The dual effect of idiosyncratic volatility on stock pricing and return

Whether a stock's expected return depends on idiosyncratic volatility is an unresolved asset pricing puzzle (Hou & Loh, 2016). Traditional asset pricing theories predict either no relation between idiosyncratic volatility and expected return under the assumption of complete and frictionless markets and perfect portfolio diversification (Black, 1972; Lintner, 1965; Sharpe, 1964) or a positive relation under the assumption of limited portfolio diversification (Levy, 1978; Merton, 1987). However, Ang, Hodrick, Xing and Zhang (2006) and several following studies find a negative relation between idiosyncratic volatility and realized return. If realized return is an adequate expected return proxy, considered an asset pricing puzzle, the finding of a negative relation between idiosyncratic volatility and realized return poses a direct challenge to those asset pricing theories.

Existing research has focused on proposing explanations for the negative relation and on evaluating the proposed explanations (Hou & Loh, 2016) ^[1]. This focus downplays the fact that quite a few studies find a positive or no relation between idiosyncratic volatility and realized return (e.g. Fama & MacBeth, 1973; Lehmann, 1990). That is, findings about the relation between idiosyncratic volatility and realized return are indeed inconsistent across studies. Moreover, some studies with inconsistent findings employ highly overlapped samples and similar – or even identical – measures and testing methods. Arguably, what drives this inconsistency is just as puzzling as the finding of a negative relation.

Evidently, research is needed to investigate what drives this inconsistency. Answers to this question help to resolve the idiosyncratic volatility puzzle. We aim to shed light on this inconsistency. To do so, we first show that the dual effect of idiosyncratic volatility on stock pricing causes its estimated relation with realized return to be an inadequate indicator of its relation with expected return. Idiosyncratic volatility has a dual effect on stock pricing because it affects both investors' expected return and the efficiency of stock price in reflecting its underlying value. Idiosyncratic volatility is positively related to arbitrage risks that deter arbitrage and thus hinder the reduction of mispricing (Pontiff, 2006). In sum, idiosyncratic volatility has a dual effect on stock pricing because it not only shapes investors' expected return but also lowers stock pricing efficiency.

Due to this dual effect, the estimated relation of idiosyncratic volatility (IVOL) with realized return (⁠$Rt→t+1$⁠) captures its relations with both expected return and with the mispricing-related component of $Rt→t+1$⁠. In principle, the mispricing-related component stems from the correction of ex ante mispricing (⁠$Ct→t+1$⁠) or from the formation of ex post mispricing (⁠$Ft→t+1$⁠) or from both. For the correction of ex ante undervaluation, $Ct→t+1$ is positive and increases with the degree of ex ante undervaluation corrected, while for the correction of ex ante overvaluation, $Ct→t+1$ is negative and decreases with the degree of ex ante overvaluation corrected; for the formation of ex post undervaluation, $Ft→t+1$ is negative and decreases with the degree of ex post undervaluation formed, while for the formation of ex post overvaluation, $Ft→t+1$ is positive and increases with the degree of ex post overvaluation formed. Because IVOL is positively related to arbitrage risks, it is positively related to the degree of mispricing, both ex ante and ex post. Taken together, evidently IVOL is positively related to $Ct→t+1$ among ex ante undervalued stocks and negatively related to $Ct→t+1$ among ex ante overvalued stocks since a higher value of IVOL is associated with greater ex ante mispricing, both undervaluation and overvaluation; and it is negatively related to $Ft→t+1$ among ex post undervalued stocks and positively related to $Ft→t+1$ among ex post overvalued stocks, since a higher value of IVOL is associated with greater ex post mispricing, both undervaluation and overvaluation. Figure 1 depicts the process through which IVOL is linked to $Ct→t+1$ and $Ft→t+1$⁠.

Figure 1 shows that IVOL's relation with $Rt→t+1$ is a potentially biased estimate of its relation with expected return due to its relations with $Ct→t+1$ and $Ft→t+1$⁠. To infer IVOL's relation with expected return from its estimated relation with $Rt→t+1$⁠, the crucial task is to calibrate the sign and magnitude of its relation with the mispricing-related component (⁠$Ct→t+1$ and $Ft→t+1$⁠). However, this task is empirically unfeasible because all estimates of value are noisy and thus “we can never know how far away price is from value” (Black, 1986, p. 533). Nevertheless, IVOL's relations with $Ct→t+1$ and $Ft→t+1$ suggest a testable hypothesis that, ceteris paribus, its estimated relation with $Rt→t+1$ decreases with the proportion of ex ante overvalued observations in the sample and increases with the proportion of ex post overvalued observations in the sample.

