2.1 Moral Hazard

Townsend (1979): Townsend’s costly state verification model, which has been adapted to finance settings by Gale and Hellwig (1985), has been used widely in applications, including macro economics in work of Bernanke and Gertler (1989) and Bernanke, Gertler, and Gilchrist (1999). The costly state verification model captures deadweight costs that outside investors might need to incur to monitor the manager. A monitoring action can be required because the manager privately observes the firm’s profits, which he may divert and refuse to pay back to investors.

Consider an agent with a profitable project, which needs an investment of I > 0. The agent does not have the full amount to invest, and needs to raise some money from an outside investor, the principal.

If the investment is made, the project has a random gross return , distributed on with CDF F. Only the agent observes the true returns. However, the principal can verify returns at a cost c. Both the agent and the principal are risk-neutral, but the agent has limited liability—he cannot be forced to pay back more than what he claims to have, or more than what he actually has if verification takes place. An optimal contract maximizes the principal’s profit subject to giving the agent a specific expected gross payoff of W₀. The value of W₀ depends on the agent’s contribution towards the up-front investment, and the relative bargaining powers of the principal and the agent. Assuming that the principal must contribute a strictly positive amount to up-front investment, .

Assuming that the principal can perfectly commit to any contract, we can use the revelation principle to consider only truth-telling contracts, in which the agent directly reports realized output (see Myerson, 1979). We can focus on contracts {V, g(x)}, where V ⊆ [0, X] is the set of reports that the principal commits to verify, and g(x) ≤ x is a transfer that the agent is required to make if he reports x and is not caught lying. If the agent is caught lying then without loss of generality we assume that the principal takes away the agent’s entire output. This transfer rule off the equilibrium path gives the agent the maximal incentive to tell the truth.

In this notation, the optimal contracting problem is written as follows:

The truth-telling constraints require that the agent be at least as well-off telling the truth rather than lying, after any output realization. Rather than writing out all truth-telling constraints explicitly, we provide a lemma that characterizes the set of all contracts that satisfy the truth-telling constraints.

The following theorem solves for the optimal contract, and shows that it takes the form of debt.

Consider a scalable project, which generates a cash flow uniformly distributed on [0, 3I] when an investment of I is made. Cash flows can be verified at the cost of cI, where . Then the principal’s payoff as a function of D is:

The debt face value that maximizes this expression is D = (3 − c)I, and so the principal’s maximal payoff is (3 − c)²I/6. Consequently, the maximum amount the firm can borrow, i.e. its debt capacity, is also (3 − c)²I/6. Since , then the project debt capacity is less than I, and so the project is infeasible unless the agent contributes to up-front investment.

Now, suppose the agent is able to contribute E > 0 towards up-front investment. Then investment greater than E/(1 − (3 − c)²/6) is infeasible, because in this case, the amount the agent must borrow, I − E, would exceed the firm’s debt capacity. Optimal investment is in the interval (0, E/(1 − (3 − c)²/6)): it maximizes the agent’s expected payoff

subject to the constraint that the principal breaks even, i.e. . Due to the scale invariance of this example, the optimal level of investment is increasing linearly in E.²

Bolton and Sharfstein (1990) presents a simple two-period model of moral hazard, which provides a useful link from static to fully dynamic infinite-horizon models. One of the implications of this model is that future investment and the probability of continuing the project can depend on past performance even when future NPV is unrelated to past performance.

A manager has an opportunity to operate the firm for 2 periods, but needs financing from outside investors. In each period, if the firm gets outside funding to make an investment of I, it gets cash flows x₁ ≥ 0 with probability θ and x₂ > x₁ with probability 1 − θ. Cash flows are i.i.d. over time, and there is no discounting between periods. Figure 3 illustrates possible outcomes if investment is always made.

As in Townsend (1979), the agency problem is that only firm manager, and not the outsiders, observe the true cash flows. The agent cannot pretend that he got a lower cash flow than x₁ ≥ 0, since that is the worst cash flow realization. However, costly state verification is not possible, so the manager can always divert the residual x₂ − x₁ for personal consumption if he receives a high cash flow.

Assume that x₁ < I, so that in a one-period version of this model investment is infeasible if the agent cannot contribute his own funds towards investment.

We also assume that

That is, if cash flows were verifiable it would be profitable to invest in every period.

By the revelation principle we can restrict attention to contracts in which the manager reports true cash flows, and transfers from the manager to the investors and the probability of continued financing depend on the manager’s report. Because in the second period the agent will tell the truth only if he is required to make the same transfer regardless of realized cash flow, we can restrict attention to contracts defined by:

An optimal contract maximizes the principal’s profit subject to giving the agent an expected gross payoff of at least W₀. The value of W₀ is determined by the amount the agent can contribute to up-front investment, and the relative bargaining powers of the principal and agent. Formally, we would like to solve

where the objective function is the principal’s gross profit (after initial investment in period 1). The constraint (IC2) guarantees that if the agent receives a high cash flow of x₂, his payoff if he reveals it truthfully, the left-hand side of (IC2), is at least as good as his payoff if he reports cash flow x₁ instead. Note also that we did not write out explicitly the analogous truth-telling constraint (IC1) for the low cash flow realization in period 1. We do not expect it to bind, as verified after we derive the optimal contract.

