Adding Investor Sentiment Factors into Multi-Factor Asset Pricing Models.

Transcription

1 Adding Investor Sentiment Factors into Multi-Factor Asset Pricing Models. Robert Arraez Anr.: Masters Finance Master Thesis Finance Supervisor: J.C. Rodriquez 1 st of December 2014

2 Table of Contents Chapter 1:Introduction Chapter 2: Literature Review..4 Multi-factor asset pricing models. 5 Investor Sentiment....7 Chapter 3: Data & Methodology.. 9 Data....9 Methodology..9 The Investor Sentiment Variables Chapter 4: Results...18 Limitations...26 Discussions. 26 Chapter 5: Conclusion Appendix...29 References

3 1. Introduction The capital asset pricing model (CAPM)(John Lintner, 1965; Jack Treynor 1961 and 1962; Jan Mossin, 1966; William Sharpe, 1964 and 1970) suggests that taking on extra risk seems to be the only way to earn above market returns. This model introduces the main concept of the market premium; the excess return of a broad portfolio over the risk free rate which essentially measures the premium investors ask in return for taking on additional risk. According to the capital asset pricing model, the market premium is the sole determinant of asset prices. While the CAPM is a strong starting point for asset pricing, others (Jensen, Black and Scholes, 1972) have shown that expected excess returns are not strictly proportional to a securities risk factor. This introduces the possibility that there are more factors at work when assessing security prices than just those that measure risk. Along these lines Fama and French (1993 and 1996) introduced their three-factor asset pricing models to better explain security pricing. Their model is able to accurately explain excess returns based on multiple factors. In the end they deduce that their factors may be proxies for various risks and certain anomalies also remain unexplained by the three-factor model such as the short run continuation phenomenon (Jegadeesh and Titman, 1993). Nonetheless, their model helps to explain various other anomalies in asset pricing and remains the best available model for researchers when assessing expected returns. Variations on multi-factor asset pricing models have also been introduced such as that of Carhart (1997) where momentum was added as a factor. It seems that there is some room for interpretation and improvement of these multi-factor asset pricing models. It is here where this paper aims to add to the research that has already been done. Various authors have pointed to investor behavior, irrational or otherwise, (Lakonishok, Shleifer and Vishny 1994; Russel and Thaler 1985; among others) as factors that may cause errors in asset pricing. If these errors are predictable then it stands to reason that based on investor behavior it is possible to explain average returns. With this in mind this paper aims to create a proxy that can accurately measure investor sentiment. Additional factors will be created based on this sentiment proxy, akin to the factors created by Fama, French and Carhart, to see whether investor sentiment has any explanatory power in determining asset prices. Finally the sentiment factors will be compared to the three-and four-factor models to assess how well it performs in explaining average excess returns. The structure of this paper is as follows; Chapter 2 will cover the existing literature on multi- 2

4 factor asset pricing models as well as investor behavior and sentiment proxies. Chapter 3 will contain the data and methodology used in creating the sentiment factors as well as some preliminary regressions. Chapter 4 will cover the results of the main regressions as well outline the limitations in this research. Additionally chapter 4 will also provide a discussion on the results as well recommendations for future research. Finally chapter 5 will contain a brief summary of the main findings of this paper. 3

5 2. Literature review In discussing asset pricing models there is no better starting point than the original capital asset pricing model (CAPM). Indeed the CAPM forms the basis of most of modern financing theory and describes the basic relationship between a securities expected return and its risk (John Lintner, 1965; Jack Treynor 1961 and 1962; Jan Mossin, 1966; William Sharpe, 1964 and 1970). The CAPM was developed based on Harry Markowtitz (1952) work where he introduced the modern portfolio theory (MPT). The modern portfolio theory attempts to optimize the expected return of portfolios for any given amount of risk. MPT essentially shows that additional risk must be taken to obtain larger returns. Sharpe (1970) explains that investments contain two types of risk; systematic risk and unsystematic risk. Systematic risks are those risks inherent in the market in which the securities are traded and thus by its nature cannot be reduced by investing in multiple and varying securities. Unsystematic risk on the other hand is firm specific and can be reduced by constructing portfolios with varying classes of securities. Based on these two categories of risks it becomes abundantly clear that unsystematic risk is the most relevant for individual investors as this can be reduced by prudent trading. As such CAPM was developed as a means for investors to measure systematic risks. The Efficient Market Hypothesis (EMH), originally described by Fama (1965), theoretically elaborates the process behind the price movement of securities. The EMH suggests that market prices will adjust accordingly to any change in a security s true value and that this occurs instantaneously. It should be noted that in this paper instantaneously implies that the actual price will overshoot the new true value as often as it will undershoot it. Fama (1970) further developed the efficient market hypothesis by further disseminating the theory into three forms; weak-form efficiency, semi-strong-form efficiency and strong-form efficiency. Weak-form efficiency suggests that the securities price will accurately reflect all historical information and semi-strong-form efficiency implies that all obviously publicly available information is reflected in the market price. Strong-form efficiency is the most extreme of the models as if the model holds it suggests that even inside information is included in the price formation process. Fama s 1970 review found that the research performed on semi-strong-form and weak-form efficiency has produced sufficient supporting evidence for the theory. A deeper explanation of each form of the efficient market hypothesis can be found in the original paper. The most important takeaway from the EMH is that market prices will always reflect all relevant information and are therefore 4

