idiosyncratic risk and the cross-section of stock returns: merton 1987 meets miller 1977

7/29/2019 Idiosyncratic Risk and the Cross-Section of Stock Returns: Merton 1987 Meets Miller 1977

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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: March 15, 2005

Abstract

Merton (1987) predicts that idiosyncratic risk should be priced when investors hold sub-

optimally diversified portfolios, but empirical research has not been supportive of the theory. Anoverlooked assumption in Merton (1987) is that the predictions are predicated on frictionless

markets, and in particular an absence of short-sale constraints. We examine the cross-sectional

effects of idiosyncratic risk (and dispersion of beliefs) while controlling for short-saleconstraints. We find that when short-sale constraints are absent, both idiosyncratic risk and

dispersion of analyst forecasts are positively correlated with future abnormal returns; a result

consistent with Merton (1987). However, when short-sale constraints are present the correlation

becomes negative: increased analyst dispersion and idiosyncratic volatility produce negativeabnormal returns, consistent with Miller (1977). This can explain the inconsistent empirical

findings in the previous literature, which casts Merton (1987) and Miller (1977) as competinghypotheses.

_______________________________

* 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 ofBusiness, University of Houston. Sorescu is from Mays Business School at Texas A&M University.

This paper has benefited from the comments of Fred Arditti, Mark Flannery, Anna Scherbina, 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 77204-6021, phone

(713) 743-4770, e-mail: [email protected]. Data on analysts forecasts was provided by I/B/E/S Inc.,

under a program to encourage academic research.


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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, but empirical research has not been supportive of the theory.

An overlooked assumption in Merton (1987) is that the predictions are predicated on

frictionless markets, and in particular an absence of short-sale constraints. We examine thecross-sectional effects of idiosyncratic risk (and dispersion of beliefs) while controlling for

short-sale constraints. We find that when short-sale constraints are absent, both idiosyncratic

risk and dispersion of analyst forecasts are positively correlated with future abnormal returns;a result consistent with Merton (1987). However, when short-sale constraints are present the

correlation becomes negative: increased analyst dispersion and idiosyncratic volatilityproduce negative abnormal returns, consistent with Miller (1977). This can explain the

inconsistent empirical findings in the previous literature, which casts Merton (1987) andMiller (1977) as competing hypotheses.


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1. Introduction

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. Investors in Mertons model suffer from extreme

information costs and only hold securities with which they are familiar. Consequently, they

hold under-diversified portfolios and demand compensation for securities idiosyncratic risk.1

Therefore, in equilibrium, cross-sectional stock returns are positively related to their

idiosyncratic risk.

Direct tests of Mertons (1987) model are rare. Mertons 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 the Mertons theory.

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 reflect the type of

idiosyncratic risk to which Merton refers. Indeed, idiosyncratic volatility and dispersion of

1 Levy (1978) and Mayers (1976) produce similar predictions in CAPM extensions where investors hold under-diversifiedportfolios. Barberis and Huang (2001) produce a prospect-theory model where idiosyncratic risk produces positiveexpected returns.


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analysts forecasts have been shown to be positively correlated.2

Their results do not support

Mertons 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 cross-sectional tests that use analysts forecast dispersion as a risk proxy, cross-

sectional results are incorrectly signed at high levels of significance compared to Mertons

(1987) predictions. Based on the empirical tests to date, the literature concludes that Mertons

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

stocks price. Thus, dispersion of investor opinion is priced at a premium when short sale

constraints are present. The implication of Millers (1977) theory is that a negative correlation

exists between risk-adjusted returns and dispersion of beliefs, ifshort sale constraints are binding.

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).3

2 Empirically, Peterson and Peterson (1982) observe a positive relationship between return volatility and the dispersion ofI/B/E/S forecasts, and we confirm their finding later in this study.

3Diamond and Verrecchia (1987) and Jarrow (1980) provide alternative theories of short-sale constraint effects. Diamondand Verrecchia model security prices in a rational expectations framework that precludes the possibility of systematicmispricing. However, changes in observed, costly short-sales are informative and result is learning that changes asset

prices. Jarrow (1980) develops a general equilibrium model which recognizes cross-effects in addition to own-effects.


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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

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 (1987). While Miller (1977) assumes that stocks are short-sale

constrained, Merton (1987) models a standard frictionless market, without borrowing and short-

selling restrictions (page 487). Absence of frictions is, in fact, crucial for Mertons (1987)

prediction of a positive relationship between equilibrium returns and idiosyncratic risk.

In this paper, we re-visit Mertons (1987) model and explicitly allow for short-sale

constraints. We find that the models predictions on the relationship of equilibrium returns to

idiosyncratic risk are ambiguous if the short-sale constraints are binding. Because investors

hold under-diversified portfolios, absence of shorting constraints is crucial in order for the

market to impound private differential information about the relative value of securities into an

unbiased, efficient estimate.

Viewed in this light, Miller (1977) and Merton (1987) are complementary. 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-sale-

constrained 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).

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

Both underpricing and overpricing for individual stocks can result from market-wide short sale prohibitions in thisframework.


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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 Millers (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. In this

paper, 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.

We analyze the cross-sectional relationship between idiosyncratic risk and equity returns

by carefully controlling for the likelihood of binding shorting constraints. For robustness, we

use several proxies, based on the recent literature, to identify the degree of short-sale

constraints.

We find that cross-sectional returns are positively correlated with idiosyncratic volatility

for stocks that are unlikely to be subject to shorting constraint, as predicted by Merton (1987).

Moreover, returns are also positively correlated with the dispersion of analyst forecasts, a

finding that is also consistent with Merton (1987), to the extent that analyst dispersion reflects

the type of idiosyncratic risk to which Merton refers (see, e.g., Diether et al., 2002).4

To clarify the role of binding short-sale constraints in the relation of returns and

idiosyncratic risk, we also examine firms that are likely subject to short-sale constraints. Not

surprisingly, perhaps, we find strong support for the Miller (1977) hypothesis here. Firms with

high dispersion of analysts forecasts earn negative abnormal returns, and firms with high

idiosyncratic volatility under-perform on a risk-adjusted basis. This result is consistent with

relatively weak support for the Miller hypothesis during the 1990s found by Diether, Malloy

and Scherbina (2002), where Millers hypothesis is tested as a competitor to Mertons

4 We note that the finding is also consistent with Varian (1985) who predicts discount pricing when dispersion of investorbeliefs exists in the absence of short-sale constraints.


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hypothesis. More recently, both Boehme, Danielsen and Sorescu (2005) and Asquith, Pathak

and Ritter (2005) find stronger support for Millers theory during the 1990s by explicitly

recognizing the important interaction of short-sale-constraints with dispersion of beliefs. Our

analysis differs from these two papers in that it reflects the importance of screening out short-

sale-constrained firms when testing Mertons hypothesis that firms with high idiosyncratic risk

will be priced at a discount.

The rest of this paper is organized as follows. Section 2 extends the Merton (1987) to

allow for short-sale constraints, and discusses the proxy variables we use for dispersion of

beliefs, idiosyncratic risk, and short-sale constraints. Section 3 discusses the testing

methodology and the data. Section 4 presents base-line cross-sectional effects without regard to

short-sale constraints. Section 5 presents empirical tests of Merton (1987) for firms that are free

of short-sale constraints. Section 6 repeats the analysis for short-sale constrained firms in

accordance with the Miller (1977) hypothesis. Section 7 summarizes and concludes with a

graph providing a more holistic analysis of the cross-sectional effects of idiosyncratic volatility

and dispersion of beliefs.

