Abstract
This paper explores the links between firms’ voluntary disclosures and their cost of capital. Existing studies investigate the relation between mandatory disclosures and cost of capital and find no crosssectional effect but a negative association in timeseries. In this paper, I find that when disclosure is voluntary firms that disclose their information have a lower cost of capital than firms that do not disclose, but the association between voluntary disclosure and cost of capital for disclosing and nondisclosing firms is positive in aggregate. I further examine whether reductions in cost of capital indicate improved risksharing or investment efficiency. I also find that high (low) disclosure frictions lead to overinvestment (underinvestment) relative to firstbest. As average cost of capital proxies for risksharing but not investment efficiency, the relation between cost of capital and ex ante efficiency may be ambiguous and often irrelevant.
Similar content being viewed by others
Notes
This can be easily illustrated with a brief example (see Lambert et al. 2007 for more details). Suppose that the economy features only two firms, A and B. If A discloses, some uncertainty relating the systematic shock will be realized, possibly leading (on average) to a reduction in the risk premium for both A and B. This does not imply (as is usually tested empirically) that the risk premium of A will be lower than that of B.
In resolving uncertainty, information also erodes risksharing opportunities when it is publicly revealed before trading. “Public information . . . in advance of trading adds a significant distributive risk” (Hirshleifer 1971, p. 568). However, my result differs from Hirschleifer (1971) in that I show how changes to voluntary disclosure may lead to greater price dispersion and study aggregate cost of capital, while Hirshleifer focuses on efficiency after price dispersion has increased.
Another important difference between Gao (2010) and my paper is that his model is one with a single firm and a single agent per generation. To my knowledge, an extension of his results to an economy with multiple firms and agents is non trivial.
Other studies investigate the interactions between voluntary disclosures and the economic and informational environment. Bertomeu et al. (2011) study whether firms’ voluntary disclosures can reduce asymmetric information in financial markets and lead to cheaper financing. Managers might also know several pieces of information and the decision to voluntary disclose their information depends on the correlation and precision about the signals (Kirschenheiter 1997) and the mandatory disclosure environment (Einhorn 2005).
Leland and Pyletype signalling considerations are beyond the scope of my analysis. This assumption is similar to the voluntary disclosure literature, which considers the manager as a person distinct from the investors. Informed trading by managers (unlike disclosure management) is explicitly prohibited by the SEC; further, in practice, managers’ trades constitute a very small portion of the total volume traded and would only marginally affect asset prices.
Under the assumptions of the capital asset pricing model (e.g., Mossin 1966), firms’ cash flows can always be decomposed into an idiosyncratic and a (suitably constructed) systematic factor, and thus such a decomposition is without loss of generality.
The restriction to \({\mathbb{E}(y)=0}\) is without loss of generality; if \({\mathbb{E}(y)\ne 0}\), one could relabel \({y'=y\mathbb{E}(y)}\), with mean zero, and \({\epsilon'=\theta+\mathbb{E}(y)}\), with no change to the results or analysis. In other words, a revision of the economy’s growth would be captured in this model by the common mean of the firmspecific factor θ.
The results and proofs are unchanged if one assumes instead that firms receive a noisy signal with probability 1 − η, say s′, on \(\epsilon\). Given that the estimation risk on \(\epsilon\) is purely idiosyncratic, it would not be priced, and thus one could relabel the model by replacing \(\epsilon\) by \({\epsilon'=\mathbb{E}(\epsilons')}\).
If α = 1, u(.) is set to \(u(x)=\ln(x)\). As the CRRA utility is not defined for negative values, I assume that if x < 0 then u(x) goes to \(\infty\). All the results of the model carry over for (i) other types of ownership (certain investors own multiple projects or share ownership), (ii) if some investors do not own a project, (iii) if all investors also have an i.i.d. personal wealth, in addition to their project. The only required assumption is that not all investors are perfectly diversified ex ante. The result on the difference in returns between disclosing or nondisclosing firms is robust to any strictly concave utility function but the CRRA assumption is required for the comparative statics and efficiency comparisons.
