The E¤ect of Information on Uncertaintyand the Cost of Capital

D.J. JohnstoneUniversity of Sydney Business School

University of SydneyNSW 2006Australia

david.johnstone@sydney.edu.au

Revised Draft: 20 July, 2014

Abstract

It is widely held that better �nancial reporting makes investors more con-�dent in their predictions of future cash �ows and reduces their required riskpremia. The logic is that more information leads necessarily to more certainty,and hence lower subjective estimates of �rm "beta" or covariance with other�rms. This is misleading on both counts. Bayesian logic shows that the bestavailable information can often leave decision makers less certain about futureevents. And for those cases where information indeed brings great certainty,conventional mean-variance asset pricing models imply that more certain esti-mates of future cash payo¤s can sometimes bring a higher cost of capital. Thisoccurs when new or better information leads to su¢ ciently reduced expected�rm payo¤s. To properly understand the e¤ect of signal quality on the costof capital, it is essential to think of what that information says, rather thanconsidering merely its "precision", or how strongly it says what it says.

Keywords: Information, uncertainty, cost of capital, CAPM, information risk

Acknowledgements: I have been greatly assisted by the editor (Steven Huddart)and two anonymous referees, and also by comments on earlier drafts by SudiptaBasu, Jeremy Bertomeu, Greg Clinch, Tom Dyckman, Bruce Grundy, MarlenePlumlee, Jim Ohlson, Robert Verrecchia, Alfred Wagenhofer, Bob Winkler andparticipants at the CAR Conference (Kingston 2013).

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Title Page of Manuscript

The E¤ect of Information on Uncertaintyand the Cost of Capital

Revised Draft: 20 July, 2014

Abstract

It is widely held that better �nancial reporting makes investors more con-�dent in their predictions of future cash �ows and reduces their required riskpremia. The logic is that more information leads necessarily to more certainty,and hence lower subjective estimates of �rm "beta" or covariance with other�rms. This is misleading on both counts. Bayesian logic shows that the bestavailable information can often leave decision makers less certain about futureevents. And for those cases where information indeed brings great certainty,conventional mean-variance asset pricing models imply that more certain esti-mates of future cash payo¤s can sometimes bring a higher cost of capital. Thisoccurs when new or better information leads to su¢ ciently reduced expected�rm payo¤s. To properly understand the e¤ect of signal quality on the costof capital, it is essential to think of what that information says, rather thanconsidering merely its "precision", or how strongly it says what it says.

Keywords: Information, uncertainty, cost of capital, CAPM, information risk

Acknowledgements: I have been greatly assisted by the editor (Steven Huddart)and two anonymous referees, and also by comments on earlier drafts by SudiptaBasu, Jeremy Bertomeu, Greg Clinch, Tom Dyckman, Bruce Grundy, MarlenePlumlee, Jim Ohlson, Robert Verrecchia, Alfred Wagenhofer, Bob Winkler andparticipants at the CAR Conference (Kingston 2013).

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Revised Manuscript

1 Introduction

This paper concerns the fundamental question of how information about the �rma¤ects its cost of capital. It is widely held that "good" �nancial reporting resolvesuncertainty and brings a lower cost of capital. The economic logic is that (i) moreinformation leads to more certainty, and hence to lower estimates of �rm "beta" orcovariance with other �rms, and (ii) the cost of capital falls when information bringsa lower estimate of beta.

The intuition is straightforward. A �rm�s cost of capital is the riskless in-terest rate plus a risk premium. Releasing more information and, in particular,more public information through �nancial reports and other public disclosuresby �rms reduces the uncertainty about the size and the timing of future cash�ows and, therefore, also the risk premium. (Christensen et al. 2010, p.817)

This common understanding is misleading on both counts. First, the best availableinformation can often leave Bayesian users less certain about future events. Second,where information does bring greater certainty, mean-variance asset pricing modelsimply that more certain estimates of expected �rm payo¤s can prompt an increasein the �rm�s cost of capital. This occurs when unfavorable information implies asu¢ ciently reduced expected (i.e. mean) future payo¤.The force of the payo¤ mean on the cost of capital is at �rst surprising. Con-

ventional wisdom has it that the cost of capital is driven only by the �rm�s returnscovariance or "beta". However, a simple step of writing the CAPM in terms of cashpayo¤s rather than returns brings to light the latent underlying e¤ects of both the�rm�s expected payo¤ and its payo¤ covariance, each of which a¤ects beta.Given that the �rm�s returns beta, and thertefore its CAPM cost of capital, react

to the estimated mean payo¤, accounting information is relevant for its "direction"(favorable or unfavorable) as well as for its precision. Favorable information, whichleads to a higher expected payo¤, pushes the �rm�s cost of capital downwards, allelse equal. Conversely, if information leads to greater certainty that the �rm�s futurepayo¤will be both low and highly dependent on market-wide factors, then its assessedcovariance will increase and its mean will fall, causing a two pronged increase in thecost of capital. In general, information can shift either or both the mean payo¤ orpayo¤covariance in either direction, and may therefore push its CAPM cost of capitalup or down. Lambert et al. (2007) set out clear proof of this result, but highlightthe role of the covariance.The Bayesian e¤ect of information on beliefs is generally greater for more precise

information, however it is wrong to assume that more precise information necessarilyreduces the cost of capital merely because it is more precise. Better information canexpose grounds for pessimism about �rms�payo¤s, or inter-dependencies between thepayo¤s of di¤erent �rms, su¢ cient in either case to increase the market risk premiumon average across all �rms, or even for all �rms individually.

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This paper formalizes important general principles regarding information and thecost of capital. I show how restrictive assumptions and overly narrow focus on thepayo¤ covariance rather than mean have led to conclusions that are appealing andconsistent with conventional wisdom, but which do not hold in general. The morevalid intuition is that, by being informative, accounting information can change users�probability beliefs and lead to either a higher or a lower cost of capital. Just as newinformation has an unpredictable e¤ect (up or down) on the share price, it also hasa generally unpredictable e¤ect on the CAPM equilibrium cost of capital.

2 The Model

My model is a generalization of Lambert et al. (2007), hereafter LLV, under whichthe �rm�s ex ante (forward looking) cost of capital is determined in equilibrium bya mean-variance CAPM, and incremental information a¤ects the cost of capital byaltering investors�Bayesian assessments of the mean and covariance parameters ofthe joint distribution of �rms�payo¤s.I treat payo¤s as the primitive uncertain quantities, rather than returns. Payo¤s

are observable and exogenous. Returns are harder to describe and comprehend. Theyare de�ned as payo¤s relative to equilibrium prices, and are therefore endogenous.My method (again like LLV) is to price assets on the basis of a joint probability

distribution of payo¤s, under an assumed utility function, and then work backwardsto what those asset prices imply about the cost of capital. The "cost of capital" isunderstood, therefore, as an ex ante expected return or discount rate. Speci�cally,if �rm j has expected terminal cash payo¤ E[Vj], then its ex ante expected returnimplied by its initial market price Pj is E[rj] = E[Vj]=Pj � 1. My equations showreturns as factors Rj � (1 + rj) = Vj=Pj, rather than as increments rj.The technical assumptions of my model are as follows:

(i) The market contains n risky �rms (j = 1; 2; :::; n.). Each �rm j has uncertainpayo¤ Vj, realized at time t = 1.

(ii) The stochastic mechanism by which (V1; :::; Vn) is generated is stationary (i.e.all �rm investment choices are set).

(iii) Asset payo¤s have a joint probability distribution f(V1; V2; :::j�) as at t = 0.

(iv) Investors are Bayesian and have the same information, and the same beliefs,f(V1; V2; :::j�).1

1Fama and Miller (1972, p.287) wrote: "It may be helpful to think of the homogeneous expec-tations assumption as a way of concentrating on the pure e¤ects of �objective�uncertainty on thepricing of investment assets."

