This factor regression tool supports factor regression analysis of individual assets or a portfolio of assets using the given risk factor model. The multiple linear regression indicates how well the returns of the given assets or a portfolio are explained by the risk factor exposures. The supported equity risk factor models include:

- The capital asset pricing model (CAPM) with market factor (MKT)
- The Fama-French three factor model with market, size, and value factors (MKT, SMB, HML)
- The Carhart four-factor model with market, size, value, and momentum factors (MKT, SMB, HML, MOM)
- The Fama-French five factor model with market, size, value, profitability, and investment factors (MKT, SMB, HML, RMW, CMA)

Additional supported equity factors include the short and long-term reversal factors (STREV, LTREV) based on Fama-French factor data, quality (QMJ) factor based on both AQR and Alpha Architect factor data, and bet against beta (BAB) factor based on AQR factor data.

For fixed income funds and balanced funds you can include the fixed income factor model to explain returns based on term risk (interest rate risk) and credit risk exposures. The fixed income factors can be further adjusted to account for the yield curve and to add high yield credit risk as an additional factor.

Name | Ticker | Start Date | End Date | Rm-Rf | SMB | HML | Alpha | Annual Alpha | R^{2} |
---|---|---|---|---|---|---|---|---|---|

Schwab International Equity ETF | SCHF | Sep 2013 | Feb 2019 | 0.99 | -0.15 | -0.10 | -0.03% | -0.40% | 97.3% |

Schwab Emerging Markets Equity ETF | SCHE | Sep 2013 | Feb 2019 | 1.08 | 0.32 | 0.12 | -0.03% | -0.41% | 63.0% |

Schwab Fundamental Intl Sm Co ETF | FNDC | Sep 2013 | Feb 2019 | 1.01 | 0.32 | -0.14 | 0.06% | 0.71% | 94.7% |

Vanguard FTSE All-Wld ex-US SmCp ETF | VSS | Sep 2013 | Feb 2019 | 1.04 | 0.47 | -0.12 | -0.09% | -1.13% | 93.3% |

Schwab International Small-Cap Eq ETF | SCHC | Sep 2013 | Feb 2019 | 1.07 | 0.42 | -0.18 | -0.11% | -1.29% | 94.8% |

Ticker | SCHF | ||||
---|---|---|---|---|---|

Time Period | Sep 2013 - Feb 2019 | ||||

Coefficient of Determination (R^{2}) |
97.3% | ||||

Adjusted R^{2} |
97.2% | ||||

Regression F statistic | 751.41 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 2.788 with p-value 1.000) | ||||

Heteroskedasticity | No heteroscedasticity confirmed (Breusch-Pagan test value is 4.017 with p-value 0.260) | ||||

Factor | Loading | Standard Error | t-stat | p-value | 95% Confidence Interval |

Market (Rm-Rf) | 0.99 | 0.021 | 46.166 | 0.000 | 0.943...1.029 |

Size (SMB) | -0.15 | 0.048 | -3.017 | 0.004 | -0.243...-0.049 |

Value (HML) | -0.10 | 0.043 | -2.364 | 0.021 | -0.188...-0.016 |

Alpha (α) | -3.33bps | 0.001 | -0.469 | 0.640 | -0.18%...0.11% |

Annualized Alpha (α) | -0.40% |

Ticker | SCHE | ||||
---|---|---|---|---|---|

Time Period | Sep 2013 - Feb 2019 | ||||

Coefficient of Determination (R^{2}) |
63.0% | ||||

Adjusted R^{2} |
61.2% | ||||

Regression F statistic | 35.16 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 1.968 with p-value 0.447) | ||||

Heteroskedasticity | No heteroscedasticity confirmed (Breusch-Pagan test value is 2.048 with p-value 0.563) | ||||

Factor | Loading | Standard Error | t-stat | p-value | 95% Confidence Interval |

Market (Rm-Rf) | 1.08 | 0.107 | 10.092 | 0.000 | 0.864...1.291 |

Size (SMB) | 0.32 | 0.242 | 1.321 | 0.191 | -0.164...0.805 |

Value (HML) | 0.12 | 0.216 | 0.568 | 0.572 | -0.309...0.553 |

Alpha (α) | -3.41bps | 0.004 | -0.096 | 0.924 | -0.74%...0.68% |

Annualized Alpha (α) | -0.41% |

Ticker | FNDC | ||||
---|---|---|---|---|---|

Time Period | Sep 2013 - Feb 2019 | ||||

Coefficient of Determination (R^{2}) |
94.7% | ||||

Adjusted R^{2} |
94.4% | ||||

Regression F statistic | 368.29 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 2.672 with p-value 0.998) | ||||

