Idiosyncratic Risk and the Cross-Section of Stock Returns: Merton (1987) Meets Miller (1977)

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1 Idiosyncratic Risk and the Cross-Section of Stock Returns: Merton (1987) Meets Miller (1977) Rodney D. Boehme, Bartley R. Danielsen, Praveen Kumar, and Sorin M. Sorescu* This version: January 5, 2006 Abstract Merton (1987) predicts that idiosyncratic risk should be priced when investors hold sub-optimally diversified portfolios, and cross-sectional stock returns should be positively related to their idiosyncratic risk. However, the literature generally finds a negative relationship between returns and idiosyncratic risk, which is more consistent with Miller s (1977) analysis of asset pricing under short-sale constraints. We examine the cross-sectional effects of idiosyncratic risk (and dispersion of beliefs) on stock returns while controlling for short-sale constraints and the level of investment recognition. When short-sale constraints are absent, both idiosyncratic risk and dispersion of analyst forecasts are positively correlated with future abnormal returns for firms with low visibility, consistent with Merton. However, stocks with higher analyst dispersion and idiosyncratic volatility have negative abnormal returns when short-sale constraints are present, consistent with Miller. Thus, rather than being competing hypotheses, Miller (1977) and Merton (1987) are actually complementary facets of a more general asset-pricing model with informational and short-sale constraints. * Boehme is from W. Frank Barton School of Business at Wichita State University. Danielsen is from the Kellstadt College of Commerce at DePaul University. Kumar is from the C.T. Bauer College of Business, University of Houston. Sorescu is from Mays Business School at Texas A&M University. We thank an anonymous referee for very helpful comments. This paper has also benefited from the comments of Fred Arditti, Mark Flannery, Matt Spiegel, Anna Scherbina, Mark Walker, Stephen C. Vogt of Mesirow Financial, Kevin Mirabile of Saw Mill Management and Research, as well as seminar participants at Texas A&M University and the University of Kansas. Please address correspondence to Kumar at the C.T. Bauer College of Business, University of Houston, Houston, TX , phone (713) , pkumar@uh.edu. Data on analysts forecasts was provided by I/B/E/S Inc., under a program to encourage academic research.

2 Idiosyncratic Risk and the Cross-Section of Stock Returns: Merton (1987) Meets Miller (1977) Abstract Merton (1987) predicts that idiosyncratic risk should be priced when investors hold sub-optimally diversified portfolios, and cross-sectional stock returns should be positively related to their idiosyncratic risk. However, the literature generally finds a negative relationship between returns and idiosyncratic risk, which is more consistent with Miller s (1977) analysis of asset pricing under short-sale constraints. We examine the cross-sectional effects of idiosyncratic risk (and dispersion of beliefs) on stock returns while controlling for short-sale constraints and the level of investment recognition. When short-sale constraints are absent, both idiosyncratic risk and dispersion of analyst forecasts are positively correlated with future abnormal returns for firms with low visibility, consistent with Merton. However, stocks with higher analyst dispersion and idiosyncratic volatility have negative abnormal returns when short-sale constraints are present, consistent with Miller. Thus, rather than being competing hypotheses, Miller (1977) and Merton (1987) are actually complementary facets of a more general asset-pricing model with informational and short-sale constraints.

3 According to the textbook capital asset pricing model (CAPM), idiosyncratic risk is not priced because investors hold efficiently diversified portfolios. However, the model makes no predictions concerning the effect of idiosyncratic risk on equilibrium returns if investors are constrained from forming diversified portfolios due to transactions costs---for example, information or trading costs. In an influential paper, Merton (1987) presents an extension of the CAPM where idiosyncratic risk plays a role in equilibrium. Merton forwards the investor recognition hypothesis which posits that, due to incomplete information on security characteristics, investors only hold securities whose risk and returns characteristics they are familiar with. Consequently, they hold under-diversified portfolios and, in the static mean-variance setting considered by Merton, demand compensation for securities idiosyncratic risk. 1 Thus, in equilibrium, cross-sectional stock returns are positively related to their idiosyncratic risk. 2 Direct tests of Merton s (1987) model are rare. Merton s predictions are cross-sectional in nature, but Ang, Hodrick, Xing and Zhang (2004) appear to be the only cross-sectional test of Merton (1987) that directly sorts stocks into portfolios ranked on idiosyncratic volatility. Observing that stocks with high idiosyncratic volatility have abysmally low average returns, they conclude their results are directly opposite to the implications of the (static) investor recognition hypothesis. Meanwhile, Diether, Malloy and Scherbina (2002) offer an indirect test of Merton (1987). Since dispersion in analysts forecasts likely indicates a more volatile, less predictable earnings stream, Diether et al. (2002) suggest that the dispersion of analysts forecasts reflects the type of 1 Levy (1978) and Mayers (1976) produce similar predictions in CAPM extensions where investors hold underdiversified portfolios. Barberis and Huang (2001) produce a prospect-theory model where idiosyncratic risk produces positive expected returns. 2 Shapiro (2002) generalizes the IRH to a dynamic setting and shows that this conclusion need not hold when the investment opportunity set is stochastically evolving, since higher volatility stocks may still provide a more effective hedge against shifting investment opportunities, compared to lower volatility stocks. 1

