## 3-Factor Regression Analysis

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Sammy_M
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### 3-Factor Regression Analysis

One of my long standing goals is to learn how to do regression analysis to derive small and value factors for funds rather than have to rely on others' work. I made my first real attempt today. I clearly need some help.

The following are the steps I took. It's pretty much as per W. Berstein's Roll Your Own article, but I don't have Morningstar Principia Pro so used Yahoo Finance.

1. Went to Yahoo Finance and downloaded monthly pricing data into Excel.
2. Calculated the monthly returns based on adjusted close values (adjusted for dividends and splits).
4. Ran regression function in Excel. Data Analysis > Regression > Input Y as fund's return series. Input X as the Rm-Rf, SmB, and HmL series. Checked yes to Labels. Output to New Worksheet Ply.

I looked at Vanguard TSM, DFA Core 2, iShares S&P600 Value (IJS), Bridgeway Ultra Small (BRSIX) and Am. Funds Wash Mutual (AWSHX). I used all of the data available within Yahoo finance for all funds and matched it up against French's data.

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``  Intercept Rm- Rf   SmB   HmLTSM   0.22   1.00   0.00   0.02Core2 0.08   1.04   0.23   0.09IJS   0.24   0.86   0.69   0.26BRSIX 0.44   0.87   0.69   0.16AWSHX 0.42   0.81   -0.26   0.28``

Looks like I've got a lot of stuff going wrong:

1. I would blame the oddities on Yahoo Finance's data, but the three factors for TSM all look correct and the R-squares are all in the high 90's, which is pretty much the only encouraging aspect of my results.
2. The intercepts, which are supposed to represent "alphas", appear all wrong. None are negative, which I would expect.
3. Core2's Rm-Rf and SmB look right, but HmL looks too low. I was expecting 0.2 or so.
4. IJS's HmL looks too low. I was expecting 0.6 or so.
5. BRSIX's SmL looks too low. I was expecting 1.0 or so.

Can anyone help me out? Where did I go wrong?
Last edited by Sammy_M on Sat Dec 31, 2011 6:57 pm, edited 1 time in total.

camontgo
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### Re: 3-Factor Regression Analysis

Did you subtract the monthly risk free rate from the monthly total returns?

The left hand side of the regression should be the excess return, not the total return. That could be why the alphas are positive and large but the factor loadings seem be to about right.

I have some example code using R here:

http://www.calculatinginvestor.com/2011 ... -tutorial/
"Essentially, all models are wrong, but some are useful." - George E. P Box

pauliec84
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### Re: 3-Factor Regression Analysis

It looks like you did it correctly with one exception. You need to subtract the risk free rate from the fund return each month. Failing to do this is why you get positive alphas for all the funds.

Otherwise the HML & SMB numbers seam okay. BRSIX always as a oddly low SMB when I have run the FF on it.

Good work. Knowing how to run the 3 Factor model is incredibly helpful. I would also play around with some different time periods just to see how much the factors are time period dependent.

Let me know if this works.

Paul

Sammy_M
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### Re: 3-Factor Regression Analysis

Thanks both for pinpointing my error. Looks like I'm almost there! Now...where can one obtain the monthly tbill (Rf) return series in downloadable format? It appears to no longer be on French's website, unless I'm missing it.

pauliec84
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### Re: 3-Factor Regression Analysis

If you can't find it on Fama French, you could always just download the return of a 1-Month T-Bill Fund from Yahoo.

camontgo
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### Re: 3-Factor Regression Analysis

It is the last column in the Fama/French factors file.

Example:

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``        Mkt-RF     SMB     HML      RF192607    2.62   -2.16   -2.92    0.22192608    2.56   -1.49    4.88    0.25192609    0.36   -1.38   -0.01    0.23192610   -3.43    0.04    0.71    0.32192611    2.44   -0.24   -0.31    0.31192612    2.77   -0.01   -0.10    0.28192701   -0.11   -0.30    4.79    0.25192702    4.32   -0.24    3.35    0.26192703    0.32   -1.87   -2.58    0.30192704    0.41    0.29    0.95    0.25192705    5.39    1.53    5.07    0.30192706   -2.30    0.65   -2.04    0.26192707    7.27   -3.54   -0.86    0.30192708    2.42   -0.75   -4.14    0.28``
"Essentially, all models are wrong, but some are useful." - George E. P Box

Sammy_M
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### Re: 3-Factor Regression Analysis

I believe that is the last piece of the puzzle. At the top of the page is the file I should have used, under the heading:

Fama/French Benchmark Factors
Fama/French Factors

Instead I orginally used this one that is further down the page:

Historical Benchmark Returns
Fama/French Benchmark Factors

...and it did not contain the column for Rf returns. The HmL factors were also significantly different.

