Difference of opinion and the cross-section of equity returns: Australian evidence

Pacific-Basin Finance Journal 19 (2011) 435–446

Contents lists available at ScienceDirect

Pacific-Basin Finance Journal j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p a c f i n

Difference of opinion and the cross-section of equity returns: Australian evidence Philip Gharghori a,1, Quin See b, Madhu Veeraraghavan c,⁎ a

Department of Accounting and Finance, Monash University, Clayton Campus, Victoria 3800, Melbourne, Australia Technology, Media & Telecom-Asia, Investment Banking, UBS AG, 52/F 2 International Financial Centre, 8 Finance Street, Hong Kong c Department of Accounting and Finance and Corporate Finance Cluster, Monash University, Clayton Campus and Centre Associate, Australian Centre for Financial Studies, Melbourne, Australia b

a r t i c l e

i n f o

Article history: Received 23 April 2007 Accepted 31 March 2011 Available online 19 April 2011 JEL classification: G10 G12 G15 Keywords: Difference of opinion Analysts' earnings forecasts Abnormal turnover Idiosyncratic volatility

a b s t r a c t This paper examines the relationship between difference of opinion among investors and the return on Australian equities. The paper is the first to employ dispersion in analysts' earnings forecasts, abnormal turnover and idiosyncratic volatility as proxies for difference of opinion. We document a negative relationship between difference of opinion and stock returns when dispersion in analysts' forecasts and idiosyncratic volatility are employed as proxies. This result provides support for Miller's (1977) model and is consistent with the findings of Diether et al. (2002). In contrast, we find mixed results when using abnormal turnover to proxy difference of opinion. © 2011 Elsevier B.V. All rights reserved.

1. Introduction In an efficient stock market, future payoffs on assets cannot be predicted on the basis of available information. At least three decades ago financial economists believed that this assumption was true. Now, it is widely accepted that stock returns are at least partly predictable (see, for example, Fama, 1991). A large body of empirical research over the past three decades has provided evidence against the prediction of the traditional Capital Asset Pricing Model (CAPM). This previous work shows that the cross-section of expected stock returns is not sufficiently explained by their market betas. It is well documented that factors

⁎ Corresponding author. Tel.: + 61 3 99034553; fax: + 61 3 9905 5475. E-mail addresses: [email protected] (P. Gharghori), [email protected] (Q. See), [email protected] (M. Veeraraghavan). 1 Tel.: + 61 3 9905 2947; fax: + 61 3 9905 5475. 0927-538X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.pacfin.2011.03.004

436

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

such as firm size (Banz, 1981), earnings yield (Basu, 1983), leverage (Bhandari, 1988) and the firm's book value of equity to its market value (Chan et al., 1991) explain the cross-section of average stock returns better than the beta of a stock. More recently, Diether et al. (2002) document that firms with more uncertain earnings do worse. Specifically, they show that high (low) dispersion in earnings expectations can predict low (high) stock returns in the future. In short, Diether et al. (2002) suggest that dispersion in analysts' earnings forecasts can help explain future stock returns. In a controversial paper, Miller (1977) hypothesised that stock prices will reflect the valuations of optimists but not of pessimists, as pessimistic investors do not participate in the market due to short-sales constraints. Miller (1977)theorised that stocks that are subject to short-sale constraints become overvalued as pessimists are restricted to owning zero shares and thus the price is set by optimistic investors. Miller's hypothesis was based on the assumption that the most optimistic investors set stock prices. However, Diamond and Verrecchia (1987) challenge Miller (1977) by stating that short-sale constraints do not lead to overvaluation. Diamond and Verrecchia (1987) introduce a rational expectations model to analyse the effects of short-sale constraints on the speed of adjustment of stock prices. They report that although short-sale constraints can eliminate some informative trades, they do not have an upward bias on stock prices. They also document that if traders have rational expectations, short-sale constraints do not lead to biassed prices. In a recent paper, Berkman et al. (2009) point out that Miller's (1977) hypothesis that stock prices reflect an optimistic bias cannot persist indefinitely as periodic announcements reduce the differences of opinion among investors and thus stock prices move closer to their fundamental values. As far as empirical support for Miller's (1977) hypothesis is concerned, Ackert and Athanassakos (1997) document that high earnings forecast dispersion is associated with lower stock returns. This evidence is consistent with Miller's price-optimism model in that the greater the disagreement about a stock's value the higher its market price and hence the lower its future return. In a similar vein, Dische (2002) documents that a portfolio of high dispersion stocks yields a return of 0.80 percent per month while a portfolio of low dispersion stocks yields a return of 1.74 percent per month. Diether et al. (2002) also document a negative relationship between difference of opinion and stock returns. They show that stocks with higher dispersion in analysts' earnings forecasts earn lower future returns than otherwise similar stocks. Specifically, they document that a portfolio of stocks in the highest quintile of dispersion underperforms a portfolio of stocks in the lowest quintile by 9.48 percent per year. They also report that this effect is most pronounced in small stocks and stocks that have performed poorly over the past 12 months. Their findings provide support for Miller (1977) and are contrary to the claim that dispersion in forecasts can be viewed as a proxy for risk. Doukas et al. (2002) advance difference of opinion as a plausible explanation for the abnormal return on value stocks. They document that value stocks display higher forecast errors and larger downward forecast revisions than growth stocks indicating that investor's are not excessively optimistic about growth stocks. In their 2004 paper, they document that dispersion in analysts' earnings forecasts is considerably higher for high book-to-market stocks than for low book-to-market stocks. That is, value stocks are associated with greater disagreement than growth stocks. Their findings suggest that difference of opinion represents risk not captured by the CAPM or by the multifactor model of Fama and French (1996) and that it plays an important role in explaining why value stocks generate superior returns to growth stocks. In their 2006 paper, they examine whether divergence of opinion is priced at a premium or discount. They report a positive and significant relationship between divergence of opinion and future stock returns. Their findings contradict Miller (1977) but are consistent with Varian (1985) who argues that divergence of opinion proxies for risk. Garfinkel and Sokobin (2006) examine the relationship between post earnings announcement returns and unexplained trading volume (proxy for divergence of opinion) and report that unexpected trading volume at the earnings announcement positively correlates with future stock returns. Specifically, they show that high divergence of opinion at the earnings announcement date is associated with positive returns during the post announcement period. This finding is consistent with Varian (1985) who documents that asset prices will be lower when opinions are dispersed. Hintikka (2008) tests the difference of opinion hypothesis for equities listed in seven European countries and reports that stocks with lower dispersion in analysts' earnings forecasts generate superior returns than stocks with higher dispersion. In a similar vein, Leippold and Lohre (2010) find that high

