A Costless Way to Increase Equity

Transcription

1 A Costless Way to Increase Equity Raphael Flore October 27, 2016 Abstract This paper complements stanar theories of optimal capital structure by allowing firms to invest in the financial markets in which their own capital is issue. I show that, uner certain conitions, issuing new equity in orer to buy securities from other firms increases the equity level above the suppose optimum without causing any loss of firm value. This result inicates a way how capital requirements for banks can be increase without any private costs. The increase equity level provies aitional loss-absorbing capital an reuces the probability of bankruptcy for the firm/bank as long as the securities maintain some positive value in states in which the purchasing firm has insufficient revenues. JEL coes: G30, G32, G28, G10 I thank Martin Hellwig, Felix Bierbrauer, Viral Acharya, Alexaner Kempf, Jan P. Krahnen, Alexi Savov, an Kermit L. Schoenholtz, as well as seminar participants at Cologne, Frankfurt, an the NYU Stern for valuable comments an suggestions. Financial support by the CGS as well as the DAAD is gratefully acknowlege. flore@wiso.uni-koeln.e, University of Cologne an Max Planck Institute for Research on Collective Goos

2 1 Introuction 1.1 The Basic Iea an the Main Results The key argument against capital requirements for banks are the social costs that might arise from a reuction of lening by banks, which incur private costs ue to an enforce increase of their equity level. Amati et al. (2013) point out that many arguments for such costs are flawe an they instea emphasize the social benefits from an increase of capital requirements. This paper aresses arguments for private costs of equity increases that are usually taken serious, because they are base on stanar theories of corporate finance an banking that give reasons for the existence of an optimal capital structure. A common characteristic of these theories an the corresponing moels is their focus on single firms that invest in a firm-specific set of projects. The key innovation of this paper is to enlarge these moels in a simple but realistic way, namely by allowing for the possibility that firms can purchase bons or stocks of other firms. The key result is that uner certain conitions the equity level of a bank (or a firm, more generally) can be raise above the supposely optimal state without any loss of value, which means without any private costs. This is possible by issuing aitional stocks an using the procees to buy securities form other firms in the same capital market. Figure 1: The firm portfolio with prouctive assets A, which is finance with ebt D an equity E, is enlarge by issuing aitional equity E + in orer purchase securities from other firms. These integrate funs (illustrate in Fig. 1) provie aitional loss-absorbing capital, as long as not all purchase securities lose all of their value exactly in the same states in which the purchasing firm woul become insolvent (i.e. when the losses of A excee E). Even in the extreme case that a firm issues stocks worth E + = $ 100 to purchase equity of other firms whose value eclines by 90% in a crisis, the firm still has $ 10 more capital 1 than before to absorb losses from the firm assets A. The analysis of integrate funs is inspire by the liability holing companies (LHCs) that have been propose by Amati et al. (2012) in orer to increase capital buffers of banks 2. Although they suggest LHCs as a new an separate legal structure, their eco- 1 The example an the iscussion at the en of this section will illustrate why integrate funs ecrease the risk in the entire economy, even though the net amount of equity is not increase an the unerlying risk of the assets remains the same. 2 They motivate LHCs as an attempt to counteract the negative incentives of the limite liability of equity in a non-istortionary way. While it is unfeasible to access private funs of owners of highly isperse 1

3 nomic function is equivalent to the one of the integrate funs iscusse here 3. This paper aopts the iea of LHCs/integrate funs an evelops it in ifferent ways: First, I analyze the effect of integrate funs on the firm value by aing them to several stanar theories of corporate finance an banking an stuying the potential istortions of the optimal capital structure. Amati et al. (2012) iscuss the effect of LHCs on the firm value only with respect to agency costs of equity. Secon, while Amati et al. (2012) only make a qualitative argument, I systematically stuy the impact of integrate funs in an economic moel with an equilibrium of optimizing firms. (One result is that my conclusion concerning the agency costs of equity iffers from the one of Amati et al. (2012).) The following theories of capital structure are stuie in this paper: first, the trae-off theory between tax avantages of ebt an bankruptcy costs, see e.g. Moigliani an Miller (1963) or Kraus an Litzenberger (1973); secon, the agency cost theory escribe in Jensen an Meckling (1976); thir, the theory about the isciplining role of emanable ebt, as argue in Diamon an Rajan (2000), for instance; an fourth, the argument for a high leverage of banks ue to a premium for safe money-like ebt, which has recently been emphasize by DeAngelo an Stulz (2013) an Gorton an Winton (2014). While the first an secon theory have been evelope for generic firms, the last two are suppose to explain why banks have particularly high leverage. I neglect, however, another (probably important) reason for the strong leverage in the banking sector: the subsiies provie by implicit bailout guarantees. I chose this focus, because private costs that solely arise from the loss of these subsiies are usually not regare as serious argument against the increase of capital requirements. All theories mentione so far escribe a static trae-off between avantages an isavantages of leverage. A secon class of moels about the choice of financing focuses on the process of capital increases. A classic example is the pecking orer theory escribe by Myers an Majluf (1984). More recently, Amati et al. (2015) ientifie a leverage ratchet effect, which also leas to a ebt bias in the choice of financing. The problems ientifie by these theories are not relate to the size of the capital requirements, but to the way in which they are introuce. I will iscuss this issue in Section 6.1. The central result of stuying the impact of integrate funs on the firm value is: there is a way how equity levels can be raise above the supposely optimal level (an bankruptcy probabilities can be ecrease) without a loss of value, irrespective of which trae-off theory of capital structure applies. This is possible by issuing new equity claims in orer to purchase securities of other firms. With respect to banks, this result implies that the probability of their bankruptcies can be reuce without any costs an without an impairment of their economic function an prouctivity. stocks in orer to hol them liable, LHCs enable an extene liability by recourse to aitional, centralize funs that can easily be liquiate. 3 Theses LHCs shoul, first, be the exclusive owners of their corresponing banks, secon, hol securities from other firms, thir, be finance by equity only, an fourth, ensure the cash flow to ebt holers as long as either the bank equity or the purchase securities have some value. 2

