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Investor sentiment and volatility; Does it solve

the idiosyncratic volatility puzzle?

Abstract: This paper extends on the current literature regarding the negative relation

between idiosyncratic volatility and stock returns. Previous literature has proposed several other measures to evaluate the problem, but the idiosyncratic volatility puzzle still remains unsolved. This paper proposes adding investor sentiment as a factor to explain the

discrepancy of increasing idiosyncratic volatility and lower returns. Using monthly portfolio data ranging from July 1965 to September 2015 we show that high idiosyncratic portfolios show significantly lower (higher) returns during bullish (bearish) markets. Adjusting for economic drivers shows even more significant results. We also show that the investment factor suggested by Fama and French is irrelevant as it performs worse than the original three factor model. The results hold when accounting for total variance.

Supervisor: E. Eiling

Name: Tom Cornelissen

Student number: 10608397

MSc: Finance

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Statement of Originality

This document is written by Tom Cornelissen who declares to take full responsibility for the contents of this document.

I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.

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Table of Content

1. Introduction ... 4

2. Literature Review ... 6

2.1 The existing problem ... 6

2.2 The positive relationship ... 7

2.3 The negative relationship ... 8

2.4 How to solve the puzzle ... 9

2.5 Research question and hypotheses ... 11

2.5.1 Research question ... 11

2.5.2 Hypotheses ... 12

2.5.3 Research contribution ... 12

3. Methodology ... 14

3.1 CAPM ... 14

3.2 Fama French Three factor model ... 14

3.3 Fama French Five factor model ... 15

3.4 Sentiment Measurement ... 15

3.5 Dataset ... 18

4. Results and discussion ... 19

4.1 Descriptive statistics dataset ... 19

4.2 Beta exposures ... 20

4.2.1 CAPM estimates ... 20

4.2.2 Fama and French three factor model ... 21

4.2.3 Fama and French five factor model ... 23

4.2.4 Fama and French 3 factor model with separate additions ... 25

4.3 The factor premiums ... 26

4.3.1 Pricing premiums among existing models ... 26

4.3.2 Sentiment risk premiums ... 27

4.4 Idiosyncratic volatility and the Sentiment premium ... 29

4.4.1 CAPM and sentiment ... 29

4.4.2 The Three factor model ... 30

4.4.3 The five-factor model ... 31

4.4.4 The four factor models ... 32

4.5 Robustness check ... 33 5. Conclusion ... 34 6. Reference list ... 37 7. Appendix ... 40 Robustness Check ... 53 Codes used ... 59

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1. Introduction

The volatility of a stock consists of two elements; the systematic and the idiosyncratic part. The former is the volatility due to the market risk varying over time and the latter reflects the volatility that’s specific to the individual asset. Karolyi and Stulz (2002) shows that the market factors are determined internationally, whereas Campbell, Lettau, Malkiel and Xu (2001) find that systematic volatility is fairly stable in their 1962-1997 dataset. Due to these findings, systematic volatility is a lesser complicated measure. Also, the consistency and the still fluctuating volatility shows us systematic is less important compared to idiosyncratic

volatility. Therefore, this paper will focus on explaining what explains idiosyncratic volatility. Previous research on the topic shows the idiosyncratic volatility puzzle still exists and remains an unsolved question within finance literature. Research regarding the topic is mixed as conclusions show mixed signs. Several papers show a positive relationship between stock returns and the volatility measure (see; Merton (1987); Brockman, Schutte and Yu (2007); Fu (2009)), whereas others find a negative relation between the two (see; Ang, Hodrick, Xing and Zhang (2006, 2013); Guo and Savickas (2006)) or even no relation at all (Bali and Cakici, 2008). Due to these mixed findings, the idiosyncratic volatility puzzle still is an interesting topic within asset pricing. When reflecting on the developments in the idiosyncratic volatility, Campbell et al. (2001) conclude that the volatility is upward trending in the years nearing 2000. This again illustrates the potential of accounting for idiosyncratic risk when estimating stock returns.

While many papers attempt to solve the idiosyncratic risk puzzle, they all differ in the methodology and datasets used. This makes it difficult to compare the results which is also concluded by Hou and Loh (2016) when reviewing the idiosyncratic risk puzzle. Therefore, they focus their research on evaluating the existing explanations. They find that while improving the explanatory power of the established models explaining the idiosyncratic volatility puzzle, a large portion is still left unexplained.

This paper will evaluate whether idiosyncratic volatility affect the stock returns. It will also look at possible explanations for the mixed results found in previous research by reflecting on several measures to help explain the idiosyncratic volatility effect on stock returns. I also add another measure, investors’ sentiment to improve upon the existing current literature. This suggested measure incorporates many aspects noted in previous literature and bundles them in an insightful measure to track willingness to invest. Baker and Wurgler (2006) show that sentiment is an important measure and although at times irrational due to its link with emotions, they still find value in applying it as a measure to help explain stock returns.

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Since sentiment is based on emotions, which would suggest sentiment is irrational, we will have to review the literature regarding sentiment to help make an initially irrational factor rational. To resolve this issue, factors related to sentiment yet graspable and rational should be used. Factors which are used in forecasting reversions, predict overvaluation and reflect the industry’s confidence in the market. These factors have been shown to reflect sentiment (as shown by Zweig (1973), Baker and Stein (2004), Stigler (1964), Baker and Wurgler (2000, 2004) and Ritter (1991)) and by adjusting them for overall economic drivers such as consumption, production and unemployment the rationality behind market upswings (and downturns) has been accounted for, while the factors themselves still account for investor’s confidence (or lack thereof).

In this paper I show that sentiment plays an important part in explaining the lower returns found among high idiosyncratic volatility portfolios. Similar to Ang et al (2009) we find that the highest volatility portfolios perform worse than the lower ones and show that this is mostly due to their exposure to investor sentiment. Stocks with high volatility show negative exposures to sentiment whereas low volatility stocks have a positive exposure to investors’ sentiment. We show that sentiment is a priced factor in addition to the three-factor model and remains priced when adding the Investment factor. This paper will also provide proof that the Profitability factor solely accounts for the additional explanatory power of the five-factor model. Due to sentiments positive premium, we show that, with exception of the highest level of idiosyncratic volatility, all portfolios show a compensation for the exposure to investor’s sentiment.

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2. Literature Review 2.1 The existing problem

This section will reflect on the previous literature regarding volatility, idiosyncratic volatility and underlying stock returns. Following this, findings showing a positive relation between idiosyncratic volatility and stock will be reviewed after which the negative relationship

between these two topics is evaluated. Since the results are mixed, the idiosyncratic volatility puzzle remains an interesting topic in research. Different approaches to solving the puzzle are evaluated before we state why further analysis on the topic is needed. Concluding I will evaluate the negative relation by looking at investor sentiment and the idiosyncratic volatility sorted stock returns to show the importance of this research.