2. Capital Market Equilibrium with Incomplete Information

and Shorting Constraints

We extend the Merton (1987) model to allow for constraints on short-selling. For

consistency and ease of reference, we maintain Mertons basic model structure and notation.

There is one investment period and n firms in the economy. The equilibrium return per

dollar from investing in firm k, ,kR is given by the one-factor model:

,)( kkkkk YbRER ++= (1)


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where Y is a random common factor whose unconditional expectation is zero and variance is

unity, and .0),,...,,,...,()( 111 == + YEE nkkkk There is a riskless security (the (n+2)

security) with sure return per dollarR and another security (the (n+1) security) that combines

the riskless security with cash settlement on the observed factor Y. Taking the standard

deviation of equilibrium return on this security to be unity, we can write this return as,

.)( 11 YRER nn += ++ (2)

Both the riskless and the forward contract security are inside securities; i.e., their aggregate

demand sums to zero.

There are no taxes and transactions costs, and that investors can borrow and lend without

restriction at the same rate. However, and by contrast to Merton (1987), we disallow short-

selling of the firms shares. There are N price-taking and risk-averse investors who select

optimal portfolios according to the Markowitz-Tobin mean-variance criterion. Hence, the

preferences of investorj = 1,N, can be represented as,

),(2

)( jjjjjj

j WRVarW

WREU

= (3)

where jW is the investors investible wealth (equal to his or her initial endowment of shares in

the firms evaluated at equilibrium prices); jR denotes the return per dollar on the portfolio;

and, .0>j

While the riskless return and the mean and variance of the return of the forward contract

security are common knowledge, a typical investor only knows the parameters of the factor

return model, given in (2), for a subset of securities. Specifically, an investor is said to know

about security k= 1,,n, only if he or she knows the triple ).,),(( 2kkk bRE For each investor,j

= 1,N, there is associated a collection of integers ,jJ such that k belongs to ,jJ only if the


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investor knows k (in the sense specified above). As in Merton (1987), the key behavioral

assumption is that investorj includes security konly if .jJk

To set up the typical investors portfolio optimization problem, let jk (k= 1,2,n+2)

denote the faction of investible wealth allocated to security kby investorj. Then, using (1) and

(2), the portfolio returns are,

,)( jjjjj YbRER ++= (4)

where, ,11j

nk

n j

k

j bb ++ ,)( 221 kn j

k

j and ./1j

kk

n j

k

j Since the

portfolio weights must sum to one, with judicious substitutions, we can write the variance and

expected return on the portfolio as (see, Merton (1987)),

,)()()( 22 jjj bRVar += (5)

,))(()(1

1 k

nj

kn

jj RREbRRE ++= + (6)

where, ).)(()( 1 RREbRRE nkkk +

Then the optimal portfolio is the solution to the constrained maximization problem,

.)(2

)(1 1},{

+ n n jkjkjkjkjjjb RVarREMax jj

(7)

Here, jk is the Kuhn-tucker multiplier reflecting the constraint that the investorj cannot

include security k in his or her portfolio if k does not belong to ,jJ whilej

k is the Kuhn-

Tucker multiplier reflecting the constraint that the investor cannot short-sell security k. We note

that 0=jk if .jJk But if ,jJk then 0=j

k and .0=j

k

The first order conditions for (7) are, fork= 1,2,n.

j

jn bRRE = + )(0 1 (8.a)


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.0 2 jkj

kk

j

kjk += (8.b)

From (8.a) and (8.b), the optimal ,/])([ 1 jnj RREb = + and the optimal portfolio

weights for assets in the known set, i.e., ,jJk are:

,2

kj

j

kkj

k

+= (9)

with 0=jk for ,jJk ,11 kn j

k

jj

n bb =+ and ).1(1 12 = + kn j

k

jj

n bb If we

assume identical preferences and wealth across agents, then investors will choose the same

exposure to the common factor ,...,1, Nbbj = and hence, in equilibrium,

.)( 1 bRRE n +=+ (10)

Next, we can aggregate security demands across agents to yield the aggregate demand for

firms shares; i.e., fork= 1,n,

,)(

2

1

k

N j

kkk

k

WND

+= (11a)

where kN denotes the number of investors who know security k. Furthermore,

k

n

kn bDNWbD =+ 11 , .1

12 +

+ =n

kn DNWbD (11b)

We note that the aggregate share demand given in (11) differs from Merton (1987) due to

the presence of shorting constraints. As seen in (11a), if the shorting constraints for security k=

1,n, are binding, then the aggregate demand is strictly larger (in algebraic terms) compared to

the situation where there are no shorting constraints.

To derive the equilibrium returns on firms share, letMdenote the aggregate wealth, and

let kx denote the fraction of aggregate wealth invested in security k = 1,n. Finally, put


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,N

Nq kk = i.e., the fraction of investors who know about security k. Then, market clearing

requires that,

.

)(

2 1

k

N j

kkk

k

q

x

+

= (12)

Using (10), (12), and the definition of k in (6), we can write the equilibrium expected return

for (6b) as, kkk bbRRE ++= )( . Using the equilibrium condition (12), we can substitute for

k to write,

=++=

N

j

j

kk

kk

kk q

xbbRRE

1

2

)(

(13)

The last term in (13) can be interpreted as the Miller (1977) effect: if shorting constraints are

binding, then the expected return on a security falls since the current price reflects the beliefs of

the most optimistic investors.5

Moreover, (13) shows that the relationship between equilibrium

returns and idiosyncratic risk (i.e., 2k ) is ambiguous. The reason is that whether the shorting

constraints bind or not (i.e., whether 0>j

k or 0=j

k ) itself depends on2

k . Indeed, one can

argue that shorting constraints are more likely to bind for new securities, since the supply of

such securities is usually curtailed (see, e.g., Miller (1977) and Houge, Loughran, Suchanek,

and Yan, (2001)). But we would also expect the idiosyncratic risk on these securities to be

higher. Hence, the last two terms in (13) would both increase with 2k , and hence the net effect

of an increase in 2

k

on equilibrium returns would be ambiguous.

5 Of course, and exactly as in Merton (1987), we have not formally incorporated heterogeneity of beliefs for notational ease.But the foregoing analysis is readily extended to the case where expectations are investor specific.


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3. Hypotheses, Explanatory Variables, and Test Design

The foregoing analysis, along with a reading of Miller (1977), suggests the following

complementary hypotheses that we test in this study:

1(a). The Merton Hypothesis. Cross-sectional differences in idiosyncratic volatility

are positively correlated with subsequent returns when short-sale constraints are

likely to be non-binding.

1(b). A cross-sectional pattern similar to 1(a) will exist for I/B/E/S forecast

dispersion with high dispersion firms earning higher risk-adjusted returns because

idiosyncratic risk and dispersion of analyst forecasts is highly correlated.

2(a). The Miller Hypothesis. Cross-sectional differences in I/B/E/S forecast

dispersion are negatively correlated with subsequent returns when short-sale

constraints are binding. Moreover, the value premium accompanying dispersion

of beliefs will increase as the level of short-sale constraints increase.

2(b). A cross-sectional pattern similar to (2a) will exist for idiosyncratic volatility

because idiosyncratic risk and dispersion of analyst forecasts is highly correlated.

We now describe the independent (or explanatory) variables used in the analysis, as well as

our methodology for measuring stock returns.

A. Explanatory Variables

A.1 Short Sale Constraint Proxies

Previous research suggests several proxies for the degree to which stocks are short sale

constrained. These include the presence of exchange-traded options, and the level of short

interest relative to shares outstanding.