4 idiosyncratic risk to which Merton refers. Indeed, the literature shows that idiosyncratic volatility and dispersion of analysts forecasts are positively correlated. 3 Their results do not support Merton s theory, and they note our results clearly reject the notion that dispersion can be viewed as a proxy for risk, since the relation between dispersion and future returns is strongly negative (page 2139). Similarly, when Gebhardt, Lee and Swaminathan (2001) use forecast dispersion as a risk proxy for estimating cost of capital, they are surprised to find the wrong sign on the variable at statistically and economically significant levels. Thus, both in direct tests using idiosyncratic volatility as a proxy for idiosyncratic risk, and in indirect tests that use analysts forecast dispersion as a risk proxy, cross-sectional results are incorrectly signed at high levels of significance compared to Merton s (1987) predictions. Based on the empirical tests to date, the literature concludes that Merton s hypothesis, while both intuitively appealing and theoretically well grounded, is not supported by the data. In fact, Diether et al. (2002) argue that their results, along with those of Gebhardt et al. (2001), are more consistent with predictions by Miller (1977). Miller (1977) argues that dispersion of opinion, in the presence of short sale constraints, leads to systematic security overvaluation because the most optimistic market participants set a stock s price. Thus, dispersion of investor opinion is priced at a premium when short sale constraints are present. Miller s (1977) theory implies that if short sale constraints are binding, then there is a negative correlation between risk-adjusted returns and dispersion of beliefs. 4 The observed negative correlation between returns and analyst forecast dispersion (Diether et al. (2002)) is interpreted as supportive of Miller (1977), but contrary to the 3 Empirically, Peterson and Peterson (1982) observe a positive relationship between return volatility and the dispersion of I/B/E/S forecasts, and we confirm their finding later in this study. 4 Indeed, several other theoretical papers derive similar asset pricing predictions. A non-exhaustive set includes Figlewski (1981), Morris (1996), Viswanathan (2002), Chen, Hong and Stein (2002), Danielsen and Sorescu (2001), and Duffie, Garleanu and Pedersen (2002). Diamond and Verrecchia (1987) and Jarrow (1980) provide alternative theories of the effects of short-sale constraints on security prices. 2

5 predictions of Merton (1987), because the dispersion of analysts forecasts and idiosyncratic volatility are positively correlated. However, there are strong reasons to question whether Merton (1987) should be viewed as a competing theory to Miller (1977). While Miller (1977) assumes that stocks are short-sale constrained, Merton (1987) models a standard frictionless market, without borrowing and shortselling restrictions (see, page 487). Absence of frictions is, in fact, crucial for Merton s (1987) prediction of a positive relationship between equilibrium returns and idiosyncratic risk, as we show in this paper. We consider a static investor recognition model with heterogeneous beliefs (see Appendix A), where there are transaction costs in shorting, and binding short-sale constraints are possible. 5 The imposition of short-sale constraints alters the relationship of equilibrium returns to idiosyncratic risk. Returns are no longer unambiguously positively correlated with idiosyncratic risk. Instead, a Miller-like effect counteracts Merton s result, and the impact of increased idiosyncratic risk may be positive or negative. Intuitively, more risky stocks may have higher equilibrium prices (or lower subsequent returns) if they are short-sale constrained and pessimistic investors can not fully counterbalance the optimistic investors. Viewed in this light, Miller (1977) and Merton (1987) are complementary facets of a more general asset-pricing framework with informational and short-sale constraints. Miller (1977) applies to short-sale constrained firms, and Merton (1987) is applicable to stocks where the short-sale constraint is not binding. Empirical tests that fail to distinguish between short-saleconstrained firms and firms that are unlikely to be subject to such constraints are neither appropriate for tests of Merton (1987), nor for tests of Miller (1977). 5 See, e.g., Jones and Lamont (2002), Asquith, Pathak, and Ritter (2005), and Boehme, Danielsen and Sorescu (2006). 3

6 We analyze the cross-sectional relationship between idiosyncratic risk and equity returns by controlling for the likelihood of binding shorting constraints. Boehme, Danielsen and Sorescu (2006) posit that short interest ratios are positively correlated with the extent of short sale constraint, as measured by the stock shorting fees. In Appendix A, we provide theoretical support for this relation (see Proposition 3), and accordingly focus our search for Merton s hypothesized pricing among firms with low levels of short interest. In addition to short sale constraints, visibility is another critical ingredient in Merton s investor recognition hypothesis since incomplete information plausibly applies only for firms with low visibility. Falkenstein (1996) and Aggarwal, Klapper and Wysocki (2005) link institutional ownership (positively) to firm visibility. We utilize this relationship to screen out firms with high levels of institutional ownership (high-visibility) for which the investor recognition hypothesis should not apply. 6 Using data on short interest on NYSE and NASDAQ stocks from and institutional ownership data for the same period, we find that cross-sectional returns are positively correlated with idiosyncratic volatility, as predicted by Merton, for less visible stocks with low shorting constraints. When low institutional ownership portfolios are sorted according to short interest, there is a significant and positive correlation between returns and idiosyncratic risk for low short interest portfolios. These portfolios have low intrinsic visibility (consistent with the investor recognition hypothesis), and low short sale constraints. 6 Our screening method is consistent with that used by Asquith, Pathak, and Ritter (2005). These authors posit that short interest ratios are a proxy for short sale demand and institutional ownership is a proxy for lendable supply (page 244). Thus, according to Asquith, Pathak and Ritter, only stocks with high levels of short interest and low levels of institutional ownership are likely to be short-sale constrained, while the remaining stocks are left unconstrained. These remaining stocks include (1) those with high levels of short interest and high levels of institutional ownership, (2) those with low levels of short interest and high levels of institutional ownership, and (3) those with low levels of short interest and low levels of institutional ownership. From these three remaining subsets, only the third one includes stocks that are also likely to have low visibility, a necessary condition for Merton s investor recognition hypothesis. This third subset therefore includes stocks that are not short-sale constrained in the Asquith, Pathak and Ritter sense, and that in addition have low visibility. This is precisely the subset of stocks we utilize in our study. 4

