A behavioral explanation of the value anomaly based on time-varying return reversals

Journal of Banking & Finance 37 (2013) 2367–2377

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A behavioral explanation of the value anomaly based on time-varying return reversals Soosung Hwang a,⇑, Alexandre Rubesam b a b

School of Economics, Sungkyunkwan University, 25-2, Sungkyunkwan-ro, Jongno-gu, Seoul 110-745, South Korea Itaú-Unibanco, São Paulo, Brazil

a r t i c l e

i n f o

Article history: Received 9 February 2012 Accepted 20 January 2013 Available online 15 February 2013 JEL classification: G12 Keywords: Overconfidence Self-attribution bias Value anomaly Return reversals

a b s t r a c t We investigate the dynamics of the value anomaly in order to identify the driving forces of the anomaly. We show that the large positive value-minus-growth portfolio returns are explained by an over-reaction (under-reaction) to the positive (negative) market movements in short, specific time periods, during which the average returns of value-minus-growth portfolios are more than 2% a month. We propose an explanation based on behavioral biases: the dynamics of the value anomaly reflect the increased speed of return reversals subsequent to overreaction. Two conditions that increase the return reversals are proposed: when investors respond to public signals asymmetrically or when public signals become noisy. Our empirical results reveal that the value anomaly is explained by either one of these two channels. Ó 2013 Elsevier B.V. All rights reserved.

1. Introduction Various explanations have been proposed for the value anomaly since it was identified by Rosenberg et al. (1985). Despite these efforts, however, studies that empirically try to explain the value anomaly are only partly successful; for example, Jagannathan and Wang (1996), Petkova and Zhang (2005), Lewellen and Nagel (2006), and Chen et al. (2008), find only weak evidence to suggest that the increased risk of value firms during recessions is responsible for the value anomaly. The value anomaly remains positive and significant after controlling for time-varying risk. We revisit the value anomaly by focusing on investors’ behavioral biases. According to the behavioral finance literature, price-to-fundamental ratios represent measures of mispricing, and the mispricing is subsequently corrected (Lakonishok et al., 1994; Daniel et al., 1998). The speed of the correction subsequent to mispricing, however, may not be time-invariant. We argue that the dynamics of the value anomaly reflect the increased speed of return reversals subsequent to overreaction, and propose two conditions under which the correction intensifies: when investors respond to public signals asymmetrically or when public signals become noisy. Our explanation of the value anomaly based on behavioral biases is motivated by stylized patterns of the anomaly, which we identify using a model that switches between three different ⇑ Corresponding author. Tel.: +82 (0)2 760 0489; fax: +82 (0)2 744 5717. E-mail addresses: [email protected] (S. Hwang), alexandre.rubesam@itau-uni banco.com.br (A. Rubesam). 0378-4266/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jbankfin.2013.01.030

conventional risk measures: i.e., CAPM beta (Rosenberg et al., 1985; Fama and French, 1992), higher moments (Harvey and Siddique, 2000), and downside beta (Petkova and Zhang, 2005). The empirical results from the regime-switching model clearly show that none of the risk measures support the claim that value is riskier than growth. Instead we find significant average return differences among the three regimes. The average value-minus-growth (VMG) portfolio returns are more than 2% per month during the downside beta regime, whereas they are close to zero and around 1% per month during the CAPM beta and the higher moments regimes, respectively. The large positive VMG portfolio returns in the downside beta regime come from asymmetric responses of value stocks to market movements: in this regime, value stocks are more sensitive to positive market returns than to negative market returns; thus, upside beta is larger than downside beta by at least 0.6 for these stocks. The high returns from the asymmetric responses in the downside beta regime are not well explained rationally since assets with large upside betas should not be compensated with high expected returns (Ang et al., 2006). As an explanation, we investigate if the dynamics of the value anomaly come from changes in the correction of mispricing (Lakonishok et al., 1994). We extend the Bayesian optimization model proposed by Daniel et al. (1998) (DHS) by allowing a non-zero payoff in the prior mean and heteroskedasticity in the noise term of public signals. The non-zero payoff reflects the mispricing (i.e., positive average returns) of VMG portfolios created by overconfident investors (Lakonishok et al., 1994), whereas the

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heteroskedasticity reflects the increase of VMG portfolio return volatility in bear markets (Chen et al., 2008; Gulen et al., 2011). Our model suggests two cases in which the speed of the correction increases: when public signals become noisy, or when investors’ prior beliefs are confirmed by noisy public signals. When public signals are noisy, they are not informative for updating the prior distribution, and thus investors rely on their prior beliefs that value (growth) stocks are underpriced (overpriced). On the other hand, when public signals are not too noisy, investors may asymmetrically react to public signals because of their self-attribution bias. They may overreact to positive signals for value stocks since the signals confirm their prior beliefs that underpriced value stocks would show positive returns, whereas they underreact to positive signals for growth stocks since the signals are inconsistent with their beliefs that overpriced growth stocks would show negative returns. We find empirical evidence that lends support to the model. The value anomaly is explained by either the volatility of the noise term in public signals or by public signals that confirm what the investors believe, and these two channels operate separately. Idiosyncratic volatility, which we use as a proxy for the volatility of the noise term in public signals, is positively related to value and VMG portfolio returns. Value and VMG portfolio returns also increase (growth portfolio returns decrease) with sentiment, which we use as a proxy for the public signal. We also find that when sentiment or idiosyncratic volatility is high, the downside beta regime is more likely to be observed than the other two regimes. Therefore, the large and positive average VMG portfolio returns come from the correction of mispricing, which is accelerated during the downside beta regime. The correction is more severe for value stocks than for growth stocks, because stocks with higher uncertainty (e.g., idiosyncratic volatility) are subject to greater mispricing than those with less uncertainty (DHS, 2001; Baker and Wurgler, 2006). This could be an explanation for why approximately 80% of the value anomaly comes from value rather than growth stocks. Our results differ from the studies of Zhang (2005), Petkova and Zhang (2005), Chen et al., (2008) and Gulen et al. (2011), who argue that the countercyclical pattern of the value anomaly reflects the increased risk of value stocks during bad times. In our study, the dynamics of the value anomaly are interpreted as changes in the speed of return reversals. Value stocks appear to outperform growth stocks during bad times since the underpricing of value stocks is corrected faster during bear markets when volatility increases. The remainder of this study is organized as follows. We introduce the regime-switching model and present the empirical results in the following section. In Section 3, we propose our explanation of the value anomaly based on behavioral biases. Finally, Section 4 concludes the paper. 2. The dynamics of risk measures for VMG portfolio returns In this section, the dynamics of the value anomaly are investigated with respect to three well-known risk measures in a regime-switching model. The regime-switching model helps us discover stylized patterns in the value anomaly, which we use to develop a model based on behavioral biases in the next section. 2.1. The regime-switching model Using a method similar to that of Guidolin and Timmermann (2008), we propose a regime-switching model as follows:

r pt ¼ a þ S1t ½br mt  þ S2t ½bþ r þmt þ b r mt  þ S3t ½b1 r mt þ b2 ðRmt  EðRmt ÞÞ2 þ b3 ðRmt  EðRmt ÞÞ3  þ

