Financial Statements Insurance

The largest corporate bankruptcy filed in the United States, that of Enron in 2001, was preceded by a string of disclosures about audit failures, and errors in financial statements.1 The presence of such errors highlights the fact that market participants face two inter-related problems when pricing securities based on financial statements. First, they must assess the quality of the information contained in financial statements. Second, they must make projections about future cash flows and fold these projections back into an appropriate value for the security. Even if one assumes that accurate models are available for projecting cash flows and valuing securities, uncertainty about the quality of financial statements can lead to pricing distortions and inefficient market allocations of capital. The objective of this paper is to develop a market-based mechanism that can lead to a timely disclosure of financial statement quality and, thereby, a more efficient allocation of capital. We show that our proposed mechanism improves social welfare.2

The cascade of recent audit failures has given rise to a regulatory initiative, the Sarbanes-Oxley Act of 2002 (the Act), and to an ever growing commentary on ‘corporate governance’. A major theme of this literature is the role of ‘gatekeepers’ and, in particular, the failure of auditors to fulfill their role as independent gatekeepers.3 Indeed, the issue of auditor independence (or its absence) has occupied a major place in the debate over the failure of corporate governance and is the focus of much attention in the Sarbanes-Oxley Act.4 The Act seeks to address the problem by increased regulation and penalties, empowerment of audit committees, and reduction of the auditor's involvement with the client.5 But the Act does not untie the auditor/management knot: auditors continue to be hired and paid by the firms they audit.

Without joining the debate about the effectiveness of the Act,6 and as an alternative to, or supplement to, its mandates, this paper introduces a market-based financial statements insurance scheme (herafter, FSI)7 designed to eliminate conflicts of interest that inhere in the auditor–client relationship and, at the same time, to signal credibly the quality of financial statements. The model developed in this article shows that such a scheme would allow more accurate inferences regarding future cash flows to be drawn from financial statements, and thus permit more efficient resource allocations.

The social value of ex-ante self reporting has been recognized in the literature (Kaplow and Shavell, 1994). In essence, the FSI mechanism involves induced truthful ‘self reporting’ through the auditor's attestation of the quality of the financial statements even when such quality is poor and expected to trigger market sanctions when made public.8 In contrast, the current structure of incentives driving auditors' behaviour may not elicit unbiased reports. Auditors are paid by the companies they audit; this creates an inherent conflict of interest that is endemic to the relationship between the manager (the principal) and the auditor (the agent). We analyze, first, the financial statement quality equilibrium under the existing legal and regulatory regime governing auditing when managers obtain significant benefits both direct and indirect when they successfully raise capital. Defining quality as the inverse of the probability of overstatement in financial statements, we show that the natural equilibrium is one where lowest quality financial statements are chosen (resulting in a high probability of overstatement in the financial reports). Under these circumstances, firms that potentially yield a low rate-of-return (low fundamental value) cannot be identified easily and are overfunded relative to firms characterized by a high rate-of-return (high fundamental value). We show analytically that the introduction of FSI can lead to a better assessment of financial statement quality resulting in a much more accurate identification of low value firms.

The basic structure of FSI can be described as follows (details may be found in Appendix B, based on Ronen (2002)).9 Instead of appointing and paying auditors, companies purchase FSI that provides coverage to investors against losses suffered as a result of misrepresentation in financial reports. The premiums paid for that coverage are publicized. The insurance carriers appoint and pay the auditors, who attest to the accuracy of the financial statements of the prospective insurance clients. Those firms announcing lower premiums distinguish themselves in the eyes of the investors as companies with higher quality financial statements. In contrast, those with higher premiums reveal themselves as firms with lower quality financial statements. Every company will be eager to get higher coverage and pay lower premiums lest it be identified as the latter. A sort of Gresham's law would be set in operation, resulting in a flight to quality.

According to sound principles of corporate governance, auditors are supposed to be the agents of the shareholders. However, in practice, although shareholders (and audit committees) vote on management's recommendation of which auditor to hire, it is the management of the company that effectively engages the auditor and ultimately pays for the services. The fact that CEOs and CFOs control the fees paid for auditing and consulting services allows them to elicit actions, including opinions and assurances, that it desires from the auditor. The risk of losing fees from a long-term audit engagement—even in light of the limitations on non-audit services imposed by the Sarbanes-Oxley Act of 2002—may secure auditor compliance with management's objectives. We argue that the imperfect alignment of interests between managers and shareholders and the intractable conflict of interest imposed on auditors can be rectified through an agency relationship between the auditor and an appropriate principal, whose economic interests are aligned with the goal of promoting the quality of the financial statement.10 Within a free market mechanism, insurers can serve the role of such an intermediary.

The critical features of the FSI scheme underlying this study are: (a) the effect of publicizing the premium charged to different firms; and (b) the shift of control over the auditor's compensation and, hence, incentive structure from management to the insurer. We seek to formalize these two features and to demonstrate that FSI, when linked with appropriate disclosure provisions, leads firms to improve the quality of their disclosures voluntarily. A key economic feature underlying this result is the fact that the insurer's primary business is providing coverage and insurers are primarily asessed on whether their polices are generating profits. In contrast, an auditor's primary business is providing the audit service and the allocation of fees across services and coverage can be quite arbitrary. For this reason, the insurer sets the premium at a break-even level at least, whereas the auditor wishes to break even across the joint payments for audit services and indirect litigation coverage.

