Managerial leverage and risk-taking incentives in the case of endogenous balance sheet size

Transcription

1 Managerial leverage and risk-taking incentives in the case of endogenous balance sheet size Elisabeth Megally January 15, 2016 Abstract A potential shortcoming of the celebrated Merton (1974) framework is that it does not apply to cases where changes in capital structure actually modify firm size. For instance, the Merton model does not account for the fact that a change in equity volatility induced by a change in the firm s leverage and which results in an expansion of balance sheet size necessarily modifies the initial size of underlying asset rather than keeping it exogenously determined. This paper develops a sequential agency framework in continuous time that endogenizes balance sheet size by financial intermediaries through their joint control of firm financing and investment decisions. We relax the assumption of exogenous initial assets in the Merton (1974) model by allowing financial intermediaries financing and investment decisions to affect the stochastic process governing firm dynamics from both sides of the bank s balance sheet, i.e. through the asset side of the firm s balance sheet (by the choice of riskiness of assets) and through its liability side (by the choice of capital structure). While financial intermediaries that are residual claimants to the firm s assets have incentives to take on leverage because limited liability limits only downside risk, we show that contractual incompleteness induces greater risk-taking incentives through their discretion of both the moneyness and the convexity of the option in equity. 1 Introduction How are financial intermediaries incentives stemming from their equity ownership modified when balance sheet size is endogenously determined by intermediaries discretion of firm policies? Since the seminal contribution of Merton Swiss Finance Institute and University of Zurich. elisabeth.megally@bf.uzh.ch. I am particularly thankful and grateful to my PhD advisor, Michel Habib, for invaluable guidance and suggestions. I also thank Marc Chesney, Jérome Detemple, Thomas Geelen, Jean-Charles Rochet, Antoinette Schoar, Javier Suarez, Jean Tirole, Alexandre Ziegler as well as seminar participants at the SFI Workshop in Finance at Gerzensee (2015) and the SFI Academic Job Market Workshop (2015) for very helpful comments. Finally, I thank Yacine Barhoumi for mathematical assistance. All remaining errors are mine. 1

2 (1974), which draws on the isomorphic relationship between firm equity and a standard call option on assets, managerial risk-taking incentives have been derived making use of the contingent claim framework of firms balance sheets. The derivation of incentives assumes that the underlying asset of claims is exogenously determined. However, since equity volatility is endogenously determined by the degree of riskiness of assets and the firm s capital structure, this assumption may not necessarily hold. Indeed, assuming an exogenously-determined firm size entails that any changes in the firm s capital structure must imply either an equity buy back through the issuance of debt, or a debt repurchase through the issuance of equity. As a result, Merton s framework does not apply to cases where changes in capital structure actually modify firm size. For instance, the Merton model does not account for the fact that a change in equity volatility induced by a change in the firm s leverage and which results in an expansion of balance sheet size necessarily modifies the initial size of underlying asset rather than keeping it exogenously determined. The aim of this paper is to modify the Merton model to allow for financial intermediaries determination of balance sheet size through their discretion of firm policies. Allowing for leverage-driven balance sheet size expansion necessarily implies a departure from the Modigliani-Miller setting of irrelevance of capital structure. Since leverage allows financial intermediaries to expand the initial asset size, capital structure does affect firm value. In essence, firm value now depends on both the asset side and the liabilities side of balance sheets. Drawing on the contingent claims analysis, we develop a theoretical framework that allows us to replicate the behavior of financial intermediaries. To account for leverage decisions that modify asset size, we design an agency framework in continuous time whereby cash-endowed investors delegate the portfolio management of their capital to financial intermediaries. Delegated management with endogenous balance sheet size involves financial intermediaries acting as agents to both shareholders (whose capital they manage) and to creditors (whose funds they borrow in order to lever up shareholders initial capital), our model involves a finite-horizon common agency setting in continuous time that endogenizes firm decisions by value-maximizing intermediaries. The sequential decision-making process has (unskilled) wealthy investors first contract with skilled financial intermediaries in order to the delegate the management of their initial capital, and financial intermediaries then contract with creditors in order to lever up the investors initial capital. Compared to the extant literature, we explicitly allow financial intermediaries decisions to affect the stochastic process governing firm dynamics from both sides of the bank s balance sheet. We assume that financial intermediaries decisions affect firm value not only through the asset side of the firm s balance sheet (by the choice of riskiness of assets), but also through its liability side (by the choice of capital structure). In comparison to models of controlled-drift and/or controlled-diffusion, our model entails that the value of the underlying asset at the initial date is also controlled by managerial discretion of firm policies. 2