3 X Sjt r2j et ; j¼1

ð1Þ

where rmt = Rmt  Rft, the market return in excess of the risk-free rate, and rj is the idiosyncratic volatility in regime j, and et  N(0, 1).  Note that rþ mtþ1 ¼ r mtþ1 Iðr mtþ1 P r target Þ and r mtþ1 ¼ r mtþ1 Iðr mtþ1 < rtarget Þ are the positive and negative components of the excess market return with respect to the target return rtarget, I(.) is the indicator variable, and b and b+ are functions of semi-variance and semicovariance, as in Ang et al. (2006). Following Hamilton (1989), we allow the state variable St to be governed by a first-order Markov chain with a transition probability matrix P = {pij}, where pij = P(St = j|St1 = i) is the probability that regime i at time t  1 is followed by regime j at time t. Three different risk measures matter in the regime-switching model depending on the dummy variable, i.e., Sjt, j = 1, 2, 3. When S1t = 1 and S2t = S3t = 0 (CAPM beta regime), the CAPM beta (b) is the risk measure. On the other hand, when S2t = 1 and S1t = S3t = 0 (downside beta regime), the downside beta (b) matters for VMG portfolio returns (Petkova and Zhang, 2005). Finally, when S3t = 1 and S1t = S2t = 0 (higher moments regime), higher moment betas (b1, b2, and b3) become the risk measures (Harvey and Siddique, 2000). The regime-switching model reduces mis-specification problems because of its flexibility and at the same time provides answers to the questions of whether and when the three risk measures matter in practice. It allows three different types of risk over time, and thus, it is not the same as other conditional models that allow timevarying risk within a type of risk, e.g., conditional CAPM that allows time-varying beta. Unlike our approach, other studies in the value anomaly literature first identify market states, and then investigate the difference in risk between market states (e.g., Petkova and Zhang, 2005; Gulen et al., 2011). These studies may not identify the risk measures that matter over different periods.1 The regime-switching model (1) is estimated using a Bayesian Markov Chain Monte Carlo (MCMC) Gibbs sampling approach. Diffuse priors are used for the parameters since our purpose in this study is to investigate the way in which the three risk measures are selected over time.2 All results are calculated with 10,000 draws after 30,000 burn-in iterations. The details of the estimation method can be obtained from the authors upon request.

2.2. Is value riskier than growth? The regime-switching model is applied to various VMG portfolios for the sample period from July 1963 to December 2008. The main results are reported for Fama and French’s (1992) VMG portfolio formed on the two-by-three sort on size and book-to-market (VMG_SBM), together with four other VMG portfolios formed on the top and bottom 30% of earnings-to-price (VMG_EP), cash flow-to-price (VMG_CP), dividends-to-price (VMG_DP), and bookto-market (VMG_BM). We also analyze VMG portfolios formed on the top and bottom decile portfolios (DVMG portfolios) using these ratios. The excess market return is the CRSP value-weighted portfolio return minus the one-month Treasury bill rate. All the monthly data are obtained from Kenneth French’s data library. 1 Despite the advantages of our regime-switching model, a few limitations are worth noting. Our results are specific to the regime-switching model since we only focus on the three conventional risk measures only and do not directly test other rational approaches in the value anomaly literature. Examples are real growth options and the inflexibility of value firms (Berk et al., 1999; Zhang, 2005), or the difference in the duration of cash flows between value and growth firms (Lettau and Wachter, 2007). Another concern may be that our model is over-parameterized with the latent variable (St) in addition to the parameters (a, b, b+, b1, b2, b3, r1, r2, and r3). The results may also be affected by assumptions such as the first-order Markov chain or others that are required for the estimation of the model. 2 The downside and upside betas (b and b+) in the downside beta regime are cov ðr pt ;rmt jrm
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Table 1 Estimation results for the regime-switching risk measure model for the post-1963 period. We estimate the following regime-switching model with a Gibbs-sampling MCMC method:

r pt ¼ a þ S1t ½br mt  þ S2t ½bþ r þmt þ b r mt  þ S3t ½b1 r mt þ b2 ðRmt  EðRmt ÞÞ2 þ b3 ðRmt  EðRmt ÞÞ3  þ

3 X Sjt r2j et ; j¼1

where rpt is the value-minus-growth portfolio return or the excess return on either a value or growth portfolio, Sjt = I(St = j) is a dummy-variable indicating the occurrence of regime j = 1, 2, 3 (where regime 1 = CAPM beta, regime 2 = downside beta, regime 3 = higher moments), rmt denotes excess market return, r  mt ¼ r mt Iðr mt < 0Þ is the negative component of the market return, r þ mt ¼ r mt Iðr mt > 0Þ is the positive component of the market return, Rmt is the raw market return and et  N(0, 1). All results are generated with 10,000 iterations after 30,000 burn-in iterations. Given that the MCMC chains are likely to be positively autocorrelated, we store only every 5th draw after the burn-in iterations and then apply the Geweke (1992) diagnostic test of convergence to the stored 10,000 draws. We calculate the standard errors (numerical standard errors) using 4% tapering of the periodogram window in order to consider the non-iid nature of the draws. We report the results when the equality of the means is not rejected for all parameters at the 1% significance level. The coefficients b2 and b3 are multiplied by 100 for purposes of visualization. VMG_SBM

VMG_EP

Mean

Std. deviation

Mean

Std. deviation

Mean

Std. deviation

Mean

Std. deviation

Mean

Std. deviation

0.110 0.032 0.088 0.074 0.088 0.748 0.059 0.028 0.025 0.017 0.034 0.044 0.031 0.021 0.032 0.036 0.106 0.159 0.240

0.131 0.001 0.628 0.356 0.138 3.090 0.134 0.942 0.023 0.036 0.035 0.768 0.197 0.040 0.178 0.782 1.952 3.297 2.600 0.220 2659.9 2709.7

0.128 0.047 0.098 0.106 0.068 0.593 0.037 0.026 0.015 0.020 0.020 0.047 0.045 0.021 0.042 0.045 0.121 0.239 0.209

0.249 0.153 0.553 0.362 0.244 3.686 0.100 0.849 0.043 0.108 0.056 0.747 0.197 0.105 0.179 0.717 1.745 3.339 2.633 0.241 2662.7 2712.5

0.125 0.043 0.103 0.110 0.069 0.619 0.048 0.040 0.022 0.034 0.027 0.050 0.046 0.034 0.042 0.048 0.127 0.232 0.191

0.109 0.353 0.557 0.189 0.626 3.616 0.143 0.920 0.038 0.042 0.057 0.720 0.223 0.039 0.234 0.727 1.814 3.331 3.028 0.353 2720.9 2770.7

0.128 0.044 0.078 0.099 0.097 1.002 0.101 0.025 0.019 0.019 0.025 0.049 0.046 0.020 0.045 0.048 0.116 0.250 0.258

0.130 0.112 0.657 0.427 0.233 3.630 0.028 0.930 0.026 0.045 0.058 0.712 0.230 0.042 0.232 0.726 1.934 2.976 2.879 0.251 2639.6 2689.4

0.123 0.041 0.097 0.099 0.079 0.688 0.051 0.025 0.015 0.020 0.026 0.052 0.049 0.023 0.048 0.049 0.116 0.240 0.224

A. VMG portfolios with top and bottom 30% a 0.113 CAPM b 0.141  LCAPM b 0.429 b+ 0.307 HCAPM b1 0.621 b2(100) 1.681 b3(100) 0.104 Transition probability p11 0.900 p12 0.064 p13 0.036 p21 0.100 p22 0.822 p23 0.079 p31 0.040 p32 0.087 p33 0.873 Idiosyncratic r1 1.681 volatility r2 1.988 r3 3.493 Adj R2 0.250 Hannan-Quinn 2599.8 Schwarz 2649.6

DVMG_BM Mean B. VMG portfolios with top and bottom decile portfolios a 0.248 CAPM b 0.159 LCAPM b 0.670 b+ 0.768 HCAPM b1 0.093 b2(100) 0.047 b3(100) 0.000 Transition probability p11 0.945 p12 0.021 p13 0.034 p21 0.038 p22 0.773 p23 0.189 p31 0.023 p32 0.175 p33 0.802 Idiosyncratic volatility r1 2.814 r2 5.066 r3 3.893 2 Adj R 0.184 Hannan-Quinn 3124.5 Schwarz 3174.3