Model

We develop a model in which firms try to attract capital through their financial reports.11 The firm's management benefits from obtaining capital, but there is a social waste if firms with low rates-of-return are funded.12 We consider an economy with N firms, where each firm is of type i, i = 1, … , L where a type i firm will earn return r_i with r₁ < r₂ < … < r_L. The type of each firm f is drawn randomly by nature at the start of the period and is independent of other firms. The true type drawn by nature is unobservable at the start of the period but the managers of each firm f obtain a private signal, ω_f, about their firm's type where ω_f ∈ {ω₁, ω₂, … , ω_L} represents the set of L possible signals observable by the firm. We denote by P(i|ω_l) the probability that the end-of-period rate-of-return will be r_i for a firm f that receives a private signal, ω_l.

The strategic tool for obtaining capital is an audited financial report that is issued to investors. Although this report may not be directly falsified, it can be manipulated indirectly through a reduction in the quality of the statements.13 Based on his or her private signal, ω_f, the firm's manager chooses accounting policies that determine the internal quality of the reporting system, denoted by q, where we assume . The overall financial statement quality, x, is determined both by the firm's choice of internal quality, q, and the auditor's effort, e which lies in some real interval . The overall quality is determined as the function x = V(q, e) where V is strictly increasing in both q and e. We shall use x to denote the ordered pair {q, e} and x to denote the value V(q, e). When dealing with ordered pairs we use the natural coordinate order as follows: and e′ ≤ e. In contrast, the statement y ≤ x is an assertion that V(q′, e′) ≤ V(q, e) and does not indicate that each component is lower. Clearly, but the converse may not be true. For a given e, V (q, e) is a strictly increasing function of q and therefore, there is only one value of q such that the given value x is the same as V (q, e). For this reason, any function of q and e can also be treated as a function of x and e.

After firm f's type, i, and associated rate-of-return r_i, is realized, a financial report θ_f ∈ {1, … , L} is disseminated to investors. Associated with each report θ_f = j is the rate-of-return r_j, which we shall refer to as firm f's reported rate-of-return. r_j may, of course, be different from the actual rate-of-return of firm f, r_i. However, investors will not blindly accept r_j; instead, they will use this implied (by the report) rate as well as their perceptions about audit quality to ‘infer’ an expected rate-of-return for firm f.

To characterize this inference process, let P(i|j, x) denote the probability that the realized rate-of-return is r_i given that the reported rate-of-return is r_j and overall quality x. Investors observe the report θ_f = j but they cannot observe the choices q and e and hence x. Therefore, investors make conjectures about x and use these beliefs to infer a rate-of-return based on a report θ_f = j. Denoting investor beliefs about the overall quality of firm f by ν_f, we write P(i|j, ν_f) for the posterior probability assigned by investors to the event that the true rate-of-return is r_i given that the reported rate-of-return is r_j and the conjectures about the level of x. The inferred rate-of-return associated with this posterior probability distribution, :

()

We assume that there is a minimum threshold rate, r*, such that funding firms with rates of return less than r* results in a social loss. r* is a random variable with a distribution G(r*) that represents the social cost of capital. r* is assumed to be independent of firms' rates of return and G is assumed to be convex.To exclude the trivial cases (a) where all firms should be funded (and there is no social loss resulting from misleading statements) and (b) where even the highest-type firm does not merit funding (and capital providers will not enter the market) we assume that r* is distributed over some interval where and . The simplest example of a convex G satisfying these requirements is when r* is uniformly distributed over the interval .

In the first-best scenario, in which the true rate-of-return is perfectly observed, only firms with rates of return higher than r* will obtain capital. In a second-best scenario, in which investors do not know each firm's type, they analyze the report, θ_f = j, and fund firm f if the inferred rate-of-return, , is greater than the threshold rate r*. For simplicity, we assume that whenever , investors contribute one unit of capital to the firm.14

The managers of a firm typically benefit in both pecuniary and non-pecuniary ways from capital inflows.15 We represent the (portion) of the value of the firm appropriated by management by B (for benefits). In other words, by ensuring a capital inflow, the firm's management generates both a return which is passed back to shareholders and a benefit B for themselves. In this setting, low quality financial statements that misdirect capital lead to two basic types of losses (viewed from the perspective of investors).

First, when a firm of type i with (r_i < r*) is funded (because the inferred rate-of-return ), the investor suffers a a loss of 1 × (r* − r_i) (recall that the investment involves one unit of capital). While such losses are straightforward, there is a second type of loss that also results from inferior accounting quality. Because investors are unable to distinguish the high type firms from low type firms, some high type firms may not be funded. So, if a firm k of high type, r_k > r* is not funded, investors lose the amount 1 × (r_k − r*). The total cost of misinvestment is the sum of these two losses (see example 3.2 for a complete loss calculation). That is, an inflated financial statement not only draws capital towards an inferior firm, it also indirectly starves superior firms (whose ‘honest’ reports are discounted in the same way as ‘dishonest’ reports) leading to both an actual loss and an opportunity loss.

