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.

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Factor Analysis Results

Factor Analysis Summary

Factor regression results
Name Ticker Start Date End Date FF-MKT-RF FF-SMB FF-HML FF-MOM FF-RMW Alpha Annual Alpha R2
Vanguard Strategic Small-Cap Equity Inv VSTCX May 2006 Nov 2020 1.07 0.72 0.24 0.05 0.14 -0.19% -2.32% 97.2%
Vanguard Small-Cap ETF VB May 2006 Nov 2020 1.04 0.64 0.13 -0.03 -0.00 -0.03% -0.31% 98.0%
Vanguard Small-Cap Value ETF VBR May 2006 Nov 2020 1.01 0.59 0.40 -0.06 0.11 -0.04% -0.49% 97.9%

Vanguard Strategic Small-Cap Equity Inv

Factor regression results for Vanguard Strategic Small-Cap Equity Inv
Ticker VSTCX
Time Period May 2006 - Nov 2020
Regression Basis 175 monthly samples
Coefficient of Determination (R2) 97.2%
Adjusted R2 97.1%
Regression F statistic 1,185.89 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 2.087 with p-value 0.704)
Heteroscedasticity No heteroscedasticity confirmed (Breusch-Pagan test value is 7.959 with p-value 0.159)
Factor Description Loading Standard Error t-stat p-value 95% Confidence Interval
FF-MKT-RF Market 1.07 0.019 55.387 0.000 1.033...1.109
FF-SMB Size 0.72 0.036 19.893 0.000 0.653...0.797
FF-HML Value 0.24 0.030 8.054 0.000 0.184...0.304
FF-MOM Momentum 0.05 0.019 2.570 0.011 0.011...0.087
FF-RMW Profitability 0.14 0.052 2.776 0.006 0.041...0.245
Alpha Alpha -19.35bps 0.001 -2.423 0.016 -0.35%...-0.04%
Annualized Alpha Annualized Alpha -2.32%  

Vanguard Small-Cap ETF

Factor regression results for Vanguard Small-Cap ETF
Ticker VB
Time Period May 2006 - Nov 2020
Regression Basis 175 monthly samples
Coefficient of Determination (R2) 98.0%
Adjusted R2 97.9%
Regression F statistic 1,632.57 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 2.215 with p-value 0.917)
Heteroscedasticity Heteroscedasticity confirmed (Breusch-Pagan test value is 21.859 with p-value 0.001)
Factor Description Loading Standard Error t-stat p-value 95% Confidence Interval
FF-MKT-RF Market 1.04 0.016 64.736 0.000 1.010...1.074
FF-SMB Size 0.64 0.030 21.019 0.000 0.577...0.697
FF-HML Value 0.13 0.025 5.331 0.000 0.085...0.184
FF-MOM Momentum -0.03 0.016 -1.916 0.057 -0.062...0.001
FF-RMW Profitability -0.00 0.043 -0.092 0.927 -0.089...0.081
Alpha Alpha -2.59bps 0.001 -0.390 0.697 -0.16%...0.11%
Annualized Alpha Annualized Alpha -0.31%  