VMG_CP

VMG_DP

DVMG_EP

VMG_BM

DVMG_CP

DVMG_DP

Std. deviation

Mean

Std. deviation

Mean

Std. deviation

Mean

Std. deviation

0.193 0.067 0.146 0.169 0.101 0.008 0.001 0.020 0.013 0.016 0.019 0.045 0.043 0.014 0.038 0.040 0.116 0.225 0.166

0.101 0.062 0.903 0.526 0.517 4.581 0.009 0.958 0.024 0.018 0.043 0.705 0.252 0.024 0.250 0.725 2.977 4.008 4.560 0.221 3063.6 3113.4

0.182 0.056 0.142 0.126 0.137 0.994 0.084 0.017 0.014 0.011 0.022 0.055 0.053 0.016 0.051 0.052 0.173 0.333 0.347

0.409 0.224 0.908 0.263 0.074 5.990 0.194 0.904 0.055 0.041 0.079 0.851 0.070 0.037 0.059 0.904 3.941 5.062 2.352 0.166 3077.7 3127.5

0.182 0.071 0.186 0.173 0.076 0.691 0.071 0.031 0.025 0.019 0.031 0.041 0.028 0.018 0.022 0.026 0.264 0.366 0.157

0.197 0.437 0.462 0.005 0.990 0.972 0.086 0.947 0.035 0.017 0.050 0.834 0.116 0.020 0.096 0.883 2.846 6.070 3.832 0.333 3195.1 3244.9

0.191 0.075 0.154 0.180 0.111 0.914 0.087 0.019 0.016 0.011 0.023 0.043 0.038 0.014 0.033 0.035 0.182 0.466 0.296

The estimates of the three risk measures in Table 1 do not show that value is riskier than growth. The CAPM beta is not positive for any of the VMG portfolios. In the downside beta regime, downside betas are all negative and significant, while upside betas are positive and significant. For example, the estimated downside beta of VMG_SBM is 0.43 with a standard deviation of 0.09, whereas

its estimated upside beta is 0.31 with a standard deviation of 0.07.3 Finally, in the higher moments regime, beta (b1) is negative 3 As a measure of dispersion, we report standard deviations of the 10,000 MCMC draws. We also calculated the 95 percent confidence interval from the 10,000 MCMC draws, but the results do not differ significantly from those we report in the tables.

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Table 2 Correlation coefficients of smoothed probabilities between various value-minus-growth portfolios and bull markets, sentiment, and conditional volatilities. We report Spearman rank correlation coefficients. The smoothed probabilities of the three regimes are calculated from the regime-switching model as explained in Section 2.1, while the smoothed probabilities of bull markets are calculated using the following two-state regime-switching model: Rmt = l1S1t + l2S2t + (r1S1t + r2S2t)et, where Rmt is the market return, and lj and rj are the expected market return and volatility of regime j, respectively. The model is estimated using the Bayesian MCMC Gibbs sampling estimation with diffuse priors. All results are calculated with 10,000 draws after 10,000 burn-in iterations. Conditional volatility is calculated from GARCH(1, 1), which we estimate with the market model. Sentiment is calculated by the differences between bulls and bears of Investors Intelligence sentiment at the last week of each month. Smoothed probabilities

Sentiment indices

SBM

EP

CP

DP

BM

Bull

VMG_EP VMG_CP VMG_DP VMG_BM Bull Market 0.122 VMG_BM3 VMG_EP VMG_CP VMG_DP VMG_BM3

0.624 0.565 0.647 0.737 0.252 0.073 0.689 0.457 0.558 0.595 0.639

0.680 0.751 0.712 0.599 0.057 0.644 0.772 0.713 0.706 0.707

0.677 0.716 0.357 0.118 0.473 0.510 0.616 0.560 0.536

0.790 0.542 0.017 0.586 0.590 0.578 0.768 0.641

0.424 0.365 0.649 0.580 0.629 0.710 0.750

0.419 0.535 0.432 0.520 0.487

VMG_EP VMG_CP VMG_DP VMG_BM Bull Market 0.233 VMG_BM3 VMG_EP VMG_CP VMG_DP VMG_BM

0.385 0.418 0.366 0.477 0.295 0.168 0.024 0.015 0.063 0.012 0.014

0.774 0.691 0.805 0.274 0.178 0.561 0.585 0.555 0.548 0.573

0.622 0.793 0.141 0.016 0.423 0.409 0.527 0.452 0.455

0.681 0.344 0.155 0.488 0.478 0.490 0.593 0.534

0.134 0.365 0.468 0.404 0.469 0.476 0.536

C. Higher moments regime Smoothed probabilities VMG_EP VMG_CP VMG_DP VMG_BM Bull Market Sentiment indices 0.123 Conditional volatility VMG_BM3 VMG_EP VMG_CP VMG_DP VMG_BM

0.375 0.395 0.523 0.629 0.383 0.016 0.680 0.397 0.442 0.582 0.595

0.670 0.563 0.617 0.439 0.091 0.303 0.463 0.409 0.391 0.378

0.542 0.677 0.260 0.134 0.229 0.219 0.312 0.219 0.297

0.708 0.348 0.064 0.387 0.385 0.369 0.551 0.445

0.355 0.365 0.462 0.416 0.425 0.495 0.542

A. CAPM beta regime Smoothed probabilities

Sentiment indices Conditional volatility

B. Downside beta regime Smoothed probabilities

Sentiment indices Conditional volatility

for all VMG portfolios; however, the coefficients on the square and cube of the market return (b2 and b3) are also negative. A negative coefficient on the second moment of market returns renders VMG portfolio returns a concave function of market movements, decreasing returns when market moves up or down. Therefore, the negative b2 increases risk. Two VMGs (VMG_EP and VMG_CP) show significant negative values of b3, but these negative b3 s would reduce risk because the market returns are negatively skewed (approximately 0.4 to 0.7, see Table 3) during the higher moments regime. Overall, the increased risk from the negative b2 is not likely to be large enough to conclude that the VMG portfolios are risky in the higher moments regime. The results of the DVMG portfolios can be explained in a similar way. Despite the result that value is not riskier than growth, the estimated alphas of the regime-switching model are not different from zero. The estimates of the alphas are significantly smaller than the CAPM alphas (not reported) for all VMG portfolios and are not different from zero at the 5% significance level in most cases, the exception being VMG_CP. The same is true for the alphas of the DVMG portfolios calculated with BM, EP and DP. The alphas calculated with CP (both the top and bottom 30% portfolios and the decile portfolios) become insignificant at the 1% level. If the regimes we identify with the three risk measures are associated with market states, then they may reflect the