Figure 4 illustrates the principal’s profit as a function of W₀. Note the special cases of and . In the former case, the agent obtains financing for the last period if he pays in the first period. Thus, he is indifferent between paying only x₁ in period 1 and paying in exchange for a round that lets him receive an expected payoff of and pay x₁ in the last period. In the latter case, the agent pays x₁ in both rounds, and receives funding for both rounds for sure. Note that on the interval , the slope of the principal’s profit is flatter than −1 because . As W₀ increases, so does the total surplus because inefficiency due to the agency problem becomes less severe. The inefficiency depends on the probability with which the project does not get funded in the second period. In Section 4, when we discuss the model of DeMarzo and Sannikov (2006), we will see how the slope of the principal’s value function changes in a dynamic setting.

However, in more complex dynamic models, the optimal contract can easily be not renegotiation-proof, as we will see in Section 4.

2.2 Adverse Selection

If a firm manager has private information about firm fundamentals, it may be difficult to sell informationally sensitive securities, especially equity, to raise funds for investment or share risk. Typically, insiders must retain some of firm’s cash flows in order to signal to the market the quality of the firm. Leland and Pyle (1977) provide a classic model in which firm insiders signal their private information about the quality of fundamentals by retaining shares of equity.

Myers and Majluf (1984) studies the effect of private information on investment decisions. It assumes that financing can be raised only in the form of equity, and that the management acts in the interest of old shareholders. It finds that “if managers have inside information there must be some cases in which that information is so favorable that management, if it acts in the interest of the old stockholders, will refuse to issue shares even if it means passing up a good investment opportunity” (p. 188).

The model is as follows. There is no discounting. At time 0 the management has an investment opportunity that requires investment I. The firm has financial slack S < I (cash + marketable securities) and must raise E = I − S by issuing new equity if it decides to go ahead. If the investment is not made at time 0, the investment opportunity evaporates.

The management knows the value y that the investment opportunity generates and the value x of assets in place. However, from the point of view of the market, these are random variables . With this information, the market requires the management to sell a fraction α of firm equity in order to raise E.

It is assumed that following a decision to issue, the old shareholders remain passive. That is, they “sit tight” if stock is issued; thus the issue goes to a different group of investors.

The following simple example illustrates equilibrium.

In general, the investment region as illustrated in Figure 5.

Assuming that the joint distribution of (x, y) over [0, ∞) × [0, ∞) is characterized by a strictly positive density, the equation

must have at least one solution. Indeed, note that the left-hand side is continuous in α, and it goes from 0 at α = 0 to infinity as α → 1 and the slope of the lower boundary of M gets steeper.

Of course, it is difficult to issue equity in the presence of informational asymmetries, because equity is very sensitive to private information. In the model of Myers and Majluf (1984), it is assumed that the firm cannot raise funds through other securities, such as debt or hybrid securities. We next discuss a model that investigates optimal security design under asymmetric information, using general securities.

DeMarzo and Duffie (1999) consider a setting in which an issuer owns assets with random cash flows, and wants to raise capital by issuing asset-backed securities. The issuer receives superior private information about the assets’ cash flows after security design stage but before sale. We saw in the overview of Myers and Majluf (1984) that it is difficult to sell equity in the presence of private information, because equity is quite information-sensitive. Here we will allow for general securities, and solve for optimal security design. The desire to issue is motivated by the assumption that the issuer is less patient than the market, e.g. because the issuer has superior investment opportunities. This is captured by assuming that the issuer discounts cash flows at the rate of δ < 1, while the market does not discount.

The model has one period, and the issuer’s assets generate a random cash flow of at the end of the period. A security is characterized by a non-decreasing function g : [0, ∞) →[0, ∞), which gives the payout to investors at the end of the period as a function of realized cash flow, x. Function g has to satisfy the feasibility constraint g(x) ≤ x. Formally the timeline is as follows:

The issuer’s expected payoff is given by

Note that the issuer does not discount the proceeds from the sale of the security, but discounts at rate δ the cash flows received upon the maturity of the assets.

The issuer would like to design a security to maximize her ex ante expected payoff. Formally, given a security g, the perfect Bayesian equilibrium consists of a map z → q such that (1) the issuer chooses q to maximize her expected payoff, (2) given q, the market forms its belief about the issuer’s type according to Bayes rule, and (3) given the belief, the market prices the security at its expected payoff.

The issuer’s optimization problem with respect to q in equilibrium is

where f = E[g(x)|z] is the expected payoff of the security. Since the issuer’s expected payoff depends on her type z only through f after the security g has been designed, it is convenient to think of f as the issuer’s type in the signaling equilibrium, in which the quantity q is chosen.