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

437

dispersion stocks not only underperform in the US but also in many European markets. They also report that in European markets most of the returns are generated in a narrow window of three years while the dispersion effect in the US displays a steady pattern. We can see from the above discussion that prior research has not generated convincing evidence on the effects of difference of opinion on stock prices. In addition, little empirical research has been conducted outside of the US on how difference of opinion affects stock prices. Berkman et al. (2009) also highlight that testing Miller's conjecture on the role of difference of opinion is important as it challenges traditional asset pricing models such as the CAPM. Given this background, the central objective of this paper is to test the relationship between difference of opinion and stock returns. We do so by examining Australian stocks. We study the Australian market for several reasons. First, asset pricing research done in Australia shows that there is much left to explain in the cross-section of equity returns (Gharghori et al., 2007). Second, the Australian equity market is quite different from the US market. With over 1700 listed stocks and a market capitalization of $1220 billion, 2the Australian market is much smaller than the US market. Further, the Australian market consists of different compositions of industries compared to the US. For example, two-thirds of the stocks listed on the ASX are represented by financials and materials companies. 3Another striking feature of the Australian market is the heavier weighting on mining and resource stocks compared to the US. 4The different size and unique features of the Australian market provide an entirely different setting than the US market. Third, we provide out of sample evidence on whether the dispersion effect is common across other markets or specific to the US market. A major contribution of this study is that we employ multiple proxies for measuring difference of opinion among investors. In a recent paper, Berkman et al. (2009) emphasise that multiple proxies are essential, as a major challenge in testing the Miller hypothesis is to determine proxies that capture differences of opinion. The three proxies employed in this paper are (a) dispersion in analysts' earnings forecasts, (b) abnormal turnover and (c) idiosyncratic volatility.5We adopt abnormal turnover and idiosyncratic volatility as proxies because dispersion in analysts' earnings forecasts is biassed toward large stocks, in that the analyst following for small stocks is thin or sometimes even non-existent. This results in an unrepresentative sample wherein the coverage for small stocks is limited. This problem was also faced by Hong and Stein (2000), Diether et al. (2002) and Doukas et al. (2004).6 The methodologies we employ are a portfolio returns analysis, where stocks are grouped into portfolios based on difference of opinion and a Fama-MacBeth (1973) regression analysis. We find a negative relationship between difference of opinion and stock returns, particularly when using analysts' forecasts and idiosyncratic volatility to proxy opinion divergence. This finding is consistent with Diether et al. (2002) and consequently provides support for Miller's (1977) model. In contrast, the results using the abnormal turnover measure are mixed. The remainder of the paper proceeds as follows. Section 2 describes the data and methods and reports some descriptive statistics. Section 3 presents the empirical findings, and Section 4 concludes.

2. Data, methods and descriptive statistics 2.1. Data Our data come from five different sources. We obtain the monthly stock returns, market capitalisations, number of issued shares, industry classifications, the market return and the risk-free return from the Centre for Research in Finance (CRIF) database. The risk-free return is proxied by the monthly return on the 13-week Treasury note and is obtained from the Reserve Bank of Australia. The market return is proxied by

2

Source: Australian Securities Exchange (ASX) website as of August 2006. Source: ASX website. 4 Source: Australian Financial Review. 5 We thank an anonymous referee for suggesting that we include abnormal turnover and idiosyncratic volatility as additional proxies for difference of opinion. 6 As an example, Diether et al. (2002) sample covers only 40.5% of stocks listed in CRSP. 3

438

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

the value-weighted market index constructed using all companies in the CRIF file. Accounting data required to calculate book equity (Net Tangible Assets) is obtained from Aspect Huntley. Standard deviation in monthly analyst forecasts and the mean of monthly analyst forecasts are collected from the Institutional Brokers Estimate System (I/B/E/S). Daily stock returns and volumes are obtained from SIRCA. Our sample covers the period from 1989 to 2005 and the test period is 1990–2005. 2.2. Methods and descriptive statistics 2.2.1. Difference of opinion proxies Our first difference of opinion proxy, dispersion in analysts' earnings forecasts is measured following Diether et al. (2002). The second, abnormal turnover is a new measure that we propose; the details of its construction will be explained shortly. The third, idiosyncratic volatility is calculated following Ang et al. (2006). It is reasonable to contend that as the idiosyncratic volatility of a stock increases (decreases) so too does the difference of opinion amongst investors about that stock. Thus, idiosyncratic volatility can be considered a proxy for difference of opinion. Dispersion in analysts' earnings forecasts is measured as the monthly standard deviation in analyst forecasts for a given stock divided by the absolute value of the mean of the monthly analyst forecasts. The reason that the standard deviation of forecasts is scaled by the mean is to make the analysis comparable across stocks, as stocks that have higher earnings could have mechanically higher levels of standard deviation. Following, Diether et al. (2002), a stock must have a minimum of two analyst forecasts in a month to be included in the sample. As a robustness test, we replicate the analysis for this measure but only for companies that have the same fiscal year-end. In our sample, approximately 70% of companies have a June financial year-end. Thus, the analysis is repeated for the subset of companies that have June fiscal year-ends. This is done to control for any seasonal variation in the standard deviation in analyst forecasts. Abnormal turnover is obtained by running monthly cross-sectional regressions of stock turnover on size, book-to-market and industry dummies. Stock turnover is defined as monthly trading volume divided by number of issued shares. Size is proxied by market capitalisation and book-to-market is defined as net tangible assets divided by market capitalisation. We use 28 dummy variables to control for the 28 industry classifications specified by S&P via GICS. Abnormal turnover is the residual from the aforementioned regression for each firm. The intuition for using abnormal turnover to proxy difference of opinion is to control for cross-sectional determinants of turnover that are unrelated to opinion divergence so as to isolate the component of turnover that captures difference of opinion. The final proxy is the idiosyncratic volatility measure of Ang et al. (2006). To calculate idiosyncratic volatility, each month a Fama–French model regression is run on the daily excess returns of a stock. Idiosyncratic volatility is the standard deviation of the Fama–French model residuals for that month for a given stock. Unreported findings show that the analyst forecast sample comprises 37,618 firm-month observations. For the June financial year-end sample for analyst forecasts, there are 25,930 firm-month observations. The abnormal turnover sample is 168,346 firm-month observations and the idiosyncratic volatility sample comprises 184,013 observations. The much larger samples for abnormal turnover and idiosyncratic volatility are a more representative cross-section of the Australian market. In contrast, the analyst forecast sample has roughly 20% of the number of observations of the other two proxies, which is consistent with the fact that it is only the largest 20% of stocks in Australia that are routinely covered by analysts. The larger and more representative samples for abnormal turnover and idiosyncratic volatility lend weight to their use as alternative proxies for difference of opinion. Nonetheless, the findings for the analyst forecast sample are important for comparability purposes with prior US research, which invariably uses dispersion in analyst forecasts to proxy difference of opinion. 2.2.2. Portfolio returns analysis There are two distinct stages in this analysis: a portfolio returns analysis and a Fama–MacBeth regression analysis. These two methods will provide complementary evidence on whether there is a relationship between difference of opinion and equity returns. We begin with the portfolio returns analysis. In each month, for each of the three difference of opinion proxies in turn, all stocks that have a valid difference of opinion measure are ranked based on this measure