The basis for important stock return drivers come from well-established financial models like the Capital Asset Pricing Model (CAPM) (Sharpe (1964) and Lintner (1965)) and the Fama and French three-factor model. These models help explain the drivers for stock returns, by adding factors based on theoretical assumptions. However, the general application of solely these factors does not account for the real world in which many more external factors influence individual stock returns. Therefore, research on the subject is continuously expanding the existing literature on stock price drivers by changing the factors used in asset pricing models. Exemplar research by Merton (1987) states that small-scale investors find more hardship in diversifying their portfolios based on the established influential factors driving returns due to lack of public available information and the lack of resources to diversify on a large enough scale to benefit from these factors. For the larger and more established firms, the data required for diversification based on these factors is available to the public. However, Merton (1987) suggest that this is not the case for small-cap stocks in which ownership tends to be concentrated to a smaller group of investors. Therefore, general information is less accessible and obtaining the required information for these stocks tends to be more difficult. In turn, this increases the costs associated with investing, especially for smaller scaled investors.

Merton (1987) suggests that firms with higher idiosyncratic (firm specific) risk require higher returns as a compensation for the costs associated with both the risk itself and the costs of determining the risk. Firms with easy accessible information allow investors to provide better price estimates of the stock and also attract more investors to the stock. Due to the higher number of owners and the multiple price assessments these stocks tend to trade within certain ranges of the perceived worth of the underlying company.

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Therefore, large fluctuation is less likely and the idiosyncratic risk will be lower which is reflected by the volatility of the stock. The share price will stabilize due to the multiple parties with the same price estimate ranges. Thus, idiosyncratic volatility is deemed a sufficient measure for estimating a firm’s idiosyncratic risk. The research performed by Merton (1987) recommends including idiosyncratic volatility within the framework of analysing asset pricing factors. More recent attempts in analysing idiosyncratic volatility show mixed results and increase the importance of finding a plausible explanation for the puzzle.

2.2 The positive relationship

Further research into the research topic provides us with evidence of a potential positive relationship between returns and idiosyncratic volatility, as Merton’s initial theory suggested. Within his research, Merton (1987) predicts that the effect of idiosyncratic volatility is

positively related to the average stock returns, which is proven by Malkiel and Xu (2002). Malkiel and Xu’s (2002) research provides evidence of the existence of a positive relation between the two variables by examining the stocks in the S&P 500 between 1963 and 1994. Apart from the positive relation between idiosyncratic volatility and average returns, they also provide evidence of a negative relation between a firm’s capitalization size and its

idiosyncratic volatility. The significance of the idiosyncratic volatility effect was also higher than the effects of market returns, book-to-market value and size (Malkiel and Xu, 2002). Additional research by Spiegel and Wang (2005) also provides evidence that higher idiosyncratic volatile portfolios provide higher returns than the lower idiosyncratic volatile portfolios.

However, more recently, Fu (2009) states that the idiosyncratic volatility is time-varying and therefore the findings of Ang et al. (2006) cannot be used to determine the relationship between idiosyncratic risk and expected returns. Subsequently Fu (2009) finds a positive relationship between estimated volatility and stock returns, therefore giving strength to the previously established theorem founded by Merton (1987). In the paper written by Brockman, Schutte and Yu (2007), the relationship is also confirmed. Their research employs the same methodology as Fu (2009) and confirms the positive relationship to be present between 1980 and 2007.

Based on these papers, the original theory of higher risks resulting in higher rewards is established, in this case higher idiosyncratic rewards resulting in higher rewards and vice versa. By accounting for this risk portfolio managers could adjust their holdings according to their clients’ risk aversion.

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2.3 The negative relationship

Merton’s initial theoretical frame has been examined and proven by several other

researchers, as aforementioned. However, more recently the positive relationship has been questioned and several papers show an opposite relationship between idiosyncratic volatility and the returns of stocks. The most established paper on this topic is written by Ang,

Hodrick, Xing and Zhang (2006) showing a significant negative relation between

idiosyncratic volatility and subsequent stock returns. Ang et al. (2006) show that the realized volatility acts as a proxy for expected volatility. They add that their robust results show that the highest quintile of idiosyncratic volatility have significantly lower total returns, an effect that is persistent in both bull and bear markets. Due to Ang et al. (2006) and their findings the original theory on idiosyncratic volatility has been debated and the basis for the idiosyncratic volatility puzzle was formed.

Guo and Savickas (2006) also find a significant negative relationship between idiosyncratic risk and stock returns after accounting for systematic risk as well. Their research also suggests that idiosyncratic volatility performs as well as the book-to-market value in explaining stock returns. In combination with the findings of Malkiel and Xu (2002), the idiosyncratic volatility factor will most likely perform as well if not better than the book-to-market factor. This shows the potential of the volatility measure in general but also raises the question whether the relationship between stock returns and volatility really is positive as previous work suggests.

More recent work from Ang et al. (2009) shows evidence that a negative relationship can also be found in international markets. This research shows an even larger difference between both extremes in the idiosyncratic volatility portfolios of -1.31% on average while accounting for the world market, size and value (the Fama and French three-factor model). The paper also mentions that the idiosyncratic volatility follows more generic factors, such as productivity, consumption and unemployment which results in a strong co-movement

between volatility groups.

Finally, Hou and Loh (2016) also find evidence for a negative relationship between the two variables. While evaluating the time period between 1963 and 2012, they expand on the time frame used by Malkiel and Xu (2002) but they do find the opposite relationship of the Malkiel and Xu paper. This difference is partly due to the different applied

methodologies, but these contradicting findings prove evidence the idiosyncratic volatility puzzle still exists and remains relevant to this date. Therefore, the actual effect of volatility on stock returns is still an issue in asset pricing.

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2.4 How to solve the puzzle

Regarding the attribution of Ang et al. (2006, 2009), Clarke, de Silva and Thorley (2013) state that the anomaly found could also be due to the systematic risk and therefore the relationship between idiosyncratic volatility and the expected stock returns remains unclear. Therefore, other approaches attempting to explain idiosyncratic volatility or risk are

evaluated in this part of the literature.

Another paper, written by Hou and Loh (2016), attempts to solve the idiosyncratic volatility puzzle by evaluating the results and methodology of previous papers which found a negative relation effect of volatility on stock returns. Their research is based on the findings of previous papers but it provides a good summary of explanatory variables used in the idiosyncratic puzzle. Within their approach they use groups of variables to explain the puzzle such as the characteristics of stocks picked by investors (so called investors’ lottery

preference) and market frictions. Their results show that the main explanatory friction is liquidity. Another paper that provides evidence of the importance of this friction is the Spiegel and Wang (2005) paper. They provide evidence that the liquidity found in the highest

volatility portfolios is significantly lower compared to the portfolios with less idiosyncratic volatility. Therefore, explaining idiosyncratic volatility using liquidity is applied by Hou and Loh (2016). However, this still doesn’t fully grasp the negative relationship as it leaves 45.5-71% left unexplained.

