Factor Regression Analysis

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:

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.

Portfolio Assets
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Factor Analysis Results

Factor Analysis Summary

Factor regression results
Name Ticker Start Date End Date Rm-Rf SMB HML Alpha Annual Alpha R2
Schwab International Equity ETF SCHF Sep 2013 Feb 2019 0.99 -0.15 -0.10 -0.03% -0.42% 97.3%
Schwab Emerging Markets Equity ETF SCHE Sep 2013 Feb 2019 1.08 0.33 0.11 -0.03% -0.42% 63.0%
Schwab Fundamental Intl Sm Co ETF FNDC Sep 2013 Feb 2019 1.01 0.32 -0.14 0.06% 0.73% 94.7%
Vanguard FTSE All-Wld ex-US SmCp ETF VSS Sep 2013 Feb 2019 1.04 0.47 -0.13 -0.09% -1.11% 93.3%
Schwab International Small-Cap Eq ETF SCHC Sep 2013 Feb 2019 1.07 0.42 -0.18 -0.11% -1.26% 94.8%

Schwab International Equity ETF

Factor regression results for Schwab International Equity ETF
Ticker SCHF
Time Period Sep 2013 - Feb 2019
Coefficient of Determination (R2) 97.3%
Adjusted R2 97.2%
Regression F statistic 751.27 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 2.779 with p-value 1.000)
Heteroscedasticity No heteroscedasticity confirmed (Breusch-Pagan test value is 4.076 with p-value 0.253)
Factor Loading Standard Error t-stat p-value 95% Confidence Interval
Market (Rm-Rf) 0.99 0.021 46.178 0.000 0.944...1.029
Size (SMB) -0.15 0.048 -3.006 0.004 -0.242...-0.049
Value (HML) -0.10 0.043 -2.383 0.020 -0.189...-0.017
Alpha (α) -3.48bps 0.001 -0.490 0.626 -0.18%...0.11%
Annualized Alpha (α) -0.42%  

Schwab Emerging Markets Equity ETF

Factor regression results for Schwab Emerging Markets Equity ETF
Ticker SCHE
Time Period Sep 2013 - Feb 2019
Coefficient of Determination (R2) 63.0%
Adjusted R2 61.2%
Regression F statistic 35.20 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 1.965 with p-value 0.443)
Heteroscedasticity No heteroscedasticity confirmed (Breusch-Pagan test value is 2.107 with p-value 0.551)
Factor Loading Standard Error t-stat p-value 95% Confidence Interval
Market (Rm-Rf) 1.08 0.107 10.093 0.000 0.864...1.291
Size (SMB) 0.33 0.242 1.352 0.181 -0.156...0.811
Value (HML) 0.11 0.215 0.525 0.601 -0.317...0.544
Alpha (α) -3.47bps 0.004 -0.098 0.922 -0.74%...0.67%
Annualized Alpha (α) -0.42%  

Schwab Fundamental Intl Sm Co ETF

Factor regression results for Schwab Fundamental Intl Sm Co ETF
Ticker FNDC
Time Period Sep 2013 - Feb 2019
Coefficient of Determination (R2) 94.7%
Adjusted R2 94.4%
Regression F statistic 367.22 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 2.680 with p-value 0.998)
Heteroscedasticity No heteroscedasticity confirmed (Breusch-Pagan test value is 1.671 with p-value 0.643)
Factor Loading Standard Error t-stat p-value 95% Confidence Interval
Market (Rm-Rf) 1.01 0.031 33.118 0.000 0.952...1.075
Size (SMB) 0.32 0.069 4.607 0.000 0.181...0.458
Value (HML) -0.14 0.062 -2.332 0.023 -0.267...-0.021
Alpha (α) 6.10bps 0.001 0.600 0.551 -0.14%...0.26%
Annualized Alpha (α) 0.73%  

Vanguard FTSE All-Wld ex-US SmCp ETF

Factor regression results for Vanguard FTSE All-Wld ex-US SmCp ETF
Ticker VSS
Time Period Sep 2013 - Feb 2019
Coefficient of Determination (R2) 93.3%
Adjusted R2 93.0%
Regression F statistic 289.78 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 2.600 with p-value 0.994)
Heteroscedasticity Heteroscedasticity confirmed (Breusch-Pagan test value is 13.034 with p-value 0.005)
Factor Loading Standard Error t-stat p-value 95% Confidence Interval
Market (Rm-Rf) 1.04 0.035 29.369 0.000 0.965...1.106
Size (SMB) 0.47 0.080 5.920 0.000 0.313...0.633
Value (HML) -0.13 0.071 -1.790 0.078 -0.270...0.015
Alpha (α) -9.24bps 0.001 -0.789 0.433 -0.33%...0.14%
Annualized Alpha (α) -1.11%  

Schwab International Small-Cap Eq ETF

Factor regression results for Schwab International Small-Cap Eq ETF
Ticker SCHC
Time Period Sep 2013 - Feb 2019
Coefficient of Determination (R2) 94.8%
Adjusted R2 94.5%
Regression F statistic 374.58 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 2.985 with p-value 1.000)
Heteroscedasticity No heteroscedasticity confirmed (Breusch-Pagan test value is 3.241 with p-value 0.356)
Factor Loading Standard Error t-stat p-value 95% Confidence Interval
Market (Rm-Rf) 1.07 0.032 33.472 0.000 1.005...1.133
Size (SMB) 0.42 0.072 5.820 0.000 0.276...0.566
Value (HML) -0.18 0.064 -2.865 0.006 -0.313...-0.056
Alpha (α) -10.54bps 0.001 -0.994 0.324 -0.32%...0.11%
Annualized Alpha (α) -1.26%  
Notes on results:
  • Past performance is no guarantee of future results, which may vary. All use is subject to terms of service.
  • Investing involves risk, including possible loss of principal. The value of the investments and the income derived from them may fluctuate over time.
  • 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.
  • 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.
  • The results are based on the total return of assets and assume that all received dividends and distributions are reinvested.
  • Refer to the FAQ section 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 Performance Attribution in Basis Points