4 The mechanisms that allow for this costless increase of equity are specific to the respective theories of capital structure. But there is a common unerlying iea. In principle, I stuy the capital structure of a hybri of a firm an a fun. A firm an a passive fun iffer in funamental aspects. The latter is mainly a set of financial contracts for the purpose of channeling cash flows from their sources to their receivers. A firm, in contrast, creates value an is a source of cash flows. It is a more complex structure in which, for instance, managers exert effort an obtain special knowlege about prouctive assets, or whose prouction can be istorte by bankruptcy proceures. As a consequence, the capital structure, which influences these processes, matters much more for firms than for funs. To put it another way, the frictions that are highlighte by the trae-off theories of capital structure an that lea to eviations from the Moigliani-Miller theorem are situate within firms. Since these frictions are absent in passive sets of financial contracts, there is no theory about the optimal capital structure of funs. When a fun is integrate in a firm, it might be affecte by the frictions within that firm, but not necessarily to the same extent as the actual firm prouction. To characterize the interaction of firm an integrate fun for each theory of capital structure is the main part of this paper. Let me briefly outline the theory-specific results. Although the conitions for a costless integration of funs vary, all of them are compatible with each other an can simultaneously be realize in an economy. Concerning the trae-off between tax benefits an bankruptcy costs, the results epen on the tax treatment of the integrate funs. One possibility is that they are treate like the actual firm prouction an thus increase the tax buren. Nevertheless, they can raise the firm value, if they are more efficient in ecreasing bankruptcy probabilities an costs than traitional equity increases. This is possible in case of an appropriate joint istribution of the cash flow of the purchase securities an the cash flow of the firm prouction. In aition to simply increasing the level of equity, integrate funs allow for an enhance iversification of cash flows 4. The secon possibility is that integrate funs are treate like inepenent funs an other passive sets of financial contracts. In that case, they are tax exempt an always increase the firm value, since they simply ecrease bankruptcy costs. This tax exemption can be implemente without a change of the taxation of the actual firm prouction an thus without a reuction of the tax revenues. It only avois a ouble taxation of cash flows that are alreay taxe on the level of the firm that issues the securities. The effect of integrate funs on a premium for safe ebt (that will be aresse in the same moel as the trae-off between bankruptcy costs an taxes) is either neutral or positive. The level of safe ebt that a firm can issue epens on its cash flow in the worst state, an this minimal cash flow can never be reuce by aitional cash flows from securities 5. 4 The relation to the literature about conglomerate mergers is iscusse at the en of Section In their iscussion of a premium for safe ebt, Gorton an Winton (2014) alreay inicate that equity increases use to purchase securities o not reuce the level of safe ebt an the relate premium. But they quickly reject this possibility, because they oubt that the purchase securities are able to ecrease 3

5 On the contrary, given an appropriate joint istribution between purchase securities an firm prouction, the minimal cash flow of the firm can actually be increase by integrate funs an the firm might be able to issue a higher level of safe ebt. Concerning the trae-off between agency costs of ebt (ue to risk-shifting) an equity (ue to reuce effort or a misuse of resources), it is important whether the performance of the purchase securities has an impact on the income of the managers. If the income epens irectly on the cash flow of the securities, the effect of an integrate fun is ambiguous: the opportunity costs of risk-shifting increase, but the incentive to exert costly effort ecreases, because the income of the managers becomes more epenent on exogenously given factors. (This fining contrasts with Amati et al. (2012), who suggest that the manager compensation shoul epen on the cash flow of the securities, because they o not account for the reuce incentives for effort.) In contrast, if the income of the managers epens only on the firm prouction to which they contribute, their incentives for effort are maintaine. The aitional cash flow from the securities reuces the bankruptcy risk, but the firm has the possibility to choose the same face value of ebt as in the absence of the fun (which is the secon eterminant of the manager behavior). The ebt simply becomes safer an more valuable. Thus, by maintaining the returns to effort an choosing the same ebt buren, the same behavior of the managers can be incentivize an the firm value oes not ecrease espite the integration of a fun. Concerning the isciplining role of emanable ebt, the effect of integrate funs epens on the joint istribution of the cash flows from the purchase securities an from the firm prouction. The firm value increases, if the cash flow of the securities is istribute in such a way that the resulting reuction of the probability of costly runs is larger than the increase of the cash flow in states in which managers can extract rents. Having obtaine these theoretical results, there might be the following concerns about the implementation of integrate funs: first, the preferences of investors might be istorte, if banks absorb securities from the market; secon, integrate funs woul be use more frequently, if they really were an avantageous way to increase equity; thir, stocks an other securities might be too volatile to provie capital buffers in crises; an fourth, the absorption of large parts of the capital markets into banks might affect the market mechanism. Let me briefly comment on these concerns: The potential istortions of the investor preferences are iscusse in more etail in the Sections A.1 - A.3, but in general the following hols: Although integrate funs mix cash flows that might satisfy ifferent preferences of investors, appropriate financial contracts can ecompose the combine cash flow into its constituents 6, as long as no value is lost. bankruptcy probabilities, base on the extreme assumption of perfect correlation between all firms. In Section 6.3, however, I will illustrate that integrate funs provie aitional loss-absorbing capital even in case of the worst realizations of aggregate risk. 6 For instance, if a security is absorbe into a bank that issues a stock to finance the purchase, the initial holers of bank equity can purchase the new stock an can sell a synthetic security to the initial holer of the security. 4