Heteroskedasticity | No heteroscedasticity confirmed (Breusch-Pagan test value is 1.868 with p-value 0.600) | ||||

Factor | Loading | Standard Error | t-stat | p-value | 95% Confidence Interval |

Market (Rm-Rf) | 1.01 | 0.031 | 33.180 | 0.000 | 0.953...1.075 |

Size (SMB) | 0.32 | 0.069 | 4.620 | 0.000 | 0.182...0.459 |

Value (HML) | -0.14 | 0.062 | -2.329 | 0.023 | -0.267...-0.020 |

Alpha (α) | 5.95bps | 0.001 | 0.586 | 0.560 | -0.14%...0.26% |

Annualized Alpha (α) | 0.71% |

Ticker | VSS | ||||
---|---|---|---|---|---|

Time Period | Sep 2013 - Feb 2019 | ||||

Coefficient of Determination (R^{2}) |
93.3% | ||||

Adjusted R^{2} |
93.0% | ||||

Regression F statistic | 287.98 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 2.592 with p-value 0.993) | ||||

Heteroskedasticity | Heteroscedasticity confirmed (Breusch-Pagan test value is 12.982 with p-value 0.005) | ||||

Factor | Loading | Standard Error | t-stat | p-value | 95% Confidence Interval |

Market (Rm-Rf) | 1.04 | 0.035 | 29.292 | 0.000 | 0.965...1.106 |

Size (SMB) | 0.47 | 0.080 | 5.902 | 0.000 | 0.313...0.634 |

Value (HML) | -0.12 | 0.071 | -1.689 | 0.096 | -0.263...0.022 |

Alpha (α) | -9.42bps | 0.001 | -0.802 | 0.426 | -0.33%...0.14% |

Annualized Alpha (α) | -1.13% |

Ticker | SCHC | ||||
---|---|---|---|---|---|

Time Period | Sep 2013 - Feb 2019 | ||||

Coefficient of Determination (R^{2}) |
94.8% | ||||

Adjusted R^{2} |
94.5% | ||||

Regression F statistic | 374.08 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 2.978 with p-value 1.000) | ||||

Heteroskedasticity | No heteroscedasticity confirmed (Breusch-Pagan test value is 3.323 with p-value 0.345) | ||||

Factor | Loading | Standard Error | t-stat | p-value | 95% Confidence Interval |

Market (Rm-Rf) | 1.07 | 0.032 | 33.459 | 0.000 | 1.005...1.133 |

Size (SMB) | 0.42 | 0.072 | 5.850 | 0.000 | 0.279...0.569 |

Value (HML) | -0.18 | 0.064 | -2.771 | 0.007 | -0.308...-0.050 |

Alpha (α) | -10.73bps | 0.001 | -1.011 | 0.316 | -0.32%...0.10% |

Annualized Alpha (α) | -1.29% |

Notes on results:

- Time frame for factor analysis is the full available data range unless a specific date interval is specified.
- Results are based on multiple linear regression against monthly factor returns.
- 3-factor model: R
_{a}= R_{rf}+ B_{mkt}× ( R_{mkt}- R_{rf}) + B_{smb}× SMB + B_{hml}× HML + α - Symbols:
- R
_{a} - Asset return
- R
_{rf} - Risk free return
- B
_{mkt} - Market loading factor (exposure to market risk, different from CAPM beta)
- R
_{mkt} - Market return
- B
_{smb} - Size loading factor (the level of exposure to size risk)
- SMB
- Small Minus Big: The size premium
- B
_{hml} - Value loading factor (the level of exposure to value risk)
- HML
- High Minus Low: The value premium
- Alpha
- Excess return over the benchmark
- t-stat
- t-statistic is a ratio of the departure of an estimated parameter from its notional value and its standard error
- p-value
- p-value measures the statistical significance of the estimated parameter
- R
^{2} - Coefficient of determination

- R
- Resources:

Monthly Factor Premiums (BPS) | 43.35 | 12.32 | -5.15 | ||||||
---|---|---|---|---|---|---|---|---|---|

