The kind of silence: managing a reputation for voluntary disclosure in financial markets

We create a continuous-time setting in which to investigate how the management of a firm controls a dynamic choice between two generic voluntary disclosure decision rules (strategies) in the period between two consecutive mandatory disclosure dates: one with full and transparent disclosure termed candid, the other, termed sparing, under which values only above a dynamic threshold are disclosed. We show how parameters of the model such as news intensity, pay-for-performance and time-to-mandatory-disclosure determine the optimal choice of candid versus sparing strategies and the optimal times for management to switch between the two. The model presented develops a number of insights, based on a very simple ordinary differential equation characterizing equilibrium in a piecewise-deterministic model, derivable from the background Black–Scholes model and Poisson arrival of signals of firm value. It is shown that in equilibrium when news intensity is low a firm may employ a candid disclosure strategy throughout, but will otherwise switch (alternate) between periods of being candid and periods of being sparing with the truth (or the other way about). Significantly, with constant pay-for-performance parameters, at most one switching can occur.


Introduction
When investors value firms, they not only base their inferences upon what news signals managers make public, but also on the likelihood that management may be hiding other news.A legal environment with high penalties reduces the chances managers will hide very bad news signals, but a constant concern for investors is: do management release early warning signals of potentially less severe bad news in a timely fashion, or do they hide it in an underhand way in the hope conditions will evolve differently and only disclose when potential legal liability arises?(Marinovic and Varas [MarV].) The Dye [Dye] model of voluntary disclosure addressed this issue in 1985 in a static setting and derives equilibrium conditions under which management adopts a sparing approach with a threshold-disclosure strategy, when deciding whether or not to voluntarily disclose new information ahead of a mandatory disclosure date.Institutional features, such as news-arrival rates and time to the mandatory disclosure date, cannot be modelled in such a static setting.In response Beyer and Dye [BeyD] (in 2012) develop a twoperiod model in which managers may make a voluntary disclosure in order to build a reputation for being candid -forthright (or 'forthcoming') -by always faithfully disclosing their updated information.By contrast managers could exploit their asymmetric endowment of information and only make voluntary disclosures if their received signal is high enough, as defined by a valuation threshold (cutoff).Such behaviour will here be termed sparing 3 .For the two-period model Beyer and Dye show why managers' concerns in the first period -for how investors form second-period inferences (based on observed first-period dividend outcomes) -affect their voluntary disclosure strategy.Their model predicts a diversity in management strategies, as in equilibrium some managers will choose a strategy which leads investors to assign a high probability that they will behave sparingly, while others choose to be candid.An insight from this model is that a voluntary disclosure strategy may be used to influence future firm-value over and above the direct effect of disclosure of any idiosyncratic signals of value.That is, establishing a reputation for being candid at times shifts firm-value upwards, over and above the direct discounted present-value of the most recent signals of value, precisely because investors now assign a reduced likelihood for management hiding bad news.
The structure of the paper is as follows.We discuss related literature in Subsection 1.1.Section 2 presents our main findings.Section 3 develops the theory of the optimal sparing-disclosure threshold in a continuous framework, for which the main optimization tool comes from control theory and relies on the Pontryagin Principle.Examples are given in Section 4. Proofs of theorems are in Section 5 with some technical results relegated to Appendix A and further, more routine, calculations to Appendix B, which ends with a symbols list.Section 6 briefly considers multi-switching, although our main interest is in single-switching.We present conclusions in Section 7. We note that all valuations are viewed as discounted .
1.1 Related Literature on the Disclosure Dynamics Earlier models of dynamic disclosure have taken various paths, as follows.Beyer and Dye comment that cheap talk models are concerned with reputational formation rather than establishing a reputation for timely disclosure.Also disclosure has the feature of a variable that must be binary and in practice managers can disclose untruthfully, which is excluded in the Dye disclosure model (and by us below).Another path, followed by Acharya et al. in [AchMK], has been to investigate clustering of corporate disclosures driven by market news.In that paper management learns only one piece of information and must decide when to disclose it if other market news events are occurring.Guttman et al. [GutKS] extend the Dye model to three periods and demonstrate a differential equilibrium response to end of first or second period disclosures.Marinovic and Varas [MarV] develop a continuous-time disclosure model, but one in which there are exogenous costs of disclosure.They show how litigation risk affects disclosure.They explain how litigation stimulates transparency in financial markets and, furthermore, how bad news can crowd out good news, leading to investors being less skeptical about corporate disclosures.Bertomeu, Marinovic, Terry and Varas [BerMTV] develop a model of dynamic concealment which is in part motivated by a desire to develop a structural estimation model applied to publicly observable annual earnings forecasts.In their dynamic disclosure model, non-disclosure corre-sponds to firms not issuing an earnings forecast (76 p.c. of cases) and so their model can be viewed as a between-fiscal periods model of disclosure.In contrast, we focus on how investors should rationally update value estimates given non-disclosure in the period between fiscal dates (years or quarters).Since the institutional setting is different for between-versus within-fiscal period settings, we provide a disclosure market-microstructure, which complements their multi-year modelling.In our intra-period (within-period) model end-effects arise in disclosure behaviour which may not be present in an inter-period (between-periods) analysis, if there is no modelling of firm exit or failure.Einhorn and Ziv [EinZ] develop a dynamic model, but one in which disclosure generates a cost, and unlike the Dye model does not naturally allow equilibrium risk-neutral pricing.The model closest in spirit to ours is the Beyer and Dye [BeyD] model in which managers choose between a strategy of either being candid (always disclosing privately observed news signals) or alternatively sparing, only disclosing when it is in their own self-interest.We extend their two-period model to a continuous-time setting and model the possibility of reputationally based switching between the two reporting behaviours, driven by news-flow, pay-for-performance, and or proximity to mandatory disclosure dates.Our model shows that a managerial strategy of being candid over some time-interval need not be a time-invariant strategy, instead it should only be interpreted as a period-specific strategy.That is, our model calls into question type-based models that assume managers have some deep intrinsic desire to be of reputable type in all periods; instead, our model shows how in equilibrium, rational, previously reputable, managers may be prepared to burn reputation (by diverging from initial expected behavior) in a later period.This suggests that it is rational for investors to condition beliefs not just on history, but also on forward-looking variables such as remaining tenure.

