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Review of Accounting Studies

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14-07-2017

A theory of risk disclosure

Authors: Mirko S. Heinle, Kevin C. Smith

Published in: Review of Accounting Studies | Issue 4/2017

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Abstract

In this paper, we consider the price effects of risk disclosure. We develop a model in which investors are uncertain about the variance of a firm’s cash flows and the firm releases an imperfect signal regarding this variance. In our model, uncertainty over the riskiness of a firm’s cash flows leads to a variance uncertainty premium in its price. We demonstrate that risk disclosure decreases the firm’s cost of capital by reducing this premium and that the market response to risk disclosure is small when the expected level of risk is high. Moreover, we find that firms acquire and disclose more risk information when their cash flow risk is greater than expected. Finally, we demonstrate that in a multi-asset setting, only risk disclosure concerning systematic risks will impact the cost of capital.

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FASB (2012). More recently, the Enhanced Disclosure Task Force issued an extensive report recommending several improvements in the risk disclosure of banks, claiming that “investors and other public stakeholders are demanding better access to risk information from banks; information that is more transparent, timely and comparable across institutions.”

See Verrecchia (2001) and Beyer et al. (2010), or Bertomeu and Cheynel (2016) for surveys of the disclosure literature.

Neururer et al. (2016) and Sridharan (2015) provide empirical studies of the variance information in disclosed earnings.

See, for example, Beyer (2009), Hughes and Pae (2004), Kirschenheiter and Melumad (2002), Penno (1996), and Subramanyam (1996). Modeling variance disclosure as a direct signal regarding the variance demands a suitable nonnegative distribution for the variance, a conjugate prior for that distribution, and a utility function that yields a closed form solution with these distributions. We believe that the prior literature assumes risk neutral or mean variance pricing for tractability purposes. While this is suitable for the settings these papers examine, our focus is on the pricing of variance uncertainty and the effect of risk disclosures.

We denote random variables with a tilde “ ˜”.

Characterizing the gamma distribution by its mean and variance creates the following restriction: $\mu _{V}=0\Longleftrightarrow {\sigma _{V}^{2}}=0$. This occurs because a zero mean implies the distribution is degenerate at zero.

The inverse gamma is widely used as a conjugate prior for the variance of a normal distribution when signals are drawn from a normal-gamma distribution (see DeGroot 1970). We choose to examine the gamma distribution rather than the inverse gamma distribution as the moment generating function for an inverse gamma does not exist.

In the ??, we show that the mean of these signals is a sufficient statistic for their individual realizations.

Technically, if the underlying Poisson signals are equal to $\left \{ \tilde {s}_{i}\right \}_{i=1}^{\tau } $ , we have that $Var(\tilde {S}|\tilde {V} ) =Var\left (\tau ^{-1}{\Sigma }_{i=1}^{\tau } \tilde {s}_{i}|\tilde {V}\right ) =\tau ^{-1}\tilde {V}$ . This is decreasing in τ for any realization of $\tilde {V}$ .

This can be seen by computing excess kurtosis, defined as the fourth standardized moment minus the kurtosis of a normal distribution (which equals 3): $\frac {E[ (\tilde {V}-\mu )^{4}]} {(E [(\tilde {V}-\mu )^{2}] )^{2}}-3=3\frac { {\sigma _{V}^{2}}}{{\mu _{V}^{2}}}$.

While the fat tails that follow from the uncertain variance seemingly map to the empirical findings in Mandelbrot (1963) and Fama (1965), those studies suggest that stock returns exhibit fat tails, whereas our result implies that cash flows themselves exhibit fat tails.

It is easily seen that $\frac {\partial ^{2}}{\partial V^{2}}E(-e^{-\rho \tilde {x}}) =\frac {\partial ^{2}}{\partial V^{2}}\left (-e^{-\rho \mu -\frac {\rho ^{2}}{2}V}\right ) <0$.

See, for example, Eeckhoudt et al. (1996), Gollier and Pratt (1996), and Noussair et al. (2014). Noussair et al. (2014) also present experimental evidence that suggests that individuals are indeed temperate.

This statement is shown in the proof of Lemma 2.

The gamma distribution with shape parameter a and a scale parameter b is only defined for a > 0 and b > 0. To derive an investor’s certainty equivalent, $b>\frac {D^{2}\rho ^{2}}{2}$ has to hold. That is, the scale parameter has to be sufficiently large or the equilibrium demand (that is, the shares per capita) has to be sufficiently small. We derive an investor’s certainty equivalent with the standard parameterization in the proof to Lemma 2.

If the per capita endowment were an arbitrary constant e rather than 1, the condition becomes $\frac {1}{2}\rho ^{2}e^{2}\frac {{\sigma _{v}^{2}}}{\mu _{v}}<1$.