7 For completeness we also examine short-sale constrained stocks expecting to find evidence of Miller-style overpricing. And we do. As in Asquith, Pathak and Ritter (2005), we classify stocks as short sale constrained if they have high levels of short interest and low levels of institutional ownership. For these stocks there is a significant and negative relation between dispersion of analysts forecasts and subsequent returns, and firms with high idiosyncratic risk under-perform on a risk-adjusted basis. Our results explain the relatively weak support for the Miller hypothesis during the 1990s found by Diether et al. (2002), where Miller s hypothesis is tested as a competitor to Merton s investor recognition hypothesis. More recently, Boehme, Danielsen and Sorescu (2006) find stronger support for Miller s theory during the 1990s by explicitly recognizing the interaction of short-sale-constraints with dispersion of beliefs. Our analysis contributes by highlighting the importance of screening out both short-sale-constrained firms and firms with high investor recognition when testing Merton s (1987) hypothesis. Furthermore, our empirical results complement the evidence Shapiro (2002) finds in support of the investor recognition hypothesis. We organize the rest of this paper as follows. In Section I, we develop the testable hypotheses and discuss the testing methodology and empirical proxies. Section II presents base-line cross-sectional effects, while section III presents empirical tests of Merton (1987) for firms that are free of short-sale constraints and have low visibility. Section IV repeats the analysis for short-sale constrained firms. Section V summarizes and concludes with a graph providing a more holistic representation of the cross-sectional effects of idiosyncratic volatility and dispersion of beliefs. 5

8 I. Hypotheses, Explanatory Variables, and Test Design A. Hypotheses Development In a static mean-variance setting, Merton (1987) uses the investor recognition hypothesis (IRH) to show that equilibrium security returns are positively related to their own idiosyncratic risk. It is important to note that Merton does not consider any market friction other than the incomplete information that underlies the IRH. In particular, Merton does not consider the effects of binding short-sale constraints on equilibrium asset pricing with the IRH. In Appendix A, we show that the relation of equilibrium asset prices to idiosyncratic risk is ambiguous even in the static mean variance setting if there are binding short-sale constraints. A simple example illustrates the potentially confounding effects of short-sale constraints in the basic IRH model. Consider an initial situation where for some security the short-sale constraints are non-binding. That is, the market demand for shorting the security, derived from aggregating the optimal shorting demand of individual investors, is less than the supply available for shorting. Now, imagine that the idiosyncratic risk of the security increases for exogenous reasons. Due to the IRH, this will increase the optimal shorting demand from the most pessimistic investors, and possibly make the short-sale constraint bind for the security. In the new equilibrium, the security s price now internalizes the beliefs of the optimistic investors (cf. Miller (1977)) and, under certain conditions, may be higher than the initial price (prior to the increase in the idiosyncratic risk). Hence, an empirical test of Merton s (1987) IRH model must identify securities with low visibility: those for which there exists some investor segment that has incomplete information on the relevant risk-return tradeoff parameters. Further, short-sale constraints should not be binding for these securities. 6

9 Meanwhile, Miller (1977) argues that with heterogeneous beliefs among investors, shortsale constraints preclude pessimistic investors from offsetting demand from optimistic investors. Consequently, the equilibrium price is positively biased and subsequent returns are low. Since Miller s hypothesis pertains only to stocks with binding short-sale constraints, its applicability is disjoint from Merton s, which applies (unambiguously) only when short-sale constraints are non-binding. It is important to note that, unlike Merton s model, Miller s analysis does not provide any equilibrium comparative statics predictions with respect to stock risk. What then is the role of idiosyncratic risk in Miller s framework? To the extent that Miller s analysis requires heterogeneity of investor s beliefs, there should be a negative association between returns and idiosyncratic risk if such risk is positively related to the unobserved dispersion of beliefs. But this suggests that returns will be negatively related to other (possibly less noisy) proxies for belief dispersion, such as analyst forecast dispersion. Later in the paper, we show that there is a positive correlation between idiosyncratic volatility and the dispersion of analyst forecasts. Based on the foregoing analysis, we can establish two basic hypotheses that restrict the relation of returns to idiosyncratic risk in the presence and absence of short sale constraints. 1. The Merton Hypothesis. Cross-sectional differences in idiosyncratic volatility are positively correlated with subsequent returns for less visible stocks (IRH) when short-sale constraints are unlikely to be binding. 2. The Miller Hypothesis. Cross-sectional differences in the dispersion of beliefs and idiosyncratic volatility are negatively correlated with subsequent returns when short-sale constraints are most likely to be binding. The refutable predictions implicit in Hypotheses 1 and 2 form the basis of our empirical tests. Transparently, in order to conduct empirical tests, we must (1) identify the two subsets of 7