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

439

and partitioned into quintiles (and tritiles). The equal- and value-weighted returns for these portfolios are calculated using returns in the following month. That is, monthly rebalancing is employed. The reason that monthly rebalancing is employed is that difference of opinion over stock value is more likely to change over a shorter period. The excess return for each portfolio is obtained by subtracting the risk-free return from the raw portfolio return. In addition to calculating excess returns for the quintile and tritile portfolios, zero-cost portfolios are created that are long high difference of opinion stocks and short low difference of opinion stocks. As well as considering the excess returns on the difference of opinion sorted portfolios, we also examine the risk-adjusted returns or alphas for these portfolios. The alphas for the portfolios are calculated using three contemporary asset-pricing models: the CAPM, the Fama–French model and the Carhart model (1997). The alphas are obtained from a regression of the monthly time-series of portfolio returns on the relevant factors for each of the models. The Fama–French factors, SMB and HML, and the Carhart momentum factor are constructed following the methodology outlined in Gharghori et al. (2007). 2.2.3. Fama–MacBeth regressions The Fama–MacBeth approach is used to test the relationship between difference of opinion and returns. Using each of the three proxies in turn, difference of opinion is measured in a given month and linked with return data in the following month. That is, we run cross-sectional regressions of next month's stock returns on difference of opinion. Following the same line of reasoning as before, we use one-month ahead returns rather than one-year ahead returns, as we conjecture that difference of opinion is more likely to have a short-term effect on stock returns. In the first instance, we run regular Fama–MacBeth regressions with no weighting of the stocks. In addition, following Ang et al. (2009), we also weight the stocks based on market capitalisation. For this analysis, the first pass is a weighted least squares regression where the weight is the inverse of the firms' market capitalisation. This is equivalent to a generalised least squares regression with a diagonal weighting matrix and where the values along the diagonal are the inverse of the firms' market capitalisation. The significance of the parameters is inferred in the usual way by computing the average and standard error of the monthly slope estimates. As Ang et al. (2009) state, the market cap-weighted Fama–MacBeth regressions are akin to value-weighted portfolios whereas the regular regressions are akin to equalweighted portfolios. As well as running regressions with difference of opinion as the sole independent variable, multiple regressions with size, book-to-market, beta, past six-month returns and past one-month returns are also run to ascertain the robustness of any observed relationship between difference of opinion and returns. The control variables specified have been chosen because prior research shows that they are useful in explaining cross-sectional variation in equity returns. To be consistent with prior Australian research by Chan and Faff (2003), the log of size and book-to-market is taken. Beta is calculated using the instrumental variable approach of Fama and French (1992) and specifically following Chan and Faff (2003) and Gharghori et al. (2009). To calculate beta, each month, pre-ranking betas are estimated for each firm using the previous 48 months of returns for firms with a minimum of 24 valid monthly returns during this period. A Dimson (1979) adjustment using one lead and one lag is employed in these regressions. Firms are then sorted into sextiles based on market capitalisation each month. Within each of the size sextiles, firms are further sorted into six groups based on pre-ranking betas. Portfolio returns are calculated for each of the 36 size-beta sorted portfolios using individual firm returns from the sorting month. Portfolio betas are then estimated using the time-series of returns from January 1990 to December 2005. Lastly, the 36 portfolio betas are assigned to each of the firms in the respective portfolios in a given month. Past six-month returns are measured as the cumulated returns over the past six-months with a onemonth lag. Past six-month returns are preferred to past 12-month returns, as Gharghori et al. (2008) show that momentum (Jegadeesh and Titman, 1993) is stronger in Australia over the past six-months. Past onemonth returns are included to control for the short-term reversal phenomenon documented by Jegadeesh (1990) in the US and by Gaunt and Gray (2003) in Australia. In the context of our earlier discussion, the past one-month return is the contemporaneous return measured in the same month in which each difference of opinion proxy is measured. These variables and the other control variables are regressed

440

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

Table 1 Returns and alphas for difference of opinion sorted portfolios. Q5

Q5 - Q1

T3 - T1

Panel A: dispersion in analysts' Excess 0.0040 Return (1.41) CAPM 0.0005 Alpha (0.29) FF − 0.0043 Alpha (− 2.33) Carhart 0.0001 Alpha (0.06)

Q1

earnings forecasts–equal-weighted 0.0052 0.0044 0.0039 (2.21) (1.62) (1.22) 0.0020 0.0009 − 0.0002 (1.70) (0.57) (− 0.11) 0.0001 − 0.0031 − 0.0068 (0.10) (− 1.66) (− 3.52) 0.0010 − 0.0013 − 0.0040 (0.59) (− 0.66) (− 2.10)

Q2

Q3

Q4

− 0.0014 (− 0.34) − 0.0059 (− 2.06) − 0.0187 (− 6.76) − 0.0129 (− 4.41)

− 0.0054 (− 2.06) − 0.0065 (− 2.49) − 0.0144 (− 4.91) − 0.0131 (− 4.34)

