Security design with interim public information

Accepted Manuscript Security design with interim public information André Stenzel

PII: DOI: Reference:

S0304-4068(18)30020-X https://doi.org/10.1016/j.jmateco.2018.02.005 MATECO 2222

To appear in:

Journal of Mathematical Economics

Received date : 4 January 2018 Accepted date : 15 February 2018 Please cite this article as: Stenzel A., Security design with interim public information. Journal of Mathematical Economics (2018), https://doi.org/10.1016/j.jmateco.2018.02.005 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Security Design with Interim Public Information∗ André Stenzel† February 22, 2018

We consider a security design problem where public information about the security’s underlying cash-flow arrives between trading periods. The optimal security minimizes less-than-full realization of gains from trade due to limited cash in the market, which may depend on the interim information. We show that the optimal security can be expressed as a convex combination of securities solving minimization problems for which the solutions share many debt-like features but exhibit endogenous tranching. We provide conditions for the non-optimality of standard debt contracts and show that implementation of the class of optimal securities can be achieved by mezzanine tranche retention, providing a public information rationale for departure from the pecking order. JEL Classification: D84, D86, E51, G14 Keywords: Security Design, Public Information Arrival, Tranching, Pecking Order

This article extends Chapter 4 of my PhD Thesis Essays in Financial Economics at the University of Mannheim and supersedes an earlier working paper version circulated under the same title. I thank the editor, Atsushi Kajii, and two anonymous referees for insightful comments. I am furthermore deeply grateful to Bruno Biais, Jana Friedrichsen, Alexander Guembel, Christian Michel, Andras Niedermayer, Volker Nocke, David Rojo Arjona, Florian Sarnetzki, Nicolas Schutz, ErnstLudwig von Thadden, Christoph Wolf, and Philipp Zahn for valuable discussions and feedback, and appreciate input from seminar audiences in Mannheim and Toulouse as well as participants of the ENTER Jamboree 2013 in Bruxelles and the SFB Tr 15 Summer School on Incomplete Contracts 2012. † University of Leicester, E-Mail: [email protected]

∗

1. Introduction In the wake of the recent financial crisis, the analysis of securitization practices and structure of traded securities has enjoyed a renewed focus by researchers and practitioners alike.1 The breakdown of previously well functioning markets for asset backed securities raised new questions about how these markets are operating, and why the securities are structured the way they are. This understanding is of particular importance given the intense regulatory efforts of policymakers aimed at preventing future crisis episodes2 – without a detailed understanding of issuers’ incentives, regulation may lead to unintended adverse consequences. One theme recurrent in the theoretical literature on security design is the optimality of standard debt contracts in a variety of settings, motivated by its low sensitivity to private information (see among others Nachman and Noe (1994), Dang et al. (2015b), Yang (2015)). Moreover, Dang et al. (2011, 2015a) show that this optimality carries over to a setting where agents may become privately informed, and public information arrives between trading periods. We build on their work by focusing exclusively on the public information issue, while generalizing the structure of public information. In our setup, optimal securities share many debt-like features, but exhibit multiple tranches and are thus distinct from standard debt. This implies a misalignment of incentives for security designers with respect to private and public information concerns. Securities which minimize the value of private information – standard debt contracts – may fail to optimally transmit value to the secondary market in the face of interim public information. Our characterization also provides a rationale both for the widespread prevalence of multiple tranches based on the same underlying pool of loans (see e.g. Gorton and Metrick (2012)), as well as for the empirically documented departure from the pecking order (see e.g. Leary and Roberts (2010)). The model setup builds on Dang et al. (2011, 2015a). An investor acquires a security 1

This is motivated both by the size of the markets for securitized assets, and their importance in the crisis. For example, U.S. asset-backed securities issuances totaled 510 billion US-$ in 2007, which plummeted to 139 billion in 2008. See Gorton and Metrick (2012) for a detailed overview of the role securitization plays in modern financial systems and in particular the recent crisis. 2 One example is the Dodd-Frank Act of 2010 in the United States, and ongoing regulatory efforts involving (partial) repeals and amendments.

1

from a primary market issuer. The security is based on the cash flow of some underlying project, such as a pool of mortgages, which is distributed according to a mixture between two distributions. The true distribution is publicly revealed after the primary market transaction. Subsequently, the investor may trade with a representative market agent, who is constrained in her endowment, in the secondary market. In this setup, the optimal security is one which maximizes expected gains from trade in the secondary market. These in turn are determined by the relation between the primary market security’s value post information revelation, and the endowment of the market. Dang et al. (2011, 2015a) show that in such a setup debt is the optimal security with regards to public information concerns if the endowment of the market does not depend on the interim information and underlying distributions are ordered according to the monotone likelihood ratio property (MLRP). In that case, the optimal security is the one whose value is most robust to the interim information, which is established to be a standard debt contract. Our contribution relaxes the underlying assumptions. First, we allow the endowment of the market to depend on the revealed information. This captures the potential interaction between the systemic relevance of a considered security and the portfolios of agents in the economy. For example, during the recent crisis, the collapse of the housing bubble had an impact both on the valuation of individual securities, and market conditions as a whole – mortgage-backed securities (MBS) lost a substantial part of their value, while the liquidity in the market for securitized assets dried up. Second, we do not put any restriction on the underlying distributions. Both the MLRP ordering, and the more general first order stochastic dominance (FOSD), allow to classify news into ‘good’ and ‘bad’. However, it is not a priori clear that news affecting the distribution of the value of an asset always needs to be inherently classifiable in such a fashion. In the context of MBS, news can heterogenously affect subgroups of the population of loans forming the pool underlying a given security, and thus not allow for an unambiguous assessment – consider as an example the recent tax reform by the Trump administration. In addition, even if the distinction into good and bad news is intended, the choice of the underlying structure matters. As such, our generalization in this direction highlights the sensitivity of the optimal security structure to 2

the assumptions in the underlying models. Both deviations individually may lead to the non-optimality of standard debt contracts. Without the MLRP ordering, debt is no longer necessarily the security whose value is most robust to interim public information, and endogenous tranching obtains in the characterization of the optimal security.3 Moreover, robustness to interim information itself may cease to be a desirable feature if the cash in the market depends on the revealed information. For example, if the market is extremely tight in one of the states, the optimal security should be sensitive to public information by maximizing the value in the other state – from an ex-ante perspective, the investor wants the security’s value to be high in the state where she can trade with a liquid market, and low if gains from trade could not be recovered due to lack of funds.4 Within our setup, we provide a structural characterization of the optimal securities. We show that the optimal security design can always be constructed as the convex combination of two securities which share many debt-like features, but exhibit endogenous tranching. These individual securities minimize the value of the security under each of the two possible distributions, subject to exhausting the investor’s initial endowment. We establish that they structurally correspond to tranched debt contracts.5 Tranched debt contracts are composed of multiple imperfect debt tranches. We provide conditions for standard debt not to be a solution to a given minimization problem. Finally, we show that implementation of tranched debt contracts can be achieved via retention of mezzanine tranches of a given cash-flow, while selling both junior and senior claims. In short, the article contributes to the understanding of security designers’ incentives, where the literature has largely focused on private as opposed to public information concerns. Our results imply that the private information and public information aspects in a generalized public information framework are not necessarily aligned. While standard debt may minimize private information acquisition incentives (as established 3

This applies even if the distinction of ‘good’ and ‘bad’ news is feasible by imposing FOSD, see Example 2. 4 A more detailed discussion of this issue is given in Example 1 following our theoretical analysis. 5 This obtains if attention is restricted to securities satisfying dual monotonicity. If the restriction is only to nondecreasing securities, the optimal contracts are contracts composed of debt-like tranches, which we analyze in detail in Appendix B. Both restrictions are common in the literature on security design, see e.g. Innes (1990), Dang et al. (2011) (nondecreasingness) or Biais and Mariotti (2005) (dual monotonicity). An earlier version of this article treated both cases in the main text; for expositional purposes, we restrict the analysis in the main text to securities satisfying dual monotonicity, and relegate the analysis with nondecreasing securities to the appendix.

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by the literature), our paper demonstrates that more complex securities exhibiting endogenous tranching are optimal with respect to interim public information. This misalignment and resulting difference between securities resolving private and public information concerns is novel to the literature. Moreover, the mezzanine tranche retention which is implied by the optimal security structure under public information provides an argument in favor of departing from the standard pecking order. It thus complements recent contributions which show that a similar departure can be rationalized by private information concerns (see e.g. Fulghieri et al. (2015)).

1.1. Relation to the Literature Our article primarily relates to two distinct strands of literature. First, it complements the literature on security design by focusing on a security’s sensitivity to public information as opposed to private information. The resulting optimal securities exhibit endogenous tranching and can be implemented via mezzanine retention. This relates to the second strand; we provide a public information rationale for departure from the pecking order. The focus of the literature on security design has by and large been on the impact of asymmetric – private – information. One central result is the optimality of standard debt due to its low information sensitivity.6 Gorton and Pennacchi (1990) explicitly model both uninformed and informed traders and show that a financial intermediary can prevent uninformed traders’ losses to insiders who hold private information by issuing (riskless) debt. Boot and Thakor (1993) show that a firm who wishes to sell an asset under asymmetric information maximizes its revenue by splitting the cash flow into an information-sensitive security promoting informed trading, and an information-insensitive one. Nachman and Noe (1994) consider a security design context at the interim stage and characterize sufficient conditions for debt to be preferred to equity. DeMarzo and Duffie (1999) analyze a security designer and issuer whose private information results in illiquidity in the sense of a downward sloping demand 6

Optimality of debt also features in settings with costly state verification or non-verifiable returns and exogenously given asymmetric information (studied for example by Diamond (1984), Townsend (1979), Gale and Hellwig (1985) and Aghion and Bolton (1992)), where standard debt is shown to be optimal for issuing a security in a primary market.

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curve. Standard debt is shown to be optimal under certain conditions, primarily the existence of a uniform worst case, because it minimizes the value of the private information the issuer holds. Biais and Mariotti (2005) consider an alternative approach to the trading game whereby the issuer commits to a price-quantity menu prior to realizing her private information. Debt is optimally issued because it minimizes the consequences of adverse selection (competitive case) and mitigates the market power of the liquidity supplier (monopolistic case). Yang (2015) analyzes a game where information acquisition is flexible and arrives at a similar conclusion: Standard debt contracts minimize incentives to acquire information and therefore maximize liquidity. Farhi and Tirole (2015) consider a security trading game with a binary state of nature and show that strategic agents tranche their assets into a safe debt and risky (leveraged) equity component. Our article directly builds on the work by Dang et al. (2011) and subsequently Dang et al. (2015a). The formally define the information sensitivity of a security and show that standard debt is least information sensitive amongst the class of nondecreasing securities. Moreover, Dang et al. (2011, 2015a) also consider the public information issue which is the focus of our paper. In their setup, the incentives of the investor align: standard debt disincentivizes potential trading partners to acquire information and is least sensitive to interim public information. This, however, is driven by the structure of the public information, which we generalize. We allow the endowment constraint, which leads to a form of cash-in-the-market pricing as in Allen and Gale (1994), to vary with the interim public information. We thus depart from Dang et al. (2011) in two distinct ways which affect the security designer’s incentives: On the one hand, when underlying distributions do not satisfy the MLRP, standard debt is not necessarily most robust to interim public information. On the other hand, robustness itself may cease to be a desirable feature once the cash in the market is correlated with the interim information – this is highly likely the case in practice, where market liquidity depends on factors which also affect individual securities’ values. We show that the solutions to the generalized security design problem can nonetheless always be expressed of a convex combination of securities which share many debt-like features but exhibit endogenous tranching. This implies that the public and private information 5

concerns do not necessarily go hand in hand. The optimal security with respect to public information concerns may be starkly different from the standard debt contract which is optimal to deter private information acquisition. As a result, the security designer’s incentives are misaligned. While the pecking order first formalized by Myers and Majluf (1984) is confirmed in a variety of theoretical contributions, several articles provide conditions such that the pecking order theory fails to hold, corroborating empirical evidence (see e.g. Leary and Roberts (2010)). Fulghieri and Lukin (2001) show that departure from the pecking order can result from considering endogenous information acquisition whereby information acquisition is beneficial as it reduces the informational asymmetry between seller and investors. Chakraborty and Yilmaz (2011) consider a framework where the issuing manager’s information advantage dissipates over time and show that the optimal security in such a setup has all the features of a convertible debt contract. Fulghieri et al. (2015) focus on the impact of asymmetric information in a security design capital raising framework. They show that a reversal of the pecking order may arise. The optimal security design depends on the location of asymmetric information – if information asymmetries have low impact on the right tail of the value of the firm, large amounts of capital are better raised via equity than debt. This reasoning is closely related to our paper: While we are concerned exclusively with public information, it is the location of states where public information is relatively unimportant and important, respectively, which shapes the optimal security. Furthermore, we provide conditions for the non-optimality of standard debt in the public information context and show that implementation of the optimal security can be achieved via mezzanine retention. While the article is written primarily in the context of securitization, the argumentation is applicable to a setting where a firm seeks funding in a primary market and investors have the ability to later on trade their holdings in a secondary market. Our analysis thus provides a public information rationale for departure from the pecking order, complementing the private information rationale present in the literature. Finally, we implicitly relate to the literature covering the extent of information available to market participants and whether such information should be disclosed. Kaplan (2006) shows that it can be efficient for a bank to commit to a policy keeping informa6

tion about its risky assets. Pagano and Volpin (2012) show that issuers of assets choose to publish coarse instead of precise ratings to enhance liquidity in the primary market, even though this reduces secondary market liquidity. Stenzel and Wagner (2015) consider the interaction between opacity and liquidity and provide a micro-foundation for the notion that opacity may deter information acquisition even if marginal units of information can be acquired. In Dang et al. (2017) and Monnet and Quintin (2017), opacity circumvents the issue of public information. The present analysis is complementary in the sense that it considers the case where public information is relevant as it leads to potential losses of gains from trade. The remainder of the paper is organized as follows: Section 2 presents the model. Section 3 structurally characterizes the optimal security. We first solve the security design problem after public information arrival, and use this to analyze the security design problem in the primary market. Section 4 illustrates the results with numerical examples. Section 5 discusses how the derived class of optimal securities can be implemented via mezzanine retention. Section 6 concludes.

