Optimal Security Design for Risk-Averse Investors†

By Alex Gershkov, Benny Moldovanu, Philipp Strack, and Mengxi Zhang*

We use the tools of mechanism design combined with the theory of risk measures to analyze how a cash-constrained owner of an asset with known, stochastic returns raises capital from a population of investors who differ in their risk aversion and budget constraints. The issuer partitions the asset’s cash flow into several¹ asset-backed securities, one for each type of investor. The optimal partition conforms to the commonly observed practice of tranching into senior debt, junior debt, and equity. Tranching arises endogenously due to the differences in risk appetites among agents and in the budget constraints they face. (JEL D81, D82, G12, G41, G51)

Figure 1. Illustration of Tranching

In the present paper we use the tools of mechanism design combined with the theory of risk measures to analyze a model where an issuer with insufficient financial resources raises capital from a population of different classes of risk-averse and budget-constrained investors by securitizing an underlying asset that has a stochastic return. The investors are heterogeneous in their risk preferences and privately informed about both their preferences and budgets. The issuer then employs a menu of securities to raise funds.

In order to clearly differentiate our explanation from the classical one based on a lemons problem where the issuer has more information than the investor about the asset’s returns, the distribution of the project’s cash flow is assumed here to be common knowledge: No agent has private information about it.

Another main methodological departure from the classical finance literature, where investors are expected utility maximizers, arises from our assumption that investors assess risk using the large and important class of spectral or coherent distortion risk measures (see, for example, Acerbi (2002) and Wang (1996)). 2 This class is obtained by taking weighted sums of average values at risk (or expected shortfalls), and it forms the building block for the entire set of law-invariant, coherent risk measures described and axiomatized in a famous paper by Artzner et al. (1999). 3 Thus, the risk measurement we use here closely follows the most recent recommendations made by the Basel Committee on Banking Supervision (2019), the primary global standard setter for the prudential regulation of financial institutions, or by the widely used Swiss Solvency Test (2024) for insurance companies. These guidelines point out the main deficiency of the ubiquitous Value of Risk (VaR) measure—specifically, that it may penalize portfolio diversification and establish the expected shortfall (ES) as the preferred way of assessing risks in financial markets.

Our main results show that in the optimal mechanism the issuer partitions and sells the underlying asset’s realized cash flow into several tranches of debt or equity, one for each type of risk-averse investor, such that more conservative investors are offered less risky securities. The optimal tranching conforms exactly to the commonly observed waterfall structure that features a subordinated payment scheme: senior debt, possibly junior debt, and equity.

In a frictionless, complete market, the nature of issued securities should be irrelevant: The field of security design aims to explain optimal financial structures given prevalent market frictions. The standard theoretical argument is that tranching occurs because it enables a decomposition of the asset’s cash flow into a component that is largely independent of a seller’s private information (debt), and an information-sensitive component with cash flows that are dependent on the seller’s information (equity). Although this theory predicts which security—debt or equity—is retained by the issuer, it does not explain how or why the various sold securities are allocated to heterogeneous investors. Meanwhile, the rise and growing significance of publicly available credit ratings for securities, combined with regulated information disclosure in sales prospectuses, helps mitigate the aforementioned lemons problem.

Our model explains the emergence of tranching without appealing to an asymmetry of information about the quality of the underlying asset. Tranching endogenously arises here in an optimal mechanism because of simpler and more basic economic forces, namely, the differences in risk appetites among investors. Indeed, a principal practical motivation for securitization is to appeal to investor groups with heterogeneous preferences: Tranching caters to both conservative and more aggressive (i.e., less risk-averse) investors since it provides a variety of product choices tailored to specific investor needs in terms of duration, risks, cash-flow patterns, and yields.4 Senior securities are designed to appeal to constrained or conservative investors who can only purchase investment-grade products and are thus more risk averse. These investors are bound either by transparent regulations (e.g., banks, pension funds, and insurance companies are often restricted in the types of assets they may hold) or by less transparent constraints, such as internal bylaw restrictions/ investment mandates or other portfolio/time-specific hedging requirements.5 Less constrained, aggressive investors—such as hedge funds, private equity funds, and sovereign funds—are less risk averse and are thus willing to purchase riskier securities. The junior securities that offer higher returns in exchange for more risk are designed to appeal to them.

Since the reasons for being constrained in the above sense are diverse and also change over time, the same financial institution may belong to either group in a specific transaction, and the issuer may not know what the current risk appetite of an individual investor is or what his present investment budget is. For example, although the current Basel Framework recommends a standard approach (based, among other factors, on ES as a measure of risk), it still allows financial institutions to use internal, private risk models for up to 40  percent of their portfolio. Even if the issuer possesses information about the various risk preferences, regulatory constraints or the possibility of resale may prevent her from effectively using it. We illustrate the last point further in Section VA, where we consider the possibility of trading among investors after the initial sale of securities by the issuer.6

As previously mentioned, we provide a complete characterization of the optimal menu of offered securities and explain how it depends on the model’s main features: the size of the financing need relative to available budgets, the relative risk aversions of all involved agents (issuer and investors), and the relative frequency of investors with different risk appetites and their budget constraints. If the issuer’s financing need cannot be fully covered by the least risk-averse (aggressive) investors, more risk-averse (conservative) investors must be attracted with less risky securities, yielding the sequential senior-junior security structure where each additional dollar of realized return is allocated to a unique class of investors, forming a waterfall structure. A higher financing need generally leads to more tranches being offered.7 If the issuer is less risk averse than all types of investors, then the offered securities are debt contracts with different seniority and risk-return profiles while the issuer then retains an equity tranche. Finally, if the issuer’s risk aversion is such that there are potential investors who are both more or less risk averse than the issuer, then the aggressive, least risk-averse investors buy the equity tranche.

Assuming the issuer is the least risk-averse party, we also find that if the issuer owns a stochastically better/safer asset (in terms of either a first-order or second-order stochastic shift), or if the issuer needs to raise less capital, then the optimally offered rate for senior debt decreases, and the issuer is better off. Interestingly, in that case the aggressive investors always get worse off. This is somewhat surprising since these investors are risk averse and yet they prefer to invest in a security that is backed by a worse/more risky asset. This phenomenon is due to the effects of screening: Aggressive investors earn information rents because they value the risky asset more than conservative investors. When the asset becomes less risky/better, the difference in valuations between the different types of investors decreases and so does the information rent. This observation has interesting policy implications: Market regulators often ban very risky assets in order to protect investors, but our results show that such restrictions may actually hurt risk-tolerant investors. We also find that, if investors become more conservative, the issued securities become riskier so that the probability of not being able to serve the outstanding debt contracts increases.

The rest of the paper is organized as follows: In the remainder of this section, we survey the relevant literature. Section I presents the security design model with risk-averse and budget-constrained, heterogeneous investors. In Section  II, we describe and characterize feasible and incentive-compatible mechanisms in this setting. Optimal security design via tranching is derived in Section III. Section IV presents several comparative statics results about changes in the optimal design when the underlying asset becomes stochastically better or safer, or less costly to implement. Section V offers several extensions to the basic model where we consider an issuer who (i) takes into account the possibility of trading among investors after the initial issue, (ii) is subject to moral hazard, (iii) is herself risk averse, (iv) faces investors whose budgets are also private information, and (v) faces more than two types of investors. Section VI concludes.

Related Literature.—The financial literature on security design is extensive. Accessible accounts of the role securitization played in the financial crises are given by Coval, Jurek, and Stafford (2009) and Hellwig (2009). Gorton and Metrick (2013) and Allen and Barbalau (2022) provide comprehensive surveys of the practice and theory of securitization. We focus below on the aspects most relevant to our paper.

Classical security design models assume that the issuer is relatively more informed than the investors (see, for example, Leland and Pyle 1977; Myers and Majluf 1984; Nachman and Noe 1994; DeMarzo and Duffie 1999; Inostroza and Tsoy 2022; Malenko and Tsoy 2023). Uninformed investors then draw inferences about the assets’ merit from the contracts proposed by the informed issuer. These models focus on securities’ sensitivity to the issuer’s private information and derive conditions under which debt dominates other securities. In an incomplete market model with noise traders, Boot and Thakor (1993) showed that the issuer’s expected revenue is enhanced by selling both equity and debt rather than selling a single claim. Such a partition is profitable because it enables the decomposition of the cash flow into an information-insensitive component and a sensitive component that is dependent on the seller’s information. DeMarzo (2005) also investigates how the asymmetry of information interacts with the ex ante incentives to pool assets before securitization.8

Frank and Goyal (2003) and Fama and French (2005) criticized the “pecking order” theory with its main driving force—the informational asymmetry between issuer and investors—and observe that firms issue much more equity than predicted by that theory (where it should be only the “last resort” security retained by the issuer). Ospina and Uhlig (2018) compare ex ante credit ratings of a large set of mortgage-backed securities with (post-financial crisis) “ideal” ratings given the observed outcomes. They conclude that ex ante ratings were relatively accurate (particularly on top AAA tranches)—the lemons problem was not that severe.

