Case 2B: The Small/Large Anomaly Maastricht University School of Business and Economics Maastricht, 15 September 2009 Maastricht University School of Business and Economics Maastricht, 15 September 2009 Table of Content Introduction3 Summary Statistics4 Spread Portfolio5 Evaluate the CAPM6 Conclusion7 References8 Introduction The Capital Asset Pricing Model (CAPM) is an equilibrium model that underlies all modern financial theory. It predicts the required rate of return of a security based on its risk, as measured by beta, and makes use of various simplifying assumptions.
Hence, equilibrium condition would evolve with all investors choose to hold the same portfolio for risky assets, the “market” portfolio. However, following Fama and French’s study in 1992, numerous studies have suggested that beta is not sufficient in accounting for risk and recommend the inclusion of other variables. Most notably, Fama and French (1996) noted that stocks of smaller firms and stocks of firms with a higher book-to-market ratio have had higher stock returns than predicted by single factor models, and thus proposed a three-factor model that adds on firm size and book-to-market ratio to the market index.
In this paper, we will focus on the small-large anomaly, and examine empirically if it still exists in today’s market. We use the Russell 1000, Russell 2000 and Russell 3000 to represent the large cap, small cap and general market respectively. Firstly, we will present the summary statistics of the 3 return series over the last 15 years. Next, we will build and analyze a spread portfolio over the same period, and discuss the effectiveness of such a strategy. Finally, this paper will test the CAPM and its accuracy in the pricing of both small and large styles, and the spread. Summary Statistics
This section will shortly introduce the different indexes used for our analysis, elaborate on the outcomes of the statistical analysis to summarize the statistics of the data sets. For the statistical analysis we choose the Russell 1000, Russell 2000 and Russell 3000 to represent the small cap, large cap and general market respectively. The Russell 1000 lists the thousand US companies with the highest market capitalization and the Russell 2000 indexes the two thousand US companies with the smallest market capitalization. Therefore, the Russell 1000 index represents approximately 90% and the Russell 2000 only 10% of the market value.
The Russell 3000 index is comparable with the Willshire 5000 or S&P 500 and represents the market portfolio. We will describe and compare the central tendency for the two return series using variables such as the mean, standard deviation, correlation coefficients and minimal and maximal returns. Over a period of 15 years (1996 – 2011), the average monthly mean return for the Russell 1000 and Russell 3000 are almost equal with 0. 43% and 0. 44% respectively. This can be explained by the dominance of approximately 90% of large cap firms in the Russell 3000.
Additionally, the Russell 2000 (small cap index) had a slightly higher mean monthly return of 0. 57%. According to Alex Kane (2008), the more a stock’s returns vary from its average return, the more volatile the stock is reflected in a higher standard deviation. The higher standard deviation of the Russell 2000 of 6. 09% compared to 4. 79% and 4. 81% of Russell 1000 and Russell 3000 respectively. These higher variations of the small cap index is reflected in the minimum and maximum returns of -353. 061 and 202,591 respectively compared to -214. 561 and 126. 812 for the Russell 1000 index.
These differences in the standard deviation can be tied to the lower liquidity levels and neglected-firm effect of small cap companies (Bodie, Kane ; Markus, 2010). Furthermore, the correlation of 0. 8313 between the Russell 2000 and Russell 1000 shows that these two indexes move largely in the same direction. Additionally, the percentage of negative month around 40% for all three data sets display the similar external influences on the each company – no surprise since, the three indexes are all based on US Companies and the legislative and external market influences are identical.
Concluding the statistics section, the standard deviation of small cap index is higher as we expected since more risk is incurred by investing in these firms. Further, the differences among minimum and maximum returns are as the theory predicted and all three indexes are influenced by the similar external factors. Spread Portfolio A spread portfolio combines a long and short position in e. g. stocks. The Aim is to exploit relative differences in return rather than general movements of the underlying stock. The spread portfolio investment strategy is a useful tool to protect the investor from the overall market volatility.
To build a spread portfolio, we would seek to go long in the shares that we expect to outperform and short in the others. In various earlier studies, there have been instances where the smallest-size portfolio outperforms the largest-firm portfolio due to neglected-firm and liquidity effects that reward the investor for the high risk of investing in small cap. (Alex Kane, 2008) Therefore, in accordance to the small firm effect CAPM anomaly, our spread portfolio is created with a long position on the Russell 2000 and short-position on Russell 1000.
