | −0.614 | 0.008 | 0.253 | (0.057) | 0.004 | 0.000 |

USIM5 | −0.059 | 0.890 | 1.269 | (0.000) | 0.009 | 0.000 |

VALE5 | 0.944 | 0.000 | −1.144 | (0.659) | 0.005 | 0.000 |

VIVT4 | −0.718 | 0.000 | 0.350 | (0.000) | 0.005 | 0.000 |

EMBR3 | 0.163 | 0.439 | 0.649 | (0.075) | 0.007 | 0.000 |

Stocks | ${\mathit{\sigma}}_{\mathit{\mu}}$ | p-Value | Φ P/VPA | p-Value | Φ P/L | p-Value |
---|---|---|---|---|---|---|

PETR4 | 0.004 | 0.198 | −0.034 | 0.503 | 0.014 | 0.163 |

ABEV3 | 0.179 | 0.016 | −0.083 | 0.322 | 0.010 | 0.207 |

BBDC4 | 0.283 | 0.003 | 0.778 | 0.001 | −0.161 | 0.034 |

BBAS3 | 0.099 | 0.052 | 0.287 | 0.240 | −0.095 | 0.144 |

BRKM5 | 0.065 | 0.010 | −0.075 | 0.327 | 0.001 | 0.669 |

ELET6 | 0.004 | 0.233 | −0.825 | 0.187 | 0.000 | 0.943 |

CMIG4 | 0.000 | 1.000 | −0.025 | 0.212 | −0.007 | 0.069 |

CSNA3 | 0.000 | 1.000 | 0.001 | 0.904 | 0.002 | 0.279 |

GGBR4 | 0.000 | 1.000 | 0.206 | 0.333 | −0.016 | 0.607 |

ITSA4 | 0.011 | 0.004 | 0.173 | 0.714 | −0.103 | 0.223 |

KLBN4 | 0.060 | 0.110 | 0.087 | 0.844 | −0.002 | 0.744 |

OIBR4 | 0.000 | 1.000 | −0.053 | 0.865 | −0.001 | 0.696 |

CRUZ3 | 0.152 | 0.009 | −0.128 | 0.046 | 0.102 | 0.040 |

USIM5 | 0.000 | 1.000 | 0.092 | 0.604 | 0.000 | 0.049 |

VALE5 | 0.009 | 0.006 | 0.014 | 0.381 | 0.006 | 0.157 |

VIVT4 | 0.035 | 0.004 | −0.827 | 0.001 | −0.071 | 0.006 |

EMBR3 | 0.759 | 0.544 | 0.054 | 0.491 | −0.007 | 0.277 |

Stocks | Fi | p-Value | Bi | p-Value | ${\mathit{\sigma}}_{\mathit{\eta}}$ | p-Value | ${\mathit{\sigma}}_{\mathit{\mu}}$ | p-Value | Φ SW360 | p-Value |
---|---|---|---|---|---|---|---|---|---|---|