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A.1.1 Option Status

Several studies have documented that firms with traded options are, in general, less short-sale-

constrained than firms without options (see, e.g., Figlewski and Webb (1993); Danielsen and

Sorescu (2001); Boehme, Danielsen and Sorescu (2005), and Evans, Geczy, Musto and Reed

(2003)). The intuition behind options relaxing short-sale constraints is that options allow

investors to take short positions in securities without short selling directly. Boehme, Danielsen

and Sorescu (2005) observe that the securities lending market for stocks is unusually opaque

with active participants paying much lower fees than less active ones. Investors who might

short-sell at a relatively high cost instead can use options to synthetically short a security.

Option market makers, as the counter-party to these synthetic short sales, are left with synthetic

long positions, which they hedge by borrowing the stock and shorting it. As frequent

participants in the short-sale market, option market makers can execute and maintain short

positions at much lower costs than infrequent short sellers. This short-sale cost advantage is

reflected in competitively priced options.6

Consistent with the idea that options facilitate short selling, Figlewski and Webb (1993)

report that short interest levels increase after option introductions. Also, Boehme et al. (2005)

report that stock lending fees for firms with traded options are lower than for non-optioned

stocks. Moreover, the difference in fees is greater between optioned and non-optioned firms

when the level of short interest is unusually high.

A.1.2. Relative Short Interest

Short interest is perhaps the longest-used and most common short-sale-constraint proxy in past

studies. Figlewski (1980) first proposed this proxy, hypothesizing that as the level of observed

6 Evans, Geczy, Musto, and Reed. (2004) and Danielsen and Sorescu (2001) provide more in-depth discussion of the optionsmarket making process and its relevance to short-sale constraints.


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short sales increases, the unobserved demand to short the security probably rises as well. Thus,

firms that are highly shorted are, at the margin, the most difficult to short.

Diamond and Verrecchia (1987) produce a theoretical justification for why observed short

interest may proxy for unfulfilled demand to short. Recently, DAvolio (2002) produces

empirical support for Figlewskis intuition and Diamond and Verrecchias theoretical

affirmation. DAvolio finds that the most heavily shorted stocks are more costly to short, on

average, and also contain a higher percentage of stocks that are on special.7

DAvolio also

reports that the differences in short-selling costs, as measured by rebate rates, are very small for

firms below the first four deciles [See DAvolio (2002) Figure 1]. DAvolios finding is

confirmed by Boehme, Danielsen and Sorescu (2005) using an alternative source for rebate rate

data.

The observations made by DAvolio (2002) and Boehme, Danielsen and Sorescu (2005)

seem to be already known to professional short sellers who describe a practice they refer to as

locking up the borrow. Locking up the- borrow involves shorting a security and

simultaneously taking a long position in the same security.8

The motivation for this behavior is

that professional short sellers fear the cost of borrowing shares will rise, or the shares will

become impossible to borrow at a later date as other parties begin to take short positions in the

security (i.e. as short interest rises). Parties who have locked up the borrow can add to their

short positions by selling part of the long position instead of by shorting additional shares.

By locking up the borrow, short sellers are actually locking in lower future short sale

costs since security lenders are reluctant to request that good customers return their shares

when share supply becomes tight. Even though a new party might be willing to borrow for

7DAvolio finds that the level of short-sale constraint is monotonically increasing in short interest, except for stocks in theleast-shorted decile, which are, on average, more costly to short than stocks in the second decile of reported short interest.

8 This practice is also known as shorting against the box.


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some short period at a very high price (i.e. low rebate), the lender values the long-term

relationship with a regular customer. In these cases, rationing, rather than price alone, may be

used to clear the market. The rationing process favors those who previously locked up the

borrow. 9

It is interesting to note that, at least in theory, a shortage of shares could arise precisely

because many parties have locked up the borrow. In any event, the practice suggests that

professional short sellers recognize marginal borrowing prices will rise as more shares are

shorted.

A.2. Idiosyncratic Risk

We measure idiosyncratic risk as the standard deviation of the error terms from the Brown

and Warner (1985) market model, estimated over the 100 days preceding the first day of the

month for which the short interest data are reported.10

This measure is analogous to 2k , the

firm-specific component of the firms return variance, in Merton (1987).

A.3. I/B/E/S Dispersion of Opinion

We measure dispersion of beliefs using the coefficient of variation for analysts annual

forecasts estimated from I/B/E/S data. 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 FY 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) use this proxy.

9 We thank Stephen C. Vogt, Managing Director, Mesirow Financial for his insights on the institutional practices ofsophisticated short sellers and market-neutral fund managers.

10 Firms are excluded from our analysis in any month if more than 10 days of returns data or 75 days of volume data areidentified as missing on CRSP in the prior 100 days or if the firm is missing RSI data. We also screen out all securitiesother than domestic common stocks.


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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. 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

computed. As noted in Diether et al. (2002), only relatively large firms have two or more

analysts providing forecasts. In fact, 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 cannot be calculated for many

small firms.

B. Measuring Subsequent Stock Returns

We adopt the standard four-factor, calendar-time 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 the two-dimensional space spanned by the severity of

the short-sale constraints and the idiosyncratic risk (or I/B/E/S analyst dispersion). 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. For robustness, we also examine monthly returns for portfolios where

firms are held for 12-month, rather than one-month, periods. For each portfolio we estimate the

following four-factor regression model:

Rp,t - Rf,t = p + p(Rm,t - Rf,t) + spSMBt + hpHMLt + upUMDt +ep,t (14)


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where Rp,t represents the raw returns of each portfolio, and Rf,t is the return of the one-month

Treasury Bill. The four independent variables are the excess return on the market portfolio

(Rm,t-Rf,t), the difference between the returns of value-weighted portfolios of small and big firm

stocks (SMBt), the difference in returns of value-weighted portfolios of high and low book-to-

market stocks (HMLt), and the difference in returns of value-weighted portfolios of firms with

high and low prior momentum (UMDt, or up minus down). The first three factors are

proposed by Fama and French (1993), while the momentum factor is proposed by Carhart

(1997).11

The intercept, p, from equation (14) is interpreted 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 on the basis of the

number of firms in a portfolio each month.

Our most important tests compare returns between the calendar time portfolios in high-

and low-idiosyncratic-risk (or dispersion-of-opinion) securities. We perform these tests by

constructing hedge portfolios with long positions in high-dispersion stocks and short positions

in low-dispersion stocks. The hedge portfolio returns are regressed on the four factors:

Rhigh-dispersion,tRlow-dispersion,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t (15)

The "hedge" intercept obtained in this manner (p) represents a measure of the relative long-

term abnormal performance of high-dispersion firms vis--vis the low-dispersion control sets

after controlling for short-sale constraint levels.12

11 The four factors are made available by Kenneth French on his website at Dartmouth College. The momentum factor isused because Fama and French (1996) and Carhart (1997) document a momentum bias for the "traditional" three-factor

model.

12 This follows a procedure similar to that proposed by Mitchell and Stafford (2000) and implemented by Boehme andSorescu (2002).


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C. Data Sources

Our sample is composed of all firms for which short interest data is electronically

available. Short interest data have been obtained from the New York Stock Exchange and the

NASD. Short interest on both NYSE and Nasdaq firms is available in electronic form

beginning with January of 1988. All short interest data are collected on a monthly basis for

transactions settling by the 15th

of each month.