A formal proof is available upon request. The twofund separation does not require CRRA but would work for any HARA class utility. However, the model of large economy would have to be “adjusted" if another utility were to be used instead. With the CARA preference, a continuum of investors for a fixed endowment would lead to an arbitrarily low risk per investor and thus would bring risk premia to the riskfree rate. This does not happen with CRRA because, as there are more investors, the wealth of each investor decreases, which increases riskaversion, so that both effects cancel, and the risk premium no longer depends on the number of investors or assets, it only depends on total endowment. Thus, if one were to write a large economy with CARA, one would have to state a finite economy with one asset per investor and then let both the number of investors and the number of assets (i.e., the total wealth in the economy) become large. This makes the CARA model more cumbersome for the purpose of defining a large economy.
While each individual asset is replicated with either negative or positive quantities of the riskfree asset, the total net supply of the riskfree asset will always be zero.
The results are unchanged if this restriction is lifted, except that there may be many economically equivalent equilibria in which some lowvalue firms choose to disclose but still do not receive financing.
For convenience, I focus on the “anonymous" or symmetric solution, in which the planner does not advantage certain investors over others. The solution \(\epsilon^{FB}\) is unchanged if one considers the complete set of Paretoefficient solutions in which the planner may favor certain investors over others.
For example, according to the model, one should not observe much time series variation in aggregate levels of disclosure as compared to, say, crosscountry or crossindustry variations. Moreover, the aggregate level of disclosure should not be related to characteristics of the overall economy, such as GDP growth or market returns.
The result also suggests some caution in interpreting singleagent models of disclosure outside of the CRRA framework. For example, CARA utility functions are rather unusual in asset pricing given that asset pricing is all about risktaking and, unlike CRRA, CARA predict empirically counterfactual risktaking (Rubinstein 1975; Cochrane 2005). Under CARA utilities, a billionaire, a millionaire, or a minimumwage worker would all hold exactly the same dollar amount of risky assets. Their investment would only differ in terms of how much riskfree asset they hold. If utility functions are CARA, for example, there will exist a representative agent; however, the preference of this representative agent will depend on the wealth of all agents, which in turn will depend on the disclosure threshold \(\epsilon^{**}\); as a result, a comparative static on the disclosure threshold would require adjusting the preferences of the representative agent, which would lead to considerable analytical difficulties.
For \(\eta\in(\eta^{under},\eta^{over})\), there exist mixedstrategy equilibria where nondisclosing firms are not financed with a positive probability. It can be shown that, in these mixedstrategy equilibria (with details available from the author), investment, the market risk premium,and investors’ ex ante welfare all increase as the disclosure friction increases.
My result should be separated from other standard models of disclosure (e.g., Verrecchia 1983 or Dye 1985) that do not incorporate systematic risk. In such models, the primary object of interest is the instantaneous response of the market price to disclosure. Such response would also exist here (the nondisclosing firm’s price would decrease), but the notion of cost of capital studied here is measured as the return for the (possibly long) period post disclosure, excluding the disclosure event. The benefit of using this approach is that it predicts longterm effects of disclosure, as observed empirically, versus a short adjustment. Further, the standard model predicts that, when not disclosing, a firm’s market price would decrease, which would lead to negative returns and/or the counterfactual empirical implication that the cost of capital of nondisclosing firms (as proxied by their market return) would be lower than the cost of capital of disclosing firms.
The average cost of capital in the economy is an unconditional expectation, whereas the average cost of capital of disclosing and nondisclosing firms is a conditional expectation.
Analyzing efficiency is useful because in theory any efficient outcomes could lead to welfare improvements in an economy with non identical agents if a planner were to make fixed transfers (secondwelfare theorem); separating efficiency from welfare considerations, in this respect, allows me to distinguish accounting from reallocative effects.
I use the standard definition of risksharing, i.e., a mutually beneficial trade in which agents exchange offsetting (idiosyncratic) sources of risk. Risksharing is impaired when more information in advance of trading limits the risksharing opportunity. I contrast risksharing with productive efficiency, defined as operating decisions that increase the total amount of resource in the economy.
A recent literature (Becka et al. 2008 and references therein) discusses whether developing economies may have more severe informational frictions than developed countries and examines implications for growth and financing decisions.
References
Barth, M., Landsman, W. R., Lang, M., & Williams, C. (2008). International accounting standards and accounting quality. Journal of Accounting Research, 46(3), 467–498.
Becka, T., DemirgüçKuntb, A., & Maksimovic, V. (2008). Financing patterns around the world: Are small firms different?. Journal of Financial Economics, 89(3), 467–487.