Funding and Managerial Benefits

Managers choose the overall quality of financial statements to maximize their own benefits. These benefits derive from a successful ability to raise capital. The manager's ultimate payoff depends both on firm value and success in raising new capital. Firm value depends on the return on investment less any potential damages that may have to be paid (to pre-existing shareholders) for inflated financial reports. In addition, managers derive a direct benefit, B, whenever they are successful in raising capital.19 We assume, for simplicity, that all firms have an investment, I, absent any new capital, or the amount I + 1 if new capital is raised. Managers may appropriate a portion B of the new capital as direct benefits.20 In this setting, denoting the probability of funding firm f by FP_f, the expected firm value when a private signal ω_f is observed, x is implemented, and audit fee F(x, e) (defined in the next section) is paid may be written as:

()

Assuming that the managers own α of the firm, and given that they expropriate B, the managers payoff function is:

()

We shall assume that the manager benefits more from direct consumption than he or she loses in firm value as a result of his or her consumption. That is,

()

If insurance is available with an associated premium π_f, then the managers' payoff may be written as:21

()

where is the overall quality reported by the auditor to the insurer (see the next section). After receiving a private signal ω_f, the managers choose x so as to maximize the payoff in equation (3) or (5) depending on whether insurance is available.

The last part of our model development concerns the role of the auditor.

Auditor Incentives and Decisions

The auditor is the principal informational intermediary in the trading of securities. In our model, we take as a given that the auditor must supply information to investors but that the quality of this information depends on the incentives provided by the auditor's employer (either the firm or the insurer). Our goal is to study firm–investor interactions and not audit contracts or auditor behavior. To this end, we focus solely on whether the auditor's incentives lead to the true overall quality, x, being revealed to investors rather than on whether auditors will ‘shirk’.

The auditor is assumed to be risk-neutral and to face a cost of effort denoted by C(e). Let F(x, e) denote the fee that ensures that the auditor breaks even, that is,

()

We assume that the fee F(x, e) will induce the auditor to exert the effort level e required by the auditor's employer. The ability to impose ex-post penalties, such as adjustment of fees or refusal to renew the auditor's contract will force the auditor to implement the level of effort e desired by the firm (or insurer) as long as F(x, e) is large enough to cover both direct, C(e) and indirect costs, .22 For a given level of overall quality, x, q*(x) and e*(x) denote the choices of q and e that minimize the total cost of .

Suppose FSI is available, a further consideration enters the picture—the report to the insurer about the choice of x. We denote this report by and write for the fees when the auditor is incentivized to report while the implemented overall quality of the firm is x. We distinguish between two cases: (a) auditor hired by the firm and (b) auditor hired by the insurer. We assume the auditor's report to the insurer is private and does not increase the auditor's exposure to litigation. We assume that the insurer can demand an ex-post adjustment if through increased future premia when the firm hires the auditor or through a transfer from the auditor when the insurer hires the auditor. We denote this adjustment by . The auditor is paid a fee commensurate with the reported to the insurer. In addition, when the auditor is hired by the insurer, the auditor's initial fee will be paid by the insurer, but reimbursed by the firm and the fee will not be affected by the overall quality level x. Gathering all these points together, the audit fee paid by the firm, , has the following structure:

(auditor hired by the firm and the fee paid directly by the firm):

(FSI: auditor hired by insurer and the fee paid by the insurer but reimbursed by the firm):

()

We note, however, that under FSI (where the insurer hires the auditor), the total payoff of the auditor , is net of the ex-post adjustment and thus depends on all three variables, , x and e. To minimize , the auditor is thus induced to report resulting in . In this way, the strategy whereby the auditor reveals x truthfully to the insurer in equilibrium is implemented. To sum up, when the firm hires the auditor, the auditor can be costlessly induced to report the overall quality level favoured by the firm (see also Proposition 1) whereas when the insurer hires the auditor, ex-post transfers induce the auditor to report quality truthfully to the insurer.23

Formalization

We now state the formal programs for the strategic choice of overall financial statement quality for a firm f receiving a private signal ω_f under several different economic regimes. There are two basic aspects of FSI that we need to examine:

• (1) the effects of the public disclosure of FSI premia;
• (2) the economic rationale for allowing the insurer to control the auditor's contract.

While both these features are part of FSI, they are driven by different incentive problems. The first incentive problem relates to the fact that the client-firm increases its likelihood of acquiring capital by implementing low x. The second relates to the fact that the client-firm would prefer, irrespective of the actual choice of x, that the auditor reports a high x implementation. While both incentive problems arise from the desire of the client-firm to acquire funding, the mechanisms required to mitigate the two problems are distinct. To clarify this point, we analyze four distinct programs (the time line for each of these programs is shown in Figure 1):

• (I) The current regime where information about x is unavailable prior to the funding decision and investors base their decisions on (ex-ante) beliefs. The auditor is hired and paid by the client-firm and may not be truly independent.
• (II) Insurance is available; the auditor is hired by the client-firm and is truly independent. That is, the auditor cannot be induced to overstate x (although he or she can be induced to lower audit effort). Investors update their beliefs about x after observing the insurance premium.
• (III) Insurance is available and the auditor is hired by the client-firm and can be induced by the client-firm to overstate x. Investors update their beliefs about x after observing the insurance premium.
• (IV) FSI is available, premia are disclosed and the auditor is hired by the insurer. Investors update their beliefs about x after observing the premium.24

Before stating the programs, it is necessary to outline the process by which investors derive their inferred returns and describe how it influences the manager's decision problem in implementing the optimal x (details are provided in Appendix A). This outline also highlights the differences in the structure of investor beliefs across Setting I where the insurance premium is not observed and Settings II–IV where it is observed.