A smaller literature assumes that the outside investors rather than the issuer have superior information about the project’s prospects9 . Several papers following DeMarzo, Kremer, and Skrzypacz (2005) study a model in which privately informed investors choose securities from a set ordered by steepness, rather than competing with securities designed by the seller as in Axelson’s (2007) model.10 Ball and Pekkarinen (2024) analyzed optimal selling mechanisms with royalties and costly verification. Figueroa and Inostroza (2023) showed that debt contract is the optimal screening contract that minimizes information rents of liquidity providers. In most of the above studies, investors are risk neutral, and the transaction occurs between the issuer and a unique winner who is the only one to make a payment. In particular, tranching based on risk aversion does not play a role in the obtained results.11 Rostek and Yoon (2021) study the design of securities in a multiasset model where assets are sold in separate uniform-price auctions. They find that, in markets with large traders, a limited number of synthetic products, customized to align with traders’ desired risk profiles, generally outperform mutual funds in welfare terms.

Somewhat surprisingly, investors’ risk aversion is not a standard feature in models of security design. For instance, it is absent in the classical models that derive the optimality of debt contracts by appealing to costly verification, bankruptcy penalties, or moral hazard.12 This is mostly due to the high technical difficulty of such an analysis within the framework of expected utility. Hellwig (2001) shows that the optimality of debt fails if the borrower is risk averse. In that case the optimal contracts have a complex mathematical form. Allen and Gale (1988) and Malamud, Rui, and Whinston (2010) study optimal security design and risk sharing with risk-averse investors. In particular, Allen and Gale show that, in a general equilibrium model with expected utility, neither debt nor equity is an optimal security. Thus, their model and analysis cannot explain the emergence of the standard securities that are observed in practice.13 Gershkov et al. (2023) focus on classical monopolistic insurance with dual risk-averse agents and derive the optimal menu of contracts for an insurer that maximizes revenue: In general these menus offer layer insurance, where each additional dollar of potential loss is either fully retained by the insuree or fully passed to the insurer.14 Contrasting the present framework, the insurance setting involves an uninformed insurer with unlimited funds who buys part of the risk faced by an agent (with known risk preferences) who is privately informed about the distribution of accidents.

A number of papers study security design problems with risk-neutral agents endowed with heterogeneous beliefs (see, for example, Garmaise 2001; Broer 2018; Simsek 2013; Ellis et al. 2022; Ortner and Schmalz 2019; and Luo and Yang 2023). Some of these models a priori assume tranching even if it is not optimal (e.g., Broer 2018), while others derive optimal tranching structures consisting of securities that are, in general, different from debt or equity. Ellis et al. (2022) study a competitive equilibrium model where both issuer and investors are risk-neutral price takers. In equilibrium, the possible realizations of future returns are partitioned into intervals: Each interval is “assigned” to a single investor with the maximal willingness to pay for that interval. Hence, this equilibrium can be also be implemented via tranching, where each trader may buy more than one tranche. Allowing for diverse risk preferences while assuming that beliefs are homogenous does not generate tranching in that model. Moreover, in that case the equilibrium-security design need not allocate risk to those most willing to bear it. Ortner and Schmalz (2019) analyze a security-design model with risk neutral agents and without the main frictions present here (the investors’ budget constraints and the issuer’s need to raise cash). The structure of the optimally issued securities is determined by the relations (conditional on the state of the world) between the optimism/pessimism of issuer versus investors and among investors. In our model the basic waterfall structure consisting of seniority-ordered standard debt tranches and one standard equity tranche is optimal, independent of whether the issuer is more or less risk averse than some or all investors—what changes is the allocation of the equity tranche.

Finally, we note that the interesting class of models with heterogenous beliefs is not fully consistent with an optimal mechanism design analysis: When it is common knowledge that market participants have diverse beliefs (i.e., when they “agree to disagree”), the issuer can arbitrage the differences in beliefs by organizing structured trades among the investors, thereby extracting the whole available surplus. In other words, in order to rationalize the use of standard securitization policies, it is necessary to impose exogenous restrictions on the class of feasible mechanisms.15 As demonstrated in Section VA, our results are also robust to allowing trading among agents after the initial issuance—a feature of many financial markets. In contrast, incorporating trading into a mechanism design model with heterogeneous beliefs may be challenging for the reasons discussed above.

I. The Model

A seller/issuer (she) has a project or asset that generates a random return with outcomes in the interval $X = \ [ 0 , \bar { x } ] \ \subseteq \ \mathbb { R } _ { + }$ . The project’s return $x$ is governed by

the distribution $H : X [ 0 , 1 ] .$ . The seller has no cash and needs to raise funds of $c \in \mathsf { \Gamma } ( 0 , 1 )$ in order to finance the project.16

Risk Preferences and Budgets of Investors.—There is a unit mass of potential, risk-averse investors/buyers/agents (he). Each one of them is described by a limited budget that is normalized here to one and a preference over lotteries.

Let $\nu$ be the set of random variables with outcomes in the interval $X = \ [ 0 , { \bar { x } } ]$ , defined on a given nonatomic probability space $( \Omega , { \mathcal { F } } , \mathbb { P } )$ . A functional $\rho : \mathcal { V } \to \mathbb { R }$ is a distortion risk measure if it satisfies the following properties (see Wang 1996):

(i) $\rho ( \nu + w ) = \rho ( \nu ) + \rho ( w ) \mathrm { i f } \nu , w \in \mathcal { V } \mathrm { a r e ~ c o m o n o t o n i c } .$ (ii) $\rho ( 1 ) = 1$ . (iii) $\rho$ is decreasing in first-order stochastic dominance (FOSD).

A distortion risk measure is coherent if it is subadditive:

Subadditivity is an appealing property because it ensures that diversification reduces risk, reflecting the benefit of spreading investments across multiple assets. Part (i) states that there is no benefit of diversification when investing in perfectly correlated assets. Throughout we assume that the risk preference of each investor is consistent with a coherent distortion risk measure—these are also called spectral risk measures.

The investors’ risk preferences over risks are consistent with a distortion risk measure if and only if there exists an increasing function $g : [ 0 , 1 ] \ \to \ [ 0 , 1 ]$ with $g ( 0 ) = 0$ and $g ( 1 ) = 1$ such that the certainty equivalent of the investor is given by17

A distortion risk measure is coherent if and only if $g$ is convex, which is also equivalent to risk aversion in the standard sense of aversion to mean preserving spreads. For $H _ { \nu } ( s ) \ = \ \mathbb { P } ( \nu \leq s )$ and $g$ , absolutely continuous integration by parts yields18

The above certainty equivalent modifies the standard expectation operator $\mathbb { E } _ { H } [ \nu ] \ : = \ : \int _ { 0 } ^ { \bar { x } } s d H _ { \nu } ( s )$ by distorting each outcome $s$ by the weight $g ^ { \prime } { \Big ( } { \underline { { \mathbb { P } } } } { \big ( } \nu \ > \ s { \big ) } { \Big ) }$ that is decreasing if $g$ is convex, that is, if the agent is risk averse. Thus, risk aversion is created here by having higher weights on less favorable outcomes. The ubiquitous channel that creates risk aversion—decreasing marginal utility of money—is absent, and the marginal utility of money is constant.

Private Information.—Each investor’s risk preferences are his private information and described by a type $\theta$ that determines the distortion function $g _ { \theta }$ . We assume for simplicity that there are two types of investors: $l$ types (conservative investors) with low risk-tolerance and $h$ types (aggressive investors) with high risk-tolerance $\theta \in \Theta = { l , h }$ . Each type occurs with probability $f _ { l } , f _ { h } > 0$ , respectively, such that $f _ { h } + f _ { l } = 1$ . In Section VE we extend our results to more than two types.

We assume that the investors are risk averse and that their risk attitudes are ordered: $g _ { l }$ is a convex transformation of $g _ { h }$ , meaning that conservative type $l$ investors are more risk averse than the aggressive type $h$ investors.19 It is noteworthy that this also implies here that $g _ { l } ( p ) \ \leq \ g _ { h } ( p )$ for all $p \in \ [ 0 , 1 ]$ , with strict inequality holding for some $p$ . We first assume, for simplicity, that the issuer is risk neutral. In Section VC we will discuss the extension to a risk-averse issuer.

Finally, to rule out trivial cases, we assume that it is technically feasible to raise the necessary funds $c$ from investors. Concretely, we assume that the conservative investors—who require a higher premium in order to purchase risk—value the project higher than its cost; that is, $\mathcal { U } _ { g _ { l } } ( \nu ) > c \mathrm { . }$ . As investors are risk averse, this condition also implies that the project’s expected return is at least $c$ .

Before concluding this section, we note that having the same budget for both types is for illustrative convenience only. Our results generalize in a straightforward manner to the case where budgets are heterogeneous and are the agents’ private information (see Section VD for details).

A. Examples of Distortion Risk Measures

Special cases of distortion risk measures are:

Value at Risk.—VaR is a commonly used risk measure, defined as $V a R ( \nu ) =$ $H _ { \nu } ^ { - 1 } ( \alpha )$ . It represents the threshold $s$ where the likelihood of achieving a return greater than $s$ is $1 - \alpha$ . This risk measure corresponds to the distortion

VaR is not coherent, hence its use is not recommended anymore by the Basel framework.