Bertomeu, J., Beyer, A., & Dye, R. A. (2011). Capital structure, cost of capital, and voluntary disclosures. The Accounting Review, 86(3), 857–886.
Bertomeu, J., & Magee, R. P. (2011). From lowquality reporting to financial crises: Politics of disclosure regulation along the economic cycle. Journal of Accounting and Economics, 52(2–3), 209–227.
Botosan, C. A. (1997). Disclosure level and the cost of equity capital. The Accounting Review, 72(3), 323–349.
Botosan, C. A., & Plumlee, M. A. (2002). Reexamination of disclosure level and the expected cost of equity capital. Journal of Accounting Research, 40(1), 21–40.
Chen, H. J., Berger, P. G., & Li, F. (2006). Firm specific information and the cost of equity. European finance association zurich meetings 2006, available at http://ssrn.com/abstract=906152 .
Christensen, P. O., de la Rosa, E. L., & Feltham, G. A. (2010). Information and the cost of capital: An exante perspective. The Accounting Review, 85(3).
Cochrane, J. (2005). Asset pricing: (Revised Edition). Princeton:Princeton University Press.
Cohen, D. A. (2008). Does information risk really matter? An analysis of the determinants and economic consequences of financial reporting quality. AsiaPacific Journal of Accounting and Economics, 15(2), 69–90.
Dye, R. A. (1985). Disclosure of nonproprietary information. Journal of Accounting Research, 23(1), 123–145.
Easley, D., & O’Hara, M. (2004). Information and the cost of capital. Journal of Finance, 59(4), 1553–1583.
Ecker, F., Francis, J., Kim, I., Olsson, P. M., & Schipper, K. (2006). A returnsbased representation of earnings quality. The Accounting Review, 81(4), 749–780.
Einhorn, E. (2005). The nature of the interaction between mandatory and voluntary disclosures. Journal of Accounting Research, 43(4), 593–621.
Francis, J., Nanda, D., & Olsson, P. (2008). Voluntary disclosure, earnings quality, and cost of capital. Journal of Accounting Research, 46(1), 53–99.
Gao, P. (2010). Disclosure quality, cost of capital, and investor welfare. The Accounting Review, 85(1), 1–29.
Hirschleifer, J. (1971). The private and social value of information and the reward to inventive activity. American Economic Review, 61(4), 561–574.
Hughes, J. S., Liu, J., & Liu, J. (2007). Information asymmetry, diversification, and cost of capital. The Accounting Review, 82(3), 705–729.
Hughes, J. S., & Pae, S. (2004). Voluntary disclosure of precision information. Journal of Accounting and Economics, 37(2), 261–289.
Jorgensen, B., & Kirschenheiter, M. (2003). Discretionary risk disclosures. The Accounting Review, 78(2), 449–469.
Jorgensen, B., & Kirschenheiter, M. (2008). Voluntary disclosure of sensitivity. Financial accounting and reporting section 2009, available at http://ssrn.com/abstract=1270977.
Jung, W.O., & Kwon, Y. K. (1988). Disclosure when the market is unsure of information endowment of managers. Journal of Accounting Research, 26(1), 146–153.
Kirschenheiter, M. (1997). Information quality and correlated signals. Journal of Accounting Research, 35(1), 43–59.
Lambert, R., Leuz, C., & Verrecchia, R. E. (2007). Accounting Information, Disclosure, and the Cost of Capital. Journal of Accounting Research, 36(2), 385–420.
Leuz, C., & Verrecchia, R. (2000). The economic consequences of increased disclosure. Journal of Accounting Research, 38, 91–124.
Liang, P. J., & Wen, X. (2007). Accounting measurement basis, market mispricing, and firm investment efficiency. Journal of Accounting Research, 45(1), 155–197.
Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica, 34(4):, 768–783.
Nikolaev, V., & Van Lent, L. (2005). The endogeneity bias in the relation between costofdebt capital and corporate disclosure policy. European Accounting Review, 14, 677–724.
Pae, S. (1999). Acquisition and discretionary disclosure of private information and its implications for firms’ productive activities. Journal of Accounting Research, 37(2), 465–474.
Rubinstein, M. (1975). The strong case for the generalized logarithmic utility model as the premier model of financial markets. Journal of Finance, 31(2), 551–571.