10 stocks that are subject to Merton s and Miller s hypotheses (i.e. the Merton stocks and the Miller stocks), (2) establish empirical proxies for idiosyncratic volatility and dispersion of beliefs, and (3) formulate an appropriate method for measuring long-term stock returns. B. Identifying Merton (1987) and Miller (1977) stocks B.1. The Merton Stocks The Merton hypothesis (cf. Hypothesis 1) applies to stocks with low visibility and nonbinding short-sale constraints. That is, the IRH applies to stocks for which there exists some investor segment that is plausibly not aware of all the relevant risk-return tradeoff parameters, and for which short position are relatively easy to establish. We use institutional ownership as a proxy for stock visibility. Falkenstein (1996) finds that institutional investors (specifically U.S. mutual funds) have a significant preference for stocks with high visibility, as measured by coverage in newspaper articles. Similarly, Aggarwal et al. (2005) find that institutional ownership is positively correlated with visibility as proxied by the number of analysts following the stock. We calculate institutional ownership (IO) as the ratio of the total or aggregate number of shares held by institutions (13F filings) to the total number of shares outstanding reported by CRSP for the end of the calendar month of the 13F filing. The 13F records provide institutional holdings on a quarterly basis. Therefore, we use the IO for the month in which the 13F s are filed and for the subsequent two months. For example, if a firm had 40% ownership by institutions at the end of March 1996, we use that 40% ownership percentage estimate also for April and May. 8

11 To identify the set of non-short-sale-constrained stocks, we use Proposition 3 from Appendix A and focus our attention on stocks with low levels of relative short interest (RSI). 7 We measure the RSI as the actual level of shares sold short scaled by outstanding shares. We obtain short interest data from the New York Stock Exchange and the NASD. Our sample is composed of all firms for which short interest data are electronically available from these sources. For both NYSE and NASDAQ firms, such data are available beginning with January of We collect all short interest data on a monthly basis for transactions settling by the 15 th of each month. We use the ticker symbols shown in the short interest reports to match each observation with the CRSP data. 8 B.2. The Miller Stocks To identify stocks for which the Miller model is most likely to apply, we follow Asquith, Pathak and Ritter (2005) and consider the intrinsic demand for shorting along with the supply of lendable shares together. Asquith, Pathak and Ritter note that low institutional ownership (IO) not only reflect low visibility, but also likely reflects a limited supply of lendable shares. However, Asquith, Pathak and Ritter posit that without high shorting demand, as reflected in high short interest, shares should not become short-sale constrained. Their empirical findings support this assumption. Thus, we identify short-sale constrained stocks as those having both high levels of relative short interest and low levels of institutional ownership. 9 We will show 7 Boehme, Danielsen and Sorescu (2006) provide empirical evidence in support of Proposition 3. 8 We noticed short interest data are occasionally missing for firms with valid CRSP data. We do not include such observations in our sample because we are unable to determine if they represent a zero level of short interest, or if short interest data are missing. 9 All other stocks are therefore classified as unconstrained. This method of classifying Miller stocks is consistent with that we use to classify Merton stocks in the previous sub-section. However, the subset of stocks we employ for testing Merton s model is narrower than the non-miller subset resulting from Asquith, Pathak and Ritter s approach. Recall that in addition to the absence of short-sale constraints, Merton s investor recognition hypothesis requires low visibility. Thus, to identify Merton stocks, we must intersect the non- Miller set with the set of stocks having low levels of institutional ownership. The resulting set (low levels of 9

12 that for these firms, the risk-return relationship is the opposite of that predicted in the Merton model. B.3. Operationalizing the Hypotheses We can now operationalize Hypotheses 1 and 2 as follows: 1(a). Cross-sectional returns are positively related to idiosyncratic risk for portfolios with low RSI and low IO. These are stocks with low visibility and non-binding constraints where Merton s theory is applicable. 2(a). Cross-sectional returns are negatively correlated with dispersion of beliefs for portfolios with high RSI and low IO. These are stocks with binding short-sale constraints where Miller s theory is applicable. Figure 1 below summarizes our predictions regarding the Merton and Miller hypotheses for portfolios sorted according to four polar combinations of RSI and IO. Figure 1 Relative Short Interest (RSI) Low High Merton s Hypothesis Applicable (No short sale constraints) Not Applicable (Binding short sale constraints) Low Institutional Ownership (IO) Miller s Hypothesis Not Applicable (No short sale constraints) Applicable (Binding short sale constraints) Merton s Hypothesis High Not Applicable (Investor recognition hypothesis not applicable due to high visibility) Miller s Hypothesis Not Applicable (No short sale constraints due to abundant supply of lendable stock) short interest and low levels of institutional ownership) is used to test Merton s model. This set is completely disjoint from the Miller set used by Asquith, Pathak and Ritter (2005). 10

