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Security design and capital structure of business groups
Accepted Manuscript Title: Security Design and Capital Structure of Business Groups Author: Alexandre Messa PII: DOI: Reference:
S1062-9769(15)00036-8 http://dx.doi.org/doi:10.1016/j.qref.2015.03.005 QUAECO 846
To appear in:
The
Received date: Revised date: Accepted date:
24-7-2014 27-1-2015 11-3-2015
Quarterly
Review
of
Economics
and
Finance
Please cite this article as: Messa, A.,Security Design and Capital Structure of Business Groups, Quarterly Review of Economics and Finance (2015), http://dx.doi.org/10.1016/j.qref.2015.03.005 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.
“Business Groups and Risk Sharing Among Firms” highlights: This paper investigates the optimal contract between a principal and an agent that manages a business group and diverts funds among its projects.
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A pyramidal ownership structure may arise endogenously. The paper provides explanations for the cross-holding of equity between firms in business groups and the contagion between the asset prices of such firms.
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It is shown that a tax on intercorporate dividends may render the organization of
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such groups infeasible and lead to the creation of conglomerates.
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Alexandre Messa1
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January 2015
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Instituto de Pesquisa Econômica Aplicada
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Security Design and Capital Structure of Business Groups
Abstract
This paper investigates the optimal contract between a principal and an agent that
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manages a business group and diverts funds among its projects. The optimal contract can be implemented by limited liability financial securities and results in a capital
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structure that provides risk sharing among the group firms. The paper provides
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explanations for the cross-holding of equity between firms in business groups, the contagion between the asset prices of such firms, and shows that a tax on intercorporate
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dividends may render the organization of such groups infeasible and lead to the creation of conglomerates.
JEL classification: D82, G30, G32, L20. Keywords: business groups; corporate governance; capital structure.
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E-mail: [email protected]. Telephone/fax number: 55-61-3315-5189/3315-5341.
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1
Introduction In several countries, firms with a diffuse investor basis and professional
management are more of an exception than the rule. In fact, in emerging countries and
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in parts of Europe, it is much more common to find firms inserted in business groups controlled by a family or the state. In a stylized setting, this control is usually exerted
which, in turn, controls
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through a pyramidal structure: a certain family controls firm
firm , making the latter indirectly controlled by the family in question.
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Firms in such groups are often managed by individuals designated to make decisions according to the controlling family interests, or even by own members of this family.2 Thus, firms belonging to business groups, as those with a diffuse investor basis,
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feature an agency conflict triggered by the divergence of interests between the manager who maximizes his utility (or, in the case of business groups, the utility of the
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controlling family) and the stockholders and creditors interested in the maximization of their securities value.
However, the ownership chain of a business group introduces an additional
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dimension to the agency conflict. This arises from the possibility that the manager of a
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certain firm will make decisions that benefit another firm of the same group, to the detriment of the one managed by him. For instance, Cheung, Qi, Rau and Stouraitis
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(2009), investigating Hong Kong publicly listed firms, show that a common practice among them is to buy and sell assets to their controlling shareholders at, respectively, above and below market prices. Also, Berkman, Cole and Fu (2009) show that another common practice among Chinese business groups concerns the issuance of a guarantee, given to a third party, by a firm for a loan taken by another firm of the same group. Although these transactions take place within the group, the stockholders and
creditors of each firm might be different. Therefore, these relationships involve wealth transfers from the stockholders and creditors of a firm to those of another one. In the corporate finance literature, these transfers of value across firms of the same business group, to the detriment of minor stockholders of one of the firms, are often referred to as tunneling.3
2 3
See, for instance, Weindenbaum (1996) for a description of this practice in Chinese business groups. For a relevant exposition on this concept, see Johnson, La Porta, Lopez-de-Silanes and Shleifer (2000).
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In order to assess the incentives behind business groups and potential frictions that may give rise to pyramidal structures, this paper introduces a principal-agent model in which the agent manages the projects of a business group. Initially, it’s assumed a firm managed by an agent with limited liability, under a contract with an investor, the principal, with unlimited liability. Afterwards, the agent is endowed with a new project.
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To implement it, he4 needs to renegotiate the previous contract with the principal.
The agent then proceeds to manage two projects, the parent (the old project) and
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the subsidiary (the new one), each one of them has its own cash reserves5. As functions of the history of the projects cash flows, the contract between the parties specifies the
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principal’s remuneration and investment in the projects. In turn, the agent has discretionary power to establish his own remuneration and observable transfers between the projects’ cash reserves6. Each project is liquidated once it is unable to make the
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payments to the principal established by the contract7.
However, the profits of both projects are random, and the principal is neither
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capable of observing their realizations nor their use by the agent. Consequently, the agent can transfer cash across projects, so as to maximize his own utility, without the principal being able to identify such practice.
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It is assumed that the agent is not able to divert cash to himself from the
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subsidiary. This assumption creates an asymmetry between the parent and the subsidiary, once the agent is compensated only through funds from the parent (by
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means of an equity share in the original firm)8. Therefore, once the agent can still distribute himself dividends if the subsidiary has been liquidated, but not if the parent did, this asymmetry gives the agent a direct incentive to transfer funds from the subsidiary to the parent. On the other hand, there is also an indirect incentive for the agent to transfer funds to the subsidiary, in case the latter is going through financial distress, in order to preserve the option of drawing funds from it later, on more 4 Throughout this paper, masculine pronouns refer to the agent, whereas feminine pronouns apply to the principal. 5 The assumption of the existence of cash reserves is made without loss of generality. Indeed, the optimal contract could establish no accumulation of them. 6 The assumption of separate cash reserves between the parent and the subsidiary is also made without loss of generality. To see this, note that the principal could determine a payment to herself of one monetary unit by one of the projects, and, at the same time, invest this unit in the other project, establishing, therefore, a simple transfer of cash between them. That being said, one should note that it is a common practice among firms to maintain separate budget accounts and controls for different projects. 7 This condition is, again, without loss of generality, once that the contract can always establish an investment, to be made by the principal in the project, equivalent to the payment to the principal. 8 This would be the case, for example, in which, for legal or regulatory reasons, the subsidiary was subject to tighter auditing procedures than the parent.
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favorable states of nature for the subsidiary’s activities. 9 This asymmetry between the projects precludes the first-best outcome from being attained10. By dynamic programming, the agent’s problem is reduced to a system of partial differential equations that allows for the identification of the strategy to be adopted by
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him. This strategy imposes the constraints that the principal must consider in her maximization problem, which is then solved using the same procedure applied in the agent’s one.
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Thereafter, the paper demonstrates that the optimal contract can be implemented
by means of limited liability securities and, under some circumstances, the creation of
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two firms. In this way, a pyramidal ownership structure may arise endogenously, when the unobservable cash transfers between the projects would involve a deadweight cost11.
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On the other hand, when there is no deadweight cost incurred by these cash transfers, the only feasible alternative is the organization of the two projects as two divisions of a single firm.
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To understand the intuition behind this condition, suppose that the subsidiary is a supplier of the parent. In this case, the agent can, for instance, transfer cash from the subsidiary to the parent through purchases, by the latter, at a price below the production
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cost for the former. As the subsidiary is a supplier of the parent, this transference may
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involve no deadweight cost. On the other hand, if the parent’s and subsidiary’s economic activities are very different from each other, this kind of trade between them
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may then lead to deadweight costs.
Therefore, as an implication of the present model, when the projects have similar
economic activities or are on the same value chain, they tend to be organized as two divisions of a single firm. Conversely, when the projects’ economic activities are very different from each other, they may be organized as two distinct firms in a business group.
The issuance of common stocks by each firm is performed allocating a given
fraction to the other one, and granting the agent full discretionary power over the 9 As a result of his maximization problem, the agent will be interested, under some states of nature, to divert cash to the subsidiary in order to smooth the effects of shocks to the parent’s activities. Indeed, as a consequence of Lemma 2 (b), even though the agent is risk-neutral, he will behave risk-averse whenever the parent firm holds low levels of cash reserves – or, in other words, whenever the probability of its liquidation is high enough. That risk aversion then induces the agent to tend to smooth the effects of shocks and, to that end, make use of the subsidiary's cash reserve. 10 The first-best outcome would be the management of the cash reserves as if the projects in question constituted a single one, turning the present problem into the one analyzed, for example, in Radner and Shepp (1996). 11 Using the notation to be introduced in subsection 2.1, when .
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dividend policies of both firms. Thus, by means of the dividends to be distributed, he will be able to determine the resource allocation across them – and in such a way that the cash transfers between the firms do not result in deadweight costs. In turn, the common stocks issued by the parent firm to be retained by the
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subsidiary must be from a distinct class of those shares held by the agent (also issued by the parent). This distinction is made so as to allow the dividend policies of each of these classes to differ between them. So, as the agent has discretionary power over the
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dividend policies of both stock classes, he can both allocate additional funds to the
subsidiary and determine his own remuneration; however, since those stocks belong to
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different classes, these two flows are determined independently.
The dividends of the stocks issued by the parent and held by the agent will be
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distributed only after both firms succeed in reducing their respective bankruptcy risks to satisfactory levels. This result somehow leads to a similar conclusion to that drawn by Biais, Mariotti, Plantin and Rochet (2007) for a firm with a diffuse investor basis.
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Nevertheless, the result shown herein differ in that the dividend policy will take into account the bankruptcy risk not only of the firm in question (in this case, the parent), but also of the other firm in the group (in this case, the subsidiary).
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On the other hand, the subsidiary’s dividend policy assumes distinct
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characteristics. In this case, the dividends can be distributed at any time, depending on the state of nature: when this is relatively more favorable to the business activities of the
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subsidiary than to those of the parent, dividends are distributed in order to make up the parent’s cash reserves. The dividend policy regarding the class of parent’s stocks held by the subsidiary features an analogous behavior. Therefore, risk sharing among the group firms occurs through two mechanisms:
cross-holding of equity between them – i.e., the holding of the parent’s equity by the subsidiary, and vice versa –, and distinct dividend policies across the firms. In fact, cross-holding allows the agent, by means of the distribution of dividends, to allocate resources across the firms of the group. In turn, distinct dividend policies allow these distributions to smooth the effects of shocks and the consequent risk sharing among the firms. In this way, this paper proposes an alternative explanation for the practice of cross-holding of equity between firms. This phenomenon is often seen by the literature as a mechanism of separation of ownership and control, differing from the strict pyramidal structure as the voting rights for control purposes are distributed across 6 Page 6 of 55
several firms, instead of being concentrated in a single holding company.12 The present paper, however, highlights the role of this cross-holding in the risk sharing among firms, enabling the resource allocation between them and the consequent smoothing of shocks.
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In addition, ever since La Porta, Lopez-de-Silanes, Shleifer and Vishny (2000) and Faccio, Lang and Young (2001), the dividend policy of business groups has been
regarded as a tool for the reduction of funds available to the controllers, thereby
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restricting their capacity to expropriate the wealth of investors. The present paper
introduces an additional function to the dividend policy, which consists in the allocation
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of losses and gains across firms within the group. As a result, the distribution of dividends by a certain firm would be subjected not exactly to its own performance, but
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more precisely to the difference between its performance and those of the other firms in the group.
This paper also shows that the market value of securities issued by a given firm
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are sensitive to shocks on the activity of another firm in the group, what helps explain the contagion between asset prices on different markets. According to the findings of this paper, this contagion would occur due to the risk sharing among the group firms, would directly affect the business
. In this regard, Johnson and Soenen (2002), investigating some
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activities of firm
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and not necessarily because the shock to firm
countries with close economic ties to Japan, show a positive correlation between foreign
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direct investment flows received from that country, and comovement between the stock markets of these economies and that of Japan. Finally, it is demonstrated that the tax on intercorporate dividends may render
the pyramidal structure infeasible, and makes the organization of the two projects as two divisions of a single firm, creating a conglomerate, the only feasible alternative. The adoption of this kind of tax by the United States in the 1930s helps explain the low frequency of these groups in that country, although they were quite common there until then. This idea is presented by Morck (2005) and is corroborated by the findings of this paper. This work is related to the literature on financial contracts in a continuous-time setting. In this literature, Holmstrom and Milgrom (1987) is a seminal work, while
DeMarzo and Sannikov (2006), Biais, Mariotti, Plantin and Rochet (2007), Cadenillas, 12
See, for instance, Bebchuk, Kraakman and Triantis (2000).
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Cvitanic and Zapatero (2007), Sannikov (2008), Biais, Mariotti, Rochet and Villeneuve (2010) and Williams (2011), among others, may be cited as recent contributions. A line of research in this literature investigates the capital structure and security design as a result of the optimal contract between an agent and a principal. DeMarzo and Sannikov
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(2006) and Biais, Mariotti, Plantin and Rochet (2007) are relevant works in this line. The present paper uses this perspective as a starting point, but considering a group of projects and investigating how the capital structure and the financial securities issued
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and purchased by the group are used for risk sharing.
Moreover, this paper is part of an effort to understand the ownership structure of
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business groups. This is usually perceived as a mechanism of separation of control from cash flow rights – see Morck, Wolfenzon and Yeung (2005) and Khanna and Yafeh
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(2007) for relevant reviews of this literature. Almeida and Wolfenzon (2006) move away from this perspective, showing that a pyramidal structure may arise for other reasons than the separation of ownership and control – in their case, due to financing
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advantages of pyramidal structures that arise as a result of certain frictions in the capital market. In the present paper, conversely, the ownership structure of the business group arises as a risk sharing mechanism between firms, providing the proper incentives to the
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manager to allocate the gains and losses in a predictable way.
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This paper is organized into four sections. Following this introduction, Section 2 introduces the principal-agent model in order to investigate the incentives related to the
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management of the business group, the constraints necessary to the alignment of interests between the parties, and the resulting optimal contract. Section 3 analyzes the consequent capital structure and the price dynamics of the issued securities. Finally, Section 4 concludes.
2
The Principal-Agent Problem and the Optimal Contract This paper assumes a firm managed by an agent with limited liability under a
contract with a principal with unlimited liability, responsible for the necessary financing. Admittedly, both are risk-neutral, and the contract between them provides the agent with a share of the firm’s equity, and requires the firm to make certain payments to the principal at each instant, in what comprises the remuneration of its debt. Thus, the agent’s compensation consists of the percentage of the dividends distributed by the firm he is entitled to. 8 Page 8 of 55
As this firm is operating normally, the agent receives another cash flow generating project. To implement it requires an irreversible investment equal to
.
This investment can, at least partially, be made by the said firm. If that happens, it represents a deviation from its activities and, consequently, the previous contract must
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be renegotiated with the principal. However, even if the implementation of the project does not involve any financial supply by the firm, the activities will imply a change in
the incentives to which the agent will be subjected. Therefore, anyway, the previous
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contract must be renegotiated between the parties before implementing the project. This renegotiation can include both changes in the terms of the original contract and
invest in each of the projects at any time she desires.
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additional financial supply by the principal. Also, using her wealth, the principal may
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The agent then proceeds to manage two projects, the parent and the subsidiary. The parent’s realized profit can initially have four allocations: the distribution of dividends, the principal’s remuneration13, cash transfers to the subsidiary, or the
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accumulation of the project’s cash reserves.
In turn, the subsidiary’s realized profit may be allocated as follows: cash
project’s cash reserves.
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transfers to the parent, the principal’s remuneration14, and the accumulation of the
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The contract between the parties specifies the principal’s remuneration as a function of the cash flows history. In turn, the agent has discretionary power to establish
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his own remuneration and cash transfers between projects. It is taken as given that each project is liquidated once it is unable to make the payments to the principal established by the contract.
Supposedly, the principal knows the probability distribution of profits and
observes the cash reserves of both projects. However, she does not observe the realized profit of any of them, allowing the agent to divert some of the profit from one of the projects to the cash reserves of the other project, without the principal being able to identify such practice. Thus, there are two types of transfers of funds between projects: those that are observed by the principal (the parent’s both remuneration and additional investments made in the subsidiary) and those that are not observed by her. Furthermore, given the 13
This remuneration can be either positive or negative. If it is negative, it represents additional investments in the project, to be made by the principal. 14 Which can also be either positive or negative.
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difficulty faced by the principal in monitoring the funds of both projects, the agent is assumed to have discretionary power to determine the distribution of dividends of the parent (and, therefore, his own remuneration) and the observable payments across projects.
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Note that, since the agent is remunerated by the parent (through a percentage of the dividends distributed to shareholders) and he has discretionary power to allocate resources between projects, he will always prevent the parent from being liquidated
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before the subsidiary. Hence, it is assumed, during the entire term of the contract between the parties, that either the subsidiary is liquidated before or that both are
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liquidated simultaneously.