Sengupta, P. (1998). Corporate disclosure quality and the cost of debt. The Accounting Review, 73(4), 459–474.
Skaife, H. A., Collins, D. W., & LaFond, R. (2004). Corporate governance and the cost of equity capital. Working paper available at http://ssrn.com/abstract=639681.
Stiglitz, J. E., & Cass, D. (1970). The structure of investor preferences and asset returns, and separability in portfolio allocation: A contribution to the pure theory of mutual funds. Journal of Economic Theory, 2(2), 122–160.
Verrecchia, R. E. (1983). Discretionary disclosure. Journal of Accounting and Economics, 5, 179–194.
Welker, M. (1995). Disclosure policy, information asymmetry and liquidity in equity markets. Contemporary Accounting Research, 11, 801–827.
Yee, K. K. (2006). Earnings quality and the equity risk premium: A benchmark model. Contemporary Accounting Research, 23(3), 833–877.
Acknowledgments
This paper has benefited greatly from discussions with my committee members Jon Glover, Steve Huddart, Carolyn Levine (chair), Pierre Liang, and Jack Stecher, who gave their time generously and helped me with their comments. I would also like to thank Jeremy Bertomeu, Carl Brousseau, Tim Baldenius, Jie Chen, Ron Dye, Pingyang Gao, Vineet Kumar, Urooj Khan, Chen Li, Russell Lundholm, Nahum Melumad, Beatrice Michaeli, Jim Ohlson, Gil Sadka, two anonymous referees, and seminar participants at Carnegie Mellon, Columbia, Northwestern, NYU, University of Illinois at UrbanaChampaign, and University of Michigan AnnArbor.
Author information
Authors and Affiliations
Corresponding author
Appendix: Technical supplements
Appendix: Technical supplements
Proof of Proposition 1:
The social planner solves the following maximization problem:
The first order condition (FOC) of the maximization problem (5) yields:
As \(h(\epsilon^{FB})>0\), one can rewrite this FOC as \(\Upphi(\epsilon^{FB})=0\) where:
Noting that U′ > 0, the fact that \(\Upphi(\epsilon^{FB})=0\) implies that \(\epsilon^{FB}={\frac{\int y f(y) U'\left(\int_{\epsilon^{FB}}^{+\infty}( \epsilon+y)h(\epsilon)d\epsilon\right)}{\int f(y) U'\left(\int_{\epsilon^{FB}}^{+\infty}(\epsilon+y)h(\epsilon)d\epsilon\right)}} dy\). To prove that \(\epsilon^{FB}\) is unique and in (0, θ), it is sufficient to show that: (i) \(\Upphi'<0\) and, (ii) \(\Upphi(0)>0\) and \(\Upphi(\theta)<0\).
Since U′ > 0 and \(U''<0, \Upphi'<0\).
where the inequality follows from the concavity of U. Moreover, θ + y > 0 implies that:
Proof of Lemma 1
The disclosure threshold satisfies \(P_{\epsilon^{over}}P_{\emptyset}=0\). This implies, by Eqs. (2) and (3) that \({\epsilon^{over}\mathbb{E}(\epsilonND)=0}\). The non disclosure conditional expectation \({\mathbb{E}(\epsilonND)}\) can be expanded by Bayesian updating to:
The equation can be rearranged as follows,
This corresponds to Eq. (7), p. 149, in Jung and Kwon (1988) and
Proof of Proposition 2
I solve the maximization problem of an investor:
The dependence on δ is due to the price P _{ δ }. But from the program \(\Upgamma_{\delta}, P_{\delta}\) is only a constant multiplicative term of the objective function, and the maximizer γ does not depend on P _{ δ }, and so does not depend on δ. Thus \(\gamma^{over}_{\emptyset}=\gamma^{over}_{ \epsilon}=\gamma^{over}\).