Conditional volatility BM

EP

CP

DP

0.003 0.044 0.095 0.122 0.012

0.704 0.720 0.723 0.904

0.842 0.714 0.788

0.720 0.815

0.766

0.419 0.535 0.432 0.520 0.487

0.003 0.044 0.095 0.122 0.012

0.704 0.720 0.723 0.904

0.842 0.714 0.788

0.720 0.815

0.766

0.419 0.535 0.432 0.520 0.487

0.003 0.044 0.095 0.122 0.012

0.704 0.720 0.723 0.904

0.842 0.714 0.788

0.720 0.815

0.766

countercyclical pattern of the value anomaly that has been documented in the literature (Zhang, 2005; Chen et al., 2008; Gulen et al., 2011). We investigate the link between the three regimes and market states by dividing the sample period into bull and bear states as in Hwang and Satchell (2010). We find that the CAPM beta regime is associated with bull markets whereas the downside beta and the higher moments regimes are likely to be observed during bear periods (Table 2). The association of the three regimes with market states, however, does not show that value becomes riskier than growth during bad times. One interesting pattern, from the analysis of the two market states, is that the idiosyncratic volatility of value increases faster than that of growth in bear markets although growth has a larger standard deviation than value. The detailed results can be obtained from the authors upon request. We summarize the results of the regime-switching model as follows. While we find that the large positive VMG portfolio returns are not associated with higher risk in any of the regimes, the alphas in the regime-switching model are not different from zero. The first result confirms what has been reported in many previous studies such as Rosenberg et al. (1985), Fama and French (1992), Harvey and Siddique (2000), and Petkova and Zhang (2005). Our results show that value is not riskier than growth even if different risk measures are allowed for different periods. What is

Table 3 Descriptive statistics of regimes. At every month, we identify the regime that has the largest smoothed probability among the three regimes in the regime-switching model, and then calculate the basic statistics of the market and VMG returns in each regime. We also calculate the statistical properties of the residuals that we obtain with the unconditional model in each regime. Market return

VMG_EP Number of months Average return Std. deviation Skewness Kurtosis Jarque–Bera Stat VMGCP Number of months Average return Std. deviation Skewness Kurtosis Jarque–Bera Stat VMG_DP Numer of months Average return Std. deviation Skewness Kurtosis Jarque–Bera Stat VMG_BM Number of months Average return Std. deviation Skewness Kurtosis Jarque–Bera Stat

CAPM beta

Downside beta

Higher moments

262

115

169

1.149 4.120 0.891 7.039 212.77

0.789 4.325 0.038 3.538 1.41

0.187 5.182 0.372 3.975 10.59

265

135

146

1.263 3.483 0.527 4.154 26.97

0.332 4.742 0.057 3.167 0.23

0.531 5.541 0.737 5.414 48.65

228

155

163

1.152 4.128 1.175 8.282 317.48

0.505 4.317 0.033 3.688 3.08

0.712 4.959 0.402 4.157 13.47

234

154

158

1.145 3.695 1.278 9.784 512.43

0.372 5.243 0.472 4.059 12.91

0.834 4.600 0.039 3.328 0.75

285

134

127

1.391 3.932 1.071 8.526 417.16

0.202 4.624 0.017 3.352 0.70

0.264 5.154 0.301 4.054 7.80

p-Value of equality

0.072 <0.001

0.087 <0.001

0.343 0.034

0.246 <0.001

0.010 <0.001

Residuals from unconditional models

CAPM beta

Downside beta

Higher moments

262

115

169

0.149 1.609 0.155 2.669 2.25

1.508 2.164 0.457 4.009 8.87

0.576 4.291 0.128 3.250 0.90

265

135

146

0.216 1.917 0.045 2.593 1.92

2.525 3.566 0.235 3.535 2.85

1.195 2.861 1.271 9.495 295.95

228

155

163

0.195 1.623 0.139 3.011 0.74

2.381 3.379 0.236 3.364 2.30

1.307 3.124 0.929 6.188 92.48

234

154

158

0.047 2.199 0.494 4.351 27.32

2.266 3.595 0.131 3.619 2.89

1.673 3.770 0.262 3.415 2.94

285

134

127

0.056 1.905 0.342 2.879 5.74

2.890 3.073 0.617 4.142 15.79

1.361 3.226 0.154 3.642 2.69

p-Value of equality

<0.001 <0.001

<0.001 <0.001

<0.001 <0.001

<0.001 <0.001

<0.001 <0.001

CAPM beta

Downside beta

Higher moments

262

115

169

– 1.527 0.164 2.462 4.33

– 1.826 0.193 3.277 1.08

– 3.319 0.022 3.963 6.54

265

135

146

– 1.916 0.051 2.592 1.95

– 3.218 0.308 3.888 6.57

– 2.269 0.046 2.352 2.61

228

155

163

– 1.517 0.061 2.338 4.31

– 3.079 0.157 2.710 1.18

– 2.402 0.330 2.568 4.23

234

154

158

– 1.757 0.070 2.980 0.20

– 3.261 0.063 3.619 2.56

– 2.562 0.014 2.505 1.62

285

134

127

– 1.858 0.294 2.754 4.83

– 2.738 0.396 3.010 3.50

– 2.618 0.234 3.094 1.21

p-Value of equality

<0.001

<0.001

<0.001

S. Hwang, A. Rubesam / Journal of Banking & Finance 37 (2013) 2367–2377

VMG_SBM Number of months Average return Std. deviation Skewness Kurtosis Jarque–Bera Stat

VMG returns

<0.001

<0.001

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Table 4 Estimates of the downside beta model during the downside beta regime. We first identify the months for which the downside beta regime has the highest smoothed probability. For each regime, we calculate the basic statistical properties of various value, growth and value-minus-growth portfolio returns together with the market returns. The unconditional downside beta model is estimated for the value, growth and value-minus-growth portfolio returns of the downside beta regime. As in Table 1, the downside and ^ r  b ^þ r þ . The t-statistics in brackets upside betas in the downside beta model are estimated without the constant, and then the alpha is calculated from the residuals; r pt  b mt mt are calculated with the Newey and West heteroskedasticity robust standard error. Bold numbers represent significance at the 5% level.

V_SBM G_SBM VMG_SBM V_EP G_EP VMG_EP V_CP G_CP VMG_CP V_DP G_DP VMG_DP V_BM G_BM VMG_BM

Number of months

Monthly return (%)

Monthly volatility (%)

Skewness

115

2.045 0.535 1.508 2.113 0.411 2.525 2.059 0.323 2.381 1.931 0.335 2.266 2.308 0.582 2.890

5.050 4.758 2.164 4.990 5.119 3.566 4.725 4.612 3.379 4.812 5.536 3.595 4.872 4.895 3.073

1.324 0.039 0.457 1.203 0.111 0.235 1.017 0.193 0.236 0.253 0.683 0.131 1.071 0.172 0.617

135

155

154

134

Kurtosis

7.575 4.817 4.009 6.102 2.758 3.535 5.910 3.323 3.364 4.634 4.076 3.619 5.971 3.015 4.142

interesting is that the alphas become insignificant for most VMG portfolios.

2.3. Which regime is responsible for the value anomaly? In order to discover which regime is responsible for the value anomaly, we investigate the properties of the VMG portfolio returns in the three regimes. The regimes are identified with the largest smoothed probability at each month. It is clear in Table 3 that the average VMG portfolio returns come from the extreme returns of the downside beta regime. The average VMG portfolio returns (excluding the VMG_SBM portfolio) range from 2.27% a month to 2.9% a month in the downside beta regime, whereas they are close to zero in the CAPM beta regime (from 0.06% a month to 0.22% a month) and are negative in the higher moments regime (from 1.67% a month to 1.2% a month). Therefore, the average return difference between the downside beta and higher moments regimes is large and significant – at least 3.7% a month.4 Therefore, the countercyclicality of the VMG returns comes from the downside beta regime, not from the higher moments regime. While both the downside beta and higher moments regimes are associated with turbulent periods, a closer look at Table 2 indicates that the higher moments regime shows stronger correlation with bear markets than the downside beta regime does. This is also consistent with the different levels of market volatility in the two regimes: the downside beta regime tends to be selected when the market is relatively less volatile. The asymmetric response to the market movement creates profound differences in the average VMG returns between the downside beta and higher moments regimes. The large and positive average returns (and positive skewnesses) of the VMG portfolios in the downside beta regime are explained by the negative downside beta for negative market returns and the positive upside beta for positive market returns. As growth is riskier than value, regardless of regimes, we would expect VMG portfolios to show negative average returns in all three regimes. The downside beta regime is 4 The average return difference between the two regimes is slightly lower for the VMG_SBM portfolio, i.e., 0.93% a month; yet it still shows the same pattern as those of the other VMG portfolios.