− 0.0035 (− 1.65) − 0.0046 (− 2.25) − 0.0113 (− 5.01) − 0.0093 (− 4.12)

Panel B: dispersion in analysts' Excess 0.0058 Return (2.32) CAPM 0.0027 Alpha (1.82) FF 0.0054 Alpha (3.26) Carhart 0.0049 Alpha (2.71)

earnings forecasts–value-weighted 0.0046 0.0044 0.0015 (1.74) (1.54) (0.43) 0.0010 0.0006 − 0.0031 (0.79) (0.41) (− 1.78) 0.0038 0.0008 − 0.0032 (2.54) (0.43) (− 1.56) 0.0052 − 0.0008 − 0.0008 (3.32) (− 0.41) (− 0.35)

− 0.0004 (− 0.09) − 0.0049 (− 2.01) − 0.0107 (− 3.63) − 0.0109 (− 3.47)

− 0.0062 (− 1.88) − 0.0076 (− 2.42) − 0.0161 (− 4.46) − 0.0158 (− 4.13)

− 0.0056 (− 2.06) − 0.0069 (− 2.65) − 0.0126 (− 4.31) − 0.0103 (− 3.22)

Panel C: dispersion in analysts' Excess 0.0054 Return (1.84) CAPM 0.0019 Alpha (1.00) FF − 0.0026 Alpha (− 1.28) Carhart 0.0016 Alpha (0.71)

earnings forecasts: 0.0045 (1.72) 0.0013 (0.79) − 0.0012 (− 0.62) − 0.0001 (− 0.05)

June financial year-end only–equal-weighted 0.0013 0.0022 − 0.0029 (0.45) (0.70) (− 0.68) − 0.0022 − 0.0017 − 0.0074 (− 1.23) (− 0.82) (− 2.19) − 0.0057 − 0.0074 − 0.0219 (− 2.60) (− 3.38) (− 7.05) − 0.0037 − 0.0042 − 0.0164 (− 1.56) (− 1.84) (− 4.75)

− 0.0084 (− 2.66) − 0.0093 (− 2.87) − 0.0193 (− 5.72) − 0.0179 (− 5.17)

− 0.0070 (− 2.98) − 0.0079 (− 3.41) − 0.0151 (− 5.90) − 0.0128 (− 4.97)

Panel D: dispersion in analysts' Excess 0.0082 Return (3.13) CAPM 0.0052 Alpha (2.92) FF 0.0094 Alpha (4.58) Carhart 0.0086 Alpha (3.84)

earnings forecasts: June financial year-end only–value-weighted 0.0024 0.0028 − 0.0001 − 0.0016 (0.85) (0.86) (− 0.04) (− 0.40) − 0.0011 − 0.0008 − 0.0047 − 0.0058 (− 0.65) (− 0.32) (− 1.90) (− 1.88) 0.0002 − 0.0010 − 0.0052 − 0.0121 (0.12) (− 0.32) (− 1.94) (− 3.40) 0.0011 − 0.0025 − 0.0015 − 0.0133 (0.51) (− 0.87) (− 0.50) (− 3.33)

− 0.0098 (− 2.61) − 0.0111 (− 2.93) − 0.0216 (− 4.98) − 0.0218 (− 4.50)

− 0.0070 (− 2.30) − 0.0084 (− 2.86) − 0.0143 (− 4.06) − 0.0132 (− 3.51)

Panel E: abnormal turnover–equal-weighted Excess 0.0137 0.0095 Return (2.58) (2.24) CAPM 0.0093 0.0059 Alpha (2.06) (1.63) FF − 0.0197 − 0.0181 Alpha (− 9.41) (− 10.76) Carhart − 0.0167 − 0.0156 Alpha (− 5.33) (− 5.89)

0.0067 (1.96) 0.0034 (1.26) − 0.0140 (− 9.31) − 0.0093 (− 5.52)

0.0085 (2.66) 0.0051 (2.16) − 0.0083 (− 5.50) − 0.0047 (− 2.45)

0.0167 (3.70) 0.0122 (3.50) − 0.0107 (− 6.43) − 0.0094 (− 3.95)

0.0030 (1.16) 0.0029 (1.15) 0.0090 (3.79) 0.0072 (2.41)

0.0010 (0.46) 0.0011 (0.49) 0.0097 (5.24) 0.0091 (3.79)

Panel F: abnormal turnover–value-weighted Excess 0.0016 0.0034 Return (0.35) (0.95) CAPM − 0.0031 − 0.0005 Alpha (− 0.86) (− 0.20) FF − 0.0171 − 0.0115 Alpha (− 5.64) (− 4.72) Carhart − 0.0167 − 0.0115 Alpha (− 4.37) (− 3.82)

0.0009 (0.29) − 0.0029 (− 1.56) − 0.0071 (− 3.44) − 0.0054 (− 2.61)

0.0015 (0.56) − 0.0021 (− 1.44) − 0.0023 (− 1.21) − 0.0003 (− 0.13)

0.0044 (1.72) 0.0006 (0.89) 0.0019 (2.44) 0.0017 (2.06)

0.0027 (0.72) 0.0036 (0.96) 0.0190 (5.77) 0.0184 (4.51)

0.0003 (0.09) 0.0008 (0.27) 0.0150 (5.41) 0.0149 (4.31)

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

441

Table 1 (continued) Q3

Q4

Q5

Q5 - Q1

T3 - T1

Panel G: idiosyncratic volatility–equal-weighted Excess 0.0161 0.0066 Return (5.68) (2.43) CAPM 0.0136 0.0032 Alpha (5.95) (2.00) FF 0.0014 − 0.0052 Alpha (0.93) (− 4.35) Carhart 0.0039 − 0.0021 Alpha (2.15) (− 1.36)

Q1

Q2

0.0070 (1.74) 0.0029 (0.95) − 0.0161 (− 9.74) − 0.0134 (− 5.62)

0.0079 (1.35) 0.0030 (0.59) − 0.0293 (− 12.99) − 0.0279 (− 7.07)

0.0144 (1.98) 0.0091 (1.41) − 0.0329 (− 12.06) − 0.0291 (− 7.16)

− 0.0018 (− 0.31) − 0.0045 (− 0.81) − 0.0343 (− 9.99) − 0.0330 (− 6.51)