6 The empirical lack of integrate funs is aresse in Section 6.2, where I point out, first, which of the conitions ientifie above might not be realize, an secon, how they coul be realize in that case. But there is also an important alternative explanation: If there are ownwar istortions of the equity levels of banks ue to implicit bailout guarantees, compliance with increase capital requirements by means of traitional equity increases might be superior to integrate funs, because they bring the firm closer to its unistorte optimal level, while the subsiies are lost in either way. The thir an fourth concern, which means the provision of loss-absorbing capital by integrate funs an their impact on the market mechanism, are iscusse in more etail in Section 6.3. At this point, I simply want to conclue the introuction with an example that illustrates how the purchase of volatile securities from other firms can reuce the bankruptcy probabilities in an economy. This is possible, although neither the unerlying risk structure of the economy changes nor the aggregate net volume of equity increases (only the volume of equity in each firm increases). Accounting for frictions that give rise to an optimal capital structure, the example also inicates how an increase of equity on the firm level is possible without a reuction of the firm value. 1.2 A Simple Example of a Costless Reuction of Risk by means of an Integrate Fun Assume that there are two firms, A an B, finance with ebt an equity, an that there are three states of the worl, I, II an III, which are equally probable. Firm A has assets which yiel 105 in state I, 90 in state II an 105 in state III. Firm B has assets which yiel 90 in state I, 100 in state II, an 110 in state III. Assume that A an B incur a loss of b A an b B in case of insolvency. Furthermore, there is a loss that is proportional to the cash flow to equity holers, with factor δ in both firms. To stuy an interesting example, consier a case with 10δ < b B < 20δ an b A > 30δ. If the firms want to maximize their firm value (given as expecte cash flows to their claim holers), the optimal face value of ebt is 7 D A = 90 for firm A an D B = 100 for firm B. The resulting state-contingent cash flows are: Firm A value of... assets ebt equity State I (1 δ)15 State II State III (1 δ)15 Firm B value of... assets ebt equity State I b B 0 State II State III (1 δ)10 7 Consier firm B, for instance: Its firm value for D B = 100 is V B = bb 1 10 δ. This is a 3 relative maximum, because an infinitesimal increase of D B causes a jump in bankruptcy costs, while a ecrease of D B increases the losses from the cash flow to equity. The are two more relative maxima, at D B = 90 an at D B = 110. Comparing these maxima, one sees that D B = 100 is the absolute maximum, since 10δ < b B < 20δ. The optimum for firm A is erive analogously. 5

7 Having introuce the benchmark case, let us now consier the purchase of securities of one firm by the other one. To keep this illustrative example simple, I o not iscuss the full interepenent problem of both firms that optimally invest into each other. Instea, I simply stuy if firm B can increase its equity level an reuce its bankruptcy probability without a loss of value by purchasing claims in firm A. Given the optimal choices of A an B in the benchmark case, firm B has to buy the fraction δ of the equity of firm A in orer to reuce its bankruptcy probability. If it oes so, the resulting state-contingent cash flows are: 1 Firm A value of... assets ebt equity State I (1 δ)15 State II State III (1 δ)15 Firm B value of... assets ebt equity State I State II State III (1 δ) δ 2 The value of the purchase equity claims from firm A is (1 δ)15 = 20 3 in terms of its expecte cash flow. It increases the expecte cash flow of firm B by 1 3 (10+b B)+ 1 3 (1 δ)10. Remember that the initial capital structure of firm B was chosen because 10δ < b B. This relation implies that the increase of the value of firm B is larger than the value of the purchase securities. As a result, the purchase of equity from firm A raises the level of equity in firm B, ecreases its bankruptcy probability, makes its investors better off an none of the investors in firm A worse off. The risk of the unerlying assets has not change. An the aggregate net amount of equity, which means the value of equity hel by investors, has not been increase. It is equal to 2 3 (1 δ) (1 δ)20 = 50 3 (1 δ) 20 3, which is even smaller than in the benchmark case, where it is 2 3 (1 δ) (1 δ)10 = 40 3 (1 δ). Nevertheless, the probability of bankruptcies in the economy has ecrease from 1/3 to 0. This simple example correspons to a trae-off between taxes an bankruptcy costs without tax exemption of the integrate fun. It shows that increasing equity by issuing new stocks in orer to purchase stocks from A has an avantage over increasing equity by reucing the ebt level. It provies an aitional iversification of the cash flow, such that an equity increase by 1 3 (1 δ)10 can reuce the bankruptcy probability to zero, while a ebt reuction of the same size ( (1 δ)10 > 90) woul not achieve this. Instea, an equity increase by means of a ebt reuction must have the size 10 (from 100 to 90) to avoi bankruptcy, which woul lea to higher losses from taxation. The avantage of the integrate fun relies on an appropriate joint istribution of the assets. Appropriate joint istribution oes not mean that the correlation has to be negative; the correlation of the unerlying assets in this example is zero, for instance. The require properties of the joint istribution are ientifie for generic cases in Section 4. The literature about conglomerate mergers has alreay inicate that the iversification of cash flows within a firm can alter the optimal capital structure which results from a 6