Estimates of the downside beta model a

t-Stat

b

t-Stat

b+

t-Stat

Idiosyncratic volatility

0.128 0.171 0.043 0.112 0.126 0.237 0.169 0.084 0.316 0.245 0.113 0.359 0.268 0.209 0.477

(0.756) (1.222) (0.259) (0.553) (1.007) (0.855) (0.965) (0.475) (1.277) (1.337) (0.891) (1.366) (1.478) (2.173) (2.016)

0.815 1.342 0.526 0.471 1.194 0.723 0.493 1.197 0.703 0.516 1.139 0.623 0.444 1.181 0.737

(12.671) (20.036) (8.634) (5.178) (32.134) (9.781) (4.823) (29.618) (8.525) (6.944) (26.119) (10.312) (4.756) (47.631) (8.211)

1.463 1.012 0.451 1.344 0.871 0.473 1.385 0.835 0.549 1.162 0.844 0.318 1.460 0.865 0.595

(16.871) (15.856) (7.464) (19.044) (26.256) (5.497) (20.906) (22.211) (6.210) (17.914) (25.562) (3.453) (24.349) (25.799) (7.228)

1.827 1.516 1.826 2.361 1.464 3.239 2.212 1.434 3.106 2.296 1.583 3.292 2.212 1.132 2.790

the most problematic since growth is riskier than value but value outperforms growth by approximately 2.4% a month in most cases. In order to explore the large positive average VMG returns in the downside beta regime, we estimate the downside beta model for the value, growth, and VMG portfolio returns of the downside beta regime.5 Table 4 confirms that in all cases, alphas are significantly less than the average VMG return, and most of the average VMG returns are explained by the asymmetric response to the market movements. The asymmetry is large for value. The difference between the upside and downside betas is at least 0.6. Compared to value, growth shows a modest asymmetric response to the market movements; i.e., the difference between the upside and downside betas of growth is only 0.3. Because of its large asymmetry, value contributes to the large and positive returns of the VMG portfolios far more than growth does; the average returns of value are over 2% a month, which is approximately 80% of the average VMG returns in the downside beta regime. 3. What drives the value anomaly? We show that the large positive value-minus-growth portfolio returns are explained by the asymmetric response to market movements in the downside beta regime. However, assets with large upside betas relative to downside betas should not be compensated with high expected returns since returns of these assets are high when investors’ wealth is already high (Ang et al., 2006). In this section, we propose an explanation of the dynamics based on investors’ behavioral biases. 3.1. Overconfidence, self-attribution bias, and mispricing Behavioral biases such as overconfidence and self-attribution bias are frequently suggested as explanations for various anomalies in equity markets. This strand of studies was first initiated by De Bondt and Thaler (1985) and further developed 5 The estimates of the VMG portfolio returns in Table 4 are similar to those of the downside beta regime of the regime-switching model in Table 1. Note that the estimates in Table 4 are obtained from the returns of the downside beta regime that we identify with the largest smoothed probability every month, while Table 1 reports the estimates of the regime-switching model, which may be affected by the other regimes.

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by Lakonishok et al. (1994) and DHS (1998, 2001) among many others. According to these studies, price-to-fundamental ratios such as book-to-market or cash flow-to-price are measures of mispricing: value (growth) stocks are underpriced (overpriced) with respect to their fundamentals. Lakonishok et al. (1994) suggest that positive VMG portfolio returns reflect the correction of the mispricing initially created by overconfident investors who extrapolate past growth rates too far into the future. DHS (1998, 2001) propose theoretical models in which asset prices increase (or decline) more than their fundamentals when investors are overconfident on good (bad) news. According to their outcome-dependent confidence model, when overconfident investors have self-attribution bias, asset prices initially rise on good news and then decline, creating positive short-term and negative long-term autocorrelations.6 These investors asymmetrically react to noisy public signals depending on the consistency between their private information and noisy public signals. Their confidence grows when noisy public signals are consistent with their information, but does not diminish by the same magnitude when noisy public signals contradict their private information.

(l > 0) is expected for VMG portfolio returns. The noisy public signal that the investor receives at time t can be written as st = rt+1 + et, where et  Nð0; r2t Þ, cov(rt+1, et) = 0, and r2t follows a time-varying volatility process. The noise term, et, is allowed to be heteroskesdastic for the countercyclical pattern in volatility (see, for example, Chen et al., 2008; Gulen et al., 2011).7 When the overconfident investor updates his expectation about the t + 1 return with the noisy public signal, his posterior mean and variance of return are

E½r tþ1 jst  ¼

r2t r þr 2 r

Var½r tþ1 jst  ¼

2 t

lþ

r2r r þ r2t 2 r

An application of Lakonishok et al. (1994) and DHS (1998, 2001) to the value anomaly suggests that return reversals subsequent to initial mispricing appear as asymmetric responses to market movements (public signals) because of investors’ self-attribution. For value stocks, investors who are confident that value is underpriced base their investment decisions more on positive signals than on negative signals. Therefore, during the correction phase, investors react more to positive signals than to negative signals. For growth stocks, investors’ beliefs that growth is overpriced lead them to base their decisions on negative signals instead of positive ones. However, these studies do not explain why the asymmetric reaction is observed only during specific time periods, i.e., the downside beta regime. To discover the conditions under which the correction intensifies, we generalize the model that DHS (1998) develop under the assumptions that investors’ prior payoff is zero and that the distribution of noisy public signals is time-invariant. Under these assumptions they show that the mispricing initially created by overconfidence is corrected partially at a constant rate over time. In our study we allow a non-zero prior payoff and heteroskedasticity in noisy public signals. 3.2.1. The model for time-varying speed of return reversals As in DHS (1998), there are two types of investors in the market: an informed investor who is overconfident and risk-neutral and an uninformed investor who is risk averse. Since price is decided by the risk-neutral informed investor rather than by the uninformed investor, we focus on the pricing by the overconfident investor. Our model is a simple one period model where the informed investor predicts t + 1 return at time t with noisy public signal and observes his payoff at t + 1. Suppose that at time t, the prior belief about the t + 1 return of an asset is represented as r tþ1  Nðl; r2r Þ, where l is not necessarily zero. As claimed by behavioral studies, if value stocks are underpriced whereas growth stocks are overpriced, a positive prior mean 6

Two psychological biases in particular, overconfidence and self-attribution, are prevalent in financial markets. For instance, the self-attribution bias could be substancial in financial markets since unsuccessful investors are likely to be driven out and successful survivors become more likely to attribute their success to themselves (Gervais and Odean, 2001). Furthermore, De Bondt and Thaler (1995) argue that overconfidence is more likely to be found among experts than among inexperienced investors.