Consistent with the hypothesis, we find robust evidence that IVOL's estimated relation with $Rt→t+1$ decreases and switches sign from positive to negative as the estimation sample consists of proportionately more ex ante overvalued observations and increases and switches sign from negative to positive as the estimation sample consists of proportionately more ex post overvalued observations. Our finding suggests that we cannot draw a reliable inference about IVOL's relation with expected return from its estimated relation with $Rt→t+1$⁠. Indeed, the sign switching for IVOL's estimated relation with $Rt→t+1$ suggests that its relations with $Ct→t+1$ and $Ft→t+1$ dominate its relation with expected return in its estimated relation with $Rt→t+1$⁠. If not, one would not observe such sign switching other than in the virtually inconceivable situation that its relation with expected return varies similarly in response to change in the proportion of ex ante (ex post) overvalued observations in the sample. One thus cannot infer the sign – let alone the magnitude – of IVOL's relation with expected return from its estimated relation with $Rt→t+1$⁠. Evidently, the documented negative relation between IVOL and $Rt→t+1$ does not necessarily contradict the predictions of traditional asset pricing theories and hence may not be an asset pricing puzzle. Moreover, we show that existing methods cannot address the bias. In sum, the estimated relation of idiosyncratic volatility (IVOL) with realized return (⁠$Rt→t+1$⁠) is an inadequate indicator of its relation with expected return.

Our findings demonstrate that IVOL's estimated relation with $Rt→t+1$ is inherently unstable due to the nonlinearity of its relations with $Ct→t+1$ and $Ft→t+1$⁠. We reason that ostensibly immaterial variation in research design choices can cause IVOL's estimated relation with $Rt→t+1$ to vary dramatically and even switch sign due to their effects on its estimated relations with $Ct→t+1$ and $Ft→t+1$⁠. First, a research design choice can change the proportions of ex ante and ex post undervalued and overvalued observations in the sample. Second, it can change the weight of ex ante significantly undervalued observations and ex post significantly overvalued observations in IVOL's estimated relation with $Rt→t+1$ due to its effect on the distribution of $Rt→t+1$⁠. A well-known regularity about $Rt→t+1$ is that it is right-skewed. We show that large positive returns driving the right-skewness of $Rt→t+1$ stem from the correction of ex ante significant undervaluation and the formation of ex post significant overvaluation to a large extent. This is not surprising because $Ct→t+1$ for the correction of ex ante undervaluation and $Ft→t+1$ for the formation of ex post overvaluation can be very large, whereas $Ct→t+1$ for the correction of ex ante overvaluation and $Ft→t+1$ for the formation of ex post undervaluation are at most −1.

We show that IVOL's estimated relation with $Rt→t+1$ is sensitive to ostensibly immaterial variation in oft-employed research design choices in a predictable manner consistent with our reasoning. Prior studies have generally excluded observations with very low price, arguing that the price movements of those stocks are susceptible to microstructure biases. However, these studies may differ regarding the price threshold for exclusion. Moreover, these studies rarely specify the timing for measuring stock price. We show that IVOL's estimated relation with $Rt→t+1$ varies dramatically and even switches sign in a predictable manner in response to ostensibly immaterial variation in the price threshold for exclusion ($0 vs $1 vs $5) and in the timing of measuring stock price (at the beginning and at the end of the holding period and average stock price for the holding period). Prior studies may also exclude small stocks, arguing that data quality of those stocks is low due to their illiquidity and ensuing noise in their stock pricing and that their economic significance is trivial. In practice, studies have great discretion over the size threshold for exclusion and rarely specify the timing of measuring size. We show that IVOL's estimated relation with $Rt→t+1$ varies dramatically and even switches sign in a predictable manner in response to variation in the size threshold for exclusion and in the timing of measuring firm size.

Our study contributes to research on the relation between idiosyncratic volatility (IVOL) and realized return (⁠$Rt→t+1$⁠) in at least two aspects. First, our study shows that because of its dual effect on stock pricing, IVOL's estimated relation with $Rt→t+1$ captures both its relation with expected return and its relations with the mispricing-correction component (⁠$Ct→t+1$⁠) and the mispricing-formation component (⁠$Ft→t+1$⁠). Hence its estimated relation with $Rt→t+1$ is an inadequate indicator of its relation with expected return. Indeed, our findings suggest that its relations with $Ct→t+1$ and $Ft→t+1$ dominate its relation with expected return in its estimated relation with $Rt→t+1$⁠. One thus cannot infer the sign of its relation with expected return from its estimated relation with $Rt→t+1$⁠. Evidently, the negative relation between idiosyncratic volatility and realized return documented in some studies does not necessarily contradict the prediction of traditional asset pricing theories and hence may not be an asset pricing puzzle.