Here are the results I am now getting. Looks much better (BRSIX does appear to have much lower SmB loadings than expected).

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``         alpha  beta    SmB     HmL   Data since  Fund name  TSM   -0.02   0.99   -0.02    0.01   1996-Jul   Vanguard Total Stock MarketAWHSX    0.08   0.84   -0.24    0.35   1994-May   Am. Funds Wash MutualAGTHX    0.15   0.95    0.11   -0.18   1993-Mar   Am. Funds Growth Core2   -0.08   1.00    0.27    0.17   2005-Dec   DFA Core II  VOE    0.01   1.01    0.25    0.26   2006-Sep   Vanguard Mid Value ETF  IJS   -0.06   0.92    0.82    0.49   2000-Aug   iShares S&P600 ValueBRSIX    0.22   0.93    0.58    0.31   1999-Sep   Bridgeway Ultra Small Mkt  RZV    0.16   1.07    1.37    1.33   2006-Apr   Rydex S&P600 Pure Value``

*Edited to add VOE, dates for data series, and full fund names.
Last edited by Sammy_M on Mon Jan 02, 2012 12:28 pm, edited 3 times in total.

nisiprius
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### Re: 3-Factor Regression Analysis

Thanks for posting this and for the pointer to the William Bernstein article, which I wasn't aware of. I'm a long way from tackling this sort of thing myself, but in principle it's on my to-do list.
Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.

Dick Purcell
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### Re: 3-Factor Regression Analysis

Verrrry interesting.

What can you say about key conclusions?

Dick Purcell

Sammy_M
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### Re: 3-Factor Regression Analysis

Dick Purcell wrote:Verrrry interesting.

What can you say about key conclusions?

Dick Purcell

My feeling is that if one truly wants to embrace the work of Fama and French, they may wish to go through this level of analysis. It may not have a material impact though. Picking a certain percentage of small and/or value would probably do just fine.

In terms of applying the data derived through this process within a porfolio setting, it's probably best that I refer to an older thread where I sought similar assistance. Robert T and SmallHi were kind enough to oblige. I hope that I am able to give at least a small portion of what I have received though this forum.

Sammy_M
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### Re: 3-Factor Regression Analysis

pauliec84 wrote:I would also play around with some different time periods just to see how much the factors are time period dependent.

Let me know if this works.

Paul

Interesting indeed. Here is what it looks like using live data from Jan 2007 to date.

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``        alpha   beta    SmB     HmL  TSM    0.09   0.96    0.01    0.03AWHSX    0.14   0.85   -0.22    0.14AGTHX   -0.02   0.98   -0.05   -0.21 Core2    0.05   1.00    0.28    0.17  VOE*   0.02   1.01    0.24    0.27  IJS    0.02   0.90    0.87    0.40BRSIX   -0.41   0.94    0.90    0.08  RZV    0.49   1.06    1.29    1.41``

Bridgeway's significant negative alpha is notable, though it is exactly what I would have expected given it's lackluster performance compared to its CSRP 10 benchmark.

I knew that Am. Funds Washington Mutual (which is the least worst option in my 401K) has done very well lately. The low beta within this tough market is likely why.

I have avoided RZV in the past due to tracking error concerns. I must admit that I would have loved to have the positive tracking error/alpha it has had over the last 5 years.

nisiprius wrote:Thanks for posting this and for the pointer to the William Bernstein article, which I wasn't aware of.

It is indeed a wonderful piece by Dr. Bernstein. My favorite line is "factor-based analysis" (FBA) is to active money managers, what a light switch is to cockroaches."
=====================================================
*Edit: I went back and added VOE in the 5 year results above. The 10 year results for funds where live data is available is as follows:

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``        alpha   beta    SmB     HmL  TSM   -0.04   0.98    0.00    0.01AWHSX    0.06   0.82   -0.16    0.25AGTHX    0.00   0.98    0.04   -0.18BRSIX    0.07   0.88    0.87    0.23  IJS   -0.16   0.92    0.83    0.42``
Last edited by Sammy_M on Mon Jan 02, 2012 11:32 am, edited 2 times in total.
US: 19 TSM ■ 6 IJS, 3 BRSIX | Dv: 13 VEA ■ 8 SFILX | Em: 4 VWO ■ 3 DGS | Bnd: 23 ST ■ 10 I-Bonds/TIPS | Alt: 8 QSPIX ■ 4 RING

Rick Ferri
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### Re: 3-Factor Regression Analysis

Notice that micro-cap isn't what it used to be. BRSIX has less SmL than IJS. I belive this is a result of the shrinking US market (see that conversation). There are only 3761 listed stocks left in the US. That's down from 4009 in 2010.