3 Providing bank managers or financial intermediaries with discretion of firm leverage may often lead to agency conflicts between firm s stakeholders when managers are incentivized to expand the firm s balance sheet size beyond the optimal size (Jensen, 1986). Indeed, financial intermediaries raising debt to expand balance sheet size may choose to allocate the funds raised to other purposes that benefit them only, giving rise to agency problems if contracts are incomplete (Dow and Hang, 2014). While Jensen argues that the issuance of debt can help alleviate problems, this arguably does not apply to the case of debt creation with retention of proceeds of the debt issue. To the contrary, managerial discretion of financing decision exacerbates, rather than mitigates, the problem of free cash when debt is issued with a view to growing firm size (balance sheet size expansion), as opposed to modifying the composition of the firm s liabilities structure for a given firm size (balance sheet optimization). We account for the fact that contracts between investors (who act as principals) and financial intermediaries (agents) are incomplete by allowing financial intermediaries discretion of firm decisions to potentially lead to outcomes that diverge from investors optimum. Contractual incompleteness is modeled as in Dow and Hang (2014) where the principal cannot contractually enforce the agent to act in his own interest. We assume that financial intermediaries derive private benefits from controlling assets and may thus be incentivized to take on more leverage than the shareholder-maximizing level. As in Dow and Hang, contractual incompleteness is modeled by assuming that the investor cannot determine whether the financial intermediary is characterized by good management (which entails maximizing equity value) or bad management (which entails both maximizing equity value and private benefits of controlling assets). The severity of agency conflicts is increasing in contractual incompleteness which prevents other stakeholders from monitoring financial intermediaries discretionary use of funds. In addition to modeling managerial private benefits from controlling assets which essentially lead to shareholder-manager conflicts, we allow for managerial diversion of funds to nonproductive assets that confer private benefits to intermediaries. Since diversion of funds raised from creditors essentially introduces potentially conflicts between managers and creditors, our theoretical framework allows us to model both types of agency issues nested in a sequential principalagent problem. We investigate the extent to which a relaxation of the assumption of exogenous assets enables us to replicate the behavior of financial intermediaries. We assume that balance sheet size is discretionarily set by financial intermediaries through their joint control of the firm s financing and investment policies. Within the contingent claims framework, this latter assumption implies that financial intermediaries have the ability to determine the value of the underlying asset of all contingent claims on the firm s assets, and can therefore affect their own expected payoff through the implicit call option embedded in equity. 1 1 Hence, by determining both firm dynamics and the notional value of debt, the intermediary s 3

4 In recent papers, Adrian and Shin (2010, 2015) have shown that banks balance sheets exhibit some equity stickiness and are instead driven by leverage dynamics. The procyclicality of leverage, they argue, provides evidence that balance sheet size evolves endogenously with banks leverage. While Adrian and Shin (2015) address the issue of how much financial intermediaries can borrow from uninsured wholesale creditors for a given initial (equity) capital, we formalize their approach by developing an agency framework that relies on option pricing theory. Besides endogenizing the stochastic process governing firm dynamics to account for managerial impact on firm value through both sides of the bank s balance sheet, we further depart from Adrian and Shin (2015) by introducing a common agency framework. Since the financial intermediary s financing decision relies on external financing, the issuance of corporate debt is necessarily subject to the creditor s participation constraint a feature that is also present in Adrian and Shin (2015). Yet, whereas they only consider the agency relationship between financial intermediaries and lenders, our framework extends their analysis to the inclusion of a second principal, i.e. wealthy unskilled investors. Given that investment and leverage decisions in risky securities require specific skills, we assume that the investor cannot retain any control of either decision, and must thus cede the full control of financing and investment decisions to the intermediary. Poor monitoring abilities by the unskilled investor in turn imply that a contract cannot be made contingent on the entrepreneur s actions, requiring the investor (principal) to compensate the financial intermediary (agent) with a performance-dependent payoff that involves both equity ownership and a fixed salary. Since delegated management with endogenous balance sheet size involves financial intermediaries acting as agents to both shareholders (whose capital they manage) and to creditors (whose funds they borrow in order to lever up shareholders initial capital), our model involves a finite-horizon common agency setting in continuous time that endogenizes firm decisions by value-maximizing intermediaries. The sequential decision-making process has (unskilled) wealthy investors first contract with skilled financial intermediaries in order the delegate the management of their initial capital, and financial intermediaries then contract with creditors in order to lever up the investors initial capital. Our model therefore draws on both contracting theory and option pricing theory. While the former allows us to modelize financial intermediaries incentive structure when these are non contractible through the financial intermediary s incentive compatibility requirement, the latter allows us to make explicit the dependence of managerial wealth on his financing and investment decisions since the face value of debt and the degree of riskiness of the firm s assets both affect the value of the option implicit in equity and enables us to account for the fact that financial intermediaries exploit the option-like feature of equity on assets when deciding how much leverage and asset volatility to take. decisions essentially affect his own expected payoff through both the value of the underlying asset (i.e. balance sheet size) of his contingent claims and the boundary conditions associated with contingent claims. 4