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(v) Investors are risk averse and have increasing quadratic utility.

(vi) The quality and arrival of information are �xed exogenously (i.e., �nancial re-porting standards are mandatory rather than voluntary).

I assume quadratic utility for two main reasons. First, quadratic utility allows thejoint payo¤distribution f(V1; V2; :::j�) to take an arbitrary continuous or discrete form,which is not only realistic but also opens up the broad question of how informationcan a¤ect beliefs f(V1; V2; :::j�), without the limitation of a speci�c parametric familyof belief distributions or likelihood functions.Second, as long as investor wealth remains within the domain for which utility is

increasing, quadratic utility is consistent with risk-averse mean-variance preferences.2

Later, in a numerical example, I show how the e¤ects of information on the cost ofcapital match up under two mean-variance setups, �rst under quadratic utility, andthen under the LLV pairing of exponential utility and joint normality.Stock market returns can be skew, fat-tailed, or depart from joint normality in

other ways. The payo¤s arising from a �rm�s activities are likely not joint normal,even if they have normally distributed components. For example, if the payo¤ from aproject or division has a normal distribution with parameters determined randomly,the probability distribution of that payo¤ is a mixture of normals, and mixture dis-tributions of normals are not normal.3

My analysis is ex post in the sense that it concerns information that has al-ready been seen by the market. The assumption is that exogenous information arisesat time t = 0, and is absorbed instantly into the market�s probability distributionf(V1; V2; :::j�) of time t = 1 payo¤s. This brings updated expectations and new assetprices at time t = 0, which together imply new expected rates of return. These arebest described following Gao (2010, p.9) as "conditional" returns, meaning that theyare expectations conditioned on the signal that has just been observed.4

It is important to distinguish between information already existing and an oblig-ation by which the �rm will disclose a certain quality of information at future times.Both my analysis and LLV examine how signals realized at t = 0 a¤ect the market�sconditional "forward looking" expected return on capital, as at t = 0, conditional onthe information observed at t = 0. Discussion on how the conditional and uncondi-tional (pre-signal) perspectives relate to one another is added in Section 8.

2Quadratic utility is often set aside, mainly because it has a point of satiation where utility startsto decrease in money, and because it is increasingly risk averse. It should be noted however thatLevy and Markowitz (1979), Markowitz (1991), Levy (2012) and others have supported portfoliochoice based on the approximate validity of risk-averse quadratic utility.

3Baron (1977, p.1692) and Liu (2004, p.233) discuss how probability mixtures of di¤erent assetsor payo¤s occur in business ventures. See also Winkler (1973, p.399).

4Beyer et al. (2010, p.308) and Gao (2010, p.21) point out that Lambert et al. (2007) is concernedwith conditional (post-signal) expected returns.

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3 Quadratic Utility CAPM

In this section, I set out an original derivation of the mean-variance CAPM, as anextension of expected quadratic utility. This derivation requires no assumption aboutthe shape or parametric family of the payo¤s distribution of the �rm or any other�rm in the market. Importantly, it allows the endogenous market risk premium to becalculated (from the utility function) rather than merely assumed. This form of theCAPM is useful for its generality and for numerical illustration of the paper�s main�ndings. Numerical examples (see Section 7) help to explain how incremental infor-mation concerning one asset a¤ects its cost of capital and also a¤ects the endogenousmarket price-weighted-average cost of capital, E[RM ].The utility function for wealth W is U(W ) =W � b

2W 2 (b > 0). Marginal utility

U 0(W ) = 1 � bW is positive for W < 1=b (this is the domain in which utility isincreasing). The investor endowed with initial wealth W maximizes expected utility,

E

�W�P

j wjRj + (1�P

j wj)Rf�� b

2

�W�P

j wjRj + (1�P

j wj)Rf��2�

;

by selecting asset weights wj = w�j , subject to asset prices Pj, where Rj = Vj=Pjis the return on asset j and Rf is the risk-free return (

Pj is shorthand for

Pnj=1

throughout). Note that Rf is a known constant and asset returns are all expressedas factors.Returns on risky assets j = 1; 2; :::; n depend on their equilibrium prices Pj. The

expected return on asset j is by de�nition E[Vj]=Pj. Importantly, Pj is a constant,set by the market, and the point of the CAPM is to �nd the theoretical value of Pj(under a given utility function and wealth endowment W ).Di¤erentiating expected utility with respect to weight wj gives the �rst order

conditionE [W (Rj �Rf )(1� bWRopt)] = 0; (1)

whereRopt =P

j wjRj+(1�P

j wj)Rf is the weighted average return on the investor�soptimal overall portfolio including both risky and risk-free assets.Using the identity cov(Rj; Ropt) = E[RjRopt] � E[Rj]E[Ropt], the �rst order con-

dition (1) simpli�es to pricing equation

E [Rj] = Rf +bW cov(Rj; Ropt)1� bWE [Ropt]

: (2)

In total, the investor allocates all her endowment W , that is Popt = W . Theexpected return on the optimal portfolio of risky and risk-free assets is thus

E[Ropt] =E[Vopt]

Popt=

Pj E[Vj] + (W �

Pj Pj)Rf

W: (3)

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Substituting (3) into (2), and remembering that Pj = E[Vj]=E[Rj], the mean-variance CAPM appears in the form of an unusual but insightful equation

Pj =1

Rf

�E[Vj]�

b cov(Vj; Vopt)1� bE[Vopt]

�(4)

=1

Rf

0@E[Vj]� b cov(Vj;P

j Vj)

1� bnP

j E[Vj] + (W �P

j Pj)Rf

o1A : (5)

Note that cov(Vj; Vopt) =cov(Vj;P

j Vj + (W �P

j Pj)Rf ) =cov(Vj;P

j Vj), since therisk-free component (W �

Pj Pj)Rf of Vopt is a constant.

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4 Which Parameters Drive Cost of Capital?

A prima facie interpretation of (2) suggests that the single �rm parameter a¤ectingits cost of capital is its returns covariance, cov(Rj; Ropt), or equivalently its returns"beta", cov(Rj; RM)=var(RM), where RM = VM=PM =

Pj Vj=

Pj Pj is the return on

the market portfolio. The problem with this rule of thumb is that it does not accountfor the logical circularity that (ingeniously) constitutes the CAPM equilibrium pricingmechanism. Speci�cally, the expected return E[Rj] depends on the returns covariancecov(Rj; RM), but Rj = E[Vj]=Pj is a function of Pj, and Pj itself depends on E[Rj].The full picture of what primitive factors a¤ect the �rm�s mean-variance CAPM-implied cost of capital is thus not super�cially evident in equation (2).

Proposition 1 Under a mean-variance CAPM, assuming an increasing risk-aversequadratic utility function, the �rm�s cost of capital E[Rj] is decreasing (increasing) inits expected payo¤ E[Vj] given positive (negative) payo¤ covariance with the marketpayo¤.

By writing E[Rj] = E[Vj]=Pj and substituting the value of Pj from (4), theexpected return is seen as a function of both the payo¤ covariance cov(Vj;

Pj Vj) and

the payo¤ mean E[Vj]

E[Rj] =E[Vj]Rf

E[Vj]� bcov(Vj;P

j Vj)=(1� bE[Vopt]): (6)

Equation (6) reveals that the cost of capital for �rm j, E[Rj], decreases (increases)with the expected payo¤ E[Vj] whenever cov(Vj;

Pj Vj) is positive (negative) This

assumes utility increasing in money, implying positive marginal utility (1� bE[Vopt]),and risk aversion, b > 0. Note that marginal utility (1 � bE[Vopt]) can be taken asconstant, provided that Vj is small relative to VM =

Pj Vj.

5For equivalent CAPM equations derived under power and log utility functions, see Satchell(2012) and Johnstone (2012).