13 C. Measuring Idiosyncratic Risk (SIGMA) We measure idiosyncratic risk (SIGMA) as the standard deviation of the error terms from the Fama-French-Carhart four-factor model, estimated over the 100 days preceding the first day of each month. The four factors are made available by Kenneth French on his website at Dartmouth College. This measure is analogous to 2 σ k, the firm-specific component of the firm s return variance, in Merton (1987). Firms are excluded from our analysis in any month if more than 10 days of returns data are identified as missing on CRSP in the prior 100 days. We also screen out all securities other than domestic common stocks. D. Measuring Dispersion of Opinion (DISPERSION) We measure dispersion of opinion using the coefficient of variation for analysts annual forecasts estimated from I/B/E/S data, now part of Thompson Financial. The coefficient of variation is estimated by dividing the I/B/E/S reported standard deviation of analyst earnings/share forecasts for the current fiscal year end (I/B/E/S Fiscal Year period 1 ) by the absolute value of the mean earnings/share forecast, as listed in the I/B/E/S Summary History file. Diether et al. (2002) also use this proxy. Diether et al. (2002) note that the standard I/B/E/S forecast file contains an error related to rounding of historical split-adjusted values. Therefore, we obtained, upon special request from I/B/E/S, a separate file containing analyst forecasts that are unadjusted for historical stock splits, and therefore do not suffer from this potentially serious rounding error. 10 We compute our coefficients of variation using this unadjusted file, which is the same as the one employed by Dither et al. (2002). We note, however, that I/B/E/S analyst dispersion data suffers from a limitation in that at least two analysts must follow the stock for a dispersion value to be 10 These unadjusted forecast data are now available on WRDS. 11

14 computed. But only relatively large firms have two or more analysts providing forecasts: Danielsen and Sorescu (2001) report that among the firms that have sufficient liquidity for traded options to be introduced, nearly one third had fewer than two analysts per the I/B/E/S database. Thus, I/B/E/S dispersion of opinion cannot be calculated for many small firms. E. Measuring Subsequent Stock Returns Stock returns are obtained from CRSP, and we adopt the standard four-factor, calendartime portfolio approach for measuring abnormal returns. For each month in the calendar during the 1988 to 2002 period, we use our explanatory variables to classify firms in portfolios. Firms with similar values on each dimension are grouped into equally-weighted portfolios. For each portfolio, we first calculate the monthly raw returns during the month subsequent to the portfolio formation. In other words, the portfolios are constructed on ex-ante information, and the returns are computed ex-post. In a sense, the tests are conducted out of sample, as in Spiegel and Wang (2005). 11 For each portfolio, we estimate the following four-factor regression model: R p,t - R f,t = α p + β p (R m,t - R f,t ) + s p SMB t + h p HML t + u p UMD t +e p,t (1) where R p,t represents the raw returns of each portfolio, and R f,t is the return of the onemonth Treasury Bill. The four independent variables are the excess return on the market portfolio (R m,t -R f,t ); the difference between the returns of value-weighted portfolios of small and big firm stocks (SMB t ); the difference in returns of value-weighted portfolios of high and low book-to-market stocks (HML t ); and, the difference in returns of value-weighted portfolios of firms with high and low prior momentum (UMD t, or up minus down ). Fama and French 11 For robustness, we also examine monthly returns for portfolios where firms are held for 12-month, rather than one-month, periods. 12

15 (1993) propose the first three factors, while Carhart (1997) proposes the momentum factor. We interpret the intercept, α p, from equation (1) as the mean monthly abnormal return of the calendar-time portfolio. Because the number of firms in a portfolio can change monthly, we use WLS procedures to weight the calendar-time portfolios based on the number of firms in a portfolio each month. 12 Our most important tests compare returns between the calendar-time portfolios of stocks having different characteristics along one of the various dimensions of interest (idiosyncratic volatility, short interest, dispersion of opinion, or institutional ownership). As in Mitchell and Stafford (2000) and Boehme and Sorescu (2002), we perform these tests by constructing zeroinvestment hedge portfolios with long positions in stocks having a certain mix of characteristics (such as low IO, low RSI, and high SIGMA) and short positions in stocks with a different mix (such as low IO, low RSI, and low SIGMA). This example is typical in that only one of the variables, in this case SIGMA, is altered between the long and short legs of the hedge portfolio. Thus, we can observe the marginal effect of SIGMA holding the other variables constant. The hedge portfolio returns are regressed on the four factors: R high-sigma,t - R low-sigma,t = α p + β p (R m,t -R f,t ) + s p SMB t + h p HML t + u p UMD t + e p,t (2) The "hedge" intercept obtained in this manner (α p ) represents a measure of the relative abnormal performance of the long-leg portfolio (low IO, low RSI, and high SIGMA) vis-à-vis the short-leg portfolio (low IO, low RSI, and low SIGMA). F. Descriptive Statistics Table 1 provides descriptive statistics for the proxy variables used. We provide a snapshot of firms in the dataset at five-year intervals beginning in 1988, the first year for which 12 Results throughout the paper also hold using ordinary least squares. 13

