OK, so as it turns out, S&P

*does* still post historical returns for their style funds, it's just hidden behind several layers of obfuscation and stupidity. It's also going away for good in a few days, so if you think you might ever want historical S&P index data, better

go grab it now!

Anyway, moving on. I found the discrepancy between my data and Robert T's, so it looks like our results agree. I expanded the regressions he did in 2008 to include data from 1997/05 (the earliest month I had returns for the S&P 600 Value fund) until June 2012. Here's what I got. Note that alphas are annual, not monthly.

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```
1997/05 - 2012/06 monthly data
Mkt-Rf SmB HmL Alpha (annual)
MSCI SMALL VALUE (VBR) 0.96 0.40 0.73 -0.24% R^2 = 0.932
Std. Error 0.02 0.03 0.03 1.32%
S&P 600 VALUE (VIOV/IJS) 0.98 0.60 0.62 -0.89% R^2 = 0.917
Std. Error 0.03 0.04 0.04 1.57%
S&P 600 PURE VALUE (RZV) 1.13 0.86 1.20 -2.05% R^2 = 0.816
Std. Error 0.05 0.07 0.08 3.11%
```

I decided to start showing confidence in terms of Standard Error instead of t-values, since to me it gives a better illustration of the confidence range. Same information though. t-value = Estimate / Std. Error. To get a feel for what these Standard Errors mean,

see here. For a large number of samples (which we have), a t-distribution approximates a normal distribution, so we can assume about 68% of the time the true value will fall within 1 standard deviation of the estimate. 95% of the time it would be within two standard deviations. (In our situation, standard deviation is equal to standard error.)

So, we do see a trend in the alphas, with the least concentrated fund looking best, and having the narrowest error band. But the size of the error on all three estimates is large enough that the true alphas for all three could very easily be the same. So why is it that we're able to get such good estimates of the average factor loadings over this period but not of the alphas? Two (related) reasons. First, the alpha estimate depends very heavily on the factor estimates. If one of our factor estimates is off slightly and there was a large (positive or negative) return for that factor, it could cause a large difference in alpha. Second, the factor loadings of funds change over time. By doing one regression over a long period, we are finding a single set of average factors, when in reality they have varied throughout that period. Since alpha depends on our factor estimates, and we're not capturing these variations in factor loadings, it's not possible to estimate alpha very precisely.

An example: say we're looking at a blend fund over a ten year period. For the first 5 years, the fund had a small positive value loading, and for the second 5 years, it had a small negative value loading, just randomly due to the nature of the companies it happens to hold. Then say for the first five years, value stocks happened to do well, and over the second five years, growth stocks happened to do well. If we do two separate regressions over the two five year periods, the results should be pretty accurate. The first period will show the value loading and alpha around zero, and the second period will show the growth loading and alpha around zero. HOWEVER, if we do a single regression over the full period, it will correctly estimate the value loading (average) to be about zero, and will find the alpha was positive! Because even though it had no value loading over the period (on average), it somehow managed to beat the market. (Or in other words, it outperformed the expectation of its average factor loadings over the period, hence alpha.) (Note: this is why we prefer to pick funds whose factor loadings don't vary too much. It removes a source of risk, in that there's less chance your fund's value loading will dip over the periods when value outperforms, and vice-versa. I imagine the best funds for this are the DFA funds, since they're designed to have consistent factor loadings. That said,

they still do vary.)

So long time periods are good because they give us lots of samples, allowing for greater precision. But they're also bad, because factor loadings change over time, so although we can get good estimates for the

*average* factor loadings over the period, we still don't know what the true alpha of the fund was, relative to is changing factor loadings over time.

I'm thinking one way to try to get around this would be to do a bunch of rolling regressions over shorter periods and average the results. I'm going to write a little script to do that at some point here, but first, there is one other way to tell how much of an effect the diversification (or lack thereof) of these funds is having.

We can build theoretically identical simple portfolios using several SV funds and compare their returns and standard deviations. To do this, I took the underlying indexes of the three funds that interest me most, VBR, VIOV/IJS, and RZV, and combined them with a combination of the MSCI Large and MSCI Small indexes, such that all three portfolios ended up with the same factor weightings, on average, over the period of May 1997 - June 2012. (Would have included JKL, but I don't believe the M* index data is available, or goes back that far.) Here they are:

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```
Portfolios MSCI Small MSCI Large MSCI SV S&P SV S&P PV t-bill | Mkt-Rf HmL SmB
P.MSCIV (VBR): 49.29% 11.72% 36.85% 2.13% | 0.95 0.4 0.4
P.SPV (VIOV): 16.32% 24.14% 59.48% 0.07% | 0.95 0.4 0.4
P.SPPV (RZV): 43.05% 26.35% 24.61% 5.99% | 0.95 0.4 0.4
```