Using a different ex ante overvaluation likelihood measure, Stambaugh, Yu, and Yuan (2015) also find that IVOL's estimated relation with $Rt→t+1$ decreases and switches sign from positive to negative as the estimation is moved from the bottom to the top quintile of their ex ante overvaluation likelihood measure. Our study complements and extends their study in at least two aspects. First, using a different but arguably superior ex ante overvaluation likelihood measure, our study affirms their major finding. More importantly, our study shows that the relation of IVOL with the mispricing-formation component also plays a significant role in shaping the overall relation between IVOL and realized return.

Second, our study sheds light on the inconsistency of findings about IVOL's estimated relation with $Rt→t+1$⁠. Our study shows that ostensibly immaterial variations in oft-employed research design choices can cause IVOL's estimated relation with $Rt→t+1$ to vary dramatically and even switch sign in a predictable manner due to their effects on its estimated relations with $Ct→t+1$ and $Ft→t+1$⁠. That is, our study shows that variations in research design choices drive the inconsistency of findings across studies. Extending prior studies that expose the sensitivity of IVOL's estimated relation with $Rt→t+1$ to variations in research design choices, our study provides a conceptual framework for understanding how and why IVOL's estimated relation with $Rt→t+1$ varies with research design choices.

The rest of the paper is organized as follows. Section 2 describes the dual effect of idiosyncratic volatility on stock pricing and develops the testable hypothesis. Section 3 presents the research design. Section 4 reports and discusses results from the main test and robustness tests. Section 5 reports and discusses results from additional analyses. Section 6 concludes.

Idiosyncratic volatility (IVOL) has a dual effect on stock pricing. First, it can shape stock pricing by affecting investors' expected return. Assuming that markets are complete and frictionless and investors are well-diversified, the classic capital asset pricing model (CAPM) implies no relation between a stock's expected return and its IVOL because no factors other than beta capture the cross-section variation in expected returns under CAPM. However, theories built on the realistic assumption of limited portfolio diversification imply a positive relation between a stock's expected return and its IVOL because total risks – including idiosyncratic risks – matter to investors with limited portfolio diversification (Levy, 1978; Merton, 1987). If investors expect a high return for holding a stock with high IVOL, the high expected return is equivalent to a low stock price given expected future cash flows available to stockholders.

Second, IVOL can shape stock pricing by affecting the efficiency of stock price in reflecting its underlying equity value. IVOL is positively related to arbitrage risks (Pontiff, 2006; Stambaugh et al., 2015). For arbitrageurs who can neutralize their exposure to benchmark risks, IVOL is more closely related to arbitrage risk than total volatility (Stambaugh et al., 2015). Pontiff (2006) shows that a mean-variance investor's desired position size for a given level of mispricing is smaller when a stock's IVOL is higher because higher IVOL is associated with a higher likelihood of substantial adverse price moves. Evidently, IVOL is positively related to the degree of mispricing, both ex ante and ex post and both undervaluation and overvaluation.

Consistent with firms with high IVOL having low stock pricing efficiency, we find that IVOL is negatively related to the informational efficiency of stock price, as gauged by the absolute value of the first-order autocorrelation of daily returns (AbsAutoCorr) ad price delay (PriceDelay) (see Figure IA1 of the  Internet Appendix). A smaller AbsAutoCorr indicates that the pricing process is closer to a random walk, making stock price more informationally efficient (Chordia, Roll & Subrahmanyam, 2008). PriceDelay captures the average delay of price movements in response to information (Hou & Moskowitz, 2005) ^[2]. A larger PriceDelay indicates greater information delay.

In sum, IVOL has a dual effect on stock pricing by affecting both investors' expected return and stock pricing efficiency. Next, we elaborate how this dual effect shapes the relation between idiosyncratic volatility and realized return.

We reason that the estimated relation of idiosyncratic volatility (IVOL) with realized return, due to its dual effect on stock pricing, is a potentially biased estimate of its relation with expected return. Our reasoning builds on two interrelated regularities. The first is that the price of a stock can deviate substantially from its value due to the speculative nature of stock price (Black, 1986; Shiller, 2014). This seems to be the norm rather than the exception, as observed and presented to members of American Finance Association by Fischer Black in his 1985 presidential address (Black, 1986) ^[3]. Fischer Black based his observation on his hands-on experience outside the academic ivory tower ^[4].