Assuming that in equilibrium, quantity sold q( f ) is a continuous function of the value of the security f, the schedule q( f ) can be characterized from the first-order condition

Using the fact that in equilibrium P_g(q) = f is the inverse of q( f ), so that , we have:

The constant of integration is pinned down by the condition that the worst type f₀ sells q = 1, and so:

From this expression, we see the trade-offs involved in optimal security design. Equity g(x) = x maximizes f₀ since it sells all cash flows of the worst type to investors. However, equity is also very informationally sensitive. Therefore, as asset fundamentals z improve, f = E[g(x)|z] rises very fast, leading to very low payoffs for issuers with good fundamentals. It may be possible to raise the expected payoff of issuers with good fundamentals by designing a less informationally sensitive security, that is, making g(x) less sensitive to the cash flow x. However, that has a cost, as the value of f₀ ends up lower. It follows that, in comparison with equity, less informationally sensitive securities give lower payoffs to issuers with low-quality assets, but may give significantly higher payoffs to issuers with high-quality assets. Below, we characterize conditions when equity is the optimal security, and when risky debt is.

Intuitively, it is optimal to sell equity if the seller’s private information is small.

If the seller’s private information is not that small, then debt turns out to be optimal under an additional condition.

Intuitively, debt is optimal because if the issuer’s private info changes from z₀ to z, then standard debt, in the class of monotone securities, is one that minimizes the increase in the issuer’s private valuation, and thus illiquidity costs.

4.1 Other Models that Involve Dynamic Moral Hazard

A number of papers adapt a continuous-time dynamic agency framework to study the interaction between agency problems and various other issues. We briefly review several of them here. We focus, for the most part, on the technical elements of these models.

Piskorski and Tchistyi (2010), who consider the problem of optimal mortgage design, have a number of important contributions. First, they adapt a model similar to that of DS to the study of mortgages. Second, they investigate what happens in the optimal contract when market conditions exogenously change. They focus specifically on changes of market interest rates and find that it is optimal to tighten the agent’s access to credit when interest rates rise.

The market interest rate in the model of Piskorski and Tchistyi (2010) is a Poisson switching process with two levels {r_L, r_H}, and the switching intensity from interest rate r_i to r_j, j ≠ i, given by δ(r_i). The cash flow process

is interpreted as the borrower’s income, which is unobservable. It is assumed that the borrower can hide income without any cost, so parameter λ in DS is set to 1.

In any incentive-compatible contract the agent’s continuation value follows:

where is the agent’s continuation value conditional on the event that the interest rate switches at time t, and the incentive constraint is β_t ≥ 1.

The principal’s value function F_i(W_t) depends on two state variables—the agent’s continuation value W_t and the current interest rate r_t = r_i. It is characterized by the system of two equations:

for i = L, H and j ≠ i, with the familiar boundary conditions

Under the optimal contract, the agent’s continuation value jumps from W_t to whenever the interest rate switches. A key equation that determines the jump in the agent’s continuation value, when the interest rate jumps from r_t− = r_i to r_t = r_j at time t, is the first-order condition

(9)

We would like to emphasize that condition (9) arises very commonly in models where a state switches via a Poisson process.

Piskorski and Tchistyi (2010) offer several implementations of the optimal contract. In particular, the variable W_t can be mapped into the balance on the homeowner’s home equity line of credit. The changes in the value of W_t in response to interest rate shifts can be linked to some properties of adjustable-rate mortgages. In particular, it is a feature of the optimal contract that the agent’s default probability rises when the interest rates increase.

Hoffmann and Pfeil (2010) consider a dynamic agency model, in which firm profitability experiences observable shocks. Their model builds upon DS, except that they allow for Poisson shocks that change the expected rate of cash flows μ. One of the key messages of Hoffmann and Pfeil (2010) is that, despite conventional intuition, the optimal contract rewards the agent for luck when it is correlated with the firm’s future profitability.

Here we review a variation of their model. Assume that the expected rate of cash flows is a Poisson switching process with values {μ_L, μ_H}, so that

The switching intensity from state μ_i to μ_j, j ≠ i, is given by δ(μ_i).⁸ The agency problem is the same as in DS: the agent can divert cash flows, and he receives benefit equal to a fraction λ ∈ (0, 1] of the diverted cash flows.

The optimal contract depends on μ_t and the agent’s continuation value W_t, which follows:

The principal’s value function solves the system of two equations

for i = L, H and j ≠ i, with the familiar boundary conditions

The most important point of this paper is that under the optimal contract the agent is rewarded for luck, i.e. when the mean of cash flows jumps from μ_L to μ_H. This conclusion seems to contradict the conventional wisdom of Holmstrom and Milgrom (1991); that it is optimal to filter out factors outside the manger’s control when evaluating the agent’s performance. An example of this involves the evaluation of fund manager performance relative a benchmark index, rather than in absolute terms. However, in the model of Hoffmann and Pfeil (2010) it is optimal to reward the manager for luck because of the dynamic feature of the model that luck is positively related to future profitability.⁹

Whenever the mean of cash flows switches from μ_i to μ_j, the jump in the agent’s continuation value satisfies the first-order condition

We have already encountered a similar condition in Piskorski and Tchistyi (2010), in the event that the interest rate jumps.