Vanguard Small-Cap Value ETF

Factor regression results for Vanguard Small-Cap Value ETF
Ticker VBR
Time Period May 2006 - Nov 2020
Regression Basis 175 monthly samples
Coefficient of Determination (R2) 97.9%
Adjusted R2 97.9%
Regression F statistic 1,603.83 (p-value = 0.000)
Autocorrelation No autocorrelation confirmed (Durbin-Watson test value is 2.265 with p-value 0.957)
Heteroscedasticity Heteroscedasticity confirmed (Breusch-Pagan test value is 12.691 with p-value 0.026)
Factor Description Loading Standard Error t-stat p-value 95% Confidence Interval
FF-MKT-RF Market 1.01 0.017 60.987 0.000 0.975...1.040
FF-SMB Size 0.59 0.031 18.880 0.000 0.526...0.649
FF-HML Value 0.40 0.026 15.295 0.000 0.345...0.448
FF-MOM Momentum -0.06 0.016 -3.382 0.001 -0.088...-0.023
FF-RMW Profitability 0.11 0.044 2.434 0.016 0.020...0.194
Alpha Alpha -4.05bps 0.001 -0.594 0.554 -0.18%...0.09%
Annualized Alpha Annualized Alpha -0.49%  
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.
  • Symbols:
    MKT-RF
    Market factor (excess return over the risk free rate)
    SMB
    Size factor (Small Minus Big)
    HML
    Value factor (High Minus Low)
    MOM
    Momentum factor (Up Minus Down)
    RMW
    Profitability factor (Robust Minus Weak)
    CMA
    Conservative investment factor (Conservative Minus Aggressive)
    ST-REV
    Short-term reversal factor
    LT-REV
    Long-term reversal factor
    QMJ
    Quality factor (Quality Minus Junk)
    BAB
    Bet Against Beta factor
    TRM
    Term factor (fixed income)
    CDT
    Credit factor (fixed income)
    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) 80.45 3.21 -38.76 3.35 25.61  
Name Ticker Start Date End Date Annual Alpha FF-MKT-RF FF-SMB FF-HML FF-MOM FF-RMW Total R2
Vanguard Strategic Small-Cap Equity Inv VSTCX May 2006 Nov 2020 -2.32% 86.18 2.33 -9.47 0.17 3.67 63.51 97.2%
Vanguard Small-Cap ETF VB May 2006 Nov 2020 -0.31% 83.82 2.05 -5.22 -0.10 -0.10 77.85 98.0%
Vanguard Small-Cap Value ETF VBR May 2006 Nov 2020 -0.49% 81.06 1.89 -15.37 -0.19 2.75 66.09 97.9%
Regression residuals
MonthVSTCXVBVBR
May 20060.0016-0.0022-0.0002
Jun 20060.01560.00170.0077
Jul 2006-0.0061-0.00510.0022
Aug 2006-0.0107-0.0072-0.0013
Sep 2006-0.0040-0.0067-0.0075
Oct 20060.01020.00320.0013
Nov 20060.00410.00600.0063
Dec 2006-0.0108-0.0097-0.0111
Jan 20070.01170.00610.0028
Feb 20070.00240.00760.0031
Mar 20070.00010.0015-0.0010
Apr 20070.00570.0017-0.0048
May 20070.00610.00620.0012
Jun 2007-0.0064-0.0003-0.0072
Jul 2007-0.0039-0.0057-0.0129
Aug 2007-0.00040.00550.0139
Sep 20070.00280.0055-0.0013
Oct 2007-0.01030.00730.0078
Nov 2007-0.00410.0002-0.0037
Dec 2007-0.00240.0010-0.0023
Jan 20080.0125-0.00130.0070
Feb 20080.01330.01240.0111
Mar 2008-0.0097-0.00410.0005
Apr 20080.00780.01880.0100
May 20080.00580.01080.0094
Jun 2008-0.00720.0024-0.0243
Jul 2008-0.0029-0.00320.0097
Aug 2008-0.0314-0.0106-0.0114
Sep 2008-0.00600.00550.0187
Oct 2008-0.0009-0.0180-0.0213
Nov 2008-0.0049-0.00350.0237
Dec 20080.01500.00930.0034
Jan 20090.00680.0004-0.0015
Feb 2009-0.0120-0.0055-0.0115
Mar 2009-0.0216-0.0161-0.0184
Apr 20090.01520.03080.0232
May 20090.00010.0144-0.0058
Jun 2009-0.0082-0.0187-0.0054
Jul 2009-0.0107-0.0051-0.0043
Aug 2009-0.00940.00230.0032
Sep 2009-0.00520.0026-0.0088
Oct 20090.0042-0.00330.0034
Nov 2009-0.0034-0.0085-0.0053
Dec 20090.00610.01260.0070
Jan 20100.0054-0.0028-0.0007
Feb 2010-0.00500.0022-0.0009
Mar 2010-0.00230.00800.0045
Apr 2010-0.01400.00130.0031
May 20100.01780.00710.0075
Jun 20100.0093-0.00240.0022
Jul 20100.0035-0.00020.0061