PETR4 | 0.073 | 0.898 | 0.971 | (0.000) | 0.004 | 0.000 | 0.068 | 0.010 | 0.824 | 0.477 |

ABEV3 | 0.259 | 0.491 | 0.394 | (0.000) | 0.005 | 0.000 | 0.152 | 0.028 | 1.833 | 0.194 |

BBDC4 | −0.657 | 0.000 | 0.939 | (0.000) | 0.003 | 0.000 | 0.138 | 0.002 | 2.564 | 0.065 |

BBAS3 | −0.083 | 0.873 | 1.091 | (0.000) | 0.005 | 0.000 | 0.149 | 0.059 | −0.315 | 0.820 |

BRKM5 | 0.904 | 0.000 | 0.837 | (0.015) | 0.012 | 0.000 | 0.089 | 0.012 | 0.610 | 0.632 |

ELET6 | 0.952 | 0.000 | 0.837 | (0.003) | 0.009 | 0.000 | 0.019 | 0.019 | 0.417 | 0.602 |

CMIG4 | −0.327 | 0.393 | 0.728 | (0.000) | 0.006 | 0.000 | 0.000 | 1.000 | −1.378 | 0.189 |

CSNA3 | 0.974 | 0.000 | 1.399 | (0.000) | 0.007 | 0.000 | 0.000 | 1.000 | −0.186 | 0.762 |

GGBR4 | −0.200 | 0.651 | 1.198 | (0.000) | 0.006 | 0.000 | 0.000 | 1.000 | −1.674 | 0.123 |

ITSA4 | 0.030 | 0.093 | 0.939 | (0.000) | 0.003 | 0.000 | 0.071 | 0.004 | 1.334 | 0.087 |

KLBN4 | −0.854 | 0.000 | 0.711 | (0.000) | 0.007 | 0.000 | 0.008 | 0.093 | 1.957 | 0.081 |

OIBR4 | 0.977 | 0.000 | 0.474 | (0.508) | 0.007 | 0.000 | 0.000 | 1.000 | −0.816 | 0.213 |

CRUZ3 | −0.652 | 0.006 | 0.473 | (0.000) | 0.004 | 0.000 | 0.127 | 0.020 | −0.703 | 0.601 |

USIM5 | N.A. | N.A. | N.A. | N.A. | N.A. | N.A. | N.A. | N.A. | N.A. | N.A. |

VALE5 | 0.972 | 0.000 | 0.653 | (0.000) | 0.005 | 0.000 | 0.001 | 0.010 | −1.639 | 0.001 |

VIVT4 | −0.701 | 0.000 | 0.411 | (0.000) | 0.005 | 0.000 | 0.018 | 0.030 | −3.886 | 0.000 |

EMBR3 | 0.189 | 0.356 | 0.683 | (0.000) | 0.007 | 0.000 | 0.763 | 0.540 | −0.947 | 0.597 |

Stocks | Fi | p-Value | Bi | p-Value | ${\mathit{\sigma}}_{\mathit{\eta}}$ | p-Value | ${\mathit{\sigma}}_{\mathit{\mu}}$ | p-Value | Φ Elet. | p-Value |
---|---|---|---|---|---|---|---|---|---|---|