ð2Þ

r2r r2t : r2r þ r2t

ð3Þ

The posterior expected return varies depending on st (the noisy public signal) and r2t which we refer to as the ‘noise volatility’ in this study. The posterior expected return can be represented as a weighted average of the prior mean and the signal: E[rt+1|st] = wt2

l + (1  wt)st, where wt ¼ r2rþtr2 . r

3.2. Time-varying speed of return reversals

st ;

t

We suggest two conditions that affect E[rt+1|st]. First, when noise volatility increases and thus the noisy public signal is dominated by the noise, the public signal is too noisy to be used for updating the prior distribution. In this case, the informed investor’s optimal prediction is the prior expected return of the asset: i.e., E[rt+1|st] ? l as r2t ! 1.8 An implication is that when r2t is large, the posterior expectation is driven by the positive (negative) prior expected return for undervalued (overvalued) assets, and the correction is accelerated towards l as r2t increases. Moreover, in this case, the posterior variance also increases. The second condition arises when the payoff volatility (r2r ) is not completely dominated by the noise volatility. Then, the second r2

component of equation (2), i.e., r2 þrr2 st , may affect the posterior r t expectation via overconfidence and self-attribution bias. When the informed investor receives a noisy public signal that confirms his prior (l > 0), i.e., st > 0, the inefficient deviation of price is corrected faster: Eb[rt+1|st]>E[rt+1|st] since r2t decreases (or the precision of the signal increases). Here, r2t is the variance of noise (et) calculated based on the investor’s confident belief, and the superscript b reflects the bias due to overconfidence and self-attribution bias. On the other hand, when the investor receives a signal that is not consistent with his prior, i.e., st < 0, he revises the estimated  2t is the upward biased  2t > r2t , where r precision downward, i.e., r variance of et. As in DHS (1998), the upward bias is not expected to be as large as the downward bias. 3.2.2. Implications and hypotheses The two conditions have several important implications for the dynamics of the value anomaly. We explain them and propose the hypotheses in order to test the time-varying speed of return reversals in VMG portfolio returns. The first channel (noise volatility) explains why VMG portfolio returns appear countercyclical. The countercyclical pattern of the value anomaly reported by the previous studies (Chen et al., 2008; Gulen et al., 2011) may simply reflect the optimal prediction under uncertainty by investors who believe that value stocks are 7 For simplicity we only allow time-variation in the noise term. However, this is not restrictive. We could allow r2r to time-vary, but then the speed of the correction is affected by the relative magnitude of r2t with respect to r2r . 8 When noise volatility is over-estimated, we have a similar result. However, in our setting neither the overconfident investor nor the uninformed risk-averse investor is assumed to over-estimate noise volatility. Noise volatility may be over-estimated by investors who lack confidence. However, the role of the investors who lack confidence in asset pricing would be minimal since they are not likely to trade actively.

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underpriced whereas growth stocks are overpriced. When public signals become noisy, investors use prior beliefs since the signals are too noisy to be used for updating their prior beliefs. Therefore, our first testable hypothesis is as follows. H1. Value and VMG portfolio returns are positively related with noise volatility whereas growth portfolio returns are negatively related with noise volatility The asymmetric response to market movements in the downside beta regime is an example of the second case. Our results in Tables 3 and 4 show that value stocks overreact to positive market movements whereas they underreact to negative market movements. According to our second channel, the asymmetric response to market movements can be explained by investors’ self-attribution bias: investors react to the signals that are consistent with their prior beliefs far more than those that are inconsistent with their prior beliefs. This asymmetric response occurs more frequently during the downside beta regime than during the other periods. Thus, our second hypothesis is as follows. H2. Value and VMG portfolio returns positively respond to positive public signals whereas growth portfolio returns negatively respond to positive public signals The above two channels are not necessarily interdependent since the first channel is driven by high noise volatility whereas the second channel is driven by the asymmetric response to the noisy public signal. For example, when investors asymmetrically respond to the noisy public signal by giving higher precision to the signal that is consistent with their prior beliefs than to the signal that is inconsistent with their prior beliefs, the weight r2 on the public signal, i.e., r2 þrr2 , increases whereas the weight on r t r2 the unconditional expected return, i.e., r2 þt r2 , decreases. Therer t fore, in this case, the speed of the return reversals is less likely to be affected by the prior beliefs than by the asymmetric response. Other cases are also possible. For example, both channels could affect the speed of return reversals when wt is close to 0.5. Even if wt is close to zero, investors may not necessarily overreact to signals that are consistent with their prior beliefs. Our empirical result, which illustrates that the value anomaly is observed only during one quarter of the entire sample period (see Table 3), indicates that for the remaining three quarters of the entire period, neither noise volatility nor asymmetric response increases the return reversals of the VMG portfolios. The following hypothesis is thus designed to test if the two channels work separately. H3. The response of value and VMG portfolio returns to noise volatility is not associated with the asymmetric response of value and VMG portfolio returns to public signals Although these two channels can be applied to both value and growth, the correction would be far more significant for value than for growth because value stocks are mispriced more than growth stocks are. As stocks with higher uncertainty are subject to greater mispricing (DHS, 2001; Baker and Wurgler, 2006), value stocks, which have higher idiosyncratic volatilities than growth stocks, are affected more by investors’ overconfidence and self-attribution bias. Our empirical results in Table 4 support this view by demonstrating that more than 80% of the value anomaly comes from value rather than growth. Thus, the final hypothesis is as follows. H4. The effects of the two channels on value portfolio returns are larger than those on growth portfolio returns

3.3. Empirical tests on the speed of return reversals In order to test the dynamics of return reversals in the value anomaly, we should investigate if value (growth) is positively (negatively) associated with noise volatility (H1) or if it responds asymmetrically to positive and negative noisy public signals (H2). We also expect the downside beta regime to be observed more frequently when public signals are noisy or when they are positive. It is not easy to choose variables for noisy public signals and noise volatility. In this study, for the noisy public signal, we use investor sentiment that represents an aggregate belief of market participants (Shefrin, 2005). According to Brown and Cliff (2004), sentiment is noisy for the prediction of future returns though it has some explanatory power for contemporaneous returns. In our model, the precision of sentiment is overestimated (underestimated) when sentiment is consistent (inconsistent) with investors’ beliefs, and thus asymmetric responses to sentiment are expected. Among many different measures of sentiment, we use a direct sentiment measure of Investors Intelligence in the US equity market, which has been publically available since the 1960s.9 The monthly Investors Intelligence sentiment is created by calculating the differences between bulls and bears at the last week of each month. As a proxy of the noise volatility, we use idiosyncratic volatility by taking out the risk (CAPM beta), which we calculate using the popular GARCH(1, 1) model. We test the first channel (H1) utilizing the following GARCH(1, 1)-in-mean process:

rp;t ¼ a þ brmt þ dHp;t þ 2 1 Hp;t1 p;t1

Hp;t ¼ c0 þ c

e

qffiffiffiffiffiffiffiffi Hp;t ep;t ;

ð4Þ

þ c2 Hp;t1 :

As return reversals increase with noise volatility, we expect d > 0 for value and VMG portfolios whereas d < 0 for growth portfolios. The second channel (H2) is tested using the following regression:

rp;t ¼ a þ brmt þ dþ Z t IZt P0 þ d Z t ð1  IZt P0 Þ þ rep;t ;

ð5Þ

where Zt is sentiment, IZt P0 is an indicator variable that is one when Zt is positive and zero otherwise, and ep,t  N(0, 1). For value (and VMG) portfolios, we expect d+ > 0 and d < 0 whereas the opposite signs are expected for growth portfolios. The two models in (4) and (5) are estimated for the five VMG portfolios as well as for value and growth portfolios for the period from July 1963 to December 2008. The empirical evidence in Tables 5–7 supports our predictions based on the time-varying speed of return reversals. The results of the regression in (4) in Table 5 demonstrate that idiosyncratic volatility is positively related with VMG portfolio returns; moreover, none of the five alphas are significant. Panels B and C show that value portfolio returns increase with idiosyncratic volatility, whereas growth portfolio returns decrease as idiosyncratic volatility increases. As predicted in the first hypothesis (H1), value is more sensitive to idiosyncratic volatility. The coefficients on the idiosyncratic volatility are all positive and significant for value stocks, and are negative but insignificant for growth portfolios (H4). Value and VMG portfolio returns are explained by sentiment, and the coefficients on the positive and negative sentiments support the asymmetric response to the noisy public signals (Table 6). 9 We also tested the Individual Investors sentiment and Baker and Wurgler’s (2006) sentiment. The results of the Individual Investors sentiment are not different from those we report in this study; however, we do not find evidence that the VMG portfolio returns are explained by the Baker and Wurgler sentiment. The difference is not surprising since the Baker and Wurgler sentiment is not correlated with the other two indices. Further investigation into the properties of these sentiment indices is beyond the scope of this study.