8 trae-off between bankruptcy costs an taxes, see for instance Lewellen (1971), Lelan (2007) or Luciano an Nicoano (2014). There is a crucial ifference, however, between this paper an that literature: while the latter has stuie mergers of two firms, I stuy a hybri of a firm an a fun. An this hybri has a ifferent problem of optimal capital structure than a merger of firms. Most importantly, although the leverage after a merger might iffer from the leverage of the initial firms, there is a strong relation between the former an the latter, whereas a fun has no optimal capital structure an allows for larger changes in leverage. Apart from this, there are two other reasons why a iversification of cash flows by means of integrate funs iffers from the merger of two firms. First, integrate funs allow for a broaer iversification by purchasing securities from many ifferent firms instea of only one (or a few); an secon, they avoi costs that might arise if firms merge for the sake of an internal iversification, although their joint operation entails frictions. Let me conclue with a general remark why integrate funs improve the absorption of losses, even if the net amount of equity in an economy oes not increase. Integrate funs allow firms to share unuse loss-absorbing capital with each other. Integrate funs increase the capital buffers of the purchasing firms in all states in which the originating firms o not consume all of their capital themselves. This iversification on the level of the firms is superior to a iversification in the portfolios of investors, because the former can prevent bankruptcies while the latter cannot. The remainer of the paper is organize as follows: Sections 2 an 3 aress capital structures that arise from a trae-off between agency costs, with Section 2 setting a benchmark case for firms without integrate funs, before Section 3 analyzes the prospects of such funs given this trae-off. This two-step analysis is repeate in Section 4 for the trae-off between bankruptcy costs, taxes, an a premium for safe ebt, an again in Section 5 for the theory about the isciplining role of emanable ebt. Section 6 iscusses issues concerning the implementation of integrate funs, an Section 7 conclues. All proofs are given in Appenix B. 7

9 2 Benchmark Moel of Agency Costs This section escribes the trae-off between agency costs for a benchmark case of a generic firm without integrate fun, before the impact of integrate funs is stuie in the next section. Both sections consier risk-neutral investors, but the results are extene to more general preferences in Appenix A.1. The assumption of rational agents with symmetric information, however, is maintaine throughout. 2.1 The Structure of the Firm Problem There is a continuum J = [0, 1] of risk-neutral owners of firms. A firm owner j J can invest I j R + in a project at t = 0 that yiels the cash flow I j X j (R j ; j, m j ) at t = 1. The cash flow per unit of investment, X j, epens on the performance R j of the investment, which is stochastic with probability ensity function f j an E f [R j ] = Rj f j (R j ) R j < for all j J. Moreover, X j epens on the capital structure of firm, represente by the value j R + of its ebt at t = 0 per unit of investment, an on the manager compensation m j. Assume that the firm has to issue either equity or ebt claims in orer to obtain the funing of I j at t = 0. These claims are purchase by risk-neutral investors in competitive capital markets, in which the t = 0 -price for one unit of expecte cash flow at t = 1 is 1/r. In orer to focus on the static trae-off theories of capital structure, assume that the firms have no outstaning ebt at t = 0. The manager compensation is represente by the share m [0, 1] of equity claims that is given to the managers of the firm in orer to incentivize effort. To sum up, firm owners solve max I j (E f [X j ] ( j, m j ) r ), j R +,m j [0,1] with E f [X j ] ( j, m j ) := X j (R j ; j, m j ) f j (R j ) R j. The firm owners simply choose ( j, m j ) = argmax ( j,m j ) E f [X j ] ( j, m j ) an a firm will be active as long as 8 E f [X j ] ( j, m j ) r. The expecte cash flow E f [X j ] ( j, m j ) will be calle firm value henceforth. The etails of the pricing are as follows: The ebt level j at t = 0 implies a face value of ebt, D j, which ensures that the expecte cash flow at t = 1 equals r j. The appropriate relation between j an D j will be given in (c) in Eq. (2). Debt claims worth j,ext are sol to investors at t = 0 in orer to raise the share j,ext of I j, while the remaining ebt, worth j j,ext, is hel by the firm owners themselves. (This is mainly relevant, when j > 1 is optimal for the firm.) In orer to get the funing for the remaining share (1 j,ext ) of I j, the initial firm owners sell a share e j,ext of the firm equity to investors. The size of e j,ext has to fulfill e j,ext E f [X e,j ] = (1 j,ext ) r, where X e,j is the cash flow to equity holers at t = 1 per unit of I j. 8 More precisely, for E f [X j] ( j, m j ) = r the firm is inifferent between being active or not. 8

10 2.2 Agency Costs Following Jensen an Meckling (1976), the optimal capital structure is etermine by a trae-off between the agency costs of ebt an equity 9. The firm has a management that chooses its effort 10 an the amount of risk-shifting uring the perio (between t = 0 an t = 1) when the face value of ebt an the manager compensation are fixe. Anticipating the behavior of its managers, the firm chooses the optimal an m at t = 0. (The firm subscript j is suppresse in this subsection for the sake of brevity.) Assume that the revenue of a firm oes not only epen on the basic performance R of the investment, but also on the effort c m [0, c m ] of the managers, such that it equals ρ(c m ) R, with c m ρ > 0. Exerting the effort c m, the managers incur the isutility h(c m ) in money-equivalent units, with c m h > 0. If a firm reuces its ebt an issues more equity to investors instea, the incentive for managers to exert costly effort ecreases, since the return to this effort is istribute to all equity holers. This is the agency cost of equity. The agency costs of ebt is ue to risk-shifting: Between t = 0 an t = 1, managers can choose a share α of I j that they shift to a risky project. In case of success, which occurs with probability p, the project yiels an extra revenue α β +. If the project fails, the firm revenue is reuce by 11 α β. Assume that the risky project has a negative NPV, which means p β + < (1 p)β. Since managers are equity holers, they yet have an incentive for risk-shifting. An this incentive grows with their share in the upsie gains, which means it increases when the equity hel by investors is ecrease an replace by ebt. To sum up, the optimization problem of the managers is ( ) max m E f [X e ] (c m, α) h(c m ) with (1) c m [0, c m],α [0,1] { }] E f [X e ] (c m, α) = E f [max 0, ρ(c m )R + αβ + 1 β + αβ (1 1 β +) D The inicator function 1 β + ientifies all states with successful outcome of the risky project. The optimal choices c m an α epen on D an m, with D being a function of an m that is implicitly given by the relation (c) in Eq. (2). Both, an m, are chosen by the firm at t = 0, which anticipates the ecisions 12 c m(d(, m), m) an α (D(, m), m) of the managers. The firm problem is the maximization of the firm value, which is the cash flow 9 I follow the mainstream of the literature by iscussing agency costs for a restricte set of contracts, which only consists of ebt an equity claims an a manager compensation base on these claims. 10 or its iscipline to limit the use of firm resources for private benefits 11 One coul also assume that β as well as β + epen on ρ R. That woul not change the further analysis. In orer to avoi teious case istinctions ue to the limite liability of ebt, assume that ρr β > 0 for all realizations of R. 12 The functions c m an α are not necessarily single-value, as it is possible that Eq. (1) has relative maxima that are equally large. Assume that each of these maxima is chosen with equal probability in these cases, an the firm accounts for the resulting istribution of c m an α, when it solves its problem. 9