Following Boehme, Danielsen and Sorescu (2005), we use the ticker symbols shown in

the short interest reports to match each observation with the CRSP data.13

As is standard practice when using short interest data, we scale each observation by the

number of shares outstanding. We refer to the scaled data as relative short interest(RSI), the

percentage of each firms outstanding shares that are held short.

D. Descriptive Statistics

Table 1 provides descriptive statistics for the proxy variables used. We provide a

snapshot of firms in our dataset at five-year intervals. The idiosyncratic risk proxy, SIGMA,

appears to have been smaller prior to 1988, but since January 1988, no clear pattern is evident

in the data. As we will discuss later, most of our tests are conducted for the post-1987 period,

due to the lack of short-sale-constraint data prior to 1988.

We possess I/B/E/S analyst dispersion data beginning in February 1976, and the table

reports statistics for DISPERSION beginning in 1978. No clear pattern exists in the

DISPERSION data. The anomalous I/B/E/S mean for December 2002 is driven by one

13 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 shortinterest data are missing. Moreover, assuming that at least some of these missing observations represent a zero level ofshort interest, DAvolios (2002) findings suggest that it is preferable to exclude these firms, because the relation betweenshort interest and short sale constraints is monotonically positive exceptfor firms in the lowest short interest decile. By

excluding these missing observations, we are reasonably confident that the empirical relation between relative short interestand short sale constraints is monotonically positive in our remaining sample, a fact that has been empirically verified byBoehme, Danielsen and Sorescu (2005).


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17

extreme outlier, which has a DISPERSION measure of 2000. Notice that the 99th

percentile

value is reasonable. Because portfolios are formed on the basis of rank-ordered SIGMA and

DISPERSION, a small number of extreme values for SIGMA or DISPERSION will not have

an out-sized influence on portfolio returns.

RSI and OPTIONS data are reported beginning in 1988, the date when RSI data

becomes available in electronic form. The proportion of firms with exchange-traded options

rises dramatically over the period, as does the mean and median RSI.

4. Baseline Cross-Sectional Tests

Table 2 presents baseline cross-sectional tests of the effects of idiosyncratic risk, without

regard to short-sale constraints. Calendar time portfolio (one and twelve month horizons)

abnormal returns are shown as a function of idiosyncratic risk (SIGMA) and analyst earnings

forecast dispersion (DISPERSION) for all CRSP listed common stocks (CRSP share codes 10

and 11) of U.S. domiciled NYSE and Nasdaq firms. SIGMA is the standard deviation of the

error term obtained from the Brown and Warner (1985) market model regression computed

over the prior 100 days of stock returns. SIGMA quartiles are reassigned each month.

DISPERSION is measured by dividing the I/B/E/S standard deviation of analyst

earnings/share forecasts for the current fiscal year end (I/B/E/S FY period 1) by the absolute

value of the mean earnings/share forecast, as listed in the I/B/E/S Summary History file.

Observations having a mean earnings forecast of zero are omitted for the purpose of assigning

the rank ordering of firms. DISPERSION quartiles are assigned for each month. Firms having

a mean forecast of zero are then assigned to the highest DISPERSION quartile. Quartiles 1 and

4 are the lowest and highest SIGMA and DISPERSION quartiles, respectively.


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18

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 (Rp,t-Rf,t) on the four factor model described above. 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. Monthly excess returns are weighted by

the square root of the number of firms in each month. The abnormal return for each sub-sample

is the intercept (p) from the regression.

The abnormal return of the hedge (or zero-investment) portfolio is also shown. This

hedge portfolio consists of long positions of stocks in the highest quartile stocks and short

positions in the lowest quartile ones. The p-value of the hedge portfolio report the statistical

difference of the measured mean return from zero.

Panel A of Table 2 presents results for the period from 1988 to 2002, which is used

throughout the rest of the paper. As can be observed in the table, hedge returns for portfolios

formed on the basis of idiosyncratic risk earn negative returns in one-month portfolios, and

positive returns in the twelve-month portfolios. These results are not statistically significant.

I/B/E/S one-month hedge portfolio returns are negative and statistically significant. This result

is consistent with Diether et al. (2002) and suggests that Millers hypothesized overpricing is

detected in the returns. The twelve-month I/B/E/S hedge portfolio returns are also negative, but

they are not statistically significant. Certainly, as others have noted, there is no evidence here

to support Mertons hypothesis. In Panels B and C we repeat the tests using longer sample

periods to include volatility measures dating back to January 1963, and I/B/E/S dispersion

measures dating back to 1976. We find no qualitative difference in the results over these longer

periods.


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19

5. The Merton Space: Short-Sale Constraints Absent

Having established a baseline for the cross-sectional effects of idiosyncratic risk, we now

turn attention to the subset of firms where Mertons theorized positive correlation between

idiosyncratic risk and return is most likely to be found; firms without short-sale constraints.

Recall that we use two proxies for the presence of short-sale constraints. First, the presence of

options, and second the relative short interest level.

The presence of exchange-traded options is binary. Firms either have exchange-traded

options or they do not. In contrast, RSI is a continuous variable. Boehme et al. (2005) observe

that although firms with options have lower costs of short selling than those without options,

even among optioned firms short-sale constraints on average are lower for less shorted

securities. Accordingly, among firms with exchange-traded options, we should expect that a

Merton effect would be more pronounced when high RSI firms are excluded from the sample.

With this in mind, we formulate the following test:

Each month, we rank all firms on the basis of RSI and sort them into twenty groups

(viciles) with 5% of firms in each vicile. Vicile 20 contains the 5% of firms having the largest

RSI that month, and Vicile 1 contains the firms with the smallest monthly RSI.

We then repeat the cross-sectional tests shown in Table 1, the baseline test, for nested

subsets of optioned firms, with the largest cross-sectional test being applied to a portfolio of

firms containing Viciles 1 through 20; in other words, to all optioned firms.

The second cross-sectional test is identical to the first, except that only RSI Viciles 1

through 19 are included in the portfolios. The third test if for Vicile groups 1 through 18, and

so on. Using this method we can observe the effect of peeling away firms that are more

likely to face shorting constraints as measured by the RSI proxy. We expect that the correlation

between idiosyncratic risk and return to be higher as the highest RSI viciles are eliminated.


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20

We focus on the returns to the hedge portfolio that is long on firms with high idiosyncratic

volatility and short on firms with low idiosyncratic volatility. These hedge returns provide a

good measure of the price effect of idiosyncratic volatility. If Mertons (1987) model is correct,

the hedge returns should be positive for stocks that are free of short sale constraints.

Figure 1 is a graphical depiction of the one-month calendar-time hedge portfolio returns,

along with p-values.

0.00%

0.20%

0.40%

0.60%

0.80%1.00%

1.20%

1.40%

1.60%

20 19 18 17 16 15 14 13 12

Size of Largest RSI Vicile in the Portfolio

MonthlyAbn

ormal

Return

0.00

0.05

0.10

0.15

0.200.25

0.30

0.35

0.40

0.45

0.50

P-Value

% Return P-Value

Figure 1: Hedge Portfolio Abnormal Returns and P-Values (Idiosyncratic Risk)

The left-most observation (20) includes all RSI viciles, and thus all firms with traded

options. The right-most portfolio (12) is composed of firms only in Viciles 1-12. As

previously discussed, Viciles 1 through 12 are relatively homogenous, and less than one-fourth

the sample are in viciles less than 12.

It is immediately obvious that, in contrast to the baseline hedge portfolio that earned

significant negative abnormal returns, Portfolio 20, the hedge portfolio for all optioned stocks

does not. The abnormal return, measured along the left axis, is -0.001% per month, a value

which is neither economically nor statistically significant.


