Factor Regression Analysis

Factor Regression Overview

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.

Regression Model

Portfolio Assets
Allocation
Asset 1
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Asset 2
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Asset 3
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Asset 4
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Asset 5
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Asset 6
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Asset 7
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Asset 8
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Asset 9
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Asset 10
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Factor Analysis Results

Factor Regression Summary

Name Ticker Start Date End Date Rm-Rf SMB HML Alpha Annual Alpha R2
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

Vanguard Total International Stock ETF

Ticker VXUS
Time Period Oct 2021 - Nov 2022
Regression Basis 14 monthly samples
Coefficient of Determination (R2) 86.0%
Adjusted R2 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%  

iShares MSCI ACWI ex US ETF

Ticker ACWX
Time Period Oct 2021 - Nov 2022
Regression Basis 14 monthly samples
Coefficient of Determination (R2) 84.5%
Adjusted R2 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%  

Dimensional World ex US Core Eq 2 ETF

Ticker DFAX
Time Period Oct 2021 - Nov 2022
Regression Basis 14 monthly samples
Coefficient of Determination (R2) 87.3%
Adjusted R2 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: Ra = Rrf + Bmkt × ( Rmkt - Rrf ) + Bsmb × SMB + Bhml × HML + α
  • Symbols:
    Ra
    Asset return
    Rrf
    Risk free return
    Bmkt
    Market loading factor (exposure to market risk, different from CAPM beta)
    Rmkt
    Market return
    Bsmb
    Size loading factor (the level of exposure to size risk)
    SMB
    Small Minus Big: The size premium
    Bhml
    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
    R2
    Coefficient of determination

Factor Returns

Monthly Factor Premiums (BPS) -57.71 -115.57 190.50  
Name Ticker Start Date End Date Annual Alpha Rm-Rf SMB HML Total R2
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%

Risk and Return Attribution

NameStart DateEnd DateCumulative ReturnAnnualized ReturnAnnualized Standard DeviationCumulative ReturnReturn ContributionRisk Contribution
MarketSMBHMLAlphaMarketSMBHMLAlphaMarketSMBHMLAlpha
Vanguard Total International Stock ETFOct 2021Nov 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 ETFOct 2021Nov 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 ETFOct 2021Nov 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%

Vanguard Total International Stock ETF Cumulative Return

iShares MSCI ACWI ex US ETF Cumulative Return

Dimensional World ex US Core Eq 2 ETF Cumulative Return

Residuals

MonthVXUSACWXDFAX
Oct 2021-0.01%-0.14%-0.65%
Nov 20210.03%-0.06%0.96%
Dec 20210.03%-0.24%0.34%
Jan 20220.88%0.86%0.97%
Feb 2022-1.57%-1.75%-1.35%
Mar 2022-0.98%-0.99%-0.87%
Apr 20220.50%0.36%0.47%
May 20220.34%0.64%-0.17%
Jun 20221.22%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 20226.46%6.91%6.22%

Vanguard Total International Stock ETF

iShares MSCI ACWI ex US ETF

Dimensional World ex US Core Eq 2 ETF