− 0.0003 (− 0.07) − 0.0026 (− 0.52) − 0.0314 (− 10.81) − 0.0318 (− 7.11)

Panel H: idiosyncratic volatility–value-weighted Excess 0.0040 0.0038 Return (1.54) (1.37) CAPM 0.0003 − 0.0001 Alpha (0.34) (− 0.09) FF 0.0030 − 0.0001 Alpha (2.73) (− 0.10) Carhart 0.0034 0.0008 Alpha (2.49) (0.57)

0.0025 (0.67) − 0.0019 (− 0.78) − 0.0094 (− 3.58) − 0.0084 (− 2.84)

− 0.0068 (− 1.21) − 0.0123 (− 2.72) − 0.0307 (− 6.47) − 0.0304 (− 4.44)

− 0.0166 (− 2.51) − 0.0224 (− 3.98) − 0.0459 (− 8.80) − 0.0417 (− 5.51)

− 0.0205 (− 3.41) − 0.0227 (− 3.74) − 0.0488 (− 8.69) − 0.0451 (− 5.46)

− 0.0188 (− 3.56) − 0.0209 (− 3.92) − 0.0448 (− 8.40) − 0.0450 (− 5.26)

This table presents returns and alphas for difference of opinion sorted portfolios. Every month, stocks are sorted into quintiles and tritiles based on difference of opinion. Portfolio excess returns are calculated based on these sorts. The time-series of portfolio returns are then used to obtain CAPM, Fama–French model and Carhart model alphas by regressing the excess returns against the respective models. The table reports the time-series average of the monthly portfolio excess returns over the sample period 1990–2005 and the model alphas over this period. T-statistics are reported directly below in parentheses. Q1 to Q5 represent the lowest difference of opinion quintile (Q1) to the highest quintile (Q5). Q5 − Q1 is a zero-cost portfolio that is long high difference of opinion stocks and short low difference of opinion stocks. Similarly, T3 − T1 is also a zero-cost portfolio but is based on tritile groupings rather than quintiles. Panels A and B report excess returns and alphas using dispersion in analysts' earnings forecasts as a proxy for difference of opinion for equal- and value-weighted portfolios, respectively. Dispersion in analysts' earnings forecasts is the standard deviation of all analyst forecasts for a given stock reported in I/B/E/S for a given month scaled by the absolute value of the mean of those forecasts. Panels C and D also use dispersion in analyst forecasts as a difference of opinion proxy. However, in Panels C and D, only stocks that have a June financial year-end are included in the sample. In Panels E and F, abnormal turnover is used to proxy difference of opinion. Abnormal turnover is obtained by running monthly cross-sectional regressions of stock turnover on size, book-to-market and industry dummies. Stock turnover is defined as monthly trading volume divided by number of issued shares. Size is proxied by market capitalisation. Book-to-market is Net Tangible Assets divided by market capitalisation. 28 dummy variables are used to control for the 28 industry classifications specified by S&P via GICS. Abnormal turnover is the residual from the aforementioned regression for each firm. In Panels G and H, idiosyncratic volatility is the difference of opinion proxy. Idiosyncratic volatility is measured as the standard deviation of daily Fama–French model residuals over a month.

against returns in the following month. The multiple regression, which includes difference of opinion and all the control variables, can be specified as follows: Ri ¼ γ0 þγ1 DOi þγ2 lnðSizeÞi þγ3 lnðB=MÞi þγ4 Betai þγ5 Ret6i þγ6 Ret1i þεi

ð1Þ

where Ri is next month's return, DO is difference of opinion, Size is size, B/M is book-to-market, Beta is beta, Ret6 is the past six-month return and Ret1 is the past month's return. 3. Empirical findings 3.1. Returns and alphas for difference of opinion sorted portfolios Table 1 reports the excess returns and alphas for difference of opinion sorted portfolios. In Panels A and B, we report equal- and value-weighted returns, respectively and the corresponding alphas using

442

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

dispersion in analysts' forecasts as a proxy for difference of opinion. Panels C and D also report equal- and value-weighted returns and the respective alphas using dispersion in analysts' forecasts but only for stocks that have a June financial year-end. In Panels E and F, portfolio returns and alphas are presented for abnormal turnover sorted portfolios. Finally, in Panels G and H, returns and alphas are reported when idiosyncratic volatility is the proxy for difference of opinion. Panel A shows that an equal-weighted zero-cost portfolio (Q5 − Q1) that is long high difference of opinion stocks and short low difference of opinion stocks generates negative returns and that the return differential is statistically significant (−0.54% per month). On closer inspection, the relationship is non-linear in that portfolios Q1 to Q4 have similar returns whereas Q5 has a considerably lower return. Thus, the stocks with the highest difference of opinion underperform. Similarly, the zero-cost tritile portfolio (T3 − T1) generates a negative return (−0.35% per month) but this return is only significant at the 10% level. The alphas for the zerocost portfolios are statistically negatively significant implying that after risk-adjustment, high difference of opinion stocks underperform relative to low difference of opinion stocks. Although the return for the quintile zero-cost portfolio is larger in magnitude for the value-weighted portfolio in Panel B relative to the equalweighted portfolio in Panel A (−0.62% versus −0.54% per month), this return differential is only significant at the 10% level. However, the return on the tritile zero-cost portfolio in Panel B is significantly negative at the 5% level. Further, in comparison to Panel A where there was a non-linear relationship between difference of opinion and returns, in Panel B there is more of an ordered decrease in the excess returns as difference of opinion increases. Similar to Panel A, all of the risk-adjusted returns on the zero-cost portfolios are significantly negative. Moreover, there is a much clearer trend in the alphas for the value-weighted portfolios. We show that the low difference of opinion portfolios significantly outperforms relative to the models, the alphas of the mid-range portfolios are insignificant and the highest difference of opinion portfolio (Q5) significantly underperforms. The results in Panels C and D where only stocks with a June financial year-end are included in the dispersion in analysts' forecast sample are consistent with the results reported in Panels A and B. If anything, the findings are even stronger, as all of the returns and alphas for the zero-cost portfolios are significantly negative. Thus, the findings are robust to any seasonal variation in analysts' forecasts. Taken together, the results in Panels A to D where dispersion in analysts' forecasts is employed as the proxy shows that there is a negative relationship between difference of opinion and returns. This finding is consistent with Diether et al. (2002) and Miller (1977). In Panel E, where abnormal turnover is the difference of opinion proxy, the equal-weighted returns on both the quintile and tritile zero-cost portfolios are insignificant. A closer observation of each quintile portfolio shows that there is a positive curvilinear (U-shaped) relationship between difference of opinion and returns. Specifically, the average portfolio return falls from Q1 to Q3 and then increases to Q5. This pattern also holds for the CAPM alphas. In contrast, there is a generally increasing trend in the Fama– French and Carhart alphas where the alphas increase from Q1 to Q4 and then decrease slightly for Q5. Moreover, all of the Fama–French and Carhart alphas are significantly negative but in spite of this, the alpha spread for the zero-cost portfolios is significantly positive. This occurs because although the alpha for Q5 is negatively significant, it is larger than the alpha for Q1 (Q5's Carhart alpha is − 0.0094 versus a Carhart alpha of − 0.0167 for Q1). The contrasting pattern in the excess returns and CAPM alphas compared with the Fama–French and Carhart alphas warrants further investigation. Unreported findings show that the average market capitalisation of the portfolios increases as abnormal turnover increases. 7This is despite the fact that size was used as a control variable when constructing abnormal turnover. Thus, it appears that the effect of size was not completely purged from the abnormal turnover measure. 8Gharghori et al. (2009) show that in Australia, the smallest and largest firms earn the highest returns; therefore, this observation coupled with the size pattern in the portfolios is a potential reason why we observe a U-shaped relationship between abnormal turnover and returns. Further unreported findings show that all five portfolios have significantly positive coefficients on SMB and HML in both the Fama–French and Carhart regressions (there is no such pattern for the momentum factor). Due to the large SMB and HML premiums in Australia (see Gharghori et al., 2007, this results in the models 7