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This �nding about the e¤ect of E[Vj] on E[Rj] matches the Proposition 1c provedby LLV (p.392), under an exponential-utility mean-variance CAPM and joint normalpayo¤s (see also the con�rmation of this result in Christensen et al. 2010).The �nding that the mean payo¤ E[Vj] helps determine the cost of capital is

fundamental to accounting standard setting. The value of the �rm can be viewedas an expected payo¤ discounted by a "risk-adjusted" discount factor, that is Pj =E[Vj]=E[Rj]. While information about the expected or mean payo¤E[Vj] is obviouslyrelevant to users�assessments of the numerator in this formula, the much less obviouspoint is that the mean also plays a large part in determining the denominator E[Rj](i.e. the cost of capital or equity discount factor).LLV not only prove this e¤ect, they give a highly agreeable intuitive explanation

for its existence. The explanation is that an increased mean without any increase incovariance amounts to adding a positive constant or risk-free element to the existingexpected payo¤, thus leading to a new "average" cost of capital for the �rm, closerto the risk-free rate. In the limit, as the risk-free component of the �rm�s cash �owincreases and approaches its mean, the cost of capital converges on the risk-free rate.

5 Direct E¤ects of Information

The purpose of this analysis is to expand on LLV by examining how, in general,estimates of the �rm�s payo¤ covariance or payo¤ mean can change with the arrivalof information, and how these changes, individually or simultaneously, can alter thecost of capital imposed on the �rm by the market. The �rm�s activities are assumed�xed, so there are no indirect e¤ects in the LLV (p.404) sense.6

5.1 Covariance E¤ects

The �rst general rule is that information does not always lead to greater certainty, ineither a variance or covariance sense, even when we assume stationarity.

Proposition 2 The variance of payo¤ Vj conditional on signal S, var(VjjS), can behigher than the unconditional or prior variance, var(Vj).

The precept that information resolves uncertainty is almost lawlike in economicsbut is not always operational. The completely general (model-free) probability ruleconcerning information and variance is the law of conditional variance,

var(Vj) = E [var (VjjS)] + var (E [VjjS]) :6LLV de�ned indirect e¤ects as those brought by the �rm reacting to its own reported information

by altering its business activities (see Beyer et al. 2010, p.308 on the LLV distinction between directand indirect e¤ects of information). See Gao (2010) for an equilibrium model that links the �rm�sreported information to its own investment choices.

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This statistical identity implies that the average posterior variance of uncertain Vjunder signal S is lower than the prior variance, that is E [var (VjjS)] < var(Vj), butnot that the posterior variance is necessarily lower (Gelman et al. 2004, p.37).7

Newly available information might directly concern Vj and yet add substantiallyto market uncertainty about Vj.8 Suppose that Vj is driven by some causal factor orparameter F , and that S is Bayesianly indicative of F , via the likelihood functionf(SjF ). The Bayesian predictive distribution of Vj is thenR

Fp(VjjF )p(F jS)dF;

assuming that Vj is independent of S when given F . Hence, any observation S thatshifts the probability distribution of F towards states of F under which Vj has highvariance, must generally add to the variance of Vj, and conversely.9

Take the hypothetical case of oil exploration. Suppose that credible drill sitesare in nature of two types, A and B, where testing is required to identify the typeof a given site. In A-type sites the known relative frequency of oil is 0:5 and in B-type sites this frequency is 0:95. The population average frequency of oil is therefore�(0:5) + (1� �)0:95, where � is the proportion of A-type sites. If from experience ortheory the geologist has � = 0:7, then his prior probability of oil in a "random" siteis 0:635. Now suppose that a technically ideal geological test reveals with certaintythe type of site he is looking at. It follows that in 70% of such tests, the geologist�sprobability of oil will fall from a prior of 0.635 to a posterior of 0.5, thus leavinggreater (maximum possible) variance, and thus uncertainty, post-test.It is reassuring nonetheless that, by corollary of the law of conditional variance,

information S produces "on "average" a reduction in variance. This is consistent withthe theoretical property of Bayes that enough data concerning Vj will "in the longrun" make Vj certain (assuming a stationary process).10

7It is well known that within the univariate Bayesian model adopted by LLV that any extra dataleads to a lower variance posterior distribution for Vj (e.g. Winkler 2003, p.150). It is apparentlyless well known that this is not true in general within Bayesian inference. Even in the very similarbeta-binomial model, where Vj 2 (0; 1) has a beta prior and beta posterior distribution, it is possible,albeit atypical, for the posterior to have a larger variance than the prior (Winkler 2003, p.141).

8Rogers et al. (2009, p.90) allow earnings forecasts to add to uncertainty: "...certain typesof earnings disclosures, especially those that are unexpected, likely cause investors to revise theirassessments of �rm or manager type, potentially increasing uncertainty about underlying �rm value."

9The archetypal Bayesian analysis of the expected value of imperfect information (EVII) waslaid out by Pratt et al. (1965) and other Bayesian statistical decision theorists in contexts such asoil exploration. These standard calculations, all built around Bayes theorem, show that a geologicalsample can leave the oil company either more or less certain of oil after seeing the realized testresult. See Clemen and Reilly (2001, pp.502-8) and Winkler (2003, pp.296-301) for expositionsof EVII demonstrating how sometimes new information leads to a posterior distribution that is"�atter" (less peaked) than the prior distribution.10Barry and Winkler (1976) hold that data loses its relevance over time in nonstationary worlds

and that uncertainty cannot be eliminated, but might wax and wane as the causal process shiftsand observation runs to catch up.

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Proposition 3 The covariance of asset payo¤ Vj and asset payo¤ Vk conditionalon signal S, cov(Vj; VkjS) can be higher than the unconditional or prior covariance,cov(Vj; Vk).

The probability identity that connects information and the covariance is the lawof conditional covariance, sometimes known as the law of total covariance,

cov (Vj; Vk) = E�cov (Vj; VkjS)

�+ cov

�E [VjjS] ; E [VkjS]

�:

Since the covariance of the conditional expectations cov(E [VjjS] ; E [VkjS]) can benegative while the prior covariance cov(Vj; Vk) is positive, it follows immediately fromthis identity that the expected conditional covariance over the possible signal eventsS, E

�cov(Vj; VkjS)

�, need not be lower than the prior covariance.

Hence, as a general rule, assuming nothing but probability theory, the CAPMmeasure of risk or uncertainty, namely cov(Vj; Vkj�), need not shift towards zero uponreceipt of imperfect information S. Moreover, prior to observing S, the investor�sexpectation may be that the covariance will increase upon revelation of signal outcomeS. This occurs where the sample space (set of all feasible values) of signal S is knownbefore S is realized, and are such that cov

�E [VjjS] ; E [VkjS]

�is negative.

Intuition and the law of conditional covariance are clearly in agreement. Imaginethat data exhibits a much stronger covariance between two variables than the priorcovariance. Data like this must logically have potential (under some formal Bayesianmodels at least) to bring an increase in the observer�s subjective assessment of thecovariance. Similarly, new information can always reveal that two variables have astronger common driver than was previously understood, and hence rational models ofstatistical inference must sometimes produce a higher subjective posterior covariancebetween two random variables.The same of course goes for the variance, where data or fundamental analysis must

have the facility under Bayesian probability to indicate that the quantity in questionis more variable or volatile than was previously appreciated. This will not surpriseportfolio managers whose subjective estimates of �rm beta or returns covariance oftenincrease with new information, rather than approaching zero.

5.2 Mean E¤ects

LLV (p.392) prove that according to the CAPM applied to typical stocks (with pos-itive covariance with the market), information prompting a lower assessment of theexpected payo¤ E[Vjj�] has an upward e¤ect on the required return (i.e. there isan inverse mathematical relationship between expected future payo¤ and CAPM re-quired return). Thus, if new information increases the estimated mean E[Vjj�] butleaves the covariance cov(Vj; VM j�) unchanged, the revised (conditional) CAPM pricePjj� = E[Vjj�]=E[Rjj�] of the �rm is higher for two reasons: (i) E[Vjj�] is higher, and(ii) the �rm�s cost of capital E[Rjj�] is lower.