6 traded at their true value. More importantly the efficient market hypothesis implies that the only way to truly earn larger returns is by taking on additional risk. While the CAPM equips investors with the necessary tools to measure risk there are other asset pricing models that include multiple factors that have proven to be a more accurate means of measuring risk and explaining the movement of market prices. Multi-factor asset pricing models Merton (1973) found that expected returns can differ from the risk free rate even if there is no systematic or even market risks. This is in stark contrast to what the CAPM assumes; remember that firm specific risk is the only relevant risk to consider in the capital asset pricing model. Merton s paper implies that it is possible that there are additional factors that can determine asset prices. Ross (1976) theorized that multiple common risk factors could be used to explain security returns. Ross Arbitrage Pricing Theory (APT) suggests that security s expected returns can be explained by security specific and macroeconomic factors. The APT differs from the CAPM as it allows for multiple factors when assessing returns, this of course allows for better explanatory models as one can more clearly see which risk factors affect the security in question. However the APT can become quite specific and certain factors may accurately explain a set of securities returns but may fail to do so with other securities. Choosing factors may also prove to be difficult as one must identify all relevant factors and in certain circumstances this may be a shot in the dark as it is not clear from the onset which factors will affect which securities. The APT does have its merits, as one can use it when assessing assets on a case-by-case basis. This can be done by calculating the risk premium by using the APT and then comparing this to the expected risk premium. If the two premiums are unequal there would be an arbitrage opportunity that savvy investors can benefit from. Conversely if the premiums are equal then one would say that the market value of the asset accurately represents its true value and accordingly there would be no arbitrage opportunities to take advantage of. The arbitrage pricing theory as well as the previously mentioned literature does introduce a key concept; the use of multiple factors to explain security returns instead of merely the market risk premium. Bollerslev, Engle and Wooldridge (1988) found that conditional covariances are significant determinants for the time varying risk premium however they also found evidence 5

7 that suggests that other factors should be taken into considerations when assessing returns. Other papers (Cochrane, 1996; Liu, 2006; Holmstrom and Tirole, 2001; Chen, Roll and Ross 1986) have also found that other factors can explain asset prices. These papers suggest that multiple factors can explain asset returns and not just the market premium. Concurrently many critics pointed toward anomalies in average returns that cannot be explained by the CAPM or the efficient market hypothesis. De Bondt and Thaler (1985, 1987) found evidence that on the long run, stocks that previously performed well tended to perform worse in the subsequent years while the opposite held for stocks that performed poorly. Their research suggests that using fundamental analyses based on past data can indeed help explain current or future stock prices. Theoretically, based on the EMH this should not be possible based in weak-form efficiency. Additionally, Jegadeesh and Titman (1993) found that following a strategy of selling stocks that performed poorly in the short-run past and buying those that performed well tended to net strong abnormal returns. Others (banz 1981; Basu 1983; Rosenberg, Reid and Lanstein 1985) found that size, E/P ratios, book-to-market ratios all have an effect on returns. In response to these critiques Fama and French (1993 and 1996) added two factors to the original CAPM market premium factor after they observed that value stocks outperformed growth stocks and that small cap stocks were also outperforming large cap stocks. The model they used is known as the Fama- French three-factor model and it uses three factors to describe the portfolio and stock returns. Using their observations they constructed two additional variables to add to the original CAPM; Small Minus Big (SMB) and High Minus Low (HML). SMB measures the excess returns of small market capitalization stocks over their large capitalization counterparts. Similarly HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. Their regressions show that their three-factor model helps to explain over 90% of portfolio returns compared to using the capital asset pricing model which explains 77% of the same returns. Additionally in their 1996 paper they set out to test if their model could explain the previously mentioned anomalies in average stock returns. These included the reversal of long-term returns described by Debondt and Thaler and the short-run continuation described by Jegadeesh and Titman. The three-factor model was able to explain all of the patterns in average returns considered to be anomalies save for the short-term continuation pattern. In summation the three-factor model presented by Fama and French seems to quite accurately explain portfolio return patterns and while it could not explain the short-term continuation anomaly it is still the 6