2. The Model Our model is based on the framework by Dang et al. (2011, 2015a). We abstract from private information acquisition, but generalize the structure of the public information problem. The model is deliberately stylized to provide the simplest possible framework which allows us to isolate the impact of public information on primary market security design. It is set up such that an investor acquires a security in the primary market to store value for the future. This is realized by trading in a secondary market, and is complicated by public information which arrives during the holding period. In what follows, we provide the details of the model setup to allow for a self-contained exposition. Agents & Preferences There are three risk-neutral agents in the economy: An institution (called bank or issuing institution) B, an investor I and a representative market agent M . In the absence of private information and associated information asymmetries, the economy is composed of agents willing to transfer utility across pe7

riods. Time is discrete and covers three periods, t = 1, 2, 3. The utility for each agent i ∈ {B, I, M } is denoted U i and depends on consumption Cti of agent i in period t, which occurs at the end of a given period:

1 B C β 2

+

C3B

+ σC2I

+

C3I

U B = C1B + UI

= C1I

UM =

C2M

(1)

+ C3M .

β > 1 and σ > 1 reflect intertemporal differences in marginal utilities of consumption. The investor prefers consumption in period 2, while the market is indifferent between consumption in periods t = 2 and t = 3, whereas the bank is indifferent between consumption at t = 1 and t = 3. The preferences are set up in such a way to enable the relevant trading pattern whereby an investor initially acquires the security and can then sell it in a second trading period.7 Endowments Agents’ endowments are common knowledge: The bank owns a pool of assets with stochastic return X distributed on X ⊆ R+ which is due at t = 3. For ease of notation, we restrict attention to an open interval of the form (xL , xH ) ⊂ R+

with xH < ∞. The inclusion of boundary points would not alter the results. The investor is endowed with ω I in cash at t = 1, while M holds a cash endowment of ω M at t = 2. Cash endowments are non-storable, and future cash endowments cannot be contracted upon. Denoting ω ˜ i = (ω1i , ω2i , ω3i ) the endowment vector of agent i and normalizing remaining endowments to zero gives ω ˜B =

(0, 0, X)

ω ˜I

(ω I , 0, 0)

=

(2)

ω ˜ M = (0, ω M , 0). Crucially, we allow ω M to depend on interim public information. Interim Public Information X is stochastic and its payoff is publicly observable and verifiable at t = 3. At t = 2, information about the distribution of X arrives. This 7

A preference for immediate consumption in t = 1 for the bank, or for consumption in period 3 for the market could be introduced and would not alter the results – the present normalization simplifies expressions as the investor’s trading partners are indifferent between consumption in the periods affected by the trade; prices paid would otherwise be multiplied by a constant which reflects the consumption preference.

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information is publicly observable. It is modeled in the following manner: Ex ante, X is distributed randomly on X with density f (x), cumulative distribution function F (x) R and finite mean X xdF (x) < ∞. To model information arrival, let f be a mixture distribution, i.e. let λ ∈ (0, 1) and

f (x) = λf1 (x) + (1 − λ)f2 (x)

(3)

where f1 and f2 are strictly positive continuous densities.8 The public signal arriving at the beginning of t = 2 perfectly reveals the true distribution, that is, there is a signal φ ∈ {φ1 , φ2 } where the conditional distribution is (4)

X|φ = φi ∼ fi

and where P r{φ = φ1 } = λ. All distributions are common knowledge, as is λ. The

endowment ω I is fixed. We impose ω I > xL to avoid issuance of riskless debt. ω M may depend on the public signal φ: ωM

  ωM 1 =  ωM 2

if

φ = φ1

if

φ = φ2

(5)

This allows for arbitrary correlations between the market endowment ω M and the underlying distributions f1 , f2 which determine the value of any security. Furthermore, note that f1 and f2 are not restricted to be ordered in any fashion. The only distinction we make is whether securities satisfy local non-proportionality or not. ∀(Ξ1 , Ξ2 ) ⊆ X : ∃ξ1 , ξ2 ∈ (Ξ1 , Ξ2 ) such that

f1 (ξ1 ) f1 (ξ2 ) 6= f2 (ξ1 ) f2 (ξ2 )

(LNP)

The local non-proportionality condition (LNP) essentially states that no interval (Ξ1 , Ξ2 ) ⊆ X exists on which the likelihood ratio of the two densities is constant.

Since the model abstracts from private information concerns, there is no disagreement about the value of the assets involved. The problem is that of a (limited) number of agents who wish to shift a known amount of consumption intertemporally. The essential friction is the limited market endowment ω M : Depending on the posterior value of the primary market security in relation to the realization of ω M , gains from trade can not

8

Strict positivity facilitates the analyses as existence and uniqueness statements can be made without accounting for f (·) = 0 on some intervals. Qualitatively, all results carry over if the assumption is relaxed.

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be realized if there is no agent liquid enough to buy the assets for their fair value.9 Essentially, there is a form of cash-in-the-market pricing as in Allen and Gale (1994). However, while the price of the asset adjusts in Allen and Gale (1994), in our setup the security is restructured to capture all available surplus. The resulting residual is held for future consumption. As in Dang et al. (2011, 2015a), the model setup and inherent assumptions are made to ensure that claims on X are traded in both periods, and that the design of these claims has relevance. Crucial for this is that future endowments are non-contractible.10 This non-contractibility breaks the Modigliani and Miller (1958) irrelevance result, and is motivated by practical considerations. It reflects that investors do not have perfect foresight as to who the counterparties with excess cash will be in future periods. In this setup, a social planner can realize gains from trade through a simple reallocation of endowments. It is imminent from the preferences that total welfare is maximized whenever I consumes ω M at t = 2. For I to consume at t = 2, she needs to trade with B at t = 1 by buying a stake of the project. She can then sell (parts of) that stake to M at t = 2. When agents trade, they exchange promises contingent on the observable realization of X. These promises are called securities. Throughout this article, securities have to satisfy the following requirements: Definition 1 A security s is a mapping from a domain D ⊆ R+ of underlying payoffs into the non-negative real numbers, s : D → R+ , satisfying the following restrictions: (i) limited liability: s(x) ≤ x for all x ∈ D and (ii) dual monotonicity: ∀x1 , x2 ∈ D : x1 ≥ x2 ⇒ s(x1 ) ≥ s(x2 )∧x1 −s(x1 ) ≥ x2 −s(x2 ) The set of securities satisfying dual monotonicity and limited liability is denoted SD . The definition implicitly restricts attention to deterministic securities. Nondecreasingness of securities is a standard assumption justified by a moral hazard opportunity 9

The problem corresponds to that of Diamond-Dybvig-type models where a limited fraction of the population is patient and therefore willing to shift a limited amount of consumption into the last period by buying assets in the interim period, see for example Diamond and Dybvig (1983), Jacklin and Bhattacharya (1988) and Chari and Jagannathan (1988). 10 If future endowments were contractible, I and M could trade prior to the realization of the public information and reach an efficient allocation. As in Dang et al. (2011, 2015a), M entering only at t = 2 would be a sufficient alternative assumption.

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of the issuer, see for example Innes (1990). The stronger restriction to dual monotonicity requires that both the security’s payoff s(x) and the payoff of the residual x − s(x) are nondecreasing in x. Both restrictions are prevalent in the literature.11 As DeMarzo et al. (2005) note, “a standard motivation for dual monotonicity is that, if it did not hold, parties would "sabotage" the project and destroy output. [...] Whether revenues can be distorted in this way depends on the context.” We solve the security design problem under both restrictions. For expositional purposes, the analysis in the main text restricts attention to securities satisfying dual monotonicity; the other case is treated in Appendix B. None of our main insights depend on the choice of restriction.12 Pooling of securities based on different projects is not addressed in this setup.13 However, the random endowment X itself can be interpreted as a collection of different assets/securities, such as MBS which are based on a collection of mortgages. With respect to tranching, note that since all agents in the model have constant marginal utilities of consumption in any given period, they can be thought of as representing an arbitrarily large number of identical agents who hold an endowment with an aggregate endowment equal to what is represented in the model by ω I (in case of I) and ω M (in case of M ) respectively. If that is the case, any tranching which overall still satisfies limited liability can also be represented by a single contract.14 In interpreting the results, a security featuring multiple tranches is best rationalized by selling individual tranches to separate, interchangeable market participants with identical intertemporal substitution rates.

Timing The timing of the game is as follows: At t = 1, the investor I makes a take-it-or-leave-it offer to the bank. This offer consists of a security s conditional on 11

Simple nondecreasingness features e.g. in Innes (1990) and Dang et al. (2011), while dual monotonicity is assumed e.g. in Biais and Mariotti (2005). 12 We establish that while discontinuities are indeed a feature of the class of optimal securities under nondecreasingness, they nonetheless share many structural characteristics with the optimal securities issued when restricted to securities satisfying dual monotonicity and hence continuity, mainly endogenous tranching which can be implemented via mezzanine retention. 13 Allowing pooling of securities depending on correlated underlying payoffs would affect the results. However, the main idea that individual securities should maximize expected gains from trade subject to the public information still factors into the security design process. 14 Consider for example the issuance of two securities contingent on X, s1 (x) and s2 (x) with s1 (x) + s2 (x) ≤ x, ∀x ∈ (xL , xH ), which are also nondecreasing. Given constant marginal utilities of consumption and the unique trading partner, this is equivalent to issuing a single security s(x) = s1 (x) + s2 (x), ∀x ∈ (xL , xH ), which will still satisfy limited liability and nondecreasingness.

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t=1

t=2

(1)

I makes take-it-or-leave-it (1) offer (s, p) to B

public signal φ ∈ {φ1 , φ2 } is revealed

(2)

B accepts contract (s, p) or not

(2)

I makes take-it-or-leave-it offer (ˆ s, pˆ) to M

(3)

M accepts contract (ˆ s, pˆ) or not

t=3 (1)

t

x publicly realized, I receives s(x) − sˆ(x), M receives sˆ(x)

Figure 1: Timeline of the Game the return of X at t = 3 which she is willing to buy, and a price p which she pays in exchange. At t = 2, a public signal regarding the distribution of X is revealed to all agents. Then, I may make a take-it-or-leave-it offer to agent M . This offer consists of a security sˆ conditional on the return of s (and hence of X), and a price pˆ. We assume that the bargaining power lies in the hand of the investor in both stages. This, coupled with the assumption that marginal utility is constant, is made to isolate the security design process with respect to arriving public information.15 The second trading stage is only relevant if trade occurred at t = 1. Furthermore, the limited liability constraint imposes that sˆ(x) ≤ s(x) for all x ∈ X. Figure 1 depicts the timeline of the game.

2.1. Concepts and Definitions There are several classes of securities which play an important role in the subsequent analysis which we briefly introduce here. One such class is that of standard debt contracts. Definition 2 A standard debt contract (SDC) is characterized by its face value D and given by sSDC (x; D) = min{x, D}. Standard debt contracts pay out according to the limited liability constraint s(x) = x up to their face value D; for realizations of X exceeding the face value, the payoff is capped. A second class of securities which is important for our analysis is the class of leveraged debt contracts. Leveraged debt contracts only pay out if the payoff of the underlying cash flow exceeds a certain threshold (L), and then pay proportional to an 15

In this particular game, the security design problem remains identical as long as the investor has at least some bargaining power in both trading stages.

12

increase in the value of the underlying collateral, up to face value (D). Definition 3 A leveraged debt contract (LD) is characterized by the leverage level L and face value D. Its payoff structure is given by sLD (x; L, D) = max{min{x − L, D}, 0}.

payoff

payoff

limited liability

leverage

sSDC (x; D)

D

limited liability

sLD (x; L, D)

D

Standard Debt Contract

Leveraged Debt Contract

x

L

L

x

Figure 2: Standard debt contract and leveraged debt contract

Figure 2 illustrates the two types of contracts. They are important as each tranched debt contract can be expressed as a combination of multiple leveraged debt tranches, each of which is a standard or leveraged debt contract. Given the strict positivity of densities, the following Lemma is imminent. Lemma 1 For each v ∈ 0,

R

X


xf (x)dx , there exists a unique D(v) such that E[sSDC (x; D(v))] =

ω. Furthermore, for each face value D such that v ≥ E[sLD (x; xL , D)], there exist unique L(v, D) such that E[sLD (x; L(v, D), D)] = v where sSDC , sLD ∈ SD .