In Setting I, investors start with prior beliefs represented by where ν_l denotes the beliefs regarding the (strategically optimal) level of x implemented by a firm f with private signal ω_l.25 Thus, in Setting I, the inferred return for firm f upon observing a report θ_f = j under beliefs is given by (see Appendix A for the details):26

()

In Settings II–IV, investors observe the premium, π_f, charged to firm f and update their beliefs. The updating process involves two steps: (a) investors will update their beliefs regarding the quality chosen by firm f to a value x_f consistent with the premium π_f; and (b) they will update their beliefs regarding the private signal ω_f based on the quality associated with π_f. For example, if firm f is observed to pay the lowest premium corresponding to the highest x, investors will infer not only that firm f has chosen the highest possible x, but may also infer that the choice of the highest x indicates a high ω_f as well. The corresponding expected return (also derived in Appendix A) is given by:

()

In either case, the inferred returns determine the funding probability contingent on choosing overall quality x. Denote this probability by in Setting I and FP_f (x|ω_f, π_f) in Settings II–IV. The decision problem for firm f's manager is the choice of quality x_f = {q_f, e_f} that maximizes the benefits of funding net of the cost of implementing x_f.

The insurer is assumed to break even through a suitable choice of ex-ante premium, π_f, and an ex-post adjustment. This assumption of ex-post break-even is common in the insurance literature and represents the ability to impose costs on the insured in the form of higher future premia. In our context, we assume that the insurer sets an initial premium based on the auditor's report regarding the overall quality of the financial statements and imposes an ex-post adjustment (on firm or auditor depending on the context) that allows the insurer to break even. However, the insurer does not wish to depend on this ex-post adjustment and tries to make it as small as possible. Lastly, we assume that the audit fee, F(x, e) in Programs I, II & III, and in Program IV, implements the audit effort e and report by the auditor.

Program I: Current Regime

maxq,e,F
subject to
		(AF)
		(FD)
	νf = x*(ωf) = (q*(ωf), e*(ωf))	(RE)

where (AF), (FD) and (RE) are respectively the Audit Fee, Funding Probability and Rational Expectations constraints.

Program II: Premia Disclosed with independent auditor hired by the firm and reports

maxq,e,F
subject to
		(AF)
		(FD)
		(BE)
	νf = x*(ωf) = (q*(ωf), e*(ωf))	(RE)

where (BE) is the insurer break-even constraint.

Program III: Premia Disclosed with a (non-independent) auditor hired by client-firm

subject to
		(AF)
		(FD)
		(IP)
		(CO)
		(BE)
	νf = x*(ωf) = (q*(ωf), e*(ωf))	(RE)

where (IP) and (CO) are respectively the insurance premium and competitive constraints.

Program IV: FSI

maxq,e,F
subject to
		(AF)
		(FD)
		(IP)
		(CO)
		(BE)
	νf = x*(ωf) = (q*(ωf), e*(ωf))	(RE)

The objective function faced by managers in all four programs is to maximize benefits less expenses. Their goal is to maximize their own perquisites by boosting the firm's capital base through their reporting strategies less the expected cost of being over-aggressive. The main differences in this equation across the programs stems from disclosing the insurance premium (absent in Program I and present in all the others) and the nature of the ex-post transfer to the insurer (absent in Program II, made by the firm in Program III and by the auditor in Program IV). The insurance premium is revealed to the market in Programs II, III and IV and affects the probability of being funded. However, masking the overall quality of the financial statements (i.e., inducing ) results in an ex-post transfer, from the firm in Program III and from the auditor in Program IV.

The (IP) equation represents the level of the premium set initially by the insurer based on the reported to them by the auditor. This premium is disclosed to investors as a signal on the overall quality. (Note that in Program II and IV, because the auditor either reports the the truth because he is independent or is induced to do so in equilibrium.) To keep the programs comparable, the cost of the audit is always borne by the firm. In other words, even when the insurer hires the auditor (under FSI), the costs of the audit are reimbursed by the client-firm.

The (BE) equation represents the break-even condition for insurers. Both this equation and the (CO) condition are motivated by an assumption of perfect competition in the insurance industry. The (CO) condition is based on the logic that the firm can, if it chooses to do so, reveal its true overall quality to the insurer and obtain a fair premium. The insurer, in turn, is not worried about undercharging on the premium because the insurance contract allows for an ex-post adjustment if the initial assessment of x is erroneous.

In all four settings, the cost to investors of false reporting has a direct component that is remedied (at least partially) through the penalties imposed by the litigation system, and an indirect one measured as misapplied investment. As we are interested only in the relative levels of the indirect cost, we hold the direct costs, that is, the litigation penalties constant across regimes and ensure that these costs are borne by the client-firm in all settings. In Program I, this cost is paid directly by the client-firm. In Programs (II)–(IV), the expected costs are paid out by the firm as an insurance premium (note that with risk neutrality, the actual uncertain cost is the same, in utility terms, as the expected cost to both client-firm and insurer). In Program II, where the auditor is truly independent, x is assumed to be known by the insurer whereas in Programs III and IV the insurer has to rely either on the firm or the auditor to make an assessment of x.

The critical difference across Programs III and IV lies in the ex-post settlement associated with the (BE) constraint. In Program III, the insurer breaks even by settling up with the client-firm through a premium adjustment, whereas in Program IV, the settling up takes place with the auditor. In fact, the structure ensures that in Program IV and ex-post transfers do not take place in equilibrium.27 In Program III, the true overall quality of the financial statements is identified only ex-post through the litigation discovery process and if benefits to funding are large enough, the firm is willing to pay these ex-post transfers (after the true quality is revealed) in exchange for the ex-ante benefits of funding.