Factor regression results
Monthly Factor Premiums (BPS) 43.41 11.82 -4.92  
Name Ticker Start Date End Date Annual Alpha Rm-Rf SMB HML Total R2
Schwab International Equity ETF SCHF Sep 2013 Feb 2019 -0.42% 42.83 -1.72 0.51 38.14 97.3%
Schwab Emerging Markets Equity ETF SCHE Sep 2013 Feb 2019 -0.42% 46.79 3.87 -0.56 46.62 63.0%
Schwab Fundamental Intl Sm Co ETF FNDC Sep 2013 Feb 2019 0.73% 43.99 3.77 0.71 54.57 94.7%
Vanguard FTSE All-Wld ex-US SmCp ETF VSS Sep 2013 Feb 2019 -1.11% 44.96 5.59 0.63 41.93 93.3%
Schwab International Small-Cap Eq ETF SCHC Sep 2013 Feb 2019 -1.26% 46.40 4.98 0.91 41.74 94.8%
Regression residuals
MonthSCHFSCHEFNDCVSSSCHC
Sep 20130.00590.00470.0093-0.0043-0.0075
Oct 2013-0.00040.00620.00160.01180.0093
Nov 20130.0007-0.0133-0.0039-0.0087-0.0112
Dec 20130.0045-0.02220.00340.00300.0181
Jan 2014-0.0083-0.0689-0.0128-0.0095-0.0134
Feb 2014-0.0055-0.01670.00410.02640.0212
Mar 20140.00360.0496-0.0029-0.00270.0051
Apr 20140.00280.0022-0.00020.01030.0032
May 20140.00160.01230.00640.0019-0.0023
Jun 20140.00280.0140-0.0032-0.00120.0001
Jul 2014-0.00030.0345-0.0042-0.0122-0.0134
Aug 20140.00640.04040.00200.01730.0117
Sep 2014-0.0045-0.0244-0.0059-0.0117-0.0111
Oct 20140.00880.04940.01770.01060.0070
Nov 2014-0.0121-0.0097-0.0216-0.0190-0.0227
Dec 2014-0.0061-0.01680.00840.00160.0046
Jan 20150.00160.00670.0072-0.00090.0018
Feb 20150.0022-0.01450.00900.01290.0075
Mar 20150.0001-0.00970.0006-0.0039-0.0006
Apr 2015-0.00320.0120-0.0205-0.0046-0.0107
May 20150.0034-0.03550.0005-0.00320.0030
Jun 20150.0011-0.00500.0049-0.00480.0019
Jul 2015-0.0064-0.0603-0.0064-0.0258-0.0129
Aug 2015-0.0033-0.02760.01180.00110.0132
Sep 20150.00340.01460.00040.01470.0094
Oct 2015-0.0040-0.0114-0.0006-0.0088-0.0128
Nov 20150.0031-0.01230.00840.00780.0040
Dec 2015-0.0095-0.0317-0.0046-0.0116-0.0092
Jan 20160.01020.02250.00880.00480.0042
Feb 2016-0.01680.0037-0.01090.00820.0002
Mar 20160.00480.04870.0083-0.00720.0014
Apr 2016-0.0044-0.0293-0.0106-0.00140.0015
May 20160.0042-0.02270.01750.00210.0046
Jun 20160.01310.08320.00110.0071-0.0070
Jul 2016-0.0066-0.00540.00700.00100.0043
Aug 20160.00750.0057-0.0006-0.0011-0.0031
Sep 20160.0016-0.00190.00680.00230.0008
Oct 20160.00010.01440.0089-0.0064-0.0103
Nov 20160.0093-0.02260.01090.00660.0156
Dec 2016-0.0028-0.0309-0.0055-0.0100-0.0069
Jan 20170.00410.0133-0.00030.0064-0.0009
Feb 20170.00230.01580.00370.0049-0.0031
Mar 2017-0.0011-0.0025-0.0074-0.0012-0.0029
Apr 2017-0.0077-0.00920.00240.00160.0038
May 2017-0.0025-0.0229-0.0131-0.0161-0.0112
Jun 20170.0063-0.00410.00430.00320.0053
Jul 2017-0.00090.01900.00080.00430.0012
Aug 2017-0.00380.0242-0.00060.00430.0022
Sep 20170.0042-0.02870.0037-0.00010.0034
Oct 20170.00370.00740.00160.0001-0.0044
Nov 2017-0.0002-0.0081-0.00010.00060.0017
Dec 20170.00010.0130-0.00090.00420.0036
Jan 20180.00170.0417-0.0080-0.0060-0.0028
Feb 2018-0.0100-0.0166-0.0048-0.0066-0.0075
Mar 20180.00590.01230.00740.01500.0096
Apr 2018-0.0046-0.0453-0.0020-0.00240.0030
May 2018-0.0030-0.0108-0.00940.00190.0004
Jun 2018-0.0057-0.0228-0.0055-0.00830.0004
Jul 20180.00410.0278-0.00350.0037-0.0031
Aug 2018-0.0029-0.0181-0.0072-0.00460.0010
Sep 20180.0012-0.0139-0.0024-0.0187-0.0054
Oct 20180.00060.0207-0.0034-0.0037-0.0012
Nov 20180.00380.04960.00030.0098-0.0065
Dec 2018-0.00520.0278-0.01130.01040.0029
Jan 20190.00410.02810.01290.00840.0163
Feb 2019-0.0032-0.0298-0.0079-0.0035-0.0041