Lemma 1 ensures that there is a unique face value (standard debt), and a unique leverage level associated with each face value (leveraged debt) respectively, for each expected value v the overall security has to match.16

16

The restriction on D in the latter case is to ensure that a leverage level exists such that the overall value v can be matched – if D is too low, even an unleveraged security would be insufficient to generate ex-ante expected value v.

13

3. Optimal Security Structure 3.1. Trading after Information Arrival At t = 2, the bank B and representative market agent M cannot profitably trade.17 Thus, trade may only occur if I possesses some security s acquired from B at t = 1. She may either sell s or use it as collateral for a new security which is offered to M at t = 2. Since all information is public, trade can only occur at a price equal to the common conditional expected value of the offered security.18 Hence, the optimal strategy for I depends on the relation of the updated value of s after public information to the realized market endowment ω M . The following Proposition characterizes trade after public information arrival. Proposition 1 Suppose I holds a security s at t = 2. Denote the realization of φ as φi , i ∈ {1, 2}. If sˆ is traded to M in equilibrium, it satisfies: (i) sˆ(x) ∈ SD (ii) If E[s(x)|φ = φi ] ≤ ωiM then sˆ(x) = s(x) for all x ∈ (xL , xH ). s(x)|φ = φi ] = ωiM and sˆ(x) ≤ s(x) for all (iii) If E[s(x)|φ = φi ] > ωiM then E[ˆ x ∈ (xL , xH ) sˆ is sold to M at its conditional expected value pˆ = E[ˆ s(x)|φ = φi ] = Efi [ˆ s(x)]. Corollary 1 The vertical slice sˆ(x) of s(x), where sˆ(x) = τ s(x) for all x ∈ (xL , xH ) with τ = min



 ωiM ,1 , E[s(x)|φ = φi ]

is one security which may be traded in equilibrium at t = 2. Proposition 1 follows immediately from the fact that I makes a take-it-or-leave-it offer to M and has preference for consumption at t = 2. Intuitively, if the asset is worth weakly less than the market endowment ωiM given public signal φ = φi , it is optimal for the investor to sell the whole security to maximize consumption at t = 2 17

B holds a (residual) claim to the payoff of the project X at t = 3, for which the two agents have the same marginal utility of consumption. M , by contrast, holds an endowment ω M in the second period, where B’s marginal utility of consumption is β1 < 1 and hence lower than that of M . 18 Recall that the bargaining power lies with I and that M is indifferent between consumption at t = 2 and t = 3.

14

(Proposition 1.(ii)). If the endowment constraint binds, I sells a security that is worth strictly less than E[s(x)|fi ]. This new security sˆ must satisfy limited liability with respect to s and E[ˆ s|fi ] = ω M to maximize consumption at t = 2 (Proposition 1.(iii)). In this case, I holds on to the residual security (s − sˆ) and consumes this remainder

at t = 3 after the realization of X becomes observable and s, sˆ pay out. A simple implementation of this is given in Corollary 1 – I can simply sell a vertical slice of s which exactly exhausts the cash in the market ω M . Given that the behavior after public information arrival is characterized by Proposition 1, we next analyze the security design problem at t = 1. For this we use that Proposition 1 implicitly characterizes the gains from trade which can be realized in the second trading stage.

3.2. Security Design in the Primary Market To address the security design problem at t = 1, we proceed in the following manner. We first establish that the investor aims to minimize the expected ’lost’ gains from trade due to a binding endowment constraint, and show that one optimal security can always be expressed as the convex combination to modified minimization problems. We then characterize the solutions to these modified problems structurally and establish that they are tranched debt by breaking the global problem on (xL , xH ) down into local problems for which the solution corresponds to a leveraged debt contract. Consider any security s designed and acquired by the investor at t = 1. It is straightforward that Ef [s(x)] ≤ ω I (the issuing institution will at most obtain ω I and not sell

anything worth more in expectation) and that – given the take it or leave it offer – the investor will pay Ef [s(x)] for it. Using Proposition 1, the investor’s expected utility can be expressed as 
EU I (s) = ω I − Ef [s(x)] + λ σ min {Ef1 [s(x)], ω1M } + max Ef1 [s(x)] − ω1M , 0  
+(1 − λ) σ min Ef2 [s(x)], ω2M + max Ef2 [s(x)] − ω2M , 0  
= ω I + (σ − 1) λ min Ef1 [s(x)], ω1M + (1 − λ) min Ef2 [s(x)], ω2M (6)

where we have used Ef [s(x)] = λEf1 [s(x)] + (1 − λ)Ef2 [s(x)]. From (6), it is clear that the initial security design reduces to minimizing the implicit losses due to imperfect 15

realization of gains from trade whenever the endowment constraint binds in either state. Inspection of (6) also reveals how the potential relation between the market endowment and the public information impacts the security design process. If the endowment were independent of public information, i.e. for ω1M = ω2M = ω M , expected utility is maximized whenever the security’s value post information revelation exhibits the lowest variance: Robustness of the security’s value to public information is desirable.19 This is no longer necessarily the case in our setup. If the market is extremely tight in one state, but extremely liquid in the other, e.g. for ω1M close to 0 and ω2M large, the investor benefits from holding a security whose value depends strongly on the public information. Ideally, she would like to have as much value as possible if φ = φ2 is revealed. From (6), it follows that it is weakly optimal for the investor to acquire a security s such that Ef [s(x)] = ω I as EU I (s) is weakly increasing in both Ef1 [s(x)] and Ef2 [s(x)]. Denote SX,ωI ≡ {s ∈ SX s.t. Ef [s(x)] = ω I }

(7)

the set of securities satisfying limited liability and dual monotonicity which exhaust the endowment ω I . Furthermore, let   S ∗ ≡ arg max{ max λ min Ef1 [s(x)], ω1M + (1 − λ) min Ef2 [s(x)], ω2M }. s∈SX,ωI | {z }

(8)

A

denote the set of securities in SX,ωI which maximize expected utility. Formally, with (6), ∀s∗ ∈ S ∗ , ∀s ∈ SX : EU I (s) ≤ EU I (s∗ ),

(9)

that is, any security exhausting the investor’s endowment ω I which minimizes the losses due to imperfect realizations of gains from trade weighted by the probability of the respective state of the world (as given by A) maximizes the investor’s utility (and is hence in S ∗ ). Before we turn to the main analysis of the paper and characterize the optimal security structure, it is convenient to introduce the following minimization

19

In Dang et al. (2011), this is achieved by minimizing the value in the ‘good’ state, which exists because of the assumed MLRP ordering. Minimizing the value there, and conversely maximizing it in the ‘bad’ state, simultaneously minimizes the variance of the value post information revelation.

16

problems: (P1) min Ef1 [s(x)]

s.t.

λEf1 [s(x)] + (1 − λ)Ef2 [s(x)] = ω I

(P2) min Ef2 [s(x)]

s.t.

λEf1 [s(x)] + (1 − λ)Ef2 [s(x)] = ω I .

s∈SX

s∈SX

A given problem (Pj) looks for the security which minimizes the value of the security in state j subject to exhausting the investor’s endowment ω I . Solutions to (Pj) exist as any (Pj) corresponds to the minimization of a continuous mapping from closed and bounded sets into the real numbers, see Appendix A.1 for a formal proof. Denote by s∗j a solution to problem (Pj) where j ∈ {1, 2}. Any s∗j minimizes the expected

value in state j subject to the security having ex-ante value ω I . We establish that one security which maximizes the investor’s expected utility can be expressed as a convex combination of solutions to problems (P1) and (P2). As such, even though the setup allows for an arbitrary correlation between the market endowment and the public information, which in principle could greatly complicate the security design problem, we show that it is sufficient to solve the two ‘extreme’ problems (P1) and (P2) which are independent of the market endowments. The resulting optimal securities can then be used to construct a security maximizing the investor’s expected utility given ω1M , ω2M . Intuitively, this is because once ω1M , ω2M determine the optimal value distribution of the security’s value across the two states, this translates into the weights put on the ‘extreme’ solutions s∗j which characterize the optimal security. Proposition 2 Consider securities s∗j , j ∈ {1, 2} as described above. Denote vjl the   expected value of s∗j in state l, vjl = Efl s∗j (x) , ∀l ∈ {1, 2}. Then (I) ∀η ∈ [0, 1] : ηs∗1 + (1 − η)s∗2 ∈ SX,ωI

(II) ∃η ∈ [0, 1] : ηs∗1 + (1 − η)s∗2 ∈ S ∗ . Proof. See Appendix A.2. Proposition 2 enables us to focus on Solutions to problems (Pj): By structurally characterizing the solutions to (Pj), we know that there exists an optimal security at t = 1 which shares the features of this structure as it can be expressed as a convex combination. Typically, there will be a large multiplicity in securities which maximize the investor’s expected utility at t = 1 and no uniquely optimal security exists. Whenever a convex combination is optimal with η ∈ (0, 1), small variations to solutions s∗j 17

do not have a large effect on the posterior values; as such, by slightly altering the weight put on the variation, a security with equal posterior values can be constructed which then is also optimal. However, there are two reasons for nonetheless focusing on solutions to (Pj). First, we establish that they share many debt-like features, but exhibit endogenous tranching, and are thus closely related to securities we observe to be traded in financial markets. Second, there are instances where η ∈ {0, 1}, that is,

where only solutions to (Pj) are traded in equilibrium (see Section 4 for examples).

To characterize solutions to (Pj), we need to introduce the modified minimization problems (P’j) (P”j)

min Efj [s(x)] s.t. E[s(x)] = ω 0 ∧ s(x) ≤ C 0 , ∀x ∈ X

(10)

min Efj [s(x)] s.t. E[s(x)] = ω 00 ∧ s(xH ) = C 00

(11)

s∈SX

s∈SX

R where C 0 and ω 0 are such that X min{x, C 0 }f (x)dx ≥ ω 0 and C 00 and ω 00 satisfy R Rx min{x, C 00 }f (x)dx ≥ ω 00 ≥ xHH−C 00 (x − (xH − C 00 ))f (x)dx. The restrictions ensure X that valid securities exist which exhaust the endowment ω 0 and ω 00 , respectively.20 We establish the following Proposition.21 Proposition 3 Consider problems (P’j) and (P”j), j ∈ {1, 2}. Then (i) If

fj f−j

is increasing in x on X, the Standard Debt Contract sSDC (x; D(ω 0 )) solves

(P’j). (ii) If

fj f−j

is decreasing in x on X, the Leveraged Debt Contract sLD (x; L(ω 0 , C 0 ), C 0 )

solves (P’j) and (P”j). (iii) If

20

fj f−j

is increasing in x on X, the contract sT T (x) solves (P”j), where sT T is

composed of two debt tranches and given by   min{x, D} if x ≤ (xH − C 00 ) + D sT T (x) = ,  x − (x − C 00 ) otherwise H

Note that the securityR smax (x) = min{x, C 0 } maximizes the ex-ante value given the restriction s(x) ≤ C 0 . If ω > X smax (x)f (x)dx, no security satisfying the restrictions exists which gives ex-ante value ω 0 . For C 00 , ω 00 the upper bound stems from the same reasoning, while the lower bound ensures that a security satisfying dual monotonicity exists which attains value ω 00 . The security smin (x) = max{x − (xH − C 00 ), 0} is the security with minimal expected value which R attains s(xH ) = C 00 and satisfies dual monotonicity. This has expected value X smin (x)f (x)dx = R xH (x − (xH − C 00 ))f (x)dx. xH −C 00 21 Note that (i) is essentially proven in Dang et al. (2011), as a standard debt contract satisfies not only nondecreasingness, but also dual monotonicity.

18

where D is implicitly defined by

R

X

sT T (x)f (x)dx = ω.

For (i)-(iii), if the likelihood ratio is strictly increasing (or decreasing, respectively), the respective securities are the unique solutions. Proof. See Appendix A.3. Corollary 2 Consider the original problem (Pj), j ∈ {1, 2}. (i) If

fj f−j

is increasing in x on X, the Standard Debt Contract sSDC (x; D(ω I )) solves

(Pj). (ii) If

fj f−j

is decreasing in x on X, the Leveraged Debt Contract sLD (x; L(ω I , ∞), ∞)

solves (Pj).

Proof. The corollary follows immediately from Proposition 3 by noting that (Pj) corresponds to (P’j) with ω = ω I and C = ∞, that is, with no constraint on the upper bound of possible payoffs. Furthermore, ω I > xL ensure that sSDC and sLD are well-defined. We illustrate the proof using the case of

fj f−j

increasing. Each security s ∈ SX with

s(x) ≤ C 0 and E[s(x)] = ω 0 = E[sSDC (x; D(ω 0 )] crosses sSDC once from below because the standard debt contract initially is at the limited liability constraint, and then flat.22

The given security and sSDC share the same unconditional expected value, i.e. the unconditional payoff difference below the crossing point (where s(x) lies below sSDC ) and above the crossing point (where s(x) lies above) even out. Due to the increasing likelihood ratio (LR), this evening out does not obtain when considering the expected value under fj , Efj . To the contrary, under the distribution characterized by fj more ‘weight’ is put on the realizations above the crossing point – as such, the standard debt contract minimizes the expected value under fj . A similar notion based on this type of single-crossing obtains also for (ii) and (iii), i.e. for sLD solving (P’j) and (P”j) if the LR is decreasing and sT T solving (P”j) with the LR increasing. Intuitively, the behavior of the likelihood ratio determines where the bulk of the security’s payoff should ideally lie. If

fj f−j

is increasing, the security should pay as little

as possible for high values of the underlying collateral to minimize the value in state 22

Formally, there exists a point ξ such that s(x) ≤ sSDC (x) for x ≤ ξ and s(x) ≥ sSDC (x) for x > ξ.