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Remark 1: The assumption that the agent is not paid by the subsidiary is an implicit consequence of the supposition that he is not able to divert funds from it. Alternatively, it could be admitted that the agent is able to make such diversions. However, this would
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simply reduce the problem to a case in which the agent manages a single project15, and is able to divert cash from it – an agency conflict analyzed, for example, in DeMarzo and Sannikov (2006) and Biais, Mariotti, Plantin and Rochet (2007).
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Differently, the assumption that the agent is not able to divert resources to
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himself from the subsidiary introduces a distinct agency problem, as a consequence of the resulting asymmetry between the projects. This would be the case, for example, in
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which, for legal reasons, the subsidiary was subject to tighter auditing procedures than the parent.
Considering any instant , suppose the cash reserves of both projects are given.
The scenario described above can be summarized in the following timeline:
(T1) The agent observes the realized profits on the interval
.
(T2) Given the cash reserves at , the contract determines the remuneration the principal will receive from each one of the projects on the interval
.
15
Using the notation that will be introduced in the next subsection, the operating profit of this project, on the interval , would be given by .
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(T3) Given the information at (T1) and (T2), the agent decides on his own remuneration, the payments between the projects and the amount to be diverted across them on the interval
.
(T4) The variations in cash reserves on the interval
are determined
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residually by the difference between the profits observed at (T1) and the cash flows .
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stipulated at (T2) and (T3). These variations will determine the cash reserves at
As a result of the above succession of events, the principal, when faced with , is not able to know whether
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certain levels of cash reserves of both projects at
these levels result from the allocation of resources by the agent at (T3) or from the profits realized at (T1). Her problem, therefore, consists in providing the agent with the
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appropriate incentives so that he can determine a dividend policy for the parent and
2.1
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transfers of funds between the projects according to her interests.
Formalization of the Problem In what follows, let
for the terms referring to the parent, and
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subsidiary. The operating profit of each one of them on the interval
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by
in which
and
are positive constants, and
probability space
for the is given
(1)
is a Brownian motion on a filtered
satisfying the usual conditions.16 Nevertheless, it
is assumed that only the agent observes
(and, consequently, the realization of
).
On each interval
profit,
, the parent has two cash flow sources: its operating
, and the payments received from the subsidiary,
.17 This cash flow
16
By usual conditions, it is meant a right-continuous filtration, and containing all sets with zero probability. 17 In what follows, any flow , represented with the right-to-left arrow, refers to resources of the , refer to flows from the parent to subsidiary allocated to the parent; those with the left-to-right arrow, the subsidiary. Thus, one should not mistake the notation introduced here for that of the vector calculus, where represents a vector. In the present paper, and will always be scalars, and the arrow serves only to help visualize the direction of cash flows within the group.
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;18 the principal’s
can be allocated as follows: the payment of dividends, remuneration,
;19 the payments made to the subsidiary,
; or the
. Thus, one has the identity
accumulation of cash reserves,
In turn, the subsidiary has the following cash flow sources: its operating profit,
,
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(2)
; or
and the investments made by the parent,
; the principal’s remuneration,
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allocations: the parent’s remuneration,
. This cash flow can initially have three
the accumulation of cash reserves,
an
. Hence, one obtains the identity
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(3)
can take on any real values. When
By definition, the flows
will be remunerated by project ; when
, she will provide additional financial
d
supply to the said project.
, the principal
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However, since the principal does not observe the realization of operating
be the amount of
unobservable payments, on the interval
, from the parent, to make up the cash
profits, the agent is able to divert some part of these profits to the cash reserves of the
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other project without the principal being aware of it. Let
reserves of the subsidiary, and
as the unobservable amount diverted in the
opposite direction. It is admitted that, of these unobservable payments, the project of each monetary unit. Thus, the share of the amount
receives a fraction
diverted from the subsidiary actually received by the parent is equal to
, while the
share of the amount diverted from the parent and received by the subsidiary is given by . In turn,
and
represent the deadweight costs incurred by
the respective diversions.
18
Let be the parent’s equity share held by the agent. Then, from the total of dividends paid a part equivalent to will go to the agent, and the other part , to the principal. 19 Besides the part of the dividends she is entitled to (given by ).
,
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Therefore, the parent obtains two payment flows from the subsidiary,
and
, on each interval, the first of which is observed by the principal, while the second one is unobservable by her. Likewise, the payments made by the parent to the and
, the first of which is observed by
the principal, and the second one is not.
2.1.1 The contract between the parties
(4)
(5)
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In this way, the identities in (2) and (3) can be rewritten as
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subsidiary may be split into two parts,
One assumes that the agent has discretionary power to establish the parent’s
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dividend policy – and, therefore, his own remuneration –, the parent’s remuneration process and the additional financial supplies made by the parent to the subsidiary. Thus, ,
, is only suggested by the contract, and it is up to the
d
the process
agent to accept or not this suggestion.
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It is also admitted that, to match the agent’s actions with the principal’s interests, – this assumption is made without
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the principal may use a pair of amplifiers
. The pair determines that: (i)
loss of generality, as it is possible to have
for each monetary unit the parent supplies to the subsidiary (by way of principal will receive
the parent (by way of
), the
units; (ii) for each monetary unit received as remuneration by ), the principal will receive
units. The purpose of these
parameters is to change the impact of cash flows between projects on their respective cash reserves, altering the incentives to which the agent will be subjected. and
Given the processes one process
, for any
that satisfies, for . Hence,
and
there exists only ,
and
are payments received by the principal
directly from the parent and from the subsidiary, respectively. Then, letting the parent’s equity share held by the principal,
be
the contract establishes 13 Page 13 of 55
as
the
principal’s instantaneous remuneration. The subsidiary project is established by a contract contract) stipulating the pair ,
and
.20 Regarding the
, the agent can divert them from the path suggested by
and
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processes
and the process
(replacing the original
choose a strategy that maximizes his expected utility throughout the contract period. The principal’s remuneration
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is determined exclusively by the
contract, and these payments must be made by the respective projects, regardless of
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their cash reserves. When these are insufficient for that, the said project is liquidated. As argued earlier, as the agent is remunerated by the parent and he can allocate resources
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between the projects, the parent shall not be liquidated before the subsidiary. Hence, the contract is assumed to be terminated when the parent’s cash reserves are insufficient to meet its obligations. The term of the contract
, in which
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.21
is then given by interval
The principal’s problem consists in determining the incentive-compatible
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contract that maximizes her expected utility. By incentive-compatible contract, it is meant any contract under which the strategy adopted by the agent is actually the one
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suggested by the contract.
To find this optimal contract, it is necessary to obtain the constraints that must
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be included so that the agent’s actions correspond to those required by the principal. With that purpose, the following section will analyze the agent’s maximization problem under a given contract
2.2
.
The agent’s problem Let
and
be the discount rates, respectively, of the agent and of the principal.
Assuming that the agent is more impatient than the principal, one has
. The
following analysis admits, without any damage to the conclusions reached, that any income flow received by the agent is immediately consumed by him. 22 20
Note that, given the terms related to , the processes and are determined residually, according to identities (4) and (5). 21 Equivalently, the subsidiary is liquidated at the instant , in which . 22 Alternatively, suppose that the agent can consume some of this income flow and save up the rest. However, supposing that these savings grow at a rate , any contract that implies positive savings by the agent at some will be weakly dominated by another one in which the principal saves in his place and
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Just for this subsection, make the following assumption about the process :
Assumption 1: For
(c) for
;
; such that
.
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,
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23
(b)
: (a)
Remark 2: The assumption above assumes a Markovian structure for
, which
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significantly simplifies the agent’s problem. Anyway, in the context of the principal’s problem, Remark 3 further ahead will discuss the appropriateness of this assumption, to .
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the detriment of a history-dependent process
In this way, besides the Markovian structure mentioned in the remark above, depends on time solely by means of the
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Assumption 1(a) assumes that the process
levels of cash reserves at each instant, but not explicitly on time itself. This assumption is made in order to guarantee the stationarity of the present problem, given its infinite
d
time horizon.
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Assumption 1(b) is intended to guarantee that the stochastic differential equations in (6) and (7) have a unique solution. Finally, Assumption 1(c) will be used in
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the proof of Lemma 2. Intuitively, it guarantees, by means of Dynkin’s formula, that for
Let
.
. Inserting (1) into (4) and (5), one sees that the cash reserves
of the parent and of the subsidiary will follow the respective processes
(6)
(7)
makes payments to the agent, through the parent, at the instants he would consume his savings. This result is formally established in DeMarzo and Sannikov (2006) and in DeMarzo and Fishman (2007). 23 represents the set of all functions times continually differentiable in .
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in which were used the definitions
and .
Define
as the set of feasible strategies available to the agent, i.e., the set of all 24
non-decreasing process
adapted to
, with values in
, and
25
Then, by letting
(8)
us
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with bounded variation. The agent’s criterion functional can be defined as
, the value function
of the present problem
an
is such that:
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(9)
Since the contract is terminated when the parent runs out of cash reserves, the function for
.
d
must satisfy
Lemma 1 below implies that the subsequent analysis can be restricted to
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26
contracts in which
. Thus, when subjected to these contracts, the agent’s
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optimal strategy will not include any unobservable (by the principal) allocation of resources between the projects. Its proof, as well as those of the other lemmas and propositions, is shown in the Appendix.
Lemma 1: (a) Let
be the set of optimal contracts, i.e., those incentive-compatible that
maximize the principal’s expected utility. Suppose that this set is non-empty. Then, there is a contract
24
under which
For any process ,
. Since, for
.
,
,
,
,
and
take on
nonnegative values, the processes , , , and shall be nondecreasing. 25 According to the notation used in the present paper, whenever a certain variable is denoted by an uppercase letter, , it will refer to the process in question. When the variable is denoted by a lower-case letter, , it represents a value assumed by the process at a specific, and clear in the context, instant (in (8), the integration parameters indicate that , i.e., ), or whenever the specific instant is not relevant. 26 Note that, being and , implies for . Obviously, the same applies to
.
16 Page 16 of 55
(b) Under an incentive-compatible contract, in order for necessary that
. Likewise, in order for
and and
, it is
, it is necessary
.
that
ip t
Item (b) of the lemma above establishes the incentive-compatibility constraints, which impose the necessary conditions so that the agent will prefer to allocate resources
cr
between the projects using processes that can be observed by the principal, to the detriment of those unobservable by her. To understand the intuition behind this result, 27
paid by the subsidiary; on the other hand, the parent
monetary unit of remuneration
for each monetary unit of
. Therefore, in order for the agent to
an
receives the fraction
for each
us
note that, on the one hand, the parent actually receives a fraction
choose to transfer resources using an observable process
, it is necessary that .
M
. The same intuition applies to the condition
As a result of Lemma 1 above, the following analysis can be restricted, without
necessary conditions for that,
. Thus, implicitly, it is assumed that the
d
any loss, to contracts that result in
and
. That is,
Ac ce p
,
te
In what follows, for any function
relative to the parent’s cash reserves ( ),
, are satisfied.
, let
,
and
represents the partial derivative of function , in relation to the subsidiary’s cash reserves
( ), and so on and so forth.
After that, define the differential operator , where
as, for any function
is a countable set of points,
(10)
in which
27
From the amount
represents the covariance between the Brownian motions
paid by the subsidiary, the parent receives the equivalent to , whereas the principal receives
and
.
.
17 Page 17 of 55
Lemma 2 below characterizes the agent’s value function. Item (a) defines the Hamilton-Jacobi-Bellman (HJB) equation to be satisfied by that function. However, since, in the present problem, the control exerts a linear influence on both the dynamics of the state trajectories and the criterion functional, one has a singular stochastic control
ip t
problem. In this case, the HJB equation involves not only a partial differential equation (represented by the first term of (11)), but actually a system of partial differential
cr
inequalities, as pointed out by (11).
Lemma 2: (a) The value function defined in (9) is a viscosity solution to the equation
.
(c)
an
;
M
is a concave function on domain
(11)
d
(b)
us
below:
To understand the intuition behind Lemma 2, first observe that the agent for each monetary unit of dividend paid by the parent. On deciding
te
receives a fraction
Ac ce p
for an additional monetary unit of cash reserve in this project, the agent incurs an opportunity cost, since this unit could alternatively be partially allocated to his own consumption.
Therefore, on the one hand,
represents the opportunity cost incurred by the
agent when he postpones his consumption (deciding on an additional monetary unit of cash reserve in the parent). On the other hand,
denotes his expected gain in terms of
future utility, obtained from that postponement. Then, by postponing his consumption, the agent’s expected gain cannot be lower than his opportunity cost. Naturally, if that were the case, the postponement would not occur and the agent would consume the parent’s cash reserves until
is satisfied, leading to the term
in (11).
Likewise, recall that, for each monetary unit of remuneration established for the parent (and, indirectly, for the principal) – in the form of actually receives a fraction
–, this project
. So, when deciding on an additional monetary 18 Page 18 of 55
unit of cash reserve in the subsidiary, the agent incurs an opportunity cost, since this unit could alternatively be allocated to the parent’s cash reserves. This opportunity cost is given by
. represents the opportunity cost
In this way, on the one hand,
ip t
incurred by the agent by postponing the parent’s remuneration (deciding on an additional monetary unit of cash reserve in the subsidiary). On the other hand,
cr
denotes his expected gain from this postponement. Hence, by postponing the parent’s remuneration, the agent’s expected gain cannot not be lower than his opportunity cost.
us
Again, if that were the case, this postponement would not occur and the agent would is satisfied, leading to the third term of
remunerate the parent until
an
(11).
Analogously, for each monetary unit of investment by the parent to the subsidiary – by way of
–, the latter receives a fraction
. When
M
deciding on an additional monetary unit of cash reserve for the parent, the agent incurs an opportunity cost, since this unit could alternatively be allocated to the subsidiary’s . Likewise, the agent’s
d
cash reserve. This opportunity cost is given by
te
expected gain from postponing this investment,
, cannot not be lower than his
, as expressed by the fourth term of (11).
opportunity cost, leading to
to represent, for a given level
Ac ce p
Therefore, define function
of
cash reserves of the subsidiary, the lowest level of the parent’s cash reserves from which the agent opts for the distribution of dividends:28
(12)
From the function above, one may obtain the set
defined below and illustrated by
Figure 1.
(13)
28
In what follows, adopt the convention that
.
19 Page 19 of 55
ip t
2
cr
1
0
us
Figure 1: The set
1
. The agent would not establish any remuneration to himself
an
whenever the cash reserves are located in this set.
Thereafter, define function
of the
M
to represent, for a given level
parent’s cash reserves, the lowest level of the subsidiary’s cash reserves from which the
Ac ce p
te
d
agent decides to remunerate the parent, i.e.,
From this function, one gets the set
defined below and illustrated by Figure 2.
(14)
20 Page 20 of 55
ip t
2
cr
2
0
us
Figure 2: The set
1
. The agent would not establish any remuneration to the parent
an
project whenever the levels of cash reserves are located in this set.
Finally, define
M
to represent, for a given level
of the subsidiary’s
cash reserves, the lowest level of the parent’s cash reserves from which the agent opts to :
Ac ce p
te
d
allocate resources to the subsidiary, under the constraint that
Let then the set
be defined as below and illustrated by Figure 3.
(15)
21 Page 21 of 55
ip t
2
cr
3
0
us
1
Figure 3: The set
. The agent would not establish any supply of funds from the
,
and
, it is possible to summarize a possible strategy to be
M
From
an
parent to the subsidiary whenever the levels of cash reserves are located in this set.
followed by the agent. First, the agent would not establish any remuneration to himself whenever the cash reserves are located in
.
, making
te
–, the agent will determine a distribution of dividends equal to these levels return to
, the
– i.e., the levels of cash reserves lie outside
d
parent’s cash reserves reach
. However, if, for a given level
Ac ce p
Likewise, the agent would not establish any remuneration to the parent
whenever the levels of cash reserves are located in
. Nonetheless, if they leave
, the
agent would establish a remuneration to the parent, making the levels of cash reserves return to
.
Finally, the agent would not establish any supply of funds from the parent to the
subsidiary whenever the levels of cash reserves are located in leave
. However, if they
, the agent will determine new financial supplies to the subsidiary, making these
levels return to
.
As a result of this policy, the cash reserves of both projects would remain in the region
29
illustrated in Figure 4. Proposition 1 below shows that this
policy actually dominates any other feasible strategy to be adopted by the agent.
29
Note that, by construction,
and, therefore,
.
22 Page 22 of 55
ip t
2
cr
4
1
. The cash reserves of both projects would remain in the region
an
Figure 4: The set
us
0
M
.
For the existence of a solution to the problem described in Proposition 1, it is still necessary to establish certain conditions necessary so that, when the process
. These necessary conditions are analogous to the ones pointed out by
te
returns to
and the agent adopts the proposed strategy, the process
d
reaches the boundary of
Remark 3.1 in Dupuis and Ishii (1993), and are described by Assumption 2 in Appendix
Ac ce p
B.