5 A major advantage of relying on option pricing theory is the possibility of explicitly introducing decreasing returns to scale with leverage. Whereas Adrian and Shin (2015) assume that balance sheet capacity expansion exhibits constant returns to scale which implies that borrowers would want to borrow as much as possible we instead model the drift component of the underlying stochastic governing firm dynamics to account for the fact that, empirically, value creation through the expansion of the firm s assets with leverage entails diminishing returns to scale. Additionally, it is important to note that while the standard approach to pricing contingent claims in option pricing theory relies on the equilibrium determination of the state price density to obtain the equivalent martingale measure necessary to price claims under risk-neutral valuation, ours is by contrast a partial equilibrium approach. Whereas in option pricing, it is standard to determine the stochastic discount endogenously such that the equivalent martingale measure exists and is unique, we take the opposite and partial equilibrium view that financial intermediaries endogenously determine firm value by taking the stochastic discount factor as exogenously determined. The latter is equivalent to stating that agents decisions do not affect the level of systematic risk in the economy. Specifically, we consider an economy characterized by two correlated sources of risk, systematic and idiosyncratic risk. In our partial equilibrium setting, the exogenous stochastic discount factor affects the valuation of the agents claims. As systematic risk cannot be diversified, it is a priced risk factor that commands a risk premium which has the effect of raising the agents (subjective) discount rate. Our first contribution is to show that the (contractual) requirement that the creditor breaks even constrains the manager s ability to expand the firm s balance sheet irrespectively of either skin-in-the-game constraints (Holmstrom and Tirole, 1997) or Value-at-Risk constraints (Adrian and Shin, 2015). Because we explicitly model decreasing returns to scale with leverage, and rely on the pricing of all corporate liabilities as contingent claims on assets, we show that balance sheet capacity expansion through leverage is contingent on the manager s investment and financing decisions, with contingency arising through the fair pricing of default risk in corporate risky debt. Moreover, we show that endogenizing the firm s balance sheet size significantly affects managerial incentives because the endogenous component (i.e. corporate risky debt given the assumption of equity stickiness) of the firm s balance sheet is a fixed point. As we show, a relaxation of Merton s assumption of exogenous firm size results in the levered balance sheet being dependent on the value of corporate risky debt as a contingent claim on assets. Technically, the latter implies in conjunction with the assumption that firm value follows an Itô process that the market value of debt is a fixed point equation. This entails that the extent to which the manager can expand the firm s balance sheet through the issuance of corporate risky debt ultimately depends on his financing and investment decisions. Our theoretical setting shows that when we allow for endogenous balance 5