The long position would enable us to reap the gains. Short position on Russell 1000 Why?? The portfolio is analyzed over the same period of 15 years. The average return of our simulative spread portfolio is rather small with 0. 14% average monthly return. However, the analysis of the spread portfolio over such a long period is less effective. If we zoom in on our observations by creating sub-periods several trading possibilities are discovered that yield very high returns. Also these opportunities require very careful examination and precise timing.
Finally, the spread portfolio can create high margin trading opportunities despite the fact that the long-run analysis discovered only small monthly returns. Evaluate the CAPM The CAPM predicts the relationship between the risk of an assets and its expected return. This relationship is expressed in this equation (Alex Kane, 2008): E(Ri) = Rf + ? i(E(Rm) – Rf) where E(Ri) is the expected return of a security i, Rf is the risk-free rate, ? i is the beta of security i which measures its risk and Rm is the expected return of the market portfolio.
Fama and French observed that small-cap stocks tended to outperform markets. This is also known as the small-firm effect anomaly. To prove the existence, we run a regression using the ordinary least squares method and the model is the following: E(Ri) – Rf = ? i + ? i(E(Rm) – Rf) + ? i where ? i is alpha, a measure of an abnormal rate of return on a security in excess of what would be predicted by the CAPM. According to Investopedia (2009), the efficient market hypothesis indicates that ? i should be zero as all stocks are fairly priced. Hence, abnormal profits cannot be gained. i would refer to diversifiable risk and is therefore near zero in a market portfolio. We run a regression to indicate the ability of the traditional CAPM to explain the changes in value. Therefore, first the small-cap (Russell 2000) is regressed, than the large-cap (Russell 1000) and finally, the created spread portfolio performance is checked. As the next step we check whether the accuracy of the CAPM changes in different market situations. The first regression result confirms the Fama ; French’ small-firm effect anomaly with a Beta of 1. 09 (at a 95% significance-level) and an insignificant alpha.
The high adjusted R^2 of 0. 739 shows that a lot if the changes is explained by the model. The insignificant alpha reflects the accuracy of the CAPM model. The highly significant beta of 0. 994 and again an insignificant alpha in combination with an R^2 close to one (R^2=0. 997) indicates that the large-cap portfolio is rather well explained by the model. Third, the beta of 0. 097 at a 90% confidence-level and an insignificant alpha show as indicated by the low expected excess return, that the risk of the spread portfolio is less compared to the market portfolio. However the adjusted R^2 is only 0. 13 and therefore ? does not explain all of the risk. The creation of two sub-periods enables us to test the CAPM accuracy more precisely. The subgroup includes the data for period I (2005-2006) and period II (2008-2009). These timeframes gather the data just before the last big financial crisis and right after the market decline. The regression analysis for the small-cap portfolio shows a high beta of 1. 697 for period I with a high significance that underlines the assumption of higher risk incurred by small-cap companies. Period II has a beta of 1. 173 (p-value = 0. 000) and insignificant alpha.
This high beta compared to the beta for the 15 year interval shows the increased risk for small companies after the . com-bubble and the higher risk of small firms during financial downturn. The large-cap regression results stay constant during the different periods compared to the whole interval. This is again consistent with the theory since the Russell 1000 index companies’ accounts for 90% of the market value. The Spread portfolio analysis shows the most differences in outcomes if you compare the two periods with the 15-year-interval. Especially period I reflects a high increase in the beta with 0. 76 (p-value = 0. 000) and a R^2 of 0. 83 which drops back to beta 0. 198 (p-value=0. 029) in period II. The regression analysis does not confirm a CAPM anomaly. The alphas of the regression were all not significant different from zero. Further, the regression underlines the theory about higher risk for small-cap investments. The large-cap index behaved like the CAPM predicts and R^2 was constant at almost 100%. Conclusion The different average returns of the Russell 1000, Russell 2000 and Russell 3000 are in line with the risk premium theory (Reference , Year) since the small-cap stocks inherit a higher risk and had a higher average return compared to the other two indexes.
The spread portfolio strategy did not outperform the other indexes. The growth / performance of the small-cap companies were not large enough to create an advantage over the traditional Russell indexes. Finally, we failed to prove the Fama ; French theory. The regression analysis did not reveal any significant difference between the CAPM predictions and the actual beta. Therefore, the chosen data did not underline the theory mentioned by Bodie et al. (2010). References