Expected Shortfall.—The main alternative used in practice and now explicitly mandated by the Basel Framework is ES (also called conditional VaR). It is a coherent risk

measure that refines VaR by considering the expected return conditional on it lying below the level that is exceeded with probability $1 - \alpha$ :

ES corresponds to the distortion

Exponential Distortion Risk Measure.—Another popular ordered class of coherent risk measures, parametrized by $\alpha \in [ 0 , 1 ]$ , corresponds to the family of exponential distortions:

Remark: Even if the investor is risk neutral but the set of feasible investments is regulatorily constrained by an upper bound on the risk measure associated with the investment, a Lagrange multiplier approach induces the maximization of a weighted sum of the expected value and the associated security equivalent $( 1 - \lambda ) \mathbb { E } [ \nu ] + \lambda$ $\mathcal { U } _ { \tilde { g } } ( \nu )$ . The corresponding distortion function is given by

Note that, as the weight $\lambda$ on the risk measure increases, $g _ { \lambda }$ becomes more convex, and the investor becomes more risk averse. Thus, investors who are more constrained in terms of how much risk (measured by a spectral risk measure) they can accept are more risk averse.

B. Alternative Interpretation

Alternatively, our investors can be viewed as risk-averse individuals with nonexpected utility functions whose risk preference arises from nonlinear probability weighting. The preferences induced by spectral risk measures coincide with the so-called dual risk-averse utility functionals from decision theory (Yaari 1987). 20 Dual utility induces first-order risk aversion which makes it appealing for many applied contexts, such as insurance decisions or investment decisions (see Segal and Spivak 1990 for a discussion of first and second orders of risk aversion). We provide a more detailed discussion of the empirical evidence on risk preferences in financial markets in Supplemental Appendix C.

Another well-known example in behavioral economics is the class of reference-dependent loss-averse preferences with linear local utility, studied by Kőszegi and Rabin (2006) and Masatlioglu and Raymond (2016), among others. These preferences correspond to the distortion function:

where $k \in \ [ 0 , 1 ]$ captures the degree of risk aversion. For $k = 0$ , preferences reduce to risk neutrality, while $k = 1$ represents the highest degree of risk aversion in this class.

II. Mechanisms: Menus of Securities

We restrict attention to deterministic direct mechanisms. Stochastic securities are rarely, if ever, used in practice—it is very hard to credibly commit to the announced randomizations—and we abstract from them here.

Since we have a continuum of agents, we look at mechanisms $( R _ { \theta } , t _ { \theta } ) _ { \theta \in \Theta }$ consisting of a menu of asset-backed securities $R _ { l } , R _ { h }$ and their prices $t _ { l } , t _ { h }$ . For each $\theta , R _ { \theta } ( x )$ is the payoff of the security $R _ { \theta }$ if the underlying assets return equals $x$ , and $t _ { \theta } \geq 0$ is the price of this security. We restrict attention to monotonic contracts having nonnegative returns. That is, for any $\theta$ and $x .$ , $R _ { \theta } ( x ) ~ \geq ~ 0$ and $R _ { \theta } ( x )$ is nondecreasing in $x _ { \ast }$ . We also assume that for each $\theta$ the payoff of the security $R _ { \theta } ( x )$ is absolutely continuous in the project’s return, and we denote by $R _ { \theta } ^ { \prime } ( x )$ its generalized derivative. In addition, the promised return of all offered asset-backed securities cannot exceed the value of the underlying asset. That is, for any $x \in X$ , the following feasibility constraint must hold:

It directly follows that $R _ { \theta } ( 0 ) = 0$ for all $\theta$ .

Fix any mechanism $\left( R _ { \theta } , t _ { \theta } \right) _ { \theta \in \Theta }$ . An agent with type $\theta$ who reports to be of type $\theta ^ { \prime }$ obtains utility

Remark: Payments made to investors in the context of asset-backed securities are exclusively sourced from the asset’s returns. Alternatively, one can extend the model by allowing the issuer to add the capital collected from investors in order to increase some of the payments. This modifies the feasibility constraint to

However, as demonstrated in Supplemental Appendix A, it is never strictly optimal for the issuer to adopt this reinvestment approach. It is thus without loss of generality to focus on purely asset-backed securities.

Implementable Mechanisms.—As the conservative $l$ type of investor assigns a strictly higher value to the project than its cost, the issuer can always offer a single security that will be bought by all types and that has a price greater than $c$ . Doing so yields a strictly positive profit, so it cannot be optimal not to finance the project. This yields that, in any optimal menu, the collected revenue exceeds the financing cost $c$ :

Therefore, we only consider below the mechanisms in which the project is successfully financed, and thus (FC) needs to hold. For a type $\theta$ agent not to deviate and claim to be of type $\theta ^ { \prime }$ , it needs to hold that $U ( \theta , \theta ) \stackrel { } { \geq } U \big ( \theta , \bar { \theta } ^ { \prime } \big )$ . This is the same as having the difference in the security equivalents of the assets exceed the difference in their prices:

Similarly, in order to make a type $\theta$ agent purchase the security offered to him instead of pursuing an outside option that is normalized here to yield zero utility (e.g., acquiring a risk-free government bond with a fixed interest rate), it must be the case that

The feasibility constraint requires that the promised payments from the asset cannot exceed the value of the asset: $f _ { l } R _ { l } ( x ) + f _ { h } R _ { h } ( x ) \ \leq \ x$ for each $x \in X$ . That is equivalent to $R _ { \theta } ( 0 ) = 0$ for $\theta \in \Theta$ and for all $x > 0$ :

(Feasibility)

In addition, recall that we require that the return of the security is increasing in the return of the underlying asset

for $\theta \in \Theta$ and for all $x$ , and that each type $\theta$ has a limited budget of 1:

III. Optimal Security Design

The issuer’s profit when financing the project via the mechanism (Rθ(x), tθ) is given by:

Our first Lemma demonstrates that the issuer never wants to raise more than $c$ : Raising funds that are not strictly needed is too costly in terms of the forgone returns that can be obtained by retaining a higher portion of the underlying asset. This happens because all investors demand here a risk premium in order to acquire risk, while the seller is risk neutral.

LEMMA 1: Fix any optimal menu $( R _ { \theta } , t _ { \theta } ) _ { \theta \in \Theta }$ . It must hold that

By the above Lemma, and since the asset’s expected return $\mathbb { E } _ { H } [ x ]$ is fixed, the seller’s objective function reduces to

subject to constraints (IR), (IC), (Feasibility), (M), (BC), and (FC′). In other words, the issuer wants to minimize the loss of potential cash flow from the asset caused by the sale of securities to investors, subject to the constraints that she needs to raise a sum $c$ from them, and that the mechanism is implementable.

We next derive the optimal security design, distinguishing between two cases. In the first case, the project can be entirely financed by selling securities solely to aggressive investors. The solution to the first case offers a building block for the second, more complex case, where it is necessary to design securities for both types of investors.

Investors’ Preferences over Securities.—For later use, we first recall a fundamental result from the insurance literature (see, for example, Van Heerwaarden, Kaas, and Goovaerts 1989), adapted to our security design framework. Let $\mathcal { R } _ { \gamma }$ denote the set of all feasible, monotonic securities $R$ that yield the same expected payment $\gamma = \mathbb { E } R [ ( x ) ]$ . Define $x ^ { d }$ by solving $\mathbb { E } \Big [ \operatorname* { m i n } \big { x , x ^ { d } \big } \Big ] = \gamma .$ and note that $R _ { d e b t } ^ { \gamma } ( x ) =$ $\operatorname* { m i n } { x , x ^ { d } }$ represents a debt contract in $\mathcal { R } _ { \gamma }$ .

THEOREM  1: Consider any security $R \in \mathcal { R } _ { \gamma }$ Then $R ( x ) \ \preceq \ R _ { d e b t } ^ { \gamma } ( x )$ where $\preceq$ denotes second-order stochastic dominance $( S O S D )$ .

In other words, among all monotonic securities with a given expected return, debt is the least variable security and is therefore the best choice for any risk-averse investor.

A. Aggressive Investors Are Sufficient to Finance the Project

Suppose first that the project can be financed by raising money only from the high types, that is, $c \leq f _ { h } \leq 1 .$ . In this case, the most efficient way of raising the necessary funds is to offer securities only to the aggressive investors so that it is optimal to set $t _ { h } = ~ c / f _ { h }$ and $t _ { l } = 0$ . An alternative interpretation of this basic scenario is one in which a single investor with a sufficiently large budget finances the entire project, as in a specially tailored, single-tranche collateralized debt obligation (CDO).

The resulting maximization problem for the issuer becomes

(Problem P)

subject to

The first equality in the above set of constraints is the binding participation constraint for the aggressive types. This is an isoperimetric constraint. The second inequality is the monotonicity constraint. The last two constraints ensure that the payouts to the agents do not exceed the return of the security (i.e., the feasibility) where the second-last constraint is a majorization constraint and the last constraint is a boundary condition.

The result below shows that the optimal security in this case is a debt contract targeting the aggressive investors. The debt’s interest rate is chosen so that the aggressive type’s participation constraint is binding.

PROPOSITION 1: Let $x ^ { * }$ denote the solution to

and note that $c < x ^ { * } < \bar { x } . { } ^ { 2 1 }$ The optimal security is given by

and by $t _ { h } ^ { * } = c / f _ { h } .$ . That is, a debt contract with interest rate $x ^ { * } / c - 1 > 0$ is optimal.

The convexity of $g _ { h }$ is equivalent here to the assumption that the function $z \mapsto$ $z / g _ { h } ( z )$ is decreasing or that the function $x \mapsto \left[ \bar { 1 } - H ( x ) \right] / g _ { h } { \big ( } 1 - H ( x ) { \big ) }$ is increasing. Intuitively, when this last condition holds, it is beneficial for the issuer to make $R _ { h } ^ { \prime } ( x )$ , the security’s slope, as large as the majorization constraint allows for small realizations of the asset’s return $x$ and as small as possible for larger realizations obtained as soon as the isoperimetric constraint is satisfied. Instead of a direct proof based on this observation, we utilize Theorem 1: Among securities with a given cost, a debt contract yields the highest utility to a risk-averse agent. Equivalently, to secure a fixed utility level for such an agent—as required by (IR-h)—the issuer’s most cost-effective strategy is to offer a debt contract. Notably, this conclusion holds for any risk-averse agent regardless of whether their preferences align with EU or non-EU models.