PETR4 | 0.082 | 0.900 | 0.969 | (0.000) | 0.004 | 0.000 | 0.077 | 0.009 | −0.331 | 0.913 |

ABEV3 | 0.296 | 0.376 | 0.415 | (0.000) | 0.006 | 0.000 | 0.109 | 0.021 | −4.220 | 0.486 |

BBDC4 | −0.678 | 0.000 | 0.931 | (0.000) | 0.003 | 0.000 | 0.152 | 0.005 | −1.002 | 0.799 |

BBAS3 | −0.070 | 0.894 | 1.087 | (0.000) | 0.005 | 0.000 | 0.153 | 0.055 | 1.174 | 0.765 |

BRKM5 | 0.892 | 0.000 | 0.867 | (0.014) | 0.012 | 0.000 | 0.088 | 0.008 | −1.630 | 0.777 |

ELET6 | 0.043 | 0.960 | 0.796 | (0.000) | 0.010 | 0.000 | 0.081 | 0.212 | 4.051 | 0.437 |

CMIG4 | −0.256 | 0.535 | 0.750 | (0.000) | 0.006 | 0.000 | 0.000 | 1.000 | −3.715 | 0.179 |

CSNA3 | 0.987 | 0.000 | 2.040 | (0.359) | 0.007 | 0.000 | 0.000 | 1.000 | −1.618 | 0.530 |

GGBR4 | −0.160 | 0.774 | 1.197 | (0.000) | 0.006 | 0.000 | 0.000 | 1.000 | 1.819 | 0.721 |

ITSA4 | 0.022 | 0.958 | 0.933 | (0.000) | 0.003 | 0.000 | 0.068 | 0.007 | 0.220 | 0.951 |

KLBN4 | −0.723 | 0.022 | 0.692 | (0.000) | 0.006 | 0.000 | 0.076 | 0.101 | 2.716 | 0.527 |

OIBR4 | −0.023 | 0.959 | 0.902 | (0.000) | 0.007 | 0.000 | 0.000 | 1.000 | −4.320 | 0.352 |

CRUZ3 | −0.914 | 0.000 | 0.527 | (0.000) | 0.005 | 0.000 | 0.002 | 0.091 | −5.140 | 0.010 |

USIM5 | 0.905 | 0.000 | 1.501 | (0.000) | 0.009 | 0.000 | 0.000 | 1.000 | −1.571 | 0.409 |

VALE5 | 0.929 | 0.000 | 0.772 | (0.000) | 0.004 | 0.000 | 0.022 | 0.000 | 0.416 | 0.820 |

VIVT4 | −0.255 | 0.266 | 0.416 | (0.000) | 0.004 | 0.000 | 0.317 | 0.002 | −0.286 | 0.955 |

EMBR3 | 0.360 | 0.021 | 0.753 | (0.000) | 0.007 | 0.000 | 0.555 | 0.178 | −11.344 | 0.091 |

Stocks | Fi | p-Value | Bi | p-Value | ${\mathit{\sigma}}_{\mathit{\eta}}$ | p-Value | ${\mathit{\sigma}}_{\mathit{\mu}}$ | p-Value | Φ P/VPA | p-Value |
---|---|---|---|---|---|---|---|---|---|---|

PETR4 | −0.171 | 0.738 | 0.920 | (0.007) | 0.004 | 0.000 | 0.074 | 0.009 | −0.149 | 0.528 |

ABEV3 | 0.101 | 0.803 | 0.645 | (0.096) | 0.005 | 0.000 | 0.170 | 0.014 | −0.085 | 0.256 |

BBDC4 | −0.454 | 0.007 | 0.994 | (0.010) | 0.003 | 0.000 | 0.250 | 0.002 | 0.478 | 0.104 |

BBAS3 | −0.186 | 0.642 | 1.364 | (0.000) | 0.005 | 0.000 | 0.103 | 0.047 | 0.278 | 0.255 |

BRKM5 | 0.925 | 0.000 | 1.756 | (0.196) | 0.012 | 0.000 | 0.063 | 0.011 | −0.071 | 0.346 |

ELET6 | 0.894 | 0.000 | 3.201 | (0.000) | 0.009 | 0.000 | <0.0001 | 1.000 | −0.825 | 0.071 |

CMIG4 | 0.936 | 0.000 | 1.975 | (0.000) | 0.005 | 0.000 | <0.0001 | 1.000 | −0.033 | 0.176 |

CSNA3 | 0.888 | 0.000 | 0.818 | (0.012) | 0.007 | 0.000 | <0.0001 | 1.000 | 0.009 | 0.464 |

GGBR4 | 0.891 | 0.000 | 0.806 | (0.007) | 0.006 | 0.000 | <0.0001 | 1.000 | 0.036 | 0.465 |

ITSA4 | −0.890 | 0.000 | 1.243 | (0.000) | 0.003 | 0.000 | 0.014 | 0.006 | 0.095 | 0.844 |

KLBN4 | −0.789 | 0.000 | 0.628 | (0.036) | 0.006 | 0.000 | 0.027 | 0.122 | 0.113 | 0.773 |

OIBR4 | −0.067 | 0.874 | 0.931 | (0.005) | 0.007 | 0.000 | <0.0001 | 1.000 | −0.037 | 0.915 |

CRUZ3 | −0.904 | 0.000 | 0.447 | (0.012) | 0.005 | 0.000 | 0.005 | 0.024 | −0.047 | 0.358 |

USIM5 | 0.525 | 0.151 | 1.342 | (0.000) | 0.009 | 0.000 | <0.0001 | 1.000 | 0.032 | 0.697 |

VALE5 | 0.968 | 0.000 | −2.383 | (0.207) | 0.004 | 0.000 | <0.0001 | 0.100 | 0.013 | 0.138 |

VIVT4 | −0.664 | 0.000 | 1.445 | (0.000) | 0.004 | 0.000 | 0.008 | 0.233 | −0.747 | 0.001 |

EMBR3 | 0.308 | 0.063 | 0.796 | (0.041) | 0.007 | 0.000 | 0.577 | 0.257 | 0.024 | 0.723 |

Stocks | Φ P/L | p-Value | Φ SW360 | p-Value | Φ Elet. | p-Value |
---|---|---|---|---|---|---|