16 short interests data is available in digital form. Proxies are estimated for all U.S.-domiciled common stocks listed on the NYSE and Nasdaq. Both IO and RSI have increased substantially between 1988 and No clear trend in SIGMA or DISPERSION is noticeable. However, the anomalous DISPERSION mean for December 2002 is driven by one extreme outlier, which has a DISPERSION measure of Notice that the 99 th percentile value is reasonable. Because we form portfolios based on monthly rank-orderings of each of these variables, neither a small number of extreme values, nor non-stationarity in the time series of a variable introduces any bias to the analysis. II. Baseline Cross-Sectional Tests Table 2 presents baseline cross-sectional tests of the effects of idiosyncratic risk, without regard to short-sale constraints for the universe of all CRSP listed common stocks of U.S. domiciled NYSE and NASDAQ firms. Panel A shows the calendar time portfolio abnormal returns as a function of idiosyncratic risk (SIGMA) for the one- and twelve-month holding period horizons. The SIGMA deciles are reassigned each month by sorting the continuous SIGMA estimates obtained from the Fama- French-Carhart four-factor model during the previous 100 trading days. The abnormal returns are computed by forming equally-weighted calendar-time portfolios, of either one-month or twelve-month horizons, and regressing the excess portfolio returns (R p,t -R f,t ) on the four Fama- French-Carhart monthly factors. Calendar time portfolios are re-balanced each month, and the portfolios include only firms that entered the portfolio during either the prior month or the previous twelve months. We weight monthly excess returns by the square root of the number of 14

17 firms in each month. 13 The abnormal return for each sub-sample is the intercept (α p ) from the regression. On the right side of the panel, we also show the abnormal return of the hedge (or zeroinvestment) portfolio that takes long positions of stocks in the highest SIGMA decile, and short positions in stocks in the lowest SIGMA decile. The p-value of the hedge portfolio reports the statistical difference of the measured mean return from zero. There is no evidence in Panel A of any relation between abnormal returns and SIGMA: the abnormal return point estimates across the ten SIGMA deciles do not appear to follow any particular trend, and the hedge portfolio returns are not statistically significant. In short, as others have noted, there is no evidence here to support Merton s hypothesis. Because we focus on the complementary nature of Miller (1977) and Merton (1987), we emphasize the high level of correlation between dispersion of opinion (DISPERSION), as proxied by the standard deviation of analyst forecast, and idiosyncratic risk (SIGMA). Notice that in the last two lines of Panel A we report the mean values of SIGMA and DISPERSION for each of the deciles. Since firms are sorted into deciles each month based on SIGMA, by construction, the mean SIGMA value is monotonically increasing across deciles. However, the mean DISPERSION value is also rising in a monotonic manner, reflecting the close correlation between SIGMA and DISPERSION. As further evidence, the mean monthly correlation between individual-security SIGMA and DISPERSION values is Panel B shows the abnormal returns as a function of analyst earnings forecast DISPERSION, measured as the standard deviation of analyst earnings per share forecasts divided by the absolute value of the mean forecast, obtained from the I/B/E/S Summary History 13 Results in this table and elsewhere in the paper are not sensitive to the use of this weighted least square approach; using OLS regressions results in highly similar point estimates and significance values. 15

18 file. DISPERSION deciles are assigned for each month. Observations having mean earnings forecast of zero are omitted for assigning the rank ordering of firms, but are assigned to the highest DISPERSION decile. The relation between abnormal returns and DISPERSION is visibly negative and the hedge portfolio returns are significant at the one-month horizon. This result is consistent with Diether et al. (2002) and suggests that Miller s hypothesized overpricing is detected in returns for the one-month horizon even without conditioning on the level of short-sale constraints. A much weaker negative trend is visible for the twelve-month horizon and the hedge portfolio abnormal returns remain negative, but are no longer statistically significant. The final two lines of Panel B report the mean values of SIGMA and DISPERSION for each decile of stocks constructed based on the DISPERSION metric. By construction, the values for mean DISPERSION increase monotonically. Mean SIGMA values are monotonically increasing, except for the first decile, which is larger than the following three deciles. In summary, DISPERSION appears to be highly correlated with SIGMA in general, but not at very low levels of DISPERSION. In untabulated tests, we have repeated these tests using longer sample periods to include volatility measures dating back to January 1963, and I/B/E/S dispersion measures dating back to We find no qualitative difference in the results over these longer periods. III. The Merton Space: No Short-Sale Constraints and Low Visibility Having established a baseline for the cross-sectional effects of idiosyncratic risk, we now turn our attention to the subset of firms where Merton s theorized positive correlation between idiosyncratic risk and return is most likely to be found; low-visibility firms without short-sale constraints. Recall that these stocks reside in the space where both RSI and IO are low. 16

19 Table 3 depicts the hypothesized Merton effect for the one-month investment horizon. While our Merton tests require the use of three variables (IO, RSI, and SIGMA), we are constrained to reporting the results in two-dimensional tables. For expositional efficiency, we present Panel A of Table 3 using two explicit dimensions: IO and RSI, which are the first two variables used to parse the data. As explained below, the third dimension, SIGMA, is captured implicitly in each cell as the difference between the abnormal returns of stocks with high and low values of SIGMA. For each calendar month, we first sort all firms available in the CRSP database into quartiles based on the Institutional Ownership percentage (IO). Within each IO quartile, we again sort firms into quartiles based on Relative Short Interest (RSI). This produces 16 IO/RSI groups each month, with an approximately equal number of firms in each group. Finally, within each of these 16 equal-sized IO/RSI groups, we sort firms into ten subgroups based on SIGMA. This final sorting produces 160 (4x4x10) subgroups for each month, with each grouping containing approximately the same number of firms each month. Within each cell in Panel A, we report the long-term abnormal return of a hedge (or zeroinvestment) calendar-time portfolio consisting of long positions in stocks with the highest SIGMA decile and short positions in stocks with the lowest SIGMA decile. The abnormal return for each sub-sample is the intercept (α p ) from the following regression: R sigma 10, t - R sigma 1, t = α p + β p (R m,t -R f,t ) + s p SMB t + h p HML t + u p UMD t + e p,t. We expect to find support for Merton s hypothesis in the region of the space where visibility is low and short-sales constraints are also low; this would correspond to the uppermost left cell of the Panel. 14 The hedge portfolio in this cell earns an abnormal return of 2.268% per 14 Only 8.1 % of the firms in the low-io and low RSI cells are followed by two or more analysts. This is testimony to the fact that we are examining low-visibility firms, as a test of the IRH would require. 17