The (RE) equation expresses the rational expectations constraint that the beliefs about the x implemented by a firm with private signal ω_f, ν_f coincides with the actual equilibrium choices, {q*, e*}. In contrast, the objective function of the manager is maximized holding beliefs constant. This structure is chosen to incorporate the following two economic features: (a) managers have the ability to set low x's without being detected ex-ante but (b) market beliefs will stabilize in equilibrium at x levels that will not be undercut by the manager.

We now turn to our main analysis concerning the equilibrium levels of x that are chosen by firms under each of the Programs (I)–(IV). In Programs (II)–(IV), the insurance premium is observable by investors. When the firm implements its optimal level of x, it has to take into account the reaction of investors to the financial report that will eventually be issued, and this reaction depends either on prior beliefs regarding x (Program I) or posterior beliefs formed after observing the insurance premium charged to the firm (Programs (II)–(IV)). x is assumed to be observable to the insurer in Program II but not observable in Programs (III)–(IV).

Results

To explain the dynamics of investor–firm interactions arising from the introduction of FSI, we analyze the equilibrium in all four programs. Our goal is to show that the implemented x is highest when insurance premia are revealed ex-ante to investors and the auditor is an agent of the insurer rather than the firm. We start with the simpler setting where the auditor is truly independent and reports the level of x implemented by the firm to the insurer. Note that this does not affect the audit effort or the probability of errors in the financial report. All it does is to ensure that the premium truly reflects the x implemented by the firm. We show that in this case (Program II), a simple disclosure of the FSI premium leads to a race to the ‘top’ and all firms implement high x. In contrast, if the firm can induce the auditor to misreport x to the insurer as in Program(III), all potential benefits of providing insurance are lost providing a rationale for the auditor to become an agent of the insurer as in Program (IV).

We begin with a lemma that lists the basic properties of the inference process. Before stating the lemma, it is useful to summarize the notation about funding probabilities. FP_f (j|ν_f) has been defined earlier as the probability that a firm f with report θ_f = j will be funded (under posterior beliefs ν_f). Given these probabilities, firm f, by choosing overall quality x, will have an (expected) funding probability, , given by:

()

In other words, the probability of being funded on a report j depends on investor beliefs ν_f and this is multiplied by the probability of getting a signal j which depends both on the firm's private signal and on the overall quality choice. For a given firm f, let denote the inferred rate-of-return under beliefs and observed report θ_f = j (we suppress the dependence on ω_f and for notational clarity). In addition, in the special case when the beliefs are that the firm has chosen quality x (i.e., ν_f = x), we denote the vector of inferred returns by where . With this notation, the next lemma lists a number of results that link inferred returns and funding probabilities using the information structure in Assumption 1.

Lemma 1. (Reports, inferences and funding probabilities) Let ν_f denote the posterior beliefs about the implemented x and private signal type of firm f, and denote the inferred rate-of-return under belief ν_f.

• (1) A higher reported rate-of-return implies a (strictly) greater inferred rate-of-return that is, is increasing in j.
• (2) Let ω_f denote the private signal observed by firm f. Then the firm's expected probability of funding by choosing quality x is given by:
  ()
• (3) Suppose that σ_f and ν_f are two different posterior beliefs about firm f with the following properties:
  • a) Under ν_f, investors believe that x_f = x and ω_f = ω_l with the prior probability P(ω_l).
  • b) Under σ_f, investors believe that x_f = x and ω_f = ω₁ (with probability 1). (i.e., under belief ν_f, all types pool at x whereas under σ_f, only the ω₁-type picks x).

Then the inferred return E[r|j, ν_f] > E[r|j, σ_f] for every report j and FP(x|ν_f) > FP(x|σ_f).

• (4) Let ω_f denote the private signal observed by firm f and be two distinct levels of overall quality. Let R^x, R^y, denote the associated vectors of inferred returns (under beliefs ν_f = x and ν_f = y respectively) and P^x, P^y the probability vectors of observing a report j under quality x, y respectively. Then there is a column-stochastic matrix Γ^xy satisfying both the following (vector) equalities:
  
• (5)

  Let r* be given and be two levels of quality. Let j, k denote the lowest reports such that and . Then

  

Proof See Appendix A. □

The next lemma shows how the probability of funding for firm f changes in the x choices.

The results in the two lemmas above help to derive the equilibria under the current regime when investors only discover x ex-post through litigation and in the setting where the disclosure of the insurance premium provides ex-ante information to the market.

Proposition 1. (Equilibrium in Program I (current regime)) If the benefits from funding, B, are such that for every y < x:

()

(i.e., B is large enough to ensure that the benefits from funding exceed the incremental cost from increasing overall quality for each private-signal type), then the equilibrium overall quality level chosen by the firm is , that is, the lowest possible level of overall quality. Consequently, as benefits to the manager from funding increases, capital is allocated to low rate-of-return firms with greater probability.