So you see the three portfolios, the percent weighting they have to each index, and the fact that all the portfolios have identical factor weightings. (A bit of t-bills is used because the betas (Mkt-Rf factors) of the indexes are slightly different.) Now, this isn't perfect, because the MSCI Small and Large cap indexes don't necessarily have exactly zero alpha, so they will affect the results somewhat. Still though, this does give an idea of how these funds might perform as part of a real-world portfolio.

Here are the annualized returns and standard deviations for the three portfolios over the May 1997 - June 2012 period, along with the MSCI total investible market, for comparison.

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```
P.MSCIV P.SPV P.SPPV TSM
Return (annual): 8.81% 8.21% 8.30% 6.09%
SD (monthly): 5.39% 5.45% 5.41% 4.93%
SD (annualized): 18.67% 18.87% 18.73% 17.07%
Sharpe Ratio: 0.33 0.29 0.30 0.20
```

So, the main thing is, the returns and standard deviations of all three portfolios are very similar... Which is exactly what the 3F model says should happen. All three portfolios beat TSM on a risk-adjusted basis, not surprising given the overall performance of SV over the period. The winner in this test is the MSCI Small Value index. However, any variation in returns over this period is subject to the same issue I described with the alphas above. It is possible that the difference is due to actual systematic inefficiencies in one index vs another, but it's also possible that the MSCI index just happened to fluctuate its value loading at opportune times. There is

*no* guarantee that the higher return will persist.

What's more interesting to me are the Standard Deviation numbers. All three portfolios are nearly identical, meaning the lesser diversification of the pure value index does not result in greater volatility, which surprised me. The logic behind that is that you need much less of it to reach the same overall factor loadings, but I still expected there to be some non-systematic risk from its level of concentration that would show up in the SD of the portfolio. Instead we also see essentially no difference between the standard deviations of all three portfolios.

So one should be able to build a portfolio with the same factor loadings and the same risk using funds tracking any of these indexes. That said, I still prefer VBR, but not as much due to that historical outperformance as for two other reasons:

First, I used the *index* returns, not the fund returns, for this comparison to allow for a longer period to be studied. VBR has had tracking error very near zero, while IJS has consistently been at -0.2%, and so far VIOV looks very close to that at -0.19%. RZV so far has averaged -0.1%, but has varied a bit more, and evidence from the mid and large pure value funds, as well as RZV's management expenses, suggests that somewhere in the -0.2% to -0.3% range would be a more likely result going forward.

Second, the regressions to determine the factor loadings of the funds was done over the

*the same period* that we tested the results over. That guarantees that the factor weights of our test portfolios are the same over the period we're looking. In the real world, we have to estimate factor loads for funds using

*past* data, then get our returns from the future. If the factor weights change in the mean time, it can alter our risk and return. This is bound to happen to some extent, but the more they vary, the more further our results might be from our expectations. As I mentioned (way) above, that's why it's safest to choose funds with relatively stable factor weightings. (In this case that's VBR, but we do have to remember it's the factor weights of the entire portfolio that matter, so the variations of RZV will have less effect since we'll own less of it.) When I'm able to write that script to do rolling regressions we should be able to get a better idea how much the weights for each index have changed over time.

All that said, I am somewhat more convinced now than before that if there were other reasons to choose a different fund - particularly tax efficiency and/or limited portfolio space in which to achieve tilt - RZV (or VIOV) could be a good choice.

WHEW! Hope this example helped someone! I know I found it interesting.