Rick Ferri
The views expressed by Rick Ferri are strictly his own as a private investor and author and do not reflect the views of any entity or other persons.

RaleighStClaire
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### Re: 3-Factor Regression Analysis

Great job getting it to work. It's a good feeling when you finally figure it all out and the numbers come out right. The great part is once you get it to work once you'll have no problems plugging in data later to update your findings.

What I'd love to see is a site that auto-updated factor loads each month for a whole host of funds. If someone out there had Bloomberg access this should just involve writing a fairly simple macro to dump the data each month.

On to your data though: I use IJS and was pleased with how it performed relative to other SCV funds this year. It looks like it's HML stayed around where it normally is but the SMB went up some from analyses past.
Where's that red one gonna go?

pauliec84
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### Re: 3-Factor Regression Analysis

What I'd love to see is a site that auto-updated factor loads each month for a whole host of funds. If someone out there had Bloomberg access this should just involve writing a fairly simple macro to dump the data each month.

Monthly updates are probably unnecessary. On a short term basis you are probably capturing statistical noise in the changes. A less noisy place to look is at changes in the composition of the fund via the average market cap and book to market.

In the same vain I wouldn't focus on alpha to much. There are some instances like with the Russel 2000 were persistent negative alpha may be the result of poor index construction (ie front running). However, in most cases you are just capturing short term tracking error, and expense ratio + turnover are going to be better predictors of alpha going forward.

Consider running 4 factor models with the momentum factor (also available in the French data library). This will add more explanatory power to the model. As an example RZV in longer data regressions just about always comes out with a giant negative alpha (contrary to your finding, maybe due to shorter time period you used?). However this is almost entirely due to the fact it has a -0.54 loading on the momentum factor. The other pure value fund RPV as a -0.43 momentum loading. This is a HUGE red flag, that you would miss if you didn't include the momentum factor. I think most people on boglehead want to maximize there exposure to momentum, so ignoring it, when it is easily accessible and implementable makes no sense.

Rick Ferri
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### Re: 3-Factor Regression Analysis

RaleighStClaire wrote:What I'd love to see is a site that auto-updated factor loads each month for a whole host of funds. If someone out there had Bloomberg access this should just involve writing a fairly simple macro to dump the data each month.

Mornginstar has this information. I've tried to get them to put it on the risk page of their basic mutual fund analytics. Why they don't publish it is mystery to me.

Rick Ferri
The views expressed by Rick Ferri are strictly his own as a private investor and author and do not reflect the views of any entity or other persons.

Sammy_M
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### Re: 3-Factor Regression Analysis

pauliec84 wrote:Consider running 4 factor models with the momentum factor (also available in the French data library). This will add more explanatory power to the model. As an example RZV in longer data regressions just about always comes out with a giant negative alpha (contrary to your finding, maybe due to shorter time period you used?). However this is almost entirely due to the fact it has a -0.54 loading on the momentum factor. The other pure value fund RPV as a -0.43 momentum loading. This is a HUGE red flag, that you would miss if you didn't include the momentum factor. I think most people on boglehead want to maximize there exposure to momentum, so ignoring it, when it is easily accessible and implementable makes no sense.

Good point. I hadn't bothered to include the momentum factor because my plan doesn't contemplate a specific momentum target like it does SmB and HmL, but your point about not ignoring momentum within the various index funds is a good one.

richard
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### Re: 3-Factor Regression Analysis

pauliec84 wrote:Consider running 4 factor models with the momentum factor (also available in the French data library). This will add more explanatory power to the model. As an example RZV in longer data regressions just about always comes out with a giant negative alpha (contrary to your finding, maybe due to shorter time period you used?). However this is almost entirely due to the fact it has a -0.54 loading on the momentum factor. The other pure value fund RPV as a -0.43 momentum loading. This is a HUGE red flag, that you would miss if you didn't include the momentum factor. I think most people on boglehead want to maximize there exposure to momentum, so ignoring it, when it is easily accessible and implementable makes no sense.