20 month (more than 30% per year annualized), which is significant at close to the one percent level (p-value=0.0166). In addition, this hedge abnormal return is clearly the largest among the 16 hedge returns reported. The two bordering cells [(IO quartile = 2 with RSI quartile = 1) and (IO quartile = 1 with RSI quartile = 2)] report the second and third highest hedge portfolio alphas among the 16 IO/RSI groups, but only one of the alphas in these two adjacent cells attains statistical significance (p-value=0.0508). Among the highest-rsi quartile firms, the alphas are consistently negative and generally insignificant. We should not observe Merton s predicted positive alphas here, and we do not. In panel B of Table 3, we examine the cross-section of SIGMA-sorted portfolios for the Merton corner of Panel A: the upper leftmost cell and the two bordering cells. The first column of Panel B presents the alpha values for all 10 SIGMA-sorted subgroups within the upper leftmost cell of Panel A (lowest IO and lowest RSI quartiles). Here, we observe small, but slightly positive alphas for the lowest-sigma portfolios, which may be due to low intertemporal variation in their returns. In the first column, consistent with Merton s prediction, SIGMA portfolio 10 has the highest abnormal return. Portfolios 9 and 8 produce the second and third highest alphas, respectively. For the reader s convenience, the bottom row of Panel B reproduces the hedgeportfolio results previously reported in Panel A of the table. We also report in Panel B the average number of firms in each calendar-time portfolio. Because the holding period for each firm is only one month, only approximately 28 firms are included in each portfolio each month. The relation between abnormal returns and SIGMA for portfolios adjacent to Panel A s, uppermost-left cell are presented in the second and third columns of Panel B. As in the first column, we observe the highest abnormal return in each column in the portfolio belonging to the 10 th SIGMA decile, which is once again consistent with Merton s hypothesis. 18

21 Table 4 repeats the analysis in Table 3 for the one-year investment horizon instead of one month. Each of the cells in Table 4 mimics the Table 3 contents except that we retain firms in the portfolio for 12 months. However, as in Table 3, we report returns in Table 4 on a monthly basis. Notice that a single firm can be a component of multiple portfolios when a one-year horizon is used because the firm will remain in a portfolio for 12 months even if the firm s rank-ordering on one or more sorting variables changes in a subsequent month. The results of Panel A in Table 4 are similar to those shown in Panel A of Table 3. The uppermost-left cell has the largest abnormal hedge returns at a high level of statistical significance, but adjacent cells are also positive. In fact, we find statistical significance not only in the Merton Corner (i.e., lowest IO and lowest RSI firms), but in three other cells in a wider area where the IRH seems plausible. Panel B examines the cross-section of SIGMA-sorted portfolios for the Merton corner of Panel A: the upper leftmost cell and the two bordering cells. The results reported here are qualitatively very similar to those reported for one-month calendar-time portfolios in Panel B of Table 3. This panel also provides a clue as to why we may more often observe greater statistical significance in the twelve-month hedge portfolios. Notice in Panel B that the average number of firms in each portfolio is much greater for twelve-month than for one-month holding-period portfolios. The larger portfolios produce less noisy monthly returns so that the standard errors in the hedge-portfolio regressions are smaller at the twelve-month horizon. A visual depiction of the abnormal returns for first column in Panel B is shown below in Figure 2. 19

22 % Abnormal Return (IO quartile 1, RSI quartile 1) Monthly Abnormal Return. 2.00% 1.80% 1.60% 1.40% 1.20% 1.00% 0.80% 0.60% 0.40% 0.20% 0.00% Idiosyncratic Risk (SIGMA) decile Figure 2: Twelve-month Abnormal Returns as a function of SIGMA for stocks belonging to the intersection of lowest IO and lowest RSI quartiles Overall, the evidence in Tables 3 and 4 provides relatively strong support for Merton s investor recognition hypothesis. Abnormal returns that increase with SIGMA are concentrated in the locality of low relative short interest and low institutional ownership. As either RSI or IO rises, returns to SIGMA diminish. 15 IV. The Miller Space: Short-Sale Constraints Present As we have previously discussed, extant research casts Merton (1987) as a competing theory to Miller (1977). Leading examples include Diether et al. (2002) and Gebhardt et al. (2001): these papers find evidence in support of Miller (1977) and reject Merton (1987). 15 For robustness, we repeat the analysis in Tables 3 and 4 for two non-overlapping sub-periods ( and ), and the results do not change. 20