These conditions, intuitively, require that, in the first place, if the boundaries of
and
have decreasing regions, they must have, in absolute terms, a maximum slope
(i.e., they should not be excessively vertical). In the second place, in the corners formed by30
and
(points
and
in Figure 4), the set , and
extremely long horns. More formally, denote equidistant points in
from
(but with
in
, and
in
and
as
). Assumption 2 then
implies a minimum necessary distance (dependent upon ) between
30
should not have
and
.
represents the boundary of the set .
23 Page 23 of 55
Proposition 1: Let the sets
,
and
as defined by (13), (14) and (15), and the set , it is known31 that there exists a
. Given an initial state process
that results in a trajectory
(i)
, almost surely with respect to ; ,
and
are nondecreasing;
ip t
(ii)
, such that:
cr
(iii) one has
us
(16)
an
(17)
M
(18)
which
denotes
of
the
set
in
question.
Then,
.
The principal’s problem
Ac ce p
2.3
interior
te
,
the
d
in
For this problem, define the agent’s continuation value as the present value of
the promised flows of utility by the contract, from to the end of its term, i.e.,
in which
Lemma 3: For
. Lemma 3 below establishes the dynamics of the process
(19)
.
,
31
Items (i), (ii) and (iii) describe a Skorokhod problem in a domain with corners. The existence of a solution to this problem was shown by Dupuis and Ishii (1993). The problem of the statement corresponds to Case 2 of the referenced paper.
24 Page 24 of 55
(20)
,
Let
, satisfies
.
be the set of all nondecreasing stochastic process , with values in
adapted to
ip t
in which
and with bounded variation. In her maximization
cr
problem, the principal will be subject to three state variables, namely, the cash reserves of both projects and the agent’s continuation value. Then, the principal’s criterion
us
functional is given by
M
an
(21)
So, it is possible to define the value function
of the principal’s problem:
te
d
(22)
Ac ce p
Define the differential operator where
as, for any function
,
is a countable set of points,
(23)
Lemma 4 below, analogously to Lemma 2 of the agent’s problem, characterizes the principal’s value function. Again, respect to, respectively,
and
and
, and
represent the partial derivatives of
, the partial derivative relative to
with
.
Lemma 4: (a) The value function defined in (22) is a viscosity solution to the system of equations below:
25 Page 25 of 55
(24)
(c)
.
;
ip t
is a concave function on domain
cr
(b)
Figure 5 shows the implications of Lemma 4, assuming, for the sake of .
us
and
M
an
illustration only, fixed values for
te 0
Ac ce p
0
d
> −1
= −1
Figure 5: The value function , assuming fixed values for larger than
Then, define
whenever
and
, and equal to that value when
. Its slope is .
as the lowest level of the agent’s continuation value in which the
principal’s expected gain from making a direct transfer to the agent is equal to its cost, i.e.:
(25)
The following strategy summarizes the possible terms of the optimal contract. By the definition of
in (25), for every
, the reduction of one unit in the agent’s
26 Page 26 of 55
continuation value will cause
to increase by less than one. Therefore, the principal
will not establish any remuneration to the agent whenever his continuation value is . However, at
value, the increase in
, for the decrease by one unit in the agent’s continuation
will have the same magnitude. Thus, when the agent’s , the principal will determine an instantaneous
continuation value reaches
ip t
smaller than
, making that value returns immediately to
remuneration to the agent equal to
.
cr
Proposition 2 below shows that the strategy above dominates any other that
Proposition 2: Assume process
and parameters
as given. Let the set
is taken as defined in (25). Given
an
, in which an initial state
, there is a process
that leads to a trajectory
M
such that:
, almost surely with respect to ;
(i)
is nondecreasing;
Ac ce p
te
(iii) one has
d
(ii)
us
could be adopted by the principal.
Then,
Remark 3: Given
.
, the principal will be indifferent to the possible
specifications that the processes
and
may assume. This result has two
implications. Firstly, this indifference allows assuming that these processes are Markovian, as proceeded in the previous subsection (from Assumption 1). In fact, by assuming a cost associated with the verification and enforceability related to the establishment of
from a history of realizations of
, and being the principal
indifferent to that process, it is natural to imagine that she accepts to minimize that cost through a Markovian structure (in which she needs to take account of only the current
27 Page 27 of 55
value of
). Secondly, the optimal contract will not be necessarily unique, i.e., the one
derived in this paper, as resulting from that assumption, can be just one among others possible from different specifications (which not necessarily Markovian) for the
2.4
and
.
ip t
processes
The optimal contract
In order for the contract considered here to be incentive-compatible, the
cr
promised flows of utility by the contract to the agent must match those that result from
the contract value from the cash reserves (11),
,
it
is
an
and
, i.e.,
must be equal to
us
his maximization. In other words, the agent’s continuation value at
. By Ito’s lemma
known
that
. Comparing this result
with (20), it may be concluded that, given an incentive-compatible contract, one must and
.
M
have
By Proposition 2, the principal will remunerate the agent only from
.
Therefore, in an incentive-compatible contract, the cash reserves from which the .
In other words, the
curve in Figure 4 must belong to the level curve
.
Then, the terms
of the contract must be such that they satisfy the conditions:
Ac ce p
te
d
principal would be willing to remunerate the agent are those that satisfy
(C1)
;
(C2)
;
(C3)
, in which
is the function defined in (12),
in addition to Assumption 1. Conditions (C1) and (C2) result from Lemma 1. There may be several contracts that satisfy the indicated conditions. Anyway, the proposition below demonstrates the existence of at least one optimal contract for each feasible value assumed by
Proposition 3: Suppose that (C2). Given
.
takes on values that satisfy conditions (C1) and
, there exists a nonnegative function
, such that 28 Page 28 of 55
, which causes Assumption 1 to be satisfied, and implies (C3).
For the full characterization of the optimal contract, it remains to determine the and
, which will depend on the bargaining power of the
ip t
values of
parties. To match this choice with the analysis in the subsequent section, suppose that
represents the irreversible
cr
the agent has full bargaining power. As defined earlier,
investment needed for the implementation of the new project. Since
is the
us
subsidiary’s cash reserves in the initial period, the total amount necessary for the . This amount can be, at least partially,
establishment of this project is given by
, whereas the remaining
amount,
an
covered by an initial financial supply by the principal,
, is provided by the parent’s internal resources (in the present case, by its
cash reserves).32 Let
be the parent’s cash reserves at the instant immediately before the value of the original contract for
M
the establishment of the subsidiary, and
, the agent
te
solves the problem below:
and
d
the principal. Then, to determine the values of
Ac ce p
(26)
subject to:
32
Anyway,
can be negative. This will be the case when is allocated to the parent’s cash reserves.
(27)
, and the amount
29 Page 29 of 55
in which the last constraint is given by Lemma 1 and by the definition that . In first place, since in the initial instant the agent is able to allocate resources between the projects without deadweight cost, the cash reserves of each project shall be such that their respective marginal values are equivalent, i.e.,
and
, with at least one strict inequality. Then, for .
us
,
cr
Lemma 5: Let
ip t
. Consider then the lemma below:
According to the lemma above, the smaller the value of
, for the same level of
larger value implies a higher influence
an
cash reserve, the larger the value of the contract for the agent. Notwithstanding, this (by the fourth constraint in (27)) that, in turn, will
and
(second constraint). Looking at Figure 5, it can be , that larger
will yield a smaller value of
and,
consequently, a smaller
. This will have to be compensated for by smaller
M
perceived that, if
. Hence, the levels of
d
which will lead to a reduction in
and
or
,
to be chosen shall be
te
those in which the gain for the agent from a smaller value for them exactly offsets the
Ac ce p
loss caused by a smaller resulting and
. In other words, suppose that
are continuous functions. Then, the values of
shall be such that noted that
, since, if
and increase
3
or
33
and
. Also, by means of Figure 5, it can be , the agent could concomitantly reduce
.
Capital market
The optimal contract derived in the previous section can be implemented through the issuance of financial securities to be negotiated in a competitive capital market. For this purpose, the design of these securities reproduces the ownership
33
Except, evidently, in cases of corner solutions, in which the optimal values for .
or
will be zero or
30 Page 30 of 55
structures of firms resulting from the optimal contract, as well as the remuneration of the parties.
3.1
Implementing the optimal contract
ip t
Define the financial securities below:
Common stocks: securities whose instantaneous payoff is not determined beforehand,
cr
but recurrently at each instant. It is assumed that (i) the stocks issued to the public are held by a diffuse investor base, what makes the company decisions be controlled by the
us
restricted stockholders – in the present model, only the agent; and (ii) there is one or more classes of stock, and these classes may differ from one another with respect to the
an
dividends to be received;
Fixed-coupon debt securities: perpetuities that pay their holders at an instantaneous
M
fixed coupon rate.
Variable-coupon debt securities: perpetuities that pay their holders at an instantaneous
te
d
variable coupon rate.
of
For simplification, suppose that the parent is a public firm, with a fraction
Ac ce p
its common stocks held by the agent, and the remaining fraction, diffuse investor base.34 Define
, held by a
as the total payments to be made by the parent for
remuneration of its outstanding debt. Then, the optimal contract derived in the previous section can be implemented as follows:
Optimal Financing Strategy. If the group issues financial securities in the capital market in order to finance the establishment of the subsidiary firm, a financing strategy may be adopted such that:
Parent firm:
34
This assumption is made only to refrain from any possibility of control over the firm by some investor.
31 Page 31 of 55
Common Stocks: issuance of a second class of stocks, of which a fraction
is
, is held by the subsidiary.
sold to investors, and the remaining,
Fixed-coupon debt securities: issuance of perpetuities that pay an instantaneous coupon
perpetuities, each of them being
cr
Variable-coupon debt securities: purchase of
).
ip t
(or their purchase, if
equal to
remunerated by an instantaneous coupon equivalent to
.
us
,
Subsidiary firm:
, held by the controlling firm.
M
sold to investors, and the remaining fraction,
an
Common stocks: issuance of a single class of stocks, with a fraction equal to
Fixed-coupon debt securities: issuance of perpetuities that pay an instantaneous coupon .
d
equal to
te
The financing strategy above refers to the implementation of the contract identified by Proposition 3 and by the maximization in (26). Although this contract is
Ac ce p
not necessarily the unique optimal one, the financing strategy above allows drawing some conclusions that can be generalized. In this regard, the cross-ownership of firms – i.e., the holding of the parent’s equity by the subsidiary, and vice versa – is targeted at reproducing the incentives provided for by the optimal contract. Each of the firms will be remunerated by the dividends distributed by the other firm to its stockholders. Thus, through the respective dividend policies, the agent can determine the resource allocation between them. Nevertheless, as each firm holds a fraction stocks, each monetary unit received by the firm will represent
of the other firm’s units of the total
dividends to be distributed. Likewise, through the distribution of dividends by the parent, the agent can determine his own remuneration. However, if the parent’s stocks owned by him belong to a different class from those held by the subsidiary, both payment flows made by the parent (to the agent and to the subsidiary) are independent of each other.
32 Page 32 of 55
Finally, the debt securities of both firms are targeted at reproducing the flows resulting from functions payments
and
introduced in Proposition 3, after considering
to be made by the parent, due to its pre-existing debt.
Therefore, two remarks can be made about the group’s capital structure. The
ip t
first one is concerned with the different functions attributed to each security. In fact, the cross-holding of equity between the firms allows the agent to transfer resources between them. Debt issues, however, aim to provide appropriate incentives so that the agent
cr
postpones his remuneration (and, therefore, cash outflows) until both firms succeed in accumulating certain levels of cash reserve and, consequently, reducing their respective
us
bankruptcy risks to satisfactory levels.
Secondly, it shall be noted that the dividend policies differ among the firms of
an
the business group. On the one hand, the dividends of the parent’s stocks owned by the agent (and by the public) will be distributed only when both firms succeed in reducing their respective bankruptcy risks to satisfactory levels. On the other hand, the
M
subsidiary’s dividend policy behaves differently. In this case, the dividends can be distributed at any time, depending on the state of nature: when the state of nature is
d
relatively more favorable to the subsidiary’s activities than to those of the parent, the distribution of dividends is determined. The class of stocks issued by the parent and
Market price dynamics
Ac ce p
3.2
te
held by the subsidiary has a similar dividend policy.
Based on the financing strategy pointed out above, the present problem can have
three types of common stocks:
(S1) common stocks issued by the parent and held by the agent; (S2) common stocks issued by the parent and held by the subsidiary; (S3) common stocks issued by the subsidiary and held by the parent.
Let
be the market values of all securities, such that ,
and . Likewise, let
be the market
values of the parent’s and subsidiary’s net debts, respectively, such that
33 Page 33 of 55
and
. The
proposition below determines the behavior of these functions over time.35
and
, one has
ip t
Proposition 4: For
in
which
us
cr
(28)
,
an
.
and
sources of random shocks (
and
M
According to the proposition above, each market value is influenced by two ). In this way, the prices of securities of each
firm are subject not only to uncertainty over the activity of the firm in question, but also
d
to that related to the other firm in the business group. In other words, regardless of the
te
geographical or sectoral proximity between the markets of the parent and of the subsidiary, shocks to the activity of either of them shall have an impact on the prices of
Ac ce p
securities of the other firm, leading to a contagion effect between the markets.
3.3
Tax on intercorporate dividends Suppose that, in the economy in question, a tax at a rate
is levied on the flows
of dividends between the firms. Therefore, for each monetary unit of dividend distributed by the stocks (S2), the subsidiary actually receives a fraction . Likewise, for each monetary unit of dividend distributed by the . The last
stocks (S3), the parent actually receives the fraction constraint in (27) must be rewritten as follows:
35
Again, applies to
and and
represent the partial derivatives of .
relative to
and
, respectively. The same
34 Page 34 of 55
By the constraint above, note that, if
, the only feasible value for . Hence, if
the parameters in question is
, the new project
must necessarily be financed only by debt issues, without new issuances of common
ip t
stocks for investors. In this case, the parent and the subsidiary will constitute two divisions of a single firm, forming a conglomerate.
cr
It may then be concluded that, if the tax rate is high enough, it renders the
organization of the business group unviable, making the organization of the projects
us
under a conglomerate the only feasible option. This result goes in line with the exceptional situation of the US economy, which has a low frequency of those groups. In fact, in the US, intercorporate dividends have been taxed since 1935. For Morck (2005),
an
the levy of this tax helps explain the fact that, while business groups were quite common in the US economy until the 1930s, after then, the importance of such groups
4
M
significantly decreased.36
Conclusions
d
This paper sought to investigate the risk sharing mechanisms between firms of a
te
business group and the resulting capital structure. In fact, it was shown that, even with the agency conflicts involved, the optimal contract can be implemented by the issuance
Ac ce p
of financial securities for a diffuse investor base, situation in which stockholders and creditors of each firm might be different. The issuance of these securities is possible due to the fact that the resulting capital structure generates incentives so that the agent allocates losses and gains between the firms in a predictable way, allowing investors to adequately assess their securities. The results of this paper confirm Morck’s (2005) idea that the taxes levied on
intercorporate dividends can help explain the fact that business groups are less frequent in the US economy. In this sense, Morck (2005) shows that, considering 33 countries for which data are available, the US, in 1997, was the only country that taxed
36
Other measures implemented in the same time period which, according to Morck (2005), also proved to be relevant for the breakup of business groups were: elimination of consolidated group corporate income tax filings, changes to the capital gains tax rules regarding the elimination of controlled subsidiaries, and restrictions on pyramidal groups in utilities industries.
35 Page 35 of 55
intercorporate dividends.37 At the same time, the US and the UK were the only ones where none of their top ten firms were controlled by a business group.38 In the model of this article, when the agent diverts cash between the firms, it may result in a deadweight cost. Hence, the group's capital structure must be such that
ip t
makes the agent prefer to transfer resources between firms through the distribution of dividends than through unobservable payments. For that purpose, the percentage of the subsidiary dividends to be received by the parent (and vice versa) shall be greater than
cr
the receivable percentage of the unobservable payments (that is, net of deadweight costs). In this way, if the tax rate on intercorporate dividends is high enough, there will
us
be no capital structure with two separate firms satisfying this condition. In this case, the organization of the two projects as two divisions of a single firm would be the only
an
capital structure able to induce the agent to not divert cash between the firms. Exclusively under the model proposed in this paper, the conglomerate which is formed in consequence of this tax is a suboptimal result. One must bear in mind,
M
however, that this suboptimality is due only to the controller’s preferences. In fact, the level of frequency of business groups in a certain economy gives rise to different distributions of corporate control in society. These distributions, in turn, can exert
d
significant influence on the public policies to be implemented and on the investments in
te
innovation. Therefore, the final outcome for social welfare from the presence of business groups in the economy might even be negative. The insertion of the framework
Ac ce p
proposed in this paper in a general equilibrium model is a possible way to investigate these issues.