Hence realized return – the outcome of speculative stock price movements – consists of a mispricing-related component in addition to expected return (Black, 1986; Elton, 1999; Shiller, 2014). The mispricing-related component stems from the correction of ex ante mispricing, the formation of ex post mispricing, or both. Realized return (⁠$Rt→t+1$⁠) from time t to time t + 1 thus can be decomposed into four components: expected return, a mispricing-correction component (⁠$Ct→t+1$⁠), a mispricing-formation component (⁠$Ft→t+1$⁠), and anything else (⁠$Ot→t+1$⁠).

where $Rt→t+1$ is realized return from time t to time t + 1; $Et(Rt→t+1)$ is expected return given information at time t; $Ct→t+1$ is the mispricing-related component that stems from the correction of ex ante mispricing; $Ft→t+1$ is the mispricing-related component that stems from the formation of ex post mispricing; and $Ot→t+1$ is the component of $Rt→t+1$ other than $Et(Rt→t+1)$⁠, $Ct→t+1$⁠, and $Ft→t+1$⁠. For the correction of ex ante undervaluation, $Ct→t+1$ is positive and increases with the degree of ex ante undervaluation corrected, while for the correction of ex ante overvaluation, it is negative and decreases with the degree of ex ante overvaluation corrected; for the formation of ex post undervaluation, $Ft→t+1$ is negative and decreases with the degree of ex post undervaluation formed, while for the formation of ex post overvaluation, it is positive and increases with the degree of ex post overvaluation formed.

The second regularity is that because IVOL is positively related to the degree of mispricing, both ex ante and ex post and both undervaluation and overvaluation, its estimated relation with $Rt→t+1$ captures both its relation with $Et(Rt→t+1)$ ant its relations with $Ct→t+1$ and $Ft→t+1$⁠. Without loss of generality, we formalize the second regularity in the following equations:

where $plim$ is the probability limit operator, $COV(⋅)$ denotes covariance, and $Var(⋅)$ denotes variance. In Equation (2), we presume that there is a linear relation between IVOL and $Et(Rt→t+1)$⁠, which is captured by $β$⁠. $εt→t+1$ is the residual with a mean of 0. We get Equation (3) by expanding $Et(Rt→t+1)$ in Equation (1) according to Equation (2). Equation (3) can be viewed as a simplified version of the model specification that we estimate to examine the relation (⁠$β$⁠) between IVOL and expected return $Et(Rt→t+1)$⁠. In Equation (4), $βˆ$ is the OLS estimate of $β$⁠.

Assuming that $COV(Ot→t+1,IVOL)=0$⁠, we get

Equation (5) shows that $βˆ$ is a potentially biased estimate of $β$ due to $COV(Ct→t+1,IVOL)$ and $COV(Ft→t+1,IVOL)$⁠. Worse, the sign of $COV(Ct→t+1,IVOL)$ and $COV(Ft→t+1,IVOL)$ is indeterminate. Have $LnPt$ denote the natural logarithm of market value of equity at time t and $LnVt$ denote the natural logarithm of the intrinsic value of equity at time t. Then, $|LnPt−LnVt|$ measures the degree of ex ante mispricing at time t and $|LnPt+1−LnVt+1|$ measures the degree of ex post mispricing at time t + 1. Because IVOL is positively related to the degree of mispricing, both ex ante and ex post, we have $COV(IVOL,|LnPt−LnVt|)>0$ and $COV(IVOL,|LnPt+1−LnVt+1|)>0$⁠. Therefore,

If $LnPt−LnVt<0$ (i.e. ex ante undervalued), $Ct→t+1↑(>0)$⁠. Therefore, $COV(Ct→t+1,IVOL|LnPt−LnVt<0)>0$.

If $LnPt−LnVt>0$ (i.e. ex ante overvalued), $Ct→t+1↓(<0)$⁠. Therefore, $COV(Ct→t+1,IVOL|LnPt−LnVt<0)<0$.

and

If $LnPt+1−LnVt+1<0$ (i.e. ex post undervalued), $Ft→t+1↓(<0)$⁠. Therefore, $COV(Ft→t+1,IVOL|LnPt+1−LnVt+1<0)<0$.

If $LnPt+1−LnVt+1>0$ (i.e. ex post overvalued), $Ft→t+1↑(>0)$⁠. Therefore, $COV(Ft→t+1,IVOL|LnPt+1−LnVt+1>0)>0$.

where $↑$ denotes larger; $↓$ denotes smaller; and $|x|$ denotes the absolute value of x.

It is so far evident that IVOL's covariance with $Ct→t+1$ and with $Ft→t+1$ causes its estimated relation (⁠$βˆ$⁠) with $Rt→t+1$ to be a potentially biased estimate of its relation (⁠$β$⁠) with expected return (⁠$Et(Rt→t+1)$⁠). To draw a reliable inference about IVOL's relation (⁠$β$⁠) with $Et(Rt→t+1)$ from its estimated relation (⁠$βˆ$⁠) with $Rt→t+1$⁠, we need to calibrate the sign and magnitude of the bias resulting from its covariance with $Ct→t+1$ (⁠$COV(Ct→t+1,IVOL)$⁠) and with $Ft→t+1$ (⁠$COV(Ft→t+1,IVOL)$⁠). However, it is empirically unfeasible to do so, since “we can never know how far away price is from value” (Black, 1986, p. 533). Nevertheless, Equation (5) suggests a testable hypothesis that, ceteris paribus, IVOL's estimated relation with $Rt→t+1$ decreases with the proportion of ex ante overvalued observations in the sample and increases with the proportion of ex post overvalued observations in the sample.