Moreover, Stambaugh, Yuan and Yu (2015) show that negative effects caused by idiosyncratic volatility are attributable to higher investor sentiment. While reviewing their data, they find that in times of high (low) sentiment the high and low idiosyncratic volatility stocks range between -1.06% and 0.26% (-0.33% and -0.10%). When the sentiment is high (low), the difference between the categories is -1.32% (-0.23%). The intuition behind these findings can be summarized as follows; High sentiment leads to more differences in

performance of stocks at different idiosyncratic volatility levels. This effect still holds when removing small stocks, which in general have more idiosyncratic volatility. The aim of this paper also takes into account the other measures as these factors explain idiosyncratic volatility albeit partially.

Baker and Wurgler (2006a) show that although traditional finance models do not apply investor sentiment to explain stock returns, there’s a solid base for arguing the opposite and to include such a measure in asset pricing. Their results show high volatile stocks have higher returns when investor sentiment is low and when sentiment is high the more volatile stock perform worse compared to their lesser volatile counterparts.

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Liquidity for example has proven to be related to investors’ sentiment (Baker and Stein, 2004). However, this research depends on requirements which are not met in reality, such as the non-existence of short sales and limited variety in the types of agents. Since liquidity is both linked to volatility and sentiment, I deem it necessary to evaluate the effect of sentiment itself as well. Regarding the link between idiosyncratic volatility and sentiment, Baker and Wurgler (2006b) show that high volatility stocks tend to be sensitive to sentiment developments. This is mainly attributed to the fact that these stocks are new and have not yet been proven recession resistant.

There have been several variables that are applied to explain idiosyncratic risk; liquidity, human capital and sentiment. Spiegel and Wang (2005) find that stocks with low liquidity show higher idiosyncratic risk in turn resulting in a more volatile stock price. They state that idiosyncratic risk compensates investors for their exposure to the risk, but lower liquidity decreases, or cancels out the effect. Others such as Eiling (2013) link the

idiosyncratic risk to other factors. This paper shows that the idiosyncratic risk appears to be priced when not taking into account the human capital within the overall benchmark. The paper finds that including industry-specific human capital removes the effect idiosyncratic risk has on the cross-section of returns.

In addition to the suggested factors, I deem idiosyncratic risk to be illustrated by a stocks idiosyncratic volatility. In order to reaffirm existing theories, I will evaluate the different findings by analysing a broad dataset. In order to explain parts of what causes differences among volatility sorted stock returns, I suggest adding another broadly ranging factor to cover general market statistics. Ang et al. (2009) mentioned that the co-movement between sorted idiosyncratic portfolios are largely due to generic drivers, as mentioned before.

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2.5 Research question and hypotheses 2.5.1 Research question

In this paper, I will first establish whether idiosyncratic volatility should be priced and how it affects stock returns. This analysis will look at the exposures to several factors and comment on differences within them among several portfolios. Following up on these findings, I will reflect on whether Sentiment should be a priced factor, and whether it bears importance for investors to account for this type of risk. Furthermore, I will then establish whether

idiosyncratic volatility can be explained by several factors.

To help solve the idiosyncratic volatility puzzle, a measure of overall market sentiment will be used to check whether the unexplained inconclusive findings. As mentioned, the addition of sentiment shows promise in explaining negative returns found among several datasets. The applicable measure for sentiment has already been

established by Baker and Wurgler (2006a) and accounts for components of the overall economy, such as consumption, production and employment, while still providing insight to the stock return performance and investor’s confidence in the stock market. Based on the combination of findings of Hou and Loh (2016) and Stambaugh, Yuan and Yu (2015) we expect high (low) volatility portfolios to have a negative (positive) exposure to sentiment. If this is the case, and the risk premium for sentiment is positive it would help explain the anomaly shown by Ang et al.

The existing literature on the topic shows aspects and the overall importance of such a measure to help dissect the idiosyncratic volatility puzzle. According to Dumas, Kurshev and Uppal (2009), there are two types of agents; overconfident and rational investors. Their results show that overconfident investors overreact to public signals, which results in extreme volatility. These investors react to current sentiment to make their decisions and thus also are a potential factor influencing idiosyncratic volatility. Lee, Liang and Indro (2002) also show that bullish (bearish) changes in sentiment lead to downward (upward)

adjustments of volatility measures. Since the overall volatility is affected by two factors, the benchmark should be included in the analysis as it will account for the systematic volatility in general.

Using previous literature, factors such as market risk, size, value, profitability and investment sorted portfolio return differences will be added to test the strength of accounting for overall investor sentiment. This will allow for statements to be made regarding sentiment and its different effect on increasing levels of idiosyncratic volatility levels. This enables us to state whether sentiment affects the idiosyncratic volatility puzzle found by Ang et al (2006).

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Logically, the research question in light of these statements and findings is; Does investor sentiment help explain the idiosyncratic volatility puzzle?

2.5.2 Hypotheses

In order to answer this question, the following hypotheses will be tested in order to help solve the overall question. Based on the reviewed literature, the following statements are expected to reflect the reality in the best possible way:

1. High idiosyncratic stocks have a negative exposure to sentiment

2. All stocks face exposure to sentiment and the associated risk should be priced

Based on the previous literature, we assume to find high volatility level stocks to show negative exposure to investor sentiment. This would confirm the findings of Stambaugh, Yuan and Yu (2015) and will confirm the process in establishing the link to the negative anomaly found by Ang et al (2006). Stocks with high volatility that can be specifically linked to the stock, i.e. idiosyncratic volatility, show lower returns due to investor sentiment

(Stambaugh, Yuan and Yu, 2016). This leads us to assume that the exposure among the higher volatility levels is negatively related to investor sentiment.

The second hypothesis is based on classical financial theory, in which holding risky assets results in higher expected compensation. Therefore, I expect that a higher perceived risk should compensate the investor for the additional risk. Investors are faced with multiple opportunities to invest. Investors who are willing to take on more risky assets require sufficient compensation for their exposure to these risks. Therefore, stocks with higher idiosyncratic volatility should provide higher returns to properly compensate investors. If these high idiosyncratic volatility stocks do not provide these returns, they will be faced with lower demand, resulting in lower prices which in turn provides the higher returns if the value of the company remained constant.

2.5.3 Research contribution

While reviewing literature (Hou and Loh (2016)) explaining idiosyncratic volatility, we find several stock-specific factors that help explain, albeit partly, the existence of a negative relationship between idiosyncratic volatility and stock returns. Investors sentiment is one determinant (Baker and Wurgler, 2006a) that helps drive investors to take more risk, and for that reason should be included when attempting to solve the puzzle. To illustrate the

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a good driver in predicting stock returns due to its effect on idiosyncratic volatility. Cuong and Ishaq (2015) show that idiosyncratic volatility and its influence on stock returns significantly relies on investors sentiment. Based on these findings it is likely that adding a measure for this factor can help explain the unexplained part of the idiosyncratic volatility puzzle.

