Don’t compare Apples to Oranges: Why do portfolio construction details matter?

Background

When Banz (1981) wrote about the size premium in his Ph.D. Dissertation, it was “the” news, at least in the academic community. Soon, after that practitioners followed as well. A range of small-cap funds started to appear on the street promoting their strategy. Fast forward to the present time, there are thousands of different variations of the small-cap funds around the global markets. The size factor was indeed the first anomaly to be discovered. The research by Banz (1981) started kind of a revolution in the academic finance community in that there was a sudden surge of motivation among researchers to look harder into the CRSP database (this was and is still is the go-to database for security price data) and try to find factors that would potentially explain the cross-section of expected returns. After Fama and French (1992) decided to include size and value along with the market factor in their seminal paper thereby introducing the three-factor model, the size along with value became default factors on which the practitioners based their fund's strategy.

You might be now thinking, “So what’s the catch here? Everything here looks well and good.”

Well, sadly everything is not well and good for the size factor. It has come under huge scrutiny from academics and practitioners over the last 2 decades. A number of arguments (with evidence) against the size factor have been put forth which makes everybody curious about this seemingly intuitive anomaly. A few of these points against size are as follows:

With regards to performance, the size effect/factor is the weakest among all other major factors that have been discovered.

The post-publication decay of excess returns is quite vivid in size factor.

Almost all of the significant returns from the size factor are linked with the January effect.

Having said that, rather than abandoning the factor, some researchers have also found a new way to implement size. The idea of size adjusted for quality by Asness et al. (2018) helped revive the size premium and delivered significant returns in the full sample period. This was a short background on how the size factor’s evolution has been in the markets.

Now, let’s move on to the main objective of this article. The main goal of this article is to show why the fund construction matters for its long-term outperformance. So, let’s dive right into it. There are many small-cap funds/ETFs who write in their investment policy statement that their main objective is to harvest the size premium. Okay, that sounds good! But, there are many ways to skin the cat. In other words, although all these funds shall be grouped in a small-cap category, the details of how each fund is constructed and benchmarked might be the determining factor that decides whether the fund will generate excess returns.

Data

To show the evidence for this, I have arbitrarily selected 4 funds on which I could get the data on through publicly available sources. These are:

               I.            DFA U.S. Small Cap Portfolio Institutional Class (DFSTX)

            II.            Bridgeway Ultra-Small Company Market Fund (BRSIX)

         III.            iShares Core S&P Small-Cap ETF (IJR)

         IV.            Vanguard Small-Cap Index Fund Admiral Shares (VSMAX)

I am also going to use the Center for Research in Security Prices (CRSP) data through Prof. Kenneth French's database. I am going to use the monthly return observations for the analysis. The sample period we are looking at starts from January 2001 and ends in December 2021 which brings our data to 21 years. The only reason that I did not go further back in this study is because of the lack of data for each of these funds before 2001.

For the regression analysis, I have decided to go for a 5-factor model where the factors involved are size (SMB), value (HML), market (Rm-Rf), momentum (MOM), and quality (QMJ). Now that the study design is clear, we can go on to the actual analysis. Let’s first get some descriptive stats and returns for the size quintiles/deciles and the funds.

Analysis

Funds Return Summary Statistics (2001-2021)

	Vanguard	iShares	DFA	Bridgeway
Count	252	252	252	252
Mean	0.94	0.98	1.02	1.04
Std.deviation	5.54	5.49	6.11	7.18
Minimum	-22.10	-22.91	-22.40	-26.26
1st quartile	-1.73	-2.11	-2.34	-2.59
Median	1.37	1.60	1.13	1.23
3rd quartile	4.34	4.32	4.53	4.80
Maximum	18.37	18.48	21.24	32.10

1The returns are expressed in percentage terms

CRSP Deciles Returns: Summary statistics (2001-2021)

	Dec 1-2	Dec 2-4	Dec 4-6	Dec 6-8	Dec 8-10
Count	252	252	252	252	252
Mean	1.07	1.03	0.99	0.98	0.76
Std.deviation	6.29	6.01	5.51	5.14	4.24
Minimum	-23.13	-23.21	-21.24	-21.25	-15.89
1st quartile	-2.58	-2.51	-2.11	-1.53	-1.34
Median	1.46	1.86	1.51	1.47	1.21
3rd quartile	5.19	4.74	4.55	4.22	3.26
Maximum	22.42	19.52	17.91	15.48	13.25

2The returns are expressed in percentage terms

From the two tables above, we can immediately notice one thing which seems quite explicit. Both the average returns and the standard deviations have a monotonic relationship among the deciles. In other words, as we move downwards from Dec 8-10 to Dec 1-2, almost always the average returns increase but this increase in average returns comes at a cost because, with it, the standard deviations also increase. This touches upon a very important point that has caused a huge debate for the last 4 decades in the academic finance community. This debate is about whether the factors deliver excess returns because of mispricing in the market or because it is just compensation for the risk taken. The literature that surrounds this topic is so large that it is beyond the scope of this article. Maybe in the future, I might write something about it.

For now, let’s also calculate the annualized/compounded returns for these size deciles:

CRSP Deciles: Annualized Returns

	Dec 1-2	Dec 2-4	Dec 4-6	Dec 6-8	Dec 8-10
Annualized Return (in %) 1926-2021	11.99	12.16	12.02	11.59	10.07
Annualized Return(in %) 2000-2021	11.03	10.76	10.64	10.75	8.43

We also know that the factors are constructed as a long-short portfolio. For example, the size factor, as defined by Fama and French (1992), is constructed by sorting all the stocks by their market equity (Total number of shares multiplied by the Market price of a share) into 10 deciles and then taking the weighted average annual return of deciles 1 to 5 (Small-cap stocks) minus the weighted average annual return of deciles 6-10 (Large-cap stocks). In other words, each decile here represents a portfolio and by subtracting the upper half (6 to 10 decile portfolios) from the lower half (1 to 5 decile portfolios) of the dataset, we are essentially constructing a long-short portfolio that goes long on the small stocks and shorts the large caps. The return that we get from these long-short portfolios is called premium hence, the word size premium.