Next, we test this hypothesis. Finding evidence supporting the hypothesis will cast doubt on the appropriateness of inferring IVOL's relation with expected return from its estimated relation with $Rt→t+1$⁠. Moreover, if we ever find sign switching for IVOL's estimated relation with $Rt→t+1$ – from positive (negative) to negative (positive) as the estimation sample consists of proportionately more ex ante (ex post) overvalued observations – we know that IVOL's relations with $Ct→t+1$ and $Ft→t+1$ dominate its relation with expected return in its estimated relation with $Rt→t+1$⁠. If not, we would not observe such sign switching other than in the virtually inconceivable situation that IVOL's relation with expected return varies and even switches sign similarly in response to change in the proportion of ex ante (ex post) overvalued observations in the sample. Hence one cannot infer the sign – let alone the magnitude – of IVOL's relation with expected return from its estimated relation with $Rt→t+1$⁠.

The crucial task for the hypothesis testing is to measure the overvaluation likelihood of observations. Following Rhodes-Kropf, Robinson, and Viswanathan (2005), we use the difference between the natural logarithm of the market value of equity and the natural logarithm of the estimated intrinsic value of equity as our primary measure of overvaluation likelihood (hereafter $LnP/V$⁠). Detailed in  Appendix 1, this estimated intrinsic value of equity (log) is a function of accounting items; the coefficients are rolling time-series averages of annual estimates. Rhodes–Kropf et al. (2005) provide initial evidence for the validity of $LnP/V$ as an overvaluation likelihood measure, showing that the pattern of merger and acquisition activities varies with $LnP/V$ as theoretically predicted. We further show that observations in the top and bottom $LnP/V$ quintiles differ systematically along dimensions that prior studies find vary with observations' valuation status (i.e. undervalued or overvalued).

Any estimation of a stock's valuation status is subject to the misclassification problem, since “all estimates of value are noisy” (Black, 1986, p. 533). But that may not be a concern in our research setting since misclassification only runs against finding evidence for the hypothesis. More importantly, what we need in our setting is an instrument that reasonably captures the relative overvaluation likelihood rather than the exact valuation status. $LnP/V$ seems well suited, since the comparison results reported in  Appendix 1, together with Rhodes–Kropf et al. (2005) finding, demonstrate that observations with larger $LnP/V$ are more likely to be overvalued than observations with smaller $LnP/V$⁠. Moreover, we show that our inference is robust to the use of two alternative measures of overvaluation likelihood.

Our primary measure of idiosyncratic volatility (IVOL) is the standard deviation of residuals from a regression that takes daily excess returns as a function of daily excess market returns and daily returns to the small-minus-big, high-minus-low, momentum, robust-minus-weak, and conservative-minus-aggressive factors. IVOL is computed using daily data from 07/01 of t − 1 to 06/30 of t, t = 1966 to 2015. Moreover, we show that our inference is robust to two alternative ways of computing IVOL.

The regression model for the hypothesis testing is:

where i is firm i and t is year t; $Ri,t→t+1$ is stock return over 07/01 of t through 06/30 of t + 1, t = 1966 to 2015; IVOL is the idiosyncratic volatility measure; $LnP/V(t)$ (⁠$LnP/V(t+1)$⁠) is the difference between the natural logarithm of the market value of equity on 06/30 of t (t + 1) and the natural logarithm of the estimated intrinsic value of equity obtained using the latest accounting information available by 06/30 of t (t + 1) (see  Appendix 1); and $LnP/V(t):Qj$ (⁠$LnP/V(t+1):Qj$⁠) is an indicator variable that equals 1 if $LnP/V(t)$ (⁠$LnP/V(t+1)$⁠) is in the j-th quintile (0 otherwise), j = 1 to 5, in which we sort observations with no missing values for $Ri,t→t+1$⁠, IVOL and $LnP/V(t)$ (⁠$LnP/V(t+1)$⁠) into five equal groups by $LnP/V(t)$ (⁠$LnP/V(t+1)$⁠) on 06/30 of t (t + 1). Consistent with the hypothesis, the construction of $LnP/V(t+1)$ uses the latest information available by 06/30 of t + 1. This is not an issue since we are proposing any trading strategy. Instead, we examine what underlies the estimated relation between idiosyncratic volatility and realized return.

Controls stands for control variables. We refer to Fama and French (2008) to identify them. Specifically, we control for these equity attributes: firm size (Size), the book-to-market ratio (B/M), momentum (Momentum), net stock issues (NetStkIssue), zero net stock issues (ZeroNetStkIssue), negative total accruals (NegTtlAcc), positive total accruals (PosTtlAcc), asset growth (AssetGrowth), positive profitability (PosIB/BE) and loss (NegIB). Definitions of these control variables are provided in  Appendix 2.