PETR4 | 0.040 | 0.117 | 1.073 | 0.412 | −0.841 | 0.794 |

ABEV3 | 0.010 | 0.127 | 1.844 | 0.224 | −2.005 | 0.748 |

BBDC4 | −0.112 | 0.195 | 2.870 | 0.083 | 1.297 | 0.763 |

BBAS3 | −0.096 | 0.144 | 0.137 | 0.917 | 1.669 | 0.694 |

BRKM5 | 0.001 | 0.693 | 0.522 | 0.663 | −0.348 | 0.951 |

ELET6 | 0.000 | 0.811 | 0.330 | 0.695 | 5.342 | 0.036 |

CMIG4 | −0.006 | 0.146 | −0.029 | 0.966 | 1.835 | 0.478 |

CSNA3 | 0.003 | 0.311 | −0.446 | 0.609 | −3.315 | 0.328 |

GGBR4 | −0.001 | 0.652 | −0.097 | 0.867 | −0.005 | 0.998 |

ITSA4 | −0.095 | 0.257 | 0.647 | 0.435 | 0.376 | 0.883 |

KLBN4 | −0.001 | 0.850 | 2.499 | 0.064 | 3.284 | 0.489 |

OIBR4 | −0.002 | 0.561 | −0.528 | 0.730 | −5.123 | 0.280 |

CRUZ3 | 0.039 | 0.401 | −0.013 | 0.985 | −5.430 | 0.015 |

USIM5 | 0.000 | 0.096 | 0.713 | 0.657 | −1.780 | 0.724 |

VALE5 | 0.005 | 0.051 | −1.621 | 0.004 | −0.410 | 0.858 |

VIVT4 | −0.049 | 0.032 | −3.692 | 0.000 | −3.074 | 0.374 |

EMBR3 | −0.006 | 0.343 | −1.858 | 0.276 | −11.623 | 0.081 |

Stocks | ${\mathit{\sigma}}_{\mathit{\eta}}$ | p-Value | ${\mathit{\sigma}}_{\mathit{\mu}}$ | p-Value |
---|---|---|---|---|

PETR4 | 0.004 | 0.000 | 0.001 | 0.000 |

ABEV3 | 0.006 | 0.000 | 0.001 | 0.000 |

BBDC4 | 0.005 | 0.000 | 0.000 | 1.000 |

BBAS3 | 0.006 | 0.000 | 0.001 | 0.000 |

BRKM5 | 0.012 | 0.000 | 0.043 | 0.000 |

ELET6 | 0.009 | 0.000 | 0.012 | 0.000 |

CMIG4 | 0.005 | 0.000 | 0.004 | 0.000 |

CSNA3 | 0.007 | 0.000 | 0.001 | 0.002 |

GGBR4 | 0.006 | 0.000 | 0.001 | 0.000 |

ITSA4 | 0.003 | 0.000 | 0.000 | 1.000 |

KLBN4 | 0.007 | 0.000 | 0.000 | 1.000 |

OIBR4 | 0.007 | 0.000 | 0.002 | 0.000 |

CRUZ3 | 0.006 | 0.000 | 0.000 | 1.000 |

USIM5 | 0.009 | 0.000 | 0.001 | 0.002 |

VALE5 | 0.005 | 0.000 | 0.010 | 0.000 |

VIVT4 | 0.005 | 0.000 | 0.002 | 0.000 |

EMBR3 | 0.011 | 0.000 | 0.005 | 0.000 |

## Appendix B

Information Criteria | |||||||
---|---|---|---|---|---|---|---|

No Variables | P/BV & P/E | SW Fixed | Energ. Consump. | All | Random Walk | ||

BBDC4 | Akaike | −2.476 | −2.456 | −2.494 | |||

Schwarz | −2.402 | −2.344 | −2.413 | ||||

Hannan-Quinn | −2.446 | −2.411 | −2.456 | ||||

BRKM5 | Akaike | −1.332 | −1.321 | ||||

Schwarz | −1.258 | −1.284 | |||||

Hannan-Quinn | −1.308 | −1.306 | |||||

ELET6 | Akaike | −1.683 | −1.677 | ||||

Schwarz | −1.598 | −1.640 | |||||

Hannan-Quinn | −1.693 | −1.662 | |||||

CRUZ3 | Akaike | −2.288 | −2.290 | −2.328 | −2.176 | ||

Schwarz | −2.213 | −2.179 | −2.235 | −2.139 | |||

Hannan-Quinn | −2.258 | −2.245 | −2.291 | −2.161 | |||

VALE5 | Akaike | −2.333 | −2.365 | −2.333 | |||

Schwarz | −2.258 | −2.297 | −2.296 | ||||

Hannan-Quinn | −2.302 | −2.327 | −2.318 | ||||

VIVT4 | Akaike | −2.326 | −2.317 | −2.243 | |||

Schwarz | −2.214 | −2.224 | −2.206 | ||||

Hannan-Quinn | −2.280 | −2.279 | −2.228 |

## Appendix C

## Appendix D

## Appendix E

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1 | The reference exchange rate for the US dollar, known in the market as the PTAX rate, which is the arithmetic average of four daily requests from foreign exchange dealers for bid/offer rates. |