Unreported findings discussed in the paper are available on request. However, the dispersion in size for the abnormal turnover sorted portfolios is much less than the dispersion in size for corresponding market cap sorted portfolios. 8

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

443

generating a high expected return for the portfolios and subsequently a negative and significant alpha. Further, Q5 has a lower exposure to SMB and HML than Q1. This explains the larger alphas on Q5 relative to Q1 and the significantly positive alphas for the quintile and tritile zero-cost portfolios. Panel F presents the returns and alphas for value-weighted portfolios sorted by abnormal turnover. There is no pattern of note in the returns and CAPM alphas of the quintile portfolios and the return and CAPM alpha spread for the zero-cost portfolios are insignificant. In contrast, the Fama–French and Carhart alphas increase as difference of opinion increases and as a result, the spreads in the alphas for the zero-cost portfolios are significantly positive. A positive alpha spread was also observed in the equal-weighted portfolios (Panel E) however, the finding is stronger in the value-weighted portfolios (Panel F). The reason for this is that Q1 (Q5) has a significantly negative (positive) Fama–French and Carhart alpha for the valueweighted portfolios whereas with the equal-weighted portfolios, although the spread was significant, both extreme portfolios had negatively significant alphas. It is again pertinent to examine the unreported coefficients to obtain further insight on the conflicting findings. The positive alpha spread from the Fama–French and Carhart models for the value-weighted portfolios (Panel F) is driven by lower betas and lower SMB and HML coefficients for Q5 relative to Q1. Moreover, Q1 (Q5) has a significantly positive (negative) coefficient on SMB and whereas the HML coefficient on Q1 is insignificant, the HML coefficient on Q5 is significantly negative. The higher beta and the positive coefficient on SMB for Q1 coupled with the strong SMB premium results in a high expected return from the Fama–French and Carhart models and subsequently a significantly negative alpha from these models for Q1. In contrast, the lower beta and the negative coefficients on SMB and HML for Q5 along with the strong SMB and HML premiums generate the significantly positive alpha for Q5. In summary, we observe mixed results with the abnormal turnover proxy. First, we see a U-shaped relationship in the equal-weighted portfolio returns but no relationship in the value-weighted returns. Further, the alpha spread for the Fama–French and Carhart models are significantly positive whereas the return and CAPM alpha spreads are insignificant. The Fama–MacBeth analysis will help provide clarity on these findings because variables such as size and book-to-market will be directly controlled for in the regressions. Panel G presents the results for equal-weighted idiosyncratic volatility sorted portfolios. The zero-cost quintile and tritile portfolio returns are negative but insignificant (− 0.0018 and −0.0003, respectively). On closer inspection, there is a U-shaped relationship between idiosyncratic volatility and excess returns. This relationship is also observed in the CAPM alphas. These findings are similar to those observed for the equal-weighted abnormal turnover sorted portfolios. In contrast, there is a monotonic downward trend in the Fama–French and Carhart alphas as idiosyncratic volatility increases. Further, the Fama–French and Carhart alpha spreads are negatively significant. An observation of the unreported regression coefficients shows that the betas and SMB coefficients increase as idiosyncratic volatility increases. This is driving the negative relationship between idiosyncratic volatility and the Fama–French and Carhart alphas. In Panel H for the value-weighted portfolios, the findings are much clearer. There is a monotonic decrease in the excess returns and the risk-adjusted returns from all three models as idiosyncratic volatility increases. Moreover, all of the return spreads for the zero-cost portfolios are statistically negatively significant. On balance, the findings in Panels G and H suggest a negative relationship between idiosyncratic volatility and returns. However, in the equal-weighted portfolios, the relationship between idiosyncratic volatility and both the excess returns and the CAPM alphas is U-shaped. 3.2. Fama–Macbeth regressions Table 2 reports the results of Fama–MacBeth regressions of future stock returns on each difference of opinion proxy and a number of control variables. Correspondingly with Table 1, Panels A and B present the findings for equal- and value-weighted estimations, respectively where dispersion in analysts' forecasts is the proxy. Panels C and D report the findings for the dispersion in analysts' forecasts sample where all firms have June financial year-ends. In Panels E and F, abnormal turnover is the difference of opinion proxy. Finally, Panels G and H report the results for idiosyncratic volatility. For both the equal- and value-weighted estimations in Panels A and B, a significant negative relationship is observed between dispersion in analysts' earnings forecasts and returns. This finding is observed when difference of opinion (DO) is the sole variable in the regression and when size, book-to-