The previously mentioned research shows the importance of sentiment on stock returns. However, the sentiment measure has never been directly implemented with regards to the idiosyncratic volatility puzzle. The most elaborate paper written by Hou and Loh (2016) shows evidence that including market friction and lottery preferences helps explain part of the found puzzle. However, by reflecting solely on frictions related to stock specific

characteristics, the paper does not account for any outside factors. We argue that stocks are liable to change due to a number of non-stock related factors, which are not accounted for in the Hou and Loh paper. Ang et al. (2009) reaffirm the doubts regarding the Hou and Loh approach by mentioning that high idiosyncratic volatility stocks show low returns even after accounting for firm specific factors and go as far as suggesting that the observed effect is a worldwide phenomenal. Therefore, the addition of a broader factor incorporating both internal and external factors could prove to help solve the puzzle. This is why the sentiment measure is the main focus for this research.

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3. Methodology 3.1 CAPM

Before starting the analysis in order to reflect on the research question, the paper will briefly explain the models used and their relevant factors. The methodology on which the portfolios are sorted will be elaborated on after reflecting on the CAPM originating from the findings of Sharpe (1964) and Lintner (1965). They argue that investors choose the identical efficient stocks which maximize returns for the perceived risk, this combination of stock was deemed the market factor, MKT, which is used in the formula 3.1;

This formula shows that the returns of asset i at time t are affected by the stock’s exposure, , to the market factor. Note that the MKT reflects the markets excess return and the risk-free rate, , is retracted from the market returns to reflect the excess compensation. For investors to bear holding risky assets, the asset’s returns should compensate for the risk in addition to matching the assumed risk-free rate, , at which investors can hold their portfolios without bearing any risks.

3.2 Fama French Three factor model

Following this basic model, Fama and French (1993) proved that several additional factors are important in determining the required compensation for holding assets. Formula 3.2 shows the factors added by Fama and French (1993)

In which MKT, SMB and HML are measures provided by Fama and French as control variables for the market excess returns, size and value factors. The MKT, SMB and HML signs represent the exposure factors of the firms to these factors as suggested by Fama and French (1992). Ang et al (2006) have shown that the main estimator for idiosyncratic

volatility is the residual variance after accounting for the three-factor suggested by Fama and French. Ang et al (2006) determine that , based on daily stock returns of t-1.

Within this model the size factor, and its exposure 𝛽𝑆𝑀𝐵𝑖 , shows the risk associated with the increasing differences in returns between small and big firms. The model uses the differences in earnings for several small and big sized firms to establish the average

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difference between small and big firms. The combination shows the required reward for the perceived size risk. Continuing, the value factor shows the average return for high book valued portfolios minus the growth portfolios. The respective beta, 𝛽𝐻𝑀𝐿𝑖 , shows the exposure of the firm to the risk factor. The combination shows the expectations of the risk associated with realized earnings versus potential future earnings and the risk associated with this.

3.3 Fama French Five factor model

The most recent model suggested by Fama and French (2015) expands the previous model with two additional factors. Formula 3.3 shows the addition of the Profitability (denoted RMW) and the Investment (CMA) factors. The former shows the differences between earnings of portfolios formed based on robust operating profitability and the weaker

operating profitability firms. The latter shows the difference between a conservative investing portfolio and the aggressive investment portfolios.

𝑟𝑡𝑖 = 𝛼 + 𝛽0+ 𝛽𝑀𝐾𝑇𝑖 𝑀𝐾𝑇𝑡+ 𝛽𝑆𝑀𝐵𝑖 𝑆𝑀𝐵 + 𝛽𝐻𝑀𝐿𝑖 𝐻𝑀𝐿 + 𝛽𝑅𝑀𝑊𝑖 𝑅𝑀𝑊 + 𝛽𝐶𝑀𝐴𝑖 𝐶𝑀𝐴 + 𝜀𝑡𝑖 (3.3) 3.4 Sentiment Measurement

For the investors sentiment I will use the index for market-wide investor sentiment1. These measurements consist of two measures; a stock market specific measure and the macro-economic business cycle adjusted measurement.

The stock market specific measurement reflects on 6 factors which the paper deems of high importance to the overall stock market’s performance. Baker and Wurgler (2006a) account for several stock market factors; number of IPOs (Initial Public Offerings), IPO first day returns, overall stock turnover ratio, dividend premiums, equity and debt issuance and closed-end discounts. These factors are used to establish the confidence in stock market levels, with respect to the US market. Since this research reflects on the US stock market performance, these indicators are deemed great estimators for US investor stock-market sentiment.

Several papers have provided factors which were used in the creation of the sentiment measures. One such factor are the closed-end fund discounts. These are the average differences between closed-end stock net asset values and the market price for which the shares trade at. Higher differences in pricing influence the first sentiment measure negatively, indicating that mispricing of closed-end fund stock prices reflects on reluctance to

1 The index and its deconstruction can be found on the Wurgler data library retrievable from http://people.stern.nyu.edu/jwurgler/

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invest and therefore less confidence in the market in general (Zweig (1973) and Lee et al (1991)). The data used for this measure is taken from Neal and Wheatley (1998) and the Wall Street Journal turn-of-the-year issues.

Secondly, NYSE share turnover rates are used as an indicator of stock liquidity. Baker and Stein (2004) suggest its importance for establishing a sentiment index. High liquidity is an indication of stock overvaluation and a high turnover rate forecast lower stock returns (Jones, 2001). This factor is based on the natural log of the raw turnover ratio which is de-trended by the 5-year moving average.

Thirdly, the number of IPOs and their first-day returns reflect higher investor enthusiasm and the market timing of firm issuing IPOs. Based on data from Stigler (1964), Ritter (1991), Ibbotson, Sindelar and Ritter (1994) this measure accounts for these factors. Baker and Wurgler (2006a) notice that when sentiment is positive, investors focus on

younger (recently available to the public) stocks, which makes the demand and therefore the climate for IPOs better. Therefore, the addition of this measure to determine investor

sentiment seems highly applicable.

Continuing the fourth factor used in the first sentiment measure, Baker and Wurgler (2006a) argue using the share of equity issuance and the debt issues of firms to capture firms’ financing activity. Using data from the Federal Reserve Bulletin2, the gross equity divided by the long-term debt issuance could forecast lower market returns. This indicates that firms deem their stock prices at more comfortable levels than the risk-free lending rate they are exposed to.

The final factor is the dividend premium. This measure is the log difference of average market-to-book ratios of dividend payers and nonpayers. This proxies for the value measure used by Fama and French (2001) and assumes payers are larger and more profitable firms with lower growth opportunities (Baker and Wurgler, 2006a).

The second sentiment measure takes into account several macroeconomic factors such as: service, durable and non-durable consumption indices, consumer price index, industrial production and employment. Baker and Wurgler (2006a) orthogonalized the data to the growth in these factors and a flag for NBER-determined recessions. The addition of this second measure is argued by Baker and Wurgler (2006a) to distinguish the measure from the common business cycles that occur in economies. The number of IPOs for example doesn’t follow the same timing as the overall business cycles do.

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The following graph (Graph 1) portrays the development in sentiment from 165-2015. We find that the peaks within the graphs are just before times of recession, which explains the dramatic drops thereafter. Based on the National Bureau of Economic Research3 (NBER, 2018), we denote the following years as periods of recession which the sentiment measure also uses. Some examples of the crises marked in the graph are: The Oil crises (1969-1970, and 1980), the Dotcom crash (2001) and the financial crisis (2007-2009). Within the graph we notice that the sentiment measures are affected by recessions, but primarily when investors are extremely bullish before the recession occurs. The second oil crisis shows unaffected investor sentiment even though the NBER has marked this as a business cycle recession.