Model generalities and findings
Consider a firm whose financial state X t evolves in continuous time according to a Black-Scholes model with periodic mandatory disclosure dates and with interim intermittent capability of voluntary disclosures of the next mandatory expected financial report.This would be based on partial observation Y t of the financial state X t .
Consider two possible reporting behaviours executed by the firm manage-ment at any time t when Y t is observed: candid (faithful) reporting -reporting the observed value, as seen, i.e. unconditionally; sparing (threshold) reporting -reporting only the value observed when above a time-t dependent threshold, i.e. conditionally.
These behaviours are both capable of being applied at any one time, i.e. leading to their use in some combination, and are assumed to be both truthful and prompt (i.e.without delay).We also assume that managers cannot credibly assert absence of information arriving at time t (i.e.absence of knowledge of Y t ).Furthermore, no evidence of an undisclosed observation is retained.We admit no further sources of information about Y t ; modelling with the inclusion of further sources is touched on in [GieOS].
Sparing here is used in the sense of being economical with information delivery as in 'economical with the truth' or 'actualité'4 , sometimes called strategic.It is of course a foundational question whether it is suboptimal to withold information.An early finding in the disclosure literature, provided by Grossman and Hart [GroH] (in 1980) and Grossman [Gro] (in 1981), has become known as the unravelling result.It suggested that withholding information would lead investors to discount the valuations, thus incentivising a firm to make a full disclosure in order to restore the value.
The contribution of Dye [Dye] (in 1985) was to provide, in a discrete framework with one interim date (say at some time s between two mandatory disclosure dates of 0 and 1), a rationale for why this 'full disclosure unravelling' result might not occur at the interim date s, and to supply an equation uniquely determining the resulting market discount in value in an equilibrium framework.The market discount is an appropriate weighted average that combines the possibility that management lacks fresh information with the possibility that management may hide information which if disclosed would have led to an even larger discount (i.e.below the weighted average).
Dye's paradigm for valuing a non-disclosing firm may be characterized by an amended statement of the Grossman-Hart paradigm as follows Minimum Principle ([OstG], cf.[AchMK] -their Prop.1).In equilibrium the market values the firm at the least level consistent with the beliefs and information available to the market as to its productive capability.
See also Section 3.This result carries some detailed implications to which we return later.But the principle already suggests intuitively that if the management reporting behaviour is believed by the market to be at times candid, then in equilibrium the weighted average valuation may at times move upwards, by placing less weight on the chance of poor observation being witheld.
We will demostrate the validity of such a suggestion in the continuoustime context of the firm as described above, by creating a continuous analogue of Dye's argument in which the Poisson arrival rate of the observation time of Y t is λ and assuming management can report in a sparing mode (relative to an optimally generated threshold) with a probability π t at time t when simultaneously the market believes (in equilibrium) that the selected probability is indeed π t .
Management choice of π t is motivated through the maximization of an appropriate objective function rewarding in proportion to a factor α t the instantaneous firm value and penalizing in proportion to a factor β t the instantaneous value-differential (value relative to that derived from sparing behaviour executed throughout all time).The parameter κ t = 1 − α t /β t emerges as significant (see Section 3.3).
The aim of the penalty term is to provide a tension between sparing and candid reporting: adopting candour throughout would require more frequent disclosure of potentially bad news, which could result in larger falls in firm value, i.e. over and above falls that resulted from continued sparing silence (non disclosure).
We discuss our findings in this section, leaving details of the optimization and proofs to the next and later section.Our first surprising finding is that an optimal disclosure behaviour is of bang-bang type which will always switch (alternate) between intervals of constancy with only π = 0 or π = 1, i.e. a mixed strategy is ruled out in the following theorem.(See Appendix A for the stronger statement in Theorem 1S.) Theorem 1 (Non-mixing Theorem).When α t , β t are constant: A mixing control with π t ∈ (0, 1) is non-optimal over any interval of time.
The theorem agrees with empirical findings (due to Grubb [Gru]) that after an announcement management is observed to follow initially either candid or sparing behaviour but not a mixture.
Accordingly, we study π t ∈ {0, 1}.In particular, we study the possible occurrence of an initially candid (candid-first) equilibrium in which management at first adopt candid behaviour out of which they switch after some time θ, the switching time, in favour of a sparing policy, and also initially sparing (sparing-first) equilibrium in which management at first adopt sparing behaviour out of which after some time they switch in favour of candid behaviour.This provides a model framework for empirical detection of regime change (disclosure policy change): cf.Løkka [Lok].Theorems 2a and 2b identify both the location of the switching time of such an equilibrium policy and the attendant necessary and sufficient existence conditions guaranteeing an equilibrium.These theorems are followed by a clarifying discussion concerning the location conditions.
We stress that having merely a characterization of the location condition is not adequate.The technical nature of the existence conditions emerges from a Hamiltonian analysis (Section 3.4 below) in which the Pontryagin Principle relies on Theorem 1 (the non-mixing) in supplying a necessary and sufficient optimality condition.