Observation: The above proposition shows how the optimal security’s interest rate depends on the risk aversion of the aggressive investors, on the project’s return profile, and on the financing cost $c$ . It includes several very intuitive predictions that can be tested empirically: Ceteris paribus, a better distribution of returns $H$ (in the sense of FOSD) yields a lower interest rate, a higher degree of risk-aversion yields a higher interest rate, and a higher financing need $c$ also yields a higher interest rate.22

B. Both Types of Investor Are Needed to Finance the Project

Consider now the case where both types of investor are needed to finance the project, that is, $c > f _ { h } > 0$ . For this setting, the participation constraint of a low type, (IR-l), must be binding. If not, then the seller could increase her profit by extracting higher payments from both types. Analogously, the incentive constraint of a high type, (IC-h), must also bind.

To derive the optimal mechanism, we consider a relaxed problem where we ignore the IC constraint of the low type and the participation constraint of the high type and later verify that the ignored constraints are not binding. We then establish that in any solution the issuer must completely exhaust the budget of all agents with high risk tolerance in order to finance the project. As it is never optimal to raise more money than the cost of the project, this determines the amount of money raised from both types of investors. We then establish that the optimization problem can be decoupled across investor types: One can first maximize over the security offered to the high type (fixing the low-type security) and then over the security offered to the low type (fixing the high type’s security). As each of these two problems have a single type of investor, an argument similar to the one of Proposition 1 yields that both offered securities are debt contracts.

THEOREM 2: Suppose that $c > f _ { h }$ , and let $x _ { l } ^ { * }$ , $x _ { h } ^ { * }$ denote the solutions to

respectively. The optimal menu of securities is given by $t _ { l } ^ { * } = ( c - f _ { h } ) / f _ { l } , t _ { h } ^ { * } = 1$ and by

The proof of the theorem can be found in the Appendix. It establishes that the optimal mechanism is a menu of two contracts:

(i) Senior debt with interest rate

that is determined by the participation constraint of the conservative investors (i.e., (IR-l)).

(ii) Junior debt with interest rate

that is determined by the incentive compatibility constraint of the aggressive investors (i.e., (IC-h)).

Furthermore, the amount of money that any agent can invest in senior debt is limited to $( c \mathrm { ~ - ~ } f _ { h } ) / f _ { l }$ .

To gain some intuition on why this waterfall structure is optimal, note first that a similar structure of securities (i.e., senior/junior debt and equity) is also optimal under complete information. With complete information the designer only needs to induce an efficient risk sharing among agents (subject to participation constraints). As the investors’ risk preferences are ordered, the efficient way to share risk is to allocate the safest possible asset (i.e., senior debt) to the most risk-averse investor type, to allocate the second-safest possible asset (i.e., junior debt) to the more risk-tolerant investors, and to keep the remaining riskiest asset (i.e., equity). In this case, the interest rate for the senior debt is still determined by the (IR-l) constraint, and it is equal to that in the incomplete information setting. However, under complete information the interest rate for junior debt is determined by the (IR-h) constraint, and it is lower than that under incomplete information where it is determined by the (IC-h) constraint.

The above optimal allocation presents an atypical structure compared to standard screening models where the allocation of the low type is distorted downwards from its efficient level in order to reduce the information rent extracted by the high type. This is not the case here since conservative investors (low types) obtain exactly the same product and pay the same price as in the complete information benchmark. By contrast, the interest rate offered to the aggressive investors (high types) is distorted upwards to meet their incentive compatibility constraint. This happens because the efficient risk allocation to the low types—senior debt with the minimal interest rate to guarantee their participation—also minimizes the high type’s information rent. Since the high type earns a rent because he values the risky asset more than the low type, the information rent is minimized when the low type receives the safest asset, that is, in an allocation that coincides with the solution to the complete information benchmark. Thus, the most cost-effective way to provide the high type the required information rent is to maintain the waterfall structure that is optimal under complete information while increasing the interest rate for junior debt. We will further illustrate this point in Section V.

Remark (Expected Utility): Even under complete information, a waterfall structure is not optimal if investors have expected utility preferences. This happens because in such cases the marginal utility of money is not constant, leading to a much more complex risk-sharing problem (see, for example, Allen and Gale 1988).

Remark (Inefficiency of Investment): Suppose that the seller can choose any investment level $c > 0$ , each associated with a return distribution $H _ { c }$ . It follows from Theorem 2 that if both investor types are needed to raise funds of $c$ , then in addition to compensating all investors for risk taking, the agressive investors earn an information rent when the investment level $c$ is chosen. Thus, as the issuer does not fully internalize the benefits of increasing the investment level, she will in general underinvest in the project.

IV. Comparative Statics

In this section we present several comparative statics results. These results are markedly different from those obtained in the same model under complete information. We focus here on the more interesting case where both types of investor are needed for financing the project (e.g., assume $f _ { h } < c )$ ). We first investigate the effects of having a better/safer asset in the sense of an FOSD or SOSD shift, respectively. We then discuss the effect of a decrease in the financing cost.

To simplify notation let

denote the interest rates offered by the issuer to the conservative and aggressive investors, respectively, in the optimal menu of asset-backed securities.

A. Who Prefers a Better or a Safer Asset?

Our main finding here is that the issuer always prefers a stochastically better/safer asset, while the aggressive investors always prefer a worse/riskier asset. Recall that $H$ denotes the distribution of returns of the asset.

PROPOSITION 2: Either an SOSD or FOSD improvement of $H$ results in:

(i) a decrease in the interest rate $i _ { l }$ for conservative investors, (ii) a decrease in the expected cost of financing $\int _ { 0 } ^ { x _ { h } ^ { * } } \bigl [ 1 - H ( z ) \bigr ] d z ,$ (iii) a decrease of the surplus $\int _ { x _ { l } ^ { * } } ^ { x _ { h } ^ { * } } g _ { h } \big ( 1 - H ( z ) \big ) d z - 1$ obtained by the aggressive investors.

Furthermore, an FOSD shift of H leads to a decrease in the interest $i _ { h }$ obtained by aggressive investors.

In the optimal mechanism conservative investors are indifferent between purchasing and not purchasing the asset. With a stochastically better/safer asset, the issuer offers a lower interest rate to conservative investors, gives up a smaller share of the asset, and leaves less information rent to the aggressive investors. With a stochastically better asset, the interest rate offered to aggressive investors will also decrease. Intuitively, the comparative advantage of aggressive investors lies in their higher tolerance for risk. If the asset becomes less risky, this advantage decreases, and the rents of aggressive investors decrease as they can be more easily substituted by conservative investors.

B. The Project’s Cost

We find that, as the project becomes less costly to implement, both the offered interest rates and the financing cost decrease. Conservative investors are indifferent, but the aggressive investors are worse off.

PROPOSITION 3: In the optimal contract that finances asset x and raises c, the following hold:

(i) Both interest rates, $i _ { l }$ and $i _ { h }$ , increase in c;

(ii) Aggressive investors prefer to finance costlier projects since their utility $\mathcal { U } _ { g _ { h } } \big ( R _ { h } ( x ) \big ) - t _ { h }$ increases in c.

As the cost of the project increases, the issuer needs to forgo a larger share of the security to raise sufficient funds. Consequently, the riskiness of both senior and junior debt rises, necessitating higher interest rates. Moreover, as the asset sold to the conservative investors becomes riskier, the aggressive investors can capture a higher information rent.

V. Extensions

In this section we offer several extensions to our basic model. In each of these extensions, we introduce only one modification at a time, keeping all other aspects identical to the baseline model, and we focus on the more interesting case where multiple investor types are needed for financing. In particular we consider:

(i) the security design problem where the issuer cannot impose purchase limits;
(ii) security design where the issuer can also undertake an action that affects the asset’s return (moral hazard);
(iii) security design by a risk-averse issuer;

(iv) the case where investors’ budgets are also their private information; and finally (v) the generalization of our basic framework to more than two types of investors.

A central result that emerges across these extensions is that the waterfall structure of subordination remains optimal under a wide range of modifications to the original model.

A. No-Purchasing Limits

We characterize here the optimal mechanism for the case where purchasing limits, as featured in the benchmark model, cannot be imposed. While above we explored the optimal mechanism design problem where the issuer fully controls both the allocation and monetary transfer of every type, here we consider a simpler and more prevalent selling procedure where every investor decides how many units of each security to acquire. The main change lies in the incentive constraint of the aggressive investor type who must now be deterred from buying multiple units of the security intended for the conservative types. This influences the offered interest rates but does not alter the waterfall structure of the optimal menu.

For an alternative motivation, note that if agents can trade among themselves after the initial issue, aggressive, less risk-averse investors may be able to purchase securities from the conservative, more risk-averse investors at a mutually beneficial price. Foreseeing this possibility, the aggressive investors will refrain from buying at the initial issue, causing a loss of revenue to the issuer. The incentive constraint imposed in this section is precisely constructed in order to prevent this possibility: There will be no incentives for trading among the various investors after the initial issue. We note that, even under complete information about the investors’ risk types, the issuer cannot effectively use it in order to eliminate the agents’ information rent: Conservative investors could purchase additional units of securities with the intent of resale to aggressive investors, thereby undermining the issuer’s ability to enforce the intended allocation.