Name | Ticker | Start Date | End Date | Annual Alpha | Rm-Rf | SMB | HML | Total | R^{2} |

Schwab International Equity ETF | SCHF | Sep 2013 | Feb 2019 | -0.40% | 42.75 | -1.80 | 0.53 | 38.14 | 97.3% |

Schwab Emerging Markets Equity ETF | SCHE | Sep 2013 | Feb 2019 | -0.41% | 46.72 | 3.94 | -0.63 | 46.62 | 63.0% |

Schwab Fundamental Intl Sm Co ETF | FNDC | Sep 2013 | Feb 2019 | 0.71% | 43.94 | 3.95 | 0.74 | 54.57 | 94.7% |

Vanguard FTSE All-Wld ex-US SmCp ETF | VSS | Sep 2013 | Feb 2019 | -1.13% | 44.90 | 5.83 | 0.62 | 41.93 | 93.3% |

Schwab International Small-Cap Eq ETF | SCHC | Sep 2013 | Feb 2019 | -1.29% | 46.33 | 5.22 | 0.92 | 41.74 | 94.8% |

Month | SCHF | SCHE | FNDC | VSS | SCHC |
---|---|---|---|---|---|

Sep 2013 | 0.0058 | 0.0050 | 0.0092 | -0.0044 | -0.0076 |

Oct 2013 | -0.0003 | 0.0059 | 0.0017 | 0.0117 | 0.0094 |

Nov 2013 | 0.0008 | -0.0134 | -0.0039 | -0.0088 | -0.0112 |

Dec 2013 | 0.0045 | -0.0223 | 0.0032 | 0.0027 | 0.0178 |

Jan 2014 | -0.0083 | -0.0688 | -0.0128 | -0.0095 | -0.0135 |

Feb 2014 | -0.0054 | -0.0169 | 0.0041 | 0.0264 | 0.0214 |

Mar 2014 | 0.0036 | 0.0495 | -0.0027 | -0.0025 | 0.0053 |

Apr 2014 | 0.0027 | 0.0020 | -0.0002 | 0.0103 | 0.0032 |

May 2014 | 0.0016 | 0.0123 | 0.0063 | 0.0019 | -0.0024 |

Jun 2014 | 0.0028 | 0.0142 | -0.0032 | -0.0011 | 0.0001 |

Jul 2014 | -0.0004 | 0.0345 | -0.0043 | -0.0123 | -0.0136 |

Aug 2014 | 0.0064 | 0.0403 | 0.0017 | 0.0170 | 0.0114 |

Sep 2014 | -0.0046 | -0.0245 | -0.0058 | -0.0116 | -0.0110 |

Oct 2014 | 0.0088 | 0.0493 | 0.0176 | 0.0106 | 0.0070 |

Nov 2014 | -0.0121 | -0.0099 | -0.0218 | -0.0191 | -0.0227 |

Dec 2014 | -0.0061 | -0.0167 | 0.0082 | 0.0014 | 0.0044 |

Jan 2015 | 0.0017 | 0.0065 | 0.0074 | -0.0007 | 0.0021 |

Feb 2015 | 0.0021 | -0.0145 | 0.0089 | 0.0128 | 0.0074 |

Mar 2015 | 0.0003 | -0.0097 | 0.0005 | -0.0041 | -0.0007 |

Apr 2015 | -0.0032 | 0.0122 | -0.0206 | -0.0047 | -0.0109 |

May 2015 | 0.0034 | -0.0353 | 0.0005 | -0.0032 | 0.0031 |

Jun 2015 | 0.0011 | -0.0049 | 0.0049 | -0.0049 | 0.0018 |

Jul 2015 | -0.0064 | -0.0603 | -0.0067 | -0.0260 | -0.0131 |

Aug 2015 | -0.0035 | -0.0274 | 0.0115 | 0.0009 | 0.0129 |

Sep 2015 | 0.0034 | 0.0148 | 0.0002 | 0.0145 | 0.0092 |

Oct 2015 | -0.0042 | -0.0115 | -0.0008 | -0.0091 | -0.0130 |

Nov 2015 | 0.0032 | -0.0121 | 0.0083 | 0.0078 | 0.0039 |

Dec 2015 | -0.0095 | -0.0314 | -0.0046 | -0.0116 | -0.0092 |