Our findings refer to a decreasing discount function h(t) (responsible for the rate of fall in values when continued absence of disclosure is attributed to sparing behaviour -see Section 3.2 equation (cont-eq)) and to its integral Theorem 2a (Single switch equilibrium location and existence for an initially candid strategy).Assume that α t , β t are constant and 0 < κ < 1 for κ := 1 − α t /β t .In an equilibrium, if such exists, in which π = 0 on [0, θ) = 0 and π = 1 on [θ, 1], the uniquely optimal switching time θ solves For given λ this equation is solvable for large enough κ, in fact iff 1) . (cand) Such an equilibrium exists iff the unique switching time θ satisfies In such an equilibrium, the unique switching time θ satisfies So larger news-arrival rates λ create shorter periods of initial candid behaviour.
Remark.The left-hand side term of the existence condition above is monotonically decreasing from 1 down to g(1)/h(0), as in the red graph in Figure 1 below; its lowest value is dictated by σ, a volatility measure.The righthand side ranges monotonically from 0 to 1; thus a fixed value of κ supplies a value to the right-hand side and is illustrated in green below for a choice which allows all values θ to satisfy the inequality here (with other choices restricting the θ range).Matters are more complicated in an equilibrium that is initially sparing.
Theorem 2b (Single switch equilibrium location and existence for an initially sparing strategy).
In an equilibrium, if such exists, in which π = 1 on [0, θ) = 0 and π = 0 on [θ, 1], the uniquely optimal switching time θ solves the first-order condition For given λ, this equation is solvable for large enough κ, in fact iff Such an equilibrium exists iff the unique switching time θ satisfies > λ (a bound on λ in terms of θ).
In such an equilibrium, the unique switching time θ satisfies So larger news-arrival rates λ create longer periods of initial sparing behaviour.
The theorems expose a fact of direct relevance to empirical study: that with the same parameter values it may happen that both equilibrium types coexist, as in Figures 2a and 3. (Conditions on parameter values κ, λ permitting this can be derived numerically from the conditions of Theorems 2a and 2b.)In particular, the two conditions (cand) and (spar) on κ with fixed λ, may both hold simultaneously: indeed, since the map λ → e λg(1) is convex, there is a unique λ = λ crit > 0 such that e λg(1) = 1 + λh(0), and e −λg(1) ≶ (1 + λh(0)) −1 according as λ crit ≶ λ.
In contrast to the coexistence of equilibria in which switching occurs, there does exist a constantly candid equilibrium (i.e. with no switching): Qualitative Corollary.For small enough λ, candid (unconditional) disclosure throughout the period of silence is an equilibrium policy.
For proof: see Corollary 2 in Section 5.The location of optimal switches, assuming they correspond to an equilibrium (requiring additional conditions), can be pursued in generality.We identify the consequent generalization as this leads to yet another surprising finding stated in the Corollary below.For the proof of Theorems 3a and 3b see Appendix B.
Theorem 3b (Multiple switching locations).Assume that α t , β t are constant and 0 < κ < 1 for κ := 1 − α t /β t .The sequence of solutions to the switching equation defines the switching times according to the recurrence If θ 1 is a right endpoint of an interval where π = 0, then Furthermore, the sequence κi is (weakly) decreasing with alternate members strictly decreasing.
Corollary (Candid-first single switching).Assume that α t , β t are constant and a contradiction to θ 2 < θ 3 .Consequently, there cannot be a further switching from sparing to candid mode.
A similar result appears to be supported by numerical analysis for a sparing-first equilibrium policy, albeit Theorem 3b (on its own, i.e. without invoking equilibrium conditions) implies by a similar argument that if π = 1 on [0, θ 1 ), then θ 4 = θ 3 , i.e. at most two switchings can occur.
In summary, this section has characterized how tractable single-switching conditions can be derived.The issue of equilibrium selection (which of the sparing-first or the candid-first) must rest on the underlying assumption that the market has found its way to one or other of the two by some evolutionary game-theoretic mechanism; for a standard textbook view of the latter, see e.g.Weibull [Wei].
Remark.Optimal multi-switching becomes possible when a time-varying κ t replaces the constant κ; this is particularly easy to arrange in the case of a piecewise-constant κ t with constancy on each inter-switching interval (interval between successive switching points): see Example 3 in Section 3.
We have seen above that single-switching should be regarded as natural in the constant κ context and not just a stylized model choice.Moreover, single-switching provides the pragmatic, preferred equilibrium choice by an appeal to focal-point (Schelling-point) equilibrium selection -for a standard textbook view of which, see e.g.[FudT].
3 The sparing disclosure threshold For X s the financial state and Y s its observation assume the regression function m s (y) := E t=0 [X 1 |Y s = y] is increasing.Then the opimal threshold γ s is uniquely determined and has three properties, the first of which in (i) below implies the Minimum Principle of Section 2.
(i) Minimum Principle: [OstG].The valuation function has a unique minimum at γ = γ s ; (ii) Risk-neutral Consistency Property: γ s is the unique value γ such that with τ D is the (time t) probability of disclosure occuring at s.This is highly significant, in that the valuation at time 0 anticipates the potential effects of a voluntary disclosure at the future interim date s.In brief, the approach is consistent with the principles of risk-neutral valuation; for background see Bingham and Kiesel [BinK], Chap.6.In particular, the risk-neutral valuation is a martingale, constructed via iterated expectations from (iii) Interim discount From the perspective of time 0, in a model with only three dates: 0, s, 1, the Dye equation at time s may be written: and has the interpretation of a protective put-option with strike g against a fall in value at t.Here q s is the probability that Y s is observed, and The argument leading to these results is also sketched in the next section.