He (2009) considers a dynamic agency model in which both the agent’s hidden actions and shocks affect the scale of the firm, rather than the current cash flow. Apart from this distinction, the model has many similarities to that of DFHW.

Below we present an extension of the model of He (2009), which incorporates investment decisions that affect the scale of the firm. This model can be a convenient option for applications, as it has different moment properties from DFHW (e.g. firm cash flows are highly correlated in the model of He (2009)—they follow a random walk—but uncorrelated in the model of DFHW).

Consider a firm, whose cash flow is given by

where ι_t is the investment rate and K_t is the firm’s capital, both of which are observable. The firm’s capital follows:

if the agent works and

if the agent shirks, where . Neither the shocks Z_t nor the agent’s actions are observable. The agent gets a private benefit of BK_t from shirking, where B ≥ 0. Both the agent and the principal are risk-neutral, and the agent’s discount rate is γ ≥ r, where r is the principal’s discount rate.¹⁰ In the event of termination, the agent’s outside option is 0, and the value of the assets to the principal is given by , where

Assume that the difference and parameter B are such that it is optimal to implement working at all times until termination.

For a given contract, if the law of motion of the agent’s continuation value is given by

then the incentive constraint to ensure that it is optimal for the agent to work is β_t ≥ λ, where λ = B/(). The optimal contract sets β_t = λ.

The principal’s value function F(W, K) satisfies the HJB equation

As in DFHW, the scale-invariance properties of the model imply that the solution must have the form F(W, K) = f(w)K, where w = W/K. By Ito’s lemma, w_t = W_t/K_t follows:

and the HJB equation, in the region where dC_t = 0, reduces to

If γ > r, then the relevant boundary conditions that determine the function f(w) as well as the point where the agent gets paid are

as in DFHW. Point is a reflecting boundary of the process w_t.

If γ = r, then and f(w) must satisfy

and the process w_t becomes absorbed when it hits . When that happens, inefficiency completely disappears, and the solution is first-best. In both cases, termination occurs when w_t hits 0.

Zhu (2011) provides a general solution to the variation of the DS model with costly effort. Recall that in this model the principal observes cash flows

The agent gets a private benefit of B = λA if he shirks. This model is similar to the baseline cash flow diversion model of DS, except that the rate, at which the agent can divert cash flows, is bounded by A.

Here we review the results of Zhu (2011), specializing them to the case when the agent’s outside option is R = 0. The form of the optimal contract depends on the region in the space of payoff pairs of the agent and principal, in which the point (λA/γ, (μ − A)/r) lies.

Denote by F(W) the principal’s value function in the baseline model of DS. Recall that this function solves Eqn (6) with boundary conditions , and (or, equivalently, ). Furthermore, let

Function g(w) bounds region A in Figure 10.

The boundary between regions B and D is the locus of points, where solutions to Eqn (6) with boundary conditions and reach slope 0. The boundary between regions C and D is the locus of points, where solutions to Eqn (6) with boundary conditions and , for some , reach slope 0.

The principal’s value function in the optimal contract can be found as follows:

In both cases B and C, function is maximized at a point where it satisfies Eqn (6). Thus, if the principal can start the contract at point W₀ where his profit is maximized, then the agent will be initially putting effort, and his continuation value will evolve according to:

In case B, the agent is putting in effort while ; at the left endpoint of this interval the agent is fired, and the right endpoint the agent is allowed to shirk temporarily. Point is a sticky reflecting boundary of the process W_t. In case C, the agent is putting in effort when . At point the agent is paid so that W_t reflects at . Point is a sticky reflecting boundary: at that point the agent shirks and consumes private benefits temporarily (which is not so attractive for the agent because in region C, λA/γ is low).

Biais, Mariotti, Rochet, and Villeneuve (2010) (hereafter BMRV) adapt the agency framework to settings where the agent’s action, e.g. negligence, may lead to large losses. These situations are common in practice, and often the damage is significantly greater than what the agent can cover. BMRV study the question of optimal incentive provision in these settings. They find that losses often require the downsizing of operations.

Formally, BMRV assume that performance-related information takes the form of a Poisson process, rather than a Brownian one. Like DFHW, they also allow for investment (and disinvestment) that changes the scale of the project. They assume a production technology that uses capital K_t to produce a cash flow of

The term M dN_t in the expression for dY_t represents possible losses (which translate to cash flow losses of the size K_tM). A loss arrives when the counting Poisson process N_t jumps up by 1, and the intensity of losses Ψ_t depends on the agent’s unobservable action. The agent has two actions: working and shirking. The impact of these actions on the intensity of losses, as well as the agent’s private benefit, are summarized as follows:

The investment function Φ is increasing and concave. BMRV take δ = 0 and consider a particular form of Φ represented in Figure 11.¹¹ That is, it is possible to costlessly destroy arbitrary amounts of capital, and build new capital at cost c, as long as the growth rate does not exceed g < r.