In any implementable mechanism where both types are needed to finance the project, an aggressive investor derives strictly positive utility from pretending to be a conservative type. As our risk preferences are homogenous, the aggressive type’s best deviation without purchase limits is to purchase $1 / t _ { l } > 1$ units of $R _ { l }$ . (IR-l) is then still given by

but the incentive constraint of the high type (IC-h) changes to

THEOREM 3: Suppose that $c > f _ { h } ,$ and let $\hat { x } _ { l }$ , $\hat { x } _ { h }$ denote the solutions to

and

respectively. The optimal menu of securities is given by $\hat { t } _ { l } = \left( c - f _ { h } \right) / f _ { l } , \hat { t } _ { h } = 1$ and

and

The above theorem establishes that the optimal mechanism is still a menu of two contracts: one of them senior debt with interest rate $\hat { x } _ { l } / ( c - f _ { h } ) \mathrm { ~ - ~ } 1$ and the other one junior debt with interest rate $\left( \hat { x } _ { h } - \hat { x } _ { l } \right) / f _ { h } - 1 .$ . Relative to the benchmark model with a purchasing limit, the new menu offers the same interest rate for senior debt holders but a higher interest rate for the junior-debt holders. This happens because the more aggressive investors can now earn more by deviating and buying $\frac { 1 } { t _ { l } }$ units of senior debt. Thus, in order to ensure incentive compatibility, the interest rate offered for the junior debt must increase.

Remark (Interest Rates without Purchase Limits): It is intuitive that, in the present framework where investors are not constrained in their purchases, the interest rate for junior debt should always be higher than the one for senior debt, and we establish in Lemma 5 in the Supplemental Appendix that this is indeed the case.

Remark (Risk Aversion and Debt Default Probability): The above result offers many intuitive implications that can be tested empirically. For example, in the aftermath of the financial crisis of 2007–2008, it has been argued that investors became more risk averse with possible negative consequences on the viability of tranched CDOs.23 Our model makes clear predictions about how the optimal securities change in such a case.

Let us assume, for example, that the conservative investors become even more risk averse and thus use a more convex distortion $\tilde { g } _ { l } .$ . It follows directly that $\tilde { g } _ { l } ( p ) \ \leq$ $g _ { l } ( p )$ for any $p \in \ [ 0 , 1 ]$ . To meet the (IR-l) constraint, the new relevant cutoff $\tilde { x } _ { l }$ that solves

must exceed $\hat { x } _ { l } .$ . This implies the interest rate, given by $\tilde { x } _ { l } / ( c - f _ { h } ) \mathrm { ~ - ~ } 1$ , offered to these (more) conservative investors increases. This change also has implications for the optimal security $\tilde { R } _ { h } ^ { * }$ presented to aggressive investors, whose characteristics have not changed. Given that (IC-h)

remains binding, the relevant new cutoff $\tilde { x } _ { h }$ , and thus also the interest rates offered to the aggressive investors, must increase. The rise in interest rates, without a corresponding shift in the asset’s return distribution, indicates an increased probability of failing to fulfill the obligations of the outstanding debt contracts. This finding aligns with the observations by Ospina and Uhlig (2018), who pointed out that mortgage-backed securities issued around the 2007–2008 financial crisis—a time when investors had become notably more risk averse—were in fact more susceptible to defaults compared to those from earlier periods.24

B. Moral Hazard and “Skin in the Game”

We next consider a variant of the basic design problem where the securities’ issuer needs to exert effort in order to (stochastically) increase the asset’s returns. Chemla and Hennessey (2014) and Malamud, Rui, and Whinston (2013), among others, analyze the issuer’s incentives to affect the asset’s returns, modeled as a moral hazard problem. In contrast to those authors, we do not assume a specific security design: we do derive the optimal structure of securities taking into account the moral hazard problem.

We maintain the assumption that the issuer needs to raise $c > 0$ to finance the project, has no funds, and is protected by limited liability. In this subsection and the next one, we restrict attention to doubly monotonic securities. A menu of securities, $\left( R _ { \theta } \right) _ { \theta = l , h }$ is doubly monotonic if, in addition to $R _ { \theta }$ being monotonic for $\theta \in { l , h }$ , the function

is also monotonic; that is, the issuer’s own tranche is also monotonic in the asset’s return. This assumption, commonly adopted in the finance and insurance literatures, serves as a safeguard against issuers’ potential manipulation of asset cash flows—either through external lending or interproject transfers—with the intent of diminishing payouts to investors.

For simplicity, suppose that the issuer can take two actions $a \ \in \ { 0 , 1 }$ , where $a = 1$ corresponds to exerting effort and where $a = 0$ corresponds to shirking. For example, the founder of a start-up might work hard to ensure its success, or she might not work given that most of the returns were promised to investors. In another example, a bank needs incentives to carefully screen the loan takers behind a pool of mortgages.25

Clearly, the incentives to exert effort in such cases crucially depend on the precise structure of the issued securities: based on this intuition, the Dodd-Frank Act of 2010 requires issuers to retain an equity share of no less than 5 percent of the aggregate credit risk of assets they securitize.

We denote the cost associated with each action by $\kappa _ { a }$ , and without loss of generality we assume that $0 ~ = ~ \kappa _ { 0 } < ~ \kappa _ { 1 } .$ . The distribution of the project’s return depends on the issuer’s effort and is denoted by $H _ { a } ,$ , with density $h _ { a } = H _ { a } ^ { \prime }$ . We assume that the action $a \ : = \ : 1$ leads to a higher return than the action $a = 0$ in the sense of the hazard rate stochastic order.26 This assumption implies that $H _ { 1 }$ and $H _ { 0 }$ are ordered in (strict) FOSD.

Theorem  4 below shows that the optimal solution to the problem with moral hazard takes a simple form: For each possible action consider the securities that are optimal without moral hazard. We show that, if the high action is incentive compatible for the seller given the retained profit under its associated distribution of returns, then the same securities as without moral hazard remain optimal. Otherwise, the low action is optimal combined with its associated menu of securities that were optimal without moral hazard.

For a given pair of securities $R _ { l } , R _ { h } : \mathbb { R } _ { + } \to \mathbb { R } _ { + }$ purchased by low- and high-risk-tolerance investors, the expected surplus retained by the issuer equals

Denote by $\bar { R } ( x ) = x - \bigl [ f _ { h } R _ { h } ( x ) + f _ { l } R _ { l } ( x ) \bigr ]$ the share retained by the issuer if the asset’s return equals $x$ . It is optimal for the issuer to exert high effort if and only if the retained profit increases more than the cost when taking the high action:

We first analyze an auxiliary problem. Let

be the set of doubly monotonic, absolutely continuous securities that lead to a profit of $\pi$ when taking the action $a \ \in \ { 0 , 1 }$ (ignoring the effort cost). Denote by $\bar { \pi } ~ = ~ \int x d H _ { 1 } ( x )$ the return of the asset under high effort. In the auxiliary problem we fix the issuer’s profit at some level $\pi \in \mathsf { \Gamma } ( 0 , \bar { \pi } )$ , given that $a = 1$ is taken, and finding the retained profit function $\bar { R }$ that minimizes the issuer’s incentive to deviate to $a = 0$ , that is, we solve

Finally, denote by $R _ { e q u i t y } ^ { \pi } ( x ) \ : = \ : \operatorname* { m a x } { x - p , 0 }$ a call option security with strike price $p$ (or equity) such that its expected value under the distribution $H _ { 1 }$ equals $\pi$ .

PROPOSITION 4: $R _ { e q u i t y } ^ { \pi }$ is the unique solution to the auxiliary problem (1).

Intuitively, as a call option is the security that is most sensitive to high returns, it maximizes the issuers incentives to exert effort. We can now define $\kappa ^ { * } ( \pi )$ as the highest cost $\kappa _ { 1 }$ such that the issuer can be incentivized to take action $a = 1$ while being promised an expected retained profit of $\pi$ :

As $H _ { 0 }$ is first-order stochastically dominated by $H _ { 1 }$ and, as we have shown above that the solution to the above problem is a call option, it follows that $\kappa ^ { * }$ is a strictly increasing function: More profit retained by the issuer leads to better incentives to exert effort. We are now able to state our main result of this section: The moral hazard problem has no influence on the basic structure of the optimal securities when the issuer is risk neutral.

THEOREM 4: Let $\left( R _ { l } ^ { a } , R _ { h } ^ { a } \right)$ denote the optimal menu of senior-junior debt issued for the distribution of returns $H _ { a }$ in the design problem without moral hazard. Let $\bar { R } ^ { a } ( x ) = x - f _ { l } R _ { l } ^ { a } - f _ { h } R _ { h } ^ { a }$ denote the equity part kept by the issuer, and let

(i) If $\kappa _ { 1 } > \kappa ^ { \dagger }$ , then the optimal contract specifies issuer’s action $a = 0$ and securities $\left( R _ { l } ^ { 0 } , R _ { h } ^ { 0 } \right)$ for investors.

(ii) If $\mathbf { \dot { \omega } } _ { \kappa _ { 1 } } < \kappa ^ { \dagger }$ , the optimal contract specifies issuer’s action $a \ : = \ : 1$ and securities $\left( R _ { l } ^ { 1 } , R _ { h } ^ { 1 } \right)$ for investors for the buyers.