Derivation of the Dye threshold equation
Relocating the dates to t < s < 1, the interim discount γ s , which is also the threshold for announcements in equilibrium at time s, is the value γ = γ s which satisfies with Q = market's probability measure for all relevant events, F t = market information at time t; ND s (γ) = event at time s that no disclosure occurs when the information is below γ; RHS = market's expectation of value conditional no-disclosure ND s (γ s ).
If management observe a value γ s at time s, then they are indifferent between disclosing the valuation as γ s and withholding said information.
If the probability of information reaching management at time s is q s = q, assumed exogenous and independent of the state of the firm, then for p : as where the subscript indicates conditioning on F t .Equivalently, we have Here we may routinely evaluate this put using the Black-Scholes formula.
As above, rearrangement will show incorporation of future information: as in risk-neutral valuation, where D s (γ) = event of time s when values above γ disclosed τ t D = market's evaluation at time t of disclosure probability at time s.
The presumption this far precluded the use of a candid strategy.If management restricts application of the sparing (threshold-generated) strategy to act with probability π and the market likewise believes (in equilibrium) that this probability is π, then in a period of silence: . (cond-π) For π = 1 (sparing) this reduces to the Dye equation.

Equilibrium condition: continuous-time analogue
We embed the three dates t < s < 1 of the Dye model into the unit interval to provide a continuous-time framework in which any future date s > t can be interpreted as a time at which the management have the opportunity to disclose a forecast of value to the market.As in the Dye model, key here is the creation of an ambiguity at time s, so that the market knows that absence of a disclosure is caused either by absence of fresh endowment of private information or by a management decision to withhold the private information arriving at moment s.With this aim we introduce a Poisson process with intensity λ whose jump at time s, when privately observed by management, determines that an observation of Y s occurs.The market does not observe the jumps.Thus every moment now takes on the character of an interim disclosure date and, depending on the disclosure policy believed by the market to be implemented by management, absence of a disclosure can mean no new observation or a withheld observation.
With the Poisson process in place, for t < s take q = q ts = λ(s − t) + o(s − t), employing the Landau little-o notation.Passage to the limit as s ց t yields: Dividing by −λ(s − t): ignoring errors of order o(s −t)/(s −t).With economic activity and the noisy observation in a standard Black-Scholes setting, this yields with σ the (aggregate) volatility (aggregating productive and observation vols.); for the proof see [GieO].This ODE is our continuous-time disclosure-equilibrium condition in any period of silence (i.e. when the management make no disclosures).It equilibrates in a period of silence between the market's ability to downgrade the valuation below γ t and the management's potential ability to upgrade the valuation were they to observe a greater value of Y t (cf. the weighted average discussed in Section 2).We refer to this as the equilibrium ODE.
The market valuation of the firm γ t is thus a piecewise-deterministic Markov processes in the sense of Davis [Dav84,Dav93].