444

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

Table 2 Fama–MacBeth regressions of monthly stock returns on difference of opinion and a number of control variables. Constant

Beta

Ret 6

Ret 1

Panel A: dispersion in analysts' earnings forecasts–equal-weighting 0.0055 − 0.0063 (2.06) (− 2.37) 0.0067 − 0.0073 0.0005 0.0032 (0.47) (− 2.80) (0.72) (3.20)

DO

Size

B/M

− 0.0159 (− 2.85)

0.0372 (5.89)

0.0279 (2.16)

Panel B: dispersion 0.0051 (1.86) 0.0070 (0.42)

in analysts' earnings forecasts–value-weighting − 0.0066 (− 2.37) − 0.0079 0.0005 0.0032 (− 2.96) (0.62) (2.95)

− 0.0166 (− 2.81)

0.0358 (5.42)

0.0253 (1.90)

Panel C: dispersion 0.0044 (1.56) 0.0178 (1.16)

in analysts' forecasts: June financial year end only–equal-weighting − 0.0084 (− 1.62) − 0.0097 0.0003 0.0027 − 0.0258 (− 1.92) (0.33) (1.56) (− 3.93)

0.0451 (6.64)

0.0334 (2.40)

Panel D: dispersion 0.0040 (1.38) 0.0159 (0.97)

in analysts' forecasts: June financial year end only–value-weighting − 0.0084 (− 1.55) − 0.0099 0.0004 0.0028 − 0.0266 (− 1.94) (0.46) (1.58) (− 3.89)

0.0445 (6.43)

0.0332 (2.33)

Panel E: abnormal turnover–equal-weighting 0.0017 0.0019 (0.45) (0.46) 0.0342 0.0037 − 0.0002 (2.07) (0.91) (− 0.24)

0.0054 (5.77)

− 0.0339 (− 6.48)

0.0143 (3.85)

− 0.0739 (− 10.10)

Panel F: abnormal turnover–value-weighting 0.0025 0.0017 (0.61) (0.39) 0.0508 0.0043 − 0.0010 (2.75) (1.04) (− 1.05)

0.0060 (5.82)

− 0.0372 (− 6.50)

0.0103 (2.70)

− 0.0859 (− 11.28)

Panel G: idiosyncratic volatility–equal-weighting 0.0076 − 0.1906 (2.82) (− 3.37) 0.0530 − 0.1473 − 0.0010 (2.94) (− 4.29) (− 1.28)

0.0046 (5.24)

− 0.0304 (− 6.48)

0.0120 (3.37)

− 0.0786 (− 10.65)

Panel H: idiosyncratic volatility–value-weighting 0.0073 − 0.1598 (2.49) (− 2.96) 0.0683 − 0.1350 − 0.0016 (3.41) (− 4.00) (− 1.89)

0.0049 (5.14)

− 0.0328 (− 6.64)

0.0092 (2.55)

− 0.0884 (− 11.53)

This table reports equal- and value-weighted Fama–MacBeth regression estimates using individual firm data for all months of our sample period — 1990 to 2005. For the equal-weighted regressions, in each month, a cross-sectional regression is estimated using OLS wherein the stock's return is regressed on the variables specified. For the value-weighted regressions, in each month, a crosssectional regression is estimated using WLS where the weight is the inverse of the firms' market cap. The values reported in the table are the average time-series slope estimates. The associated t-statistics are reported in parentheses directly under the relevant mean slope estimate. DO is difference of opinion, Size is the log of the firm's market capitalisation, B/M is the log of the book-to-market ratio, Beta is the market model beta, Ret 6 is the past six-month return with a one-month lag and Ret 1 is the prior month's return. Panels A and B use dispersion in analysts' earnings forecasts as a proxy for difference of opinion for the equal- and value-weighted estimations, respectively. Panels C and D also employ dispersion in analysts' earnings forecasts as the difference of opinion proxy but only includes firms that have a June financial year-end. In Panels E and F, abnormal turnover is the difference of opinion proxy. In Panels G and H, idiosyncratic volatility is the proxy for difference of opinion.

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

445

market, beta, past six-month returns and past one-month returns are controlled for. In Panels C and D for the June financial year-end sample, a negative relationship is also observed. However, when dispersion in analysts' forecasts is the sole variable in the model the relationship is not significant (t-statistic of − 1.62 in Panel C and − 1.55 in Panel D). In the regressions where the control variables are included, the average coefficient is close to significance at the five percent level (t-statistic of −1.92 in Panel C and −1.94 in Panel D). On balance, these findings suggest a negative relationship between dispersion in analysts' forecasts and returns. This result is consistent with the portfolio returns analysis. In Panels E and F, the relationship between abnormal turnover and returns is insignificant. In the portfolio returns analysis for the equal-weighted portfolios, a U-shaped relationship was observed between abnormal turnover and returns. To account for this potential non-linearity, a squared term for abnormal turnover was included in the Fama–MacBeth regressions. Unreported findings show that this coefficient is insignificant for both the equal- and value-weighted estimations and when abnormal turnover is regressed in isolation or with the control variables. Thus, the Fama–MacBeth analysis suggests that there is no relationship between abnormal turnover and returns. Finally, in Panels G and H, there is a significant negative relationship observed between idiosyncratic volatility and returns. This relationship is robust to the equal- and value-weighted estimations and to the inclusion of the control variables in the regressions. This result is generally consistent with the portfolio returns analysis. In summary, the findings indicate that there is a negative relationship between difference of opinion and returns when dispersion in analysts' forecasts and idiosyncratic volatility are employed as proxies for difference of opinion. There are some instances where a negative relationship is not observed, such as in the excess returns of the equal-weighted idiosyncratic volatility sorted portfolios, but in the majority of cases there is a statistically significant negative relationship. This finding is consistent with prior empirical research by Diether et al. (2002) and with Miller's (1977) predictions. In contrast, the findings for the abnormal turnover measure are mixed. There is a U-shaped relationship in the equal-weighted portfolio returns, a significantly positive return spread in the Fama–French and Carhart alphas but no relationship in the Fama–MacBeth regressions. Thus, one cannot reach a clear conclusion on the relationship between abnormal turnover and returns.