Graph 1: The sentiment measures across time

This graph shows the sentiment and its development across time. Positive developments indicate bullish markets and low sentiment displays investors’ lack of confidence. The shaded areas display the recessions as established by the National Bureau of Economic Research.

-3 -2 -1 0 1 2 3 4 1970 1975 1980 1985 1990 1995 2000 2005 2010 2015 Sentiment 1 Sentiment 2

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3.5 Dataset

The dataset that will be used for this analysis consist of data ranging from January 1965 up until September 2015, due to the availability of the data regarding the investors sentiment measure established by Baker and Wurgler (2006a). The portfolios used in this research are retrieved from French’s data library4. In order to conduct a proper analysis, 25 portfolios sorted on size and the residual variance of the stocks will be used. The idiosyncratic volatility is calculated by applying a three factor analysis on 60 days of lagged returns4. The residual from this analysis is denoted as the idiosyncratic volatility.

The main explanatory factors used by the CAPM, three factor model and the five-factor model are also imported from the French data library4. Since these models all expand on CAPM, this simplified model will be reflected on first. After these analyses, we add the measures found on the Baker and Wurgler library5 to check for the effect of adding sentiment among different levels of idiosyncratic volatility.

4 The data library established by French can be found on

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html 5 As mentioned, this data is retrieved from http://people.stern.nyu.edu/jwurgler/

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4. Results and discussion 4.1 Descriptive statistics dataset

Before starting the analysis, I review the provided data from both Baker and Wurgler (2006a), the sentiment measures, and Fama and French (1993), the three and five factor model data. A simple, yet effective way to cancel out any early stage problems, is reviewing the correlations between the used factors.

Table 1 (see Appendix) shows that, in general, the correlations show no cause for concern, the exception being the two different sentiment measures. The high correlation between these two measures is explained by the fact they use the exact same data, the difference between the two being the latter using orthogonal matrices. Also, note the

negative correlation between the market coefficient and the Sentiment measures, indicating that higher sentiments resulted in lower market returns, which opposes the axiom that markets go up in bull markets.

For further analysis I will be using the portfolios provided by French, which are sorted on the size of the firm and the variances for the individual stocks. The portfolios are

summarized in order to provide a quick view as to whether the means might differ as the volatility increases. Based on the findings in table 2, the average mean return decreases as volatility increases.

As the size of the firm increases, the average returns seem to decrease and the differences in idiosyncratic volatility between the lowest (1st row in column 5) and the highest (5th row in column 5) variance stocks is less than the difference within the smaller sized firms. Based on the minimal and maximum returns, we can reaffirm that higher idiosyncratic volatility

portfolios indeed have lower minimums and higher peaks every month. Instead of the premium suggested by Merton (1987) to compensate for idiosyncratic volatility, we find that the results show similarity to those found by Ang et al. (2006). Higher idiosyncratic risk has lower returns, therefore short selling the higher volatile portfolio may benefit investors.

After reviewing the standard statistics of the portfolios, the research reflects on the summary statistics of the applicable factors which will be used in the analysis. In table 3, one can immediately notice a mean of 0.00 for the sentiment measures. Due to the min, max and the standard deviation statistics, the bullish and bearish markets seem to cancel out a

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4.2 Beta exposures 4.2.1 CAPM estimates

After the initial summary of the dataset, I apply the same approach as Fama and MacBeth (1973) in attempt to calculate the pricing in of several factors, with the main focus on the sentiment measure. By applying a simple statistical analysis, the portfolios betas are

estimated. These estimates, betas, are in turn evaluated to check whether size and portfolio variance result in different exposures to the following factors: market returns, firm size, value, investments and firm profitability. After reflecting on the portfolio betas, the next section will use these estimates and the returns to estimate the factor premiums as is custom in a Fama MacBeth (1973) analysis.

For these estimates, the dataset used will consist of the stock returns from the NYSE, AMAX and NASDAQ with the timeframe ranging from July 1965 to September 2015. The monthly stock returns are filtered and sorted into five portfolios based on the firm’s market equity and five portfolios based on the variance of the stock returns, for which 20-60 days of data is used. This leaves a total of 25 portfolios to be reflected and analysed in the following procedures.

Reflecting on the portfolios beta estimates will help determine the exposure of each portfolio to these factors and shows differences across the different volatilities. Our analysis starts with a simple exposure to the capital asset pricing model (CAPM). This will allow us to check whether the more volatile portfolios tend to react more to moves among the overall market benchmark.

Table 4 Panel A show us the most volatile stocks (fifth row) portray way higher exposures to the overall market’s performance. This finding worsens the IVOL puzzle even more, as it increases the gap between return for the different levels. The differences

between the lowest volatile portfolios and the highest volatility portfolios range from 0.600 to 0.851. The biggest difference found in the second sized portfolios, whereas the smallest difference occurs among the biggest sized portfolios. The highest (lowest) volatile portfolio in second column of portfolios, shows us that for each 1% movement of the overall benchmark, the stock within this portfolio change by 1.612% (0.711%). Since CAPM only accounts for market performance, the remainder (alpha) of the regression show high significance levels, indicating that the simplified analysis lacks several explanatory variables (average R2=0.75).

Next, the first additions to the regression will be the Baker and Wurgler (2006a) sentiment measures. Panels C and D (Table 4) show the addition of these respective

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measures. The former, Panel C showing evidence that adding the first sentiment measure (Sentiment 1) to the CAPM partially helps explaining return of primarily high volatile

portfolios. The exposures are negative, indicating that high volatile portfolios underperform less volatile portfolios. All of the highest volatile portfolios show significant negative effects of sentiment on the portfolio returns. Notice that the medium sized firms also show a significant effect of Sentiment 1 on the returns of the low idiosyncratic portfolio, the effect positive and significant at a 99% level.

Reflecting on the second measure (Sentiment 2) found in Panel D we find the same results; the high volatility portfolios showing significant negative exposures to sentiment among all portfolio sizes. The effect declining as the size of firms among the portfolios increases, in numbers from -0.819 to -0.255 (69% decrease). So, we can conclude that although the market exposure partly accounts for the IVOL puzzle, the exposure to sentiment also plays an important part in the differences found between high and low idiosyncratic volatility portfolios.

4.2.2 Fama and French three factor model

As Fama and French (1993) suggest in their paper, the addition of two factors; Value and Size helps explain cross-sectional average returns. Therefore, the next step in this analysis is expanding the CAPM by adding these factors. Adding new factors to the first models allows for a more realistic view on the effect sentiment has on the idiosyncratic volatility puzzle.