The agent’s and principal’s preferences over consumption streams are identical to those of DS: both the principal and the agent are risk-neutral, the agent’s discount rate is given by γ > r, and the agent can consume only non-negative amounts. Moreover, the agent’s outside option is assumed to be 0 as in DFHW and He (2009).

BMRV mainly focus on parameters, for which it is optimal to give the agent incentives to work at all times, the maximal risk prevention case. We focus on this case here.

The optimal contract is based on two state variables: the agent’s continuation payoff W_t and the size of the firm K_t. Denote the principal’s value function by F(W_t, K_t).

The agent’s continuation value follows:

The incentive constraint is

If this constraint holds, then the benefit from shirking is not greater than the negative impact of shirking on the agent’s continuation payoff, i.e. AK_t ≤ β_tΔψMK_t. Naturally, the optimal contract sets β_t = λ.

Because information about the agent’s performance arrives via a Poisson process, one has to take into account that it is impossible to give the agent incentives to work when W_t ∈ [0, λMK_t). The reason is that a required punishment in the event of a loss would reduce the agent’s continuation payoff by at least λMK_t, below the agent’s outside option.¹² As a result, if the agent’s continuation payoff ever falls below λMK_t, the optimal contract prescribes either randomization that lowers W_t to 0 or boosts it to λMK_t, or downsizing that reduces the firm’s capital to W_t/(λM).

Due to the scale-invariance properties of the model,

When W_t ≤ λMK_t, i.e. w ≤ λM, f is a linear function of the form . The two-dimensional Bellman equation for F(W, K) can be reduced to a one-dimensional equation by a method similar to that used in DFHW. The equation takes the form

on the interval , and it must satisfy the boundary conditions

Figure 12 illustrates the form of function f in this model.

The optimal investment rate ι solves

as in DFHW. Because f is a concave function, it follows that is increasing in w; thus the rate of investment is increasing in the firm’s financial slack.

For the specific piecewise linear form of function Φ(ι) assumed in BMRV, the optimal investment rate is

Effectively, investment happens when w_t exceeds a critical level of wⁱ, where

BMRV has an important implication about how the contract gives incentives to the agent to work to prevent losses. Once w_t reaches , the agent is paid a continuous stream of payments until the next loss. If a loss occurs, payments are suspended and the agent has to wait a fixed amount of time before the payments resume. If another loss occurs, the agent has to wait longer. Too many losses in a row lead to a partial downsizing or liquidation.¹³

5.1 Static Contracts in Dynamic Settings

A number of new issues come to light once we think about the time dimension. Below we review several models that highlight these issues, starting with models that incorporate static contracts in dynamic settings.¹⁴Lucas and McDonald (1990) consider a model in which firms can raise money for investment opportunities only by issuing equity. Investment opportunities can be postponed, and inside information about assets in place is time varying. In equilibrium, managers of overvalued firms issue equity to finance the investment right away, while managers of undervalued firms wait to issue until their private information becomes public. Stock prices tend to drop upon the announcement of issuance, and generate high abnormal returns prior to issuance.

The model of Lucas and McDonald (1990) motivates us to think about further issues that are likely to arise in dynamic settings with adverse selection. First, while in the model of Lucas and McDonald (1990) managers carry private information for only one period before it becomes public, in more elaborate models managers can signal the quality of their information by waiting. Second, the distribution of private information can change over time: if managers of undervalued firms wait to issue, their proportion in the market can rise over time, alleviating the problem of asymmetric information. This can lead to hot markets when the dilution problem due to asymmetric information is less severe as more high-quality firms sell securities, and cold markets, during which high-quality firms wait and only low-quality firms, or firms with dire capital needs, issue. Third, firms can issue securities other than equity and can build up financial slack during times when the informational problem is less severe.

The model of Daley and Green (2012) sheds light on some of these issues. Specifically, in their setting the seller/issuer, who has persistent private information about asset quality, can signal by waiting. Market belief about the seller’s private information tends to rise during the periods of non-issuance. An improvement of market belief about asset quality leads to a hot market, in which sellers issue regardless of private information. Interestingly, the equilibrium also features a regime where the market freezes, i.e. issuance becomes suspended until further arrival of news.

The models of Lucas and McDonald (1990) and Daley and Green (2012) only give a flavor of many interesting issues that arise in dynamic adverse selection models with simple contracts. A number of other papers look at these issues both empirically and theoretically, including Baker and Wurgler (2002) and Hennessy, Livdan, and Miranda (2008). Market dynamics with adverse selection is also a fruitful area for future research, as a lot of interesting questions have not been answered.

It turns out that many adverse selection frictions become significantly less severe once dynamic contracts are allowed. We finish this section by quickly revisiting Lucas and McDonald (1990) and Daley and Green (2012) with the perspective of dynamic contracts, and review several insights about optimal dynamic contracting under adverse selection.