Proof Before proceeding to the proof, we note that Lemma 1 (1) implies that the term in parentheses on the left-hand-side is positive and that 4 implies that the right-hand-side is positive. The mean-value-theorem of calculus ensures that for any x = {q, e} and y = {q′, e′}:

()

As a result, the condition in 12 ensures that

()

However, the inequality in 14 ensures that the manager's payoff (given in equation 3) is higher with the choice of y than with x. Since x and y were arbitrary choices, the manager maximizes utility by choosing the lowest possible overall quality . Note also that B* is any value satisfying 12 with equality, then any B > B* also sets off a race to the bottom with regard to overall quality. □

Proposition 1 shows that if the benefits to funding are large enough, it sets off a race to the bottom in terms of overall quality. We now proceed to analyze the effects of introducing FSI. Before presenting that argument, we note that if then the break-even premia associated with these quality levels, π(x) and π(y) satisfy π(x) < π(y). This is an immediate consequence of Assumption 1.

Assume that each firm purchases insurance and that the premiums charged to firms are observable. Suppose now that in equilibrium some firm f sets x_f < x_L, where x_L is the quality choice of a firm receiving the highest possible private signal ω_f. Then, the premium charged to firm f, π_f, is strictly larger than π_L—and investors will infer that firm f is of some type other than L. Thus, the inferred rate-of-return conditional on observing π_f will be different from that based on the prior beliefs, . We will show that the disclosure of π_f and attendant changes in the inferred rate-of-return lead to an equilibrium where all firms pool at the highest level of internal quality.

Proposition 2. (Equilibrium with revelation of premia (Program II)) If the benefits from funding, B, are such that for every y < x:

(i.e., B is large enough to ensure that the benefits from funding exceed the incremental cost from increasing overall quality for each private-signal type), then the equilibrium overall quality levels chosen by the firm is to set , that is, the highest possible level of overall quality. The association between management benefits (perquisites) and misallocation of capital is negated through the provision of insurance premia that are publicly disclosed (note, however, that in this program, we assume that the auditor will not misreport x to the insurer).

Proof Before proceeding to the proof, we note that x and y have been interchanged in the term in parentheses on the left-hand-side, and it is positive from Lemma 1 (2). Let denote the highest quality choice of q and e. Denote the associated break-even premium by . A rational expectations equilibrium is given by the following beliefs:

• (1) whenever , then (conditional probability after observing equals unconditional probability at the highest level of quality corresponding to beliefs that all firm-types choose the quality level .);
• (2) whenever , then E[r|θ_f = j, π_f] = E[r|θ_f = j, ω₁, x_f] where x_f is the quality level corresponding to π_f.

First, by Lemma 1 (3), . Next for any , the associated quality, . Next, by Lemma 1 (2), . Putting these two inequalities together, (under the belief structure outlined above). Therefore, the funding probability declines whenever is observed, that is, whenever, is implemented by firm f. As in Proposition 1, the condition on B ensures that the manager's utility is strictly greater with x rather than y. Therefore, every firm will implement . □

The fact that defections from high quality are detected and immediately penalized results in the ‘flight to quality’ documented in Proposition 1. Specifically, high-type firms gain from setting high x. If low-type firms can muddy investor perceptions through low x, high-type firms are also driven to exaggerate their own outcomes, leading to the result in Proposition 1. In contrast, in Proposition 1, by staying with high x, good firms force others to follow suit or be identified as low types. Thus, low-type firms either abandon their quest for capital or accept a much lower probability of being able to mislead investors in equilibrium.

We note that a key feature of the equilibrium derivation stems from the payoff structure of the insurer. Insurers do not benefit from firms obtaining funding. In contrast, the firm benefits directly from raising capital and high-type firms and low-type firms have different preferences over quality. If the firm controls the audit effort, alternative communication mechanisms for conveying quality (such as publicizing the audit fee) may fail (see Appendix C). The conflicting incentive structures (with regard to quality) across types makes it harder to make accurate inferences about quality based on the audit fee as compared with inferences based on the insurance premium.

The choice of audit effort (as opposed to the firm's type) constitutes an endogenous hidden action. Signaling about endogenous hidden action that is strategically chosen by some participants but unobservable to others is qualitatively different from signaling exogenous hidden information (DeGroote, 1990). Intuitively, if the hidden information is exogenous (such as the firm's type), then the sender chooses the best signal while holding type constant. However, if the hidden information is an action, the sender of the signal may change both the action and the signal. Under these circumstances, it is much harder to set up separating equilibria. For example, if a particular signal-action combination yields the highest payoff, all senders will choose that particular signal-action combination. In our context where there is both hidden exogenous and endogenous information, for any internal quality choice q of the firm, the insurer's payoff is always maximized at the cost minimizing level of audit effort; in contrast, even while holding internal quality fixed, the firm's payoffs are maximized either at high or low audit effort depending on the firm's type. For this reason, shifting the control of the audit effort to the insurer is critical in deriving the equilibrium in Program IV.

In general, one can obtain economically unintuitive sequential equilibria by specifying implausible off-equilibrium beliefs. The standard device to rule out ‘bad equilibria’ is to impose restrictions on such off-equilibrium beliefs. In the simple case of two rates of return discussed in the example (see below), a direct proof can be given that ‘pooling-at-the-top’ is the only equilibrium that meets the universal divinity test.

Auditor as an Agent of the Insurer

In this section, we provide an economic rationale for shifting the responsibility for engaging an auditor from the firm to the insurer. We first discuss the situation where the firm purchases FSI but continues to hire the auditor (Program III).We emphasize that this is not an implementation of FSI—under FSI, the auditor is an agent of the insurer. Rather, we analyze this situation to highlight why FSI requires that the auditor stop being an agent of the firm. Let denote the cost to the firm of getting the auditor to exert effort e and report the quality as when the true quality is x. The auditor can be incentivized to report the true x but the firm may not desire the true x to be revealed.28

When the auditor continues as an agent of the firm truthful revelation of x may be impossible ex-ante. However, litigation will typically reveal the true x ex-post. The insurer will find it possible to make an adjustment with the firm (because the insurer has a contractual relationship only with the firm). The key point is that this transfer is made ex-post and will not be known at the time of trading. Hence the firm does not internalize the cost of the low quality, ex-ante.