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5.3 Relative E¤ects of E[Vjj�] and cov(Vj; VM j�)In �nance it is generally established that portfolio optimization is more sensitive tothe estimated mean return than the estimated returns variance or covariance (e.g.Best and Grauer 1991). For this reason, Chopra and Ziemba (1993) argue that thebulk of e¤ort should be spent on estimating mean returns rather than covariances.11

The relative sensitivity of optimal portfolio weights to the mean return ratherthan the covariance suggests that perhaps the cost of capital is similarly sensitive tothe mean payo¤. Partial derivatives of the quadratic utility CAPM cost of capital (6)support this suspicion. To see this, �rst de�ne

pcov(Vj; VM) as the square root of

the payo¤ covariance, thereby constructing a variable with dimension dollars ratherthan dollars-squared. Now examine the respective derivatives

�E[Rj]

�E[Vj]=

�bcov(Vj; VM)(1� bE[Vopt])Rf(E[Vj]� b(cov(Vj; VM) + E[Vj]E[Vopt]))2

; and

�E[Rj]

�pcov(Vj; VM)

=2bpcov(Vj; VM)E[Vj](1� bE[Vopt])Rf

(E[Vj]� b(cov(Vj; VM) + E[Vj]E[Vopt]))2:

It can be seen from these derivatives that the ratio of the change in cost of capitalbrought by a (small) $� increase in E[Vj] versus the change brought by a $� increaseinpcov(Vj; VM) is equal to

�qcov(Vj; VM)

.2E[Vj];

and is independent of the risk aversion parameter b.It appears that for �rms with material positive payo¤correlation with the market,

this ratio will commonly be very large (i.e. large negative), indicating that the meanpayo¤ is a signi�cantly stronger force than the payo¤ covariance. Given su¢ cientpositive correlation between the stock and market payo¤s, and a su¢ ciently largemarket payo¤ variance E

�(VM � V M)2

�, the square root of the payo¤ covariance,q

E�(Vj � V j)(VM � V M)

�, will tend to be a very large quantity in absolute terms

relative to the expected payo¤ E[Vj] of a single stock.12

This conclusion rests on how strongly positively correlated stocks are in generaland how large is the market payo¤ variance and the stock�s payo¤ variance. Ul-timately, therefore, the relative sensitivity of the cost of capital to the mean is anempirical property of any given market, since it depends on whether, for typicalstocks, the cov(Vj; VM) is large relative to the mean E[Vj]. Note for instance thatfrom the �rst of the partial derivatives above, the cost of capital is highly insensitive

11Verrecchia (2001, p.174) makes general comments regarding the usual dominance of �rst-momentover second-moment e¤ects.12Thinking empirically, with positive correlation the average value of the period-by-period (Vj �

V j)(VM � VM ) will be generally be very large when (VM � VM ) is commonly large.

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to the mean payo¤ when the payo¤ covariance itself is low (i.e the stock and markethave near zero correlation). That is sensible, since from (6) the cost of capital is aconstant (the risk-free rate) when cov(Vj; VM) = 0, regardless of E[Vj].

6 Lambert et al. (2007)

The results proved by LLV have been widely misinterpreted. In this section I makenote of how the literature has understood LLV and compare LLV with my moregeneral results. The main point is that the Bayesian conclusions found by LLV, iftaken as universals rather than as artifacts of just one subclass of Bayesian probabilitymodels based on narrow assumptions, are unjusti�ed.

6.1 Interpretations of Lambert et al. (2007)

LLV has been extremely in�uential, and is taken as theoretical grounds for believingthat more information leads economically to a lower cost of capital. The followingpassages show succinctly how LLV is widely interpreted in accounting research:

...in Lambert et al.�s (2007) model, accounting information is not anindependent risk factor, but does a¤ect the cost of capital by in�uencinginvestors�assessments of the covariance of �rm cash �ows with those ofthe market, and thereby a¤ecting beta. (Core et al. 2008, p.21)

Lambert et al. (2007) examine whether and how public accounting re-ports and disclosures a¤ect a �rm�s cost of equity capital in the presenceof diversi�cation. Using a model that is consistent with the CAPM andallows for multiple securities whose cash �ows are correlated, they demon-strate that the quality of accounting information ...has a direct e¤ect onthe assessed covariance of a �rm�s cash �ows with other �rms�cash �ows,suggesting that earnings quality can a¤ect the cost of capital via a �rm�sbeta. (Beyer et al. 2010, p.308)

Lambert et al. (2007, 2009) suggest that �rms with more preciseinformation about future cash �ows have lower conditional covarianceswith the market, and, as a consequence, lower conditional betas and lowexpected returns. (Ogneva 2012, p.1418)

Similarly, see Leuz and Wysocki (2008, p.9) and Leuz and Schrand (2011, p.7)and many similar passages in the empirical research literature.Contrary to the common understanding of LLV shown in these quotes, Gao (2010)

observes correctly that although it is taken in the literature that more or betterdisclosure reduces the �rm�s cost of capital, by reducing its payo¤ covariance, thefact is that LLV show that more or better disclosure can increase the cost of capital

11

via its e¤ect on the estimated mean payo¤. Gao concludes by noting that LLV donot carry this point further in relation to the question at issue:

A common theme in the previous literature is that disclosure reducesthe cost of capital by reducing the conditional variance (or covariance) ofthe �rm�s future cash �ow... One exception is Lambert et al. (2007)...They point out that cost of capital may increase with disclosure qualityif disclosure changes both the mean and variance of the �rm�s cash �ow.However, they do not link this result directly to disclosure quality. (Gao2010, p.20)

It is apparent from the clash between Gao�s reading of LLV and the more universalview, expressed in those several quotes above, that the implications of LLV are subjectto some confusion. In what follows I summarize LLV and explain how this confusionhas arisen.

6.2 Summary of Lambert et al. (2007)

The asset pricing model derived by LLV assumes a Bayesian joint normal posteriordistribution f(V1; V2; :::; Vnj�) of �rm payo¤s Vj, where � is the available information,along with an exponential utility function. The resulting asset pricing model (LLVpp.413-4) is of the form

Pj =1

Rf

�E[Vjj�]�

1

�cov(Vj;

Pj Vjj�)

�; (7)

where � > 0 speci�es the market�s risk tolerance under the assumed exponential

utility function U(W ) = ��1� exp[� 1

�W ]�, 1�=

PjE(Vj j�)�Rf

PjPj

var(P

jVj j�)

is the market

price of risk in equilibrium, and Rf is the risk-free return factor.13 Using their CAPMequation (7), the LLV �ndings are that the cost of capital E[Rjj�] is (i) increasingin the covariance between asset j and "the market" (the sum of all asset values),cov(Vj;

Pj Vjj�), and (ii) decreasing (increasing) in the expected asset value E[Vjj�]

when cov(Vj;P

j Vjj�) is positive (negative). Of these, �nding (i) is extremely wellknown, and (ii) is very interesting and rarely if ever raised (see Johnstone 2014). Thesame �ndings hold under my quadratic utility CAPM, regardless of the asset payo¤distribution, as shown above and conjectured by LLV (p.393).At this point LLV (p.393) note correctly that simultaneous changes in E[Vjj�],

and cov(Vj;P

j Vjj�), driven by some change in information �, are not likely tobe "proportionate" in the sense that their separate e¤ects cancel out, such as toleave the cost of capital unchanged. The implication, therefore, is that changes in

13Note that � = N� where in LLV�s proof there are N traders in the market each with CARAutility U(W ) = �

�1� exp[� 1

�W ]�and individual risk tolerance � > 0.

