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I´m currently working on a project where I basically want to compare historically returns between large and small-capitalization stocks in a given time period.

I want to approach this problem by test the perfomance with fama-french regressions on the excess return on two portfolios consisting of small and large-cap stocks.

I have retrieved the factors (monthly), and when summarized, I notice that the SMB-factor generated negative returns in my time period. This I interpret as large stocks have outperformed small ones.

Now to my problem. While SMB < 0, I found that the smallcap-index have generated approximately 50% higher return than the largecap-index. Isn´t this contradictory?

Also, if I want to investigate if there is a size-effect on the market (in the given time period), which approach is appropriate for this purpose? Is it enough to run a Fama-French regression on the excess return on both portfolios and to check if the SMB-factor exhibits statistical significance.

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  • $\begingroup$ Welcome on Quantitative Finance SE! Could you please mention what capital market/country you are looking at and did you exclude certain stocks from your analysis? It would be very helpful for giving a detailed answer, if you would also edit your question and additionally show some summary statistics, i.e. what is your SMB return per month and its statistical significance, etc. $\endgroup$ – skoestlmeier Dec 23 '18 at 9:56
  • $\begingroup$ Thanks for the reply. I´m investigating the swedish capital market. The average SMB return per month is -0.06%. Though the small cap index has outperformed the large cap index significantly in this time period. $\endgroup$ – Ole G S Dec 23 '18 at 10:46
  • $\begingroup$ And when I regress the Large cap excess return on the FF3F-model the SMB is significant (at 0.05 level). The same procedure is done with the Small cap portfolio and here the SMB-factor becomes insignificant. The alphas and HML-factor are insignificant for both portfolios. When I expand the model with Momentum factor, this will become insignificant as well. $\endgroup$ – Ole G S Dec 23 '18 at 10:56
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There are several issues to address your question, so let me first summarize some empirical facts on the SMB portfolio and then give you some hints for your analysis.


SMB-return

Using Stefano Marmi's data library for the Swedish stock market, we observe a monthly SMB-return of -0.32% (Newey/West adjusted t-value 1.24, using lag of four) for the period July 1991 - March 2013. This result is supporting your comment, that you also found an insignificant, negative SMB-return. I observe the following summary statistic for the monthly Swedish SMB-return:

> summary(smb_sweden)
    Min.  1st Qu.   Median     Mean  3rd Qu.     Max. 
-22.3000  -2.5500  -0.4500  -0.3185   2.1600  19.1000

> sd(smb_sweden)
[1] 4.483592

> reg <- lm(smb_sweden~1)
> coeftest(reg, NeweyWest(reg, lag = 4))

   t test of coefficients:

            Estimate  Std. Error  t value   Pr(>|t|)
(Intercept) -0.31854    0.25784   -1.2354   0.2178

However, as the SMB-return is statistically insignificant, there is no evidence that it is different from zero. This contradicts your first interpretation, that large stocks have outperformed small stocks.

SMB-return and Indices

Despite comparing your SMB-return series with certain listed stock-indices (like ETFs), a meaningful analysis has to use the entire stock universe you are looking for. That means, that you should carefully look which stock's are contained in your reference index. I doubt, that these (e.g. ETF-) indices fully cover your stock universe you used for calculating the SMB-return, so your listed reference index may omit many stocks you used for your SMB-return calculation (or define other breakpoints, etc.). Setting up your stock universe is written precise and clear in Fama/French (1992) or Fama/French (1993), so follow their approach and

  • exclude banks / financial services (SIC-code 6,000 - 6,799)
  • exclude stocks where the firm has a negative book-value
  • ...

However, if your reference index of small stocks (whether listed ETF-indices or your own created index of stocks) outperforms the index of large stocks, there is no contradiction to our results. Just take a look at the summary statistic above on how volatile the SMB-return is. There are certainly many periods of time, where your large-index outperforms the small-index, but, their difference in returns over a large period of time is statistically insignificant from zero (at least for the period 1991 - 2013 as stated above).

Interpretation

In the US, the CAPM alpha of the SMB-return for the entire period of July 1926 - December 2012 is an insignificant 0.10% per month with a Newey/West (1987) t-stat of only 1.12. These results seem to indicate that the excess returns of the SMB portfolio can be explained by the sensitivity of the SMB portfolio to the market portfolio.

Despite these facts, Fama/French(1993) show that the SMB portfolio has the ability to explain the abnormal returns of portfolios whose returns are not explained by the market factor. Although, there are substantial risks associated with this portfolio, and poor performance may persist for extended periods of time. They conclude (very precisely), that the difference in expected returns between large and small stocks must be due to exposures to a latent risk factor that are cross-sectionally correlated with market capitalization.

Tests

Tests, if there is a size-anomaly in a certain stock market is quite straightforward, and you have mainly two options described here:

  1. portfolio sorts
  2. regression techniques

In general:

Portfolio sorts are applied by sorting stocks based on their (here:) market capitalization into separated portfolios and calculating their subsequent portfolio returns until your portfolios are re-assembled. The "high-low" portfolio return gives insight, if a significantly risk-adjusted return can be obtained following this strategy.

Regression techniques like Fama/MacBeth (1973) use e.g. one-month ahead stock return as the dependent variable and the logarithm of market capitalization with further control-variables like a stock's beta-factor, etc. as independent variables. If the coefficient of $ln(marketcap)$ is statistically significant, there is strong evidence for a size-anomaly in this market.

For these techniques and their details (like winsorizing, Newey/West adjustment, etc.) i strongly recommend you to take a look at Bali/Engle/Murray (2016), chapter 5 and 6.


References:

Bali/Engle/Murray (2016), Empirical Asset Pricing: The cross-section of stock returns, 1. ed.

Fama/French (1992), The Cross-Section of stock returns, The Journal of Finance 47(2)

Fama/French (1993), Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33(1)

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