Table 5 shows the exposures of the portfolios with regards to the three-factor model established by Fama and French (1993). We find that the smallest sized firms show higher exposures to each of the factors as the volatility increases. The market factor for example shows that portfolios of the smallest size have an increasing exposure to the market’s performance. Among the other sized portfolios, the results show that the lowest and highest IVOL portfolios have an exposure of greater than 1 with regards to the movements of the market. The portfolios in between show exposure ranging from 0.707-1.107 (indicating a 0.7%-1.1% move for each percentile the overall market shifts), whereas the high volatile portfolios show exposures of 1.228-1.341 (1.2%-1.3% co-movement with the market), excluding the exposure of the smallest and lowest volatile portfolio which shows a significant exposure of 0.687.

When reviewing the other factors, we find that the biggest portfolios show negative exposures to size, as displayed in panel B of table 5. This follows the basic logic that whenever the difference between smaller and bigger firm’s performance increases, the

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bigger firm show lower returns, whereas smaller firms increase their returns. Panel B indeed shows this by showing increasingly lower returns as the portfolio size increases.

Finally, the addition of the value factor shows the effect of the difference between high book-to-market value compared to low book-to-market value portfolios. Higher value stocks tend to outperform so called growth stocks and an increasing difference between these two, the following effect on the applied sorted portfolios is portrayed in table 5 panel C. The only negative betas are found among the lowest IVOL portfolios, excluding the smaller portfolios (Smallest and the second size portfolio in the first row of panel C). These tend to underperform if the difference between value and growth stock performance increases. The results range from -0.246% to -0.353% for each point the value factor increases.

Overall table 5 shows that increases in volatility result in a higher exposure to the separate factors. This indicates that the respective returns are affected more by the

portfolio’s exposure to each risk. The reaction to movements in the market exposure shows rather consistent results as the portfolio’s size increases. For the added factors, size and value, we find decreasing effect for both of these measures as the size of the firms

increases. The value factor also shows less change compared to the size factor when the volatility increases.

As a result of the previous analysis the next logical step is adding the sentiment measures to the suggested model. This will allow better reflection on the effects of sentiment on more volatile portfolios and will help eliminate overestimation caused by excluding

important variables. Table 5 panel E shows the results of the first measure added to the model including market, size and value factors. The first of these factors shows lower exposures to these measures among the lowest volatile portfolios. In contrast to the normal three factor model, the exposures to market performance in the low volatility portfolios have changed from betas higher than 1 to betas smaller than 1, with the exception of the smallest size portfolio. This beta remains unchanged by the addition of the sentiment measure to the estimation model.

Finally, the reflection of the Sentiment factor shows that whenever idiosyncratic volatility increases, the higher IVOL stocks perform worse. In times of bullish stock markets, these portfolios therefore underperform compared to the lower idiosyncratic volatility

portfolios. The effect of sentiment on these portfolios in turn shows significance when the idiosyncratic volatility increases. The highest volatility portfolios for any firm size show that whenever sentiment increases (high optimism among investors) the stock perform worse, ranging from -0.197% and -0.381% for each one-point increase in the measure established

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by Baker and Wurgler. Within the second and third column we also find rather significant effects of sentiment on portfolio returns, 3/5 and 3/5 of the portfolios showing significant exposure to the factor.

Remarkably, the exposures of several low and high idiosyncratic volatility are significant, the lower idiosyncratic volatility portfolios show a positive effect of overall

sentiment on the returns whereas the higher idiosyncratic volatility portfolios show significant negative exposure to overall sentiment. Ang et al (2009) show that higher idiosyncratic volatility stocks on average have lower returns than their counterparts. These findings remain even after accounting for the three factors used by Fama and French (1993). The fact that the high idiosyncratic volatility portfolios show significant exposure estimates to sentiment, reinforces the idea that sentiment might play a part in explaining the previously unexplained puzzle.

Continuing, analysing the addition show the results of adding sentiment measure 2 to the three-factor model. Due to the factors considered in the establishment of this measure, the expectation is that this analysis will show similar results to the previous added

measurement. This indeed is the case when we look at the beta estimates displayed in the panels. For four out of the five highest idiosyncratic volatility portfolios, we find the effects increased, although the found estimates remain within the standard deviation of the original estimate. The increasing negative effect of sentiment shows consistency among all sizes and therefore reaffirms the findings that sentiment in general affect higher IVOL stocks more than lower IVOL portfolios, which usually benefit from higher investor sentiment.

Concluding, if we take into account the sentiment measures, as depicted in panels E and F, we can notice that the sentiment measures have a significant effect on the returns among the highest volatile portfolios. Recall that high positive (negative) numbers for sentiment are characterized as bullish (bearish) stock markets. Based on the findings in table 5, we can state that, initially, these portfolios perform significantly worse in market upswings whereas they tend to outperform other portfolios in bearish market swings. The initial, unexplained, lower stock returns among high IVOL stocks found by Ang et al (2009) and the negative effect of sentiment on IVOL sorted portfolios shows promise in solving the puzzle.

4.2.3 Fama and French five factor model

In addition to the previous analysis, we continue this research by adding additional factors to the analysis. The first step will be reflecting on the portfolio exposures to the five-factor model as established by Fama and French (2015), which adds Profitability and Investment to the previous three factors. The former factor assumes that the difference found between

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firms with robust and weaker earnings. The latter assumes, based on evidence found by Fama and French (2015), that firms which dedicate more resources in investment

opportunities create higher returns than firms that invest less. Table 6 shows the exposures of the portfolios with regards to these and the other 3 factors.

Reflecting on the first factor shows us that, compared to the original three factor model, the lower IVOL portfolios show less exposure to the market. For four out of five low IVOL portfolios, we find higher exposures to the market’s movement. The already

disproportionate co-movement only increases as the number of reviewed factors increases. We continue the reflection of this model by reviewing the exposure to the size factor. The first notable difference is seen in the first row of Panel B, the lowest IVOL portfolios show significantly higher coefficients compared to the original model, the biggest portfolio sign turns negative in accordance to most of the other portfolios within this size sorting. Value shows lower exposures all around and diminishing the significance of betas among the largest sized portfolios. The effect likely driven out by either/both of the newly added factors.

The newly added factors are displayed in panels D and E. The results show that the profitability factor has significant effect on a portfolio’s exposures and shows a positive effect on the lesser volatile portfolios. The highest portfolios however, show large negative

exposures to profitability. This is likely due to the less robust earnings within these portfolios, the volatility of the stocks is most likely (partly) affected by the unreliable earnings among these portfolios. Regarding the investment factor, we find that as firm size increases, the effects of different investment strategy increases the underlying returns (panel E).

The lower the firm’s IVOL, the higher the effect of its investments on the overall returns. As IVOL increases, we find evidence that the underlying effect of Investments on the portfolio diminishes. Whenever the highest IVOL quintile is reached the results show that the

exposure decreases drastically and returns are affected as much as -0.286% to -0.503% for each one-point increase of the Investment factor.

As we review the newly added factors we find higher volatile portfolios have significant negative exposures. These signs indicate that despite a firm’s size, stocks with high idiosyncratic volatility which are exposed to more profitability risk than their peers, underperform lower volatility firms. This could indicate that costly failed investments have increased the firms among this portfolio leading to higher volatility and lower earnings.