19

j as the high states are relatively more likely under fj than f−j compared to the low ones. Dual monotonicity and limited liability, however, require the slope of the security to be at most 1, and to not exceed the value of the underlying collateral. As such, if the likelihood ratio is increasing, the optimal security is standard debt in the unconstrained problem (Pj) (Corollary 2.(i)). The security pays according to the limited liability line up to its face value – dual monotonicity then requires a constant payoff thereafter. In the constrained problem (P”j), a similar logic applies. However, as the problem requires the security to match an upper bound C 00 at xH , the solution features a second tranche so that the security just reaches that payoff (Proposition 3.(iii)) – by doing so, it minimizes the ‘exposure’ to the high states which are relatively likely under fj . The rationale for whenever the likelihood ratio

fj f−j

is decreasing is similar, with

the optimal security in this case putting as little weight as possible on low realizations of the underlying collateral. This results in leveraged equity solving the unconstrained problem ((Corollary 2.(ii)); the leveraged debt contract sLD (x; L(ω I , ∞), ∞)

corresponds to leveraged equity), and leveraged debt solving the constrained problems (Corollary 2.(ii)).

Proposition 3 and Corollary 2 are important for various reasons: First, the Corollary provides solutions to the security design problem at t = 1 under relatively strict conditions. In particular, if the monotone likelihood ratio property holds, which is the case e.g. in Dang et al. (2011), Standard Debt minimizes the value of a security under ’good’ information, while leveraged equity maximizes it. However, the core contribution is the characterization of solutions to the constrained problems (Pj’) and (Pj”). We will show that the solution to an arbitrary security design problem (Pj) irrespective of any assumptions on the densities can be obtained by breaking it down into problems on intervals where the likelihood ratio is decreasing (increasing) and transforming these problems into global ones such that the Proposition applies. The constrained problems (Pj’) and (Pj”) need to be considered as the piecewise construction of the optimal security requires that dual monotonicity is never violated.23 Before we proceed as discussed above, we first provide sufficient conditions such that a standard debt contract is not a solution to (Pj). 23

The piecewise construction itself is reminiscent of the ironing procedure as in e.g. Guesnerie and Laffont (1984).

20

Proposition 4 Denote D(ω I ) the face value of the standard debt contract sSDC (x; D(ω I )) with Ef [sSDC (x; D(ω I ))] = ω I . Let Gj (x) ≡

1−F−j (x) 1−Fj (x)

∃ξ ∈ (D(ω I ), xH ) : Gj (ξ) >

ˆ j (x) ≡ and G inf

x∈(xL ,D(ω I ))

ˆ j (x) G

F−j (D(ω I ))−F−j (x) . Fj (D(ω I ))−Fj (x)

If

(12)

then sSDC (x; D(ω I )) is not a solution to (Pj). Proof. See Appendix A.4. If the condition is satisfied, a shift in payoffs away from low realizations of the payoff of the underlying collateral (where the limited liability constraint binds, i.e. x < D(ω I )) to high realizations (i.e. x > D(ω I )) is beneficial for minimizing the value of the security in state j as the relative likelihood of these realizations occurring is ˆ j ) measures how much of the value change due to lower under fj . The function Gj (G a marginal increase (marginal decrease) in payoff at x is attributed to the distribution f−j – as such, if (12) is satisfied, standard debt cannot be optimal to minimize the value under fj . Increasing the payoff marginally in the flat part would result in more value being attributed to f−j and thus less to fj compared to a corresponding decrease in the part where limited liability is binding. Proposition 4 provides an easy-to-evaluate conditions for the non-optimality of standard debt. Some security design problems (e.g. whenever distributions can be ordered by First Order Stochastic Dominance and market endowment is independent of public news, see Example 2 in Section 4) can be reduced to finding the solution to a particular (Pj) and, if only one such (Pj) needs to be considered, condition (12) can be used to rule out the standard debt contract as a solution to the security design problem. This is particularly relevant whenever the likelihood ratio does not exhibit monotone behavior; if

fj f−j

is increasing, (12) is never satisfied and standard debt solves (Pj)

as in Dang et al. (2011, 2015a). Ruling out standard debt as a solution is crucial as non-optimality of standard debt with respect to interim public information implies a misalignment in the security designer’s incentives – even if standard debt minimizes incentives for private information acquisition and thus protects liquidity, this needs to be traded off with utility losses due to interim public information and imperfect realization of gains from trade. Before stating the main Proposition of the article, we formally define the class of 21

tranched debt contracts. We then show that solutions to (Pj) belong to this class. Definition 4 A tranched debt contract is characterized by a strictly increasing sequence N {xi }N i=1 ∈ (xL , xH ) of points and a strictly increasing sequence {Di }i=0 ∈ R+ of face

values where D0 = 0 and ∀i ≥ 1 : xi − Di−1 > Di−1 . The contract sT D is then

characterized by the following payoff structure:

sT D (x) =

    

for x ≤ x1

D0

min{x − Di−1 , Di }     min{x − D N −1 , DN }

if x ∈ (xi , xi+1 ], i < N if x ∈ (xN , xH )

where (xN , DN ) are cutoff point and face value of the most junior tranche if N is finite, and xN ≡ supj xj , DN ≡ supj
D2 Junior (Leveraged) Debt Tranche sLD2 D1

Senior (Leveraged) Debt Tranche sLD1

xL x1

x2

xH

x

Figure 3: Tranched Debt Contract In terms of the securities’ structure, tranched debt contracts are composed of multiple debt tranches which differ in their seniority. Junior tranches are necessarily leveraged. The condition xi − Di−1 > Di−1 ensures this and implicitly restricts the contract structures to be the ones composed of the minimal number of tranches amongst a continuum of securities with the same payoff structure.25 In the following, we show that there always exists a solution to a given (Pj) which is a tranched debt contract. Moreover, it is sufficient that underlying densities f1 , f2 are never proportional to ensure that all solutions exhibit this structure. 24

The allowance of an infinite number of tranches is for completeness only and would matter in equilibrium only if the densities f1 and f2 are such that the likelihood ratio ff12 oscillates – that is switches between being locally increasing and decreasing – infinitely often. 25 To see this, note that adding a tranche characterized by (xi0 , Di0 ) with Di0 = Di and xi0 ∈ (xi , xi+1 ), the payoffs would not be affected.

22

Proposition 5 Consider problem (Pj), j ∈ {1, 2}. There always exists a security

D D s∗,T which solves (Pj.d), where s∗,T is a tranched debt contract. If (LNP) holds, all j j

solutions s∗j to (Pj) are tranched debt contracts. Proof. See Appendix A.5. The intuition for the result is the following and also outlines the proof. (xL , xH ) can be partitioned into intervals where the likelihood ratio is decreasing or increasing, respectively. On each such interval, for a given security structure on the remaining support and a given mass of unconditional payoffs, the local security design problem can be transformed into a global problem such that Proposition 3 applies. Crucial for this is that the transformed problem is such that it adheres to dual monotonicity on the global scale, which gives the constraints in (P’j),(P”j). A locally optimal security exists such that it is locally consistent with tranched debt – it is either a contract composed of two tranches (likelihood ratio increasing), or a leveraged debt contract (likelihood ratio decreasing). Moreover, if (LNP) holds, the likelihood ratio is always strictly increasing or decreasing, implying that the locally optimal security is necessarily of the above structure. Overall, these local solutions can be pieced together to yield a solution to (Pj) which is a tranched debt contract. To illustrate this, consider a candidate security s which is inconsistent with tranched debt on some interval (ξ1 , ξ2 ) where the likelihood ratio

fj f−j

is increasing. Denote

C = s(ξ2 ) − s(ξ1 ). We can then rescale the problem and obtain densities gj , g−j

which preserve the increasing likelihood ratio, and where Gj (ξ1 ) = 0 = G−j (ξ1 ) and Gj (ξ2 ) = 1 = G−j (ξ2 ), i.e. where the local problem on (ξ1 , ξ2 ) now has the characteristics of a global problem (because the entire probability mass characterized by Gj , G−j is on (ξ1 , ξ2 )).26 From Proposition 3, we then know that there exists a security composed of two tranches which minimizes the value under Gj amongst the securities matching sloc (ξ2 ) = C. Because the process is reversible, we know that by replacing the candidate security s with our tranched debt contract on (ξ1 , ξ2 ) we preserve dual monotonicity, while weakly decreasing the value of the modified security under distribution Fj – the local behavior of the likelihood ratio is thus sufficient to ensure that there exists an optimal security locally consistent with tranched debt, irrespective of the 26

This also requires an adjustment of λ by a constant factor to ensure reversibility of the process, for details please refer to the proof.

23

behavior elsewhere. Because this obtains for any interval where the likelihood ratio is increasing, and can similarly be constructed using a leveraged debt contract whenever it is locally decreasing, there always exists a security globally consistent with tranched debt. Moreover, the entire argument holds strictly provided that the likelihood ratio is strictly increasing or decreasing, respectively; if (LNP) holds, a candidate security s being inconsistent with tranched debt would thus lead to a contradiction in terms of optimality of s. Proposition 5 thus yields a structural characterization of the optimal security issued at t = 1: If (LNP) is satisfied, any solution to (Pj) is a tranched debt contract. As the solution to the security designer’s problem can be constructed as a convex combination of solutions to (P1) and (P2), this greatly reduces the complexity of the security design problem. If there are finitely many changes in the sign of the slope of the likelihood ratio (that is, there is a partition of X into finitely many intervals such that the likelihood ratio is increasing or decreasing on each interval), the security design problem is reduced to finding a combination of face values and cutoff points such that (i) the security exhausts ω I at t = 1 and (ii) minimizes the value after the public signal is φ = φj for each j. While in principle standard debt could be such a solution even if the likelihood ratio changes sign more than once, this can be ruled out using Equation (12) – in that case, the security solving (Pj) necessarily exhibits leverage. The following Section provides examples which illustrate the result, in particular by showing how standard debt fails to be optimal, and what can nonetheless be said about the structure of an optimal security in light of Proposition 5.

4. Examples 4.1. Densities satisfy MLRP Suppose that densities satisfy the monotone likelihood ratio property (MLRP), that is, without loss of generality, let

f1 f2

be decreasing in x on X. Note that this implies

several things: First, if the MLRP is satisfied, so is first order stochastic dominance (FOSD), that is, F1 (x) ≥ F2 (x) , ∀x ∈ X and any nondecreasing (and hence also any

dual monotone) security s satisfies Ef1 [s(x)] ≤ Ef [s(x)] ≤ Ef2 [s(x)]. Second, this 24

implies that the public signal can be denoted as either good news or bad news. Proposition 3 establishes that if the likelihood ratio of the underlying densities

f1 f2

is weakly decreasing, the standard debt contract sSDC (x; D(ω I )) solves (P2) as

f2 f1

is weakly increasing. Furthermore, the leveraged equity contract sLD (x; L0 (ω I , ∞), ∞) solves (P1). If

f1 f2

is strictly decreasing, (LNP) holds and these are the (up to point-wise

deviations) unique solutions. We discuss two different cases regarding the correlation between public news and market endowment. 4.1.1. Endowment unaffected by public news Suppose that ω1M = ω2M = ω M . In this case, the standard debt contract is the optimal security issued at t = 1, as established by Dang et al. (2011). The reasoning is straightforward: expected utility of the investor (6) is maximized whenever losses due to imperfect realization of gains from trade at t = 2 are minimized. However, this imperfect realization is more relevant after good news (φ = φ2 ) as any nondecreasing security has larger value than after bad news (φ = φ1 ). Relative to ω M , the goal is thus to minimize value after good news to maximize overall gains from trade. This corresponds to a setup where robustness of the security’s value is desirable, and standard debt exhibits maximal robustness. 4.1.2. Endowment correlated with public news Suppose instead that ω1M 6= ω2M . In this case, standard debt is no longer necessarily

optimal. To see this, take the extreme case where the market is illiquid after bad news, that is, ω1M = 0. In this case, it is clear that the optimal security in fact minimizes the value after bad news: This is achieved by the solution to (P1), i.e. by leveraged

debt. However, even if ω1M > 0, standard debt may fail to be optimal. We illustrate this with a parametric example. Example 1 Let X = (0, 2), ω I = 1.5, f1 (x) = 2 − x, f2 (x) = x. Furthermore, let λ = 0.5 which gives f (x) = 1.