Note that adding factors will often increase the explanatory power (in the statistical sense) of a model. But beware, on this path lies data mining

RaleighStClaire
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### Re: 3-Factor Regression Analysis

Rick Ferri wrote:
RaleighStClaire wrote:What I'd love to see is a site that auto-updated factor loads each month for a whole host of funds. If someone out there had Bloomberg access this should just involve writing a fairly simple macro to dump the data each month.

Mornginstar has this information. I've tried to get them to put it on the risk page of their basic mutual fund analytics. Why they don't publish it is mystery to me.

Rick Ferri

They compare funds to SML and HML data? That would seem unlikely but sounds great if they do have it.

Where can I find the info? Can you put in time periods and see what the factor load was over a specific time period?
Where's that red one gonna go?

empb
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### Re: 3-Factor Regression Analysis

Giving this a little tinker myself...presumably for international small stocks you could (crudely) take the difference between the returns of EAFE and EAFE Small. But does anyone know a work-around way to get monthly returns for value stocks? I suspect taking the return of EAFE Value less EAFE Large would result in artificially high HmL loadings considering the generally low HmL exposure of LV funds.

EDIT: If, as Robert has stated, EAFE Value has a .30ish load on HmL, could you:
1. Take EAFE Value less EAFE Large (isolating the returns to a .30 HmL loading),
2. Multiply the resultant series by 1/.3 (giving the returns to a 1.0 HmL loading),
3. Regress your fund against the results of Step 2? [obviously along with EAFE (mkt) and EAFE Large less EAFE Small (SmB)]

I'm likely missing something entirely fundamental.

Sammy_M
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### Re: 3-Factor Regression Analysis

empb wrote:Giving this a little tinker myself...presumably for international small stocks you could (crudely) take the difference between the returns of EAFE and EAFE Small. But does anyone know a work-around way to get monthly returns for value stocks? I suspect taking the return of EAFE Value less EAFE Large would result in artificially high HmL loadings considering the generally low HmL exposure of LV funds.

I bookmarked this post from Robert T where he gives his step by step instructions for international factor load estimates. International HmL data is on the French data library website.

I gave it a run using only the international HmL's and omitting the SmB's for now. I may include SmB data later, but wanted to see how it looked using only "pure" benchmark data. R squared for all are still high.

This is what I'm getting using all data available in Yahoo finance through Dec 2010 (most recent available on the French website)

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``      alpha   beta   HmL  Data since Fund nameEFA    0.04   1.03   0.02  2001-Sep  iShares EAFEEFV   -0.20   1.02   0.35  2005-Sep  iShares EAFE ValueSCZ    0.21   1.12   0.08  2008-Jan  iShares EAFE SmallPDN    0.52   1.03   0.24  2007-Oct  Powershares RAFI Sm-MidDLS*   0.24   1.07   0.11  2006-Jul  Wisdomtree Small Div``

Here are the results looking at all of these funds in the relatively short time span from Jan 2008 to Dec 2010. (SCZ and PDN are both very new)

Code: Select all

``      alpha   beta   HmLEFA   -0.14   1.03   0.09EFV   -0.47   1.01   0.40SCZ    0.21   1.12   0.08PDN    0.61   1.03   0.23DLS    0.27   1.06   0.08``

*R squared for DLS was 0.97. The remainder of the R squared's were above 0.98. The alphas were not statistically significant, with the exception of PDN's t-Statistic of 2.43 and P-value of 0.02.

It's interesting that the HmL loadings on PDN and DLS are both low, but they have positive alphas (PDN's statistically significant). I suspect some of that is due to : (1) my not including SmB in this analysis, (2) lucky timing of their re-balancing, and/or (3) "Value" being delivered through their weighting approaches which is not explained well by the high minus low BtM data series.

If you give the international factors a try, please let me know how it goes.

A Devout Indexer
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### Re: 3-Factor Regression Analysis

The dwindling # of publicly traded stocks is not having any impact on a small/micro cap strategy's ability to deliver high size loadings. First, the FF indexes that are used to calculate the factor sensitivities are restricted to the same set of public companies as the funds are (FF Research indexes can only buy public companies). Secondly, we are still talking about > 2000 tradable micro cap stocks and 2500 small/micro cap stocks...more than enough.