Additionally, we reflect on the effects of Sentiment on the stock performance of the portfolios. Panels G and H show the effect of sentiment in addition to the five-factor suggested by Fama and French (2015). Panel G adds the first sentiment measure and

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shows that the beta exposures of the portfolios are barely changed by the addition, and the changes amount to less than a hundredth of a percent. The more notable change is that the new factor shows insignificant results, with the exception of six portfolios, and the effect seems random and incomprehensible as the effect are among different levels of IVOL and portfolio sizes. These findings show that the effect of Sentiment found in addition to the three-factor model disappears when Profitability and Investment are added. This assumption is confirmed by the findings in panel H, in which the second measurement shows the same results, that adding sentiment doesn’t affect the returns significantly when the other five factors are reviewed.

Regarding the remainder of the factors, we find small negative exposures among the portfolios with higher volatilities in regards to their exposure to stock market sentiment as established by Baker and Wurgler (2006a). Although the results for both measurements are insignificant, they are in accordance with the significant results found in the earlier part of this part of the analysis. The conclusion based on the Profitability and Investment factors are reaffirmed for the high volatility portfolios. These statements hold for both of the applied sentiment measures.

4.2.4 Fama and French 3 factor model with separate additions

As noted in the previous section, the five-factor model removes the significance of the Sentiment measures effect. In order to check which of the newly added factors removes the significance of this effect, tables 7 and 8 shows the exposures found when adding each term separately to the three-factor model.

The addition of the Investment factors (table 8) shows us that the Sentiment measure has a significant negative effect on the returns of the portfolios with the highest level of

idiosyncratic volatility. We find that 11/25 portfolios are significantly influenced by the level of Sentiment for the first measure and 12/25 for the second measure. Based on these findings, the profitability factor should be the factor that lowers significance levels and shows lower exposure estimates in general. Table 7 (panels F and G), shows that 6/25 and 8/25 portfolios show a significant exposure to the respective measures. We also find that the persistent negative exposures found among the highest idiosyncratic volatility portfolios are no longer present. Therefore, we conclude that Profitability is the factor that removes the effect of adding the sentiment measures to the five-factor model.

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4.3 The factor premiums

4.3.1 Pricing premiums among existing models

The second step in the Fama MacBeth procedure requires the estimated betas to be regressed against the portfolio returns at each separate point in time to estimate the premiums. Table 9 shows the results of the second step in the Fama MacBeth procedure. The table gradually adds factors to determine which factors should be priced in and which factors do not show significant premiums.

The first row in the table shows that if we account for market risk only, the portfolios show a significant constant and a less significant market risk premium. The interpretation for these findings is that market risk should be priced in when evaluating investment strategies. The significant constant shows us that due to the restrictions of the number of factors included in the analysis the evaluated portfolios have a non-risk related priced-in factor. Since the constant estimate is positive (1.233%), in a world with only perceived market risk, these portfolios provide higher returns compared to the overall market.

Secondly, we add more factors to the incomplete CAPM and reflect on the priced factors when accounting for size and value. We find that the risk associated with market exposure and the value factor have positive premiums. The former shows a 0.596%

premium on average whereas value affects shows a premium of 1.266% (Table 9, row 3 and 4).

Before we start evaluating the five-factor model, we add the new factor separately since section 4.2 shows us that the exposures of the portfolios to the risk factors disappears when adding these factors. Rows 5 and 6 (Table 9) show that the newly added Profitability factor should be priced in based on the findings as it shows an average 1.256% premium for exposure to this factor. Also, note the high increase in the model’s explanatory power

compared to the previous models. Granted, the former models are rather simplistic and non-reflective of reality the still provide some insight in the way idiosyncratic portfolio returns can be explained. The increase in explanatory power, shows us that the new factor shows greater importance in estimating relevant risk premiums.

Continuing, we add the investment factor to check whether this factor also shows a significant risk premium. Which in turn, could prove to be important for investors’ decision making in reflecting on investment strategies. The investment factor has an insignificant effect in returns and the exposure to this risk is not priced. Also note that the explanatory power decreases compared to the Three factor model, which confirms earlier findings in the

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non-importance of including this factor in asset pricing. Increasing differences in returns made on aggressive vs conservative portfolios shows no priced effect when reflecting on idiosyncratic volatility and size sorted portfolios.

Finally, when accounting for the additional factor suggested by Fama and French (2015) we find that all factors should be priced in. The market, value and investment risk factors show negative premiums while both the size and the investment factors show significant positive premiums. We also find a significant constant premium, indicating that part of the puzzle remains unexplained.

4.3.2 Sentiment risk premiums

Next, we add the sentiment measures suggested by Baker and Wurgler to the models. Starting with the CAPM model, the results show that the constant significantly differs from zero, indicating that the analysis likely lacks several measures to help explain the returns.

The first sentiment measure shows both a significant constant and a significant premium for exposure to investor’s sentiment. The positive premium indicates that exposure to this risk factor is rewarded by an additional 0.56% at a 90% confidence interval. When comparing the model to the original CAPM, we find a higher R2 for the model including the new measure. The second measure shows similar results with an even higher, and more significant, premium (0.786%) on average and an even higher increase (5.85%) to explanatory power. Based on these findings, exposure to investor’s overall sentiment

measures should provide a 0.56% to 0.789% premium for the average idiosyncratic volatility sorted portfolio.

Additionally, the three-factor model is combined with the two measures. The addition to the model (row 15 table 9) shows significant (at a 1% level) risk premiums for the following factors; Value and Sentiment. The market factor shows significance at a 95% confidence level. These premiums are positive and therefore exposure to these risks provides investors with the following average premiums: 0.68% for market exposure, 0.782% for exposure to difference in value sorted returns and a 1.094% premium for exposure to overall investor sentiment.

The significance of the market factor premium increases when adding the second measure (row 17 table 9). The results also show increases in the risk premiums for both Market (a change of 0.105%) and Sentiment (0.383%), while the value premium decreases with (0.147%). The explanatory power of the model increasing 7.65%. investors are thus compensated for their exposure to these risks when picking portfolios sorted on size and

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idiosyncratic volatility. Compared to the original three factor model, we find it likely that a large part of the sentiment premium is reflected in the value factor, as this premium decreases most when adding investor sentiment.

Finally, the results for the five-factor model show the disappearance of the premiums related to sentiment. Rows 26 and 28 (table 9) provide us with several insights. The first conclusion is the insignificant premiums found for both S1 and S2. Next, we notice the decrease in explanatory power of the model. The adjusted R drops with each addition of a sentiment related factor. This raises the question which of these factors removes the

previously found premiums. When adding the Profitability and Investment factors separately, we find that the investment factors decreases the explanatory power of the three-factor model including the sentiment measures (rows 22 and 24 compared to rows 14 and 16). The Investment risk factor also shows insignificant results. Rows 18 and 20 of table 9 show a significant change in the sign of the sentiment pricing premiums. Adding the first measure shows a negative premium for sentiment opposed to the previously found positive premiums. Although the significance persists, we find that the second measure shows insignificant results and lowers the explanatory power of the model. We can therefore assume that the effect of Sentiment is overtook by the Profitability factor added in the five-factor model.