Lucas and McDonald (1990) consider an infinite-horizon model in which firms have assets in place and may have investment opportunities. For simplicity, the risk-free rate is assumed to be 0. In each period t, the market perceives that the value of the firm’s assets in place is A_t. At the same time, the manager learns the next period’s value of the firm’s assets, which is

It is assumed that u > d, so that if A_t+1 = uA_t, the firm is undervalued on the market, and if A_t+1 = dA_t, the firm is overvalued.

In addition, the manager knows whether the firm has an investment opportunity in that period. If an opportunity exists, it requires an investment of KA_t, and generates value in the next period (net of the underwriting fees). It is assumed that a firm can raise money for investment only by issuing equity. As in Myers and Majluf (1984), the manager cares about the firm’s old shareholders.

In addition, the following assumptions are made. Between periods t − 1 and t the firm is liquidated for exogenous reasons with probability 1 − ρ, and generates a payoff of A_t to shareholders. Investment opportunities are random: if the firm has an investment opportunity in period t that it does not take, it still has an investment opportunity in period t + 1. If the firm took its opportunity in period t, or did not have one altogether, it gets an investment opportunity in period t + 1 with probability q.

Lucas and McDonald (1990) focus on an equilibrium, in which firms with an investment opportunity issue equity to finance it only if they are overvalued, and undervalued firms postpone the opportunity. Parameter restrictions are imposed to ensure that overvalued firms without an investment project and undervalued firms do not want to issue (e.g. due to high underwriting costs). Then upon issuance, new investors will infer that A_t+1 = dA_t and that the firm has an investment opportunity. Based on this belief, investors then demand in exchange for capital KA_t an appropriate fraction of the firm s, such that they break even.

Due to scale invariance, the manager’s valuation of the firm is of the form V(a, b)A_t, where a = u or d depending on the value of A_t+1, and b = 0 or β, depending on whether the firm has an investment opportunity or not. The value function satisfies the following recursive equations in equilibrium:

for a = u, d

since K = s(dβ + K + V(d, β)) to ensure that the firm’s new investors break even.

While V(a, b)A_t is the manager’s valuation of the firm based on his private information, market valuation of the firm will depend on its belief about manager’s information. In particular, in period t, prior to issue announcement, market will believe that a = u with probability p and d with probability 1 − p. In addition, the probability that the firm has a project is increasing in the number of periods n its stock has gone up (and thus the firm had been undervalued in the prior period, and unable to raise funding for the project). The probability that the firm has a project is:

and so market valuation of the firm’s stock prior to issue announcement is:

Since V(a, β) > V(a, 0), the value of the firm is increasing in the number of periods its asset value has gone up.

If the firm decides to issue, its market value immediately changes to V(d, β). The decision to issue reveals two pieces of news to the market: bad news that the firm’s asset value will go down in the next period, and good news that the firm has an investment opportunity. If n is large, then P(n)  pV(u, β) + (1 − p)V(d, β) > V(d, β), and the price of the firm drops for sure upon announcement. In this case, the good news that the firm has a project is not really news. If n is small, then stock price reaction to equity issue announcement may be ambiguous.

The model predicts that the firm generates a positive abnormal return prior to issue. This happens because the decision to not issue is related to good private information about the firm’s asset value. The model also predicts that, usually, the firm’s share price should drop upon the announcement of issuance, particularly after long periods of stock price increases.

Daley and Green (2012) consider a dynamic asset market with asymmetric information. They investigate how trade patterns change over time as information is gradually revealed to buyers, who continuously update their beliefs about asset quality. The equilibrium they derive has a very interesting feature of market breakdown: a period when trade stops even though there are beneficial trading opportunities, and sellers have incentives to strategically wait for improved market conditions.

The model has one seller of the asset and a continuum of competitive buyers. The seller has one asset whose quality θ ∈ {L, H} is privately known to the seller. The seller derives a payoff flow of K_θ from the asset until a trade occurs, and any buyer who purchases the asset is able to derive a higher payoff flow of V_θ > K_θ after the time of the trade. Quality H generates more value, i.e. V_H > V_L and K_H > K_L. Buyers continuously make offers to the seller until a sale occurs. After the sale, the purchaser holds the asset in perpetuity.¹⁵ Denote by W_t/r the highest offer that the buyers make to the seller at time t in equilibrium, where r is the common discount rate of the seller and all buyers.

Buyers initially believe that the seller is type H with probability π₀ ∈ (0, 1). They update their belief from the signal X_t that follows

where μ_L < μ_H, as well as from the seller’s decisions. Ignoring the information about the seller’s decisions, Bayes rule implies that the buyers’ belief about the seller’s type would evolve according to¹⁶

The variable z_t ∈ (−∞, ∞) is more convenient to work with than the belief π_t ∈ (0, 1).

Furthermore, Daley and Green (2012) show that if we take into account the seller’s decisions, then the absence of trade is always good information about the seller’s type. That is, type H would never trade—i.e. he strictly prefers to wait for future offers—whenever type L prefers to wait or is indifferent. Intuitively, this is true because type H gets a higher payoff flow and more favorable news than type L from holding on to the asset. If no trade is good news, then conditional on the absence of trades, the process z_t that determines the buyers’ beliefs about the seller’s type follows:

(11)

where Q_t is a non-decreasing process.