8 best available model to approximate returns with. Carhart (1997) added a momentum factor into the three-factor model to better explain the persistence of a securities returns. The momentum factor measures the tendency for security returns to continue increasing if they were on the raise or conversely continue to decrease if it was on the decline. The results indicated that the momentum factor is able to explain away this variance in the returns. The literature review suggests that multi-factor asset pricing models are able to explain a considerable amount of movement in average returns and that these factors improve the understanding of these returns by adding factors to the original CAPM and its market premium. Investor sentiment The capital asset pricing model and multi-factor asset pricing models are valuable tools in assessing the movement of asset returns, however as the previously mentioned papers suggest there seems to be more factors affecting returns than only risk factors. Lakonishok, Shleifer and Vishny (1994) found that sub-optimal behavior by the average investor can be exploited to earn greater returns and that these returns in particular were not a result of taking riskier positions in the market. Russel and Thaler (1985) explain that rational behavior can only be observed in highly efficient markets, however the more efficient the market is the less discipline the market provides. This theoretically makes sense as the in more efficient markets there would be little to no need for discipline. This implies that in less efficient markets there is room for quasi rational behavior. On the other hand if the market is efficient there would be relatively little penalty which may also encourage irrational behavior. To understand how then these decisions, irrational or otherwise, are taken, a look must be taken into Tversky and Kahneman s prospect theory (1973 and 1979). This theory posits that people tend to take decisions based on the value of possible losses or gains rather than the outcome of the decision itself. Further complicating the issue they also find that these possible losses (gains) were evaluated using heuristics. These heuristics are used to more effectively make decisions under uncertainty however they also lead to biases and predictable errors. From the surveys done in De Bondt s paper (1998) it can be ascertained that investors observe naïve patterns in price movements, are not well diversified, they also shared popular models of evaluation and traded sub optimally. This paper should be taken with a grain of salt as it was mainly done with students 7

9 as the primary respondents. While De Bondt s paper does provide valuable insight the results may be biased as students may not yet have sufficient knowledge to make sound investment choices. Bradshaw (2002 and 2004) found that even analysts were not immune to the use of heuristics. His research showed that analysts used price-multiple heuristics and that their recommendations correlated more with heuristic valuations than they do with present value models. The research shows that investors themselves seem to be quasi-rational, or at the very least rely on heuristics to simplify the decision making process. Couple this with the behavior of analysts and how they also make use of heuristics one might assume that there are quite a few human errors resulting in asset mispricing where their market price may not accurately represent their true value. If heuristics and irrational behavior do indeed lead to predictable errors then it may also be possible to capture these errors in the form of factors much like what the multifactor asset pricing models do for various types of risks. However to model individual investor behavior would be an increasingly difficult and a near impossible task to complete. This is where the research tends to favor models where assumptions are made on investor behavior in aggregate. Jackson (2003) found that individual investor trades on the aggregate exhibit systematic patterns. He concludes that the assumption that investor decisions aggregate does seem to be plausible. Baker and Wurgler (2007) also take this aggregate view and find that investor sentiment does have an effect on individual firms and the stock market. Additionally they found that investor sentiment has a larger effect on securities that are difficult to evaluate. Their research shows that it is possible to use investor sentiment as a means of explaining stock return movement and that it can provide valuable information on the stock market in general. If an economic wants to explain the price movement of a product they would look at the supply and demand of said product. If the product is a stock than the suppliers are the firms of those stocks and the buyers are the investors. Often the emphasis is placed on the firm and its stock s characteristics while it may be beneficial to also look at the buyers and what they would want to hold as investors. 8