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S. Hwang, A. Rubesam / Journal of Banking & Finance 37 (2013) 2367–2377 Table 5 Noise volatility and value anomaly. The GARCH(1, 1)-in-mean model is estimated for each of the VMG portfolios with excess market return for the period from July 1963 to December 2008.

r p;t ¼ a þ br m;t þ h1 Hp;t þ 2 1 H p;t1 p;t1

e

Hp;t ¼ c0 þ c

pffiffiffiffiffiffiffiffi Hp;t ep;t ;

Table 6 Sentiment and value anomaly. The following regression equation is estimated for each of the VMG portfolios with excess market return and positive and negative sentiments for the period from July 1963 to December 2008.

rp;t ¼ a þ brmt þ dþ Z t IZt P0 þ d Z t ð1  IZt P0 Þ þ rep;t ;

þ c2 Hp;t1 ;

where rm,t represents excess market return (EMR), Zt is the sentiment, IZ t P0 is an indi-

where rm,t represents excess market return (EMR), and ep,t  N(0, 1). The numbers in brackets are the Newey and West heteroskedasticity robust standard errors. Bold

cator variable that is one when Zt is positive and zero otherwise, and ep,t  N(0, 1). We use the Investors Intelligence sentiment index. The monthly Investors Intelligence sentiment index is created by taking weekly differences between bulls and

numbers represent significance at the 5% level.

bears at the end of each month. The numbers in brackets are the Newey and West BM

EP

CP

DP

BM3

portfolio 0.332 (0.299) 0.103 (0.021) 0.097 (0.039)

0.248 (0.347) 0.149 (0.023) 0.083 (0.045)

0.159 (0.292) 0.332 (0.023) 0.059 (0.035)

0.425 (0.251) 0.091 (0.019) 0.100 (0.033)

GARCH(1, 1) process Constant 0.313 (0.179) ARCH 0.133 (0.042) GARCH 0.827 (0.057) Adjusted R2 0.160

0.317 (0.159) 0.082 (0.023) 0.882 (0.034) 0.051

0.297 (0.151) 0.072 (0.021) 0.896 (0.030) 0.080

0.538 (0.236) 0.120 (0.033) 0.827 (0.049) 0.225

0.356 (0.143) 0.121 (0.027) 0.842 (0.028) 0.049

B. Value (V) portfolio Intercept 0.125 (0.162) EMR 0.959 (0.023) GARCHM 0.134 (0.041)

0.082 (0.153) 0.947 (0.014) 0.121 (0.046)

0.025 (0.120) 0.920 (0.014) 0.081 (0.037)

0.027 (0.132) 0.773 (0.013) 0.070 (0.035)

0.023 (0.119) 0.944 (0.013) 0.100 (0.034)

GARCH(1, 1) process Constant 0.133 (0.070) ARCH 0.114 (0.027) GARCH 0.865 (0.004) Adjusted R2 0.791

0.162 (0.059) 0.101 (0.024) 0.855 (0.030) 0.807

0.102 (0.043) 0.123 (0.028) 0.858 (0.026) 0.792

0.216 (0.084) 0.149 (0.033) 0.803 (0.045) 0.683

0.163 (0.044) 0.152 (0.027) 0.818 (0.021) 0.785

C. Growth (G) portfolio Intercept 0.072 (0.133) EMR 1.175 (0.016) GARCHM 0.012 (0.055)

0.172 (0.222) 1.049 (0.015) 0.177 (0.148)

0.085 (0.160) 1.050 (0.010) 0.099 (0.103)

0.019 (0.123) 1.098 (0.012) 0.015 (0.062)

0.033 (0.113) 1.044 (0.014) 0.072 (0.101)

GARCH(1, 1) process Constant 0.255 (0.095) ARCH 0.144 (0.068) GARCH 0.769 (0.070) Adjusted R2 0.907

0.052 (0.033) 0.035 (0.015) 0.932 (0.027) 0.935

0.047 (0.022) 0.040 (0.017) 0.933 (0.027) 0.931

0.126 (0.040) 0.128 (0.027) 0.818 (0.033) 0.916

0.025 (0.009) 0.022 (0.014) 0.958 (0.017) 0.947

A. Value Minus Growth (VMG) Intercept 0.098 (0.213) EMR 0.176 (0.028) GARCHM 0.093 (0.036)

heteroskedasticity robust standard errors. Bold numbers represent significance at the 5% level.

As predicted in the second hypothesis (H2), the coefficients on the positive sentiment are much larger than those on the negative sentiment because the positive signal is consistent with the prior beliefs that value stocks are underpriced and thus would show high returns, supporting the idea that the correction is accelerated when sentiment increases. On the other hand, the coefficients are negative for growth portfolios, but only half of the coefficients are significant. These results also confirm that the correction for growth is not as strong as that for value (H4). One channel does not subsume the other (H3). In Table 7, we report the results for a model combining both channels, i.e., noise volatility and self-attribution bias to noisy public signals (sentiment). The results demonstrate that the coefficients on

BM

EP

A. Value Minus Growth (VMG) portfolio Intercept 0.039 0.063 (0.226) (0.242) EMR 0.267 0.162 (0.043) (0.043) + Positive sentiment (d ) 3.045 2.849 (0.981) (1.134) Negative sentiment 0.019 1.252 (d) (1.734) (1.975) 0.165 0.061 Adjusted R2 B. Value (V) portfolio Intercept EMR Positive sentiment (d+) Negative sentiment (d) Adjusted R2 C. Growth (G) portfolio Intercept EMR Positive sentiment (d+) Negative sentiment (d) Adjusted R2

CP

DP

BM3

0.059 (0.247) 0.214 (0.042) 3.374 (1.141) 0.827

0.170 (0.241) 0.383 (0.038) 2.485 (1.014) 1.417

0.140 (0.266) 0.167 (0.049) 3.571 (1.214) 0.070

(2.034) 0.100

(2.024) 0.234

(2.078) 0.071

0.072 (0.198) 0.929 (0.040) 3.101 (0.919) 0.182

0.118 (0.163) 0.901 (0.036) 1.846 (0.758) 1.163

0.016 (0.152) 0.852 (0.034) 2.273 (0.732) 0.388

0.017 (0.158) 0.710 (0.035) 1.613 (0.599) 0.488

0.056 (0.188) 0.891 (0.039) 2.564 (0.886) 0.300

(1.428) 0.796

(1.194) 0.803

(1.156) 0.795

(1.166) 0.681

(1.390) 0.787

0.112 (0.124) 1.197 (0.019) 0.057 (0.478) 0.165

0.055 (0.108) 1.063 (0.015) 1.004 (0.468) 0.089

0.075 (0.118) 1.066 (0.016) 1.100 (0.506) 0.439

0.153 (0.119) 1.093 (0.022) 0.872 (0.514) 0.929

0.084 (0.099) 1.057 (0.015) 1.007 (0.408) 0.370

(0.989) 0.907

(1.019) 0.935

(1.052) 0.932

(1.140) 0.916

(0.886) 0.948

idiosyncratic volatility and sentiment are positive in all VMG portfolios and are significant in most portfolios. Comparing the results in Tables 5 and 6, the coefficients in Table 7 do not change significantly in the presence of the other variable. Moreover, these two variables are not correlated (Table 2), although both can explain the VMG portfolio returns. Among the two channels, the adjusted R2 values indicate that the asymmetric response to sentiment explains VMG portfolio returns slightly better than noise volatility. Idiosyncratic volatility and sentiment are associated with the downside beta regime. We calculate rank correlations between the sentiment, the idiosyncratic volatility of VMG portfolios from the GARCH(1, 1) model, and the smoothed probabilities of regimes from the regime-switching model. The correlation coefficients in Table 2 show that the smoothed probability of the downside beta regime is the only one that is positively correlated with sentiment. When sentiment is high, asymmetric response is likely to be observed. Idiosyncratic volatility, on the other hand, increases in the downside beta and higher moments regimes, whereas it decreases in the CAPM beta regime. Therefore, idiosyncratic volatility identifies turbulent periods but does not provide further information on the difference between the downside beta and higher