12

extant information � can lead to re-assessments of E[Vjj�] and cov(Vj;P

j Vjj�) thatjointly drive the CAPM cost of capital either up or down. LLV do not develop thisbroader argument. Instead they target the e¤ect of information on the covariance,cov(Vj;

Pj Vjj�), and conclude that more information leads necessarily to a lower

(i.e. nearer zero) assessment of the �rm�s covariance with the market, and thereforea lower cost of capital.

. . . ..we show that higher quality accounting information and �nancial dis-closures a¤ect the assessed covariances with other �rms, and this e¤ect unam-biguously moves a �rm�s cost of capital closer to the risk-free rate. (Lambertet al. 2007, p.387)

6.2.1 Information and the Variance

The LLV model of information is an elementary Bayesian standard (e.g. Winkler2003, pp.149-150). Uncertain quantity Vj, representing �rm payo¤, has an assumednormal prior distribution Vj � N(�0; �0). A random sample of m iid observations(x1; x2; :::; xm) is drawn from a normal distribution N(Vj; �x), where �x is taken asa known constant. Each draw xj is thus an unbiased estimate of Vj with knownprecision 1=�2x. The posterior belief distribution f(VjjS) of payo¤ Vj is N(�1; �1),where

�1 =(1=�20)�0 + (m=�

2x)x

(1=�20) + (m=�2x)

and1

�21=1

�20+m

�2x:

It follows that no matter what the sample evidence, or how small the sample, theposterior variance of uncertain future payo¤ Vj is always less than its prior variance.That is, �21 < �20 for any m > 0. LLV (p.395) hold that this one Bayesian model"formalizes the notion that accounting information and disclosure reduce the assessedvariance of the �rm�s end-of-period cash �ow". By that way of thinking, certaintycan be accumulated monotonically - merely by generating more data, contrary to thegeneral Bayesian law of conditional variance.

6.2.2 Information and the Covariance

LLV consider the e¤ect of information concerning Vj on the investor�s assessment ofthe covariance, cov(Vj;

Pj Vjj�), separately from its e¤ect on the variance var(Vjj�).

Ultimately, their aim is to show that like the assessed variance, the assessed covarianceapproaches zero whenever new information is disclosed.One way to test this proposition would have been to extend their univariate

Bayesian model of inference about Vj to the matching case of multivariate normalf(V1; V2; :::; Vnj:). Under this model, the unknown payo¤ vector V = (V1; V2; :::; Vn)has a conjugate multivariate normal prior distribution V � N(�0;�0), where �0 isthe vector of prior means and �0 is the prior covariance matrix. The data or infor-mation about V might then be conceived as a random sample of m iid observations

13

f(x1; x2; :::; xn)1; :::; (x1; x2; :::; xn)mg, where each observation is drawn from a multi-variate normal distribution N(V ;�x) with some known covariance matrix �x. As inthe univariate case, each sample observation (x1; x2; :::; xn)i is an unbiased estimateof the parameter vector (V1; V2; :::; Vn), with �xed and known precision �x.By a standard result, the Bayesian posterior distribution of V is then multivariate

normal V � N(�1;�1) with

�1 =(1=�0)�0 + (m=�x)x

(1=�0) + (m=�x)and

1

�1

=1

�0+m

�x;

where x = (x1; x2; :::; xn) is the vector of observed sample means. Note that in thismodel the posterior covariance, like the posterior variance above, is in�uenced onlyby the precision of the sample observation, not by its actual realized value, x.14

The main implications of this model from the LLV standpoint are (i) the posteriorvariance var(Vjjx) is always less than the corresponding prior variance var(Vj), asin the univariate model, and (ii) given that the prior covariance cov(Vj; Vk) and(known) sampling covariance cov(xj; xk) are of the same sign, the posterior covariancecov(Vj; Vkj�) is closer to zero than the corresponding prior covariance, for all k 6= j.Note that in regard to (ii), if the prior and sampling covariances cov(Vj; Vk) andcov(xj; xk) are of opposite signs, then the posterior covariance cov(Vj; Vkj�) can befurther from zero than the prior covariance cov(Vj; Vk).Rather than continuing via a multivariate Bayesian model, LLV set out a model

that is not explicitly Bayesian but which leads to a parallel result. The LLV modelbegins by presuming a joint normal probability distribution f(Vj; Vk; Zj; Zk), whereVj and Vk are the uncertain payo¤s of assets j and k respectively, and Zj = Vj + �iand Zk = Vk+ �k are unbiased noisy measurements of Vj and Vk. These estimates arecharacterized by normally distributed errors �j and �k (independent of Vj and Vk) with�xed variances and covariance, var(�j), var(�k), and cov(�j; �k). Taking advantage ofthe covariance properties of the joint normal distribution, LLV show that

cov(Vj; VkjZj) = cov(Vj; Vk)var(�j)

var(Vj) + var(�k):

On this basis, LLV conclude that the covariance of asset j with asset k, conditionalupon observation of signal Zj, is always closer to zero than the unconditioned (prior)covariance of assets j and k. This e¤ect, which is stronger the smaller the observationerror variance var(�j), holds for asset j with respect to all other single assets k, andhence the covariance of asset j with the market sum of all assets,P

j cov(Vj; VkjZj) + var(VjjZj);

shifts toward zero, not only because the conditional variance var(VjjZj) goes towardzero, but also because each of the covariance terms is impacted the same way.14This is a recognized convenience of such models (Christensen and Feltham 2003, p.78; Verrecchia

2001, p.174, Zhang 2013). See caveats in Christensen et al. (2010, footnotes 3 and 16).

14

At this point in the LLV analysis, any discrete observation Zj leads to a lower(nearer zero) covariance for asset j, that is jcov(Vj; VkjZj)j < jcov(Vj; Vk)j for allk. But this conclusion does not remain in the more general case of a multivariateobservation, such as (Zj; Zk). LLV recognize, and show by a longer proof (pp.416-7), that within their model the joint signal (Zj; Zk) leads necessarily to a reducedposterior covariance between Vj and Vk only when the covariances of cash �ows andmeasurement errors, cov(Vj; Vk) and cov(�j; �k), are of the same sign. When this is notthe case, LLV (p.402) �nd that it is di¢ cult to "sign" the e¤ect of information (Zj; Zk)on the covariance between Vj and Vk (i.e. the covariance can increase or decrease).This latter concession is an internal weak spot in the LLV argument, particularlygiven that signals about di¤erent individual �rms rarely exist in isolation.The essence of LLV�s argument is captured by their second and �nal proposition

(Proposition 2, p.399) which holds that cov(Vj; VkjZj) is closer to zero than cov(Vj; Vk)for any new information Zj (and all k). This proposition is taken as grounds toconclude that all new information leads to a covariance closer to zero, and thus ceterisparibus to a lower cost of capital for �rms with positive covariance with the market.15

As in the case of the variance, this conclusion is not consistent with the more generallaw of conditional covariance. It has the implication that any accounting informationthat leaves the estimated payo¤mean higher or unchanged must lead to a lower �rmbeta.

7 Numerical Illustration

The following example serves several purposes. First it exhibits the consistency be-tween my quadratic utility CAPM and the common textbook mean-variance CAPM.Second, it involves multivariate normal payo¤s, so that mean-variance asset pricesand returns can also be calculated under the LLV exponential-utility CAPM. Third,and most importantly, it gives some intuitive insight into how information about �rmfundamentals alters both �rm and market costs of capital.In the LLV set up, information is modelled as if the �rm�s unknown future business

payo¤ Vj can be randomly sampled, like draws from an urn. This is intended as anabstract depiction of information rather than as an attempt to model the causationof �rm payo¤ Vj. An alternative approach, designed to resemble an economic model,is set out below. The point of this analysis is to demonstrate how under a simplecausal model, incremental information about fundamentals can drive the �rm�s costof capital either up or down, and can have the same e¤ect on the market risk premiumas a whole.For simplicity, imagine a market o¤ering just two risky assets, with uncertain

payo¤s, V1 and V2 respectively. From (5) the general equations for the prices under

15The fact that this relationship no longer holds necessarily within the LLV model once there aretwo or more concurrent pieces of information (Zj ; Zk; :::), is acknowledged in discussion (pp.402-3).