23 Viewing Miller (1977) as complementary to Merton (1987), rather than as a competitor, suggests a set of empirical tests that distinguish between firms which are likely to be short-sale constrained from those which are not. Our tests in the previous section do this and we find evidence of Merton s investor recognition hypothesis in the expected region. 16 Having found evidence in support of the IRH, we should expect to find evidence of Miller s short-sale-constraint-induced overpricing in the lowermost-left corner of Panel A in both Tables 3 and 4 This intersection of low IO and high RSI is exactly the region where Asquith Pathak and Ritter (2005) document Miller-style overvaluation. In the Miller space, dispersion of beliefs should result in premium pricing (negative hedge portfolio alphas), in contrast to the discount pricing in the Merton space. The most intuitive proxy for dispersion of opinion is the dispersion of analyst forecasts. The literature commonly uses this proxy (e.g., Boehme et al. (2005), Diether et al. (2002), and Danielsen and Sorescu (2001)). However, as we have seen in Table 1, DISPERSION and SIGMA are correlated. In fact, Jones, Kaul and Lipson (1994), among others, note that volatility is related to the dispersion of beliefs. Thus, we may find a Miller-like correlation between SIGMA and returns in the low-io, high-rsi space. When we examine the lower-left corner of Table 3(A), we find that the hedge portfolio of high-sigma minus low-sigma stocks earns negative returns, but they are not statistically significant. To the extent that SIGMA proxies for dispersion of investor beliefs, one would 16 Johnson (2004) provides an option-theoretic interpretation of the observed negative relationship between returns and idiosyncratic risk. For a levered firm, equity returns will decrease with idiosyncratic asset risk due to convexity. Because analyst dispersion is correlated with idiosyncratic asset risk, equity returns will decrease with higher analyst forecast dispersion also. As Johnson notes (page 1958), the option-theoretic model and the Miller s (1977) short-sale-constraint theory produce parallel predictions. However, because these theories are not mutually exclusive, they are not in conflict. Both effects can exist simultaneously. We focus on resolving the apparent contradiction between Merton (1987) and Miller (1977) which we argue to be spurious and use a short-sale constraint filter, rather than the leverage filter used by Johnson. 21

24 expect to find significantly negative abnormal returns here. In Table 4(A), the twelve-month hedge portfolio returns in the lower-left corner are actually positive, which conflicts with what we expect to observe. To better understand the nature of these hedge results, we examine the cross-section of the ten SIGMA-sorted sub-portfolios, which comprise the low-io, high-rsi space. Table 5 presents the cross-section of SIGMA-sorted portfolio returns in the Miller space for both onemonth and twelve-month calendar-time portfolios. The format for Table 5 is identical to that shown in the B Panels of Tables 3 and 4. A careful examination of the one-month horizon individual portfolio returns reveals that decile one has the highest returns of all ten portfolios, and decile eight has the lowest returns. In fact, between deciles one and eight, the returns fall in a near-montonic manner; only decile seven modestly violates the series of monotonic declines. While decile nine has negative returns that are similar to those in decile eight, the highest-sigma decile (ten) has noticeably less-negative returns. The results can be well summarized as follow: the SIGMA-return relation is negative and nearly monotonic, except for decile ten. Because deciles ten and one are used for the long and short legs of the hedge portfolio, respectively, the hedge portfolio fails to capture the more general negative correlation between SIGMA and abnormal returns. We also report the twelve-month horizon individual portfolio returns in Table 5. The hedge portfolio for these twelve-month results is identical to that shown in the lowermost-left cell of Table 4, Panel A. We find that the negative SIGMA-return relation is even less pronounced for the twelve-month portfolios than for the one-month portfolios. Still, the alphas for portfolios five through nine are all less than for portfolios one through four. However, the tenth decile has the highest alphas. In summary, a weakly negative SIGMA-alpha relation 22

25 exists in the bottom nine deciles, but the tenth decile, which is the long leg of the reported hedge portfolio, clearly violates the more general relation. Overall, we find weak support for Miller s hypothesis when we use SIGMA as the proxy for dispersion of opinion. However, as we have noted, the dispersion of analyst forecasts (DISPERSION) is arguably a better proxy for the investors dispersion of beliefs. We now turn to testing Miller with the DISPERSION measure rather than SIGMA. These tests will mirror the SIGMA-based tests of Miller. For each calendar month, we first sort all firms in the sample into quartiles based on the Institutional Ownership percentage (IO). Within each IO quartile, we again sort the firms into quartiles based on Relative Short Interest (RSI). This produces 16 IO/RSI groups for each month with an equal number of firms in each group. These are the same 16 IO/RSI portfolios used previously. Finally, within each of these 16 equal-sized IO/RSI groups, we sort each of the 16 groups into ten subgroups based on DISPERSION. This final sorting again produces 160 (4x4x10) subgroups for each month, with each grouping containing approximately the same number firm-months. In Panel A of Table 6, we report the abnormal return of 16 hedge portfolios mirroring the method of Panel A, Table 3. Each hedge portfolio consists of a long position in stocks in the highest DISPERSION decile and a short position in stocks in the lowest DISPERSION decile. Based on the analysis of APR (2005), we expect that short-sale constraints are more likely to bind where RSI is high and IO is low, and this is precisely where we find the most negative alpha (-1.519% per month compounding to -16.8% annually). Thus, using the more appropriate dispersion of beliefs proxy, rather than the idiosyncratic risk proxy, we find Miller s hypothesized overpricing at high levels of statistical significance. Our contribution here is that we are able to reconcile these results with Merton s hypothesis and that we are able to find Miller s effect with a different parsing of the RSI data. For example, one of the results reported 23

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