Because firms differing in idiosyncratic volatility may differ in their exposure to systematic risk factors, we control for firms' sensitivity to six risk factors identified in Carhart (1997) and Fama and French (2015). Specifically, we control for Beta-MktRf, Beta-SMB, Beta-HML, Beta-MOM, Beta-CMA, and Beta-RMW; these are factor loadings on the market factor, the small-minus-big factor (SMB), the high-minus-low factor (HML), the momentum factor (MOM), the conservative-minus-aggressive factor (CMA), and the robust-minus-weak factor (RMW), respectively. Because expected return varies across industries (Fama & French, 1997), we also control for industry fixed effects, Industry FE. We define industry membership according to the Fama-French 49 industries.

We apply the Fama-MacBeth regression to estimate Equations (6a) and (6b), since it is “standard in tests of asset pricing models” (Fama, 2014, p. 1478). Our Fama-MacBeth regression estimate is the time-series average of annual OLS coefficient estimates (Fama & MacBeth, 1973). We use standard errors adjusted for Newey-West autocorrelations of three lags to compute T-statistics.

The hypothesis predicts that in Equation (6a), $λ5<0$ and $λ1>(λ1+λ5)$⁠, while in Equation (6B), $λ5>0$ and $λ1<(λ1+λ5)$⁠. We focus on the contrast between observations in the top and the bottom $LnP/V$ quintiles because $nP/V$ , as a noisy measure of the price-to-value ratio, is arguably better able to differentiate the valuation status for observations in these two quintiles than for observations in the middle three quintiles. While we do not expect a monotonic transition from $λ1$ to $λ1+λ5$⁠, we expect $λ1+λj,j=2,3,4,$ to be between $λ1$ and $λ1+λ5$⁠.

We obtain accounting data from Compustat; equity data from CRSP; and the Fama-French industry group classifications and factor return data, including risk-free rates, from Kenneth R. French's online data library. To mitigate the concern about data snooping and mining, we use all firm-year observations with the required variables available.

Table 1 reports descriptive statistics of variables for the sample used in the main test. The sample consists of 180,717 firm-year observations from 1966 through 2015 ^[5]. We winsorize all continuous variables except $Rt→t+1$ at the 1st and 99th percentiles of their cross-sectional distributions each year. Summary statistics reported in Panel A are comparable with those reported in prior studies.

Panel B reports Pearson and Spearman correlations. Three sets of correlations deserve attention. First, the correlations of $Rt→t+1$ with control variables are consistent with findings of prior studies. Second, $LnP/V(t)$ and $LnP/V(t+1)$ are significantly positively correlated: 0.73 (Pearson) and 0.70 (Spearman), suggesting that a firm's overvaluation likelihood is persistent over time. Third, consistent with $LnP/V(t)$ measuring the ex ante overvaluation likelihood $LnP/V(t)$ and $Rt→t+1$ are negatively correlated: −0.10 (Pearson) and −0.11 (Spearman); consistent with $LnP/V(t+1)$ measuring the ex post overvaluation likelihood $LnP/V(t+1)$ and $Rt→t+1$ are positively correlated: 0.30 (Pearson) and 0.33 (Spearman).

Table 2 presents results from the main test of the hypothesis. Table 2 shows that the coefficient estimates for $LnP/V(t):Qj$ (⁠$LnP/V(t+1):Qj$⁠), $j=2to5$⁠, are significantly negative (positive) and increase in magnitude. The transition pattern of these coefficient estimates is consistent with the notion that observations with larger $LnP/V$ are more likely to be overvalued than observations with smaller $LnP/V$⁠.

When the model specification ignores that IVOL's estimated relation with $Rt→t+1$ varies with the proportion of ex ante (ex post) overvalued observations in the sample, the overall estimated relation is positive but statistically insignificant ^[6]. This appears to be consistent with the finding of some prior studies (e.g. Fama & MacBeth, 1973). Panel A shows that with(out) control variables, IVOL's estimated relation with $Rt→t+1$ monotonically decreases from 2.9027 (t = 3.47) (2.9829 (t = 2.52)) to −1.9359 (t = −1.85) (−1.6024 (t = −1.14)) as the estimation is moved from the bottom $LnP/V(t)$ quintile to the top one; Panel B shows that with(out) control variables, IVOL's estimated relation with $Rt→t+1$ monotonically increases from −1.8086 (t = −3.95) (−1.8627 (t = −2.49)) to 9.1157 (t = 6.11) (9.9579 (t = 5.84)) as the estimation is moved from the bottom $LnP/V(t+1)$ quintile to the top one. These results support the hypothesis.