There are two primary factors that are driving the surprisingly low size loading (meaning it's not delivering as much of the size premium as you'd expect) on the Bridgeway Ultra Small Fund:
(1) the index that it attempts to track (via its prospectus objective), the CRSP 10 Index and its very few holdings have not loaded very heavily on the size factor since 2007 (CRSP 10 SmB = 0.79, whereas CRSP 9-10 SmB = 0.93 and Russell Micro Cap Index = 0.99)
(2) BRSIX uses an extreme sampling method (about 600 stocks) with profitability and other screens that have led to extreme negative tracking error in 2009, polluting the entire sample since 2007

When attempting to capture the size premium, you need #s, #s, #s, as in broad diversification. All small/micro funds and non-CRSP indexes suffered in 2008 due to CRSPs frequent reconstitution process that is next to impossible to replicate in real world funds, but BRSIX seems to be the only one that isn't venturing into the 0.9 to 1.0 SmB territory, which is what you hope to see from really small strategies. Below are the SmB loadings on various small and micro cap funds and indexes since 2007:

MSCI 1750 Index = 0.72
S&P 600 Index = 0.85
Russell 2000 Index = 0.86
CRSP 6-10 Index = 0.88
Russell Micro Cap Index = 0.99
CRSP 9-10 = 0.93
DFA Micro Fund = 1.0
DFA Micro Index = 0.97
DFA Small Fund = 0.92
DFA Small Index = 0.89

Note that the R^2s on all indexes are 0.98 or 0.99 (running a 3F regression), but is only 0.90 on the CRSP 10 Index, and I'd guess even lower on BRSIX. By the way, since 2007, the size premium has been concentrated in the mid cap segment of the market, so you will see mid cap indexes or small cap indexes with a heavy mid cap weighting will have large positive (and statistically significant in many cases) alpha's where as these indexes/funds above will be mostly negative or 0. Over time, of course, we'd expect to see a more constant distribution (mid beats large, small beats mid, etc.)

yessiree83
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### Re: 3-Factor Regression Analysis

A Devout Indexer wrote:The dwindling # of publicly traded stocks is not having any impact on a small/micro cap strategy's ability to deliver high size loadings. First, the FF indexes that are used to calculate the factor sensitivities are restricted to the same set of public companies as the funds are (FF Research indexes can only buy public companies). Secondly, we are still talking about > 2000 tradable micro cap stocks and 2500 small/micro cap stocks...more than enough.

There are two primary factors that are driving the surprisingly low size loading (meaning it's not delivering as much of the size premium as you'd expect) on the Bridgeway Ultra Small Fund:
(1) the index that it attempts to track (via its prospectus objective), the CRSP 10 Index and its very few holdings have not loaded very heavily on the size factor since 2007 (CRSP 10 SmB = 0.79, whereas CRSP 9-10 SmB = 0.93 and Russell Micro Cap Index = 0.99)
(2) BRSIX uses an extreme sampling method (about 600 stocks) with profitability and other screens that have led to extreme negative tracking error in 2009, polluting the entire sample since 2007

When attempting to capture the size premium, you need #s, #s, #s, as in broad diversification. All small/micro funds and non-CRSP indexes suffered in 2008 due to CRSPs frequent reconstitution process that is next to impossible to replicate in real world funds, but BRSIX seems to be the only one that isn't venturing into the 0.9 to 1.0 SmB territory, which is what you hope to see from really small strategies. Below are the SmB loadings on various small and micro cap funds and indexes since 2007:

MSCI 1750 Index = 0.72
S&P 600 Index = 0.85
Russell 2000 Index = 0.86
CRSP 6-10 Index = 0.88
Russell Micro Cap Index = 0.99
CRSP 9-10 = 0.93
DFA Micro Fund = 1.0
DFA Micro Index = 0.97
DFA Small Fund = 0.92
DFA Small Index = 0.89

Note that the R^2s on all indexes are 0.98 or 0.99 (running a 3F regression), but is only 0.90 on the CRSP 10 Index, and I'd guess even lower on BRSIX. By the way, since 2007, the size premium has been concentrated in the mid cap segment of the market, so you will see mid cap indexes or small cap indexes with a heavy mid cap weighting will have large positive (and statistically significant in many cases) alpha's where as these indexes/funds above will be mostly negative or 0. Over time, of course, we'd expect to see a more constant distribution (mid beats large, small beats mid, etc.)

^^ Get this folks, to earn the "size" premium you now have to hold midcaps as opposed to small or microcaps.

Just goes to show that all this slice and dice stuff is really just quantitative active management with false labels attached to make it seem like it has something to do with "inherent/fundamental risk". If you believe it in it, that's fine, but don't confuse it with passive investing or the investment philosophy of John Bogle.