Probabilistic strategy optimization
The governing equation of our continuous-time version of the Dye model, the equilibrium ODE, is based on the assumption that the manager's objective is to achieve the highest possible valuation at all times t preceding the subsequent mandatory disclosure date.However, if management follow the conditional threshold rule with probability π t and otherwise disclose the observation candidly with probability 1 − π t , then, as in (cond-π), for an equilibrium strategy π the corresponding valuation γ t = γ π t satisfies: where t = 0 corresponds to the last public disclosure (after a change of origin here, mutatis mutandis).We rescale the valuation so that γ 0 = 1.Consistently with this last equation, we will employ the notation: γ 1 t for its solution when π ≡ 1 (sparing policy applied throughout), so that With comparison against this solution in mind, the manager is now induced to maximize an objective in selecting π so as to yield As before, t = 0 denotes the most recent time of disclosure and unit time is left to the mandatory disclosure (time to expiry).This objective includes a penalty proportional to (γ t − γ 1 t ).The amended unravelling principle of Section 2 implies that introduction in a market equilibrium of candour (candid reporting) will cause the valuation γ t to exceed γ 1 t and the aim of the penalty is to motivate management into protecting the value of the firm from potential falls in value if a candid strategy is followed for too long (i.e. from excessive use of a candid position).
In equivalent form, the objective may be rewritten as It is thus natural to demand that for some proper interval of time so we make the blanket assumption which enables discounting of γ t by κ t to a level below γ 1 t .

Hamiltonian analysis: Pontryagin Principle
We approach the maximization problem via the Pontryagin Maximum Principle, PMP, for which see [BreP] (esp.Ch. 7 on sufficiency conditions for PMP), or the more concise textbook sketches in [Lib], [Sas], or [Tro].It is also possible to establish the results below by solving the Bellman equation along the lines of Davis [Dav93,p. 165], a matter we hope to return to elsewhere.
In a period of silence, the valuation is deterministic and so we formulate optimisation in Hamiltonian terms.We apply a standard Hamiltonian approach from control theory to maximizing the objective of the preceding section by treating γ t as a state variable and π t as a control variable.Denoting the co-state variable by µ t , the Hamiltonian is by construction linear in π t .So with µ t continuous and piecewise smooth: where we follow the càdlàg convention that π t right-continuous with left limits and satisfies 0 ≤ π t ≤ 1.Thus We now apply the Pontryagin Principle.Evidently, concentrating only on terms involving π t below, the Hamiltonian is maximized by setting π t at 0 or 1 according as It emerges that µ t ≤ 0 (see Appendix A, Proposition 3) consistent with its being interpreted as a penalty term in H, so This gives rise to an optimal switching curve and associated optimality rule: Proposition 1 (Optimality Rule).A necessary and sufficient for π to be optimal is given by the rule: Proof.By the Non-mixing theorem, π t can only take the values 0 and 1 and so by the Pontryagin Principle the optimality condition above is necessary and sufficient: the strong form of Theorem 1 (see Appendix A) asserts that if γ t = γ * t on an interval of time, then π t = 1 on that interval.A corollary of the above form of γ * t now follows.
Qualitative Corollary.A large enough valuation γ t allows sparing reporting, a small enough valuation γ t encourages candid revelation.
Remark.Evidently, the value of π t is not instantly observable, so management may at any instance of bad news (however defined) hide it and so deviate from their prescribed equilibrium strategy.However, systematic deviation of this sort is statistically observable and so deviation leads to loss of reputation, removing the very means by which the firm maintains a higher valuation, which in turn hurts the deviating agent.We therefore assume that managers hold themselves to their prescribed equilibrium strategy.For further background on the Bayesian persuasion aspect here, see Kamenica and Gentzkow [KamG].

Examples of equilibrium behaviour
In this section we give three examples of different equilibrium behaviour in the form of graphs which include the switching curve derived in the preceding section.The role of the switching curve is to confirm, by Proposition 1, the optimality of the equilibrium valuation.

Example 1
Here π = 0 initially (candid).Figure 2a below and Figure 3 later share the same parameters: κ = 0.799432, λ = 0.940489, σ = 4; here θ = 0.260229.Figure 2b illustrates a more pronounced switching curve with κ = 3/13, λ = 9, σ = 4 and θ = 0.175.Commentary to Example 1.Here under silence, initially the switching curve γ * t (shown in red) is above the starting firm value γ 0 = 1 and so candour (π = 0) is initially optimal; it is rational to infer that silence here means managers have received no information, hence the valuation remains unchanged, until the switching time is reached, as signified by the stationarity of the switching curve.Thereafter, γ * t falls below γ = 1 and so the optimal strategy yields a superior valuation to that given by γ 1 t (which would have resulted from a policy of being sparing-throughout, i.e. π ≡ 1); here the valuation ignores the kind of silence that hides bad news.In this time interval γ * t and γ t coalesce, as predicted by the Non-mixing Theorem 1S.With a higher intensity-value λ of the private managerial news-arrival, the switching time would come earlier, thus absorbing the higher chances of ensuing privately received bad news, which strategically wants to be withheld.The figures above graphically depict the reputational benefits to a firm following a candid-first strategy.Starting at t = 0 investors do not downgrade the value of the firm when they see no disclosure, since they infer this follows from non-observation of updated information.The blue curve in both figures remains flat.This is in contrast to how investors treat a firm applying a sparing strategy, for which they continually downgrade firm value in the presence of silence.Thus the distance between the blue and green lines reflects the reputational benefit of following a candid strategy at times.In summary, the reputational benefit is in the fact that investors do not downgrade firm value quite so heavily in the presence of continuing silence.