Evidently, to draw a reliable inference about IVOL's relation with expected return (⁠$Et(Rt→t+1)$⁠) from its estimated relation with $Rt→t+1$⁠, we need to calibrate the sign and magnitude of its relations with $Ct→t+1$ and $Ft→t+1$⁠, which is empirically unfeasible (Black, 1986). Importantly, the sign switching for IVOL's estimated relation with $Rt→t+1$ suggests that its relation with the mispricing-related component (⁠$Ct→t+1$ and $Ft→t+1$⁠) dominates its relation with expected return (⁠$Et(Rt→t+1)$⁠) in its estimated relation with $Rt→t+1$⁠. If not, we would not observe the sign switching other than in the virtually inconceivable situation that IVOL's relation with expected return varies similarly between ex ante (ex post) undervalued and overvalued observations. We thus cannot infer the sign – let alone the magnitude – of IVOL's relation with expected return from its estimated relation with $Rt→t+1$⁠. That is, we have no way to know whether the negative estimated relation between IVOL and $Rt→t+1$ documented in some studies contradicts the prediction of classic asset pricing theories.

We try two alternative overvaluation likelihood measures. One is the market-to-book ratio (MTB(t)), in which the market value of equity is measured on 06/30 of t and the book value of equity is computed using the latest accounting information available by 06/30 of t. The book-to-market ratio (BTM(t)), the inverse of MTB(t), has consistently been found to be positively related to realized return. Piotroski and So (2012) find that BTM(t) has predictive power for realized return only for firms for which the expectation implied by BTM(t) is incongruent with the strength of the firm's fundamentals. Their finding is consistent with the view that observations with larger MTB(t) are more likely to be overvalued than those with smaller MTB(t).

The other is the price-to-value ratio, based on the residual income valuation model (P/V‒F&L(t)); the numerator (P) is the market value of equity on 06/30 of t and the denominator (V‒F&L) is the estimated intrinsic value of equity obtained by incorporating model-based earnings predictions and the industry-specific cost of equity into Frankel and Lee's (1998) empirical implementation of the residual income valuation model introduced in Ohlson (1995). We adopt Hou, van Dijk, and Zhang's (2012) model-based approach to forecasting earnings ^[7]. Following Frankel and Lee (1998), we apply Fama and French's (1993) three-factor model to estimate the industry-specific cost of equity. Frankel and Lee (1998) find a statistically reliable positive relation between their V/P estimate, the inverse of P/V-F&L, and realized return, suggesting that observations with larger P/V-F&L are more likely to be overvalued than those with smaller P/V-F&L.

We report results based on MTB and P/V-F&L respectively in Tables IA2 and IA3 of the  Internet Appendix. IVOL's estimated relation with $Rt→t+1$ is significantly smaller (larger) in the top quintile than in the bottom quintile of MTB or P/V-MPEG measured on 06/30 of t (t + 1) and is in between for the middle three quintiles. Evidently, results based on MTB and P/V-F&L support the hypothesis.

We try two alternative ways of computing IVOL. Our primary measure of IVOL is computed using daily data from 07/01 of t − 1 to 06/30 of t. To alleviate the potential concern that this measure may not well capture investors' expectation about future idiosyncratic volatility on 06/30 of t since it seems to rely on distant data, we use data from 04/01 of t through 06/30 of t to compute idiosyncratic volatility (IVOL3MON). We use monthly data from 07/01 of t − 5 through 06/30 of t with at least 12 observations to compute IVOL (IVOL5Year).

We report results based on IVOL3MON and IVOL5Year respectively in Tables IA4 and IA5 of the  Internet Appendix. Results based on IVOL3MON and IVOL5Year also well support the hypothesis.

Prior studies propose various explanations for the negative estimated relation between IVOL and realized return that is first documented by Ang et al. (2006). It is hard to conceive how these explanations account for the nonlinear relation between IVOL and realized return documented in this study. However, to show that our results cannot be accounted for by these explanations, we control for return skewness and stock liquidity. Hou and Loh (2016) show that explanations based on investors' lottery preferences and market frictions are most promising in explaining the negative estimated relation between IVOL and realized return. Return skewness captures investors' lottery preferences and stock liquidity measures market frictions (Hou & Loh, 2016).

We compute return skewness (RetSkewness) using daily return data from 04/01 of t to 06/30 of t and compute stock liquidity (StkLiq) as −1 × the natural logarithm of Abdi and Ranaldo's (2017) effective bid-ask spread estimate using daily close, high, and low prices from 07/01 of t − 1 to 06/30 of t. Table IA6 of the  Internet Appendix shows that controlling for RetSkewness and StkLiq has no material impact on our inference.

We run four more robustness analyses. In the first robustness analysis, we follow Brennan, Chordia, and Subrahmanyam (1998) and use the risk-adjusted return as the dependent variable. In the second robustness analysis, we use monthly realized return as the dependent variable. Table IA7 the  Internet Appendix shows that our inference remains the same under these two alternative quantifications of realized return.