In this case, it is straightforward to calculate D(ω I ), L(ω, ∞) and we obtain the solutions to (Pj) and their values given φ = φi . This is depicted in Table 1. 25

Table 1: Optimal Securities and expected values in Example 1 Security

Ef [·]

Ef1 [·]

Ef2 [·]

sSDC (x; 1) = min{x, 1}

1.5

1.167

1.833

1.5

0.866

2.134

sLD (x; 2 −

√

3, ∞) = max{0, 2 −

√

3}

As expected given that FOSD holds, standard debt is more robust to the interim information than leveraged equity. Nonetheless, which security is optimally issued in equilibrium will depend on the correlation of ωjM and the public information. Consider for example the case of ω2M = 2. This implies that the standard debt contract sSDC does not exhaust the market endowment after good news, while the leveraged debt contract sLD can not be fully sold if good news materialize. Which security performs better from an ex-ante perspective depends on ω1M . If the market is extremely tight after bad news, ω1M < 0.866, irrespective of the security issued at t = 1, the endowment constraint binds after bad news. Hence, the focus is on realization of gains from trade after good news – the leveraged equity-like contract sLDL yields more gains from trade after good interim news and is hence better from an ex-ante perspective than standard debt sSDC . If instead the market is very liquid even after bad news, ω1M > 1.167, the opposite rationale applies. As any security will be fully sold after bad news, the focus is on minimizing losses due to imperfect realization of gains from trade after good news – as standard debt performs better than the leveraged equity contract, it is better to issue sSDC at t = 1.27 The purpose of this example is to illustrate that robustness in terms of posterior values need no longer be desirable if the asset-specific information also has an impact on the market conditions (or vice versa). Nonetheless the individual securities which solve (P1) and (P2) are still simple and not composed of multiple tranches. They are a standard debt and leveraged equity contract, respectively.

27

Note that the above argument simply compares the two extreme contracts – as previously discussed, in most situations there is multiplicity in the optimal security design which maximizes expected utility of the investor at t = 1.

26

4.2. Densities do not satisfy MLRP Another reason to depart from Standard Debt Contracts as optimally issued securities is that the densities themselves do not satisfy the MLRP – in this case, even if market endowment is independent of the interim public information, i.e. ω1M = ω2M = ω M , incentives to issue tranched debt may prevail. We outline such a case in Example 2 below. Example 2 Let X = (0, 2π), ω I = 3, f1 (x) = let λ = 0.5 which gives f (x) =

sin(x)+2 , f2 (x) 4π

=

sin(x−π)+2 . 4π

Furthermore,

1 . 2π

Example 2 satisfies First order Stochastic Dominance (FOSD) as F1 (x) > F2 (x), ∀x ∈

(0, 2π), but not the MLRP. Moreover, (LNP) is satisfied. Thus, while news can still be strictly classified into being either ’good’ (φ = φ2 ) or ’bad’ (φ = φ1 ), we can no longer conclude that standard debt minimizes Ef2 . The densities, cdfs and likelihood ratio are plotted in Figure 4.

F (x) F1 (x) f2 (x) f (x)

f1 (x) f1 (x) f2 (x)

F2 (x) x

(a)

x

x

(b)

ξ1

ξ2

(c)

Figure 4: Densities, CDFs and Likelihood Ratio (Example 2)

Suppose that ω M is independent of public news and that ω M > ω I . In this case, it is straightforward that the optimal security issued at t = 1 minimizes the expected value after good information, that is, solves (P2). However, as the MLRP does not hold, this solution is not necessarily the standard debt contract sSDC (x; D(ω I )). We know that the local non-proportionality condition (LNP) is satisfied and hence that the solution to (P2) is consistent with tranched debt. Given that the likelihood ratio is first strictly increasing, then strictly decreasing, and finally strictly increasing again, we know that the optimal security is a contract composed of two debt tranches sT T on (xL , ξ1 ), a leveraged debt contract sLD on (ξ1 , ξ2 ), and a standard debt tranche on 27

(ξ2 , xH ). Structurally, we know that the optimal security thus looks as depicted in Figure 5, Panel (a).

s∗ (x)

sT T

sLD

s∗ (x)

sSDC

xL

G2 (x)

xH D2

D2 C1

ˆ 2 (x) G

D1

ˆ 2 (x) inf x∈(xL ,D(ωI )) G x

x ξ1 ξ1 − C1 + D1

ξ2

I

D(ω )

L1

(b)

(a)

ˆ 2 -functions (Example 2) Figure 5: Solution s∗ to (P2.d) & G2 /G

This in principle does not rule out a standard debt contract being optimal. If C1 = ξ1 = L1 , this corresponds to a single debt tranche with face value D2 . However, we can use the condition for the non-optimality of standard debt captured in Proposition 4. For ω I = 3, we obtain D(ω I ) = 4.949. Figure 5, Panel (b) depicts G2 (x) on [0, 4.949) = ˆ 2 (x) on (4.949, 2π] = (D(ω I ), xH ] and illustrates that condition (12) [xL , D(ω I )) and G is satisfied.28 This is because ω I is sufficiently high such that the standard debt contract which exhausts ω I requires that the limited liability constraint is binding even where realizations of X are relatively likely to occur after good information (f = f2 ). By shifting payoffs from there to the very right tail of the distribution, where realizations are almost equally likely to come after good and bad news, the value of the security after good information can be reduced.

28

In principle, this reduces the problem to (numerically) solving for the optimal security s∗ which needs to satisfy  min{x, D1 } if x ≤ (ξ1 − C1 ) + D1    x − (ξ1 − C1 ) if x ∈ ((ξ1 − C1 ) + D1 , ξ1 ] ∗ I ∗ Ef [s (x)] = ω and s (x) = , C1 if x ∈ (ξ1 , L1 ]    min{x − (L1 − C1 ), D2 } if x > L1 π 3 where ξ1 = , ξ2 = π , D1 ≤ C1 ≤ ξ1 , L1 ∈ [ξ1 , ξ2 ] , C1 ≤ D2 . 2 2

28

5. Debt Tranches and Mezzanine Retention We have shown above that the solution to (Pj) corresponds to a tranched debt contract (under dual monotonicity). This section discusses how such contracts can be implemented in practice via retention of mezzanine debt tranches. Figure 6 illustrates the mezzanine retention for a stylized example where the solution to (Pj) corresponds to a tranched debt contract composed of a senior standard debt tranche and a junior leveraged debt tranche.

payoff

payoff D2 + (x2 − D1 ) Junior (Leveraged)

sT D (x)

D2 Junior (Leveraged) Debt Tranche sLD2

D1

Debt Tranche sˆLD2

x2

Retained Mezzanine Tranche

D1

Senior Debt Tranche sLD1 xL

x2

Senior Debt Tranche sLD1 xH

x

xL

x2

xH

x

Figure 6: Implementation of Tranched Debt Contracts

The key feature to notice is that any type of tranched debt-contract, which is composed of imperfect standard debt tranches, is payoff equivalent to tranching the payoff from the underlying pool of returns X into perfect debt tranches, that is, standard debt tranches which fully exceed the limited liability constraint irrespective of the return of the underlying collateral. By retaining a mezzanine tranche and selling off the senior and junior tranches, the original sT D is sold while x − sT D is retained by the originator.

Formally, any junior tranche sLDi characterized by its face value Di and starting

point xi > Di−1 can be obtained by incorporating the standard debt mezzanine tranche min{x, xi } − min{x, Di−1 }, retaining it, and selling the standard debt junior tranche sˆLD2 (x) = min{x, Di + (xi − Di−1 )} − min{x, xi }. Proposition 6 Any tranched debt contract which is not standard debt can be implemented via a security structure involving retention of mezzanine debt. Proof. Follows from the construction outlined in the previous paragraph. 29

The main takeaway is that in the presence of public interim information and limited market depth, mezzanine retention which implies selling off both senior and (leveraged) junior claims may be optimal – the pecking order fails to hold. Note that for this we only require that the optimal security is not a standard debt contract. Any tranched debt contract which is not standard debt involves leverage and, in terms of implementation, retention of senior tranches while selling off more junior ones. As such, any of the two deviations from the literature we consider can be sufficient for mezzanine retention.

6. Conclusion This article analyzes a security design framework where public information arrives between trading periods and limited market endowment may prevent full realization of potential gains from trade. While abstracting from private information acquisition, the public information framework is generalized compared to the existing literature to allow for arbitrary distributions and arbitrary interdependence between market endowment and public information. We show that the optimal security can be derived as a convex combination of contracts which share many debt-like features, but exhibit endogenous tranching. The theoretical findings are consistent with the issuance of multiple tranches of different seniorities based on a single pool of collateral, which is frequently observed in financial markets. Crucially, this implies a misalignment in the security designer’s incentives between private and public information concerns, which is novel to the literature. A prevailing theme of the literature is the optimality of standard debt contracts with respect to minimizing private information acquisition incentives. However, we show that under a generalized public information structure standard debt is not necessarily most robust to public information. Moreover, robustness itself may cease to be a desirable feature depending on the correlation between the cash in the market and the interim public information. While this setup has abstracted from private information, a model which tractably combines both private and generalized public information concerns in a single setting and hence allows for an explicit assessment of how the trade-off plays out is an interesting avenue for future research. It may shed additional light on the securitization

30

practices observed both in the run-up and aftermath of the recent financial crisis and inform policymakers in their regulatory activities aimed at preventing future crises. We furthermore show that the optimal security structure under public information can be implemented through mezzanine retention. While the present framework is primarily motivated by security design concerns in the context of asset-backed securities, it can also be applied to security issuance in a primary (capital-raising) market when a secondary market is present and susceptible to interim information. In this context, our contribution can be seen as a public information rationale for departure from the pecking order, complementing research motivating this departure by private information considerations.

References Aghion, P. and P. Bolton (1992): “An incomplete contracts approach to financial contracting,” Review of Economic Studies, 59, 473–494. Allen, F. and D. Gale (1994): “Limited market participation and volatility of asset prices,” American Economics Review, 933–955. Biais, B. and T. Mariotti (2005): “Strategic liquidity supply and security design,” Review of Economics Studies, 72, 615–649. Boot, A. and A. Thakor (1993): “Security design,” The Journal of Finance, 48, 1349–1378. Chakraborty, A. and B. Yilmaz (2011): “Adverse selection and convertible bonds,” The Review of Economic Studies, 78, 148–175. Chari, V. and R. Jagannathan (1988): “Banking panics, information, and rational expectations equilibrium,” Journal of Finance, 749–761. Dang, T. V., G. Gorton, and B. Holmström (2011): “Financial Crises and the Optimality of Debt for Liquidity Provision,” working paper. Dang, T. V., G. Gorton, and B. Holmström (2015a): “Ignorance, debt and

31

financial crises,” Yale University and Massachusetts Institute of Technology, working paper. Dang, T. V., G. Gorton, and B. Holmström (2015b): “The Information Sensitivity of a Security,” working paper. Dang, T. V., G. Gorton, B. Holmström, and G. Ordonez (2017): “Banks as Secret Keepers,” American Economic Review, 107, 1005–1029. DeMarzo, P. and D. Duffie (1999): “A liquidity-based model of security design,” Econometrica, 67, 65–99. DeMarzo, P., I. Kremer, and A. Skrzypacz (2005): “Bidding with Securities: Auctions and Security Design,” American Economic Review, 95, 936–959. Diamond, D. (1984): “Financial intermediation and delegated monitoring,” Review of Economics Studies, 51, 393. Diamond, D. and P. Dybvig (1983): “Bank runs, deposit insurance, and liquidity,” Journal of Political Economy, 91, 401–419. Farhi, E. and J. Tirole (2015): “Liquid bundles,” Journal of Economic Theory, 158, 634–655. Fulghieri, P., D. Garcia, and D. Hackbarth (2015): “Asymmetric information and the pecking (dis) order,” UNC Kenan-Flagler Research Paper. Fulghieri, P. and D. Lukin (2001): “Information production, dilution costs, and optimal security design,” Journal of Financial Economics, 61, 3–42. Gale, D. and M. Hellwig (1985): “Incentive-compatible debt contracts: The oneperiod problem,” Review of Economic Studies, 52, 647. Gorton, G. and A. Metrick (2012): “Securitization,” Tech. rep., National Bureau of Economic Research. Gorton, G. and G. Pennacchi (1990): “Financial intermediaries and liquidity creation,” Journal of Finance, 45, 49–71. 32

Guesnerie, R. and J.-J. Laffont (1984): “A complete solution to a class of principal-agent problems with an application to the control of a self-managed firm,” Journal of public Economics, 25, 329–369. Innes, R. (1990): “Limited liability and incentive contracting with ex-ante action choices,” Journal of Economic Theory, 52, 45–67. Jacklin, C. and S. Bhattacharya (1988): “Distinguishing panics and informationbased bank runs: Welfare and policy implications,” Journal of Political Economy, 568–592. Kaplan, T. (2006): “Why banks should keep secrets,” Economic Theory, 27, 341–357. Leary, M. and M. Roberts (2010): “The pecking order, debt capacity, and information asymmetry,” Journal of Financial Economics, 95, 332–355. Modigliani, F. and M. H. Miller (1958): “The cost of capital, corporation finance and the theory of investment,” American economic review, 48, 261–297. Monnet, C. and E. Quintin (2017): “Rational Opacity In Private Equity Markets,” Review of Financial Studies. Myers, S. and N. Majluf (1984): “Corporate financing and investment decisions when firms have information that investors do not have,” Journal of Financial Economics, 13, 187–221. Nachman, D. and T. Noe (1994): “Optimal design of securities under asymmetric information,” Review of Financial Studies, 7, 1–44. Pagano, M. and P. Volpin (2012): “Securitization, transparency, and liquidity,” Review of Financial Studies, 25, 2417–2453. Stenzel, A. and W. Wagner (2015): “Opacity and Liquidity,” CEPR Working Paper. Townsend, R. (1979): “Optimal contracts and competitive markets with costly state verification,” Journal of Economic Theory, 21, 265–93.

33

Yang, M. (2015): “Optimality of debt under flexible information acquisition,” SSRN working paper.