# | Bovespa Code | Type | Company |
---|---|---|---|

1 | ABEV3 | ON | AMBEV SA |

2 | BBDC4 | PN | Banco Bradesco SA |

3 | BBAS3 | ON | Banco do Brasil SA |

4 | BRKM5 | PNA | Braskem SA |

5 | ELET6 | PNB | Centrais Elétricas Brasileiras SA |

6 | CMIG4 | PN | Cia. Energética de Minas Gerais SA |

7 | CSNA3 | ON | Cia. Siderúrgica Nacional SA |

8 | GGBR4 | PN | Gerdau SA |

9 | ITSA4 | PN | Itaúsa—Investimentos Itaú SA |

10 | KLBN4 | PN | Klabin SA |

11 | OIBR4 | PN | Oi SA |

12 | PETR4 | PN | Petróleo Brasileiro SA |

13 | CRUZ3 | ON | Souza Cruz SA |

14 | VIVT4 | PN | Telecomunicações de São Paulo SA |

15 | USIM5 | PNA | Usinas Siderúrgicas de Minas Gerais SA |

16 | VALE5 | PNA | Vale SA |

17 | EMBR3 | ON | Embraer SA |

Models with all the Coefficients with 90% of Significance | ||||||
---|---|---|---|---|---|---|

No Variables | P/BV & P/E | SW Fixed | Energ. Consump. | All | Random Walk | |

PETR4 | OK | |||||

ABEV3 | OK | |||||

BBDC4 | OK | OK | OK | |||

BBAS3 | OK | |||||

BRKM5 | OK | OK | ||||

ELET6 | OK | OK | ||||

CMIG4 | OK | |||||

CSNA3 | OK | |||||

GGBR4 | OK | |||||

ITSA4 | OK | |||||

KLBN4 | OK | |||||

OIBR4 | OK | |||||

CRUZ3 | OK | OK | OK | |||

USIM5 | OK | |||||

VALE5 | OK | OK | OK | |||

VIVT4 | OK | OK | OK | |||

EMBR3 | OK |

Share | Result | Share | Result |
---|---|---|---|

PETR4 | Random W. | ITSA4 | SWAP360 |

ABEV3 | Random W. | KLBN4 | SWAP360 |

BBDC4 | SWAP360 | OIBR4 | Random W. |

BBAS3 | Random W. | CRUZ3 | Energy Cons. |

BRKM5 | No Variables | USIM5 | Random W. |

ELET6 | No Variables | VALE5 | SWAP360 |

CMIG4 | Random W. | VIVT4 | P/VPA & P/L |

CSNA3 | Random W. | EMBR3 | Random W. |

GGBR4 | Random W. |

RMSE (Root Mean Squared Error) Average | |||||||
---|---|---|---|---|---|---|---|

No Variables | P/BV & P/E | SW Fixed | Energ. Consump. | All | Random Walk | OLS | |