Example 2
Here π = 1 initially (initially sparing behaviour).In Figure 3 Commentary to Example 2. Here, under silence, initially the switching curve γ * t (red) is below the firm valuation γ t , albeit close, and γ t = γ 1 t is consequently the dynamic disclosure-threshold (sparing policy threshold) curve.Thereafter, γ * t is above the γ 1 t (green) curve, so it is optimal to switch to candour, which yields a constant equilibrium valuation under silence (shown in blue).The valuation subsequently omits to account for the kind of silence that hides bad news.After the disclosure policy switch from π = 1 to π = 0, it is rational to infer that silence means managers have received no new information.With a higher intensity λ of private managerial news-arrival, the switching time would come later, thus absorbing the higher chances of bad news arrival needing strategically to be withheld.

Example 3 (non-constant κ)
Here π = 0 on [0, θ 1 ) with θ 1 = 0.175 and κ = 0.594;followed by π = 1 on [θ 1 , θ 2 ) with θ 2 = 0.85 and κ = 0.533 and finally by π = 0 on [θ 2 , 1].Throughout λ = 3.2 and σ = 2.A choice of piecewise constancy may at first sight seem specious.However, this is an arrangement of a pre-determined managerial reward capable of being agreed by the shareholders, as the switching times are not dynamically selected.Moreover, our analysis with constant κ can be adapted (by reference to the first mean-value theorem of integration) to the general case of continuous κ t by replacing within any inter-switching interval a proposed varying κ t by some appropriate constant value (along the lines of a 'certainty equivalent' relative to the Poisson jumps), that value being intermediate between those taken by κ t on that interval.

Proofs
The proof of Theorem 1 (actually in a stronger form) is in Appendix A.Here we consider Theorems 2a and 2b.We recall that below α t and β t are assumed constant.We begin with some preliminary observations.Proposition 2. (i) On any interval [θ ′ , θ] where π = 1, the co-state equation and solution take the form: so that µ t , being negative, is decreasing with K < 0 and (ii) On any interval where π = 0, Thus here µ t is increasing.
In particular, in both cases µ t is either non-constant or zero.
Proof.Since the co-state equation asserts that ), the conclusions are immediate from the form of the differential equation.
Corollary 1 (Final switching conditions).If the last two intervals of π constancy are given by π = 0 switching at θ to π = 1, then µ t = 0 on [θ, 1] and near and to the left of θ : For the reversed strategy, if the last two aforementioned intervals are given by π = 1 switching at θ to π = 0, then on [θ, 1] Corollary 2. Being candid at all times ( π ≡ 0) is optimal for λ sufficiently low, i.e. below a threshold depending on κ (equivalently, depending on α/β).
We need some details about the function h: Note that h(1) = 0, as Φ(0) = 1/2, and further that h(0) > 0 as Φ(σ/2) > 1/2 for σ > 0. Finally note that h(t) < 1, since h(0) < 1, the latter because If in equilibrium there is just one optimal switching θ, it is straightforward to derive its location property using a first-order condition on θ, as given in Theorems 2a and 2b above.The existence conditions for the equilibria are derived from the general Hamiltonian formulation; this requires the explicit derivation of the switching curve, which is the content of a technical lemma and carries all the work for theorems 2a (and likewise for 2b).

Lemma 2a.
If π t = 0 for t < θ, and π t = 1 for t > θ, is optimal, then Here the switching curve is given by Here, γ * t = γ t , for t > θ.Remark.The graph of γ * t is stationary at t = θ, for, writing ∂ t for the time derivative, Proof of Lemma 2a.We will deduce µ t directly from the formal solution (Appendix A) via the integrating factor ϕ there, which reduces to For consistency we need at t = θ = θ(λ) that As regards the coalescence, note that if θ solves e −λg(θ) = κ, as above, then, for t ≥ θ, rather than the expected inequality (>); so as in Theorem 1S (see Appendix A), and since γ ′ t = −λh(t)γ t , it follows that π = 1.
Finally we compute the rate of change of θ w.r.t.λ from the location condition expressed as So the larger is λ the smaller is θ.
From where the horizontal blue line λ = 10 in the figure intersects the green curve one may drop vertically to the red curve to obtain a value of λ which lies below the green and on the red curve and lower blue line.
We again begin with a technical Lemma.
Lemma 2b.Assume that α t , β t are constant.In an equilibrium where π t = 1 for t < θ and π t = 0 for t > θ, one has Here This curve defines the switching time θ as the intersection time t = θ of γ * t with γ 1 t when Proof.As in Lemma 2a, we again deduce µ t directly from the integrating factor (see Appendix A), which here is t) , t ≤ θ, e −λg(θ) , θ < t < 1.
For t > θ, as s > θ and π = 0 below, For t < θ we have We may now prove Theorem 2b.
Proof of Theorem 2b.We begin with the location condition.Since π t = 1 on [0, θ), γ t = γ 1 t ≡ e −g(t) on [0, θ).Here the objective function reduces to since there is a zero contribution to the objective function on [0, θ].Differentiation w.r.t.θ yields the first-order condition which on re-arrangement yields the claim.We turn to the existence condition.By Lemma 2b, intersection at t = θ of γ * t with γ 1 t occurs iff Combining this with the requirement that γ * 0 < 1 yields This holds for some κ iff yielding a bound on λ in terms of the switching time θ (as illustrated by a green graph in Figure 6 below).From here we obtain in turn a lower bound on θ (as illustrated by a red graph in Figure 6).Finally we compute the rate of change of θ w.r.t.λ from the location condition.Here for θ = θ(λ) we have, as h ′ (θ) < 0, Remark.In Figure 6 above the red curve traces possible values of the function in the first condition (1) above (and in Theorem 2b earlier); the green curve corresponds to the function in the second display (2) above.The blue curve identifies the θ value given a horizontal λ value.Thus λ must lie on the portion of the blue curve lying in between the red and green.