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Table 7 Sentiment, noise volatility and value anomaly. The following GARCH(1, 1)-in-mean model is estimated for each of the VMG portfolios with excess market return and positive and negative sentiment for the period from July 1963 to December 2008.

r p;t ¼ a þ br mt þ dþ Z t IZt P0 þ d Z t ð1  IZt P0 Þ þ h1 Hp;t þ

pffiffiffiffiffiffiffiffi Hp;t ep;t ;

Hp;t ¼ c0 þ c1 Hp;t1 e2p;t1 þ c2 Hp;t1 ; where rm,t represents excess market return (EMR), Zt is the sentiment, IZt P0 is an indicator variable that is one when Zt is positive and zero otherwise, and ep,t  N(0, 1). We use the Investors Intelligence sentiment index. The monthly Investors Intelligence sentiment index is created by taking weekly differences between bulls and bears at the end of each month. The numbers in brackets are the Newey and West heteroskedasticity robust standard errors. Bold numbers represent significance at the 5% level. BM A. Value Minus Growth (VMG) portfolio Intercept 0.373 EMR 0.186 + Positive sentiment (d ) 2.454 Negative sentiment (d) 0.055 GARCHM 0.079 GARCH(1, 1) process Constant ARCH GARCH Adjusted R2 B. Value (V) portfolio Intercept EMR Positive sentiment (d+) Negative sentiment (d) GARCHM GARCH(1, 1) process Constant ARCH GARCH Adjusted R2 C. Growth (G) Portfolio Intercept EMR Positive sentiment (d+) Negative sentiment (d) GARCHM GARCH(1, 1) process Constant ARCH GARCH Adjusted R2

EP

CP

DP

BM3

0.258 0.019 0.785 1.374 0.035

0.328 0.121 1.789 2.369 0.074

0.318 0.023 0.892 1.702 0.039

0.400 0.171 3.003 1.742 0.055

0.335 0.025 0.936 1.749 0.042

0.432 0.345 1.996 0.160 0.056

0.336 0.023 0.860 1.648 0.036

0.719 0.104 3.159 0.270 0.079

0.286 0.022 0.869 1.669 0.032

0.302 0.135 0.826 0.173

0.121 0.028 0.030

0.336 0.089 0.871 0.064

0.180 0.027 0.041

0.290 0.079 0.889 0.099

0.155 0.023 0.034

0.522 0.119 0.830 0.229

0.225 0.032 0.047

0.347 0.129 0.836 0.070

0.144 0.029 0.030

0.320 0.947 1.818 0.854 0.124

0.182 0.014 0.636 1.153 0.037

0.127 0.939 0.962 1.376 0.106

0.180 0.015 0.533 0.998 0.047

0.059 0.913 1.652 0.908 0.050

0.148 0.015 0.560 0.835 0.038

0.108 0.769 1.012 0.123 0.065

0.178 0.013 0.546 1.145 0.036

0.216 0.939 1.636 0.137 0.088

0.169 0.015 0.593 1.090 0.034

0.145 0.113 0.862 0.797

0.051 0.025 0.026

0.181 0.104 0.847 0.809

0.068 0.025 0.035

0.111 0.135 0.844 0.795

0.048 0.031 0.030

0.217 0.149 0.804 0.684

0.083 0.033 0.045

0.164 0.157 0.814 0.789

0.047 0.029 0.024

0.115 1.177 0.113 0.674 0.010

0.178 0.014 0.450 0.944 0.055

0.275 1.057 1.024 0.333 0.149

0.200 0.010 0.407 0.714 0.133

0.301 1.060 1.320 0.709 0.123

0.161 0.010 0.416 0.750 0.093

0.109 1.104 0.864 0.195 0.010

0.148 0.013 0.410 0.699 0.062

0.250 1.054 1.156 0.268 0.109

0.154 0.009 0.367 0.593 0.120

0.259 0.145 0.767 0.907

0.100 0.027 0.055

0.051 0.040 0.927 0.935

0.030 0.020 0.035

0.043 0.050 0.924 0.932

0.022 0.020 0.029

0.125 0.127 0.819 0.916

0.042 0.029 0.035

0.026 0.035 0.946 0.948

0.013 0.016 0.021

moments regimes. Although both sentiment and idiosyncratic volatility can explain the VMG portfolio returns, only sentiment can identify the periods when the asymmetric response matters. Finally, return reversals are expected to increase following turbulent periods. Asset prices are affected more seriously by overreaction when markets are noisy than when markets are quiet, because of investors’ overconfidence and self-attribution bias (DHS, 2001). This means that in our regime-switching model, the downside beta regime is more likely to follow the volatile higher moments regime rather than the quiet CAPM beta regime. The estimates of the transition matrix in Table 1 show that for all VMG portfolios, the probability that the higher moments regime is followed by the downside beta regime (p32) is larger than the probability that the CAPM beta regime is followed by the downside beta regime (p12): p12s are approximately 0.05 whereas p32s are approximately 0.2. The differences between p12 and p32 are significant at the 1% level for most VMG portfolios.

two conditions under which the price correction intensifies: when noise volatility increases and when noisy public signals are consistent with prior beliefs. These explanations are indeed consistent with our empirical results. A growing number of recent studies take rational approaches to explain the value anomaly. Some seek answers from real growth options and the inflexibility of value firms (Berk et al., 1999; Zhang, 2005). Others focus on the difference in the duration of cash flows between value and growth firms (Lettau and Wachter, 2007). Guo et al. (2009) take a different perspective and argue that the value premium reflects investment opportunities in Merton’s (1973) intertemporal capital asset pricing model (ICAPM). None of these studies, however, explains what we find in this paper as to why large positive average returns are observed via the asymmetric response of value to market movements in the downside beta regime.

Acknowledgements 4. Conclusions This work contributes to the debate about the value anomaly in several ways. We propose an explanation based on investors’ psychological biases, such as overconfidence and self-attribution bias. By generalizing the model introduced by DHS (1998), we suggest

This paper supersedes an earlier version titled ‘‘Is value really riskier than growth?’’ We would like to thank participants in the AFA Annual Meeting, the Meeting of the Brazilian Finance Society, the Annual Meeting of the Korea Money and Finance Association, and seminar participants at Sir John Cass Business School and at

S. Hwang, A. Rubesam / Journal of Banking & Finance 37 (2013) 2367–2377

the School of Economics, Sungkyunkwan University. We thank Min Hwang, Aneel Keswani and Caio Ibsen Rodrigues de Almeida for their helpful comments and special thanks to Beatriz Singer and Ralitsa Petkova.

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