10 3. Data & Methodology This chapter will start with a brief overview of the data collected. Subsequently the methods used in developing the sentiment variables will be discussed along with some summary statistics and preliminary regression results. Data The dataset used contains the daily stock prices and trading volume of firms that form part of the S&P500 index. The dataset has a time period of thirty years with the earliest observations starting in January 1984 and ending with the final observations in January The index constituents database on Compustat was used to obtain the identifiers of the firms that were part of the S&P500 over the selected time period. These identifiers were then used to obtain the daily stock prices and trading volume of the firm s stocks. Daily returns were then calculated using the daily stock prices and these were then used to calculate monthly variances for every stock in the dataset. The final dataset including daily returns, after dropping missing variables, contained 1,048,148 observations. While this is an extensive dataset it should be noted that after creating the new variable there is only one monthly observation for the variable over the 30 year period which translates to 361 observations for the final variable. Methodology Before discussing the methods used to create the investor sentiment variable it would be prudent to have a brief review of the theoretical assumptions behind the investor sentiment concept. Taking a top-down view of the topic the first assumption taken is that investor sentiment can be broadly defined in two categories; optimism for the market and pessimism for the market. These investor sentiment categories are based on the perceived market conditions. If the market conditions are perceived to be positive then it is expected that investors are more optimistic and are more willing to take on riskier investments. Conversely if the perceived market condition is negative then the opposite is expected and investors will likely make use of safer investment strategies. If this is the case then it is expected that the volume traded in riskier stocks will increase, and consequently the volume traded in relatively safe stocks will decrease, if investor sentiment is optimistic and that the opposite will occur if a more pessimistic sentiment is 9

11 prevalent. These relatively straightforward assumptions are what govern the creation method of the investor sentiment variable. The investor sentiment variables With the previously mentioned assumptions in mind monthly variances were calculated for every stock in the dataset over the entire thirty-year period. These were then ranked every month from low to high and then split into deciles. The tenth decile then contained the riskiest stocks for that month while the first decile contained the least risky stocks for said month. Then the total volume traded in the stocks that belong in the tenth decile were summed to reflect the volume traded in the riskiest stocks for that month. This was also done for the stocks that belong in the first decile to reflect the volume traded in the safest stocks for that month. The total volume traded in the tenth decile stocks was then divided by the total volume traded in the first decile to create the investor sentiment variable. Theoretically when this ratio is large it indicates that the investor sentiment is optimistic as they are trading more in risky stocks and when the ratio is low this would indicate a pessimistic investor sentiment. This ratio was then normalized on a scale of 0 to 1 for ease of interpretation, in this case the closer the variable is to 1 the more optimistic the investor sentiment is and the opposite holds if the variable is close to 0. To summarize, the steps taken are as follows; 1. Calculate monthly variances 2. Rank the variances from low to high and split into deciles for every month in the dataset 3. Sum the total volume traded in the tenth decile of stocks (the riskiest). The same is done for the first decile of stocks (the least risky). 4. Divide the total volume traded in the tenth decile by the total volume traded in the first decile. For ease of interpretation this ratio is then normalized on a scale of 0 to 1. Figure 1 graphs the investor sentiment against time. This is done for an eye test of the variable to assess whether it is a suitable proxy for investor sentiment. Peaks and upward trends are indicative of an optimistic investor sentiment while dips and downward trends are indicative of a pessimistic investor sentiment. Notably the two largest concentrations of peaks are around the dot-com bubble period, 1991 to 2001, and the period before the 2008 crisis hit, Additionally even though it is not as clearly visible in the graph as the other incidents one can 10

12 1/1/1984 3/1/1985 5/1/1986 7/1/1987 9/1/ /1/1989 1/1/1991 3/1/1992 5/1/1993 7/1/1994 9/1/ /1/1996 1/1/1998 3/1/1999 5/1/2000 7/1/2001 9/1/ /1/2003 1/1/2005 3/1/2006 5/1/2007 7/1/2008 9/1/ /1/2010 1/1/2012 3/1/2013 also see the effects of the Black Monday crash; in the period around 1986 there is a visible peak leading to a sharp decline the following year. Figure 1: Monthly Investor Sentiment Investor Sentiment