Appendix A: Proofs of Lemmas 1-2 Proof of Lemma 1: (a) Consider initially process
contract
may have
that establishes ,
and any process or
. Suppose that there is a . Under such contract, one . Below, it will be
37
See Table 2 in Morck (2005). Regarding the UK, as Morck (2005) reports, Franks, Mayer and Rossi (2005) highlight the London Stock Exchange Takeover Rule of 1968 as the main determinant for the low frequency of business groups in that country. According to that rule, any acquisition of 30% or more of a listed company must be an acquisition of 100%. In this way, the parent firm is required to own either 100% or less than 30% of the subsidiary, creating a barrier to the development of pyramidal ownership. 38
36 Page 36 of 55
shown that, in any of these three cases, there is a contract
in which
. In
any of the cases, the proof consists in demonstrating that every optimal contract in which there are unobservable (by the principal) transfers between projects is weakly dominated by another one in which such transfers do not occur.
ip t
By convenience, equations (6) and (7) are rewritten below, but only highlighting
(A.1)
us
cr
the terms that will be relevant in the following demonstration.
refers to the terms of the contract
an
In what follows, the superscript
. Let
be the contract with the same terms as
differences: (i)
; and, (ii)
. Suppose
. Considering (ii) in (A.2), one realizes that
d
such that
.
, but with two
M
Case 1:
Also,
multiplying
te
.
(A.2)
(ii)
by
, leading, by way of (A.1), to
In this way, one concludes by induction that
Ac ce p
,
for
. , yielding
.
Case 2:
. Let
; (ii)
(i)
be the contract with the same terms as
, except for:
; (iii)
; and (iv) .
Suppose
such
. Considering (iii) in (A.2), one notes that multiplying (iii) by
. Also,
,
, yielding
(when considering (iv)) and,
by
(A.1),
that
, .
Again,
by
induction,
it
is
concluded
37 Page 37 of 55
for
, yielding
. The inequality is for
due to the fact that, according to (iv),
Case 3:
. Let ;
(i)
.
be the contract with the same terms as
(ii)
; .
Suppose
,
(ii)
(iii)
such
that
. Also,
one
has
us
multiplying
and
cr
. Considering (ii) in (A.2), one realizes that
, except for:
ip t
that
, yielding (when considering (i) and , and, by (A.1),
an
(iii)) . Again, by induction, it is concluded that
M
.
yielding
, assuming there is a contract
Consider now process
,
that establishes
. Rewriting again (6) and (7) and highlighting only the terms
d
and any process
for
te
that will be relevant for the demonstration:
Ac ce p
(A.3)
Case 1´: and
into
Case 2´: (i)
(A.4)
. The proof is similar to that of Case 1, only substituting
,
,
and , respectively.
. Let
be the contract with the same terms as ;
(ii)
; ;
.
, except for:
Suppose
. Considering (iii) in (A.4), one notes that
(iii) and such
(iv) that . Also, 38 Page 38 of 55
multiplying (iii) by
(=
),
yielding , and, by (A.3),
(when considering (iv))
yielding
.
Case 3´:
. Let
for
be the contract with the same terms as
, except for:
cr
; (ii)
(i)
,
ip t
. Again, by induction, one concludes that
.39 Suppose
such that
us
(iii)
. Considering (ii) in (A.4), one realizes that
an
,
multiplying (ii) by (when considering (iii))
. Also, yielding , and, by (A.3), for
,
M
. Again, by induction, it is concluded that .
yielding
; and
and
, but
(contradicting the first assertion of the
te
such that
d
(b) Step 1: Suppose that an incentive-compatible contract determines a control
Ac ce p
statement). Let then the alternative control
in which, for
(ii)
; . Denote
obtained under control
.
Suppose
such that
.
and
Multiplying
By
(ii)
by
induction,
and,
for
for
by ,
when
In this case, it shall be noted that, being, by (ii), . Thus, by (iii),
,
, yielding (when considering (i) and
. However, since
one has
(iii)
. Considering (ii) in (A.2),
, .
;
as the state trajectory
(iii))
39
: (i)
(A.1), yielding , this ,
.
39 Page 39 of 55
inequality is strict. So, strategy
is dominated by
, implying that the
contract in question cannot be incentive-compatible.
Step 2: Suppose that an incentive-compatible contract determines a control and
, but
(contradicting the second assertion of
ip t
such that
the statement). Let then the alternative control
in which, for
cr
; (ii) . Denote .
Suppose
such that
.
as the state trajectory
. Considering (ii) in (A.4),
Multiplying
(ii)
an
obtained under control
;
us
and (iii)
: (i)
by
,
one
obtains
M
, yielding (when considering
(i) and (iii)) induction,
for
yielding
when
is dominated by
, implying that the contract in
te
. Again, since is strict. So, strategy
,
.
d
By
, and, by (A.3),
, this inequality
Ac ce p
question cannot be incentive-compatible.
Proof of Lemma 2: (a) Consider the auxiliary lemma below:
Lemma A.1: Suppose that constants
is a polynomial growth function, i.e., that there are
such that
(A.5)
in which
represents the Euclidean norm. Admit also that
programming principle such that, for any stopping time
satisfies the dynamic
,
40 Page 40 of 55
(A.6)
is a viscosity solution to (11).
Proof: See Theorem 5.1, chapter VIII, on Fleming and Soner (2006).
ip t
Then,
cr
To use the lemma above, it is necessary to prove (A.5) and (A.6). The auxiliary
Lemma A.2: The function
is continuous in
and satisfies the dynamic
an
programming principle.
us
lemma below proves the second condition.
It remains then to show that ,
d
such that, for
has polynomial growth. With this goal, define . From (6) and (7),
Ac ce p
te
process
M
Proof: See Lemma 2.1, chapter V, on Fleming and Soner (2006).
Applying the generalized Ito’s formula to
in which
,
and
,
represent the continuous parts of the respective processes.
Taking the conditional expectation to the above equation, 41 Page 41 of 55
By the definitions of
and
, the last summation of the above equation takes on and
. Note that Assumption 1(c) implies that
us
nonnegative values, just as such
cr
ip t
(A.7)
that
.
since
M
an
,
Thus,
the
above
equation
and
Ac ce p
,
letting
. Since
, , and given
, one has, after doing
the arbitrariness of the control As
and
te
d
Rearranging
,
.
, one concludes that
(A.8)
satisfying condition (A.5).
(b) Let
and
, such that . Define also
from initial conditions
,
and, for ,
and
are optimal controls under trajectories
and
as the state trajectories
. Suppose then that and
, let
and
, respectively.
42 Page 42 of 55
In turn, let
. is given by
us
cr
ip t
From this control, the state trajectory
one
notes
and of the initial state
an
By the definitions of the control
for
that (with
and
in
which
defined analogously). Also by the definition of
d
M
, one has40
the control
,
,
Ac ce p
te
(A.9)
in which the third row used the definition of controls
and
. However, by . Substituting (A.9)
the definition of the value function,
, which proves
into this inequality, it is concluded that the concavity of .
(c)
By
the
concavity
of
,
for
,
one
must
have
. 40
To obtain the second equality, define respectively. By the definition of , conditional expectation, definition of
, one also has
and
as the filtrations generated by trajectories and , . Since process is -measurable, by the properties of . Naturally, from an analogous .
43 Page 43 of 55
Suppose
for a given
One notes that
is a local maximum of
,
, with
and
.
, by the viscosity subsolution
cr
Since
, consider the smooth test function
ip t
. For
, and let
For
a
sufficiently
small
an
us
property of , one obtains
,
there
is
a
M
. Identically, one proves that
contradiction.
Therefore,
is continuous.
te
d
Appendix B: Assumption 2 One notes that, in ,
and
and
Ac ce p
vectors
,
, the process
is reflected according to
, respectively. Let
,
and
be
the unit vectors with the same direction of such vectors, i.e.,
44 Page 44 of 55
In addition, for and
, define
as the inward normal in relation to
defined analogously. Finally, for
in , with
, let
, in
represents the interior of the given set. Consider then Assumption 2 below:
, there are scalars
,
, such that
in which
us
cr
Assumption 2: For each
ip t
which
represents the scalar product.
an
Appendix C: Proofs of Lemmas 3-5 and Propositions 1-4
M
Proof of Proposition 1: Step 1: Suppose stopping time formula to
and define the
. Applying the generalized Ito’s
Ac ce p
te
d
, one gets
in which notes
,
and
(C.1)
represent the continuous parts of the respective processes. One that
45 Page 45 of 55
, and an analogous reasoning applies to
. Substituting these
ip t
results in (C.1),
us
cr
(C.2)
, and analogously for
such that
. Subtracting
concavity of ,
, and, by (11),
. By (A.8),
on both sides of the
, one obtains
. Then, by the
M
and letting
inequality, dividing it by
and
an
in which
. Thus, taking the conditional expectation
Ac ce p
te
d
to the equation above and rearranging the resulting expression,
(C.3)
By definition of the control
, one knows that
Suppose that the optimal control
implies
, with strict
inequality in at least one dimension. By (11) and by the construction of that,
for
, . Then, letting
,
.
,
, one knows and
in (C.3),
46 Page 46 of 55
which violates the dynamic programming principle in (A.6). Hence, optimal control in
Suppose
ip t
2:
.
,
and
define
cr
Step
is the
. Following the same procedures used in the
us
previous step, one obtains the same equation (C.3).
, one knows that
By definition of the control
implies
, with strict
an
Suppose that the optimal control
.
inequality in at least one dimension. In the present case, by construction, one has , letting again
,
and
te
d
M
in (C.3),
. Then,
Ac ce p
once more violating the dynamic programming principle in (A.6). Therefore, the . Moreover, by the concavity of
optimal contract implies knows that, for
which implies the optimality of optimal control in
and
, one
,
,
. It may then be concluded that
is the
.
Step 3: Performing an analogous procedure to that used in Step 2, one concludes that is the optimal control also in
41
and in
41
.
As done in Step 2, one shall note that, by the concavity of , for
, which implies the optimality of
in
. In turn, for
and
and
,
,
47 Page 47 of 55
Since
and in
one obtains the optimality of the control
Proof of Lemma 3: Define
,
.
as the contract value for the agent
us
cr
ip t
throughout the term of the contract, such that
is a martingale. Then, by the martingale representation
M
, which allows concluding that
and
such that
te
d
theorem, one knows that there are two predictable processes
. Calculating the differential
Ac ce p
in which
,
an
For
(C.4)
(C.5)
from (C.4) and (C.5), equating the
resulting expressions and considering the homogeneity with respect to time of the present problem, one gets (20).
Proof of Lemma 4: (a) This proof is quite similar to that of Lemma 2. Likewise,
one uses Lemmas A.1 and A.2. It is then necessary to show that
has a polynomial
growth.
To do that, consider the same process
used in the proof of Lemma 2, and
rewrite (A.7) taking into account that, in the present problem, the process
, leading to the optimality of
in
is a control:
.
48 Page 48 of 55
ip t cr
Since
,
.
given
that
the
above
equation
Ac ce p
Rearranging
te
d
M
an
us
,
Also,
. Once
has, after letting
and
letting
,
, and given the arbitrariness of the control
, one
. Since
, one
,
and
concludes that
(C.8)
proving that
has a polynomial growth.
49 Page 49 of 55
The proofs for items (b) and (c) are similar to those of the respective items of Lemma 2.
Proof of Proposition 2: Suppose
and define the stopping time
ip t
. Applying the generalized Ito’s formula to
(C.9)
M
an
us
cr
, one gets
that
and
an
analogous reasoning
applies
to
. In turn,
Ac ce p
,
te
d
Note
. Substituting these results in (C.9),
By (C.8),
such that
. Since . Dividing this inequality by
, and letting
50 Page 50 of 55
, one obtains
. Then, by the concavity of ,
. Therefore,
taking the conditional expectation to the above equation and rearranging the resulting
ip t
expression,
By the definition of the control
. Consider then a control
. By (24) and by the construction of
, one knows that, for
an
that implies
, one knows that
us
cr
(C.10)
and
. Then, by (C.10),
d
M
,
te
which violates the dynamic programming principle. Thus,
Ac ce p
.
In addition, by the concavity of is the optimal control in
Hence,
is the optimal control in
, one knows that
.
Proof of Proposition 3: Consider a nonnegative function ; and (ii) for
that: (i)
functions
and
.
,
such that
such . From the
proposed by the statement, one notes that Assumption 1 is satisfied.
Admit also that, for
By (11), one knows that, for
,
,
51 Page 51 of 55
and
. Since, according
ip t
By construction, , one has
to Lemma 2(c),
. One then concludes that
Proof of Lemma 5: Suppose and define
and
and
for
, and
, yielding
By the definition of
.
, and, by (8),
M
,
and
be defined analogously. By
. Since
. Given the arbitrariness of
,
d
,
be the optimal
as the state trajectories under
, respectively. Let (6) and (7),
. Let
an
control under
us
cr
.
.
Ac ce p
te
Still by (8), one knows that
If
, then
(C.11)
, yielding, by the result shown earlier,
(C.12)
If
, by (6) and (7),
and ,
and,
, one concludes that, if
. Since, according to (9), by ,
the
previous
result, . Inserting 52 Page 52 of 55
this result and (C.12) into (C.11) and considering that .
for
Letting
one
. The proof for the case in which
, obtains
and
is
ip t
analogous.
Proof of Proposition 4: Consider initially the function
cr
lemma below:
that satisfies the ordinary
us
Lemma C.1: (a) Consider a function
an
differential equation
; (ii) for
M
subject to conditions: (i)
,
.
te
for
; and (iv)
(C.13)
,
and
; (iii) for bounded on
. Then,
d
,
and the auxiliary
,
Ac ce p
Proof: Applying the generalized Ito’s formula to
Considering (C.13) and the conditions (ii), (iii) and (iv) in the above equation, one obtains, after taking the conditional expectation,
53 Page 53 of 55
in the above equation, after considering condition (i) and rearranging the .
resulting equation, one obtains
,
in (28) is immediate. The proofs for ,
however,
the
equations
are similar. In the case of in
(C.13)
are
us
and
and
and
for
given
by
, respectively, .
an
and conditions (ii) and (iii) are replaced by
the result
cr
From the lemma above, by applying the generalized Ito’s formula to
ip t
Letting
M
References
d
ALMEIDA, H. V., and D. A. WOLFENZON (2006): “A Theory of Pyramidal Ownership and Family Business Groups”. Journal of Finance, 61, 2637-2680.
Ac ce p
te
BEBCHUK, L., R. KRAAKMAN, and G. TRIANTIS (2000): “Stock pyramids, crossownership, and dual class equity”, in Concentrated Corporate Ownership, ed. by R. K. MORCK. University of Chicago Press, Chicago, IL. BERKMAN, H., R. COLE, and L. FU (2009): “Expropriation through loan guarantees to related parties: Evidence from China”. Journal of Banking and Finance, 33, 141-156. BIAIS, B., T. MARIOTTI, G. PLANTIN, and J. ROCHET (2007): “Dynamic Security Design: Convergence to Continuous Time and Asset Pricing Implications”. Review of Economic Studies, 74, 345-390. BIAIS, B., T. MARIOTTI, J. ROCHET, and S. VILLENEUVE (2010): “Large Risks, Limited Liability, and Dynamic Moral Hazard”. Econometrica, 78, 73-118. CADENILLAS, A., J. CVITANIC, and F. ZAPATERO (2007): “Optimal risk-sharing with effort and project choice”. Journal of Economic Theory, 133, 403-440. CHEUNG, Y. L., Y. QI, P. R. RAU, and A. STOURAITIS (2009): “Buy high, sell low: How listed firms price asset transfers in related party transactions”. Journal of Banking and Finance, 33, 914-924.
54 Page 54 of 55
DeMARZO, P., and Y. SANNIKOV (2006): “Optimal security design and dynamic capital structure in a continuous-time agency model”. Journal of Finance, 61, 2681-2724. DUPUIS, P., and H. ISHII (1993): “SDEs with oblique reflection on nonsmooth domains”. The Annals of Probability, 21, 554-580.
ip t
FACCIO, M., L.H.P. LANG, and L. YOUNG (2001): “Dividends and expropriation”. American Economic Review, 91, 54-78.
cr
FLEMING, W. H., and H. M. SONER (2006): Controlled Markov Processes and Viscosity Solutions, Springer, NY.
us
HOLMSTROM, B., and P. MILGROM (1987): “Aggregation and Linearity in the Provision of Intertemporal Incentives”. Econometrica, 55, 303-328.
an
JOHNSON, S., R. LA PORTA, F. LOPEZ-DE-SILANES, and A. SHLEIFER (2000): “Tunneling”. American Economic Review Papers and Proceedings, 90, 22-27.
