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)
- The q-factor model with market, size, investment, return on equity, and expected growth factors (MKT, ME, I/A, ROE, EG)

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.

Additionally you can freely construct the factor model using any combination of existing factors or other time series under the custom factor returns model.

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

Vanguard Total International Stock ETF | VXUS | Oct 2021 | Nov 2022 | 0.96 | 0.31 | 0.28 | -0.49% | -5.93% | 86.0% |

iShares MSCI ACWI ex US ETF | ACWX | Oct 2021 | Nov 2022 | 0.95 | 0.20 | 0.28 | -0.60% | -7.16% | 84.5% |

Dimensional World ex US Core Eq 2 ETF | DFAX | Oct 2021 | Nov 2022 | 0.98 | 0.48 | 0.40 | -0.41% | -4.90% | 87.3% |

Statistically significant factor exposures in bold |

Ticker | VXUS | ||||
---|---|---|---|---|---|

Time Period | Oct 2021 - Nov 2022 | ||||

Regression Basis | 14 monthly samples | ||||

Coefficient of Determination (R^{2}) |
86.0% | ||||

Adjusted R^{2} |
81.8% | ||||

Regression F statistic | 20.52 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 1.971 with p-value 0.505) | ||||

Heteroscedasticity | No heteroscedasticity confirmed (Breusch-Pagan test value is 3.928 with p-value 0.269) | ||||

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

Market (Rm-Rf) | 0.96 | 0.124 | 7.682 | 0.000 | 0.679...1.234 |

Size (SMB) | 0.31 | 0.490 | 0.634 | 0.540 | -0.781...1.402 |

Value (HML) | 0.28 | 0.186 | 1.529 | 0.157 | -0.130...0.700 |

Alpha (α) | -49.44bps | 0.009 | -0.569 | 0.582 | -2.43%...1.44% |

Annualized Alpha (α) | -5.93% |

Ticker | ACWX | ||||
---|---|---|---|---|---|

Time Period | Oct 2021 - Nov 2022 | ||||

Regression Basis | 14 monthly samples | ||||

Coefficient of Determination (R^{2}) |
84.5% | ||||

Adjusted R^{2} |
79.9% | ||||

Regression F statistic | 18.24 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 1.930 with p-value 0.471) | ||||

Heteroscedasticity | No heteroscedasticity confirmed (Breusch-Pagan test value is 3.841 with p-value 0.279) | ||||

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

Market (Rm-Rf) | 0.95 | 0.132 | 7.221 | 0.000 | 0.658...1.246 |

Size (SMB) | 0.20 | 0.519 | 0.380 | 0.712 | -0.959...1.353 |

Value (HML) | 0.28 | 0.197 | 1.406 | 0.190 | -0.162...0.717 |

Alpha (α) | -59.67bps | 0.009 | -0.649 | 0.531 | -2.65%...1.45% |

Annualized Alpha (α) | -7.16% |

Ticker | DFAX | ||||
---|---|---|---|---|---|

Time Period | Oct 2021 - Nov 2022 | ||||

Regression Basis | 14 monthly samples | ||||

Coefficient of Determination (R^{2}) |
87.3% | ||||

Adjusted R^{2} |
83.4% | ||||

Regression F statistic | 22.84 (p-value = 0.000) | ||||

Autocorrelation | No autocorrelation confirmed (Durbin-Watson test value is 1.935 with p-value 0.475) | ||||

Heteroscedasticity | No heteroscedasticity confirmed (Breusch-Pagan test value is 3.841 with p-value 0.279) | ||||