15

quadratic utility of the two assets are

P1 =1

Rf

�E[V1]�

bcov(V1; V1 + V2)1� b(E[V1 + V2] + (W � P1 � P2)Rf )

�(8)

and

P2 =1

Rf

�E[V2]�

bcov(V2; V1 + V2)1� b(E[V1 + V2] + (W � P1 � P2)Rf )

�(9)

Solving (8) and (9) simultaneously and then eliminating either P1 or P2 leads to aquadratic equation for Pj (j = 1; 2), with two possible solutions

Pj =1

Rf

24E[Vj]� �j;Mn(1� bRfW )�

p(1� bRfW )2 � 4b2(�1M + �2M)

o2b(�1M + �2M)

35 ;where for notational convenience �j;M denotes cov(Vj; V1 + V2). Unfortunately, itis impossible to generalize about which of these solutions is economically sensible,although commonly only

Pj =1

Rf

24E[Vj]� �j;Mn(1� bRfW )�

p(1� bRfW )2 � 4b2(�1M + �2M)

o2b(�1M + �2M)

35lies in the sensible range between 0 and E[Vj].Suppose that underlying V1 and V2 there are two causal factors, X � N(1; 1) and

Y � N(1; 1). These are independent normal random variables with unit means andvariances. Now imagine that the risky asset payo¤s are known to be V1 = cX + Y ,and V2 = X + dY respectively, where c and d are real constants. The quantitiesV1 and V2 are therefore bivariate normal, and their parameters are E[V1] = c + 1,E[V2] = d + 1, var(V1) = c2 + 1, var(V2) = d2 + 1, and cov(V1; V2) = c + d. Itfollows then that cov(V1; VM) =var(V1)+cov(V1; V2) = c2 + 1 + c + d, and similarlycov(V2; VM) = d2 + 1 + c+ d.We now plug these values into equations (8) and (9). For example, letting c = 1,

d = 3, W = 10; Rf = 1:10 and b = 0:025, the resulting prices are P1 = 1:6254and P2 = 3:1865, implying expected return factors E[R1] = E[V1]=P1 = 2=1:6254 =1:2305 and E[R2] = E[V2]=P2 = 4=3:1865 = 1:2553. The resulting endogenous returnon the "market portfolio" of risky assets is the price-weighted average of these twoindividual expectations, or equivalently E[V1 + V2]=(P1 + P2) = 1:2469 (i.e. 24.69%).It is reassuring to test these numerical results for consistency with the usual form

of the CAPM as seen in most textbooks. Importantly, note that it is not possible tocalculateRM from the usual textbook CAPM. However, having obtainedRM under anassumed quadratic utility function, it is possible to check all results for consistencywith the usual CAPM equation. Relevant calculations are as follows. First, the

16

variance of the market return is �2M =var(VM)=P 2M = 0:8638, where VM = V1 + V2,PM = P1 + P2 and var(VM) =cov(V1; VM)+cov(V2; VM) = c2 + d2 + 2(c+ d+ 1). The"returns beta" of asset 1 is �1 =cov(R1; RM

)=�2M = cov(V1 ; VM )=(P1PM�2M) = 0:8881.

Similarly, �2 = cov(V2; VM )=(P2PM�2M) = 1:05706.

Hence, the price-weighted average returns beta equals one, and, by the usual formof CAPM, E[Rj] = Rf + �j(E[RM ] � Rf ), we have E[R1] = 1:10 + 0:8881(1:2469 �1:10) = 1:2305, and similarly E[R1] = 1:10 + 1:05706(1:2469� 1:10) = 1:2553: Thesereturn factors match those calculated above.To observe the possible e¤ects of information regarding the parameters c and d,

consider the results plotted in Figure 1. This �gure shows the required rates of returnon each of the two assets, and on the market sum of these assets, all as functions ofparameter c (holding d = 3 constant).

Figure 1Plot of Expected Return Factors of Individual Assetsand the Market as Functions of c (with 0 6 c 6 5)Under Quadratic Utility U(W ) =W � b

2W 2

(with W = 10, b = 0:025; d = 3 and Rf = 1:10)

These results are merely for example, but are su¢ cient to show how new infor-mation about the unknown c can shift the cost of capital of an individual asset, or ofthe market as a whole, up or down. Underlying these changes in the cost of capital,information about c in�uences estimates of both asset mean E[V1] and covariancecov(V1; VM).Figure 2 is a repeat of Figure 1 except that it is based on the LLV exponential

utility function (see above) rather than my assumed quadratic utility function. Thetwo markets exhibit di¤erent expected returns for the same value of c, due to theirdi¤erent utility functions. The asset prices and implicit expected returns found usingLLV�s pricing equation (7), derived under exponential utility and joint normal payo¤s,were found to be consistent with the usual textbook CAPM. As above, this cross-check involves taking the endogenous market return calculated using the LLV model,

17

plugging this into the usual CAPM, and showing that the resulting CAPM individualasset returns are the same as those produced endogenously by the LLV model.

Figure 2Plot of Expected Return Factors of Individual Assetsand the Market as Functions of c (with 0 6 c 6 5)

Under Exponential Utility U(W ) = ��1� exp

�� 1�

�W�

(with � = 20; d = 3 and Rf = 1:10)

Figure 3Plot of Expected Return Factor for Asset 1

as a Function of its Payo¤ Mean and Payo¤ CovarianceUnder Exponential Utility U(W ) = �

�1� exp

�� 1�

�W�

(with � = 20; d = 3 and Rf = 1:10)

Figure 3 is another plot of Asset 1 expected returns. These are the same returnsas shown in Figure 2, but are plotted as a function of the asset�s mean payo¤ E[V1]and payo¤ covariance cov(V1; VM), rather than of the underlying constant c. Justas in Figure 2, the plotted points are found by holding d = 3 constant and lettingc take a range of possible values (here between 0 and 4). Note that the expectedreturn on Asset 1 increases monotonically with its payo¤ covariance and decreasesmonotonically with its expected payo¤, as per LLV.

18

8 Ex Ante Disclosure Obligations

Christensen et al. (2010) and Gao (2010) emphasize that, for the purpose of evalu-ating accounting standards, it is essential to consider how prices and costs of capitalmight react to both the promise of better information (the ex ante perspective) andto its realization (the ex post perspective). In principle, choice of one informationsystem over another is made on the basis of its ex ante expected utility, which is aweighted average of the conditional expected utilities of the user�s best actions (in-vestment portfolios) under each of the possible signal realizations (weighted by theprior probabilities of each possible observation).16 Ex ante evaluation of an infor-mation system depends therefore on �rst understanding how beliefs and investmentchoices will be altered upon each possible signal outcome. The ex ante (pre-signal)and ex post (post-signal) perspectives are thus inseparable.Consider speci�cally the probability relationship between future signal S and cur-

rent assessments of the CAPM parameters of the �rm. By the law of iterated ex-pectations, E

�E[VjjS]

�= E[Vj]. This means that although it is understood that the

conditional expectation E[VjjS] can di¤er greatly from the current expectation E[Vj],and that the calculation of E[VjjS] hinges intrinsically on the error properties of S,the current (ex ante) expected value of E[VjjS] is E[Vj], and is una¤ected by the errorproperties (likelihood function) of future signal S.17 The less obvious but equivalentpoint is that the current unconditional (ex ante) covariance assessment, cov(Vj; VM),is also una¤ected by the attributes of the future signal, notwithstanding that theexpected conditional (on S) covariance does not equal the unconditional covarianceand is a¤ected by properties of S. Demonstration of this point is provided in theAppendix.