The aggregate demand of all investors in the riskfree asset AD is equal to:
I rewrite expression \(\int_{\epsilon^{over}}^{+\infty} \epsilon h(\epsilon)d\epsilon\) by integrating by parts as follows:
Substituting this expression into Eq. (7)
The mean θ can be rewritten as:
Finally expression (8) is equal to \({\gamma^{over}(\theta+{\frac{\theta}{\mathbb{E}(R_{m})}}\theta)=\gamma^{over} {\frac{\theta}{\mathbb{E}(R_{m})}}}\). As the net supply is equal to zero, it yields γ ^{over} = 0. I determine next the expression of the expected market portfolio return in this economy. Simplifying the maximization problem \((\Upgamma_{\delta})\), it yields:
The FOC from the investor’s maximization problem with respect to γ is equal to:
As from the market clearing condition, I showed that γ ^{over} = 0., the FOC is reduced to:
Simplifying,
I prove next that Q ^{over} is negative where \(Q^{over}=\int {\frac{y f(y)U'(\theta+y)}{\int f(y)U'(\theta+y)dy}} dy\). By additivity of the integral,
Moreover \(\int_{0}^{+\infty} y f(y)U'(\theta+y)dy<\int_{0}^{+\infty} y f(y)U'(\theta)dy\) as U is strictly concave. Likewise \(\int_{\underline {y}}^{0} y f(y)U'(\theta+y)dy<\int_{\underline {y}}^{0} y f(y)U'(\theta)dy\). By assumption \(\int_{\underline {y}}^{+\infty} y f(y)dy=E(\tilde{y})=0\), and it yields
In the overinvestment equilibrium, the nondisclosing price \( P_{\emptyset}(\epsilon^{over})\) is equal to \(P(\epsilon^{over})=\epsilon^{over}+Q^{over}\). As Q ^{over} is independent of η, the derivative of \(P(\epsilon^{over})\) w.r.t to η is equal to \(\frac{\partial{\epsilon^{over}}}{{\partial \eta}}\geq 0\). Thus the nondisclosing price is increasing in the disclosure friction η. Further it implies that there exists a unique η ^{over} such that \(P_\emptyset(\epsilon^{over}(\eta^{over}))=0\) and \(\epsilon^{over}\) is always positive.
Proof of Proposition 3:
If the investor has a nondisclosing firm, its wealth is equal to zero, and it cannot invest in the riskfree asset nor in the market portfolio. However if an investor has a disclosing firm then he solves the maximization problem \((\Upgamma_{\epsilon})\).
Simplifying,
The maximizers of an individual investor having a disclosing firm do not depend on \(P_{\epsilon}\) and thus \(\epsilon\). Therefore \(\forall \epsilon, \gamma^{under}_{\epsilon }=\gamma^{under}\). The aggregate demand of all investors for the risk free asset is equal to:
As the net supply is equal to zero, it yields γ ^{under} = 0. Taking the FOC of the investor problem, it yields
By the market clearing condition, γ ^{under} = 0, and the FOC is then reduced to:
I now turn to the determination of the disclosure threshold \(\epsilon^{under}\). The firm observing \(\epsilon^{under}\) has a cash flow:
Its price is then \(\epsilon^{under}{\frac{\int_{\epsilon^{under}}^{+\infty} \epsilon h(\epsilon)d\epsilon}{ 1H(\epsilon^{under})}} +{\frac{1}{(1\eta)(1H(\epsilon^{under}))}}P_{m}^{under}=0\) as by definition this firm has a market price of zero, where
Replacing P ^{under}_{ m } by its expression and simplifying it yields \(\epsilon^{under}={\frac{\int y f(y)(\int_{\epsilon^{under}}^{+\infty}(\epsilon+y)h(\epsilon)d\epsilon)^{\alpha}dy}{\int f(\tilde{y})(\int_{\epsilon^{under}}^{+\infty}(\epsilon+\tilde{y})h(\epsilon)d\epsilon)^{\alpha}d\tilde{y}}}\). The disclosure threshold \(\epsilon^{under}\) is independent of the disclosure friction η.
I now prove that Q ^{under} is negative. This is similar to the proof of Q ^{over} < 0.
As the utility function is strictly concave,
and,
By assumption \({\int_{\underline {y}}^{+\infty} y f(y)dy=\mathbb{E}(\tilde{y})=0}\), which yields
Further in the underinvestment region, the turning point η ^{under} is determined by Eq. (4) such that the price for nondisclosing firms is equal to zero. Given that Q ^{under} is not equal to Q ^{over}, there exists η ^{under}≠ η ^{over} such that \(\epsilon^{under}=Q^{under}\).