The cost $\kappa ^ { \dagger }$ is the largest possible one such that the issuer does not want to deviate to the low action given $\left( \hat { R } _ { l } ^ { 1 } , R _ { h } ^ { 1 } \right)$ , the optimal menu for $H _ { 1 }$ without moral hazard. Since moral hazard poses an additional constraint on the issuer, and since the issuer achieves the same value as without it, these are clearly also the optimal securities for any $\kappa < \kappa ^ { \dagger }$ given $a \ = \ 1$ . In the Supplemental Appendix we prove that deviating to low effort $a = 0$ and offering the corresponding optimal securities also reduces the issuer’s profit.

If the effort cost exceeds $\kappa ^ { \dagger }$ , then the securities $\left( R _ { l } ^ { 1 } , R _ { h } ^ { 1 } \right)$ do not provide sufficient incentives for the issuer to take the action $a \ : = \ : 1$ .27 As the retained profit forms a call option, Proposition 4 implies that there is no other security that provides sufficient incentives to implement $a = 1$ . Thus, the optimal contract must implement $a = 0$ , and we establish in the Supplemental Appendix that the constraint coming from the moral hazard problem is slack, implying that the optimal securities are $\left( \hat { R } _ { l } ^ { 0 } , R _ { h } ^ { 0 } \right)$ . In either case, the optimal securities are the same as they would have been in the absence of the moral hazard problem if the action of the issuer was exogenously fixed.

Remark: The above “irrelevance” of the moral hazard problem relies on the fact that in the benchmark problem without moral hazard it is already efficient for the issuer to retain equity. By Proposition 4 this is also the optimal way of providing incentives to exert effort. If the issuer is more risk averse than some investors, then he would not retain equity in the absence of a moral hazard problem, and its introduction may change the design of the optimally issued securities—we leave for further research this interesting topic and the examination of the related regulatory scope and impact.

C. Security Design by a Risk-Averse Issuer

Consider now the possibility that the issuer is also risk averse, and her risk preference is represented by a convex distortion function $g$ . In this subsection, we also restrict attention to doubly monotonic securities. The analysis utilizes a counterpart to Theorem 1 (see Gershkov et al. 2023) showing that equity is the most variable security among double monotonic securities with a given expected return.28 Let $\mathcal { R } _ { \gamma } ^ { D }$ denote the set of all feasible, doubly monotonic securities with an expected payment $\gamma _ { : }$ and let $x ^ { o }$ solve $\mathbb { E } \big [ \big ( x - x ^ { o } \big ) _ { + } \big ] = \gamma .$ . Note that $R _ { e q u i t y } ^ { \gamma , D } ( x ) \ = \ ( x \bar { ~ - ~ } x ^ { o } ) _ { + }$ represents an equity contract included in $\mathcal { R } _ { \gamma } ^ { D }$ , and let $R _ { d e b t } ^ { \gamma , D } ( x )$ denote a debt contract in the same set.

LEMMA 2 (Gershkov et al. (2023), Theorem 2): Consider any security $R \in \mathcal { R } _ { \gamma } ^ { D }$ Then $R _ { e q u i t y , } ^ { \gamma , D } ( x ) \ \preceq \ R ( x ) \ \preceq \ R _ { d e b t } ^ { \gamma , D } ( x )$ , where $\preceq$ denotes SOSD.

We distinguish two cases: (1) The case where the issuer is more risk averse than the aggressive investors, and (2) the case where the issuer is more risk averse than the conservative investors.

Case 1: The issuer is less risk averse than the conservative investors; that is, $g$ is less convex than $g _ { l }$ . If the issuer is also less risk averse than the aggressive investors—that is, if $g$ is less convex than $g _ { h }$ —then our previous analysis for a risk-neutral issuer applies. Novel findings arise for the case where the issuer is more risk averse than the aggressive investors, that is, if $g$ is more convex than $g _ { h }$ . Thus we focus below on the case where the seller’s risk aversion is intermediate between those of the two types of investors.

PROPOSITION 5: Suppose that $c > f _ { h }$ and that $g$ is more convex than $g _ { h } .$ . Let x̃l, $\tilde { x } _ { h }$ denote the solutions to

respectively. The optimal menu is given by ${ \tilde { t } _ { l } } = \left( { c - f _ { h } } \right) / f _ { l } , { \tilde { t } _ { h } } = 1$ , and

The conservative investor still receives the safest part of the asset, that is, a senior debt. However, the aggressive investor, who is now the least risk-averse party, receives the equity, which, according to Lemma 2 represents the riskiest part of the asset. It follows that the part of the asset that is kept by the designer, $x - f _ { l } R _ { l } ( x ) \ : - \ : f _ { h } R _ { h } ( x )$ , now takes the form of junior debt and is given by:

Case 2: If the issuer is more risk averse than the conservative investors, the maximization problem is slightly different but can still be solved in an analogous manner. Here we focus on the case where the gain from trade is sufficiently large, ensuring that the issuer always seeks to extract as much money from investors as possible.

If the project is sufficiently profitable, the issuer can extract the full amount from the investors; that is, $t _ { l } = t _ { h } = 1$ . In this case, the issuer’s problem reduces to minimizing

subject to the same set of constraints as in Case 1 but with different values of $t _ { h }$ and $t _ { l }$ . Following an analysis analogous to Case 1, it can be verified that the designer will offer equity to the aggressive investors, offer junior debt to the conservative investors, and retain the senior debt. The corresponding cutoff points are jointly determined by the binding constraints (IR-l) and (IC-h), as in Case 1.

If collecting the full amount proves infeasible given the assumption of sufficient gain from trade, the issuer will aim to sell the entire asset—that is, $f _ { l } R _ { l } ^ { \prime } ( x ) +$ $f _ { h } R _ { h } ^ { \prime } ( x ) \ = \ 1$ —in order to maximize her revenue. Using similar arguments as those in Lemma 3, one can show that the optimal menu still features $t _ { h } = 1 .$ . The problem is then reduced to maximizing the amount $t _ { l }$ collected from the low type, subject to all relevant constraints.

This problem is essentially the same as the baseline model case where only one type is needed to finance the project (see Section IIIA, Proposition 1). The solution involves offering a debt contract to the conservative type and allocating the remaining asset (in the form of equity) to the aggressive type. The issuer, being the most risk averse, sells all of the assets to the investors and retains only cash. A more detailed analysis of this problem is provided in the Supplemental Appendix.

D. Private Budgets

Here we consider investors who have two possible types of budgets: $b = \beta < 1$ with probability $p$ , and $b = 1$ with probability $1 - p$ . The individual budgets are privately known, and for each agent the perceived distribution of his budget is independent of the distribution of the agent’s risk type. Under this assumption the analysis for more budget types remains essentially the same. For later use, we also define the average budget as $\bar { \beta } = p \beta + 1 - p$ . To ensure that the project can be financed, we assume $\bar { \boldsymbol { \beta } } > c$ .

Let $x _ { l , 1 } ^ { * }$ denote the solution to the participation constraint of the conservative investor, who has a budget of 1:

Let $x _ { h , 1 } ^ { * }$ denote the solution to the IC constraint of the aggressive investor, who has a budget of 1, under the assumption that he can only pretend to be a conservative investor with the same budget type:

THEOREM 5: If $\hat { \beta } \ge c > f _ { h } \hat { \beta } .$ , the menu of securities described below is optimal:

Moreover, the price of securities is given by $\begin{array} { r } { t _ { l , 1 } = \frac { c - f _ { h } \bar { \beta } } { f _ { l } \bar { \beta } } , t _ { h , 1 } = 1 , t _ { l , \beta } = \beta t _ { l , 1 } . } \end{array}$ and $t _ { h , \beta } = \beta t _ { h , 1 }$ .

It is still optimal for the issuer to offer senior debt to the conservative investor with a high budget of 1 and junior debt to the aggressive type with a high budget 1. However, the issuer now introduces an additional option for the more budget-constrained investors: These individuals can purchase a share $\beta < 1$ of the corresponding debt at a share $\beta$ of the price charged to high-budget investors.

The details of the proof can be found in the Supplemental Appendix. Here we provide an outline: We first derive the optimal mechanism for the scenario in which agents have heterogeneous yet publicly known budgets (while risk preferences remain private to the agents). Next, we demonstrate that this mechanism remains implementable even when the budget is private information for each agent; thus it must be optimal in this context as well. This result underscores the notion that in a large market small investors do not derive any informational advantage/rent by keeping their budget information private.