6 sheet size in an agency framework using the contingent claim analysis, financial intermediaries optimally refrain from issuing excessive debt and investing in excessively risky assets. Arguably, because the contractual framework between lenders and creditors gives rise to an endogenous boundary on balance sheet expansion, value-maximizing financial intermediaries holding equity are forced to internalize their firm decisions if they want to maximize the value of their claims. We demonstrate that our findings are inherently explained by the short put option implicit in the creditor s claim, the pricing of which actually reflects the manager s firm policy choices through the creditor s participation constraint. We argue that when financial intermediaries simultaneously control the financing and investment policies, they are forced to internalize the effect of one policy on the other. By contrast, we show that leverage-related decisions and risk-taking incentives of bank managers at too-big-to-fail banks are significantly different from those of financial intermediaries because of the implicit or explicit government insurance that only applies to the former. We show that those incentives increase when the creditor s claim is partially insured and are considerably higher when his claim is fully insured. Our model therefore shows that the finding of moral hazard behavior implied by implicit government insurance arises in the case of endogenous balance sheet size through the pricing of debt and results from a modification of the extent to which the short put option in the creditor s participation constraint is borne by the bank manager. In particular, we show that the modified creditor s participation constraint in the case of deposit insurance is consistent with Gray, Merton and Bodie s (2010) paper. They assume that government deposit insurance is akin to an assetlabeled government guarantee amounting to a fraction of the short put option in creditors claim, with the remainder of the fraction representing credit risk remaining in the debt and deposits of the financial sector. Indeed, we show that with (partial, resp.) government insurance, the bank s levered balance sheet size following the issuance of debt is decreased by precisely (a fraction of, resp.) the put option value that the creditor is short to the manager. If the manager has to bear the entire decreasing effect from the short put option, the reduction effect of the balance sheet size is accordingly greater. In addition, we argue that our modeling framework could provide an analytical characterization of the claim that the too-big-to-fail status of large banks prompts them to behave as though their borrowing cost does not fully reflect their policy choices and thereby affects these banks risk taking. If investors, creditors in particular, believe that certain banks are too big to fail, they will discount risk when providing those banks with funding. This insensitivity of financing costs to risk will encourage too-big-to-fail banks to take on greater risk. (Santos, 2014.) A number of studies have indeed documented that too-big-to-fail banks enjoy an effective subsidy, which essentially allows them to benefit from lower spreads on the bond market (See Santos (2014) for an overview). Acharya et al. (2013), for instance, report the reduced sensitivity of bond credit spreads enjoyed by the biggest financial institutions to underlying risk. These banks are shown to take on greater risk than smaller banks which do not qualify for the 6

7 too-big-to-fail status. By endogenizing both the bank s policies and balance sheet size, we show from an option pricing framework that excessive policies resulting from deposit insurance implicit in the too-big-to-fail status can potentially arise because both financial intermediaries and creditors rationally anticipate governments deposit insurance, and thus implicitly behave as if the value of the short put option embedded in the levered firm s balance sheet were multiplied by a coefficient less than unity. If financial intermediaries expect government guarantees to occur ex-post, they will perceive the risk to be very low and will thus behave as if they are not internalizing the full impact of their investment policy on the financing cost. They will thus be able to take considerable risk without bearing the cost on the balance sheet expansion. In the limit, full insurance amounts to removing the entire short put option thereby nullifying the impact of the pricing of default risk on the balance sheet size. We show that the extent to which the manager bears the cost of his financing and investment policies significantly affects his ability to expand the firm s balance sheet capacity for given equity, which in turn determines his risk-taking behavior. Whether the short put option is borne in full or partially by the manager actually determines his investment decision. If the manager has to bear the full short put option implicit in the creditor s claim, his ability to expand balance sheet capacity is severely constrained if he chooses excessively risky and highly leveraged policies. Endogenous balance sheet growth with no deposit insurance therefore induces the manager to adopt more conservative policies in order to maximize balance sheet capacity expansion for given equity. In other words, when the manager is required to internalize his firm decisions, balance sheet expansion is reduced by the whole amount of the put option, which is increasing in both the face value of debt and the degree of riskiness of assets. Full internalization therefore entails that balance sheet is increased by less than the face value of debt issued (discounted at the riskfree rate) due to the presence of the short put option implicit in the creditor s claim. By contrast, if the manager does not internalize his firm decisions, his capacity to expand the firm s balance sheet size for given equity is not affected by his financing and investment decisions. No internalization implies that the balance sheet is increased by exactly the face value of debt issued (discounted at the riskfree rate) because the creditor s claim is insured against potential default by the manager. Finally, because our analytical framework relies on the valuation of contingent claims within an agency framework, our results can justifiably be interpreted in light of both option pricing theory and contracting theory. Option pricing provides a means of estimating managerial incentives by partial derivatives of the call option in equity and the short put option implicit in corporate risky debt. We have shown that the value of the manager s (junior) claim on assets is a function of the value of the creditor s (senior) claim on assets by the accounting identity of the levered firm s balance sheet. Hence, bank managers can be viewed as trading off the positive vega associated with the long option in equity for the negative vega associated with the short put option on assets. 7