PETR4 | 0.0662 | 0.0658 | 0.0659 | 0.0662 | 0.0650 | 0.0677 | 0.0660 |

ABEV3 | 0.0777 | 0.0769 | 0.0769 | 0.0770 | 0.0760 | 0.0794 | 0.0776 |

BBDC4 | 0.0672 | 0.0701 | 0.0658 | 0.0672 | 0.0672 | 0.0738 | 0.0704 |

BBAS3 | 0.0767 | 0.0751 | 0.0767 | 0.0768 | 0.0751 | 00778 | 0.0764 |

BRKM5 | 0.1168 | 0.1163 | 0.1167 | 0.1166 | 0.1161 | 0.1191 | 0.1189 |

ELET6 | 0.0986 | 0.0963 | 0.0986 | 0.1008 | 0.0943 | 0.0995 | 0.1005 |

CMIG4 | 0.0747 | 0.0715 | 0.0744 | 0.0743 | 0.0712 | 0.0744 | 0.0778 |

CSNA3 | 0.0870 | 0.0858 | 0.0867 | 0.0863 | 0.0849 | 0.0879 | 0.0855 |

GGBR4 | 0.0792 | 0.0785 | 0.0789 | 0.0789 | 0.0780 | 0.0802 | 0.0799 |

ITSA4 | 0.0566 | 0.0544 | 0.0562 | 0.0566 | 0.0541 | 0.0575 | 0.0565 |

KLBN4 | 0.0832 | 0.0832 | 0.0823 | 0.0831 | 0.0821 | 0.0847 | 0.0834 |

OIBR4 | 0.0851 | 0.0866 | 0.0844 | 0.0864 | 0.0862 | 0.0869 | 0.0864 |

CRUZ3 | 0.0712 | 0.0713 | 0.0721 | 0.0684 | 0.0681 | 0.0783 | 0.0750 |

USIM5 | 0.0949 | 0.0940 | N.A. | 0.0932 | 0.0929 | 0.0957 | 0.0948 |

VALE5 | 0.0759 | 0.0696 | 0.0715 | 0.0704 | 0.0666 | 0.0711 | 0.0736 |

VIVT4 | 0.0759 | 0.0705 | 0.0715 | 0.0759 | 0.0677 | 0.0764 | 0.0762 |

EMBR3 | 0.1062 | 0.1050 | 0.1060 | 0.1044 | 0.1031 | 0.1144 | 0.1072 |

CPE (Composite Pricing Error) | ||||||
---|---|---|---|---|---|---|

No Variables | P/BV & P/E | SW Fixed | Energ. Consump. | All | Random Walk | OLS |

0.6851 | 0.5475 | 0.5782 | 0.6545 | 0.5310 | 0.5797 | 0.7334 |

Test | CAPM Models | |||||||
---|---|---|---|---|---|---|---|---|

No Variables | P/VPA & P/L | SWAP360 | Energy Consump. | |||||

VaR 95% | VaR 99% | VaR 95% | VaR 99% | VaR 95% | VaR 99% | VaR 95% | VaR 99% | |

Unconditional Convergence Test (Kupiec) | 76% | 53% | 71% | 53% | 76% | 59% | 76% | 65% |

Independence Test (Christoffersen) | 59% | 47% | 59% | 47% | 65% | 53% | 65% | 53% |

Conditional Convergence Test (combined) | 59% | 53% | 65% | 53% | 59% | 59% | 65% | 59% |

Test | CAPM Models | |||||||
---|---|---|---|---|---|---|---|---|

All | Random W. | OLS | Historical VaR | |||||

VaR 95% | VaR 99% | VaR 95% | VaR 99% | VaR 95% | VaR 99% | VaR 95% | VaR 99% | |

Unconditional Convergence Test (Kupiec) | 71% | 47% | 82% | 71% | 100% | 100% | 100% | 100% |

Independence Test (Christoffersen) | 59% | 47% | 71% | 59% | 82% | 88% | 82% | 88% |

Conditional Convergence Test (combined) | 59% | 47% | 71% | 65% | 82% | 100% | 82% | 100% |

© 2017 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**

Ronzani, A.R.d.P.; Candido, O.; Maldonado, W.F.L.
Goodness-of-Fit versus Significance: A CAPM Selection with Dynamic Betas Applied to the Brazilian Stock Market. *Int. J. Financial Stud.* **2017**, *5*, 33.
https://doi.org/10.3390/ijfs5040033

**AMA Style**

Ronzani ARdP, Candido O, Maldonado WFL.
Goodness-of-Fit versus Significance: A CAPM Selection with Dynamic Betas Applied to the Brazilian Stock Market. *International Journal of Financial Studies*. 2017; 5(4):33.
https://doi.org/10.3390/ijfs5040033

**Chicago/Turabian Style**

Ronzani, André Ricardo de Pinho, Osvaldo Candido, and Wilfredo Fernando Leiva Maldonado.
2017. "Goodness-of-Fit versus Significance: A CAPM Selection with Dynamic Betas Applied to the Brazilian Stock Market" *International Journal of Financial Studies* 5, no. 4: 33.
https://doi.org/10.3390/ijfs5040033