8.1 Christensen et al. (2010)

Christensen et al. (2010) consider both ex ante and ex post e¤ects of exogenousinformation by assuming a two period model in which the uncertain payo¤ d from asingle-asset market is paid at t = 2 and an unbiased signal y concerning d occurs atmidpoint t = 1. As in LLV, the signal and the uncertain asset payo¤ are assumedjoint normal, hence (contrary to my more general model) all new information reducesinvestor uncertainty about payo¤ d. Findings, in the case of homogeneous beliefs andpublic signal y can be reconstructed and summarized as follows.First, the precision of the t = 1 signal makes no di¤erence to beliefs formed at t =

0, consistent with Bayesian logic, as explained above. More speci�cally, introducingsubscripts to denote time t and taking the risk-free interest rate as zero, the price P016In Bayesian theory, this is sometimes called preposterior analysis (O�Hagen 1994, p.84; Winkler

2003, p.267), and is the framework for choice between competing experiments.17Put succinctly, today�s expectation of tomorrow�s expectation equals today�s expectation.

19

of the asset at t = 0 isP0 = E0[d]� c var0(d); (10)

where c is a positive constant capturing the level of market risk aversion. Neither theexpected payo¤ E0[d] nor the variance var0(d) is a¤ected by the (known) precisionof the forthcoming t = 1 signal, as explained above.Second, the expected return at t = 0 for the period t = 0 to t = 2, written as a

factor, isE0[d]

P0=

E0[d]

E0[d]� c var0[d];

recon�rming, as in LLV and Gao (2010, p.20), that the forward-looking cost of capitalis decreasing in mean payo¤ E0[d].Third, although the return expected at t = 0 over the two-period life of the asset

is not altered by anticipation of a signal of given quality at t = 1, the time intervalsover which that return is expected to accumulate are a¤ected. A good way to showthis is to consider P0 in terms of the t = 1 payo¤, P1

P0 = E0[P1]� c var0(P1); (11)

where P1 = E1[d]� c var1(d). From there

var0(P1) = var0�E1[d]� c var1(d)

�= var0(E1[d]);

since in this model var1(d) is a known constant at t = 0. Hence, the return expectedat t = 0 over period t = 0 to t = 1 becomes

E0[P1]

P0=

E0�E1[d]� c var1(d)

�E0[P1]� c var0(P1)

=E0[d]� c var1(d)

E0�E1[d]� c var1(d)

�� c var0

�E1[d]� c var1(d)

�=

E0[d]� c var1(d)E0[d]� c var1(d)� c var0(E1[d])

: (12)

In Bayesian analysis, the term var0(E1[d]) is known as the "preposterior variance",and describes the variance (before observing signal y) of the mean E1[d] � E[djy] ofthe posterior distribution of d conditioned on y. Using results provided by Rai¤a andSchlaifer (1961, p.297), and mirrored by Christensen et al. (2010, pp.826-7),18

var0(E1[d]) = var0(d)� var1(d): (13)

Hence, from (12) the return expected at t = 0 over period t = 0 to t = 1 is

18It follows from (13) that the two expressions (10) and (11) for asset price P0 are equivalent.

20

E0[d]� c var1(d)E0[d]� c var0(d)

: (14)

It follows therefore that if signal y is completely uninformative (var1(d) =var0(d)),the return factor (14) expected over period t = 0 to t = 1 equals one, which is therisk-free rate, implying that all of the return expected at t = 0, over the asset�s two-period life, is expected to accrue in the second period. At the other extreme, wherey reveals d with certainty (var1(d) = 0), all of the lifetime expected return E0[d]=P0is expected in the �rst period, and the return expected in the second period is justthe risk-free rate (merely as reward for waiting). Between these two extremes, thegeneral rule is that a signal y with higher (known) precision brings a higher expectedreturn over the �rst period, and a commensurately lower expected return thereafter.This model suggests that if better accounting (e.g. higher "earnings quality")

adds to the rate at which uncertainty is resolved, higher returns are expected to ariseearlier. It follows on this basis that �rms which commit to better accounting can beexpected to show higher average realized returns, at least in the short term.It is important to remember that the Christensen et al. (2010) model di¤ers from

my model in two ways, due to its assumption of joint normality between payo¤ andsignal. This assumption, in keeping with most models in the noisy rational expec-tations literature, implies that (i) it is known at t = 0 that all new information willresolve some (known) amount of uncertainty (i.e. will reduce the perceived varianceof d, by a known amount), and (ii) the conditional variance, upon receiving the signalat t = 1 is in�uenced only by signal quality (precision), and not by signal realizationy (see my footnote 14).19 A more general Bayesian model should allow the signalarising at t = 1 to sometimes heighten uncertainty about the payo¤ d coming att = 2, in a variance or covariance sense.20 In economic reality, uncertainty about anyfuture payo¤ d can go up or down, depending on what news is received, in which caseexpected future returns rise and fall, and are reallocated in time, as each new signalappears.Paradoxically, it is possible to de�ne signal characteristics such that it is known

before observing the t = 1 signal that whatever the realization of that signal (i.e.no matter what is actually observed at t = 1), the conditional (at t = 1) expectedreturn for the ensuing period between t = 1 to t = 2 can only be higher than theunconditional (at t = 1) expected return for that same period. This can be understoodintuitively once we allow for the instantaneous e¤ect of the t = 1 signal on the assetprice at t = 1. Speci�cally, it can be known in advance that the signal due at t = 1has feasible values that will lead to either a higher price and a lower expected return,or a lower price and a lower expected return, where the new expected return is relativeto the new t = 1 price.21

19The importance of point (ii) is stressed by Gao (2010, pp.10, 12).20Christensen et al. (2010) consider only variance, because theirs is a single asset market.21A proof is available from the author.

21

9 Empirical Considerations

A given �rm has share price Pt at time t and produces mandatory accounting reportsat the end of each �xed time period, t. Let the report or signal arising at time t be St,so Pt is conditioned on St. The question that arises in empirical accounting researchis whether the �rm�s cost of capital over period t, represented by Rt = Pt=Pt�1, isa¤ected by properties of :::St�1, St; St+1; St+2; ::: Assume that Pt�1 is ex-dividend andPt is cum-dividend, as usual in empirical returns.Under rational expectations, any theoretical proposition that empirical returns

can be expected to be decreasing in accounting signal quality requires that ex anteexpected returns are decreasing in signal quality. In the following analysis, I showthat the ex ante expected return over any period t is not necessarily decreasing in thequality of information already received, ...St�2, St�1, nor in the quality of informa-tion yet to be received, St; St+1; St+2::: . For notational simplicity, I treat informationSt�1 as the conjunction of all signals received to date t � 1, and similarly St as theconjunction of all future signals.

(i) At t�1, the end-of-period payo¤ is the stock price Pt. Expected returns Et�1[Rt] =Et�1[Pt]=Pt�1 will be lower when St�1 leads to a higher estimated expected payo¤Et�1[Pt] or a lower estimated covariance between Pt and the market, covt�1(Pt; PM;t).There is no necessary positive or negative association between the quality of St�1

and the direction or magnitude of its e¤ect on Et�1[Pt]. Instead, the signal St�1can bring good or bad news with respect to the expected future price Et�1[Pt], re-gardless of its quality. Similarly, covt�1(Pt; PM;t) can also be higher or lower, nomatter what the quality of signal St�1. For instance, accounting information St�1,whatever its attributes, might alter beliefs about whether earnings are predominatelymarket-dependent or more �rm-speci�c and idiosyncratic, thus changing the assessedcovariance, covt�1(Pt; PM;t), upwards or downwards. It follows therefore that ex-pected returns Et�1[Rt] are not linked in any one direction to the quality of signalSt�1.