A. Proof of Lemmas and Propositions A.1. Proof of Existence Proposition A.1 Consider I = (a, b) ⊆ X and ω with 0 < ω < ∃s∗ ∈ SI,ω : Efi [s∗ (x)] ≥ Efi [s(x)] for all s ∈ SI,ω .

R

D xf (x)dx

< ∞. Then

Proof: Existence is established in the following way. We wish to maximize Efi [s(x)] over the set SI,ω ≡ {s ∈ SI such that Ef [s(x)] = ω}. The objective Efi [s(x)] corresponds to a mapping h : SI,ω → R where h(s) = Efi [s(x)]. By showing that SI,ω is closed and bounded, and that h is continuous, existence of a maximum of h on the set SI,ω follows. To establish this, first note that X = (xL , xH ) with xH < ∞ ensures that all permissible securities are bounded functions, as limited liability ensures s(x) ≤ x ≤ xH for all x. We can hence work in the space of bounded functions on I with the sup-norm ||f || = supx∈S |f (x)| and induced distance d(f1 , f2 ) = supx∈I |f1 (x) − f2 (x)|. First, consider boundedness of SI,ω . For any s ∈ SI,ω we have 0 ≤ s(x) ≤ x due to limited liability for all x, and hence that d(s, s0 ) ≤ supx∈S |x| ≤ xH , where s0 (x) = 0 , ∀x ∈ I. As such, SI,ω is contained in a ball of size r = xH + ,  > 0 about s0 and hence bounded. i→∞ Closedness follows from the fact that for any converging sequence si −−−→ s where si ∈ SI,ω for all i, the limit s is contained in SI,ω . We will show that s ∈ SI,ω by establishing that it satisfies dual monotonicity, non-negativity and has an expected value Ef [s(x)] = ω. First, Ef [s(x)] = ω is established. Suppose Ef [s(x)] > ω (the contradiction for Ef [s(x)] < ω i→∞ works analogously). Let δ ≡ Ef [s(x)] − ω. Then si −−−→ s implies for any  > 0,  < δ: ∃N : ∀i ≥ N : si (x) ≥ s(x) −  for all x ∈ I.

(A.1)

Hence, for all i ≥ N it follows that Ef [si (x)] ≥ Ef [s(x)] −  > Ef [s(x)] − δ = ω.

(A.2)

This is a contradiction to si ∈ SI,ω . Next, suppose that s does not satisfy limited liability, i.e. that s(ξ) > ξ for some ξ ∈ I. By i→∞ si −−−→ s, this implies that ∃N : ∀i ≥ N : si (ξ) > ξ.

(A.3)

This contradicts si ∈ SI,ω for i ≥ N . In the same manner, non-negativity of s is established.

34

Finally, suppose that s violates nondecreasingness, i.e. that ∃x1 , x2 ∈ I such that x1 < x2 ∧ s(x1 ) > s(x2 ).

(A.4) i→∞

However, si is nondecreasing for all i. Hence, si (x1 ) ≤ si (x2 ) for all i. Since si −−−→ s, a contradiction again follows. s(x1 ) > s(x2 ) requires si (x1 ) > si (x2 ) for all i ≥ N for some N . Similarly, we get nondecreasingness of the residual x − s(x) and thus dual monotonicity. Finally, continuity of h is established. Recall h(s) = Efi [s(x)] =

Z

xH

s(x)dF1 (x).

(A.5)

xL

i→∞

Now take si −−−→ s. It follows that Z xH lim si (x)dF1 (x) i→∞ x Z xH L s(x)dF1 (x) =

lim h(si ) =

i→∞

xL

= h(s),

(A.6)

which yields continuity of h, and where the second step follows from the Dominated ConRx vergence Theorem (sM (x) = x is a point-wise upper bound for any si , and xLH xf (x)dx is finite). We have thus established that SI,ω is closed and bounded and that h is continuous. Hence, h(s) = Efj [s(x)] attains a maximum on SI,ω and a solution to (Pj) exists. 

A.2. Proof of Proposition 2 Recall the definition of S ∗ , i.e.   S ∗ ≡ arg max{ max λ min Ef1 [s(x)], ω1M + (1 − λ) min Ef2 [s(x)], ω2M . s∈SX,ωI

We proceed through the cases of Proposition 2 one by one. Statement (I) is straightforward as Ef [s∗1 (x)] = Ef [s∗2 (x)] = ω I ⇒ ∀η ∈ [0, 1] : Ef [ηs∗1 (x) + (1 − η)s∗2 (x)] = ω I and both nondecreasingness and dual monotonicity respectively are preserved when considering convex combinations of two securities. For (II), we prove the following: (a) If v11 ≤ ω1M ∧ v12 ≤ ω2M then s∗1 ∈ S ∗ .

(b) If v21 ≤ ω1M ∧ v22 ≤ ω2M then s∗2 ∈ S ∗ . (c) If v21 > ω1M ∧ v12 > ω2M then

(i) If v11 ≥ ω1M then s∗1 ∈ S ∗ .

(ii) If v22 ≥ ω2M then s∗2 ∈ S ∗ .

35

(iii) If v22 < ω2M ∧ v11 < ω1M then v21 −ω1M ∗ s v21 −v11 1

+

ω1M −v11 ∗ s v21 −v11 2

∈ S ∗ and

v12 −ω2M ∗ s v12 −v22 2

+

ω2M −v22 ∗ s v12 −v22 1

∈ S∗.

(d) If v11 > ω1M ∧ v12 ≤ ω2M then s∗1 ∈ S ∗ .

(e) If v21 ≤ ω1M ∧ v22 > ω2M then s∗2 ∈ S ∗ . For (a), recall that v11 = Ef1 [s∗1 (x)] ≤ ω1M ∧ v12 = Ef2 [s∗1 (x)] ≤ ω2M

(A.7)

and hence λ min{Ef1 [s(x)], ω1M } + (1 − λ) min{Ef2 [s(x)], ω2M }

∀s ∈ SX,ωI : ≤

λEf1 [s(x)] + (1 − λ)Ef2 [s(x)]

=

ω =

(A.7)

=

I

λEf1 [s∗1 (x)]

+ (1 −

(A.8)

λ)Ef2 [s∗1 (x)]

λ min{Ef1 [s∗1 (x)], ω1M } + (1 − λ) min{Ef2 [s∗1 (x)], ω2M }

s∗1 ∈ S ∗ .

⇒

The proof for (b) is identical for s∗2 and hence omitted. Consider thus (c). For (c).(i), recall that v12 > ω2M ∧ v11 ≥ ω1M (A.9) and hence λ min{Ef1 [s(x)], ω1M } + (1 − λ) min{Ef2 [s(x)], ω2M }

∀s ∈ SX,ωI : ≤

(A.9)

=

⇒

λω1M + (1 − λ)ω2M

λ min{Ef1 [s∗1 (x)], ω1M }

s∗1 ∈ S ∗ .

+ (1 −

(A.10)

λ) min{Ef2 [s∗1 (x)], ω2M }

Again, the proof for (c).(ii) is identical for s∗2 and hence omitted. Finally, consider (c).(iii). Consider an arbitrary s ∈ SX,ωI and consider sˆ ≡

v21 − ω1M ∗ ω1M − v11 ∗ s + 1 s . v21 − v11 1 v2 − v11 2

Note that Ef1 [ˆ s(x)] = = =

v21 − ω1M ω M − v11 Ef1 [s∗1 ] + 11 Ef1 [s∗2 ] 1 1 v2 − v1 v2 − v11

v21 − ω1M 1 ω1M − v11 1 v + 1 v v21 − v11 1 v2 − v11 2

v21 · v11 − ω1M · v11 + ω1M · v21 − v11 · v21 ω M (v 1 − v 1 ) = 1 1 2 1 1 = ω1M . 1 1 v2 − v1 v2 − v1

Denote v 1 ≡ Ef1 [s(x)] and v 2 ≡ Ef2 [s(x)]. Consider the following cases: • Case 1: Ef2 [ˆ s(x)] > ω2M

36

Then λ min{Ef1 [ˆ s(x)], ω1M }+(1−λ) min{Ef2 [ˆ s(x)], ω2M } = λω1M +(1−λ)ω2M and sˆ ∈ S ∗ follows by inequality (A.10). • Case 1: Ef2 [ˆ s(x)] ≤ ω2M

Then λ min{Ef1 [ˆ s(x)], ω1M }+(1−λ) min{Ef2 [ˆ s(x)], ω2M } = λEf1 [ˆ s(x)]+(1−λ)Ef2 [ˆ s(x)] = I ∗ ω and sˆ ∈ S follows by inequality (A.8).

In the same manner it can be established that s˜(x) ≡ S∗.

v12 −ω2M ∗ ω2M −v22 ∗ s + v2 −v2 s1 v12 −v22 2 1 2

satisfies Ef2 [˜ s(x)] =

and hence gives s˜(x) ∈ Finally, consider (d). Note that by construction of s∗1 , we have ∀s ∈ SX,ωI

ω2M

Ef1 [s(x)] ≥ Ef1 [s∗1 (x)] > ω1M and Ef2 [s(x)] ≤ Ef2 [s∗1 (x)] ≤ ω2M .

(A.11)

Thus, ∀s ∈ SX,ωI :

(A.11)

=

(A.11)

λ min{Ef1 [s(x)], ω1M } + (1 − λ) min{Ef2 [s(x)], ω2M } λω1M + (1 − λ)Ef2 [s(x)]

≤

λω1M + (1 − λ)Ef2 [s∗1 (x)]

=

λ min{Ef1 [s∗1 (x)], ω1M } + (1 − λ) min{Ef2 [s∗1 (x)], ω2M }

(A.11)

s∗1 ∈ S ∗ .

⇒

(A.12)

The proof for (e) and s∗2 is identical and omitted. To complete the proof of Proposition 2, it remains to show that Cases (a)-(e) are exhaustive. To see this, note that 



  ¬ v21 > ω1M ∧ v12 > ω2M  | {z } (c)

⇔

v21 

≤

ω1M

∨

v12

≤ ω2M



  ⇔ v21 ≤ ω1M ∧ v22 > ω2M ∨ v21 ≤ ω1M ∧ v22 ≤ ω2M  ∨ | {z } | {z } 

(e)

(b)



 2 M 1 M 2 M 1 M v1 ≤ ω2 ∧ v1 > ω1 ∨ v1 ≤ ω2 ∧ v1 ≤ ω1  . | {z } | {z } (d)

(A.13)

(a)

Thus, if (c) does not apply, (a), (b), (d) or (e) necessarily does. As a side note, observe that all the steps of the proof work identically for nondecreasing securities, which we will use in Appendix B.

A.3. Proof of Proposition 3 To facilitate the proofs of (i)-(iii), we first establish the following Lemma. Lemma A.1

37

Consider securities s1 , s2 with Ef [s1 (x)] = Ef [s2 (x)], X = (xL , xH ) ⊂ R+ and suppose that ∃ξ ∈ X : ∀x ∈ X : x ≤ ξ ⇒ s1 (x) ≥ s2 (x) ∧ x ≥ ξ ⇒ s1 (x) ≤ s2 (x).

(A.14)

R R (a) If ff21 is increasing in x on X, then X s1 (x)f1 ≤ X s2 (x)f1 . Furthermore, if s1 and s2 R are not equal almost everywhere, i.e. if x∈X:s1 (x)6=s2 (x) 1f (x)dx > 0, and if ff12 is strictly R R increasing, then X s1 (x)f1 < X s2 (x)f1 . R R (b) If ff12 is decreasing in x on X, then X s1 (x)f1 ≥ X s2 (x)f1 . Furthermore, if s1 and s2 R are not equal almost everywhere, i.e. if x∈X:s1 (x)6=s2 (x) 1f (x)dx > 0, and if ff12 is strictly R R decreasing, then X s1 (x)f1 > X s2 (x)f1 .

Proof. We only prove (a) as the proof for (b) follows along identical lines. Denote k ≡ for some x ˆ which satisfies (A.14). As

f1 f2

f1 (ˆ x) f2 (ˆ x)

is increasing in x on X, we have that

f1 (x) ≤ k ⇔ f1 (x) ≤ kf2 (x) f2 (x) f1 (x) ∀x ∈ X, x ≥ x ˆ: ≥ k ⇔ f1 (x) ≥ kf2 (x). f2 (x) ∀x ∈ X, x ≤ x ˆ:

(A.15)

Thus, 1 k Z

1 k

Z

x ˆ

(s1 (x) − s2 (x))f1 (x)dx ≤

xL xH

(s2 (x) − s1 (x))f1 (x)dx ≥

x ˆ

Z

x ˆ

(s1 (x) − s2 (x))f2 (x)dx

xL xH

Z

(s1 (x) − s2 (x))f2 (x)dx,

x ˆ

(A.16)

where integrands are weakly positive by the definition of x ˆ. From Ef [s1 (x)] = Ef [s2 (x)], 0

=

Z

xH

xL

= =

λ λ

Z

(s1 (x) − s2 (x))f (x)dx

xL Z xˆ

xL

+ =

(s1 (x) − s2 (x))f1 (x)dx + (1 − λ) (s1 (x) − s2 (x))f1 (x)dx + λ

(1 − λ) λ

Z

x ˆ

Z

x ˆ

xL

Z

x ˆ

− (A.16)

≥

=

xH

(s1 (x) − s2 (x))f2 (x)dx

xL

xH

(s1 (x) − s2 (x))f1 (x)dx

(s1 (x) − s2 (x))f2 (x)dx + (1 − λ)

(s1 (x) − s2 (x))f1 (x)dx + (1 − λ)

xL xH

Z

Z

xH

Z

x ˆ

xL

Z

Z

x ˆ

xH

(s1 (x) − s2 (x))f2 (x)dx (A.17)

(s1 (x) − s2 (x))f2 (x)dx xH

(s2 (x) − s1 (x))f1 (x)dx − (1 − λ) (s2 (x) − s1 (x))f2 (x)dx (A.18) x ˆ x ˆ  Z xˆ  Z xH 1 λ + (1 − λ) (s1 (x) − s2 (x))f1 (x)dx − (s2 (x) − s1 (x))f1 (x)dx (A.19) k x x ˆ   Z xHL 1 λ + (1 − λ) (s1 (x) − s2 (x))f1 (x)dx. (A.20) k xL

λ 

38

However, this directly implies

⇔

Z

0 ≥ xH

xL

s2 (x)f1 (x)dx ≥

Z

xH

x Z LxH

(s1 (x) − s2 (x))f1 (x)dx s1 (x)f1 (x)dx.