Increases in the mean holding the variance fixed reduce the degree of positive skew in the distribution.

¹⁸To understand more generally how prices respond to shifts in the distribution, consider changes in the variance distribution in the sense of first- and second- order stochastic dominance (FSD and SSD respectively). We should expect that distributional shifts in $\tilde {V}$ in the sense of FSD reduce price, and distributional shifts in $\tilde {V}$ in the sense of SSD increase price. Ali (1975) derives the following necessary and sufficient conditions for FSD and SSD for the gamma distribution characterized by shape and rate a and b:

$$\begin{array}{@{}rcl@{}} \!\!\!\!\tilde{V}_{1}\underset{FSD}{\succ} \tilde{V}_{2}\text{ when}~ a_{1} &\geq &a_{2}\text{ and}~ b_{1}\leq b_{2}\text{ with one equality strict;} \end{array} $$

(8)

$$\begin{array}{@{}rcl@{}} \;\tilde{V}_{1}\underset{SSD}{\succ} \tilde{V}_{2}\text{ when} \frac{a_{1}}{ a_{2}} &\geq &Max\left( 1,\frac{b_{1}}{b_{2}}\right) \text{.} \end{array} $$

(9)

Expressing prices in terms of a and b, we find:

$$ P=\mu -\frac{a}{b}\rho -\frac{\rho^{2}}{2b-\rho^{2}}\frac{a}{b}\rho \text{.} $$

(10)

A shift in the distribution of $\tilde {V}$ in the sense of FSD involves either increasing a or decreasing b; in either case, price falls. A shift in the distribution of $\tilde {V}$ in the sense of SSD is achieved by either increasing b and increasing a by at least the same percentage or by decreasing b and weakly increasing a. In either case, price increases as expected. The comparative static with respect to ${\sigma _{V}^{2}}$ in effect increases b while increasing a at the same rate. Equation 10 indicates that this increases prices only through its impact on the variance uncertainty premium.

Setting the mean of the signals equal to their prior mean isolates the uncertainty reduction effect of information from any effect due to a change in the posterior expectation of the variance distribution. Although not apparent from the diagram, all three distributions have the same mean; the skewness of the gamma distribution obscures this fact.

An exception to this is the work of Zhou (2016), where the firm’s discretionary disclosure decision depends on past realizations. In the model, the market does not know the true expected value, which causes the perceived distribution and therefore the discretionary disclosure threshold to be a function of past disclosures.

To see this formally, consider the setup of our model with no uncertainty over the variance of cash flows and consider the sequential disclosure of mean signals $\tilde {m}_{\tau } $. In particular, let $\tilde {m}_{\tau } = \tilde {x}+\tilde {\varepsilon }_{\tau } $ , where $\tilde {\varepsilon }_{\tau } \thicksim N(0,\eta ) $ , $Cov(\tilde {\varepsilon }_{i}, \tilde {\varepsilon }_{j}) =0$ ∀i,j, and $Cov(\tilde { \varepsilon }_{\tau } ,\tilde {x}) =0$ . Let $\tilde {M}_{\tau } $ be the mean of the first τ disclosed signals, and let V equal the known variance of cash flows. Then, after disclosing τ − 1 signals with mean $ \tilde {M}_{\tau -1}$, the expected benefit from receiving another signal is:

$$E(P(\tilde{M}_{\tau}) |\tilde{M}_{\tau -1}) -P(\tilde{M}_{\tau -1}) =\delta \text{,} $$

where $\delta =\rho \left (\frac {1}{\tau \eta ^{-1}+V^{-1}}-\frac {1}{(\tau +1) \eta ^{-1}+V^{-1}}\right ) $. Since the benefit is not a function of $\tilde {M}_{\tau -1}, $the decision to disclose an additional signal never depends on prior disclosures.

These conditions mirror the condition from the single asset case and imply that investorsare willing to hold shares at any finite price.

Mathematically, modeling variance disclosure by multiple firms in our setup is not straightforward since the systematic components of firms’ disclosures likely overlap. In particular, multiple firms may aggregate signals that contain the same $\tilde {s}_{ik}$. Nevertheless, it is intuitive that firms’ information disclosures can be jointly used to assess the uncertain variance of the factor, and thus we assume that investors can tease apart the novel information in a firm’s disclosures.

A mean-variance-kurtosis utility function is consistent with the fourth-order development of the Arrow-Pratt expression for the risk premium (see Le Courtois 2012).

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Title: A theory of risk disclosure
Authors: Mirko S. Heinle
Kevin C. Smith
Publication date: 14-07-2017
Publisher: Springer US
Published in: Review of Accounting Studies / Issue 4/2017
Print ISSN: 1380-6653
Electronic ISSN: 1573-7136
DOI: https://doi.org/10.1007/s11142-017-9414-2

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