E. More Than Two Types

Suppose now that there are $N$ different kinds of investors. Each one is characterized by a type $\theta _ { n } \in \Theta = \big { \theta _ { 1 } , \theta _ { 2 } , \dots , \theta _ { N } \big }$ that determines the distortion function $g _ { n }$ . Each type occurs with a probability $f _ { n } > 0$ such that $\textstyle \sum _ { n = 1 } ^ { N } f _ { n } = 1 .$ We further assume that the investors’ risk attitudes are ordered: For each $n > 1$ , $g _ { n - 1 }$ is a convex transformation of $g _ { n }$ . We consider the case where all $N$ types are needed to finance the project; that is, $1 - f _ { 1 } < c$ . This is without loss of generality: If only $m \ < \ N$ types are needed, the analysis is the same as the one for the $m$ least risk-averse types.29

We examine direct mechanisms $\left( R _ { n } , t _ { n } \right) _ { n = 1 } ^ { N }$ . By using arguments similar to those for Lemma 3, one can verify that, in the optimal mechanism, $t _ { n } = 1$ for all $n \geq 2$ and $t _ { 1 } = ( c - 1 + f _ { 1 } ) / f _ { 1 }$ . The optimal mechanism has a waterfall structure with several “mezzanine” tranches:

THEOREM 6: Suppose that $c > 1 - f _ { 1 }$ , and let $x _ { 1 } ^ { * }$ denote the solution to

$x _ { 2 } ^ { * }$ denote the solution to

and $x _ { n } ^ { * }$ , $n \geq 3$ , denote the solution to

respectively. The optimal menu of securities is given by $t _ { 1 } ^ { * } = \left( c - 1 + f _ { 1 } \right) / f _ { 1 }$ $t _ { n } ^ { * } = 1$ for any $n \geq 2$ , and by

Local incentive compatibility of the above structure follows from the same arguments as in the two-type case. Global incentive compatibility is ensured because by assumption, for any $i ~ < ~ j , ~ g _ { i }$ is more convex than $g _ { j } ,$ which implies that an investor with type $\theta _ { i }$ is always willing to pay more for a risk reduction than one with type $\theta _ { j }$ . Moreover, the structure of the issued securities establishes a complete ordering of asset riskiness: the most risk-averse agent receives the safest portion of the asset (the most senior debt), the second-most risk-averse agent receives the safest portion of the remaining asset, and so on for each subsequent type. Together, these two conditions ensure that if a type $\theta _ { i + 1 }$ has no incentive to deviate to purchase $R _ { i } ^ { * }$ , the same holds for any higher investor type. Thus, local incentive compatibility implies global incentive compatibility. The detailed proof is provided in the Supplemental Appendix.

VI. Conclusion

We have analyzed a novel security design model where an issuer raises capital from a population of heterogeneous, risk-averse, and budget-constrained investors. The issuer sells securities backed by an underlying asset with stochastic returns. Investors assess risk according to the important class of spectral risk measures, including ES. Investors differ in their risk appetites and in their budgets, both of which are their private information. All agents (issuers and investors) possess the same information about the distribution of the asset’s returns.

In this environment, we used the tools of mechanism design in order to derive the optimal security design. We found that the optimal mechanism partitions the asset’s realized cash flow into several securities conforming to the commonly observed practice of tranching, where senior claims are fully paid before subordinate ones. We explicitly derived the structure of the optimal menu of offered securities, such as senior debt, junior debt and equity, and described how it depends on the model’s main features, such as the asset’s volatility, the financing need, and the investors’ heterogenous degrees of risk aversion.

Appendix

PROOF OF LEMMA 1:

Suppose that there exists an optimal menu $\left( R _ { \theta } ( x ) , t _ { \theta } \right) _ { \theta \in \Theta }$ for which $f _ { h } t _ { h } + f _ { l } t _ { l }$ $> c$ and $t _ { l } > 0$ . This implies that $R _ { l } ( x ) \neq ~ 0$ on a set of strictly positive measures. Then we can construct another menu $\left( \stackrel { \cdot } { R } _ { \theta } ( x ) , \tilde { t } _ { \theta } \right) _ { \theta \in \Theta }$ such that $\tilde { R } _ { h } ( x ) \ : = \ : R _ { h } ( x )$ for all $x$ and $\tilde { R } _ { l } ( x ) = ( 1 - \varepsilon ) R _ { l } ( x )$ for all $x$ and for some $\varepsilon > 0$ .

Let $\tilde { t } _ { h } = t _ { h } ,$ and

There clearly exists a sufficiently small $\varepsilon$ such that $f _ { h } \tilde { t } _ { h } + f _ { l } \tilde { t } _ { l } \ge c$ . It is then easy to verify that the newly constructed mechanism is implementable as long as the original mechanism is implementable. Moreover, as

we can conclude that the new mechanism is strictly more profitable than the original one. Thus the original mechanism could not have been optimal, yielding a contradiction. The case where $t _ { l } = 0$ can be proven in a similar way—we omit here the details. ∎

PROOF OF PROPOSITION 1:

Let

denote an aggressive investor’s utility from holding security $R _ { h }$ , and let

denote the cost to the issuer of providing such a security (in terms of forgone cash flow from the asset). Suppose that the debt contract $R _ { h } ^ { * }$ defined in the statement of Proposition 1 is not optimal. Then there exists another feasible mechanism $\left( \tilde { R } _ { h } , \right.$ $t _ { h } )$ such that $V \big ( \tilde { R } _ { h } \big ) \ = \ V \big ( R _ { h } ^ { * } \big ) \ = \ c / f _ { h }$ so that (IR-h) binds and such that $C \big ( \tilde { R } _ { h } \big )$ $< \ C \left( R _ { h } ^ { * } \right)$ .  It is clear that ${ \tilde { R } } _ { h }$ cannot be another debt contract. Then by Theorem 1 there exists a debt contract $R _ { h } ^ { D }$ with cutoff $x ^ { * * }$ such that $R _ { h } ^ { D }$ second-order stochastically dominates any security $R _ { h }$ having the same provision cost $C ( R _ { h } ) \ = \ C \big ( \tilde { R } _ { h } \big )$ . Since the investor is risk averse, we obtain that

where the equality follows from the construction of $( \tilde { R } _ { h } )$ . The above inequality, together with the observation that both $R _ { h } ^ { * }$ and $R _ { h } ^ { D }$ are debt contracts, implies that the

interest rate offered in $R _ { h } ^ { D }$ must be higher than the one offered by $R _ { h } ^ { * }$ , so that $x ^ { * * } \geq$ $x ^ { * }$ . This also implies that the cost of provision is higher $C \big ( R _ { h } ^ { D } \big ) \ge \overline { { C \big ( R _ { h } ^ { * } \big ) } } > C \big ( \tilde { R } _ { h } \big )$ , yielding a contradiction to the construction of ${ \tilde { R } } _ { h }$ . ∎

PROOF OF THEOREM 2:

In order to prove Theorem  2, we first derive the optimal mechanism for the relaxed problem where we impose neither the incentive constraint for a low type, (IC-l), nor the participation constraint for a high type, (IR-h). We later check that the obtained solution to the relaxed problem indeed satisfies these omitted constraints. Formally, the relaxed problem is

subject to

The first equality is the (IR-l) constraint, the second one is (IC-h), the third line is (FC′), and the last two lines ensure feasibility. The following Lemma demonstrates that, in any solution to the relaxed problem, all aggressive, high types invest their whole budget; that is, $t _ { h } = 1$ .

LEMMA 3: Suppose that $( R _ { \theta } , t _ { \theta } ) _ { \theta \in \Theta }$ is a solution to (Problem $Q$ ). If $t _ { l } > 0$ , then $t _ { h } = 1$ .

It follows from the above lemma that in the optimal mechanism the seller raises $f _ { h }$ (in total) from the aggressive investors and still needs to raise $\begin{array} { r } { f _ { l } t _ { l } = \textit { c } - f _ { h } } \end{array}$ from the conservative investors. That is, $t _ { l } = ( c - f _ { h } ) / f _ { l }$ . Given this insight, the relaxed problem $Q$ can be further simplified into

subject to

Below, we outline the solution to (Problem $\boldsymbol { Q } ^ { \prime }$ ), which also serves as a solution to the designer’s original problem.

PROOF 1OF LEMMA 3:

Suppose that there exists a solution to the relaxed problem such that $t _ { l } > 0$ and $t _ { h } ~ < ~ 1 .$ . Then we can construct another menu $\left( \tilde { R } _ { \theta } ( x ) , \bar { \tilde { t } } _ { \theta } \right) _ { \theta \in \Theta }$ that transfers a share $\varepsilon$ , $0 \ < \ \varepsilon \ < \ 1$ , of the asset designed for conservative investors to the asset designed to aggressive investors; that is,

for all $x$ . Further, let

For sufficiently small $\hat { \mathsf { z } } , \tilde { t } _ { h } < 1$ and $\tilde { t } _ { l } > 0$ . It is easy to verify that the new mechanism is implementable as long as the original mechanism is. Moreover, the change in the seller’s profit is given by

Therefore, the original mechanism cannot be the solution to (Problem $Q$ ), yielding a contradiction. ∎

PROPOSITION 6: Let $x _ { l } ^ { * }$ , $x _ { h } ^ { * }$ denote the solutions to

and

respectively. The solution to (Problem $Q ^ { \prime }$ ) is given by

PROOF OF PROPOSITION 6:

For notational convenience, let

denote the slope of the offered aggregate securities, and observe that

Intuitively, $\phi ( x )$ is the share of an additional dollar of the project’s return that is allocated to investors if the current return equals $x$ . Then (Problem $\boldsymbol { Q } ^ { \prime }$ ) can be rewritten as

(Problem Q″)

subject to

We further relax the above problem by ignoring the third constraint that says that the share given to both investor types must exceed the share given to conservative investors. The problem then becomes

subject to

If the solution to the above problem satisfies all the constraints of Problem $Q ^ { \prime \prime }$ , then it is also a solution to Problem $Q ^ { \prime \prime }$ . For solving Problem $Q ^ { \prime \prime \prime }$ we proceed in two steps.