The initial premium observed by the market (Constraint (IP)) reflects the overall quality reported by the auditor to the insurance company rather than the true x. Under these circumstances, the firm will always have incentives to set low x to increase the probability of funding while incentivizing the auditor to over-report the x. Market participants will anticipate this and set funding strategies based on low x precipitating a race to the bottom as shown in the next proposition.

Proposition 3. (Equilibrium with the auditor as firm's agent) Assume that the benefits from funding, B, are such that for every y < x:

(i.e., B is large enough to ensure that the benefits from funding exceed the incremental cost from increasing overall quality for each private-signal type), and that the auditor is hired by the firm and may misrepresent the overall quality of the audit to the insurer. Then the equilibrium overall quality levels chosen by the firm is to set , that is, the lowest possible level of overall quality.

Proof We begin by noting that the insurer is indifferent across breaking even ex-ante or ex-post. From the insurer's perspective, the premium can be set based on the auditor's assessment of x, , and adjusted later through the transfer . As explained in the discussion preceding equation 7, the firm can (with no extra cost) incentivize the auditor to report by setting fees as follows:

Program III is therefore essentially equivalent to Program I: all firms induce the auditor to report (with off-equilibrium beliefs that any firm choosing , as manifested in , received signal ω₁). Therefore, all firms pay the premium corresponding to and, hence, investors learn nothing from observing the premium. For this reason, the ex-ante beliefs, , are not updated after observing the client-firm's insurance premium and the equilibrium in Program III then becomes the same as in Program I, that is, one where all firms set the lowest possible level of x. Summarizing, the equilibrium is one where all firms choose the lowest level but pay premia corresponding to the highest level . The equilibrium beliefs of investors is that the overall quality is the lowest level resulting in the same equilibrium as in Program I. □

In contrast, Program IV leads to the same equilibrium as in Program II where all firms pool at the highest overall quality level. The key step again is the updating of beliefs by investors after observing the premium. We assume that the insurer offers a schedule and sets the auditor's transfer function in such a way as to ensure that the auditor reveals x truthfully (see details in proposition 1). As a consequence, the equilibrium is the same as in Program II.

Proposition 4. (Equilibrium with the Auditor as the Insurer's Agent) Assume that the benefits from funding, B, are such that for every y < x:

(i.e., B is large enough to ensure that the benefits from funding exceed the incremental cost from increasing overall quality for each private-signal type). Under the FSI regime where the auditor is hired by the insurer (and premia are publicly disclosed), there is an equilibrium where all firms set . In addition, if L = 2, this is the only equilibrium meeting the divinity criterion.

Proof Given that the auditor bears the cost of misreporting x, he or she would typically choose to underreport x. However, the competition (CO) constraint ensures that the firm can always get quoted a fair premium elsewhere and so the insurer does not wish x to be under-reported. Under these circumstances, the ex-ante premium π_f correctly reflects the overall quality x chosen by the firm. The inferences drawn from π_f are the same as in Program II resulting in the same equilibrium where all firms pool at the highest level of overall quality. □

An Example

We provide an example that demonstrates the effects of FSI in ensuring a flight to quality. In order to keep the example as simple as possible, we simplify the strategic role of the auditor and assume that the insurer can observe the x levels.

We assume that there are N firms each of which can have two possible rates of return, that is, that there are two types (L = 2). In addition, we assume that the internal quality choice q = q, is one-dimensional and lies in , where . Next, we assume that the private information is perfect, and that ω_l = 1, 2 reveals the expected rate-of-return as r_l. We set V(q, e) = q (that is, x = q) and P(i|i, q) = q for i = 1, 2. So with a quality level q, the probability that the firm's type is reported correctly is q, and, as there are only two types, the probability of the firm being misclassified is 1 − q. Let the beliefs of investors be represented by ν_i; the firm that receives a private signal that its rate-of-return is r_i is expected to set its q at ν_i. Assuming that each firm-type is equally probable, the inferred rates-of return are as follows:

()

()

The funding probability from selecting q for a firm that receives private signal ω_l = 1 is given by:

()

(recall that G denotes the distribution function of r*). Because , it follows that the funding probability is strictly decreasing in q for every type 1 firm. In contrast, a firm with private signal ω_l = 2 has a realized rate-of-return 2 and its funding probability is unaffected by the choice of q (because it's type cannot be overstated).

Assuming that the benefits of funding are sufficiently high, the firm with a low private signal will set . The implied type of such a firm has l = 2 always. That is to say, such a firm is invariably misclassified. Thus the only report observable by investors is θ_f = 2 for every firm. It follows that when all firms are funded and when no firm is funded. As there are a total of N firms in the economy, N/2 of them (in expectation) are of high-type.

• (1) Therefore, when all firms are funded, an expected amount of (N/2) units of capital are wrongly allocated and the associated loss is: . 29
• (2) When no firms are funded, a total of (N/2) firms may wrongly be denied capital with associated loss: .