4. Conclusions Previous empirical research conducted in the US on the relationship between difference of opinion and stock returns has yielded mixed results. This paper provides out of sample evidence on the relationship between difference of opinion and stock returns. The study employs multiple proxies (dispersion in analysts' earnings forecasts, abnormal turnover and idiosyncratic volatility) for measuring difference of opinion among investors. Overall, the results document a significant negative relation between difference of opinion (using dispersion in analysts' forecasts and idiosyncratic volatility as proxies) and stock returns. This result provides support for Miller's (1977) hypothesis and is consistent with the findings of Diether et al. (2002). For the abnormal turnover proxy, the findings are mixed and no clear conclusion can be reached. Thus, the study highlights the importance of using a number of difference of opinion proxies.

Acknowledgements We thank Henk Berkman, Robert Faff, John Doukas and Jon Garfinkel for helpful comments and discussions. We are also grateful to an anonymous referee and the editor (Professor Kalok Chan) for many insightful comments and suggestions. The helpful comments of participants at the 2007Australasian Finance and Banking Conference, the 2008 Finsia-MCFS conference, the 2009 Paul Woolley Centre workshop at the University of Technology Sydney, the 2009 European Financial Management Association annual meetings, the 2010 Mid-West Finance Association conference and the 2010 Multinational Finance Society conference are gratefully acknowledged. Particular thanks to an anonymous referee for many helpful comments and suggestions. We are thankful to Linh Pham and Paul Dou for research assistance. The financial assistance of the Australian Centre for Financial Studies (1779320) is gratefully acknowledged. Gharghori and Veeraraghavan are centre associates of the Australian Centre for Financial Studies.

446

P. Gharghori et al. / Pacific-Basin Finance Journal 19 (2011) 435–446

References Ackert, L.F., Athanassakos, G., 1997. Prior uncertainty, analyst bias and subsequent abnormal returns. Journal of Financial Research 20, 263–273. Ang, A., Hodrick, R.J., Xing, Y., Zhang, X., 2006. The cross-section of volatility and expected returns. Journal of Finance 61, 259–299. Ang, A., Hodrick, R.J., Xing, Y., Zhang, X., 2009. High idiosyncratic volatility and low returns: international and further U.S. evidence. Journal of Financial Economics 91, 1–23. Banz, R.W., 1981. The relationship between return and market value of common stocks. Journal of Financial Economics 9, 3–18. Basu, S., 1983. The relationship between earnings yield, market value, and return for NYSE common stocks: further evidence. Journal of Financial Economics 12, 129–156. Berkman, H., Dimitrov, V., Jain, P.C., Koch, P.D., Tice, S., 2009. Sell on the news: difference of opinion, short-sales constraints and returns around earnings announcements. Journal of Financial Economics 92, 376–399. Bhandari, L.C., 1988. Debt/equity ratio and expected common stock returns: empirical evidence. Journal of Finance 43, 507–528. Carhart, M., 1997. On persistence in mutual fund performance. Journal of Finance 52, 57–82. Chan, H.W.H., Faff, R.W., 2003. An investigation into the role of liquidity in asset pricing: Australian evidence. Pacific-Basin Finance Journal 11, 555–572. Chan, L.K.C., Hamao, Y., Lakonishok, J., 1991. Fundamentals and stock returns in Japan. Journal of Finance 1739–1789. Diamond, D., Verrecchia, R., 1987. Constraints on short-selling and asset price adjustment to private information. Journal of Financial Economics 18, 277–311. Diether, K., Malloy, C., Scherbina, A., 2002. Differences of opinion and the cross section of stock returns. Journal of Finance 57, 2113–2141. Dimson, E., 1979. Risk measurement when shares are subject to infrequent thin trading. Journal of Financial Economics 7, 197–226. Dische, A., 2002. Dispersion in analyst forecasts and the profitability of earnings momentum strategies. European Financial Management 8, 211–228. Doukas, J., Kim, C., Pantzalis, C., 2002. A test of the error-in-expectations explanation of the value/glamour stock returns performance: evidence from analysts' forecasts. Journal of Finance 57, 2143–2165. Doukas, J., Kim, C., Pantzalis, C., 2004. Divergent opinions and the performance of value stocks. Financial Analysts Journal 60, 55–64. Fama, E.F., 1991. Efficient capital markets II. Journal of Finance 36, 1575–1617. Fama, E.F., French, K.R., 1992. The cross-section of expected stock returns. Journal of Finance 47, 427–465. Fama, E.F., French, K.R., 1996. Multifactor explanations of asset pricing anomalies. Journal of Finance 51, 55–84. Fama, E.F., MacBeth, J., 1973. Risk, return and equilibrium: empirical tests. Journal of Political Economy 81, 607–636. Garfinkel, J.A., Sokobin, J., 2006. Volume, opinion divergence and returns: a study of post-earnings announcement drift. Journal of Accounting Research 44, 85–112. Gaunt, C., Gray, P., 2003. Short-term autocorrelation in Australian equities. Australian Journal of Management 28, 97–117. Gharghori, P., Chan, H.W.H., Faff, R.W., 2007. Are the Fama–French factors proxying default risk? Australian Journal of Management 32, 223–249. Gharghori, P., Lee, R., Veeraraghavan, M., 2008. Are stock returns related to short-term and long-term past returns? Australian evidence. Applied Financial Economic Letters 4, 277–282. Gharghori, P., Chan, H.W.H., Faff, R.W., 2009. Default risk in equity returns: Australian evidence. Pacific-Basin Finance Journal 17, 580–593. Hintikka, M., 2008. Market reactions to differences of opinion. Working Paper, The Swedish School of Economics and Business Administration. Hong, H., Stein, J., 2000. Bad news travels slowly: size, analyst coverage, and the profitability of momentum strategies. Journal of Finance 55, 265–295. Jegadeesh, N., 1990. Evidence of predictable behaviour of security returns. Journal of Finance 45, 881–898. Jegadeesh, N., Titman, S., 1993. Returns to buying winners and selling losers: implications for stock market efficiency. Journal of Finance 48, 65–91. Leippold, M., Lohre, H., 2010. The dispersion effect in international stock returns. Working Paper, Imperial College, London. Miller, E.M., 1977. Risk, uncertainty, and divergence of opinion. Journal of Finance 32, 1151–1168. Varian, H.R., 1985. Divergence of opinion in complete markets: a note. Journal of Finance 40, 309–317.