In the third robustness analysis, we examine whether the occurrence of economic recessions affect our results since economic recessions and ensuing crises may affect market-wide mispricing. As shown in Table IA8, our results hold regardless of whether economic recessions occur before and after the portfolio formation.

Finally, we run size-weighted Fama-MacBeth regressions to test the hypothesis. In untabulated results, we find that the hypothesis holds. This suggests that the results are not unduly influenced by microcap stocks.

Our hypothesis development builds on the regularity that IVOL and the degree of mispricing are positively related. $|LnPt−LnVt|$⁠, the absolute difference between $LnPt$ and $LnVt$⁠, measures the degree of mispricing where $Ln(⋅)$ is the natural logarithm transformation operator, $Pt$ is the market value of equity at time t, and $Vt$ is the intrinsic value of equity at time t. A larger value of $|LnPt−LnVt|$ means greater mispricing. Because IVOL and the degree of mispricing are positively related, we have $COV(IVOL,|LnPt−LnVt|)>0$ where $COV(⋅)$ denotes covariance. For overvalued stocks, $|LnPt−LnVt|= LnPt−LnVt$ = $LnPt/Vt$ while for undervalued stocks, $|LnPt−LnVt|=−(LnPt−LnVt)$ = $−LnPt/Vt$⁠. Therefore, for overvalued stocks, $COV(IVOL,LnPt/Vt)>0$ while for undervalued stocks, $COV(IVOL,LnPt/Vt)<0$⁠. This suggests a testable hypothesis that the greater the proportion of overvalued observations in the sample, the larger the estimated relation between IVOL and $LnPt/Vt$⁠.

$LnP/V(t)$⁠, the overvaluation likelihood measure, can be taken as the sum of $LnPt/Vt$ and a value estimation error ^[8]. We therefore expect to observe that IVOL's estimated relation with $LnP/V(t)$ is larger for firms in the top $LnP/V(t)$ quintile than for firms in the bottom $LnP/V(t)$ quintile, which Figure 2 shows to be the case. To generate Figure 2, we run the Fama and MacBeth (1973) regression to estimate the following equation for each $LnP/V(t)$ quintile: $Pct= γ+β∗IVOL+ε$⁠, where $Pct$ is the annual rank of $LnP/V(t)$ scaled to have a minimum of 0 and a maximum of 100. Figure 2 shows that IVOL's estimated relation with $Pct$ monotonically increases from −62.1516 (t = −5.16) to 54.9086 (t = 5.96) as the estimation is moved from the bottom $LnP/V(t)$ quintile to the top one.

We substitute $M/B(t)$ or $P/V−F&L(t)$ for $LnP/V(t)$ and redo the analysis. We report the results in Figure IA2 of the  Internet Appendix. Panel A shows that IVOL's estimated relation with the annual rank of $M/B(t)$ monotonically increases from −50.7943 (t = −7.02) to 48.2048 (t = 5.71) as the computation is moved from the bottom $M/B(t)$ quintile to the top one. Panel B shows that IVOL's estimated relation with the annual rank of $P/V−F&L(t)$ monotonically increases from −50.9163 (t = −3.84) to 82.2379 (t = 7.76) as the computation is moved from the bottom $P/V−F&L(t)$ quintile to the top one.

Collectively, Figure 2 and Figure IA2 provide direct evidence that idiosyncratic volatility and the degree of mispricing are positively related.

Research on the relation between idiosyncratic volatility (IVOL) and $Rt→t+1$ is characterized by inconsistent findings. IVOL's estimated relation with $Rt→t+1$ is inherently unstable, due to the nonlinearity of IVOL's relation with $Ct→t+1$ and with $Ft→t+1$⁠. We reason that variation in research design choices across studies may drive the inconsistency by shaping IVOL's estimated relations with $Ct→t+1$ and $Ft→t+1$⁠.

We can think of at least two reasons for which research design choices affect IVOL's estimated relations with $Ct→t+1$ and $Ft→t+1$⁠. First, a research design choice can change the proportions of ex ante and ex post undervalued and overvalued observations in the sample. Second, it can change the weight of ex ante significantly undervalued observations and ex post significantly overvalued observations in IVOL's estimated relation with $Rt→t+1$ due to its effect on the statistical properties of $Rt→t+1$⁠. A well-known regularity of $Rt→t+1$ is that it is right-skewed. $Ct→t+1$ for the correction of ex ante undervaluation and $Ft→t+1$ for the formation of ex post overvaluation can be very large, but $Ct→t+1$ for the correction of ex ante overvaluation and $Ft→t+1$ for the formation of ex post undervaluation are at most −1. Therefore, large positive returns driving the right-skewness of $Rt→t+1$ high likely stem from the correction of ex ante significant undervaluation and the formation of ex post significant overvaluation.