The total negative returns associated with the misallocation of capital may then be written as:

()

In contrast, consider the case where the firm's choice of q is known to the insurer, which then sets a premium based on the level of coverage. We shall assume that the cost of coverage is strictly decreasing in quality, that is, π(q) is decreasing in q. As in Proposition 1, the following is seen to be an equilibrium:

• both Firm types choose , i.e., the highest quality financial statement, and pay the associated (low) insurance premium ;
• any firm that is observed to have a premium is classified as a type-1 firm.

To see that this represents an equilibrium, let represent the equilibrium quality choice of Firm f that has received private signal ω₁; either or . If, , the premium rate for firm f, , and its inferred rate-of-return is r₁ irrespective of the report. Therefore, it's probability of funding is G(r₁). If, however, Firm f mimics the high private-signal firm and sets , the inferred rates of return are given by

and the probability of funding increases by (i.e., by the higher probability of funding when the firm is reported as ‘type-2’). Note that the total social loss now becomes:

• (1) because low-type firms are funded; and
• (2) because high-type firms are not funded.

with attendant total social loss:

()

Because , 18 and 19 shows that the social loss is reduced through the provision of FSI.

Notice in this example that if a firm with private signal ω_l = 1 sets , then the best response for the firm with the high private signal is to set —this separating policy leads to funding with probability G(r₂) at a minimum insurance cost. If, however, the firm with the high private signal sets , the best strategy for a firm with low private signal is to ‘mimic’ and set . Mimicry increases the funding probability (and leads to the equilibrium described above). In contrast, when the type-1 firm sets and the type-2 firm sets the situation is untenable in equilibrium because by increasing q slightly, the type-2 firm reduces insurance costs but still separates itself from the type-1 firm. Thus, , should never be an equilibrium. Hence, the only plausible equilibrium is for both firms to set (see Lemma 1). □

This example does not incorporate a role for the auditor, but a little reflection shows that the core intuition survives in a more complex setting where reports are influenced by an auditor acting under moral hazard. In particular, if the fee of the auditor is determined by the insurer, sufficient incentives may be provided to elicit truthful revelation regarding the auditor's assessment of the firm's x. Once x is known (perhaps imperfectly) to the insurer, premium levels reveal this information to investors. In particular, when firms defect from the anticipated level of x, that is, if a firm has been charged a higher than anticipated premium π_f, investors find out about this before trading. This allows investors to alter their funding strategies and we are then essentially back in the situation discussed in the example.

The example has the characteristics of a signaling model where firms are separated out through the level of the insurance premium but the cost associated with signals has a special structure that should be clarified. In a standard signaling model, there is a difference in cost for a given signal across types (arising from an exogenous factor related to type). This difference deters the low type from choosing the same signal as the high type. In contrast, in the setting of the example, the cost of financial statement quality is the same for all firms. The differential cost arises because the choice of high quality reveals the firm's true type and is thus indirectly more costly for the low-type firm.

Let represent the beliefs of investors. In this example, the expected return on the high report, θ = 2, is some weighted average of r₂ and r₁ with the weights depending on ; in addition, because firm 2 always issues report θ = 2, the weight on r₂ is strictly positive. In contrast, the report θ = 1 necessarily implies that the rate-of-return is r₁. Thus, rational investors would fund the report θ = 2 with a greater probability than the report θ = 1. In general, ensuring that higher reports are funded with greater probability requires some form of regularity assumption; in our formulation, this is the role of Assumption 1.

The analysis in this example assumes that the insurance premium is based on the actual level of x chosen by firms. This was the setting analyzed in Proposition 1. More generally, the insurer relies on the auditor's assessment of x to set the premium. In such a setting, the auditor reports x strategically in order to maximize his or her own payoffs, and the revelation of the premium to investors does not break the race to the bottom with regard to overall quality so long as the auditor functions as an agent of management (Proposition 1). That is, firms would prefer to get the auditor to report a higher level of x to the insurance company and thereby to investors than the one that has been chosen. As rational insurers and investors will anticipate this ‘bias’, the equilibrium unravels to the lowest choice of x. In contrast, when the auditor functions as an agent of the insurer, the pooling at the highest quality again becomes the rational equilibrium (Proposition 1).

Conclusion

Several causes have been advanced in the media for the ‘accounting’ meltdown: irrational exuberance, infectious greed, moral turpitude of executives, unethical accountants, misleading financial statements and related ‘ills’. We have argued that the inherent conflict of interest in the auditor–client relationship and the unobservability of financial statement quality, coupled with incentives to ‘cook the books’ are among the potential culprits. FSI, as developed here, provides a market-based solution that acts as an effective check on the issuance of overly biased financial statements. First, the publicization of the insurance premium will credibly signal the quality of the insured's financial statements and direct investments toward better projects. Second, by transferring the auditor hiring decision to the insurer, FSI eliminates the auditor's inherent conflict of interest. At the same time, the ability to signal the quality of financial statements will provide companies with incentives to improve the quality of their financial statements. Hence, FSI will result in fewer misrepresentations and smaller shareholder losses.

Under the present regime, auditors' legal liability is not an effective tool for inducing truth telling in financial statements because the costs of such liability are essentially covered by the client-firms. As mentioned, the FSI scheme effectively eliminates the conflict of interest that came to light in the aftermath of accounting scandals. Yet FSI has other important benefits: the credible signalling of financial statement quality leads to an improvement of such quality, and consequently, decreases in shareholder losses, and the better channelling of savings to socially desirable projects.

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