(A.21)

xL

For the strict inequality, note that strictly increasing ff21 implies that (A.15) holds strictly for x x ˆ). Together with s1 not equal to s2 almost everywhere, this gives that at least one of the two inequalities in (A.16) holds strictly, which gives a strict inequality at (A.19) and therefore (A.21). Using Lemma A.1, the proof of the Proposition is straightforward as we only need to show that any arbitrary s (subject to the considered restrictions) intersects sSDC , and sLD respectively, at most once. If this is shown, Lemma A.1 can be applied. For (i), consider the standard debt contract sSDC (x; D(ω)) and any arbitrary s ∈ SX,ω with s(x) ≤ C , ∀x ∈ X. Then ∃ξ1 ∈ X such that ∀x ∈ X : x ≤ ξ1 ⇒ sSDC (x; D(ω)) ≥ s(x) ∧ x ≥ ξ1 ⇒ sSDC (x; D(ω)) ≤ s(x).

(A.22)

To see this, suppose otherwise, i.e. that ∃x1 , x2 ∈ X such that x1 < x2 , s(x1 ) > sSDC (x1 ; D(ω)), but s(x2 ) < sSDC (x2 ; D(ω)).29 Due to s satisfying limited liability, it needs to be the case that s(x1 ) > D(ω). Nondecreasingness then gives s(x2 ) ≥ s(x1 ) > D(ω) ≥ sSDC (x2 ; D(ω)), which is a contradiction. Given (A.22), Lemma A.1 immediately yields (i). For (ii), consider the leveraged debt contract sLD (x; L(ω, C), C) and an arbitrary s ∈ SX,ω with s(x) ≤ C , ∀x ∈ X. Then ∃ξ2 ∈ X such that ∀x ∈ X : x ≤ ξ2 ⇒ sLD (x; L(ω, C), C) ≤ s(x) ∧ x ≥ ξ2 ⇒ sLD (x; L(ω, C), C) ≥ s(x). (A.23) To see this, suppose otherwise, i.e. that ∃x1 , x2 ∈ X such that we have x1 < x2 and s(x1 ) < sLD (x1 ; L(ω, C), C), but s(x2 ) > sLD (x2 ; L(ω, C), C). s(x1 ) < sLD (x1 ; L(ω, C), C) implies sLD (x1 ; L(ω, C), C) > 0 and thus sLD (x1 ; L(ω, C), C) = min{x1 − L(ω, C), C}. If x2 − L(ω, C) ≥ C, we have sLD (x2 ; L(ω, C), C) = C < s(x2 ) which is a contradiction to s(x) ≤ C. If instead x2 − L(ω, C) < C it follows that sLD (x2 ; L(ω, C), C) = x2 − L(ω, C) and sLD (x1 ; L(ω, C), C) = x1 − L(ω, C). But then a contradiction obtains in that s violates dual monotonicity since x1 − s(x1 ) > x1 − sLD (x1 ; L(ω, C), C) = L(ω, C) = x2 − sLD (x2 ; L(ω, C), C) > x2 − s(x2 ). (A.24) Given (A.23), application of Lemma A.1 yields (ii). Finally, consider (iii). The proof is analogous to the previous ones in that we only need to show that any arbitrary s ∈ SX,ω00 with s(xH ) = C intersects sT T only once (or is identical to sT T , that is, we need to establish that ∃ξ5 such that ∀x ∈ X : x ≤ ξ5 ⇒ sT T (x) ≥ s(x) ∧ x ≥ ξ3 ⇒ sT T (x) ≤ s(x). 29

If such x1 , x2 do not exist, ξ1 = inf x∈X {x : s(x) > sSDC (x; D(ω))} trivially satisfies (A.22).

39

(A.25)

Suppose otherwise, that is, ∃x1 , x2 ∈ X such that x1 < x2 and s(x1 ) > sT T (x1 ), but s(x2 ) < sT T (x2 ). We obtain a contradiction by noting that s(x2 ) < sT T (x2 ) implies x2 < xH −C 00 +D. If x2 ≥ xH − C 00 + D, we have that sT T (x2 ) = x − (xH − C 00 ) and hence xH − s(xH ) = xH − C 00 = x − sT T (x2 ) < x − s(x2 ),

(A.26)

which violates dual monotonicity of s. But if x2 < xH −C 00 +D, we know s(x2 ) < min{x2 , D}. Taking into account x1 < x2 and nondecreasingness and limited liability of s immediately yields s(x1 ) < min{x1 , min{x2 , D}} ≤ min{x1 , D} = sT T (x1 ). We have thus obtained a contradiction. Using (A.25) and applying Lemma A.1 yields (iii). Finally, uniqueness (up to point-wise deviations) follows by applying the strict version of Lemma A.1. 

A.4. Proof of Proposition 4 We construct s with Ef [s(x)] = Ef [sSDC (x; D(ω I ))] = ω I and Efj [s(x)] < Efj [sSDC (x; D(ω I ))]. As condition (12) is satisfied, take some such ξ. Due to continuity of f1 , f2 , we know that ∃ˆ x ∈ (xL , D(ω I )) such that in an -neighborhood U around x ˆ we have ˆ j (ˆ ∀x ∈ U (ˆ x) : Gj (ξ) > G x).

(A.27)

Construct s such that

s(x) =

          

for x ∈ (xL , x ˆ) for x ∈ [ˆ x, x ˆ + ] for x ∈ (ˆ x + , ξ) for x ∈ (ξ, xH )

x x ˆ min{x − , D(ω I )} min{x − (ξ − D(ω I )); D(ω I ) + κ}

(A.28)

where κ and  are such that Ef [s(x)] = ω I . Note that s is identical to sSDC everywhere but on (ˆ x, D(ω I ) + ) and (ξ, xH ). Figure 7 illustrates the construction. payoff D(ω I ) + κ

κ

D(ω I )

sSDC (x)

ξ+κ

x ˆ+ x ˆ

s(x)

ξ

D(ω I ) + 

x

Figure 7: Illustration: Inclusion of Junior Debt Tranche By continuity of f1 , f2 , we know we can choose  and κ sufficiently small such that R D(ωI )+

[sSDC (x; D(ω I )) − s(x)]f−j (x)dx x ˆ R D(ωI )+ [sSDC (x; D(ω I )) − s(x)]fj (x)dx ξ

ˆ j (ˆ is arbitrarily close to G x) < Gj (ξ), while

40

(A.29)

R xH

[s(x) − sSDC (x; D(ω I ))]f−j (x)dx Rξ xH SDC (x; D(ω I ))]f (x)dx j x ˆ [s(x) − s

(A.30)

is arbitrarily close to Gj (ξ). Thus, for , κ sufficiently small, (12) ensures that R D(ωI )+

[sSDC (x; D(ω I )) − s(x)]f1 (x)dx

x ˆ R D(ωI )+ [sSDC (x; D(ω I )) ξ

− s(x)]f2 (x)dx

Applying Lemma A.1 immediately gives Z

xH

R xH

ξ < R xH x ˆ

[s(x) − sSDC (x; D(ω I ))]f1 (x)dx [s(x) − sSDC (x; D(ω I ))]f2 (x)dx

sSDC (x; D(ω I ))dFj (x) >

xL

Z

xH

s(x)dFj (x).

. (A.31)

(A.32)

xL

Thus, sSDC (x; D(ω I )) cannot solve (Pj.d).

A.5. Proof of Proposition 5 We first establish (i). Let s∗j be a solution to (Pj) and s∗−j be a solution to (P-j). We here fj is initially weakly increasing, that is30 consider the case where f−j ∃ > 0 : ∀x1 , x2 ∈ (xL , xL + ) : x1 > x2 ⇒

fj (x2 ) fj (x1 ) ≥ . f−j (x1 ) f−j (x2 )

(A.33)

f

j Partition X in the following manner. Given that f−j is initially weakly increasing, denote by f j Φ1 the largest interval (xL , φ¯1 ) such that f−j is weakly increasing on Φ1 . Formally, this yields

Φ1 = (φ1 , φ¯1 ) ≡ (xL , φ¯1 ) ¯ where31

fj (x1 ) fj (x2 ) φ¯1 s.t. ∀x1 , x2 ∈ (xL , φ¯1 ) : x1 > x2 ⇒ ≥ f−j (x1 ) f−j (x2 ) fj (x) fj (φ¯1 ) ∧ ∃δ > 0 s.t. ∀x ∈ (φ¯1 , φ¯1 + δ) : < . f−j (x) f−j (φ¯1 )

Unless Φ1 = (xL , xH ) (in which case largest subsequent interval on which

fj f−j

fj f−j

(A.34)

is globally weakly increasing), denote by Ψ1 the

is weakly decreasing. Formally,

Ψ1 = (ψ1 , ψ¯1 ) ≡ (φ¯1 , ψ¯1 ) ¯ where

30

fj (x1 ) fj (x2 ) ψ¯1 s.t. ∀x1 , x2 ∈ (φ¯1 , ψ¯1 ) : x1 > x2 ⇒ ≤ f−j (x1 ) f−j (x2 )

f

j The case of f−j being initially increasing allows the same partitioning except for initially having an interval with a weakly decreasing likelihood ratio. 31 The second condition can be expressed in this form due to continuity of f1 , f2 .

41

fj (x) fj (ψ¯1 ) ∧ ∃δ > 0 s.t. ∀x ∈ (ψ¯1 , ψ¯1 + δ) : > . f−j (x) f−j (ψ¯1 )

(A.35)

fj Subsequently, unless ψ¯1 = xH , denote by Φ2 the largest subsequent interval on which f−j is ¯ weakly increasing, and, unless φ2 = xH , let Ψ2 be the largest subsequent interval on which fj f−j is weakly decreasing. Proceeding with this iterative process yields a partition of X such that Z [ [ 1f (x)dx = 0. (A.36) X= Ψm ∪ Φn ∪ P where m

P

n

Pick an arbitrary Ψm . Denote B ≡ s∗j (ψm ) and C ≡ s∗j (ψ¯m ). We then know from dual ¯ monotonicity (and thus nondecreasingness) that ∀x ∈ Ψm : B ≤ s∗j (x) ≤ C ⇔ 0 ≤ s∗j (x) − B ≤ C − B. Define gl (x) ≡

thermore,

gj g−j

fl (x) Fl (ψ¯m )−Fl (ψm ) ¯

for l ∈ {j, −j}. Note that this implies

behaves identically to ˜= λ

fj f−j ,

that is,

gj g−j

R

Ψm

(A.37) gl (x)dx = 1. Fur-

is weakly increasing on Ψm . Define

λ(F1 (ψ¯m ) − F1 (ψm )) ¯ λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) ¯ ¯

(A.38)

which implies (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) . ¯ ¯ λ(F1 (ψm ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) ¯ ¯ i R h ∗ Finally, let ω = Ψm sj (x) − B f (x)dx and ˜ = (1 − λ)

ω . λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) ¯ ¯ R Note that by construction ω ≤ Ψm min{x − B, C − B}f (x)dx and hence ω ˜=

(A.39)

(A.40)

ω λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) Z ¯ ¯ λf1 (x) + (1 − λ)f2 (x) ≤ min{x − B, C − B} dx λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) Ψm Z ¯ ¯ ˜ 1 (x) + (1 − λ)g ˜ 2 (x)dx. = min{x − B, C − B}λg (A.41)

ω ˜ =

Ψm

Consider the minimization problem ˜ g [s(x)] + (1 − λ)E ˜ g [s(x)] = ω (P) min Egj [s(x)] s.t. λE ˜ ∧ s(ψ¯m ) = C − B, ∀x ∈ Ψm 1 2 s∈SΨm

(A.42)

42

and it follows from Proposition 3.(iii) that there exists a contract sT T (x) which solves (P). Thus, Egj [sT T (x)] ≤ Egj [s∗j (x) − B] Z Z TT (s∗j (x) − B)gj (x)dx (A.43) s (x)gj (x)dx ≤ ⇔ Ψm Ψm Z Z fj (x) fj (x) dx ≤ (s∗j (x) − B) dx sT T (x) ⇔ ¯ ¯ F ( ψ ) − F ( ψ ) F ( ψ m j m m ) − Fj (ψm ) j j Ψm Ψm Z Z ¯ ¯ (s∗j (x) − B)fj (x)dx. (A.44) sT T (x)fj (x) ≤ ⇒ Ψm

Ψm

Furthermore, ˜ g [sT T (x)] ˜ g [sT T (x)] + (1 − λ)E λE 2 1 λ(F1 (ψ¯m ) − F1 (ψm )) Ef1 [sT T (x)] = ¯ λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) F1 (ψ¯m ) − F1 (ψm ) ¯ ¯ ¯ (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) Ef2 [sT T (x)] + (A.45) ¯ λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) F2 (ψ¯m ) − F2 (ψm ) ¯ ¯ ¯ λEf1 [sT T (x)] + (1 − λ)Ef2 [sT T (x)] . (A.46) = λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) ¯ ¯ With ˜ g [sT T (x)] + (1 − λ)E ˜ g [sT T (x)] ω ˜ = λE 1 2 ω = λ(F1 (ψ¯m ) − F1 (ψm )) + (1 − λ)(F2 (ψ¯m ) − F2 (ψm )) ¯ ¯

(A.47)

we thus obtain TT

λEf1 [s

TT

(x)] + (1 − λ)Ef2 [s

(x)] = ω =

Z

Ψm

 ∗  sj (x) − B f (x)dx.