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21

Consistent with Merton (1987), as we begin to exclude firms that are likely to be subject

to short-sale constraints, the hedge portfolio abnormal returns become more positive, and the p-

values rise in significance. Excluding firms with RSI in the highest 10% for the month (viciles

19 and 20), the abnormal return jumps to 0.61% per month, and the p-value is 14.6%. Including

only firms with RSI viciles 1 though 15 produces abnormal returns of 1.1% per month (p-value

= 0.049). This is a very high abnormal return in an economic sense, more than 13% annually.

For portfolio 12 the returns is even higher at 1.42% per month (p-value = 0.024).14

These results are completely consistent with the predicted positive correlation between

idiosyncratic risk and abnormal returns as predicted in Merton (1987). As firms likely to have

the highest levels of short-sale constraint are excluded from the portfolio, the effect of

idiosyncratic volatility on abnormal returns becomes more positive.

Table 3 presents the data in Figure 1 along with individual quartile results. The format is

the same as that used in the baseline results in Table 2. Notice in particular that the increase in

hedge portfolio returns for Portfolio 12, relative to Portfolio 20, is driven entirely by increased

returns for the high idiosyncratic risk quartile. The low idiosyncratic risk quartile does not

evidence falling returns. As firms that are more likely to be short-sale constrained are excluded

from the sample, the firms with higher idiosyncratic risk exhibit the behavior predicted by

Merton (1987).

We next turn to an analysis of the I/B/E/S dispersion data. Diether et al. (2002) suggest

that the dispersion of analysts forecasts reflect the type of idiosyncratic risk to which Merton

refers. Using I/B/E/S forecast dispersion, they reject Merton (1987) in favor of Miller stating,

our results clearly reject the notion that dispersion can be viewed as a proxy for risk, since the

14 Because we examine only firms with traded options, the lowest RSI viciles are very thinly populated. Optioned firms tendto be larger and to have higher levels of short interest. [See Figlewski and Webb (1977)] Less than 25% of firms withexchange-traded options fall into Viciles 1 through 12.


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22

relation between dispersion and future returns is strongly negative. Likewise, Gebhardt et al.

(2001) note that forecast dispersion has the wrong sign to be a risk proxy useful in estimating

the cost of capital.

It certainly seems appropriate to suspect that the dispersion of analyst forecasts would be

related to idiosyncratic risk levels. Numerous authors present theoretical models correlating

belief dispersion with asset time-series volatility. For example, Shalen (1993) and Harris and

Raviv (1993) develop models that specifically investigate the role of belief dispersion on

volatility and other trading characteristics. Empirically, Peterson and Peterson (1982) observe a

positive relationship between return volatility and the dispersion of I/B/E/S forecasts. In our

sample, we find the correlation between I/B/E/S dispersion and idiosyncratic risk has a

correlation of 0.3521.

Repeating the methodology depicted in Figure 1, we present Figure 2 showing the hedge

portfolio abnormal returns and p-values for I/B/E/S forecast dispersion.


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23

0.00%

0.10%

0.20%

0.30%

0.40%

0.50%

0.60%

20 19 18 17 16 15 14 13 12

Size of Largest RSI Vicile in the Portfolio

MonthlyAbn

ormal

Return

0.00

0.05

0.10

0.15

0.200.25

0.30

0.35

0.40

0.45

0.50

P-Value

% Return P-Value

Figure 2: Hedge Portfolio Abnormal Returns and P-Values (I/B/E/S Dispersion)

In a manner similar to the results depicted in Figure 1, I/B/E/S dispersion is positively

correlated with hedge portfolio alphas. Ignoring the RSI viciles for the moment, but

restricting the analysis to firms with traded options only, the abnormal returns reported for

Portfolio 20 are positive, but not statistically significant. This is in contrast to the findings in

Table 2 where a negative and statistically significant correlation is observed between IBES

dispersion and abnormal returns. Stripping away the highest RSI viciles, hedge portfolio

returns and p-values rise in a manner similar to that found in the idiosyncratic risk tests.

Portfolio 12 reports monthly abnormal returns of 0.51% (p = 0.054), which annualizes as

6.3% per year. These results, reported numerically in Table 4, strongly support Mertons

hypothesis. They stand in keen contrast to previous tests pitting Merton (1987) as a

competing hypothesis to Miller (1977).


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24

We have also conduct the tests reflected in Figures 1 and 2 using one-year, as opposed

to one-month, calendar-time portfolios. These results are not significantly different from

those presented above. We infer the Merton risk premium is not rapidly dissipated, but

persists for an extended time-period.

6. The Miller Space: Short-Sale Constraints Present

We now turn to examining the price effect of idiosyncratic risk when short-sale

constraints are present: the Miller space. We first examine the cross-sectional effects of

dispersion-of-opinion as proxied by I/B/E/S estimates. In an explicit sense, Merton (1987)

hypothesized that idiosyncratic volatility is priced at a discount. Diether et al. (2002) and other

empiricists infer that the dispersion of analysts forecasts reflects a type of idiosyncratic risk to

which Merton refers. In contrast, Miller (1977) explicitly considers the dispersion of investor

beliefs. While Jones, Kaul and Lipson (1994), among others, note that volatility is related to the

dispersion of beliefs, I/B/E/S forecast dispersion more intuitively reflects the dispersion of

beliefs to which Miller refers.

In the Miller space, dispersion of beliefs should result in premium pricing, in contrast to

the discount pricing in the Merton space. The following tests mirror those conducted in the

previous subsection with the following modifications. As a first screen, because we wish to

examine short-sale constrained securities rather than unconstrained ones, we examine firms

without traded options. From this set of option-less firms, we further refine the screen using the

RSI vicile classifications previously described.

Table 5 presents the results, which are also depicted in Figure 3. The format is similar to

the one used before, except that we are now looking at a series of portfolios with increasing

short-sale constraint, so the numbers along the x-axis refer to the lowest RSI vicile in the


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25

portfolio. Therefore, Portfolio 1, the left-most portfolio on the x-axis, contains all non-optioned

firms in Viciles 1 through 20. Portfolio 2 contains Viciles 2 through 20. Portfolio 3 contains

Viciles 3 through 20, and this pattern continues.

-1.10%

-1.05%

-1.00%

-0.95%

-0.90%

-0.85%

-0.80%

1 2 3 4 5 6 7 8 9 10 11 12 13

Size of Smallest RSI Vicile in the Portfolio

MonthlyAbnormal

Return

0.00

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0.04

0.05

0.05

P-Value

% Hedge Return P-Value

Figure 3: Hedge Portfolio Abnormal Returns and P-Values (I/B/E/S Dispersion)

In Figure 3, we observe that the monthly hedge portfolio abnormal return for Portfolio

1 is -0.83% with a p-value better than 0.001. Thus, solely by restricting the analysis to firms

with traded options, the result depicted in Figure 2 is reversed, and hedge portfolios

constructed on the basis of dispersion of beliefs have negative returns. Although the p-values

of these returns are already very high, restricting the sample to higher-RSI viciles makes the

point estimate more negative, as we would expect.

Miller Portfolio 13 in Figure 3 is a mirror image of Merton portfolio 12 in Figure 2.

Portfolio 12 in Figure 2 is comprised of firms with traded options having RSI below the 13th

vicile. Portfolio 13 in Figure 3 is comprised of firms without traded options having RSI

greater than the 12th

vicile. In the first case, short sale constraints are likely absent, and


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26

returns are an increasing function of idiosyncratic volatility. In the second, short sale

constraints are present and the relation is reversed.