M
JOHNSON, R. and L. SOENEN (2002): “Asian Economic Integration and Stock Market Comovement”. The Journal of Financial Research, 25, 141-157. KHANNA, T. and Y. YAFEH (2007): “Business groups in emerging markets: paragons or parasites?” Journal of Economic Literature, 45, 331-372.
te
d
LA PORTA, R., F. LOPEZ-DE-SILANES, A. SHLEIFER, and R. VISHNY (2000): “Investor protection and corporate governance”. Journal of Financial Economics, 58, 3-27.
Ac ce p
MORCK, R. (2005): "How to Eliminate Pyramidal Business Groups: The Double Taxation of Intercorporate Dividends and other Incisive Uses of Tax Policy", in Tax Policy and the Economy, 19, ed. by J. POTERBA. Cambridge and London: MIT Press. MORCK, R., D. WOLFENZON, and B. YEUNG (2005): "Corporate Governance, Economic Entrenchment, and Growth". Journal of Economic Literature, 43, 655-720. RADNER, R., SHEPP, L. (1996): “Risk vs. profit potential: a model for corporate strategy”. Journal of Economic Dynamics and Control, 20, 1373-1393. SANNIKOV, Y. (2008): “A Continuous-Time Version of the Principal-Agent Problem”. Review of Economic Studies, 75, 957-984. WEINDENBAUM, M. L. (1996): “Greater China: the next economic superpower?”, in The Dynamic American Firm, ed. by R. A. BATTERSON, K. W. CHILTON, and M. L. WEINDENBAUM. Springer US. WILLIAMS, N. (2011): “Persistent Private Information”. Econometrica, 79, 1233‒ 1275. 55 Page 55 of 55
S1062-9769(15)00036-8 http://dx.doi.org/doi:10.1016/j.qref.2015.03.005 QUAECO 846
To appear in:
The
Received date: Revised date: Accepted date:
24-7-2014 27-1-2015 11-3-2015
Quarterly
Review
of
Economics
and
Finance
Please cite this article as: Messa, A.,Security Design and Capital Structure of Business Groups, Quarterly Review of Economics and Finance (2015), http://dx.doi.org/10.1016/j.qref.2015.03.005 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.
“Business Groups and Risk Sharing Among Firms” highlights: This paper investigates the optimal contract between a principal and an agent that manages a business group and diverts funds among its projects.
ip t
A pyramidal ownership structure may arise endogenously. The paper provides explanations for the cross-holding of equity between firms in business groups and the contagion between the asset prices of such firms.
cr
It is shown that a tax on intercorporate dividends may render the organization of
Ac ce p
te
d
M
an
us
such groups infeasible and lead to the creation of conglomerates.
1 Page 1 of 55
Alexandre Messa1
an
January 2015
us
cr
Instituto de Pesquisa Econômica Aplicada
ip t
Security Design and Capital Structure of Business Groups
Abstract
This paper investigates the optimal contract between a principal and an agent that
M
manages a business group and diverts funds among its projects. The optimal contract can be implemented by limited liability financial securities and results in a capital
d
structure that provides risk sharing among the group firms. The paper provides
te
explanations for the cross-holding of equity between firms in business groups, the contagion between the asset prices of such firms, and shows that a tax on intercorporate
Ac ce p
dividends may render the organization of such groups infeasible and lead to the creation of conglomerates.
JEL classification: D82, G30, G32, L20. Keywords: business groups; corporate governance; capital structure.
1
E-mail: [email protected]. Telephone/fax number: 55-61-3315-5189/3315-5341.
2 Page 2 of 55
1
Introduction In several countries, firms with a diffuse investor basis and professional
management are more of an exception than the rule. In fact, in emerging countries and
ip t
in parts of Europe, it is much more common to find firms inserted in business groups controlled by a family or the state. In a stylized setting, this control is usually exerted
which, in turn, controls
cr
through a pyramidal structure: a certain family controls firm
firm , making the latter indirectly controlled by the family in question.
us
Firms in such groups are often managed by individuals designated to make decisions according to the controlling family interests, or even by own members of this family.2 Thus, firms belonging to business groups, as those with a diffuse investor basis,
an
feature an agency conflict triggered by the divergence of interests between the manager who maximizes his utility (or, in the case of business groups, the utility of the
M
controlling family) and the stockholders and creditors interested in the maximization of their securities value.
However, the ownership chain of a business group introduces an additional
d
dimension to the agency conflict. This arises from the possibility that the manager of a
te
certain firm will make decisions that benefit another firm of the same group, to the detriment of the one managed by him. For instance, Cheung, Qi, Rau and Stouraitis
Ac ce p
(2009), investigating Hong Kong publicly listed firms, show that a common practice among them is to buy and sell assets to their controlling shareholders at, respectively, above and below market prices. Also, Berkman, Cole and Fu (2009) show that another common practice among Chinese business groups concerns the issuance of a guarantee, given to a third party, by a firm for a loan taken by another firm of the same group. Although these transactions take place within the group, the stockholders and
creditors of each firm might be different. Therefore, these relationships involve wealth transfers from the stockholders and creditors of a firm to those of another one. In the corporate finance literature, these transfers of value across firms of the same business group, to the detriment of minor stockholders of one of the firms, are often referred to as tunneling.3
2 3
See, for instance, Weindenbaum (1996) for a description of this practice in Chinese business groups. For a relevant exposition on this concept, see Johnson, La Porta, Lopez-de-Silanes and Shleifer (2000).
3 Page 3 of 55
In order to assess the incentives behind business groups and potential frictions that may give rise to pyramidal structures, this paper introduces a principal-agent model in which the agent manages the projects of a business group. Initially, it’s assumed a firm managed by an agent with limited liability, under a contract with an investor, the principal, with unlimited liability. Afterwards, the agent is endowed with a new project.
ip t
To implement it, he4 needs to renegotiate the previous contract with the principal.
The agent then proceeds to manage two projects, the parent (the old project) and
cr
the subsidiary (the new one), each one of them has its own cash reserves5. As functions of the history of the projects cash flows, the contract between the parties specifies the
us
principal’s remuneration and investment in the projects. In turn, the agent has discretionary power to establish his own remuneration and observable transfers between the projects’ cash reserves6. Each project is liquidated once it is unable to make the
an
payments to the principal established by the contract7.
However, the profits of both projects are random, and the principal is neither
M
capable of observing their realizations nor their use by the agent. Consequently, the agent can transfer cash across projects, so as to maximize his own utility, without the principal being able to identify such practice.
d
It is assumed that the agent is not able to divert cash to himself from the
te
subsidiary. This assumption creates an asymmetry between the parent and the subsidiary, once the agent is compensated only through funds from the parent (by
Ac ce p
means of an equity share in the original firm)8. Therefore, once the agent can still distribute himself dividends if the subsidiary has been liquidated, but not if the parent did, this asymmetry gives the agent a direct incentive to transfer funds from the subsidiary to the parent. On the other hand, there is also an indirect incentive for the agent to transfer funds to the subsidiary, in case the latter is going through financial distress, in order to preserve the option of drawing funds from it later, on more 4 Throughout this paper, masculine pronouns refer to the agent, whereas feminine pronouns apply to the principal. 5 The assumption of the existence of cash reserves is made without loss of generality. Indeed, the optimal contract could establish no accumulation of them. 6 The assumption of separate cash reserves between the parent and the subsidiary is also made without loss of generality. To see this, note that the principal could determine a payment to herself of one monetary unit by one of the projects, and, at the same time, invest this unit in the other project, establishing, therefore, a simple transfer of cash between them. That being said, one should note that it is a common practice among firms to maintain separate budget accounts and controls for different projects. 7 This condition is, again, without loss of generality, once that the contract can always establish an investment, to be made by the principal in the project, equivalent to the payment to the principal. 8 This would be the case, for example, in which, for legal or regulatory reasons, the subsidiary was subject to tighter auditing procedures than the parent.
4 Page 4 of 55
favorable states of nature for the subsidiary’s activities. 9 This asymmetry between the projects precludes the first-best outcome from being attained10. By dynamic programming, the agent’s problem is reduced to a system of partial differential equations that allows for the identification of the strategy to be adopted by
ip t
him. This strategy imposes the constraints that the principal must consider in her maximization problem, which is then solved using the same procedure applied in the agent’s one.
cr
Thereafter, the paper demonstrates that the optimal contract can be implemented
by means of limited liability securities and, under some circumstances, the creation of
us
two firms. In this way, a pyramidal ownership structure may arise endogenously, when the unobservable cash transfers between the projects would involve a deadweight cost11.
an
On the other hand, when there is no deadweight cost incurred by these cash transfers, the only feasible alternative is the organization of the two projects as two divisions of a single firm.
M
To understand the intuition behind this condition, suppose that the subsidiary is a supplier of the parent. In this case, the agent can, for instance, transfer cash from the subsidiary to the parent through purchases, by the latter, at a price below the production
d
cost for the former. As the subsidiary is a supplier of the parent, this transference may
te
involve no deadweight cost. On the other hand, if the parent’s and subsidiary’s economic activities are very different from each other, this kind of trade between them
Ac ce p
may then lead to deadweight costs.
Therefore, as an implication of the present model, when the projects have similar
economic activities or are on the same value chain, they tend to be organized as two divisions of a single firm. Conversely, when the projects’ economic activities are very different from each other, they may be organized as two distinct firms in a business group.
The issuance of common stocks by each firm is performed allocating a given
fraction to the other one, and granting the agent full discretionary power over the 9 As a result of his maximization problem, the agent will be interested, under some states of nature, to divert cash to the subsidiary in order to smooth the effects of shocks to the parent’s activities. Indeed, as a consequence of Lemma 2 (b), even though the agent is risk-neutral, he will behave risk-averse whenever the parent firm holds low levels of cash reserves – or, in other words, whenever the probability of its liquidation is high enough. That risk aversion then induces the agent to tend to smooth the effects of shocks and, to that end, make use of the subsidiary's cash reserve. 10 The first-best outcome would be the management of the cash reserves as if the projects in question constituted a single one, turning the present problem into the one analyzed, for example, in Radner and Shepp (1996). 11 Using the notation to be introduced in subsection 2.1, when .
5 Page 5 of 55
dividend policies of both firms. Thus, by means of the dividends to be distributed, he will be able to determine the resource allocation across them – and in such a way that the cash transfers between the firms do not result in deadweight costs. In turn, the common stocks issued by the parent firm to be retained by the
ip t
subsidiary must be from a distinct class of those shares held by the agent (also issued by the parent). This distinction is made so as to allow the dividend policies of each of these classes to differ between them. So, as the agent has discretionary power over the
cr
dividend policies of both stock classes, he can both allocate additional funds to the
subsidiary and determine his own remuneration; however, since those stocks belong to
us
different classes, these two flows are determined independently.
The dividends of the stocks issued by the parent and held by the agent will be
an
distributed only after both firms succeed in reducing their respective bankruptcy risks to satisfactory levels. This result somehow leads to a similar conclusion to that drawn by Biais, Mariotti, Plantin and Rochet (2007) for a firm with a diffuse investor basis.
M
Nevertheless, the result shown herein differ in that the dividend policy will take into account the bankruptcy risk not only of the firm in question (in this case, the parent), but also of the other firm in the group (in this case, the subsidiary).
d
On the other hand, the subsidiary’s dividend policy assumes distinct
te
characteristics. In this case, the dividends can be distributed at any time, depending on the state of nature: when this is relatively more favorable to the business activities of the
Ac ce p
subsidiary than to those of the parent, dividends are distributed in order to make up the parent’s cash reserves. The dividend policy regarding the class of parent’s stocks held by the subsidiary features an analogous behavior. Therefore, risk sharing among the group firms occurs through two mechanisms:
cross-holding of equity between them – i.e., the holding of the parent’s equity by the subsidiary, and vice versa –, and distinct dividend policies across the firms. In fact, cross-holding allows the agent, by means of the distribution of dividends, to allocate resources across the firms of the group. In turn, distinct dividend policies allow these distributions to smooth the effects of shocks and the consequent risk sharing among the firms. In this way, this paper proposes an alternative explanation for the practice of cross-holding of equity between firms. This phenomenon is often seen by the literature as a mechanism of separation of ownership and control, differing from the strict pyramidal structure as the voting rights for control purposes are distributed across 6 Page 6 of 55
several firms, instead of being concentrated in a single holding company.12 The present paper, however, highlights the role of this cross-holding in the risk sharing among firms, enabling the resource allocation between them and the consequent smoothing of shocks.
ip t
In addition, ever since La Porta, Lopez-de-Silanes, Shleifer and Vishny (2000) and Faccio, Lang and Young (2001), the dividend policy of business groups has been
regarded as a tool for the reduction of funds available to the controllers, thereby
cr
restricting their capacity to expropriate the wealth of investors. The present paper
introduces an additional function to the dividend policy, which consists in the allocation
us
of losses and gains across firms within the group. As a result, the distribution of dividends by a certain firm would be subjected not exactly to its own performance, but
an
more precisely to the difference between its performance and those of the other firms in the group.
This paper also shows that the market value of securities issued by a given firm
M
are sensitive to shocks on the activity of another firm in the group, what helps explain the contagion between asset prices on different markets. According to the findings of this paper, this contagion would occur due to the risk sharing among the group firms, would directly affect the business
. In this regard, Johnson and Soenen (2002), investigating some
te
activities of firm
d
and not necessarily because the shock to firm
countries with close economic ties to Japan, show a positive correlation between foreign
Ac ce p
direct investment flows received from that country, and comovement between the stock markets of these economies and that of Japan. Finally, it is demonstrated that the tax on intercorporate dividends may render
the pyramidal structure infeasible, and makes the organization of the two projects as two divisions of a single firm, creating a conglomerate, the only feasible alternative. The adoption of this kind of tax by the United States in the 1930s helps explain the low frequency of these groups in that country, although they were quite common there until then. This idea is presented by Morck (2005) and is corroborated by the findings of this paper. This work is related to the literature on financial contracts in a continuous-time setting. In this literature, Holmstrom and Milgrom (1987) is a seminal work, while
DeMarzo and Sannikov (2006), Biais, Mariotti, Plantin and Rochet (2007), Cadenillas, 12
See, for instance, Bebchuk, Kraakman and Triantis (2000).
7 Page 7 of 55
Cvitanic and Zapatero (2007), Sannikov (2008), Biais, Mariotti, Rochet and Villeneuve (2010) and Williams (2011), among others, may be cited as recent contributions. A line of research in this literature investigates the capital structure and security design as a result of the optimal contract between an agent and a principal. DeMarzo and Sannikov
ip t
(2006) and Biais, Mariotti, Plantin and Rochet (2007) are relevant works in this line. The present paper uses this perspective as a starting point, but considering a group of projects and investigating how the capital structure and the financial securities issued
cr
and purchased by the group are used for risk sharing.
Moreover, this paper is part of an effort to understand the ownership structure of
us
business groups. This is usually perceived as a mechanism of separation of control from cash flow rights – see Morck, Wolfenzon and Yeung (2005) and Khanna and Yafeh
an
(2007) for relevant reviews of this literature. Almeida and Wolfenzon (2006) move away from this perspective, showing that a pyramidal structure may arise for other reasons than the separation of ownership and control – in their case, due to financing
M
advantages of pyramidal structures that arise as a result of certain frictions in the capital market. In the present paper, conversely, the ownership structure of the business group arises as a risk sharing mechanism between firms, providing the proper incentives to the
d
manager to allocate the gains and losses in a predictable way.
te
This paper is organized into four sections. Following this introduction, Section 2 introduces the principal-agent model in order to investigate the incentives related to the
Ac ce p
management of the business group, the constraints necessary to the alignment of interests between the parties, and the resulting optimal contract. Section 3 analyzes the consequent capital structure and the price dynamics of the issued securities. Finally, Section 4 concludes.
2
The Principal-Agent Problem and the Optimal Contract This paper assumes a firm managed by an agent with limited liability under a
contract with a principal with unlimited liability, responsible for the necessary financing. Admittedly, both are risk-neutral, and the contract between them provides the agent with a share of the firm’s equity, and requires the firm to make certain payments to the principal at each instant, in what comprises the remuneration of its debt. Thus, the agent’s compensation consists of the percentage of the dividends distributed by the firm he is entitled to. 8 Page 8 of 55
As this firm is operating normally, the agent receives another cash flow generating project. To implement it requires an irreversible investment equal to
.
This investment can, at least partially, be made by the said firm. If that happens, it represents a deviation from its activities and, consequently, the previous contract must
ip t
be renegotiated with the principal. However, even if the implementation of the project does not involve any financial supply by the firm, the activities will imply a change in
the incentives to which the agent will be subjected. Therefore, anyway, the previous
cr
contract must be renegotiated between the parties before implementing the project. This renegotiation can include both changes in the terms of the original contract and
invest in each of the projects at any time she desires.
us
additional financial supply by the principal. Also, using her wealth, the principal may
an
The agent then proceeds to manage two projects, the parent and the subsidiary. The parent’s realized profit can initially have four allocations: the distribution of dividends, the principal’s remuneration13, cash transfers to the subsidiary, or the
M
accumulation of the project’s cash reserves.