11 [ ] E f X from the firm projects minus the costs me f [X e ] of the management: max E R + f [X] (, m) =, m [0,1] with E f [X e ] (, m) = E f [max ( [ ] ) max E f X (, m) m E f [X e ] (, m) { 0, X(R; }], m) D(, m) R +, m [0,1] an X(R;, m) = ρ(c m)r + α [ 1 β +β + (1 1 β +)β ], given the constraints (a) c m = c m(d(, m), m), (b) α = α (D(, m), m), { (c) r = E f [min D(, m), X(R; }], m) (2) (3) One can show that both, firm problem an manager problem, effectively are optimizations of boune expressions over boune sets. Consequently, they have a solution 13 : Lemma 1 For each r > 0, there is an optimal choice (, m ) of the ebt level an the manager share of equity, which maximizes the firm value E f [X] an solves Eq. (2), taking into account the solution of the manager problem given in Eq. (1). 2.3 The Equilibrium of the Benchmark Case The risk-neutral investors have no preferences over the securities they purchase, as long as their expecte cash flow equals r. The capital markets can therefore be characterize by the eman an supply of generic financial claims, I an I s. The supply I s of claims at t = 0 is given by the investment volume of active firms 14 : Lemma 2 I s (r) = J I j 1 { E f [X j ]( j,m j ) r } j. (4) The supply function I s (r) is continuous 15 an monotonically ecreasing in r. Concerning the eman I for claims, let us simply assume that it is a continuous an monotonically increasing function of r: I = I (r) with r I (r) > 0 an I (0) = 0. It 13 It is possible that D(, m) given by relation (c) in the firm problem has no solution for some (, m). It epens on X(R;, m), which epens on D(, m) itself through c m an α, such that there might be problematic circularities. Those (, m), for which D(, m) has no solution, are exclue from the choice set of the firm. This set is never empty, however, since all (, m) with = 0 have the solution D(, m) = To be more precise, the supply function I s (r) can be multi-value, since the firm owners are inifferent about being active or inactive for E f [X j] ( j, m j ) = r. Consequently, I s (r) maps to all values between J Ij 1{ Ef[Xj]( j,m j )>r } j an J Ij 1{ E f[x j]( j,m j ) r } j. 15 As mentione in Footnote 14, I s (r) might be multi-value at some r. It is yet continuous at these points in the sense of multi-value functions, which means it is upper-hemicontinuous as well as lowerhemicontinuous. 10

12 coul be erive as the solution of a saving-consumption-ecision of househols, but the aitional structure woul not provie any further insights. Lemma 3 There is a unique interest rate r for which the capital market clears with I (r ) = I s (r ). To conclue, this section has introuce a benchmark moel that represents the trae-off between agency costs of equity an ebt in a very general setting. The trae-off leas to D an optimal capital structure for each firm in the economy. Denoting the ratio j ( j,m j ) E f[ X]( j,m j ) as leverage l j, the existence of an optimal capital structure implies that any eviation from the unconstraine choice lj = D j( j,m j ) E f[ X]( (for instance, ue to regulatory constraints on j,m j ) j or l j ) reuces the firm value E f [X j ]. If the firm value rops below r as consequence of such a eviation, the firm becomes inactive. If this happens for a set of firms with non-zero measure, this eviation leas to a ownwar istortion of the economic activity an of the resulting supply I s (r) of claims. This istortion eventually implies a lower expecte cash flow r for which the market clears. The next section will show that the leverage ratio can be reuce below lj without a reuction of the firm value, if one allows for equity increases by means of integrate funs. As a consequence, one can reuce the probability of bankruptcy without a reuction in the number of active firms or a reuction of the expecte cash flows in equilibrium. 3 Integrate Funs in the Presence of Agency Costs 3.1 Firm Problem with Purchase of Securities The moel shall now be enlarge by the possibility that firms can integrate a fun. This means that they can issue aitional equity in orer to buy a set S of equity an ebt claims issue by other firms in J. I will comment on the choice of S later, but let us first assume that there is a given portfolio S j of securities in which firm j can invest. The volume of purchase securities at t = 0 per unit of I j is enote as s j. The portfolio yiels a stochastic cash flow R Sj at t = 1 per unit of portfolio (measure by its value at t = 0). The joint istribution of R j an R Sj is enote ˆf j, with f j (R j ) = ˆfj (R j, R Sj ) R Sj. (The firm subscripts will again be suppresse in the remainer of this section.) A general assumption of this paper is that the firms buy these securities in the same competitive capital market as private investors. This implies E ˆf [R S ] = R S ˆf(R, RS ) R R S = r. (5) The managers have no impact on the firms whose securities constitute the portfolio of their firm. But the cash flow of the purchase securities has an impact on the behavior of the managers an hence on the cash flow from the investment project of the firm, such 11