Note that the results are not the same as those reported by Asquith and Meulbroek (1995) or

Desai, Ramesh, Thiagarajan, and Balachandran (2002). These studies find that firms with high

short interest earn low subsequent returns, but do not distinguish between high-dispersion-of-

opinion firms and low-dispersion-of-opinion firms. By contrast, Figure 3 reveals that for non-

optioned firms, at any level of RSI, firms with higher dispersion-of-beliefs earn lower risk-

adjusted returns, and this relation becomes stronger at higher levels of RSI.

Finally, we turn our attention to the effects of idiosyncratic volatility on returns when

short-sale constraints are present. Because idiosyncratic volatility is positively correlated with

I/B/E/S forecast dispersion, we expect to see hedge portfolio results similar to those observed in

Figure 3 and Table 5.

-2.00%

-1.80%

-1.60%

-1.40%

-1.20%

-1.00%

-0.80%

-0.60%

-0.40%

-0.20%

0.00%

1 2 3 4 5 6 7 8 9 10 11 12 13

Size of Smallestt RSI Vicile in the Portfolio

MonthlyAbnormalRetur

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

P-Value

% Hedge Return P-Value

Figure 4: Hedge Portfolio Abnormal Returns and P-Values (Idiosyncratic Risk)

Figure 4 (numerically presented as Table 6) provides hedge portfolio returns for portfolios

that hold long positions in high-idiosyncratic-volatility firms and short positions in low-

idiosyncratic-volatility firms. The results are as expected. Portfolio 1 reflects all firms for


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27

which options are not traded, and the abnormal returns for these firms are negative and

economically significant, although they do not reflect statistical significance at standard levels.

As the lowest RSI firms are removed from the sample, short-sale constraints rise, and the cross-

sectional effects of idiosyncratic volatility increases. Portfolio 13 reflects hedge portfolio

monthly returns of -1.94% and statistical significance at p


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28

leave a full mapping of the space to future research. Our contributions are highlighting the

complementary nature of the Merton and Miller theories, and to being the first to provide strong

empirical support for Mertons (1987) model.

Figure 5:

Merton (1987) and Miller (1977) Represented as Complementary Hypotheses

While Figure 5 details the effects of dispersion of beliefs, our tests also demonstrate why

direct tests of idiosyncratic volatility as a risk factor produce mixed results. The presence or

absence of short-sale constraints confounds the analysis. Idiosyncratic risk and dispersion of

beliefs are highly correlated; perhaps both reflecting some underlying uncertainty level.

Accordingly, abnormal returns, as a function of idiosyncratic volatility, also follow a pattern

like that in Figure 5. If tests are conducted independent of short-sale-constraint proxies, the

results will be weak, and may even be incorrectly signed.

An interesting implication of our findings relates to the area of corporate governance. In

Millers model, it may be argued that managers of short-sale-constrained firms can increase


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29

share value by undertaking activities about which investors beliefs are highly heterogeneous.

Absent short sale constraints, Merton suggests undertaking such projects would now induce

discounts rather than premia in stock prices, and managers would be strongly motivated to

reduce investor uncertainty concerning the firms prospects. When short-sale constraints are

severe, the market may temporarily reward a lack of transparency by paying a premium for

firms that create a smoke screen to obscure a firms financial position, but when short sale

constraints are eliminated, the lack of transparency will itself be penalized with a discount.

Thus, any policy aimed at reducing the cost of short selling could provide a significant social

benefit if greater firm transparency is rewarded with higher share prices in a Merton

environment rather than penalized with lower share values as in a Miller environment.


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30

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Table 1: Descriptive Statistics

A description of each proxy variable is provided for different calendar dates (conditioned upon availability for

the particular data series): January 1963, January 1968, January 1973, January 1978, January 1983, January

1988, January 1993, January 1998, and December 2002. Proxies are estimated for all U.S.-domiciled common

stocks listed on the NYSE and Nasdaq. For each calendar date, theI/B/E/S Analyst Forecast Dispersion is the

I/B/E/S standard deviation of earnings per share forecasts for the next fiscal year end. scaled by the forecast

mean. The Idiosyncratic Risk(SIGMA) is the standard deviation of the error term obtained from the marketmodel computed over the prior 100 days. The Relative Short Interest (RSI) is measured as the short interest

divided by the number of outstanding shares. The OPTIONS status reports the total number of firms in our

sample and the proportion of firms with exchange-traded options.

DependentVariable

Date N. Obs. Mean FirstPercentile

FirstQuartile

Median ThirdQuartile

99th

Percentile

196301 1130 0.0175 0.0076 0.0128 0.0162 0.0205 0.0415196801 1183 0.0193 0.0080 0.0140 0.0180 0.0231 0.0396197301 1376 0.0187 0.0076 0.0130 0.0171 0.0226 0.0452

197801 3623 0.0210 0.0042 0.0117 0.0172 0.0258 0.0791198301 4097 0.0313 0.0051 0.0173 0.0256 0.0383 0.1160198801 5437 0.0468 0.0094 0.0277 0.0407 0.0584 0.4983

199301 5027 0.0471 0.0072 0.0216 0.0355 0.0577 0.2104199801 6547 0.0389 0.0010 0.0211 0.0317 0.0465 0.1506

Idiosyncratic Risk:

SIGMA

200212 4677 0.0428 0.0095 0.0208 0.0330 0.0547 0.1608

197801 897 0.0621 0.0000 0.0174 0.0327 0.0593 0.4059

198301 1619 1.6254 0.0000 0.0296 0.0581 0.1474 4.6667198801 2153 1.6912 0.0000 0.0289 0.0593 0.1556 5.0000199301 2506 1.3904 0.0000 0.0194 0.0417 0.1059 3.1000199801 3867 1.6841 0.0000 0.0133 0.0290 0.0756 2.2143

I/B/E/S analyst

earnings forecastdispersion:

DISPERSION

200212 2566 4.4104 0.0000 0.0095 0.0233 0.0660 2.6667

198801 1145 0.0083 0.0000 0.0009 0.0026 0.0071 0.0981199301 4166 0.0105 0.0000 0.0006 0.0023 0.0086 0.1236

199801 6111 0.0157 0.0000 0.0005 0.0038 0.0163 0.1605

Relative ShortInterest(%):

RSI200212 4413 0.0266 0.0000 0.0013 0.0096 0.0213 0.2333

198801 5478 0.0785

199301 5068 0.1689199801 6665 0.2825

OPTIONS:Percent of Firmswith Exchange Traded Options 200212 4809 0.4999


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36

Table 2: Unconditional Abnormal Returns as a Function of Idiosyncratic Risk and Analyst Forecast Dispersion

Calendar time portfolio (one and twelve month horizons) abnormal returns are shown as a function of

idiosyncratic risk (SIGMA) and analyst earnings forecast dispersion (DISPERSION) for all CRSP listed

common stocks (CRSP share codes 10 and 11) of U.S. domiciled NYSE and Nasdaq firms. SIGMA is the

standard deviation of the error term obtained from the Brown and Warner (1985) market model regressioncomputed over the prior 100 days of stock returns. SIGMA quartiles are assigned for each month, beginning

with January 1963 and ending with December 2002) by sorting all NYSE and Nasdaq firms on the SIGMA.