In turn, the subsidiary’s realized profit may be allocated as follows: cash
project’s cash reserves.
d
transfers to the parent, the principal’s remuneration14, and the accumulation of the
te
The contract between the parties specifies the principal’s remuneration as a function of the cash flows history. In turn, the agent has discretionary power to establish
Ac ce p
his own remuneration and cash transfers between projects. It is taken as given that each project is liquidated once it is unable to make the payments to the principal established by the contract.
Supposedly, the principal knows the probability distribution of profits and
observes the cash reserves of both projects. However, she does not observe the realized profit of any of them, allowing the agent to divert some of the profit from one of the projects to the cash reserves of the other project, without the principal being able to identify such practice. Thus, there are two types of transfers of funds between projects: those that are observed by the principal (the parent’s both remuneration and additional investments made in the subsidiary) and those that are not observed by her. Furthermore, given the 13
This remuneration can be either positive or negative. If it is negative, it represents additional investments in the project, to be made by the principal. 14 Which can also be either positive or negative.
9 Page 9 of 55
difficulty faced by the principal in monitoring the funds of both projects, the agent is assumed to have discretionary power to determine the distribution of dividends of the parent (and, therefore, his own remuneration) and the observable payments across projects.
ip t
Note that, since the agent is remunerated by the parent (through a percentage of the dividends distributed to shareholders) and he has discretionary power to allocate resources between projects, he will always prevent the parent from being liquidated
cr
before the subsidiary. Hence, it is assumed, during the entire term of the contract between the parties, that either the subsidiary is liquidated before or that both are
us
liquidated simultaneously.
an
Remark 1: The assumption that the agent is not paid by the subsidiary is an implicit consequence of the supposition that he is not able to divert funds from it. Alternatively, it could be admitted that the agent is able to make such diversions. However, this would
M
simply reduce the problem to a case in which the agent manages a single project15, and is able to divert cash from it – an agency conflict analyzed, for example, in DeMarzo and Sannikov (2006) and Biais, Mariotti, Plantin and Rochet (2007).
d
Differently, the assumption that the agent is not able to divert resources to
te
himself from the subsidiary introduces a distinct agency problem, as a consequence of the resulting asymmetry between the projects. This would be the case, for example, in
Ac ce p
which, for legal reasons, the subsidiary was subject to tighter auditing procedures than the parent.
Considering any instant , suppose the cash reserves of both projects are given.
The scenario described above can be summarized in the following timeline:
(T1) The agent observes the realized profits on the interval
.
(T2) Given the cash reserves at , the contract determines the remuneration the principal will receive from each one of the projects on the interval
.
15
Using the notation that will be introduced in the next subsection, the operating profit of this project, on the interval , would be given by .
10 Page 10 of 55
(T3) Given the information at (T1) and (T2), the agent decides on his own remuneration, the payments between the projects and the amount to be diverted across them on the interval
.
(T4) The variations in cash reserves on the interval
are determined
ip t
residually by the difference between the profits observed at (T1) and the cash flows .
cr
stipulated at (T2) and (T3). These variations will determine the cash reserves at
As a result of the above succession of events, the principal, when faced with , is not able to know whether
us
certain levels of cash reserves of both projects at
these levels result from the allocation of resources by the agent at (T3) or from the profits realized at (T1). Her problem, therefore, consists in providing the agent with the
an
appropriate incentives so that he can determine a dividend policy for the parent and
2.1
M
transfers of funds between the projects according to her interests.
Formalization of the Problem In what follows, let
for the terms referring to the parent, and
d
subsidiary. The operating profit of each one of them on the interval
Ac ce p
te
by
in which
and
are positive constants, and
probability space
for the is given
(1)
is a Brownian motion on a filtered
satisfying the usual conditions.16 Nevertheless, it
is assumed that only the agent observes
(and, consequently, the realization of
).
On each interval
profit,
, the parent has two cash flow sources: its operating
, and the payments received from the subsidiary,
.17 This cash flow
16
By usual conditions, it is meant a right-continuous filtration, and containing all sets with zero probability. 17 In what follows, any flow , represented with the right-to-left arrow, refers to resources of the , refer to flows from the parent to subsidiary allocated to the parent; those with the left-to-right arrow, the subsidiary. Thus, one should not mistake the notation introduced here for that of the vector calculus, where represents a vector. In the present paper, and will always be scalars, and the arrow serves only to help visualize the direction of cash flows within the group.
11 Page 11 of 55
;18 the principal’s
can be allocated as follows: the payment of dividends, remuneration,
;19 the payments made to the subsidiary,
; or the
. Thus, one has the identity
accumulation of cash reserves,
In turn, the subsidiary has the following cash flow sources: its operating profit,
,
cr
ip t
(2)
; or
and the investments made by the parent,
; the principal’s remuneration,
us
allocations: the parent’s remuneration,
. This cash flow can initially have three
the accumulation of cash reserves,
an
. Hence, one obtains the identity
M
(3)
can take on any real values. When
By definition, the flows
will be remunerated by project ; when
, she will provide additional financial
d
supply to the said project.
, the principal
te
However, since the principal does not observe the realization of operating
be the amount of
unobservable payments, on the interval
, from the parent, to make up the cash
profits, the agent is able to divert some part of these profits to the cash reserves of the
Ac ce p
other project without the principal being aware of it. Let
reserves of the subsidiary, and
as the unobservable amount diverted in the
opposite direction. It is admitted that, of these unobservable payments, the project of each monetary unit. Thus, the share of the amount
receives a fraction
diverted from the subsidiary actually received by the parent is equal to
, while the
share of the amount diverted from the parent and received by the subsidiary is given by . In turn,
and
represent the deadweight costs incurred by
the respective diversions.
18
Let be the parent’s equity share held by the agent. Then, from the total of dividends paid a part equivalent to will go to the agent, and the other part , to the principal. 19 Besides the part of the dividends she is entitled to (given by ).
,
12 Page 12 of 55
Therefore, the parent obtains two payment flows from the subsidiary,
and
, on each interval, the first of which is observed by the principal, while the second one is unobservable by her. Likewise, the payments made by the parent to the and
, the first of which is observed by
the principal, and the second one is not.
2.1.1 The contract between the parties
(4)
(5)
an
us
cr
In this way, the identities in (2) and (3) can be rewritten as
ip t
subsidiary may be split into two parts,
One assumes that the agent has discretionary power to establish the parent’s
M
dividend policy – and, therefore, his own remuneration –, the parent’s remuneration process and the additional financial supplies made by the parent to the subsidiary. Thus, ,
, is only suggested by the contract, and it is up to the
d
the process
agent to accept or not this suggestion.
te
It is also admitted that, to match the agent’s actions with the principal’s interests, – this assumption is made without
Ac ce p
the principal may use a pair of amplifiers
. The pair determines that: (i)
loss of generality, as it is possible to have
for each monetary unit the parent supplies to the subsidiary (by way of principal will receive
the parent (by way of
), the
units; (ii) for each monetary unit received as remuneration by ), the principal will receive
units. The purpose of these
parameters is to change the impact of cash flows between projects on their respective cash reserves, altering the incentives to which the agent will be subjected. and
Given the processes one process
, for any
that satisfies, for . Hence,
and
there exists only ,
and
are payments received by the principal
directly from the parent and from the subsidiary, respectively. Then, letting the parent’s equity share held by the principal,
be
the contract establishes 13 Page 13 of 55
as
the
principal’s instantaneous remuneration. The subsidiary project is established by a contract contract) stipulating the pair ,
and
.20 Regarding the
, the agent can divert them from the path suggested by
and
ip t
processes
and the process
(replacing the original
choose a strategy that maximizes his expected utility throughout the contract period. The principal’s remuneration
cr
is determined exclusively by the
contract, and these payments must be made by the respective projects, regardless of
us
their cash reserves. When these are insufficient for that, the said project is liquidated. As argued earlier, as the agent is remunerated by the parent and he can allocate resources
an
between the projects, the parent shall not be liquidated before the subsidiary. Hence, the contract is assumed to be terminated when the parent’s cash reserves are insufficient to meet its obligations. The term of the contract
, in which
M
.21
is then given by interval
The principal’s problem consists in determining the incentive-compatible
d
contract that maximizes her expected utility. By incentive-compatible contract, it is meant any contract under which the strategy adopted by the agent is actually the one
te
suggested by the contract.
To find this optimal contract, it is necessary to obtain the constraints that must
Ac ce p
be included so that the agent’s actions correspond to those required by the principal. With that purpose, the following section will analyze the agent’s maximization problem under a given contract
2.2
.
The agent’s problem Let
and
be the discount rates, respectively, of the agent and of the principal.
Assuming that the agent is more impatient than the principal, one has
. The
following analysis admits, without any damage to the conclusions reached, that any income flow received by the agent is immediately consumed by him. 22 20
Note that, given the terms related to , the processes and are determined residually, according to identities (4) and (5). 21 Equivalently, the subsidiary is liquidated at the instant , in which . 22 Alternatively, suppose that the agent can consume some of this income flow and save up the rest. However, supposing that these savings grow at a rate , any contract that implies positive savings by the agent at some will be weakly dominated by another one in which the principal saves in his place and
14 Page 14 of 55
Just for this subsection, make the following assumption about the process :
Assumption 1: For
(c) for
;
; such that
.
cr
,
ip t
23
(b)
: (a)
Remark 2: The assumption above assumes a Markovian structure for
, which
us
significantly simplifies the agent’s problem. Anyway, in the context of the principal’s problem, Remark 3 further ahead will discuss the appropriateness of this assumption, to .
an
the detriment of a history-dependent process
In this way, besides the Markovian structure mentioned in the remark above, depends on time solely by means of the
M
Assumption 1(a) assumes that the process
levels of cash reserves at each instant, but not explicitly on time itself. This assumption is made in order to guarantee the stationarity of the present problem, given its infinite
d
time horizon.
te
Assumption 1(b) is intended to guarantee that the stochastic differential equations in (6) and (7) have a unique solution. Finally, Assumption 1(c) will be used in
Ac ce p
the proof of Lemma 2. Intuitively, it guarantees, by means of Dynkin’s formula, that for
Let
.
. Inserting (1) into (4) and (5), one sees that the cash reserves
of the parent and of the subsidiary will follow the respective processes
(6)
(7)
makes payments to the agent, through the parent, at the instants he would consume his savings. This result is formally established in DeMarzo and Sannikov (2006) and in DeMarzo and Fishman (2007). 23 represents the set of all functions times continually differentiable in .
15 Page 15 of 55
in which were used the definitions
and .
Define
as the set of feasible strategies available to the agent, i.e., the set of all 24
non-decreasing process
adapted to
, with values in
, and
25
Then, by letting
(8)
us
cr
ip t
with bounded variation. The agent’s criterion functional can be defined as
, the value function
of the present problem
an
is such that:
M
(9)
Since the contract is terminated when the parent runs out of cash reserves, the function for
.
d
must satisfy
Lemma 1 below implies that the subsequent analysis can be restricted to
te
26
contracts in which
. Thus, when subjected to these contracts, the agent’s
Ac ce p
optimal strategy will not include any unobservable (by the principal) allocation of resources between the projects. Its proof, as well as those of the other lemmas and propositions, is shown in the Appendix.
Lemma 1: (a) Let
be the set of optimal contracts, i.e., those incentive-compatible that
maximize the principal’s expected utility. Suppose that this set is non-empty. Then, there is a contract
24
under which
For any process ,
. Since, for
.
,
,
,
,
and
take on
nonnegative values, the processes , , , and shall be nondecreasing. 25 According to the notation used in the present paper, whenever a certain variable is denoted by an uppercase letter, , it will refer to the process in question. When the variable is denoted by a lower-case letter, , it represents a value assumed by the process at a specific, and clear in the context, instant (in (8), the integration parameters indicate that , i.e., ), or whenever the specific instant is not relevant. 26 Note that, being and , implies for . Obviously, the same applies to
.
16 Page 16 of 55
(b) Under an incentive-compatible contract, in order for necessary that
. Likewise, in order for
and and
, it is
, it is necessary
.
that
ip t
Item (b) of the lemma above establishes the incentive-compatibility constraints, which impose the necessary conditions so that the agent will prefer to allocate resources
cr
between the projects using processes that can be observed by the principal, to the detriment of those unobservable by her. To understand the intuition behind this result, 27
paid by the subsidiary; on the other hand, the parent
monetary unit of remuneration
for each monetary unit of
. Therefore, in order for the agent to
an
receives the fraction
for each
us
note that, on the one hand, the parent actually receives a fraction
choose to transfer resources using an observable process
, it is necessary that .
M
. The same intuition applies to the condition
As a result of Lemma 1 above, the following analysis can be restricted, without
necessary conditions for that,
. Thus, implicitly, it is assumed that the
d
any loss, to contracts that result in
and
. That is,
Ac ce p
,
te
In what follows, for any function
relative to the parent’s cash reserves ( ),
, are satisfied.
, let
,
and
represents the partial derivative of function , in relation to the subsidiary’s cash reserves
( ), and so on and so forth.
After that, define the differential operator , where
as, for any function
is a countable set of points,
(10)
in which
27
From the amount
represents the covariance between the Brownian motions
paid by the subsidiary, the parent receives the equivalent to , whereas the principal receives
and
.
.
17 Page 17 of 55
Lemma 2 below characterizes the agent’s value function. Item (a) defines the Hamilton-Jacobi-Bellman (HJB) equation to be satisfied by that function. However, since, in the present problem, the control exerts a linear influence on both the dynamics of the state trajectories and the criterion functional, one has a singular stochastic control
ip t
problem. In this case, the HJB equation involves not only a partial differential equation (represented by the first term of (11)), but actually a system of partial differential
cr
inequalities, as pointed out by (11).
Lemma 2: (a) The value function defined in (9) is a viscosity solution to the equation
.
(c)
an
;
M
is a concave function on domain
(11)
d
(b)
us
below:
To understand the intuition behind Lemma 2, first observe that the agent for each monetary unit of dividend paid by the parent. On deciding
te
receives a fraction
Ac ce p
for an additional monetary unit of cash reserve in this project, the agent incurs an opportunity cost, since this unit could alternatively be partially allocated to his own consumption.
Therefore, on the one hand,
represents the opportunity cost incurred by the
agent when he postpones his consumption (deciding on an additional monetary unit of cash reserve in the parent). On the other hand,
denotes his expected gain in terms of
future utility, obtained from that postponement. Then, by postponing his consumption, the agent’s expected gain cannot be lower than his opportunity cost. Naturally, if that were the case, the postponement would not occur and the agent would consume the parent’s cash reserves until
is satisfied, leading to the term
in (11).
Likewise, recall that, for each monetary unit of remuneration established for the parent (and, indirectly, for the principal) – in the form of actually receives a fraction
–, this project
. So, when deciding on an additional monetary 18 Page 18 of 55
unit of cash reserve in the subsidiary, the agent incurs an opportunity cost, since this unit could alternatively be allocated to the parent’s cash reserves. This opportunity cost is given by
. represents the opportunity cost
In this way, on the one hand,
ip t
incurred by the agent by postponing the parent’s remuneration (deciding on an additional monetary unit of cash reserve in the subsidiary). On the other hand,
cr
denotes his expected gain from this postponement. Hence, by postponing the parent’s remuneration, the agent’s expected gain cannot not be lower than his opportunity cost.
us
Again, if that were the case, this postponement would not occur and the agent would is satisfied, leading to the third term of
remunerate the parent until
an
(11).
Analogously, for each monetary unit of investment by the parent to the subsidiary – by way of
–, the latter receives a fraction
. When
M
deciding on an additional monetary unit of cash reserve for the parent, the agent incurs an opportunity cost, since this unit could alternatively be allocated to the subsidiary’s . Likewise, the agent’s
d
cash reserve. This opportunity cost is given by
te
expected gain from postponing this investment,
, cannot not be lower than his
, as expressed by the fourth term of (11).
opportunity cost, leading to
to represent, for a given level
Ac ce p
Therefore, define function
of
cash reserves of the subsidiary, the lowest level of the parent’s cash reserves from which the agent opts for the distribution of dividends:28
(12)
From the function above, one may obtain the set
defined below and illustrated by
Figure 1.
(13)
28
In what follows, adopt the convention that
.
19 Page 19 of 55
ip t
2
cr
1
0
us
Figure 1: The set
1
. The agent would not establish any remuneration to himself
an
whenever the cash reserves are located in this set.