Step 1: We first keep the function $R _ { l } ^ { \prime }$ fixed. Then we need to solve

subject to

Since $g _ { h }$ is convex, the same argument as in the proof for Proposition 1 yields that the optimal average security corresponds to a debt contract and is given by $\phi ( x ) = $ ${ \mathbf { 1 } } _ { { x \leq x _ { h } ^ { * } } }$ , where $x _ { h } ^ { * }$ solves

Step 2: Next, the seller must optimally choose the function $R _ { l }$ in order to minimize $x _ { h } ^ { * }$ (i.e., relax the isoperimetric constraint as much as possible) while satisfying all the other remaining constraints. Minimizing $x _ { h } ^ { * }$ is equivalent to the minimization problem

under the same constraints. Since $\left( 1 - c \right) f _ { h } / f _ { l }$ is a constant, the issuer’s problem in this second step reduces to

subject to

Recall that $g _ { l }$ is a convex transformation of $g _ { h }$ . By assumption, there exists an increasing and convex function $k$ such that $g _ { l } ( z ) \ : = \ : k \bigl ( g _ { h } ( z ) \bigr )$ . It must be the case that $k ( 0 ) = 0$ and $k ( 1 ) = 1$ .

Consider a new, artificial asset whose return is governed by the distribution ${ \tilde { H } } : X [ 0 , 1 ]$ defined by

The above problem can then be rewritten as

subject to

We can then apply the same argument as in Step 1 and obtain that the solution to the above problem is $\bigl ( R _ { l } ^ { * } \bigr ) ^ { \prime } ( x ) \ = \frac { 1 } { f _ { l } } { \bf 1 } _ { { x \leq x _ { l } ^ { * } } }$ , where $x _ { l } ^ { * }$ solves

It directly follows that

where $x _ { h } ^ { * }$ solves

which is equivalent to

as desired. In addition, we obtain

To conclude, we have obtained that the optimal menu of securities for (Problem $Q ^ { \prime \prime \prime } )$ is given by

In order to verify that the above menu is also a solution to (Problem $\boldsymbol { Q } ^ { \prime }$ ), we still need to check that the ignored constraint

also holds. For the last inequality to hold, it suffices to verify that $x _ { l } ^ { * } \ \leq \ x _ { h } ^ { * } .$ , and this follows from equation (A1):

PROOF OF THEOREM 2:

In order to prove that $R _ { l } ^ { * }$ and $R _ { h } ^ { * }$ as described in Proposition 6 are the optimal securities, we still need to show that the omitted constraints, namely (IR-h) and (IC-l), are also satisfied. The fact that (IR-h) holds follows directly from (IR-l) and (IC-h). (IC-l) requires that

The equivalence holds because

by the (IC-h) constraint. In order to prove (IC-l), we proceed as follows:

For any fixed $y \in X$ , consider two distributions defined on the interval [0, y] by

The ratio of the respective densities is given by

This ratio is increasing in $x$ because we assumed that $g _ { l }$ is a convex transformation of $g _ { h } .$ , which implies that $g _ { h } ( x ) / g _ { l } ( x )$ is decreasing. This further implies that $\pi _ { y h } ( x ) \ \succeq _ { L R } \ \pi _ { y l } ( x )$ , where $L R$ denotes the likelihood ratio stochastic order. It is well-known (see Theorems 1.B.1 and 1.C.1 in Shaked and Shanthikumar (2007, pp. 18, 43)) that the likelihood ratio stochastic order implies the hazard rate order and that the latter implies the usual FOSD. Hence we obtain that $\pi _ { y h } ( x ) \succeq _ { F O S D } \pi _ { y l } ( x )$ for each $y \in X$ .

Note that

is an increasing function on $\left[ 0 , x _ { h } ^ { * } \right]$ . Applying the above observation about stochastic dominance to $y = x _ { h } ^ { * }$ and recalling that the expectation of an increasing function increases under an FOSD shift, we obtain that

As $g _ { h } \big ( 1 - H ( x ) \big ) \ge g _ { l } \big ( 1 - H ( x ) \big )$ for all $x$ , we have

Together with $R _ { h } ^ { \prime } ( x ) - R _ { l } ^ { \prime } ( x ) = 0 \mathrm { f o r } x \ge x _ { h } ^ { * } .$ the two inequalities above together imply that

as desired. ∎

In order to prove Proposition 2, we first need a Lemma. Consider two assets, $x$ and $y$ , with distributions of returns $H _ { x }$ and $H _ { y }$ , respectively, such that either $x$ FOSD $y$ or $x { \mathrm { ~ S O S D ~ } } y$ .

PROOF OF LEMMA 4:

Define

$H _ { x } ^ { D } \left( H _ { y } ^ { D } ; \mathbf { \bar { z } } \right)$ , respectively) describes the distribution of a debt contract that is backed by asset $x$ ( $y$ , respectively) with cutoff $\hat { x }$ $( y ^ { * }$ , respectively). The expected values of these two debt contracts are by construction the same:

By the assumption that asset $x$ SOSD asset $y$ and by the definition of $\hat { x } .$ , we know that $\hat { x } \le y ^ { * }$ . Further, for any $s \in \mathsf { \Gamma } ( \hat { x } , y ^ { * } )$ , it holds that

For any $s \ < \ \hat { x } .$ , it holds that

which directly follows from the assumption that $x { \mathrm { ~ S O S D ~ } } y$

We can then conclude that $H _ { x } ^ { D }$ SOSD $H _ { y } ^ { D }$ . As investors are risk averse, we obtain

as desired. ∎

PROOF OF PROPOSITION 2:

We give the proof for SOSD. The proof for FOSD is similar and we omit the details.

Consider two assets, $x$ and $y$ , with distributions of returns $H _ { x }$ and $H _ { y }$ , respectively, such that either $x \mathrm { F O S D } y$ or $x$ SOSD $y$ . Let $x _ { l } ^ { * }$ , $y _ { l } ^ { * }$ , $x _ { h } ^ { * }$ , and $y _ { h } ^ { * }$ denote the solutions to

and

Then $x _ { l } ^ { * } \ \leq \ y _ { l } ^ { * }$ directly follows from the definition of SOSD and from the agents’ risk aversion. The rest of proof consists of three steps.

Step 1: Let $\tilde { x }$ denote the solution to

It follows from Lemma 4 that

Step 2: We next show that

This follows from

and from the assumption that $g _ { l }$ is more convex than $g _ { h }$ (i.e., because type $l .$ , the conservative type, is more risk averse than type $h$ , the aggressive type).

Step 3: Steps 1 and 2 imply together that

which further implies

Recall that $x _ { h } ^ { * }$ and $y _ { h } ^ { * }$ solve

It follows that $x _ { h } ^ { * } \leq \tilde { x }$ , and thus that

which proves Part 2 of Proposition 2. It also follows that

which proves Part 3 of Proposition 2.

The only remaining task is to show that under an FOSD r $\mathrm { i s k } x _ { h } ^ { * } - x _ { l } ^ { * } \le y _ { h } ^ { * } - y _ { l } ^ { * }$ Suppose not: Then

Since $x _ { l } ^ { * } < y _ { l } ^ { * }$ and $g _ { h }$ is nondecreasing, we obtain

where the last inequality follows from FOSD. This leads to a contradiction since we proved above that

We therefore conclude that $x _ { h } ^ { * } - x _ { l } ^ { * } \le y _ { h } ^ { * } - y _ { l } ^ { * }$ if $x$ FOSD y. ∎

PROOF OF PROPOSITION 3:

Take any $c$ , let $x _ { l } ^ { * } ( c )$ denote the cutoff point of the senior debt, which solves (IR-l):

Taking the derivative with respect to $c$ twice in the above expression yields

The second equation yields $x _ { l } ^ { \prime \prime } ( c ) ~ \geq ~ 0$ . Moreover, clearly $x _ { l } ( 0 ) = 0$ . Hence the corresponding interest rate, $x _ { l } ( \dot { c } ) / ( c - f _ { h } ) - 1$ is increasing in the financing need $c$ .

Similarly, $x _ { h } ( c )$ , which denotes the cutoff point for the junior debt, is given by (IC-h):

As $c$ increases, $x _ { l } ( c )$ increases. Moreover, $g _ { h } \big ( 1 - H ( t ) \big ) - g _ { l } \big ( 1 - H ( t ) \big ) \geq 0 .$ As $\begin{array} { r } { \int _ { x _ { l } ( c ) } ^ { x _ { h } ( c ) } g _ { h } \big ( 1 - H _ { x } ( z ) \big ) d z } \end{array}$ the equation increases, the left-hand side of the equation, must increase as well.

Finally, we want to prove that $x _ { h } ( c ) \ : - \ : x _ { l } ( c )$ also increases as $c$ increases. Suppose that this is not the case. Then there must exist $c _ { 1 } > c _ { 2 }$ such that $x _ { h } \big ( c _ { 1 } \big ) - x _ { l } \big ( c _ { 1 } \big ) \ <$ $x _ { h } \big ( c _ { 2 } \big ) - x _ { l } \big ( c _ { 2 } \big ) .$ . Let $\Delta = x _ { h } ( c _ { 2 } ) - x _ { l } ( c _ { 2 } )$ . Since $g _ { h } \big ( 1 - H ( t ) \big )$ is decreasing in $t$ and because $x _ { l } ( c _ { 1 } ) \ > \ x _ { l } ( c _ { 2 } )$ , we have

which contradicts the result obtained above. Therefore, it must hold that $x _ { h } ( c ) ~ -$ $x _ { l } ( c )$ increases as $c$ increases. ∎

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