DISPERSION is measured by dividing the I/B/E/S standard deviation of analyst earnings/shareforecasts for the current fiscal year end (I/B/E/S FY period 1) by the absolute value of the mean earnings/share

forecast, as listed in the I/B/E/S Summary History file. Observations having a mean earnings forecast of zero are

omitted for the purpose of assigning the rank ordering of firms. DISPERSION quartiles are assigned for eachmonth (beginning with January 1976) by sorting all NYSE and Nasdaq firms on the DISPERSION. Firms

having a mean forecast of zero are then assigned to the highest DISPERSION quartile. Quartiles 1 and 4 are the

lowest and highest SIGMA and DISPERSION quartiles, respectively.

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 (Rp,t-Rf,t) on a four factor model: thethree Fama and French (1993) risk factors (Rm-Rf, SMB, HML) and the momentum factor of Carhart (1997)

(UMD). Calendar time portfolios are re-balanced each month, and include only firms that entered the portfolio

during either the prior month or previous twelve months. The estimation method is weighted least squares;

monthly excess returns are weighted by the square root of the number of firms in each month. The abnormalreturn for each sub-sample is the intercept (p) from the following regression:

Rp,t-Rf,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.

We also report the long-term abnormal return of a hedge (or zero-investment) portfolio, consisting of

long positions of stocks with the highest SIGMA or DISPERSION quartile and short positions of stocks in the

lowest SIGMA or DISPERSION quartile. The abnormal return to the hedge portfolio is computed as the

intercept (p) from the following regression:

Rhigh-sigma,t-Rlow-sigma,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.

Results are presented in Panel A below are for the February 1988 to December 2002 period, and resultspresented in Panels B and C begin with January 1963 for SIGMA and February 1976 for DISPERSION and end

with December 2002. An abnormal return of 0.00187 below is to be interpreted as an abnormal return of 0.187

percent per calendar month. p-values reporting statistical difference from zero are shown in the rightmostcolumn for the hedge portfolio intercept and are computed from the intertemporal variation in the monthly

calendar-time portfolio returns.

Panel A: Idiosyncratic risk (SIGMA) and Analyst earnings forecast dispersion (DISPERSION), February 1988

through December 2002

Independent Variable andPortfolio Horizon

1st

quartile(lowest)

2nd

quartile

3rd

quartile

4th

quartile(highest)

Hedge

portfolio:

4th quartile

minus 1stquartile

p-value of

hedge

portfolio

SIGMA, 1-month 0.00187 0.00241 -0.00068 0.00029 -0.00159 0.7284

SIGMA, 12-month 0.00169 0.00115 -0.00025 0.00357 0.00192 0.6720

DISPERSION, 1-month 0.00240 0.00165 0.00081 -0.00233 -0.00462 0.0370

DISPERSION, 12-month 0.00167 0.00143 0.00112 0.00004 -0.00163 0.4509


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37

Panel B: Idiosyncratic risk (SIGMA), January 1963 through December 2002

SIGMA

Calendar time portfoliohorizon

1st

quartile(lowest)

2nd

quartile

3rd

quartile

4th

quartile(highest)

Hedge

portfolio:

4th quartile

minus 1st

quartile

p-value ofhedge

portfolio

1-month 0.00178 0.00230 0.00036 -0.00090 -0.00268 0.2571

12-month 0.00182 0.00125 0.00046 0.00163 -0.00014 0.9513

Panel C: Analyst earnings forecast dispersion (DISPERSION), February 1976 through December 2002

DISPERSION

Calendar time portfoliohorizon

1st

quartile

(lowest)

2nd

quartile

3rd

quartile

4th

quartile(highest)

Hedge

portfolio:4th quartile

minus 1st

quartile

p-value of

hedgeportfolio

1-month 0.00275 0.00228 0.00138 -0.00203 -0.00478 0.0025

12-month 0.00200 0.00182 0.00144 -0.00006 -0.00205 0.1759


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38

Table 3: A Test of Mertons Hypothesis:

Abnormal Returns as a Function of Idiosyncratic Risk, Conditioned on Absence of Short Sale Constraints

One-month calendar time abnormal returns are shown as a function of idiosyncratic risk (SIGMA) and Relative Short

Interest (RSI) for all CRSP listed common stocks (CRSP share codes 10 and 11) of U.S. domiciled NYSE and Nasdaq firms

having exchange traded options. SIGMA is measured as the standard deviation of the error term obtained from the Brownand Warner (1985) market model regression computed over the prior 100 days of stock returns. SIGMA quartiles are

assigned for each month, beginning with February 1988 and ending with December 2002) by sorting all NYSE and Nasdaq

firms on the SIGMA. Quartiles 1 and 4 are the lowest and highest SIGMA quartiles, respectively. The analysis below beginsby first examining firms in RSI viciles 1 through 20, and then proceeds by progressively deleting firms in the higherRSI

viciles.

The abnormal returns are computed by forming equally-weighted calendar-time portfolios of one month horizon andregressing the excess portfolio returns (Rp,t-Rf,t) on a four factor model: the three Fama and French (1993) risk factors (Rm-

Rf, SMB, HML) and the momentum factor of Carhart (1997) (UMD). Calendar time portfolios are re-balanced each month,

and include only firms that entered the portfolio during the prior month. The estimation method is weighted least squares;

monthly excess returns are weighted by the square root of the number of firms in each month. The abnormal return for each

sub-sample is the intercept (p) from the following regression:Rp,t-Rf,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.

We also report the long-term abnormal return of a hedge (or zero-investment) portfolio, consisting of long positions of

stocks with the highest SIGMA quartile and short positions of stocks in the lowest SIGMA quartile. The abnormal return to

the hedge portfolio is computed as the intercept (p) from the following regression:Rhigh-sigma,t-Rlow-sigma,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.Results are presented for the February 1988 to December 2002 period. An abnormal return of 0.00056 below is to be

interpreted as an abnormal return of 0.056 percent per calendar month. p-values reporting statistical difference from zero are

shown in the rightmost column for the hedge portfolio intercept and are computed from the intertemporal variation in the

monthly calendar-time portfolio returns.

SIGMA

Option

Status

Largest

RSI in

portfolio

1st

quartile(lowest)

2nd

quartile

3rd

quartile

4th

quartile(highest)

Hedge

portfolio:

4th quartileminus 1st

quartile

p-value of

hedge

portfolio

20 0.00056 0.00095 -0.00083 0.00320 -0.00002 0.4986

19 0.00061 0.00153 0.00139 0.00624 0.00332 0.2731

18 0.00078 0.00170 0.00248 0.00935 0.00612 0.1458

17 0.00095 0.00216 0.00423 0.01131 0.00841 0.0897

16 0.00108 0.00254 0.00523 0.01274 0.00964 0.0625

15 0.00104 0.00245 0.00563 0.01369 0.01067 0.0490

14 0.00108 0.00285 0.00598 0.01516 0.01218 0.0326

13 0.00103 0.00332 0.00519 0.01696 0.01417 0.0189

Optioned

12 0.00125 0.00392 0.00577 0.01772 0.01420 0.0240


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39

Table 4: An Alternative Test of Mertons Hypothesis:

Abnormal Returns as a Function of Forecast Dispersion, Conditioned on the Absence of Short Sale Constraints

One-month calendar time abnormal returns are shown as a function of I/B/E/S analyst earnings forecast dispersion

(DISPERSION) and Relative Short Interest (RSI) for all CRSP listed common stocks (CRSP share codes 10 and 11) of U.S.

domiciled NYSE and Nasdaq firms having exchange traded options. DISPERSION is measured by dividing the I/B/E/Sstandard deviation of analyst earnings/share forecasts for the current fiscal year end (I/B/E/S FY period 1) by the absolute

value of the mean earnings/share forecast, as listed in the I/B/E/S Summary History file. Observations having a mean

earnings forecast of zero are omitted for the purpose of assigning the rank ordering of firms. DISPERSION quartiles areassigned for each month by sorting all NYSE and Nasdaq firms on the DISPERSION. Firms having a mean forecast of zero

are then assigned to the highest DISPERSION quartile. Quartiles 1 and 4 are the lowest and highest DISPERSION quartiles,

respectively. The analysis below begins by first examining firms in RSI viciles 1 through 20, and then proceeds byprogressively deleting firms in the higherRSI viciles.