Jan 2016 | 0.0102 | 0.0226 | 0.0090 | 0.0051 | 0.0045 |

Feb 2016 | -0.0167 | 0.0037 | -0.0109 | 0.0082 | 0.0002 |

Mar 2016 | 0.0049 | 0.0488 | 0.0085 | -0.0070 | 0.0016 |

Apr 2016 | -0.0044 | -0.0293 | -0.0102 | -0.0012 | 0.0018 |

May 2016 | 0.0042 | -0.0225 | 0.0175 | 0.0022 | 0.0046 |

Jun 2016 | 0.0131 | 0.0835 | 0.0013 | 0.0075 | -0.0067 |

Jul 2016 | -0.0067 | -0.0055 | 0.0067 | 0.0007 | 0.0039 |

Aug 2016 | 0.0074 | 0.0055 | -0.0005 | -0.0012 | -0.0032 |

Sep 2016 | 0.0016 | -0.0017 | 0.0069 | 0.0025 | 0.0009 |

Oct 2016 | -0.0001 | 0.0141 | 0.0089 | -0.0067 | -0.0106 |

Nov 2016 | 0.0092 | -0.0229 | 0.0109 | 0.0065 | 0.0155 |

Dec 2016 | -0.0029 | -0.0311 | -0.0056 | -0.0102 | -0.0071 |

Jan 2017 | 0.0045 | 0.0134 | 0.0001 | 0.0067 | -0.0005 |

Feb 2017 | 0.0024 | 0.0159 | 0.0039 | 0.0052 | -0.0027 |

Mar 2017 | -0.0011 | -0.0024 | -0.0074 | -0.0010 | -0.0028 |

Apr 2017 | -0.0077 | -0.0092 | 0.0023 | 0.0016 | 0.0038 |

May 2017 | -0.0025 | -0.0228 | -0.0131 | -0.0161 | -0.0112 |

Jun 2017 | 0.0063 | -0.0041 | 0.0044 | 0.0032 | 0.0054 |

Jul 2017 | -0.0009 | 0.0187 | 0.0008 | 0.0041 | 0.0011 |

Aug 2017 | -0.0038 | 0.0244 | -0.0005 | 0.0045 | 0.0024 |

Sep 2017 | 0.0042 | -0.0287 | 0.0037 | -0.0001 | 0.0033 |

Oct 2017 | 0.0037 | 0.0074 | 0.0016 | 0.0002 | -0.0044 |

Nov 2017 | -0.0002 | -0.0082 | -0.0000 | 0.0006 | 0.0017 |

Dec 2017 | 0.0002 | 0.0131 | -0.0008 | 0.0043 | 0.0037 |

Jan 2018 | 0.0018 | 0.0416 | -0.0081 | -0.0061 | -0.0030 |

Feb 2018 | -0.0100 | -0.0164 | -0.0049 | -0.0068 | -0.0077 |

Mar 2018 | 0.0058 | 0.0123 | 0.0073 | 0.0150 | 0.0096 |

Apr 2018 | -0.0046 | -0.0456 | -0.0019 | -0.0024 | 0.0030 |

May 2018 | -0.0029 | -0.0105 | -0.0096 | 0.0019 | 0.0004 |

Jun 2018 | -0.0057 | -0.0229 | -0.0056 | -0.0083 | 0.0004 |

Jul 2018 | 0.0041 | 0.0275 | -0.0034 | 0.0037 | -0.0031 |

Aug 2018 | -0.0027 | -0.0179 | -0.0070 | -0.0042 | 0.0014 |

Sep 2018 | 0.0012 | -0.0142 | -0.0024 | -0.0188 | -0.0055 |

Oct 2018 | 0.0005 | 0.0204 | -0.0034 | -0.0038 | -0.0012 |

Nov 2018 | 0.0036 | 0.0499 | 0.0003 | 0.0099 | -0.0065 |

Dec 2018 | -0.0052 | 0.0274 | -0.0112 | 0.0104 | 0.0029 |

Jan 2019 | 0.0040 | 0.0283 | 0.0128 | 0.0084 | 0.0163 |

Feb 2019 | -0.0032 | -0.0297 | -0.0077 | -0.0031 | -0.0037 |

Rolling regressions are only shown for the first three ticker symbols entered.

Rolling returns are only shown for the first three ticker symbols entered.