13 In table 1 the initial regressions that were done with the excess returns of the 25 Size-BE/ME portfolios for the period to can be seen. This time period was used as the earliest available data points for the VIX index begin on The three- and four-factor models do an excellent job in explaining average excess returns as the intercept estimate in both models is economically small and statistically insignificant. Additionally in both models the market premium factor and the Fama-French factors are all highly significant. However the momentum factor is not significant with an average absolute t-statistic of Table 1: Three- and four-factor models, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. MOM measures a stocks tendency to continue rising or declining. All values in the table are averages from the 25 regressions. Dependent variable Coefficient Standard Error t Model 1 Model 2 M1 M2 M1 M2 M1 M2 25 Size-BE/ME ports Intercept Intercept EXM EXM *** *** SMB SMB *** *** HML HML *** *** MOM * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. Tables 2 and 3 contain the three-and four factor model regressions including the IS variable and the popular sentiment index, VIX, respectively. The regression outputs show that both the VIX and IS variables do not add any explanatory power to the either model. The intercept estimates stay statistically insignificant in all cases while the changes in the other factors are minimal. The IS variable performs only slightly better as it has the larger t-statistic in both models. 12

14 Table 2: Three- and four-factor models including IS, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. MOM measures a stocks tendency to continue rising or declining. Finally IS is the investor sentiment variable described in this paper. All values in the table are averages from the 25 regressions. Dependent variable Coefficient Standard Error t Model 1 Model 2 M1 M2 M1 M2 M1 M2 25 Size-BE/ME ports Intercept Intercept EXM EXM *** *** SMB SMB *** *** HML HML *** *** IS MOM IS * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. Table 3: Three- and four-factor models including VIX, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. MOM measures a stocks tendency to continue rising or declining. Finally VIX is the investor sentiment index popularly used by analysts. All values in the table are averages from the 25 regressions. Dependent variable Coefficient Standard Error t Model 1 Model 2 M1 M2 M1 M2 M1 M2 25 Size-BE/ME ports Intercept Intercept EXM EXM *** *** SMB SMB *** *** HML HML *** *** VIX MOM E VIX 2.93E * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. 13

15 Table 4 contains the Gibbons Ross and Shanken (1989) (GRS) tests for all the previously mentioned specifications. The GRS-test null hypothesis states that all intercepts in a multivariate regression are jointly zero. The results show that in the three-and four-factor models the null hypothesis is rejected. However when the IS variable is added to either model it manages to explain a sufficient amount of the remaining variance in average returns that these models fail to reject the GRS-test null. Additionally the models including the VIX index also fail to reject the null hypothesis. While both sentiment factors seem to be statistically insignificant, they do add some value as they allow for the failed rejection of the GRS-test null hypothesis. It is difficult to say which sentiment factor is better than the other as they both perform similarly in the regressions with the IS variable having a slightly higher t-statistic. However both factors remain statistically insignificant. It may be only a matter of preference which factor is used in showcasing investor sentiment. Note that the IS variable is recreated relatively easier than the VIX index as it mainly has stock data as its main components. This also allows for recreation of the IS variable using older stock data. The IS variable is also easier to interpret than the VIX index. Table 4: GRS-test results, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. VIX is used as a sentiment variable in this regression. MOM measures a stocks tendency to continue rising or declining. Finally IS is the investor sentiment variable described in this paper. GRS t- stat GRS P>t Ave. interc. Interc. t- stat 4 Factor Factor IS Factor VIX Factor Factor IS Factor VIX

16 The reason for discussing these results in this section is that these preliminary regressions have led to improvements in creating proxies for investor sentiment. The key takeaway from these preliminary results is that when the IS variable is added to the three- and four-factor model it allows for the failed rejection of the GRS null hypothesis. The IS variable in its current form was made using the variance of stock prices and not the variance of stock returns. Using return variances is more practical however using price variances created a variable that graphs investor sentiment quite accurately when compared to crises or bubble periods. The logic behind using price variances in this instance is that while returns give values that are comparable for various stocks, prices are more relevant when determining investor sentiment. Looking at returns per period is objective and the same for every investor however this paper s focus is on investor sentiment. Return per individual investor can vary depending on their holding period so it stands to reason that investors will react more on price movement then they would returns. Additionally it seems that based on the literature review, many researchers create portfolio strategies and calculate the returns of these strategies to create new variables or factors. The IS variable used calculated the volume traded in the portfolios of the tenth and first deciles but not their return. However this could not be easily done in one variable as the two key components of the IS variable is the stock variances and volume traded. Based on these results three new variables were created by making simple adjustments to the original. Market sentiment (MarS) - this variable is the same as the original investor sentiment variable with one small adjustment. Return variances are used to rank the portfolios from low to high for every month in the dataset. The steps in creating this variable are as follows: 1. Calculate monthly return variances. 2. For every month in the dataset, rank the return variances from low to high and split into deciles. 3. Sum the total volume traded in the tenth decile of stocks (the riskiest). The same is done for the first decile of stocks (the least risky). 4. Divide the total volume traded in the tenth decile by the total volume traded in the first decile. For ease of interpretation this ratio is then normalized on a scale of 0 to 1. 15