CRSP Deciles: Annualized Historical Premium

	50/50	30/30	20/20	10/10
Annualized Premium (in %) 2001-2021	1.11	2.87	2.82	3.50
t-statistic	0.82	1.33	1.24	1.42

The results that we get from the various combination of weighting schemes reveals that how the portfolio is constructed matters when you are evaluating a fund’s performance. After all, apples should be compared with apples.

We now shall examine the factor loadings that each of the funds in our data has when regressed upon the 5-factor model in which the size factor is constructed through various combinations of deciles. Note that in Fama and French 3 factor model, the size factor is constructed as a 50/50 decile portfolio i.e. going long on small-cap stocks and shorting the large-cap stocks.

Monthly Size Factor Loadings, Five-factor Model (2001–2021)

Fund	50/50	30/30	20/20	10/10
DFA U.S. Small Cap Portfolio Institutional Class	0.85 (15.59)	0.67 (14.49)	0.60 (13.83)	0.57 (13.58)
Bridgeway Ultra-Small Company Market Fund	0.92 (7.79)	0.81 (8.65)	0.79 (9.44)	0.86 (11.28)
iShares Core S&P Small-Cap ETF	0.82 (33.12)	0.63 (26.03)	0.54 (21.17)	0.48 (17.10)
Vanguard Small-Cap Index Fund Admiral Shares	0.64 (23.36)	0.48 (18.89)	0.39 (15.01)	0.35 (13.31)

 

I am going to ask you a question now: Do you see any pattern in the above table? If not, perhaps the plot below will definitely help you see something:

Size Factor loadings of 4 funds, (2001–2021)

I don’t know about you but to me, there is a clear pattern of the factor loadings decreasing in almost every fund aside from Bridgeway Ultra-Small Company Market Fund. So, what is going on here? Well, to see that, we have to first read the fund’s investment policy statements. We don’t have to go through everything but just their benchmarks and methodology of portfolio construction would be enough.

Fund	Construction details
DFA U.S. Small Cap Portfolio Institutional Class	Companies that belong to the lowest 10% of total market capitalization or companies whose market capitalizations are smaller than the 1,000th largest U.S. company, whichever is higher. All companies are the U.S listed. As of 12/2021, the market capitalization of a small-cap company would be below USD 10,142 million.
Bridgeway Ultra-Small Company Market Fund	Aims to approximate the total return of the Cap-Based Portfolio 10 Index published by the University of Chicago's Center for Research in Security Prices (CRSP) over long time periods. "Ultra-small companies" are defined as those companies that have a market capitalization the size of the smallest 10% of companies listed on the New York Stock Exchange, or companies with capitalizations that fall within the range of companies included in the CRSP 10 Index
iShares Core S&P Small-Cap ETF	The Fund seeks to track the investment results of the S&P Small-Cap 600. As of March 31, 2021, the Underlying Index included approximately 3% of the market capitalization of all publicly-traded U.S. equity securities. As of March 31, 2021, a market capitalization between $203 million and $12.4 billion was recorded in the portfolio.
Vanguard Small-Cap Index Fund Admiral Shares	The Fund employs an indexing investment approach designed to track the performance of the CRSP US Small Cap Index, a broadly diversified index of stocks of small U.S. companies. The CRSP US Small Cap Index includes U.S. companies that fall between the bottom 2%-15% of the investable market capitalization.

After reading the above description for each fund and reflecting on their returns and factor loadings, we can pretty much conclude that details of portfolio construction matter, and moreover, it starts to compound as time passes. Imagine if you are leaving 50 bps on the table every year, this will compound to a very big number in the long run. I want to again highlight a point to the readers that all these funds are categorized as small-cap funds by Morningstar or any such fund rating service agencies and from the above description, one thing is clear: these are very different from each other.

For example, the Bridgeway fund tracks the absolute smallest companies that reside in the smallest decile of the CRSP Index. This is exactly why the fund has a factor loading or the slope coefficient of 0.86 when regressed on the SMB 10/10 long-short portfolio and as we have learned from the earlier discussion that the smaller deciles comprised of very small firms have higher expected size premium. This is very much reflected in the returns and standard deviation of Bridgeway ultra-small cap fund.

To make the above point even clearer, we can look at the annual captured size premium by each of the funds involved in this study. For this, we are only going to take the original SMB i.e. 50/50 SMB as defined by Fama and French (1992).    

Annual Premium Vs Annual Premium Captured (2001–2021)

	Loading	Premium	Captured Premium
DFA U.S. Small Cap Portfolio Institutional Class	0.85	1.11	0.94
Bridgeway Ultra-Small Company Market Fund	0.92	1.11	1.02
iShares Core S&P Small-Cap ETF	0.82	1.11	0.91
Vanguard Small-Cap Index Fund Admiral Shares	0.64	1.11	0.71

The above table proves that the successful returns of a style-fund depend on the ability of a fund to capture the premium that is provided by the underlying factor. Hence, by the same logic, one can also infer that the return for the fund is a function of the sensitivity of the fund to its benchmark index/factor (i.e. loadings) and the premium of the factor. This analysis is not limited to the size factor. One can replace size with any other characteristic of a portfolio and run the same analysis. The tenet would still be the same.