The abnormal returns are computed by forming equally-weighted calendar-time portfolios of one month horizon andregressing the excess portfolio returns (Rp,t-Rf,t) on a four factor model: the three Fama and French (1993) risk factors (Rm-

Rf, SMB, HML) and the momentum factor of Carhart (1997) (UMD). Calendar time portfolios are re-balanced each month,

and include only firms that entered the portfolio during the prior month. The estimation method is weighted least squares;monthly excess returns are weighted by the square root of the number of firms in each month. The abnormal return for each

sub-sample is the intercept (p) from the following regression:Rp,t-Rf,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.

We also report the long-term abnormal return of a hedge (or zero-investment) portfolio, consisting of long positions ofstocks with the highest DISPERSION quartile and short positions of stocks in the lowest DISPERSION quartile. The

abnormal return to the hedge portfolio is computed as the intercept (p) from the following regression:Rhigh-sigma,t-Rlow-sigma,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.

Results are presented for the February 1988 to December 2002 period. An abnormal return of 0.00062 below is to be

interpreted as an abnormal return of 0.062 percent per calendar month. p-values reporting statistical difference from zero areshown in the rightmost column for the hedge portfolio intercept and are computed from the intertemporal variation in the

monthly calendar-time portfolio returns.

DISPERSION

Option

Status

Largest

RSI inportfolio

1st

quartile(lowest)

2nd

quartile

3rd

quartile

4th

quartile(highest)

Hedge

portfolio:4th quartile

minus 1st

quartile

p-value of

hedgeportfolio

20 0.00062 0.00122 0.00133 0.00209 0.00139 0.2973

19 0.00119 0.00227 0.00227 0.00318 0.00186 0.2351

18 0.00124 0.00272 0.00263 0.00434 0.00299 0.1212

17 0.00186 0.00271 0.00338 0.00523 0.00327 0.1050

16 0.00211 0.00315 0.00380 0.00527 0.00311 0.1154

15 0.00202 0.00349 0.00410 0.00523 0.00307 0.1272

14 0.00174 0.00415 0.00472 0.00604 0.00409 0.0714

13 0.00194 0.00466 0.00410 0.00627 0.00408 0.0867

Optioned

12 0.00185 0.00514 0.00451 0.00731 0.00511 0.0544


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40

Table 5: A Test of Millers Hypothesis:

Abnormal Returns as a Function of Forecast Dispersion, Conditioned on the Presence of Short Sale Constraints

One-month calendar time abnormal returns are shown as a function of I/B/E/S analyst earnings forecast dispersion

(DISPERSION) and Relative Short Interest (RSI) for all CRSP listed common stocks (CRSP share codes 10 and 11) of U.S.

domiciled NYSE and Nasdaq firms that do not have exchange traded options. DISPERSION is measured by dividing the I/B/E/Sstandard deviation of analyst earnings/share forecasts for the current fiscal year end (I/B/E/S FY period 1) by the absolute value

of the mean earnings/share forecast, as listed in the I/B/E/S Summary History file. Observations having a mean earnings forecast

of zero are omitted for the purpose of assigning the rank ordering of firms. DISPERSION quartiles are assigned for each monthby sorting all NYSE and Nasdaq firms on the DISPERSION. Firms having a mean forecast of zero are then assigned to the

highest DISPERSION quartile. Quartiles 1 and 4 are the lowest and highest DISPERSION quartiles, respectively. The analysis

below begins by first examining firms in RSI viciles 1 through 20, and then proceeds by progressively deleting firms in thesmaller RSI viciles.

The abnormal returns are computed by forming equally-weighted calendar-time portfolios of one month horizon andregressing the excess portfolio returns (Rp,t-Rf,t) on a four factor model: the three Fama and French (1993) risk factors (Rm-Rf,

SMB, HML) and the momentum factor of Carhart (1997) (UMD). Calendar time portfolios are re-balanced each month, and

include only firms that entered the portfolio during the prior month. The estimation method is weighted least squares; monthlyexcess returns are weighted by the square root of the number of firms in each month. The abnormal return for each sub-sample is

the intercept (p) from the following regression:Rp,t-Rf,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.

We also report the long-term abnormal return of a hedge (or zero-investment) portfolio, consisting of long positions of

stocks with the highest DISPERSION quartile and short positions of stocks in the lowest DISPERSIONquartile. The abnormal

return to the hedge portfolio is computed as the intercept (p) from the following regression:Rhigh-sigma,t-Rlow-sigma,t = p + p(Rm,t-Rf,t) + spSMBt + hpHMLt + upUMDt + ep,t.

Results are presented for the February 1988 to December 2002 period. An abnormal return of 0.00330 below is to be

interpreted as an abnormal return of 0.330 percentper calendar month. p-values reporting statistical difference from zero are

shown in the rightmost column for the hedge portfolio intercept and are computed from the intertemporal variation in the monthlycalendar-time portfolio returns.

DISPERSION

Option

Status

Smallest

RSI in

portfolio

1st

quartile(lowest)

2nd

quartile

3rd

quartile

4th

quartile(highest)

Hedge

portfolio:

4th minus

1st quartile

p-value ofhedge

portfolio

1 0.00330 0.00187 -0.00043 -0.00512 -0.00831 0.0000

2 0.00315 0.00182 -0.00057 -0.00542 -0.00843 0.0000

3 0.00294 0.00166 -0.00071 -0.00575 -0.00853 0.0000

4 0.00273 0.00142 -0.00101 -0.00607 -0.00862 0.0000

5 0.00253 0.00136 -0.00107 -0.00646 -0.00884 0.0000

6 0.00228 0.00114 -0.00139 -0.00683 -0.00898 0.0000

7 0.00196 0.00067 -0.00170 -0.00681 -0.00861 0.0000

8 0.00158 0.00022 -0.00205 -0.00728 -0.00868 0.0001

9 0.00129 0.00012 -0.00224 -0.00772 -0.00885 0.0001

10 0.00098 -0.00015 -0.00238 -0.00826 -0.00914 0.0000

11 0.00060 -0.00034 -0.00266 -0.00871 -0.00916 0.0001

12 0.00018 -0.00041 -0.00288 -0.00941 -0.00953 0.0000

Nonoptioned

13 0.00009 -0.00066 -0.00289 -0.01023 -0.01034 0.0000


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Table 6: An Alternative Test of Millers Hypothesis:

Abnormal Returns as a Function of Idiosyncratic Risk, Conditioned on the Presence of Short Sale Constraints

One-month calendar time abnormal returns are shown as a function of idiosyncratic risk (SIGMA) and Relative Short Interest

(RSI) for all CRSP listed common stocks (CRSP share codes 10 and 11) of U.S. domiciled NYSE and Nasdaq firms that do not

have exchange traded options. SIGMA is measured as the standard deviation of the error term obtained from the Brown andWa

idiosyncratic risk and the cross-section of stock returns: merton 1987 meets miller 1977

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