(ii) Anticipation of a more informative future signal St can increase or decreasecovt�1(Pt; PM;t). The main point here is that Pt is conditioned on St and can bestrongly in�uenced by St, so its anticipated statistical behavior, speci�cally its as-sessed covariance with PM;t, must be a¤ected by properties of St. As a simple example,if it were known with certainty at t � 1 that Pt is determined only by St, and alsothat St is not correlated with the market PM;t, then covt�1(Pt; PM;t) = 0.To see that the covariance covt�1(Pt; PM;t) can increase with signal quality, start

with the more straightforward proposition that vart�1(Pt) is often higher when theanticipated St is more informative. This says merely that a more informative signal isdisposed to have a bigger e¤ect on stock price. Now imagine that this variation in thefuture stock price Pt is believed at t�1 to be highly correlated with the market price

22

PM;t. Bayesianly, that could occur in various ways, one of which is that the stock ispositively correlated with the market and St has very good error probabilities underany market state. It would then be the case, for example, that a high PM;t and a"positive" signal St gives high probability of high Pt.To see that covt�1(Pt; PM;t) can decrease with signal quality, there are again many

possibilities. One obvious case is where St will provide a strong signal, positive ornegative, about the �rm�s idiosyncratic lines of business. It is then foreseen at t� 1that the �rm�s stock price Pt will be more in�uenced by St than by the market PM;t,thus reducing the assessed covt�1(Pt; PM;t).

It follows from the argument in (i) and (ii) that, in general, expected returnsEt�1[Rt] are not necessarily increasing or decreasing in signal quality, either in regardto signals already received or signals yet to be realized.22 This may help explaintheoretically why empirical work linking accounting quality to stock returns has beensaid repeatedly to o¤er "mixed" results.23

9.1 Signalling by Signal Quality

There is a plausible Bayesian rationale for an empirical connection between signalquality and the cost of capital that has not been mentioned in the literature, and whichis consistent with LLV and my general model. If accounting information quality isviewed as partly voluntary (endogenous) rather than entirely mandatory (exogenous)then adoption of more stringent or transparent accounting policies might conveycon�dence from within the �rm regarding its future cash �ows. If that positive"signal" is accepted by the market as grounds for material upward revision in the�rm�s expected cash �ows, then the relative force of the mean payo¤as a determinantof the �rm�s equity discount rate (under CAPM) may bring about an observablereduction in its cost of capital.Note that this argument departs radically from the conventional view that better

information drives the cost of capital solely via its e¤ect on risk or covariance. It saysthat the voluntary provision of better information can be grounds for a lower cost ofcapital by way of its Bayesian e¤ect on the estimated mean (expected cash �ow).There is a well known proposition that the endogenous quality of earnings reports

(and other �nancial disclosure) is in�uenced by the "true" state of underlying earn-ings and operations. It is suggested for example that there is a generally positivecorrelation between the direction of accounting signals and their quality (i.e. suc-cessful growing �rms tend to produce higher quality earnings measures; e.g. Miller

22That might be why Barth et al. (2013, pp.207, 209, 213) suggest that only information-asymmetry arguments can be relied upon to justify the widely hypothesized link between betteraccounting and lower discount rates.23Also getting in the way of such work is the practical issue that accounting information may

not be su¢ ciently informative, within the supply of all other information, to have a marginal "�rstorder" e¤ect on returns (Zimmerman 2013).

23

2002). The implicit CAPM e¤ect of the estimated mean payo¤ on the equilibriumcost of capital adds further substance to this argument. Any incentive for success-ful or pro�table �rms to volunteer "quality" accounting (e.g. to be conservative inearnings measurement) is reinforced when the favorable signals carried by that levelof disclosure lead to both higher expected future payo¤s and a lower discount rate.Similarly, the converse suggests that �rms for which things have turned negative willbe reluctant to make this any more evident through "good" accounting, since anyreduction in market assessments of their expected future cash �ows can lead to botha smaller numerator and a higher discount rate, implying a twofold reduction in shareprice.

10 Conclusion

The widely held view that better �nancial reporting leads unambiguously to a lowercost of capital is misleading on two grounds. First, information of great precision orBayesian quality can bring increased uncertainty, and hence higher required returns.Second, where incremental information does indeed lead to greater certainty aboutthe �rm�s future cash �ows, that heightened certainty can be relied upon to reducethe cost of capital only when the signal on which it rests is "positive" (i.e. goodnews). For su¢ ciently "negative" signals, the opposite generalization is true. Theresulting (conditional) cost of capital depends therefore not only on the quality of thesignal (a better signal will generally have a more pronounced e¤ect on the investor�sbeliefs) but also on the direction of that signal, or, in other words, on what it "says".Regulators should not assume that better �nancial reporting standards lead me-

chanically to lower costs of capital. The premise that accounting can of itself reducethe cost of capital has led to a �xation on that one factor rather than on the moreoverriding task of valuing the �rm. Bertomeu (2013, p.476) describes this orientationas a "current trend in the literature toward reductio cost of capital in which complexpolicy evaluations have been transformed into whether cost of capital decreases".For valuation purposes, �nancial statements should assist users to assess �rms�

expected payo¤s and payo¤ covariances. The value of the �rm, and its cost of capital,both depend on both parameters. It will be very rare if not impossible to identifytypes or instances of accounting information that are relevant to one of these pa-rameters and not the other. That would require for example a change in the jointdistribution of payo¤s f(V1; V2; :::; Vnj�) that changes the mean E[Vjj�] but does notalter the covariance, cov(Vj; VM j�).24 Remarkably, even this unlikely occurrence willa¤ect both "numerator" and "denominator" in asset valuation, since the discountrate (denominator), or CAPM cost of capital, is in�uenced by both payo¤mean andcovariance (as shown by LLV).

24Note that the payo¤ covariance can be written as cov(Vj ; VM ) � E [VjVM ]�E [Vj ]E [VM ], whichshows how a change in mean payo¤ E[Vj ] might tend to have an e¤ect on cov(Vj ; VM ).

24

Better �nancial reporting is not as a matter of principle about reducing the costof capital. From an investment-under-uncertainty perspective, the role of informationis to facilitate more accurate probability forecasts of future cash �ows. On average,more information leads to more accurate probability forecasts, and more accurateprobability forecasts will sometimes bring a rationally higher cost of capital. A goodway to understand this is to consider how the price per dollar of insurance can some-times increase when the insurer gains better information about the probability oramount of a possible claim.Conversely, a lower cost of capital can be the product of bad information or

less accurate probabilities, leading to over-pricing and, by the principles of economicDarwinism, inevitable economic decline. Accounting has an obvious role in helpingmarkets to avoid mispricing of capital. In simple terms, accounting informationshould help investors to assess the relevant statistical parameters of �rms�cash �owsmore accurately, and hence to cause some �rms a higher cost of capital, and other�rms a lower cost of capital. Think again of the insurance analogy, and how furtherinformation can lead the insurance provider to either increase of decrease a givenindividual�s life insurance premium, or indeed all premia, or the average of all premia,across the market.

AppendixProof that the covariance does not depend on the quality of future information.

A way to prove this point is to write the prior (ex ante) covariance in terms of theexpected posterior (conditional) covariance. For notational convenience, the value Vjof the given asset j is written without the subscript.

cov (V; VM) = E [cov (V; VM jS)] + cov (E [V jS] ; E [VM jS]) : (15)

Each of the two terms on the right of this equation is in�uenced by signal quality,but this in�uence is cancelled out when the two terms are added together.More speci�cally, these two terms are

E [cov (V; VM jS)] = E�E (V VM jS)� E (V jS)E (VM jS)

�= E [V VM ]� E [E (V jS)E (VM jS)] ;

and

cov (E[V jS]; E [VM jS]) = E�E (V jS)E (VM jS)

��E

�E (V jS)

�E [E (VM jS)]

= E [E (V jS)E (VM jS)]� E [V ]E [VM ] :

Hence, from (15) the covariance is

cov (V; VM) = E [V VM ]� E [V ]E [VM ] ;

and is una¤ected by the error characteristics of S.

25

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