Proof of Proposition 4
Recall that Q ^{under} is as follows
Q ^{under} can also be interpreted as Q ^{over} in an economy with an endowment process \(y+k(\epsilon^{under})\). The function z(.;a) is a welldefined probability density function. It is commonly referred to as the riskneutral probability measure. I shall argue next that the distribution decreases in the sense of the firstorder stochastic dominance when a increases. To see this, define ψ(X;a) as the cumulative distribution function:
with \(K_1=\int f(y)(y+a)^{\gamma1}dy\int_{\underline {y}}^X f(y)(y+a)^{\gamma}dy \int_{\underline {y}}^X f(y)(y+a)^{\gamma1}dy \int f(y)(y+a)^{\gamma}dy.\)
To prove firstorder stochastic dominance, I need to show that ψ _{ a } < 0, i.e., K _{1} < 0.
The term K _{2} is strictly increasing in X and is strictly negative (positive) at \(X=\underline {y} (X=\overline{y}).\) Therefore, K _{1} must be first decreasing and then increasing, which in turn implies that K _{1} < 0 for any X. It follows that ψ _{ a }(.) < 0 and thus, for two a′ > a, the riskneutral probability measure induced by the density z(.;a′) firstorder stochastically dominates that induced by the density z(.;a′). Using this and the fact that \(\theta<k(\epsilon^{under})\),
This also implies that η ^{under} < η ^{over}. Finally, comparing the risk premium in each equilibrium:
Proof of Lemma 2
The costs of capital of nondisclosing firms and disclosing firms are respectively:
As \(P_{\emptyset}=P(\epsilon^{over})=\epsilon^{over}+Q^{over}\) and \(P(\epsilon)=\epsilon+Q^{over}\), a simple rewriting of the costs of capital yields:
Looking closely at the prices, I can express them as a CAPM formulation:
I derive the beta β as an expression of the covariance between the firm’s return and the market portfolio return over the variance of the market portfolio return.
Proof of Proposition 5
Further \(P_{\emptyset}(\epsilon^{over})=P(\epsilon^{over})\). As \(\forall \epsilon\geq\epsilon^{over}, P(\epsilon^{over})\leq P(\epsilon)\), R _{∅} ≥ R _{ D }.
Proof of Lemma 3

(i)
Let us prove that if η ≥ η ^{over}, for \(\eta<\eta^{'}, \Updelta (.;\eta')\) secondorder stochastically dominates: \(\Updelta(.;\eta)\). For a price \(p<P_{\emptyset}(\epsilon^{over}(\eta))\), \(\Updelta (p;\eta)=0\) otherwise for a price \(P_{\emptyset}(\epsilon^{over}(\eta))\leq p\leq \overline{p}, \Updelta (p;\eta)=\left\{\eta+(1\eta) H(\epsilon^{over}(\eta))\right\}+(1\eta)\left\{H(pQ^{over})H(\epsilon^{over}(\eta))\right\}\). Let us define η < η′ and \(T(\overline{p})\) as the area between the two curves \(\Updelta (;\eta)\) and \(\Updelta (;\eta')\) in \([P_{\emptyset}(\epsilon^{over}(\eta)), \overline{p}]\), specifically \(T(\overline{p}) = \int_{P_{\emptyset}(\epsilon^{over}(\eta))}^{\overline{p}}(\Updelta (p;\eta)  \Updelta (p;\eta'))dp\). By additivity of the integral I rewrite \(T(\overline{p})\):
$$ T(\overline{p})=\int\limits_{P_{\emptyset}(\epsilon^{over}(\eta))}^{P_{\emptyset}(\epsilon^{over}(\eta'))}(\Updelta (p;\eta)  0)dp+\int\limits_{P_{\emptyset}(\epsilon^{over}(\eta'))}^{\overline{p}}(\Updelta (p;\eta)  \Updelta (p;\eta'))dp $$Consider η′ = η + μ, then \(T(\overline{p})\) becomes:
$$ T(\overline{p})=\int\limits_{P_{\emptyset}(\epsilon^{over}(\eta))}^{P_{\emptyset}(\epsilon^{over}(\eta+\mu))} \Updelta (p;\eta) dp+\int\limits_{P_{\emptyset}(\epsilon^{over}(\eta+\mu))}^{\overline{p}}(\Updelta (p;\eta)  \Updelta (p;\eta+\mu))dp $$Further define \(A(\mu)= \int_{P_{\emptyset}(\epsilon^{over}(k))}^{P_{\emptyset}(\epsilon^{over}(\eta+\mu))}\Updelta (p;\eta)dp\) and \(B(\mu)= \int_{P_{\emptyset}(\epsilon^{over}(\eta+\mu))}^{\overline{p}}(\Updelta (p;\eta)  \Updelta (p;\eta+\mu))dp\). I differentiate the above expressions w.r.t μ:
$$ \begin{aligned} A'(\mu)&={\frac{\partial{\epsilon^{over}(\eta+\mu)}}{\partial{\eta}}}\Updelta (P_{\emptyset}(\epsilon^{over}(\eta+\mu));\eta)\\ B'(\mu)&={\frac{\partial{\epsilon^{over}(\eta+\mu)}}{\partial{\eta}}}\left\{\Updelta (P_{\emptyset}(\epsilon^{over}(\eta+\mu));\eta)\Updelta (P_{\emptyset}(\epsilon^{over}(\eta+\mu));\eta+\mu)\right\}\\ &\quad\int\limits_{P_{\emptyset}(\epsilon^{over}( \eta+\mu))}^{\overline{p}}((1H(\epsilon^{over}(\eta+\mu)))+(1\eta\mu) h(\epsilon^{over}(\eta+\mu)){\frac{\partial\epsilon^{over}( \eta+\mu)}{\partial{\eta}}}\\ &\quadH(pQ^{over})+H(\epsilon^{over}(\eta+\mu))(1\eta\mu) h(\epsilon^{over}(\eta+\mu)){\frac{\partial\epsilon^{over}(\eta+\mu)}{\partial{\eta}}}) dp\\ &={\frac{\partial{\epsilon^{over}(\eta+\mu)}}{\partial{\eta}}}\left\{\Updelta (P_{\emptyset}(\epsilon^{over}(\eta+\mu));\eta)\Updelta (P_{\emptyset}(\epsilon^{over}(\eta+\mu));\eta+\mu)\right\}\\ &\quad\int\limits_{P_{\emptyset}(\epsilon^{over}(\eta+\mu))}^{\overline{p}}\left(1H(pQ^{over})\right) dp \end{aligned} $$When μ = 0, A′ can be simplified to \({\frac{\partial{\epsilon^{over}(\eta)}}{\partial{\eta}}}\Updelta (P_{\emptyset}(\epsilon^{over}(\eta));\eta)\geq 0\).
Likewise B′ is equal to \(\int_{P_{\emptyset}(\epsilon^{over}(\eta))}^{\overline{p}}(1H(pQ^{over}))\leq 0\).
Replacing \({\frac{\partial{\epsilon^{over}(\eta)}}{\partial{\eta}}}\) by expression (5) and simplifying, I compute A′(0) + B′(0):
$$ \int\limits_{\epsilon^{over}(\eta)}^{+\infty}(1H(\epsilon))d\epsilon \int\limits_{P_{\emptyset}(\epsilon^{over}(\eta))}^{\overline{p}}\left(1H(pQ^{over})\right) dp $$Consider the change in variable \(\epsilon=pQ^{over}\), I further simplify the above expression by:
$$ \int\limits_{\epsilon^{over}(\eta)}^{+\infty}(1H(\epsilon))d\epsilon \int\limits_{\epsilon^{over}(\eta)}^{\overline{\epsilon}}(1H(\epsilon)) d\epsilon $$(14)If \(\overline{\epsilon}\) converges to \(+\infty\) then expression (14) is equal to zero. Thus \(\forall \overline{\epsilon}\), expression (14) is positive. It also means that \(\forall \overline{p}, T(\overline{p})\geq 0\), i.e., the probability mass of \(\Updelta(;\eta)\) is more spread out than the probability mass of \(\Updelta(;\eta')\), which implies that \(\Updelta(;\eta)\) is "riskier" than \(\Updelta(;\eta')\)

(ii)
Let us prove that if η ≤ η ^{under}, for \(\eta'<\eta, \Updelta(.;\eta')\) firstorder stochastically dominates \(\Updelta(.;\eta)\). It has been proven that \(P(\epsilon^{under})\) does not depend on η. Thus the range of prices is the same for a given η or η′. Let us take the difference between \(\Updelta (p;\eta')\) and \(\Updelta (p;\eta)\) for \(p\in[0, \overline{p}]\).