It turns out that the equilibrium is characterized by three regimes:

Therefore, the process Q_t is such that α is a reflecting boundary of the system, and dQ_t = 0 on the interval (α, β).

For expositional reasons, we will assume that the equilibrium takes this form and derive equations that determine the boundaries α and β, as well as the value functions of both types of the seller. On the interval [α, β], the value functions of the two types of sellers satisfy the HJB equation (see Footnote 15):

(12)

with the “+” sign if θ = H, and the “−” sign if θ = L. The value function F_L(z) of type L reaches the level of V_L at point z = α and stays at V_L on the entire interval (−∞, α). Moreover, both functions F_H(z) and F_L(z) reach the level of

(13)

at point z = β, and satisfy Eqn (13) for all z ∈ [β, ∞). In addition, because α is a reflecting boundary of the system, .

Functions F_H(z) and F_L(z) that solve the second-order ordinary differential equation (12), together with the two boundaries α and β, can be fully determined through six boundary conditions. In addition to the five conditions that we have already identified, and F_H(β) = F_L(β) = Ψ(β), Daley and Green (2012) add a sixth condition of , which is motivated by the off-equilibrium behavior. First, for any z ∈ (α, β), it has to be the case that only a type L seller accepts any price less than Ψ(z). Otherwise, buyers have a profitable deviation for z ∈ (α, β): offering a price less than Ψ(z) that both seller types would accept, and getting a strictly positive payoff. Therefore, we cannot have , since otherwise such an offer would exist at z = β − ɛ for sufficiently small ɛ. Second, if z_t reaches β, the seller of type H should not be able to benefit by waiting a moment, instead of accepting the best offer immediately. If and if z_t keeps following Eqn (11) if the seller does not accept, for some non-decreasing process Q_t, then the seller of type H benefits from waiting.¹⁷ Thus, we must have . Note that the assumption that z_t keeps following Eqn (11) conditional on the absence of trade even if z_t ≤ β is an off-equilibrium path belief assumption.

Figure 13, reproduced from Daley and Green (2012), illustrates the value functions F_H(z) and F_L(z) in equilibrium, and compares them with the average price Ψ(z) at which the transaction would take place in the absence of the informational problem.

The equilibrium has several notable features. First, sellers of type H signal asset quality by waiting. Indeed, due to the non-negative term dQ_t in (11), the absence of trade is good news about quality. Second, the pattern of trade depends on the distribution of private information that the seller may have, which is captured by the state variable z_t. The equilibrium features a hot market, for z_t ≥ β, where both types of the seller trade, and a cold market for z_t ≤ α, where only the low-quality seller trades. Third, the equilibrium has an interesting region of (α, β) where the market freezes, i.e. trade stops until new information about asset quality is revealed.

It turns out that the conclusions of Lucas and McDonald (1990) as well as Daley and Green (2012) change drastically if it is possible to write dynamic contracts. It is a striking observation (although, of course, there are frictions in practice that may make dynamic contracting difficult). Following this observation, we review several common conclusions that the literature on optimal dynamic contracts with adverse selection delivers.

5.2 Optimal Dynamic Contracts with Adverse Selection

In both of the models that we just discussed, dynamic contracts can restore full efficiency. Consider first the setting of Lucas and McDonald (1990). Despite asymmetric information, the manager could always raise funds for investment by announcing whether A_t+1 = uA_t or A_t+1 = dA_t and issuing a security with the following characteristics: (1) it grants the holder a claim to a fraction s of the firm’s equity at time t and (2) at time t + 1 grants the holder a claim to all the remaining equity if the manager’s announcement of the value of A_t+1 turned out the be incorrect (or if it turns out that the firm did not have an investment opportunity at time t). This security gives the manager incentives to tell the truth, and fully solves the adverse selection problem.

Likewise in the setting of Daley and Green (2012), if we assume that the signal X_t reveals public information about the quality of the asset even after the transfer of ownership, then the asymmetric information problem can also be solved completely through a dynamic security. Indeed, the seller can transfer the asset to the buyer at time 0 in exchange for a payment that is contingent on the future observation of X_t. The payment can be easily designed in such a way that its expected value is V_H/r conditional on the seller’s asset being of type H, and V_L/r conditional on the seller’s asset being of type L.

When dynamic contracts are possible, there is a crucial difference between asymmetric information that exists up front at time 0, and that, which arises in the future. Speaking loosely, only the former creates distortions. The problem of future asymmetric information can be solved by a contract signed before asymmetric information materializes.