However, when adding the other factors (Investment and Profitability), suggested by Fama and French (2015), we find these premiums turn negative and insignificant. Since the addition of both (separately added) measurements shows insignificant results, we can conclude that the earlier found premium is evidently only applicable when investors do not take the factors Investments and profitability in account. This contradicts the findings of Baker and Wurgler (2006a), whom found significant negative effects of sentiment on stock returns among volatile stocks. Based on these findings, we conclude that, based on the used dataset, the unexplained part of the volatility puzzle remains an unsolved piece of the

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4.4 Idiosyncratic volatility and the Sentiment premium

Within this section, the results found in section 4.2 are combined with the findings of section 4.3 in order to show the differences between the idiosyncratic volatility sorted portfolios. This will enable us to reflect on the link between idiosyncratic volatility related returns and their relation with investor sentiment. By combining the average premiums with the portfolio specific betas, we can help explain why certain previous literature has found certain effects. Since most high volatility portfolios show negative beta exposures to sentiment while the average risk premium is found to be positive, we can state that sentiment on average affects high volatility portfolios negatively regardless of their market size. In order to check whether the negative relationship found by Ang et al (2006) is due to the beforehand mentioned findings, requires reflection on the overall exposures and their respective premiums.

4.4.1 CAPM and sentiment

We begin dissecting the analysis by reflecting on the addition of the first sentiment measure. As stated the market betas increase with the level of idiosyncratic volatility and are all

positive, therefore this factor will not drive out the negative effect found by Ang et al (2006), whereas the sentiment measure does. For the CAPM model, this research shows negative exposures among the higher levels of idiosyncratic volatility towards sentiment (row 4 and 5, panel C, table 4). The exposures range from -0.086 to -0.804 and are highest among the smaller sized firms. These figures combined with the average 0.56% premium found in the second pass of the analysis indicate that, based on this model, the negative returns found in higher idiosyncratic portfolios is due to investor sentiment. Based on these numbers the lowest and highest negatively affected portfolio would have a -0.05% and -0.45% return for the exposure to investor sentiment. The lower portfolios net positive returns for their

exposure to sentiment, possibly explaining the anomaly found by Ang et al (2006).

We confirm these findings by reflecting on the second measure (table 4 panel D and table 9). These tables show the exposures and the relevant premiums. Note, that we reaffirm the findings based on the market exposure and its premium. Higher volatility results in higher exposure to market and since the premium is positive, the higher volatility portfolios could show higher returns based on markets. Since the market premium is insignificant, these statements are merely suggestive and do not support the statement that higher volatility portfolios perform better compared than their counterparts for rewarding market risk. Consistently with the previous paragraph, the sentiment exposures are negative and are lower than those found for the first measure, ranging from -0.086 to -0.819. Combined with

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the positive average premium this results in -0.05% and -0.46% discounts for these stocks. Since the findings are significant, we can confirm that sentiment indeed explains why higher idiosyncratic portfolios would underperform their less volatile counterparts.

4.4.2 The Three factor model

Extending the simplistic CAPM, we added two more factors as used by Fama and French (1993) to provide a more realistic view of reality (see table 5). For size, the results show positive exposures regardless of volatility levels, whereas value shows negative exposure estimates among the highest levels of idiosyncratic volatility. Finally, the first measure shows both positive and negative exposures (panel E). The former found in the lower levels of idiosyncratic volatility and the negative exposures solely found in the higher levels. These exposures allow for greater disparities in the differences among high versus low idiosyncratic portfolio returns. Again, this reaffirms that sentiment helps explain the anomaly found in the literature regarding the idiosyncratic volatility puzzle.

We find that these conclusions hold when we replace the sentiment measure for its adjusted substitute (panel F, table 5). The same conclusions hold, but we do note that the exposures to the second measure are bigger for the highest level of volatility, indicating an even bigger negative risk premium for these stocks. Since the risk premium is significantly higher (an increase of approximately 35%) and the measure itself is adjusted for economic factors such as production, consumption and employment, it is likely that this measure provides us with a more realistic view of sentiment and its effect on the differences in returns found between high and low idiosyncratic volatility portfolios.

Finally, we find that the size of the portfolio greatly affects the differences found between the highest and lowest idiosyncratic portfolios. The range of exposures to the market performance tightens as the size increases, which indicates that the bigger firms are less affected in general by market movements. The effect of differences in returns found among different sized portfolios affect the bigger sized firms less as well. Their negative exposures are due to the fact that smaller stock outperform the bigger firms when the difference is positive resulting in less demand for bigger stocks. The average risk premium for Size found in table 9 is insignificant and negative and also is counterintuitive with the premises of the factor itself. Therefore, additional factors need to be added to improve the model. The value factor exposures show more interesting results as we find that the lowest idiosyncratic volatility portfolios show decreased exposure betas as size increases, whereas the higher volatility portfolios show increasing (negative) exposure estimates. If high value firms outperform the growth stocks, low volatility stocks show higher returns (as the average

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risk premium is positive) and if more volatile portfolios increase in size they show significantly increasing lower returns.

4.4.3 The five-factor model

In accordance to the previous analyses, we will review on the model including the

Profitability and Investment factors (Tables 6 and 9). Since the sentiment measures shows insignificant risk premiums and the exposures are rather infrequent and seem random, this part will primarily focus on the effect of the newest addition and their influence on the differences in returns for the sorted portfolios. Based on table 6, we find significant negative exposures for both factors in for the highest idiosyncratic volatility portfolio levels. We also noted that the exposures decrease as idiosyncratic volatility increases. The combination of the negative exposures and the positive average Profitability premium results in the highest volatility portfolios having significantly lower returns solely based on profitability, we will reflect on this measure separately in section 4.4.4. The other levels of volatility show positive exposure to the factor and increasing positive exposures toward the medium volatility level. We also note that the smaller portfolios are more affected by the different levels of

idiosyncratic volatility.

Continuing, the Investment (panel E) factor shows us that the negative exposures we found for the high idiosyncratic volatility portfolios persists when reviewed the results of adding this factor. Note that, on high levels of volatility, the range of the exposures is smaller than found in other parts of this analysis. The difference between aggressive and

conservative investment returns seems to be less dependent on the size of the firms, and the general exposure of all portfolios to Investments is rather for low compared to the other factors. Table 9 shows a negative premium for the Investment factor. Thus, the highest idiosyncratic portfolios benefit most from the increase in difference between aggressive and conservative portfolio strategies. This is most likely due to the fact that the stocks in the higher volatility portfolios also drive the more aggressive portfolios, and thus the risk in this case is their earning potential increases which drives the underlying returns in general. Although the numbers differ slightly when adding the sentiment measures, the same conclusions hold when adding either sentiment measure to the model. Since the risk premiums are insignificant and the explanatory power does not increase, we refer to the earlier conclusions on the exposures and the relevant premiums.

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