$$ \begin{aligned} \Updelta (p; \eta')\Updelta (p;\eta)&=(\eta'+(1\eta')H(\epsilon^{under}))+(1\eta')\left(H(pQ^{under})H(\epsilon^{under})\right)\cr &\left(( \eta+(1\eta)H(\epsilon^{under}))+(1\eta)\left(H(pQ^{under})H(\epsilon^{under})\right)\right) \end{aligned} $$Replacing η by η′ + μ it yields
$$ \begin{aligned} \Updelta (p;\eta')\Updelta (p;\eta'+\mu)&= (\eta'+(1\eta')H(\epsilon^{under}))+(1\eta')\left(H(pQ^{under})H(\epsilon^{under})\right)\cr &\left(( \eta'+\mu+(1\eta'\mu)H(\epsilon^{under}))+(1\eta'\mu)\left(H(pQ^{under})H(\epsilon^{under})\right)\right)\\ &=\mu(1H(\epsilon^{under}))+\mu\left(H(pQ^{under})H(\epsilon^{under})\right)\\ &=\mu \left(1H(pQ^{under})\right) \leq 0 \end{aligned} $$Therefore \(\Updelta (;\eta')\) first order stochastically dominates \(\Updelta (;\eta)\).
Proof of Proposition 6
If η ≥ η ^{over}, I will prove that R _{ D } is decreasing in η.
Differentiating R _{ D } w.r.t η, it yields
Simplifying it yields
As \(\forall \epsilon>\epsilon^{over}, P(\epsilon^{over})\leq P(\epsilon)\), and \({\frac{\partial{\epsilon^{over}(\eta)}}{\partial{\eta}}}\geq 0\) and Q ^{over} < 0, the derivative of R _{ D } with respect to η given by expression (15) is negative. Therefore R _{ D } is decreasing in η.
\(P_{\emptyset}(\epsilon^{over})=P(\epsilon^{over})=\epsilon^{over}+Q^{over}\). Differentiating \(P(\epsilon^{over})\) with respect to η yields \({\frac{\partial{\epsilon^{over}(\eta)}}{\partial{\eta}}}\geq 0\). Thus R _{∅} is decreasing in η.
If \(\eta\leq\eta^{under}, \epsilon^{under}\) and Q ^{under} do not depend on η, so R _{ D } is independent of η.
I turn to the comparative statics on the average cost of capital \(\mathcal{R}\).
If η ≥ η ^{over}, \(R_{\delta}=1{\frac{Q^{over}}{P_{\delta}}}\). Thus ordering \(\mathcal{R}(\eta)\) and \(\mathcal{R}(\eta')\) boils down to ordering 1/P _{ δ }(η) and 1/P _{ δ }(η′). I know that the function 1/P _{ δ } is a convex and decreasing function. I further proved that for η′ < η, \(\Updelta (;\eta)\) second order stochastically dominates \(\Updelta(;\eta')\). This implies that \({\mathbb{E}(1/P_{\delta}(\eta)) \leq \mathbb{E}(1/P_{\delta}(\eta'))}\). If η ≤ η ^{under} then \(\mathcal{R}=R_{D}\) as nondisclosing firms are not financed and \(R_{D}=1Q^{under}\int\nolimits_{\epsilon^{under}}^{+\infty} {\frac{h(\epsilon)}{P(\epsilon) (1H(\epsilon^{under}))}}d\epsilon\), independent of η. I now compare the average cost of capital to the risk premium. By applying Jensen’s inequality,
Proof of Proposition 7
For η ≥ η ^{over}, the investor’s indirect utility function with a firm disclosing δ is equal to
The indirect utility is concave in P _{ δ }. Further I know that for \(\eta<\eta', \Updelta (.;\eta')\) second order stochastically dominates \(\Updelta(.;\eta)\), which implies that
For η ≤ η ^{under}, the investors’ aggregate expected utility for a given η is equal to
consider η > η′
Proof of Corollary 1
By Proposition 6, it has been proven that the average cost of capital in the economy is decreasing in the disclosure friction η, if η ≥ η ^{over}. Further by Proposition 7, the investors’ expected utility increases in the disclosure friction η, if η ≥ η ^{over}.
Rights and permissions
About this article
Cite this article
Cheynel, E. A theory of voluntary disclosure and cost of capital. Rev Account Stud 18, 987–1020 (2013). https://doi.org/10.1007/s1114201392231
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s1114201392231