17 Volatility sentiment (VolaS) - The same steps in creating the original variable are followed with the only difference being that after creating the deciles, returns for the tenth and first deciles are summed instead of volume traded. The VolaS variable measures the excess returns obtained by holding a portfolio with high volatility over a portfolio that is less risky. This should measure the viability of holding volatile stocks; if this variable is positive and large then the market may be more optimistic toward holding volatile stocks relative to safe stocks. The following steps were taken to create this variable: 1. Calculate monthly return variances. 2. For every month in the dataset, rank the return variances from low to high and split into deciles. 3. Sum the total return one would obtain by holding the tenth decile stocks. The same is done for the first decile of stocks. 4. Subtract the total returns of the tenth decile stocks from the total returns of the first decile stocks. Demand sentiment (DemaS) - Instead of ranking return variances, the volume traded in stocks are ranked from low to high for every month in the dataset. Subsequently these are split into deciles and again the total returns obtained from the tenth and first decile are calculated and subtracted from each other. Essentially DemaS measures the excess return of holding popularly traded stocks over holding those stocks that are least traded on the market as such this variable should indicate which type of stocks are in demand by the general market. This variable was created as follows: 1. For every month, rank the volume traded in each stock from low to high and split into deciles. 2. Sum the total return one would obtain by holding the tenth decile stocks. The same is done for the first decile of stocks. 3. Subtract the total returns of the tenth decile stocks from the total returns of the first decile stocks. 16

18 The demand sentiment and volatility sentiment variables together should theoretically reflect what the original investor sentiment variable was trying to capture. This was done as translating the original theory into one investment strategy to create only one variable proved to be difficult. Finally, the original IS (with the minor adjustment) was kept as it seemed to be the addition to the models that led to the failure of rejecting the GRS null hypothesis. Table 5 presents the summary statistics of 2 of the new variables namely; Demand Sentiment and Volatility Sentiment. The market sentiment variable was excluded as the summary statistics do not add any valuable insight as it is a normalized variable. The demand sentiment variable suggests that holding popularly traded stocks seem to net an average return of 38% percent over that of the least traded stocks while holding volatile stocks earn an average return of 18% over that of their less risky counterparts. It would seem that the overall market was generally optimistic during the selected period trading in riskier stocks as well as in demand stocks. This is not a surprising result considering the research window covers mostly prosperous periods in investment excluding the dot-com bubble and crisis. The effect of which would be less noticeable as one event was industry specific and the other was at the tail end of the research window. Table 5: Summary Statistics. DemaS is the demand sentiment variable and measures the excess returns of the most traded portfolio over the least traded portfolio. VolaS is the volatility sentiment variable and measures the excess returns of the most volatile portfolio over that of the least volatile portfolio. Variable Observations Mean Std. Dev. Minimum Maximum Demand Sentiment Volatility sentiment