13 that X becomes 16 X(, m, s). Since the buyers of the aitional equity, which is use to purchase the portfolio, eman an expecte cash flow I s r at t = 1, the firm problem is max I (E ˆf [X] (, m, s) + se ˆf [R S ] (1 + s)r ). (6) R +, m [0,1], s R + Owing to (5), the cash flow from the purchase securities cancels out the costs to buy these securities, an the firm problem again simplifies to maximizing the value E ˆf [X] of the firm project: [ ) max E R +, m [0,1], s R + ˆf [X] (, m, s) = max (E X] ˆf (, m, s) m E ˆf [X e ] (, m, s) R +, m [0,1], s R + The value of the firm project is influence by the integrate fun ue to its effect on the behavior of the managers. The characteristics of the investment project itself, however, o not change an one can assume that R an ρ( ) are inepenent of s. Let us also assume that the characteristics of the risky project are inepenent of the purchase securities. (I will iscuss this assumption later.) Then, the cash flow from the investment project, X(R;, m, s) = ρ(c m)r + α [ 1 β + β + (1 1 β +)β ], epens on s only through the choices c m an α of the management. It is therefore important how the compensation of the managers accounts for the integrate fun. There are two possibilities: either the compensation scheme accounts for the inepenence of R S from the managers an their payment only epens on the cash flow from the investment project, or this inepenence is ignore an managers are pai base on the overall cash flow (which means X plus s R S ). Since the first possibility is arguably the more natural one, the analysis will focus on that case. A iscussion of the secon case, however, will be given at the en of this subsection. If managers are pai by equity claims on the cash flow X from the investment project, which yiel X e, their ecision problem is ( ) max m E f [X e ] (c m, α) h(c m ) with (7) c m [0, c m],α [0,1] { }] E ˆf [X e ] (c m, α) = E f [max 0, ρ(c m )R + αβ + 1 β + αβ (1 1 β +) D The problem is the same as in the benchmark case, see Eq. (1). Nevertheless, the optimal choice (c m, α ) of the managers can iffer given the same an m, since s > 0 can change the bankruptcy probability an hence the face value D of ebt. In fact, the epenence of D on s (which is implicitly efine by relation (c) in the firm problem) is the only relevant 16 For the sake of notational consistency, I coul have written X(, m, 0) instea of X(, m) in the previous section, since it simply focusse on the special case s = 0. The same hols for other functions of (, m). For the sake of a parsimonious notation, however, I avoie this an, instea, exten without further notice all functions in this section by one argument. 12

14 change in the ecision problem of the firm: max E R +, m [0,1],s R + ˆf [X] (, m, s) = [ max with E ˆf [X e ] (, m, s) = E ˆf [ ) max (E X] ˆf (, m, s) m E ˆf [X e ] (, m, s) R +, m [0,1],s R + }] { 0, X(R;, m, s) D(, m, s) an X(R;, m, s) = ρ(c m)r + α [ β + 1 β + β (1 1 β +) ], given the constraints (a) c m = c m(d(, m, s), m), (b) α = α (D(, m, s), m), [ (c) r = E ˆf min {D(, m, s), X(R; }], m, s) + s R S Despite the complexity of the problem, one can erive efinite conclusions. These partly refer to the bankruptcy probability φ(, m, s) := 1 { X(R;,m,s)+s RS <D(,m,s)} ˆf(R, R S ) R R S. Proposition 1 For given r, consier a firm with optimal ebt level 0 an maximize firm value E f [X] ( 0, m 0 ) in the benchmark case [given by the firm problem in Eq. (2)]. If this firm can also purchase securities of other firms an if it pays its managers by equity claims on the cash flow X, such that the ecision problems given in Eq. (8) an Eq. (7) apply, then E ˆf [X] (, m, s ) = E f [X] ( 0, m 0). More generally, if one fixes s by aing the aitional constraint s = s to the problem given in Eq. (8), the constraine solution ( (s ), m (s ), s ) fulfills E ˆf [X] ( (s ), m (s ), s ) = E f [X] ( 0, m 0) for any s R +. This means that the firm value oes not ecrease for any size of an integrate fun an the firm is inifferent about the size of s. The bankruptcy probability, however, ecreases in s: s φ( (s ), m (s ), s ) 0. Although the face value D of ebt an the resulting manager behavior epen on s, the optimize firm value is inepenent of s. The key point is that both, an s, affect the firm value only through the face value of ebt an its effect on the manager behavior. If a firm has chosen the optimal values D = D( 0, m 0 ) an m = m 0 in the benchmark case, it can maintain this optimal choice for any s > 0. Staying with m = m 0, it can ajust accoring to the pricing relation (c) in the firm problem, such that D(, m 0, s) = D( 0, m 0 ). Keeping the optimize values for m an D, the incentives of the managers are unchange an the optimal firm value is maintaine. (8) Although the face value of ebt remains the same, the bankruptcy probability ecreases with increasing s, because X + sr S < D becomes 13

15 less likely with D an X(R; (s), m (s), s) being constant. While the face value of ebt remains the same, its efault risk ecreases an the value of ebt at t = 0 increases. Corollary 1 Consier a firm with optimal ebt level 0 an maximize firm value E f [X] ( 0, m 0 ) in the benchmark case, which can invest in a portfolio of securities with cash flow R S, but has to comply 17 with a leverage constraint l l reg R + for l(, m, s) = E ˆf D(, m, s) [. X] (, m, s) + s E ˆf [R S ] For any l reg > 0, the firm value obtaine by the constraine optimal choice ( c, m c, s c) is equal to the unconstraine benchmark: E ˆf [X] ( c, m c, s c) = E f [X] ( 0, m 0 ). The same result hols for a constraint of the bankruptcy probability, if φ reg > E ˆf φ(, m, s) φ reg [1 {RS =0} 1 { X(R; 0,m 0,0)<D( 0,m 0,0)} ]. This implies that the bankruptcy risk can be reuce below the level of the benchmark case (by imposing a constraint φ reg ] < φ( 0, m 0, 0)) without a loss of value, as long as [1 {RS >0}1 { X(R; 0,m 0,0)<D( 0,m 0,0)} > 0. E ˆf The firm can always reuce the leverage ratio l(, m, s) to an arbitrary small (but positive) value by choosing a face value of ebt equal to the one in the benchmark case an by choosing a sufficiently large s, since E ˆf [R S ] = r > 0. Owing to the result of Proposition 1, this is possible without a reuction of the firm value. The relation between a reuction of the leverage an a reuction of the bankruptcy probability φ epens on the joint istribution of the cash flows of the firm investment an the purchase securities. The firm can reuce φ by increasing s, if the purchase securities yiel R S > 0 in some states in which the firm woul be insolvent in the benchmark case ( X(R; 0, m 0, 0) < D( 0, m 0, 0)). By choosing sufficiently large s such that s R S + X D, bankruptcy can be avoie in all these states. Before I will analyze the equilibrium of an economy with integrate funs in the next subsection, let me briefly iscuss the alternative compensation scheme for managers. Assume that the managers are pai by a share m of equity claims on the overall cash flow X +sr S, such that their payment oes not only epen on the investment project of the firm, but 17 Formally, the firm solves the problem given in Eq. (8) with an aitional constraint: () l l reg. 14