Double and multiple-switching
This section continues to study multi-switching, as a complement to our main theme that single-switching should be regarded as the preferred equilibrium type.We saw this is the case only for a candid-first equilibrium.Below we explain the numerical evidence for the absence of a second switch in a sparing-first equilibrium.
We begin with a useful observation which refers to an equilibrium condition that must hold at an optimized switching location.
Lemma 3. In an equilibrium with two optimal switching points 0 < θ 1 < θ 2 < 1 for which γ * t < γ t on [0, θ 1 ), where π = 1, it is the case that λ < η(σ) Proof.See Appendix B. .Initial sparing policy: numerical evidence for the absence of a second switching point.The existence of a second optimal switching time for a sparing-first equilibrium requires by Theorem 3b that θ = θ 1 , for θ 1 the first switching point, solve since g(θ 0 ) = g(0) = 0; this would yield a second switching point at θ = θ 2 given by .
By Proposition A below, this requires either λ to be small enough or κ large enough (otherwise T (θ) > 0).By Lemma 3, we may study the solvability of the equation T (θ) = 0 by taking λ = λ max = η(σ) = −h ′ (0)/h(0) 2 and graphing against the aggregate volatility σ the expression with κ near unity, say at 0.99, to determine whether or not T (0) > 0. It turns out that T (0) < 0 on a wide range of σ.Furthermore, the sign of is also found to be negative on a wide range of σ.Thus numerical evidence supports the hypothesis that T (θ) remains negative and bounded away from 0, so that the equation T (θ) = 0 is insolvable, and hence there does not exist a second switching point.This implies that for κ t constant there is at most one switching point in an equilibrium which initially implements a sparing policy.
Remark.In general, how many switches should there be in equilibrium?This question can be resolved by reference to the preferences either of the managers or alternatively of a representative investor in the market.The former chooses the policy with higher reward to the managers; here, the payfor-performance sides with multiple switching, since it boosts the valuation and so also the managerial payoff.Resolution by reference to investors requires the introduction of a notional cost of complexity.But the underlying assumption must remain that the market has found its way to an appropriate equilibrium -cf.§2.
We return to examining the formula which defines for any optimal switching point θ the linked (coupled) successor switching point in terms of model parameters (κ, λ and σ, the latter embedded in h).The constraint T κλ (θ) ≤ 1 holds iff placing a lower bound on κ in terms of λ (and σ).This implies that, for given σ and λ, there is a critical value κ, such that the key equation above is insolvable for κ < κ.It is also true that, for all small enough κ, there is a critical value of λ, above which the key equation T (θ) = 0 is not solvable, as clarified by the next result.
Proposition A. The following inequality holds Proof.See Appendix B. .

Remark.
To see why this last result holds, note that expansion of g near θ 1 gives a first approximation to the right-hand side of Using the approximation log(1 + x) ≃ x for x small with x = κ − 1 yields This shows both sides can be close for κ close to 1, in view of the comparison However, for small κ, 1−κ λκ is too far from 1−κ λ and so even double-switching is then not possible.
We close this section by noting a complementary equilibrating result linking the switching point of Theorem 3a to the prior switching point.Corollary 1 in §5 implies that the switching curve is made to fall on intervals where π t = 0 by a corresponding linear decreasing effect of |µ t |, helped by the decreasing nature of the instantaneous protective-put, h(t).Contrarily, it is made to rise on intervals where π t = 1 by the ensuing exponential nature, e λg(t) , embedded in |µ t |.It is not therefore surprising to see h(t)e λg(t) as the equilibrating factor in (equil) below.Since h(t)e λg(t) has a maximum at some Proposition B. As in the setting of Theorem 3a, for each switching point θ i that is a right endpoint of an interval with π = 1 constancy, the preceding switching point satisfies the equilibrating relation Proof.See Appendix B.