Thereafter, define function
of the
M
to represent, for a given level
parent’s cash reserves, the lowest level of the subsidiary’s cash reserves from which the
Ac ce p
te
d
agent decides to remunerate the parent, i.e.,
From this function, one gets the set
defined below and illustrated by Figure 2.
(14)
20 Page 20 of 55
ip t
2
cr
2
0
us
Figure 2: The set
1
. The agent would not establish any remuneration to the parent
an
project whenever the levels of cash reserves are located in this set.
Finally, define
M
to represent, for a given level
of the subsidiary’s
cash reserves, the lowest level of the parent’s cash reserves from which the agent opts to :
Ac ce p
te
d
allocate resources to the subsidiary, under the constraint that
Let then the set
be defined as below and illustrated by Figure 3.
(15)
21 Page 21 of 55
ip t
2
cr
3
0
us
1
Figure 3: The set
. The agent would not establish any supply of funds from the
,
and
, it is possible to summarize a possible strategy to be
M
From
an
parent to the subsidiary whenever the levels of cash reserves are located in this set.
followed by the agent. First, the agent would not establish any remuneration to himself whenever the cash reserves are located in
.
, making
te
–, the agent will determine a distribution of dividends equal to these levels return to
, the
– i.e., the levels of cash reserves lie outside
d
parent’s cash reserves reach
. However, if, for a given level
Ac ce p
Likewise, the agent would not establish any remuneration to the parent
whenever the levels of cash reserves are located in
. Nonetheless, if they leave
, the
agent would establish a remuneration to the parent, making the levels of cash reserves return to
.
Finally, the agent would not establish any supply of funds from the parent to the
subsidiary whenever the levels of cash reserves are located in leave
. However, if they
, the agent will determine new financial supplies to the subsidiary, making these
levels return to
.
As a result of this policy, the cash reserves of both projects would remain in the region
29
illustrated in Figure 4. Proposition 1 below shows that this
policy actually dominates any other feasible strategy to be adopted by the agent.
29
Note that, by construction,
and, therefore,
.
22 Page 22 of 55
ip t
2
cr
4
1
. The cash reserves of both projects would remain in the region
an
Figure 4: The set
us
0
M
.
For the existence of a solution to the problem described in Proposition 1, it is still necessary to establish certain conditions necessary so that, when the process
. These necessary conditions are analogous to the ones pointed out by
te
returns to
and the agent adopts the proposed strategy, the process
d
reaches the boundary of
Remark 3.1 in Dupuis and Ishii (1993), and are described by Assumption 2 in Appendix
Ac ce p
B.
These conditions, intuitively, require that, in the first place, if the boundaries of
and
have decreasing regions, they must have, in absolute terms, a maximum slope
(i.e., they should not be excessively vertical). In the second place, in the corners formed by30
and
(points
and
in Figure 4), the set , and
extremely long horns. More formally, denote equidistant points in
from
(but with
in
, and
in
and
as
). Assumption 2 then
implies a minimum necessary distance (dependent upon ) between
30
should not have
and
.
represents the boundary of the set .
23 Page 23 of 55
Proposition 1: Let the sets
,
and
as defined by (13), (14) and (15), and the set , it is known31 that there exists a
. Given an initial state process
that results in a trajectory
(i)
, almost surely with respect to ; ,
and
are nondecreasing;
ip t
(ii)
, such that:
cr
(iii) one has
us
(16)
an
(17)
M
(18)
which
denotes
of
the
set
in
question.
Then,
.
The principal’s problem
Ac ce p
2.3
interior
te
,
the
d
in
For this problem, define the agent’s continuation value as the present value of
the promised flows of utility by the contract, from to the end of its term, i.e.,
in which
Lemma 3: For
. Lemma 3 below establishes the dynamics of the process
(19)
.
,
31
Items (i), (ii) and (iii) describe a Skorokhod problem in a domain with corners. The existence of a solution to this problem was shown by Dupuis and Ishii (1993). The problem of the statement corresponds to Case 2 of the referenced paper.
24 Page 24 of 55
(20)
,
Let
, satisfies
.
be the set of all nondecreasing stochastic process , with values in
adapted to
ip t
in which
and with bounded variation. In her maximization
cr
problem, the principal will be subject to three state variables, namely, the cash reserves of both projects and the agent’s continuation value. Then, the principal’s criterion
us
functional is given by
M
an
(21)
So, it is possible to define the value function
of the principal’s problem:
te
d
(22)
Ac ce p
Define the differential operator where
as, for any function
,
is a countable set of points,
(23)
Lemma 4 below, analogously to Lemma 2 of the agent’s problem, characterizes the principal’s value function. Again, respect to, respectively,
and
and
, and
represent the partial derivatives of
, the partial derivative relative to
with
.
Lemma 4: (a) The value function defined in (22) is a viscosity solution to the system of equations below:
25 Page 25 of 55
(24)
(c)
.
;
ip t
is a concave function on domain
cr
(b)
Figure 5 shows the implications of Lemma 4, assuming, for the sake of .
us
and
M
an
illustration only, fixed values for
te 0
Ac ce p
0
d
> −1
= −1
Figure 5: The value function , assuming fixed values for larger than
Then, define
whenever
and
, and equal to that value when
. Its slope is .
as the lowest level of the agent’s continuation value in which the
principal’s expected gain from making a direct transfer to the agent is equal to its cost, i.e.:
(25)
The following strategy summarizes the possible terms of the optimal contract. By the definition of
in (25), for every
, the reduction of one unit in the agent’s
26 Page 26 of 55
continuation value will cause
to increase by less than one. Therefore, the principal
will not establish any remuneration to the agent whenever his continuation value is . However, at
value, the increase in
, for the decrease by one unit in the agent’s continuation
will have the same magnitude. Thus, when the agent’s , the principal will determine an instantaneous
continuation value reaches
ip t
smaller than
, making that value returns immediately to
remuneration to the agent equal to
.
cr
Proposition 2 below shows that the strategy above dominates any other that
Proposition 2: Assume process
and parameters
as given. Let the set
is taken as defined in (25). Given
an
, in which an initial state
, there is a process
that leads to a trajectory
M
such that:
, almost surely with respect to ;
(i)
is nondecreasing;
Ac ce p
te
(iii) one has
d
(ii)
us
could be adopted by the principal.
Then,
Remark 3: Given
.
, the principal will be indifferent to the possible
specifications that the processes
and
may assume. This result has two
implications. Firstly, this indifference allows assuming that these processes are Markovian, as proceeded in the previous subsection (from Assumption 1). In fact, by assuming a cost associated with the verification and enforceability related to the establishment of
from a history of realizations of
, and being the principal
indifferent to that process, it is natural to imagine that she accepts to minimize that cost through a Markovian structure (in which she needs to take account of only the current
27 Page 27 of 55
value of
). Secondly, the optimal contract will not be necessarily unique, i.e., the one
derived in this paper, as resulting from that assumption, can be just one among others possible from different specifications (which not necessarily Markovian) for the
2.4
and
.
ip t
processes
The optimal contract
In order for the contract considered here to be incentive-compatible, the
cr
promised flows of utility by the contract to the agent must match those that result from
the contract value from the cash reserves (11),
,
it
is
an
and
, i.e.,
must be equal to
us
his maximization. In other words, the agent’s continuation value at
. By Ito’s lemma
known
that
. Comparing this result
with (20), it may be concluded that, given an incentive-compatible contract, one must and
.
M
have
By Proposition 2, the principal will remunerate the agent only from
.
Therefore, in an incentive-compatible contract, the cash reserves from which the .
In other words, the
curve in Figure 4 must belong to the level curve
.
Then, the terms
of the contract must be such that they satisfy the conditions:
Ac ce p
te
d
principal would be willing to remunerate the agent are those that satisfy
(C1)
;
(C2)
;
(C3)
, in which
is the function defined in (12),
in addition to Assumption 1. Conditions (C1) and (C2) result from Lemma 1. There may be several contracts that satisfy the indicated conditions. Anyway, the proposition below demonstrates the existence of at least one optimal contract for each feasible value assumed by
Proposition 3: Suppose that (C2). Given
.
takes on values that satisfy conditions (C1) and
, there exists a nonnegative function
, such that 28 Page 28 of 55
, which causes Assumption 1 to be satisfied, and implies (C3).
For the full characterization of the optimal contract, it remains to determine the and
, which will depend on the bargaining power of the
ip t
values of
parties. To match this choice with the analysis in the subsequent section, suppose that
represents the irreversible
cr
the agent has full bargaining power. As defined earlier,
investment needed for the implementation of the new project. Since
is the
us
subsidiary’s cash reserves in the initial period, the total amount necessary for the . This amount can be, at least partially,
establishment of this project is given by
, whereas the remaining
amount,
an
covered by an initial financial supply by the principal,
, is provided by the parent’s internal resources (in the present case, by its
cash reserves).32 Let
be the parent’s cash reserves at the instant immediately before the value of the original contract for
M
the establishment of the subsidiary, and
, the agent
te
solves the problem below:
and
d
the principal. Then, to determine the values of
Ac ce p
(26)
subject to:
32
Anyway,
can be negative. This will be the case when is allocated to the parent’s cash reserves.
(27)
, and the amount
29 Page 29 of 55
in which the last constraint is given by Lemma 1 and by the definition that . In first place, since in the initial instant the agent is able to allocate resources between the projects without deadweight cost, the cash reserves of each project shall be such that their respective marginal values are equivalent, i.e.,
and
, with at least one strict inequality. Then, for .
us
,
cr
Lemma 5: Let
ip t
. Consider then the lemma below:
According to the lemma above, the smaller the value of
, for the same level of
larger value implies a higher influence
an
cash reserve, the larger the value of the contract for the agent. Notwithstanding, this (by the fourth constraint in (27)) that, in turn, will
and
(second constraint). Looking at Figure 5, it can be , that larger
will yield a smaller value of
and,
consequently, a smaller
. This will have to be compensated for by smaller
M
perceived that, if
. Hence, the levels of
d
which will lead to a reduction in
and
or
,
to be chosen shall be
te
those in which the gain for the agent from a smaller value for them exactly offsets the
Ac ce p
loss caused by a smaller resulting and
. In other words, suppose that
are continuous functions. Then, the values of
shall be such that noted that
, since, if
and increase
3
or
33
and
. Also, by means of Figure 5, it can be , the agent could concomitantly reduce
.
Capital market
The optimal contract derived in the previous section can be implemented through the issuance of financial securities to be negotiated in a competitive capital market. For this purpose, the design of these securities reproduces the ownership
33
Except, evidently, in cases of corner solutions, in which the optimal values for .
or
will be zero or
30 Page 30 of 55
structures of firms resulting from the optimal contract, as well as the remuneration of the parties.
3.1
Implementing the optimal contract
ip t
Define the financial securities below:
Common stocks: securities whose instantaneous payoff is not determined beforehand,
cr
but recurrently at each instant. It is assumed that (i) the stocks issued to the public are held by a diffuse investor base, what makes the company decisions be controlled by the
us
restricted stockholders – in the present model, only the agent; and (ii) there is one or more classes of stock, and these classes may differ from one another with respect to the
an
dividends to be received;
Fixed-coupon debt securities: perpetuities that pay their holders at an instantaneous
M
fixed coupon rate.
Variable-coupon debt securities: perpetuities that pay their holders at an instantaneous
te
d
variable coupon rate.
of
For simplification, suppose that the parent is a public firm, with a fraction
Ac ce p
its common stocks held by the agent, and the remaining fraction, diffuse investor base.34 Define
, held by a
as the total payments to be made by the parent for
remuneration of its outstanding debt. Then, the optimal contract derived in the previous section can be implemented as follows:
Optimal Financing Strategy. If the group issues financial securities in the capital market in order to finance the establishment of the subsidiary firm, a financing strategy may be adopted such that:
Parent firm:
34
This assumption is made only to refrain from any possibility of control over the firm by some investor.
31 Page 31 of 55
Common Stocks: issuance of a second class of stocks, of which a fraction
is
, is held by the subsidiary.
sold to investors, and the remaining,
Fixed-coupon debt securities: issuance of perpetuities that pay an instantaneous coupon
perpetuities, each of them being
cr
Variable-coupon debt securities: purchase of
).
ip t
(or their purchase, if
equal to
remunerated by an instantaneous coupon equivalent to
.
us
,
Subsidiary firm:
, held by the controlling firm.
M
sold to investors, and the remaining fraction,
an
Common stocks: issuance of a single class of stocks, with a fraction equal to
Fixed-coupon debt securities: issuance of perpetuities that pay an instantaneous coupon .
d
equal to
te
The financing strategy above refers to the implementation of the contract identified by Proposition 3 and by the maximization in (26). Although this contract is
Ac ce p
not necessarily the unique optimal one, the financing strategy above allows drawing some conclusions that can be generalized. In this regard, the cross-ownership of firms – i.e., the holding of the parent’s equity by the subsidiary, and vice versa – is targeted at reproducing the incentives provided for by the optimal contract. Each of the firms will be remunerated by the dividends distributed by the other firm to its stockholders. Thus, through the respective dividend policies, the agent can determine the resource allocation between them. Nevertheless, as each firm holds a fraction stocks, each monetary unit received by the firm will represent
of the other firm’s units of the total
dividends to be distributed. Likewise, through the distribution of dividends by the parent, the agent can determine his own remuneration. However, if the parent’s stocks owned by him belong to a different class from those held by the subsidiary, both payment flows made by the parent (to the agent and to the subsidiary) are independent of each other.
32 Page 32 of 55
Finally, the debt securities of both firms are targeted at reproducing the flows resulting from functions payments
and
introduced in Proposition 3, after considering
to be made by the parent, due to its pre-existing debt.
Therefore, two remarks can be made about the group’s capital structure. The
ip t
first one is concerned with the different functions attributed to each security. In fact, the cross-holding of equity between the firms allows the agent to transfer resources between them. Debt issues, however, aim to provide appropriate incentives so that the agent
cr
postpones his remuneration (and, therefore, cash outflows) until both firms succeed in accumulating certain levels of cash reserve and, consequently, reducing their respective
us
bankruptcy risks to satisfactory levels.
Secondly, it shall be noted that the dividend policies differ among the firms of
an
the business group. On the one hand, the dividends of the parent’s stocks owned by the agent (and by the public) will be distributed only when both firms succeed in reducing their respective bankruptcy risks to satisfactory levels. On the other hand, the
M
subsidiary’s dividend policy behaves differently. In this case, the dividends can be distributed at any time, depending on the state of nature: when the state of nature is
d
relatively more favorable to the subsidiary’s activities than to those of the parent, the distribution of dividends is determined. The class of stocks issued by the parent and
Market price dynamics
Ac ce p
3.2
te
held by the subsidiary has a similar dividend policy.
Based on the financing strategy pointed out above, the present problem can have
three types of common stocks:
(S1) common stocks issued by the parent and held by the agent; (S2) common stocks issued by the parent and held by the subsidiary; (S3) common stocks issued by the subsidiary and held by the parent.
Let
be the market values of all securities, such that ,
and . Likewise, let
be the market
values of the parent’s and subsidiary’s net debts, respectively, such that
33 Page 33 of 55
and
. The
proposition below determines the behavior of these functions over time.35
and
, one has
ip t
Proposition 4: For
in
which
us
cr
(28)
,
an
.
and
sources of random shocks (
and
M
According to the proposition above, each market value is influenced by two ). In this way, the prices of securities of each
firm are subject not only to uncertainty over the activity of the firm in question, but also
d
to that related to the other firm in the business group. In other words, regardless of the
te
geographical or sectoral proximity between the markets of the parent and of the subsidiary, shocks to the activity of either of them shall have an impact on the prices of
Ac ce p
securities of the other firm, leading to a contagion effect between the markets.