16 also on the cash flow of the purchase securities. Their ecision problem is then ( ) [ ] max m E c m [0, c m], α [0,1] ˆf X S e (cm, α) h(c m ) with (9) [ { }] [ ] E ˆf X S e (cm, α) = E ˆf max 0, ρ(c m )R + sr S + αβ + 1 β + αβ (1 1 β +) D. In this scenario, integrate funs have an ambiguous effect on the manager behavior. On the one han, there is the positive effect that the aitional cash flow sr S increases the opportunity costs of risk-shifting, because the managers lose a share of ρ R + s R S in case of ba realizations of the risky projects (instea of only losing a share of ρ R). Moreover, the aitional cash flow s R S ecreases the bankruptcy probability an hence the probability that losses of the risky projects are actually shifte to the ebt holers. On the other han, if the managers income epens on cash flows that are unaffecte by their effort, the incentive to exert costly effort is reuce 18. Statements about the sign of the overall effect are unfeasible without etaile specifications of the functional forms of the moel. Amati et al. (2012), who were the first ones to suggest equity increases combine with the purchase of securities, have not iscusse ifferent compensation schemes an the potentially ambiguous consequences of integrate funs/lhcs on agency costs. They have highlighte the possibility to suppress risk-shifting, but they have not mention the ilution of manager incentives that occurs if their payment epens on the cash flows of the securities. Let me conclue this subsection with two brief comments. First, the assumption that the characteristics of the risky project are inepenent of the purchase securities is probably too restrictive, if the managers have some iscretion over the portfolio choice or if the manager compensation epens on the cash flow of the purchase securities 19. This anger from enlarge risk-shifting opportunities provies an aitional argument for a separation of the integrate funs from the payment an the iscretion of managers. Secon, the assumption of risk-neutral investors is also probably too restrictive. It is shown in Appenix A.1, however, that the result of this section hols for generic preferences of investors, too. Although the evaluations of cash flows by agents iffer from the risk-neutral case, it is still true that funs can be integrate in such a way that they o not change the behavior of the managers or the firm value. Integrate funs lea to a mixing of cash flows from ifferent firms, which might istort the preferences of investors. This mixing, 18 The relative return that managers receive on their effort coul only be maintaine, if they were pai by a share m of equity claims on the overall cash flow X + s R S that is inepenent of s, such that their share in X is inepenent of s. This woul imply, however, that the cash flow[ to managers increases with [ ] increasing s. This woul result in a reuction of the firm value E ˆf [X] = E X] ˆf me ˆf X S e. 19 If the managers can select the portfolio, for instance, they might choose securities with an appropriate correlation with the risky project in orer to amplify their upsie gains in the risk-shifting. Or if the firm has any business relations with firms whose securities it hols, managers coul have istorte incentives when they eal with these firms ue to their interest in upsie risk. 15

17 however, can be reverse by appropriate financial contracts between the investors. The key point is that the frictions, which give rise to an optimal capital structure, are situate within the firms but not in the capital market. These allow for flexible rearrangements of cash flows. 3.2 Equilibrium with Purchases of Securities The equilibrium of the benchmark case shall serve as reference point in this section. For that purpose, all parameters of the benchmark equilibrium will be enote by a subscript 0. The iscussion of the equilibrium will focus on the case that the income of the managers epens only on the cash flow from the firm project. As shown in Proposition 1, the firms are inifferent about the size of s as well as the set of available securities an their cash flow patterns. This makes the analysis of the equilibrium rather simple. There are no irect interepenencies an the capital markets can still be characterize by the eman an supply of generic financial claims, I an I s. Each firm chooses its optimal ( j, m j, s j ) at t = 0 an the resulting supply of financial claims is I j (1 + s j ), if the firm is active. In this subsection, ( j, m j, s j ) refers either to the unconstraine choice of firms or to their constraine choice subject to the regulatory requirements escribe in Corollary 1. The analysis hols for both cases. The overall supply of claims at t = 0 is I s (r) = J I j (1 + s j) 1 { E ˆf [X j ]( j,m j,s j ) r } j. (10) The eman for financial claims by external investors is the same as in the benchmark case, an shall be enote as Iinv (r). Aitionally, there is the eman for financial claims by active firms that purchase securities. The total eman for claims is therefore I (r) = Iinv(r) + s j I j 1 { } j. (11) E J ˆf [X j ]( j,m j,s j ) r Discussing the supply an eman functions, it is useful to istinguish between the gross eman an supply state in Eq. (11) an (10) an the net eman an supply, I,n an I s,n, in which the claims hel between firms are nette out. The net supply represents the volume of the actual firm investments, an the net eman represents the volume of financial claims hel by external investors. Lemma 4 I s,n := I s I,n := I J J I j s j 1 { E ˆf [X j ]( j,m j,s j ) r } j = J I j 1 { E ˆf [X j ]( j,m j,s j ) r } j I j s j 1 { E ˆf [X j ]( j,m j,s j ) r } j = I inv (12) The net supply I s,n (r) is ientical to the supply I0 s (r) in the benchmark case (escribe in Eq. (4)): I s,n (r) = I0 s(r) for all r R+. 16