Conclusions
The disclosure model of Dye [Dye] alerts us to consider the implications of firms remaining silent between mandatory disclosure dates.The model developed here shows why management of a firm may benefit from establishing a reputation for being candid on some time intervals, voluntarily disclosing all news, good or bad.At issue is when would one expect to see such behaviour in an equilibrium and whether it is likely to be time-invariant, once established.Corollary 2 (Section 5) establishes that if the news intensity-arrival rate is sufficiently low an equilibrium exists in which managers are always candid.
The comparative statics of θ(λ) in Theorem 2a establishes that, as the newsarrival rate rises, eventually the optimal policy for management is to switch to a sparing disclosure policy; a similar effect is caused by the remaining factors in the model, namely of time-to-expiry (to the next mandatory disclosure time) and pay-for-performance ratio κ.In the model, reputation for adoption of a candid disclosure strategy is derived endogenously and we see that for higher levels of news-arrival such a strategy will not be time-invariant -managers will start to 'burn their reputation' (switching from candid reporting to sparing disclosure) the closer they get to a mandatory disclosure date.If the aim is to understand asset pricing in a continuous-time setting, then this model provides insights into how firm management will voluntarily disclose information to update markets in between mandatory disclosure dates.Litigation concerns may ensure that very negative news is always disclosed; nevertheless, as this model shows, once management switch out of candid disclosure into a sparing policy they will tend to hide "slightly" negative news, both when close to a mandatory disclosure time and when their private news-arrival rate is higher.
Proof.This follows again from the blanket assumption and from Theorem 1S (Non-mixing Theorem -Strong Form).Assume α t , β t are constant.
(i) If the state trajectory and switching curve coalesce on an interval I, then π ≡ 1 on I.
(ii) A mixing control with π t ∈ (0, 1) is non-optimal over any interval of time.
Proof.If a mixing control occurs, then, from the Hamiltonian maximisation, it follows that γ t = γ * t on an interval of time; hence (ii) follows from (i), by contradiction.To prove (i), we compute in Step 1 the corresponding control π t from the equilibrium equation and then, in Step 2, show that this control does not satisfy the co-state equation unless µ t ≡ 0, in which case π t ≡ 1.
Step 1.Since γ t and so also γ * t satisfies the equation (cont-eq-π) on I, For ease of calculations, write Then, as ( So, from the equation (cont-eq-π), Substituting for ψ and ψ ′ and writing h t for h(t) yields Step 2. We now substitute this value for π t into the co-state equation, (as κ = 1 − α/β).We compute π t from the co-state equation to be the division being valid, since µ t λh(t) ≤ 0 and βκ > 0. So, Consistency of the two formulas requires that Since the µ ′ t terms cancel on each side, this last holds iff Computing the left-hand side, using the definition of ψ t , gives So, again using ψ t , implying, as ψ t > 0, either µ t = 0 or λh(t) 2 = −h ′ (t).The latter gives But on I this contradicts So consistency requires that µ t = 0 on I.But in this case the co-state equation, Thus g(0) = 0 and g(1 So
To abbreviate the (constant) expressions independent of θ, we will write Here the impact on the objective function reduces to since there is a zero contribution to the objective function on [θ i−1 , θ], where π = 1.Differentiation w.r.t.θ yields the first-order condition at θ above), so κγ = κ i e λg(θ i ) , so a rearrangement yields the claim.

Proof of Corollary 5. Consider a pair of intervals: [θ
or equivalently ), so by Theorem 2 ′ (and since γ i = γ i−1 e λg(θ i−1 )−λg(θ i ) , as noted earlier) From here But at the same time This equation coupled with the earlier one yields the claim.
Remark.Other than the single equation in one variable when the equilibrium demands that π = 0 is applied on [0, θ 1 ), the optimization conditions yield linked pairs of simultaneous equations reduceable to a single equation involving only earlier switching points.
Remark on piecewise constant κ t .The argument proving Theorem 2 ′ about optimal switching points, allows κ to be piecewise constant with constancy between switching points.An example is illustrated in Figure 3 in Section 4.2.For suppose we have κ = κ − = κ θ i−1 to the left of θ i and κ = κ + = κ θ i to the right (i.e.càdlàg as before) that κ = κ θ i on [θ i , θ i+1 )), then the corresponding formulas remain the same with only a change to the definition of κ i , thus However, monotonicity claims concerning κi would now rely on assumptions about the sequence κ θ i .
Indeed, suppose π = 1 is intended for t > θ i , then we can interpret κ as being κ − to the right of θ i without altering the payoff (because of the factor 1 − π t ).We then get e −λg(θ i ) = γ i κ − .
Similarly, if π = 0 is intended for t > θ i , i.e. π = 1 is intended for t < θ i , then we can interpret κ as being κ + to the left of θ i again without altering the payoff.We then get .
This agrees for κ constant with the previous context.
Finally, we show that there critical pairs κ, λ with insoluble paired linking equations.Notice that for 0 < η < h(0) the map λ → κ(λ) = (1 + ηλ) −1 is decreasing in λ and also in η.For large enough λ, this map will assigns to λ a small enough kappa for which the linking equation must fail.We may thus term λ → κ(λ) = (1 + λh(0)) −1 the killing map for λ.At that level of κ there cannot be any further switching.
The proof uses compactness (equivalently uniform continuity).
So we consider the stronger condition .
By continuity at k of g(T −1 (.)) and of g(), for some δ k > 0, there is an interval The equivalent statement is clear.