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

Market (Rm-Rf) | 0.98 | 0.119 | 8.214 | 0.000 | 0.714...1.245 |

Size (SMB) | 0.48 | 0.469 | 1.021 | 0.331 | -0.567...1.525 |

Value (HML) | 0.40 | 0.178 | 2.262 | 0.047 | 0.006...0.801 |

Alpha (α) | -40.84bps | 0.008 | -0.491 | 0.634 | -2.26%...1.44% |

Annualized Alpha (α) | -4.90% |

Notes and Disclosures

- IMPORTANT: The projections or other information generated by Portfolio Visualizer regarding the likelihood of various investment outcomes are hypothetical in nature, do not reflect actual investment results and are not guarantees of future results. Results may vary with each use and over time.
- The results do not constitute investment advice or recommendation, are provided solely for informational purposes, and are not an offer to buy or sell any securities. All use is subject to terms of service.
- Investing involves risk, including possible loss of principal. Past performance is not a guarantee of future results.
- Asset allocation and diversification strategies do not guarantee a profit or protect against a loss.
- Hypothetical returns do not reflect trading costs, transaction fees, commissions, or actual taxes due on investment returns.
- The results are based on information from a variety of sources we consider reliable, but we do not represent that the information is accurate or complete.
- Refer to the related documentation sections for more details on terms and definitions, methodology, and data sources.
- The results are based on the total return of assets and assume that all received dividends and distributions are reinvested.
- Refer to the documentation regarding the list of factor data sources and methodology descriptions.
- 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

Monthly Factor Premiums (BPS) | -57.71 | -115.57 | 190.50 | ||||||
---|---|---|---|---|---|---|---|---|---|

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

Vanguard Total International Stock ETF | VXUS | Oct 2021 | Nov 2022 | -5.93% | -55.20 | -35.91 | 54.28 | -86.27 | 86.0% |

iShares MSCI ACWI ex US ETF | ACWX | Oct 2021 | Nov 2022 | -7.16% | -54.94 | -22.81 | 52.87 | -84.55 | 84.5% |

Dimensional World ex US Core Eq 2 ETF | DFAX | Oct 2021 | Nov 2022 | -4.90% | -56.53 | -55.37 | 76.90 | -75.84 | 87.3% |

Name | Start Date | End Date | Cumulative Return | Annualized Return | Annualized Standard Deviation | Cumulative Return | Return Contribution | Risk Contribution | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Market | SMB | HML | Alpha | Market | SMB | HML | Alpha | Market | SMB | HML | Alpha | ||||||

Vanguard Total International Stock ETF | Oct 2021 | Nov 2022 | -12.52% | -10.83% | 20.78% | -8.27% | -4.75% | 7.24% | -6.74% | -30.63% | -17.59% | 26.82% | -24.96% | 89.74% | -0.41% | -3.17% | 13.84% |

iShares MSCI ACWI ex US ETF | Oct 2021 | Nov 2022 | -12.33% | -10.67% | 20.93% | -8.24% | -3.03% | 7.06% | -8.12% | -31.14% | -11.47% | 26.70% | -30.70% | 88.01% | -0.40% | -2.92% | 15.30% |

Dimensional World ex US Core Eq 2 ETF | Oct 2021 | Nov 2022 | -11.24% | -9.72% | 20.85% | -8.51% | -7.35% | 10.27% | -5.65% | -26.77% | -23.13% | 32.31% | -17.78% | 90.46% | -0.46% | -2.60% | 12.61% |

Month | VXUS | ACWX | DFAX |
---|---|---|---|

Oct 2021 | -0.01% | -0.14% | -0.65% |

Nov 2021 | 0.03% | -0.06% | 0.96% |

Dec 2021 | 0.03% | -0.24% | 0.34% |

Jan 2022 | 0.88% | 0.86% | 0.97% |

Feb 2022 | -1.57% | -1.75% | -1.35% |

Mar 2022 | -0.98% | -0.99% | -0.87% |

Apr 2022 | 0.50% | 0.36% | 0.47% |

May 2022 | 0.34% | 0.64% | -0.17% |

Jun 2022 | 1.22% | 1.45% | 0.70% |

Jul 2022 | -1.92% | -2.05% | -1.86% |

Aug 2022 | -0.99% | -1.19% | -0.87% |

Sep 2022 | -0.57% | -0.40% | -0.67% |

Oct 2022 | -3.45% | -3.40% | -3.21% |

Nov 2022 | 6.46% | 6.91% | 6.22% |

The selected date range contains insufficient data for rolling regression analysis