19 4. Results Before discussing the results on the regressions with the variables presented in this paper it may prove useful to observe how well the four-factor and three-factor models do in explaining the 25 size-be/me portfolios of excess returns over the entire 30 year period. Tables 6 and 7 contain the average results over the 25 regressions, it is clear that the four-factor and three-factor model do extraordinarily well in explaining the excess returns of the 25 portfolios. In the four-factor model the intercept is economically and statistically insignificant with a value of and a t-statistic of The small intercept indicates that the model captures most of the variation in average returns of the portfolios. Additionally the GRS-test statistic is 3.6 which rejects the null hypothesis that the intercepts of the 25 regressions are jointly zero. Fama and French (1996) suggest that this is testament to the explanatory power of their regressions, however this may be a matter of perspective as failure to reject the null hypothesis would objectively be more favorable. Additionally all of the factors with the exception of the momentum factor are significant. The momentum factor in this research seems to add little explanatory power to the model. This may be due to the research window of this study as it is possible that the momentum factor is less effective in recent years. The momentum factor can be attributed to cognitive biases by irrational investors; they may be slow or unwilling to react to new information on securities causing the momentum factor (Danial, Hirshleifer and Subrahmanyam, 1998; Barberis, Shleifer and Vishny, 1998). It is quite possible that as these types of research were better studied and understood by the general public that this phenomenon slowly regressed to the point where it is no longer a significant factor in explaining asset returns. This is supported by the results in table 1 which uses a more recent time period to test the four-factor model. However, testing for preand post-crisis periods; the momentum factor is significant between the 5% and 10% levels for periods before the crisis and becomes insignificant afterwards 1. Note that the post crisis window selected here runs from to and is a relatively short testing period. It is quite possible that the crisis caused the general public to be more aware of new information surrounding securities to the point where the momentum factor is no longer significant. However one must also refrain from drawing such a conclusion based on a small testing window. 1 The pre- and post-crisis test results can be seen in tables 1 and 2 in the appendix. 18

20 Table 6: Four-factor model, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. MOM measures a stocks tendency to continue rising or declining. All values in the table are averages from the 25 regressions. Obs Parms RMSE R-sq F Dependent variable Coeff. St. Err. t 25 Size-BE/ME ports Intercept EXM *** SMB *** HML *** MOM * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. Examining the three-factor model in table 7 we again see similar results. The intercept has an absolute estimate of and is statistically insignificant. Judging by the intercept estimate, the three-factor model performs better than the four-factor model in explaining the movement of average returns. In either case both models do exceptionally well leaving little room for improvement. 19

21 Table 7: Three-factor model, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. All values in the table are averages from the 25 regressions. Obs Parms RMSE R-sq F Dependent variable Coeff. St. Err. t 25 Size-BE/ME ports Intercept EXM *** SMB *** HML *** * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. All things considered it seems that the three- and four-factor models still do an exceptional job at explaining the movement in asset returns. Whether measures for investment sentiment can do the same is another question. Table 8 shows regressions with the market premium factor and various combinations of the three variables presented in this paper. The results show that in all models the intercept is highly insignificant as well as being economically insignificant with the highest absolute estimate of the intercept being Unfortunately it seems that in all models there was failure in rejecting GRS-test null hypothesis despite the inclusion of the market sentiment variable which was expected to cause the failed rejection of the null. It should be noted that while these models produced insignificant intercepts the variables themselves seem to be insignificant as well. Although not shown here, the variable with the highest t-value across all models was the VolaS variable. The only consistently significant variable in all the models was the market premium variable. None the less this test showed the viability of the newly created variables. 20

22 Table 8: Regression alphas and t-statistics, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. DemaS is the demand sentiment variable and measures the excess returns of the most traded portfolio over the least traded portfolio. VolaS is the volatility sentiment variable and measures the excess returns of the most volatile portfolio over that of the least volatile portfolio. MarS is the market sentiment variable created by using return variances as described in the methodology section. All values except for the GRS-test values are averages from the 25 regressions. Model Intercept t-value GRS-t GRS-p EXM DemaS E-10 EXM VolaS E-13 EXM MarS EXM DemaS VolaS E-12 EXM VolaS MarS E-05 EXM DemaS MarS -9.5E EXM DemaS VolaS MarS E-05 The MarS variable performed poorly across all regressions, the original IS variable was added to the three-and four-factor model regressions seeing as the IS variable at least allowed for the failure to reject the GRS-null hypothesis in the 1994 to 2014 regressions. The results can be seen in table 9 and are similar to the earlier regressions. Table 10 contains the results of the GRS-test of the three-and four-factor models including and excluding the IS variable. The GRS-test results show that only in the four-factor model, that includes the IS, there is a failure to reject the null hypothesis. The results indicate that the original creation method for the IS variable seems to be preferable over that of the MarS variable. In other words using price variances to create the sentiment variable provides better results than using return variances. Note that the test on a longer time period was not done with the VIX index as there is no available data for the VIX prior to 1994 as well as it being difficult to replicate in the periods pre