18 The ientity of the supply curves follows from the fact that integrate funs o not change the firm value: E ˆf [X j ] ( j, m j, s j ) = E f [X j ] ( j,0, m j,0 ). The immeiate consequence is: Proposition 2 There is a unique r for which the capital market clears, I (r ) = I s (r ). This expecte cash flow r as well as the aggregate volume I s,n (r ) of firm investments are the same as in the benchmark equilibrium: r = r 0 an Is,n (r ) = I s 0 (r 0 ). These statements hol in case of unconstraine firms as well as in case of firms that are constraine by l reg or φ reg escribe in Corollary 1. Since neither the net supply nor the net eman curves change, the characteristics of the equilibrium remain the same. This means that integrate funs have no effect on the economic activity or the returns to any stake holer. Everyone is equally well off as in the benchmark case, but the increase level of equity ecreases the bankruptcy risk. 4 Taxes, Bankruptcy Costs, an a Premium for Safe Debt This section aresses the trae-off between bankruptcy costs an taxes. It also accounts for a premium for safe claims that has recently been emphasize by Stein (2012), DeAngelo & Stulz (2013) an Gorton & Winton (2014). Following the argument of Gorton an Pennacchi (1990), riskless ebt avois potential informational asymmetries an mitigates traing losses or costly information acquisition when financial claims are use as means of payment. Investors therefore accept comparably low risk-ajuste interest rates on such safe claims. The effect of this premium on the optimal capital structure is similar to the tax benefit of ebt, only restricte to a certain subtype of ebt. As in previous sections, the theory of capital structure is first introuce for a benchmark case without integrate funs, before the moel will be enlarge by the possibility of such funs. Again, the analysis is presente for the case of risk-neutral investors, while an extension to generic preferences is given in Appenix A.2. The assumption of rational agents with symmetric information, however, is maintaine throughout. 4.1 The Benchmark Case without Integrate Funs The general structure of the firm problem is the same as escribe in Section 2.1, only the secon variable iffers. This means that a firm j J = [0, 1] has to obtain the funing I j at t = 0 in a capital market with interest rate r in orer to finance a project with value I j X j at t = 1. Its ecision problem is max I j R +, s j j (E f [X j ] ( j, s j) r ) max E R+ j R +, s j f [X j ] ( j, s j), (13) R+ with E f [X j ] ( j, s j) := X j (R j ; j, s j) f j (R j ) R j, 17

19 where R j is again the state-contingent performance (istribute accoring to f j ), j is the value of ebt at t = 0 per unit of investment, an s j is the value of safe ebt at t = 0 per unit of investment. In orer to account for the special role of safe claims, I assume that there are two types of ebt: a senior one with safe cash flow at t = 1 an a junior one with efault risk φ j > 0. The volume of risky ebt per unit of investment at t = 0 is given as r j := j s j. In orer to escribe the premium for safe ebt in a concise way, let us assume 20 that investors enjoy a utility λ > 0 in money-equivalent terms per unit of safe ebt claim. As a consequence, they accept that the safe ebt yiels only r λ instea of the expecte cash flow r that risky ebt has to yiel. The value s j λ that is create per unit of I j by proviing this utility to investors contributes to 21 E f [X j ] ( j, s j ) in Eq. (13). Apart from this premium for safe ebt, firms face the classic trae-off between taxes an bankruptcy costs. The return on equity is taxe with a constant rate τ (0, 1), while both types of ebt are untaxe. Bankruptcy entails the cost b j > 0 per unit of I j an its probability φ j increases with s j an r j : (the firm subscripts are suppresse again) φ( r, s ) = D( r, s ) 0 f(r) R, (14) where D( r, s ) is the face value of ebt. It is given as D( r, s ) = D r ( r, s ) + (r λ) s, where D r ( r, s ) is the face value of risky ebt, which is implicitly given by 22 r r = ( 1 φ( r, s ) ) D r ( r, s ) + D( r, s ) 0 ( R (r λ) s b ) f(r) R. (15) In orer to focus on single value φ( r, s ) an D( r, s ) let us impose: Assumption 1 1 F (x) f(x) > b an f(x) is continuous for all x 0, with F (x) = x 0 f(y) y. The firm value is E f [X] (, s ) = E f [R] + λ s b φ( r, s ) T (, s ), with T (, s ) enoting the expecte tax payments. Owing to the tax shiel of ebt, only the cash flow X e to equity is taxe. But the tax oes not apply to the entire cash flow, but only to the return on equity. The expecte tax payments are thus τ(e f [X e ] e) with e being the value of equity (per unit of I) at t = 0. This value is given as iscounte value of the expecte cash flow to equity at t = 1: e = 1 r E f [X e ]. The expecte cash flow to equity is E f [X e ] (, s ) = E f [R] r + λ s bφ( r, s ). Owing to the linear tax, the risk-ajuste 20 As mentione, a microfounation for this utility coul be base on asymmetric information uring the perio (between t = 0 an t = 1), when nees for traing an the exchange of claims arise. In that case, safe ebt claims avoi either excepte losses from traing or costs for the acquisition of information. Such a microfounation, however, woul neither change nor contribute anything to the results of this paper. 21 This is equivalent to accounting for reuce financing costs in Eq. (13) by writing (r s j λ) = s j r λ instea of r. 22 There is the implicit assumption that negative tax payments in case of bankruptcy, which are a consequence of the linear taxation, go to the equity holers. If they were attribute to the ebt holers, some expressions in this section woul look ifferent, but the qualitative results woul not change. 18