3.3
Tax on intercorporate dividends Suppose that, in the economy in question, a tax at a rate
is levied on the flows
of dividends between the firms. Therefore, for each monetary unit of dividend distributed by the stocks (S2), the subsidiary actually receives a fraction . Likewise, for each monetary unit of dividend distributed by the . The last
stocks (S3), the parent actually receives the fraction constraint in (27) must be rewritten as follows:
35
Again, applies to
and and
represent the partial derivatives of .
relative to
and
, respectively. The same
34 Page 34 of 55
By the constraint above, note that, if
, the only feasible value for . Hence, if
the parameters in question is
, the new project
must necessarily be financed only by debt issues, without new issuances of common
ip t
stocks for investors. In this case, the parent and the subsidiary will constitute two divisions of a single firm, forming a conglomerate.
cr
It may then be concluded that, if the tax rate is high enough, it renders the
organization of the business group unviable, making the organization of the projects
us
under a conglomerate the only feasible option. This result goes in line with the exceptional situation of the US economy, which has a low frequency of those groups. In fact, in the US, intercorporate dividends have been taxed since 1935. For Morck (2005),
an
the levy of this tax helps explain the fact that, while business groups were quite common in the US economy until the 1930s, after then, the importance of such groups
4
M
significantly decreased.36
Conclusions
d
This paper sought to investigate the risk sharing mechanisms between firms of a
te
business group and the resulting capital structure. In fact, it was shown that, even with the agency conflicts involved, the optimal contract can be implemented by the issuance
Ac ce p
of financial securities for a diffuse investor base, situation in which stockholders and creditors of each firm might be different. The issuance of these securities is possible due to the fact that the resulting capital structure generates incentives so that the agent allocates losses and gains between the firms in a predictable way, allowing investors to adequately assess their securities. The results of this paper confirm Morck’s (2005) idea that the taxes levied on
intercorporate dividends can help explain the fact that business groups are less frequent in the US economy. In this sense, Morck (2005) shows that, considering 33 countries for which data are available, the US, in 1997, was the only country that taxed
36
Other measures implemented in the same time period which, according to Morck (2005), also proved to be relevant for the breakup of business groups were: elimination of consolidated group corporate income tax filings, changes to the capital gains tax rules regarding the elimination of controlled subsidiaries, and restrictions on pyramidal groups in utilities industries.
35 Page 35 of 55
intercorporate dividends.37 At the same time, the US and the UK were the only ones where none of their top ten firms were controlled by a business group.38 In the model of this article, when the agent diverts cash between the firms, it may result in a deadweight cost. Hence, the group's capital structure must be such that
ip t
makes the agent prefer to transfer resources between firms through the distribution of dividends than through unobservable payments. For that purpose, the percentage of the subsidiary dividends to be received by the parent (and vice versa) shall be greater than
cr
the receivable percentage of the unobservable payments (that is, net of deadweight costs). In this way, if the tax rate on intercorporate dividends is high enough, there will
us
be no capital structure with two separate firms satisfying this condition. In this case, the organization of the two projects as two divisions of a single firm would be the only
an
capital structure able to induce the agent to not divert cash between the firms. Exclusively under the model proposed in this paper, the conglomerate which is formed in consequence of this tax is a suboptimal result. One must bear in mind,
M
however, that this suboptimality is due only to the controller’s preferences. In fact, the level of frequency of business groups in a certain economy gives rise to different distributions of corporate control in society. These distributions, in turn, can exert
d
significant influence on the public policies to be implemented and on the investments in
te
innovation. Therefore, the final outcome for social welfare from the presence of business groups in the economy might even be negative. The insertion of the framework
Ac ce p
proposed in this paper in a general equilibrium model is a possible way to investigate these issues.
Appendix A: Proofs of Lemmas 1-2 Proof of Lemma 1: (a) Consider initially process
contract
may have
that establishes ,
and any process or
. Suppose that there is a . Under such contract, one . Below, it will be
37
See Table 2 in Morck (2005). Regarding the UK, as Morck (2005) reports, Franks, Mayer and Rossi (2005) highlight the London Stock Exchange Takeover Rule of 1968 as the main determinant for the low frequency of business groups in that country. According to that rule, any acquisition of 30% or more of a listed company must be an acquisition of 100%. In this way, the parent firm is required to own either 100% or less than 30% of the subsidiary, creating a barrier to the development of pyramidal ownership. 38
36 Page 36 of 55
shown that, in any of these three cases, there is a contract
in which
. In
any of the cases, the proof consists in demonstrating that every optimal contract in which there are unobservable (by the principal) transfers between projects is weakly dominated by another one in which such transfers do not occur.
ip t
By convenience, equations (6) and (7) are rewritten below, but only highlighting
(A.1)
us
cr
the terms that will be relevant in the following demonstration.
refers to the terms of the contract
an
In what follows, the superscript
. Let
be the contract with the same terms as
differences: (i)
; and, (ii)
. Suppose
. Considering (ii) in (A.2), one realizes that
d
such that
.
, but with two
M
Case 1:
Also,
multiplying
te
.
(A.2)
(ii)
by
, leading, by way of (A.1), to
In this way, one concludes by induction that
Ac ce p
,
for
. , yielding
.
Case 2:
. Let
; (ii)
(i)
be the contract with the same terms as
, except for:
; (iii)
; and (iv) .
Suppose
such
. Considering (iii) in (A.2), one notes that multiplying (iii) by
. Also,
,
, yielding
(when considering (iv)) and,
by
(A.1),
that
, .
Again,
by
induction,
it
is
concluded
37 Page 37 of 55
for
, yielding
. The inequality is for
due to the fact that, according to (iv),
Case 3:
. Let ;
(i)
.
be the contract with the same terms as
(ii)
; .
Suppose
,
(ii)
(iii)
such
that
. Also,
one
has
us
multiplying
and
cr
. Considering (ii) in (A.2), one realizes that
, except for:
ip t
that
, yielding (when considering (i) and , and, by (A.1),
an
(iii)) . Again, by induction, it is concluded that
M
.
yielding
, assuming there is a contract
Consider now process
,
that establishes
. Rewriting again (6) and (7) and highlighting only the terms
d
and any process
for
te
that will be relevant for the demonstration:
Ac ce p
(A.3)
Case 1´: and
into
Case 2´: (i)
(A.4)
. The proof is similar to that of Case 1, only substituting
,
,
and , respectively.
. Let
be the contract with the same terms as ;
(ii)
; ;
.
, except for:
Suppose
. Considering (iii) in (A.4), one notes that
(iii) and such
(iv) that . Also, 38 Page 38 of 55
multiplying (iii) by
(=
),
yielding , and, by (A.3),
(when considering (iv))
yielding
.
Case 3´:
. Let
for
be the contract with the same terms as
, except for:
cr
; (ii)
(i)
,
ip t
. Again, by induction, one concludes that
.39 Suppose
such that
us
(iii)
. Considering (ii) in (A.4), one realizes that
an
,
multiplying (ii) by (when considering (iii))
. Also, yielding , and, by (A.3), for
,
M
. Again, by induction, it is concluded that .
yielding
; and
and
, but
(contradicting the first assertion of the
te
such that
d
(b) Step 1: Suppose that an incentive-compatible contract determines a control
Ac ce p
statement). Let then the alternative control
in which, for
(ii)
; . Denote
obtained under control
.
Suppose
such that
.
and
Multiplying
By
(ii)
by
induction,
and,
for
for
by ,
when
In this case, it shall be noted that, being, by (ii), . Thus, by (iii),
,
, yielding (when considering (i) and
. However, since
one has
(iii)
. Considering (ii) in (A.2),
, .
;
as the state trajectory
(iii))
39
: (i)
(A.1), yielding , this ,
.
39 Page 39 of 55
inequality is strict. So, strategy
is dominated by
, implying that the
contract in question cannot be incentive-compatible.
Step 2: Suppose that an incentive-compatible contract determines a control and
, but
(contradicting the second assertion of
ip t
such that
the statement). Let then the alternative control
in which, for
cr
; (ii) . Denote .
Suppose
such that
.
as the state trajectory
. Considering (ii) in (A.4),
Multiplying
(ii)
an
obtained under control
;
us
and (iii)
: (i)
by
,
one
obtains
M
, yielding (when considering
(i) and (iii)) induction,
for
yielding
when
is dominated by
, implying that the contract in
te
. Again, since is strict. So, strategy
,
.
d
By
, and, by (A.3),
, this inequality
Ac ce p
question cannot be incentive-compatible.
Proof of Lemma 2: (a) Consider the auxiliary lemma below:
Lemma A.1: Suppose that constants
is a polynomial growth function, i.e., that there are
such that
(A.5)
in which
represents the Euclidean norm. Admit also that
programming principle such that, for any stopping time
satisfies the dynamic
,
40 Page 40 of 55
(A.6)
is a viscosity solution to (11).
Proof: See Theorem 5.1, chapter VIII, on Fleming and Soner (2006).
ip t
Then,
cr
To use the lemma above, it is necessary to prove (A.5) and (A.6). The auxiliary
Lemma A.2: The function
is continuous in
and satisfies the dynamic
an
programming principle.
us
lemma below proves the second condition.
It remains then to show that ,
d
such that, for
has polynomial growth. With this goal, define . From (6) and (7),
Ac ce p
te
process
M
Proof: See Lemma 2.1, chapter V, on Fleming and Soner (2006).
Applying the generalized Ito’s formula to
in which
,
and
,
represent the continuous parts of the respective processes.
Taking the conditional expectation to the above equation, 41 Page 41 of 55
By the definitions of
and
, the last summation of the above equation takes on and
. Note that Assumption 1(c) implies that
us
nonnegative values, just as such
cr
ip t
(A.7)
that
.
since
M
an
,
Thus,
the
above
equation
and
Ac ce p
,
letting
. Since
, , and given
, one has, after doing
the arbitrariness of the control As
and
te
d
Rearranging
,
.
, one concludes that
(A.8)
satisfying condition (A.5).
(b) Let
and
, such that . Define also
from initial conditions
,
and, for ,
and
are optimal controls under trajectories
and
as the state trajectories
. Suppose then that and
, let
and
, respectively.
42 Page 42 of 55
In turn, let
. is given by
us
cr
ip t
From this control, the state trajectory
one
notes
and of the initial state
an
By the definitions of the control
for
that (with
and
in
which
defined analogously). Also by the definition of
d
M
, one has40
the control
,
,
Ac ce p
te
(A.9)
in which the third row used the definition of controls
and
. However, by . Substituting (A.9)
the definition of the value function,
, which proves
into this inequality, it is concluded that the concavity of .
(c)
By
the
concavity
of
,
for
,
one
must
have
. 40
To obtain the second equality, define respectively. By the definition of , conditional expectation, definition of
, one also has
and
as the filtrations generated by trajectories and , . Since process is -measurable, by the properties of . Naturally, from an analogous .
43 Page 43 of 55
Suppose
for a given
One notes that
is a local maximum of
,
, with
and
.
, by the viscosity subsolution
cr
Since
, consider the smooth test function
ip t
. For
, and let
For
a
sufficiently
small
an
us
property of , one obtains
,
there
is
a
M
. Identically, one proves that
contradiction.
Therefore,
is continuous.
te
d
Appendix B: Assumption 2 One notes that, in ,
and
and
Ac ce p
vectors
,
, the process
is reflected according to
, respectively. Let
,
and
be
the unit vectors with the same direction of such vectors, i.e.,
44 Page 44 of 55
In addition, for and
, define
as the inward normal in relation to
defined analogously. Finally, for
in , with
, let
, in
represents the interior of the given set. Consider then Assumption 2 below:
, there are scalars
,
, such that
in which
us
cr
Assumption 2: For each
ip t
which
represents the scalar product.
an
Appendix C: Proofs of Lemmas 3-5 and Propositions 1-4
M
Proof of Proposition 1: Step 1: Suppose stopping time formula to
and define the
. Applying the generalized Ito’s
Ac ce p
te
d
, one gets
in which notes
,
and
(C.1)
represent the continuous parts of the respective processes. One that
45 Page 45 of 55
, and an analogous reasoning applies to
. Substituting these
ip t
results in (C.1),
us
cr
(C.2)
, and analogously for
such that
. Subtracting
concavity of ,
, and, by (11),
. By (A.8),
on both sides of the
, one obtains
. Then, by the
M
and letting
inequality, dividing it by
and
an
in which
. Thus, taking the conditional expectation
Ac ce p
te
d
to the equation above and rearranging the resulting expression,
(C.3)
By definition of the control
, one knows that
Suppose that the optimal control
implies
, with strict
inequality in at least one dimension. By (11) and by the construction of that,
for
, . Then, letting
,
.
,
, one knows and
in (C.3),
46 Page 46 of 55
which violates the dynamic programming principle in (A.6). Hence, optimal control in
Suppose
ip t
2:
.
,
and
define
cr
Step
is the
. Following the same procedures used in the
us
previous step, one obtains the same equation (C.3).
, one knows that
By definition of the control
implies
, with strict
an
Suppose that the optimal control
.
inequality in at least one dimension. In the present case, by construction, one has , letting again
,
and
te
d
M
in (C.3),
. Then,
Ac ce p
once more violating the dynamic programming principle in (A.6). Therefore, the . Moreover, by the concavity of
optimal contract implies knows that, for
which implies the optimality of optimal control in
and
, one
,
,
. It may then be concluded that
is the
.
Step 3: Performing an analogous procedure to that used in Step 2, one concludes that is the optimal control also in
41
and in
41
.
As done in Step 2, one shall note that, by the concavity of , for
, which implies the optimality of
in
. In turn, for
and
and
,
,
47 Page 47 of 55
Since
and in
one obtains the optimality of the control
Proof of Lemma 3: Define
,
.
as the contract value for the agent
us
cr
ip t
throughout the term of the contract, such that
is a martingale. Then, by the martingale representation
M
, which allows concluding that
and
such that
te
d
theorem, one knows that there are two predictable processes
. Calculating the differential
Ac ce p
in which
,
an
For
(C.4)
(C.5)
from (C.4) and (C.5), equating the
resulting expressions and considering the homogeneity with respect to time of the present problem, one gets (20).
Proof of Lemma 4: (a) This proof is quite similar to that of Lemma 2. Likewise,
one uses Lemmas A.1 and A.2. It is then necessary to show that
has a polynomial
growth.
To do that, consider the same process
used in the proof of Lemma 2, and
rewrite (A.7) taking into account that, in the present problem, the process
, leading to the optimality of
in
is a control:
.
48 Page 48 of 55
ip t cr
Since
,
.
given
that
the
above
equation
Ac ce p
Rearranging
te
d
M
an
us
,
Also,
. Once
has, after letting
and
letting
,
, and given the arbitrariness of the control
, one
. Since
, one
,
and
concludes that
(C.8)
proving that
has a polynomial growth.
49 Page 49 of 55
The proofs for items (b) and (c) are similar to those of the respective items of Lemma 2.
Proof of Proposition 2: Suppose
and define the stopping time
ip t
. Applying the generalized Ito’s formula to
(C.9)
M
an
us
cr
, one gets
that
and
an
analogous reasoning
applies
to
. In turn,
Ac ce p
,
te
d
Note
. Substituting these results in (C.9),
By (C.8),
such that
. Since . Dividing this inequality by
, and letting
50 Page 50 of 55
, one obtains
. Then, by the concavity of ,
. Therefore,
taking the conditional expectation to the above equation and rearranging the resulting
ip t
expression,
By the definition of the control
. Consider then a control
. By (24) and by the construction of
, one knows that, for
an
that implies
, one knows that
us
cr
(C.10)
and
. Then, by (C.10),
d
M
,
te
which violates the dynamic programming principle. Thus,
Ac ce p
.
In addition, by the concavity of is the optimal control in
Hence,
is the optimal control in
, one knows that
.
Proof of Proposition 3: Consider a nonnegative function ; and (ii) for
that: (i)
functions
and
.
,
such that
such . From the
proposed by the statement, one notes that Assumption 1 is satisfied.
Admit also that, for
By (11), one knows that, for
,
,
51 Page 51 of 55
and
. Since, according
ip t
By construction, , one has
to Lemma 2(c),
. One then concludes that
Proof of Lemma 5: Suppose and define
and
and
for
, and
, yielding
By the definition of
.
, and, by (8),
M
,
and
be defined analogously. By
. Since
. Given the arbitrariness of
,
d
,
be the optimal
as the state trajectories under
, respectively. Let (6) and (7),
. Let
an
control under
us
cr
.
.
Ac ce p
te
Still by (8), one knows that
If
, then
(C.11)
, yielding, by the result shown earlier,
(C.12)
If
, by (6) and (7),
and ,
and,
, one concludes that, if
. Since, according to (9), by ,
the
previous
result, . Inserting 52 Page 52 of 55
this result and (C.12) into (C.11) and considering that .
for
Letting
one
. The proof for the case in which
, obtains
and
is
ip t
analogous.
Proof of Proposition 4: Consider initially the function
cr
lemma below:
that satisfies the ordinary
us
Lemma C.1: (a) Consider a function
an
differential equation
; (ii) for
M
subject to conditions: (i)
,
.
te
for
; and (iv)
(C.13)
,
and
; (iii) for bounded on
. Then,
d
,
and the auxiliary
,
Ac ce p
Proof: Applying the generalized Ito’s formula to
Considering (C.13) and the conditions (ii), (iii) and (iv) in the above equation, one obtains, after taking the conditional expectation,
53 Page 53 of 55
in the above equation, after considering condition (i) and rearranging the .
resulting equation, one obtains
,
in (28) is immediate. The proofs for ,
however,
the
equations
are similar. In the case of in
(C.13)
are
us
and
and
and
for
given
by
, respectively, .
an
and conditions (ii) and (iii) are replaced by
the result
cr
From the lemma above, by applying the generalized Ito’s formula to
ip t
Letting
M
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d
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Ac ce p
te
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d
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