One general takeaway from the literature that investigates dynamic contracts in settings with adverse selection is that distortions that exist at time 0 decay gradually over time. For example, Pavan, Segal, and Toikka (2009) and Garrett and Pavan (2009) focus on the concept of impulse responses, which captures the extent to which private information at time 0 remains relevant at a future time t. In their settings, distortions in the optimal mechanism disappear as impulse responses decay to 0. To illustrate how distortions due to adverse selection gradually disappear, we focus below on the setting of Sannikov (2007), as it conveniently builds on a model of DeMarzo and Sannikov (2006) that we already discussed in Section 4.¹⁸

Sannikov (2007) studies adverse selection a setting similar to that of DeMarzo and Sannikov (2006) (DS). The key new assumption is that the agent has private information about the mean of cash flows, μ_H or μ_L. Only the project with cash flow μ_H > μ_L has positive NPV. The principal would like to design a contract that maximizes profits, subject to (1) giving the agent with a good project a desired expected payoff of W₀, (2) giving the agent with a good project incentives to reveal cash flows truthfully, and (3) screening out bad projects with the mean of cash flows μ_L. The agent participates in up-front investment, and is able to receive an outside value of R ∈ (0, W₀) from the resources contributed. If the project is funded and terminated, the agent’s outside option is 0.

In addition, it is assumed that (1) the agent and the principal have a common discount rate r, (2) the project has a finite time horizon T, but may be terminated early for underperformance, (3) the agent has access to a secret savings technology, with a rate of return r, and (4) the agent with a bad project may have large savings to exaggerate the project’s cash flows in the short run.

As in DS, the agent privately observes the cash flows

The agent may divert cash flows or use his private savings to boost cash flows. Moreover, the agent captures the full value of diverted cash flows, i.e. the parameter λ in the setting of DS is set to 1.

Apart from adverse selection, the setting differs from that of DS in minor ways. Therefore, we know the form that the optimal contract would take in the absence of adverse selection. With moral hazard alone, the optimal contract is based on the agent’s continuation value, which follows:

Because the agent is as patient as the principal, it is optimal to postpone payments to the agent until time T. If W_t hits zero before time T, then the project is terminated early, and otherwise the agent receives the payment of W_T at time T.¹⁹

The contract can be implemented through a credit line with balance M_t = μ_H/r − W_t and a credit limit of . Then the balance evolves according to:

If the credit limit is reached, the project is terminated. However, unlike in DS, in this implementation the agent saves excess cash flows if the credit line is fully paid off, i.e. the agent is not paid until time T.

Remarkably, it turns out that the optimal contract with adverse selection is a natural modification of this contract. The simplicity of the optimal contract is informative about the impact of adverse selection. However, while the final product is clean on the outside, it requires sophisticated engineering. In fact, the proof of optimality of the contract described below requires significantly more complex arguments than those used in the analysis of the DS model, which we discussed in Section 4. Those arguments can be found in the paper.

Observe that the optimal contract under pure moral hazard “does not work” with adverse selection. Indeed, the credit available to the agent at time 0, , exceeds the payoff of R that the agent with a bad project can obtain elsewhere if he does not pretend to have a good project. Therefore, to screen out bad projects, the credit limit at time 0 must be restricted to . In fact, the present value of the expected cash flows received by the agent with a bad project, as well as funds drawn from the credit line, cannot exceed R at any time in the future, i.e. we need

(14)

for any history of reported cash flows. Otherwise an agent with a bad project, with a sufficient flexibility to generate high short-term cash flows, can game the system and get a payoff higher than R.

Equation (14) gives an upper bound on the maximal amount of credit that can be made available to the agent,

As long as M₀ > μ_L/r, is increasing in t and reaches the level of at some future time T^*.

It turns out that the optimal contract with adverse selection takes the form of a credit line, in which the credit limit is until time T, and it is from time T to time T. It is a remarkably simple contract, as the credit limit depends deterministically on time, that is, it does not depend on the agent’s behavior. Figure 14 illustrates the form of the credit limit.

This model illustrates how the distortions due to adverse selection disappear over time. The credit available to the agent is initially restricted due to adverse selection, but the credit limit rises over time, and from time T onwards the optimal contract looks as if adverse selection were never a problem. The agent is initially in a “hot seat”. A shorter credit limit is unforgiving about losses. As a result, if the agent could affect the cash flows with effort, he would work harder up front to differentiate himself from types with a bad project.

The solution of Garrett and Pavan (2009) illustrates a similar pattern using a model that is different in many ways. One major difference from Sannikov (2007) is that the absence of adverse selection leads to a first-best outcome in the model of Garrett and Pavan (2009), as the agent is risk-neutral and does not have limited liability. Thus, the form of the optimal contract converges to first-best over time.

Several other papers have explored dynamic contracts in settings with asymmetric information. For example, Tchistyi (2006) and DeMarzo and Sannikov (2010) investigate settings where the firm’s cash flows are correlated over time. Thus, the agent has private information about the distribution of future cash flows (at least off the equilibrium path). Williams (2011) and Kwon (2012) also address the issues of persistence, and Golosov, Troshkin, and Tsyvinsky (2010) as well as Farhi and Werning (2011) focus on these issues in the context of public finance. While several common themes emerge, in general there is no unified way to analyze settings of dynamic adverse selection and moral hazard, and this area is ripe for future research.