(A.48)

We can thus conclude that, as s∗j is a solution to (Pj), so is the modified security s˜∗j (x) =

(

s∗j (x) sT T (x) + B

if if

x ∈ X\Ψm x ∈ Ψm

(A.49)

as s˜∗j satisfies dual monotonicity by construction, is identical to s∗j everywhere but on Ψm and (i) has identical unconditional expected value as s∗j on Ψm (see (A.48)) and (ii) has weakly lower expected value under distribution fj on Ψm (see (A.44)). However, s˜∗j is consistent with the definition of tranched debt on Ψm and, by (ii) must also be a solution to (Pj). Following the same steps (rescaling the problem such that Proposition 3.(ii) applies and then constructing s˜∗−j which is identical to s∗−j everywhere but on Ψm and corresponds to a leveraged debt contract on Ψm ) it can also be established that a solution to (P-j) exists which is consistent with tranched debt on Ψm . It is important to note here that due to ¯ m ) = C and therefore that the ω dual monotonicity, s∗−j (psi ˜ in (A.42) satisfies the required

43

bounds.32 Finally, for any Φn , the identical approach allows to establish that (i) Proposition 3.(ii) applies and construction of s˜∗j consistent with a leveraged debt tranche on Φn establishes that a solution to (Pj) exists which is thus consistent with debt-like tranches on Φn , as well as (ii) that construction of s˜∗−j on Φm establishes that a solution to (P-j) exists which is consistent with tranched debt on Φn . Note that while the approach is identical, roles are now reversed as for (Pj), the likelihood ratio is decreasing on Φm while it is increasing when considering (P-j). The above construction works for any Ψm , Φn and thus establishes that a security consistent with tranched debt exists which solves (Pj) and (P-j), respectively. Finally, as sT T and sLD are the unique solutions to the local minimization problem (P) whenever the local likelihood ratio is strictly increasing, a simple contradiction obtains whenever s∗j , s∗−j are not consistent with tranched debt: (A.44) holds strictly and thus s∗j (s∗−j ) can not have been optimal.

B. Security Design under nondecreasingness Instead of dual monotonicity, we can also solve the security design problem under simple nondecreasingness. The analysis proceeds similar to the one in the main text, and the resulting optimal security structure shares many features with that under dual monotonicity, but exhibits discontinuities. Definition B.1 The set of securities satisfying nondecreasingness and limited liability is denoted SDn . Formally, s ∈ SDn iff s : D → R+ satisfies (i) limited liability: s(x) ≤ x for all x ∈ D and

(ii) nondecreasingness: ∀x1 , x2 ∈ D : x1 ≥ x2 ⇒ s(x1 ) ≥ s(x2 ). When allowing for contracts to exhibit discontinuities, the role which leveraged debt contracts played in the original analysis is taken up by leveraged debt-like contracts, illustrated in Figure 8. Definition B.2 A leveraged debt-like contract (LDL) is characterized by the leverage level L and face value D. Its payoff structure is given by sLDL (x; L, D) = min{x, D} · 1x≥L . D

payoff

sLDL (x; L, D)

leverage Leveraged Debt-Like Contract

x

L

Figure 8: Leveraged debt-like contract 32

In particular, this ensures that the leveraged contract reaches C at ψ¯m and thus has the security s˜∗−j consistent with dual monotonicity.

44

Similar to leveraged debt contracts, there exists a unique leverage level associated with each face value for each expected value the overall security has to match, which follows immediately from the definition and strict positivity of densities. Lemma B.2
R For each ω ∈ 0, X xf (x)dx , and for each face value D such that ω ≥ E[sLDL (x; xL , D)], there exists a unique L(ω, D) such that E[sLDL (x; L(ω, D), D)] = ω, where sLDL ∈ SDn . Proposition 1 carries over from the main analysis, with the only exception being that the new security traded in equilibrium following the public information arrival has to be in SDn . As such, the set of securities which maximize expected utility is given by   S ∗,n ≡ arg max{ max λ min Ef1 [s(x)], ω1M + (1 − λ) min Ef2 [s(x)], ω2M }. n s∈S

(B.50)

X,ω I

Denoting

(P1.n) minn Ef1 [s(x)]

s.t.

λEf1 [s(x)] + (1 − λ)Ef2 [s(x)] = ω I

(P2.n) minn Ef2 [s(x)]

s.t.

λEf1 [s(x)] + (1 − λ)Ef2 [s(x)] = ω I

s∈SX

s∈SX

Proposition 2 carries over to the new restriction and we have l Proposition B.2 Consider securities s∗j.n , j ∈ {1, 2}, which solve (Pj.n). Denote vj.n the h i ∗ l ∗ expected value of sj.n in state l: vj.n = Efl sj.n (x) , ∀l ∈ {1, 2}. Then n (I) ∀η ∈ [0, 1] : ηs∗1.n + (1 − η)s∗2.n ∈ SX,ω I

(II) ∃η ∈ [0, 1] : ηs∗1.n + (1 − η)s∗2.n ∈ S ∗,n .

As the proof is identical to that of Proposition 2, we omit it here.33 Following Proposition B.2, we can restrict attention to characterize solutions to (P1.n) and (P2.n). As in the analysis in the main text, we first need to introduce

(P’j.n)

min Efj [s(x)] s.t. E[s(x)] = ω ∧ s(x) ≤ C , ∀x ∈ X

s∈SXn

(B.51)

and characterize the solution to these modified problems, where the bound is as in the main part. Proposition B.3 Consider the problem (P’j.n), j ∈ {1, 2}. Then f

j is increasing in x on X, the Standard Debt Contract sSDC (x; D(ω)) solves (i) If f−j (P’j.n).

33

It is contained in an earlier working paper version.

45

f

j (ii) If f−j is decreasing in x on X, the leveraged Debt-like Contract sLDL (x; L0 (ω, C), C) solves (P’j.n).

If the likelihood ratio is strictly increasing (or decreasing, respectively), the respective securities are the unique solutions in (i) and (ii) up to point-wise deviations at points of discontinuity. Proof. First note that Lemma A.1 carries over. Moreover, the proof for (i) is identical to that of part (i) in Proposition 3. For (ii), consider the leveraged debt-like contract n sLDL (x; L(ω, C), C) and an arbitrary s ∈ SX,ω with s(x) ≤ C , ∀x ∈ X. It needs to hold that ∃ξ3 ∈ X such that ∀x ∈ X : x ≤ ξ3 ⇒ sLDL (x; L(ω, C), C) ≤ s(x) ∧ x ≥ ξ3 ⇒ sLDL (x; L(ω, C), C) ≥ s(x). (B.52) Again suppose otherwise, i.e. that ∃x1 , x2 ∈ X such that we have x1 < x2 and s(x1 ) < sLDL (x1 ; L(ω, C), C), but s(x2 ) > sLDL (x2 ; L(ω, C), C). We are able to obtain a contradiction: s(x1 ) < sLDL (x1 ; L(ω, C), C) implies that sLDL (x1 ; L(ω, C), C) > 0 and thus x2 > x1 > L(ω, C). But then s(x2 ) > min{x2 , C} which is a contradiction to either limited liability or s(x) ≤ C , ∀x ∈ X. From (B.52), applying Lemma A.1 yields (ii). We can establish a non-optimality condition for standard debt similar to the one in the case of dual monotonicity. Proposition B.4 Denote by D(ω I ) the face value of the standard debt contract sSDC (x; D(ω I )) 1−F (x) with Ef [sSDC (x; D(ω I ))] = ω I . Let G(x) ≡ 1−F−j . If j (x) ∃ξ ∈ (D(ω I ), xH ) : Gj (ξ) >

inf

x∈(xL ,D(ω I ))

fj (x) f−j (x)

(B.53)

then sSDC (x; D(ω I )) is not a solution to (Pj.n). Proof. It can be established that a contract s with Ef [s(x)] = Ef [sSDC (x; D(ω I ))] = ω I and Efj [s(x)] < Efj [sSDC (x; D(ω I ))] exists whenever (B.53) holds. Consider some ξ as in (B.53). By continuity of fj , f−j , if Gj (ξ) > inf x∈(xL ,D(ωI )] f−j (x)/fj (x), there has to exist an -neighborhood U around some x ˆ ∈ (xL , D(ω I )) such that ∀x ∈ U (ˆ x) :

f−j (x) < Gj (ξ) fj (x)

(B.54)

where xL < x ˆ−
x for x ∈ (xL , x ˆ − ) x ˆ− for x ∈ [ˆ x − , x ˆ + ] s(x) = (B.55) I  min{x, D(ω )} for x ∈ (ˆ x + , ξ)     D(ω I ) + κ for x ∈ (ξ, xH ) R where κ(1 − F (ξ)) = U (ˆx) [x − (ˆ x − )]dF (x) and  is chosen sufficiently small such that κ ≤ ξ − D. The construction is illustrated in Figure 9.

46

payoff D(ω I ) + κ

κ

D(ω I )

s(x) sSDC (x; D(ω I ))

x ˆ+ x ˆ− x ˆ− x ˆ+ x ˆ

x

ξ

Figure 9: Inclusion of Junior Debt-like Tranche By construction, it holds that Ef [s(x)] = Ef [sSDC (x; D(ω I ))]. Thus, since f−j (x)/fj (x) < Gj (ξ) for all x ∈ U (ˆ x), R

RU (x)

R

(sSDC (x; D(ω I )) − s(x))dF−j (x)

< κGj (ξ) = κ

SDC (x; D(ω I )) − s(x))dF (x) j U (x) (s

U (x) ⇒ R

(sSDC (x; D(ω I )) − s(x))dF−j (x)

SDC (x; D(ω I )) U (x) (s Z xH SDC

⇒

⇒

s

xL

Z

xH

<

− s(x))dFj (x)

(x; D(ω I ))dF−j (x) <

sSDC (x; D(ω I ))dFj (x) >

xL

R xH

1 − F−j (ξ) 1 − Fj (ξ)

(B.56)

(s(x) − sSDC (x; D(ω I ))))dF−j (x) R xH (B.57) (s(x) − sSDC (x; D(ω I )))dFj (x) ξ

ξ

Z

xH

s(x)dFj (x)

(B.58)

s(x)dFj (x)

(B.59)

xL xH

Z

xL

where the second to last implication follows from Lemma A.1 and the fact that s and are identical outside of U (ˆ x) and (ξ, xH ). sSDC (x; D(ω I )) cannot be optimal. Intuitively, the above construction does not affect the unconditional expected value. However, the decrease at all points in U (ˆ x) yields a decrease in both the expected value under fj and f−j information. Of the overall change in expected value, less is attributed to the expected value under f−j (as compared to that under fj information) than in the increase of the payoff by κ at all points in (ξ, xH ). Thus, since the unconditional expected value changes exactly offset each other, s has a higher expected value under f−j than sSDC (x; D(ω I )), and, thus a lower expected value under fj . Hence, sSDC (x; D(ω I )) cannot solve (Pj.n). The resulting optimal security is no longer composed of ‘perfect’ debt tranches, but may exhibit discontinuities. Formally, we define the class of securities composed of debt-like tranches. sSDC (x; D(ω I ))

Definition B.3 A security composed of debt-like tranches is characterized by a strictly inN creasing sequence {xi }N i=1 ∈ (xL , xH ) of points and a strictly increasing sequence {Di }i=1 ∈ R+ of face values where ∀i ≥ 2 : xi > Di−1 ∀i ≥ 2. The contract sT DL is then characterized by the following payoff structure:

sT DL (x) =

  

0 min{x, Di }   min{x, DN }

for x < x1 if x ∈ (xi , xi+1 ), i < N if x ∈ (xN , xH )

where (xN , DN ) are cutoff point and face value of the most junior tranche if N is finite,

47

and xN ≡ supj xj , DN ≡ supj
payoff

payoff D1 + x2 TD

D2

s

Retained Quasi-Equity

(x)

Junior (Leveraged)

D1 + (x2 − D2 )

Debt-like Tranche sLDL2

Retained Mezzanine Tranche

D1

D1

Senior Debt Tranche sLDL1

Senior Debt Tranche sLDL1 xL

D2

x2

Junior (Leveraged) Debt-like Tranche sLDL2

xH

x

xL

x2

Figure 10: Implementation of Contracts composed of debt-like tranches

48

xH

x