Aug 20100.01320.00530.0065
Sep 2010-0.00270.0005-0.0006
Oct 20100.0036-0.00210.0007
Nov 20100.00060.0039-0.0000
Dec 2010-0.0143-0.0031-0.0082
Jan 20110.01280.0014-0.0014
Feb 20110.00870.00800.0010
Mar 20110.01240.00590.0023
Apr 20110.01270.00520.0056
May 20110.00240.00040.0014
Jun 20110.0072-0.0003-0.0017
Jul 2011-0.0017-0.0014-0.0005
Aug 20110.00640.00200.0026
Sep 2011-0.0025-0.0075-0.0070
Oct 20110.03080.01270.0119
Nov 2011-0.00930.00140.0041
Dec 2011-0.0086-0.0028-0.0020
Jan 20120.01120.00270.0088
Feb 20120.0006-0.0019-0.0089
Mar 2012-0.0122-0.0055-0.0059
Apr 2012-0.00270.00360.0051
May 2012-0.00900.00000.0031
Jun 2012-0.0054-0.0001-0.0002
Jul 20120.0142-0.00020.0046
Aug 20120.01240.0041-0.0026
Sep 2012-0.0147-0.0059-0.0073
Oct 20120.00940.00640.0048
Nov 20120.01100.00250.0029
Dec 2012-0.00010.0031-0.0004
Jan 20130.00440.00160.0039
Feb 20130.00770.00230.0071
Mar 20130.00360.00080.0022
Apr 20130.00040.0018-0.0023
May 2013-0.0069-0.0074-0.0155
Jun 2013-0.0019-0.0049-0.0062
Jul 20130.0104-0.00290.0011
Aug 2013-0.0064-0.0022-0.0058
Sep 20130.00530.00350.0054
Oct 2013-0.0005-0.0032-0.0011
Nov 20130.0001-0.0130-0.0099
Dec 2013-0.00340.00010.0037
Jan 20140.01610.01180.0141
Feb 20140.00490.00170.0042
Mar 20140.0031-0.0018-0.0036
Apr 20140.01040.00210.0054
May 20140.00250.00410.0084
Jun 20140.00140.00420.0044
Jul 20140.0107-0.0016-0.0029
Aug 20140.00740.00530.0100
Sep 2014-0.0031-0.0066-0.0088
Oct 20140.0034-0.00570.0061
Nov 20140.00060.00180.0084
Dec 2014-0.0029-0.0052-0.0027
Jan 20150.02560.02070.0162
Feb 2015-0.0141-0.0075-0.0008
Mar 20150.01180.00940.0102
Apr 2015-0.0071-0.0118-0.0122
May 2015-0.00050.00370.0050
Jun 2015-0.0069-0.0082-0.0117
Jul 20150.02260.01620.0217
Aug 2015-0.0014-0.0022-0.0022
Sep 20150.01510.00460.0098
Oct 2015-0.0101-0.0119-0.0020
Nov 2015-0.0093-0.0097-0.0075
Dec 20150.00340.00340.0013
Jan 20160.00760.00390.0043
Feb 20160.00290.00290.0075
Mar 2016-0.00370.00400.0067
Apr 2016-0.0117-0.0025-0.0054
May 20160.00350.00510.0072
Jun 2016-0.00970.00310.0049
Jul 20160.0037-0.0070-0.0066
Aug 2016-0.0196-0.0112-0.0171
Sep 20160.0005-0.0102-0.0045
Oct 20160.00380.00350.0007
Nov 20160.0133-0.0118-0.0131
Dec 20160.0046-0.0052-0.0054
Jan 20170.00030.00500.0041
Feb 2017-0.00020.00050.0017
Mar 2017-0.0067-0.0070-0.0041
Apr 2017-0.0011-0.0046-0.0050
May 2017-0.0137-0.0005-0.0053
Jun 20170.0031-0.00260.0003
Jul 20170.00620.00220.0027
Aug 2017-0.01000.00270.0037
Sep 2017-0.0040-0.0158-0.0140
Oct 2017-0.00510.0052-0.0020
Nov 2017-0.01130.0012-0.0004
Dec 2017-0.00130.0003-0.0012
Jan 2018-0.0003-0.0095-0.0114
Feb 2018-0.0022-0.0017-0.0074
Mar 20180.00440.00850.0076
Apr 2018-0.0051-0.0083-0.0058
May 20180.0064-0.00610.0032
Jun 20180.0018-0.0045-0.0024
Jul 2018-0.0029-0.0035-0.0003
Aug 20180.01520.00790.0003
Sep 2018-0.00230.00000.0008
Oct 2018-0.00380.0030-0.0021
Nov 2018-0.01220.00640.0083
Dec 20180.00420.00610.0041
Jan 20190.00980.00700.0060
Feb 2019-0.00620.00210.0031
Mar 2019-0.00640.00410.0031
Apr 2019-0.00490.0006-0.0032
May 2019-0.00750.01190.0076
Jun 2019-0.0097-0.0049-0.0030
Jul 20190.01210.01150.0066
Aug 20190.01230.01030.0080
Sep 2019-0.0009-0.0076-0.0030
Oct 2019-0.0015-0.00500.0022
Nov 20190.0072-0.0030-0.0123
Dec 2019-0.0095-0.0162-0.0143
Jan 20200.00070.00970.0141
Feb 20200.0013-0.0027-0.0092
Mar 2020-0.0380-0.0272-0.0247
Apr 2020-0.0076-0.0137-0.0226
May 2020-0.00330.0097-0.0032
Jun 2020-0.0108-0.0140-0.0115
Jul 20200.00370.0040-0.0033
Aug 2020-0.0337-0.0338-0.0214
Sep 20200.00300.01600.0125
Oct 2020-0.01360.00910.0103
Nov 2020-0.0029-0.01360.0043