23 Table 9: Three- and four-factor models including IS, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. MOM measures a stocks tendency to continue rising or declining. Finally IS is the investor sentiment variable described in this paper. All values in the table are averages from the 25 regressions. Dependent variable Coefficient Standard Error t Model 1 Model 2 M1 M2 M1 M2 M1 M2 25 Size- BE/ME ports Intercept Intercept EXM EXM *** *** SMB SMB *** *** HML HML *** *** IS MOM IS * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. Table 10: GRS-test results, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. MOM measures a stocks tendency to continue rising or declining. Finally IS is the investor sentiment variable described in this paper. GRS t-stat GRS P>t Ave. interc. Interc. t-stat 4 Factor Factor IS Factor Factor IS

24 Table 11 shows the regression results of the DemaS and VolaS variables on the excess returns of the 25 size-be/me portfolios. Both variables are significant and do well in explaining the variance in average returns as the absolute intercept estimate is and statistically insignificant. However this result was achieved by excluding the market premium factor which; suggests that these variables proxy for what the market premium measures. To further test this table 12 contains the three-factor model including the most statistically significant variable DemaS. Indeed the results show that while the intercept estimate remains small and statistically insignificant, the DemaS variable becomes insignificant. Table 11: Sentiment model regressions, to The dependent variables are the 25 size-be/me portfolios of excess returns. This model excludes the EXM variable. DemaS is the demand sentiment variable and measures the excess returns of the most traded portfolio over the least traded portfolio. VolaS is the volatility sentiment variable and measures the excess returns of the most volatile portfolio over that of the least volatile portfolio. All values in the table are averages from the 25 regressions. Obs Parms RMSE R-sq F Dependent variable Coeff. St. Err. t 25 Size-BE/ME ports Intercept DemaS *** VolaS *** * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. 23

25 Table 12: Three-factor model including DemaS, to The dependent variables are the 25 size-be/me portfolios of excess returns. EXM is the excess of the market portfolio over the risk free rate. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. DemaS is the demand sentiment variable and measures the excess returns of the most traded portfolio over the least traded portfolio. All values in the table are averages from the 25 regressions. Obs Parms RMSE R-sq F Dependent variable Coeff. St. Err. t 25 Size-BE/ME ports Intercept EXM *** SMB *** HML *** DemaS * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. Finally table 13 contains a model with only the Fama-French (FF) factors and one with the FF factors and the DemaS variable 2. The output indicates that the FF factors explain variation in excess returns better in cohesion with the market premium factor. Alone these factors produce statistically significant intercept estimates. Additionally it seems that adding the DemaS variable reduces the explanatory power of the HML variable as well as produces statistically insignificant intercepts. Similar results are achieved using specifications with the FF and Carhart factors 3.This along with the results in table 12 suggests that the DemaS variable captures variations for what HML proxies for, however DemaS seems to be correlated with the market premium factor as it becomes statistically insignificant when it is included. 2 The results of the specifications including the VolaS factor can be seen in table 3 in the Appendix. 3 The results of these specifications can be seen in table 4 in the appendix. 24

26 Table 13: Regressions with SMB, HML and Demas; to The dependent variables are the 25 size-be/me portfolios of excess returns. SMB measures the excess returns of small cap firms over big cap firms. HML measures the excess returns of low price-to-book ratio stocks of that of high price-to-book ratio stocks. DemaS is the demand sentiment variable and measures the excess returns of the most traded portfolio over the least traded portfolio. All values in the table are averages from the 25 regressions. Dependent variable Coefficient Standard Error t Model 1 Model 2 M1 M2 M1 M2 M1 M2 25 Size- BE/ME ports *** Intercept Intercept SMB SMB *** *** HML HML DemaS *** * = Significant at the 10% level. ** = Significant at the 5% level. *** = Significant at the 1% level. In summation; multiple tests were conducted using various combinations of the sentiment variables 4. In the end the sentiment model presented in table 11 proved to be the most effective in explaining the movement of average excess returns of the 25 size-be/me portfolios. Additionally the DemaS variable seems to reduce the explanatory power of the HML factor however it seems that the DemaS variable is also correlated with the market premium factor. The three-factor model is still the best explanatory asset pricing model out of all the specifications. The sentiment variables created here did produce interesting results and alone were able to produce economically small and statistically insignificant intercept estimates however they seem to also be proxies for the market premium factor. Finally the momentum factor seems to have little explanatory power in all of the specifications that were run except for the pre-crisis specification with a time period between and Tables 6 and 7 in the appendix contains the results of specifications with all the FF/Carhart factors including the sentiment factors. 25