8 Whereas the latter appears to dominate the former in the no-insurance case resulting in asset risk-taking disincentives, we show that government deposit insurance amounts to dwarfing the coefficient associated with the short vega while keeping the long vega unchanged, thereby distorting bank managers risk-taking incentives. From a contracting perspective, we show that whereas the shadow price associated with the Value-at-Risk constraint in the no-insurance case is very close to zero, indicating that the constraint is slack, it is only when the government (partially) insures the creditor s claim that the shadow price of the Value-at- Risk constraint is highly positive, indicating that absent this constraint, the manager would have optimally chosen more excessive financing and investment policies. The remainder of the paper is organized as follows. Section 2 provides a description of the modeling assumptions (2.1) and of the underlying contractual framework (2.2). Managerial incentives in the case of endogenous balance sheet size growth are derived in the case of market completeness (section 3.1) and in the case of market incompleteness (section 3.2). Section 4 reports numerical solutions whereas section 5 extends the previous analysis to the case of government deposit insurance. 2 Contracting framework for endogenous balance sheet size In order to capture potential agency conflicts associated with financial intermediaries discretion of firm size, we construct a simple two-period contingent claims model in continuous time. There are three types of agents in the model: (i) wealthy investors endowed with capital to invest but no financial skills, (ii) financial intermediaries with financial skills but no wealth of their own, and (iii) creditors (available in large supply) with sufficient funds to lend. Wealthy investors distinguish themselves from creditors in that the former are expectedwealth maximizers in search of the most profitable investment opportunities (i.e. investors), whereas the latter can be viewed as passive providers of capital (i.e. lenders). Contractually, investors seek to maximize their expected wealth whereas creditors only require that their expected discounted payoff equals their initial investment. If investors invest directly in financial securities, they can only earn a given expected return. Delegating the management of their capital to financial intermediaries enables them, by contrast, to earn a higher expected return. However, investment and leverage decisions in (risky) financial securities require sophisticated management skills which preclude the investor from retaining the control of financing and investment decisions. Delegated portfolio management thus necessitates full delegation to the intermediary. Yet, contractual agreements between investors and financial intermediaries are by definition incomplete as investors cannot design a contract that makes in- 8

9 termediaries financing and investment decisions contractible. Poor monitoring abilities by the unskilled investor indeed imply that a contract cannot be made contingent on the intermediary s actions, requiring the investor to compensate the intermediary with a performance-dependent payoff. Contractual incompleteness, therefore, entails that delegated investment management is prone to agency conflicts between investors (who act as principals) and financial intermediaries (who act as agents) as these may take financing and investment decisions that deviate from investors optimal choices. We model two types of agency conflicts. First, we assume that financial intermediaries have empire-building motivations and thus derive private benefits of controlling assets (see Dow and Hang (2014)). In addition, we also allow for intermediaries potential diversion of funds arising from their discretion of firm policies when the latter are noncontractible. Delegated management occurs despite potential agency conflicts because portfolio management requires sophisticated skills, which we model through intermediaries ability to generate a higher drift term. Since delegated management with endogenous balance sheet size involves financial intermediaries acting as agents to both shareholders (whose capital they manage) and to creditors (whose funds they borrow in order to lever up shareholders initial capital), our model involves a finite-horizon common agency sequential decision-making process in continuous time. The optimal contract is determined by value-maximizing investors whereas firm policies are set by valuemaximizing managers. The sequential decision-making process has (unskilled) wealthy investors first contract with skilled financial intermediaries in order to delegate the management of their initial capital, and intermediaries then contract with creditors in order to lever up the investors initial capital. The model has two periods. In the first period, the investor hires a financial intermediary to manage its capital, which entails leveraged investments in (risky) financial securities through the issuance of risky debt. In the second period, the contract expires and all claims are settled. Provided no default occurs, the firm s assets are liquidated. Debt is first paid off to the creditor, and equity-holders receive the liquidation dividend, if any, as residual claimants. 2.1 Modeling assumptions We consider an economy whose underlying probability space (Ω, F, P) is enriched with some filtration F = (F t ) t0 t T. Time is continuous and all contracts are characterized by a finite horizon date T > 0. All agents are assumed to be risk averse and assign the same probabilities to the different outcomes of risky claims. 2 Although the model is characterized by arbitrage opportunities, because trading is restricted, no agent can attempt to exploit such opportunities. Uncertainty in the economy is described by a two-dimensional correlated Brownian motion, with the former referring to idiosyncratic risk and the latter 2 Alternatively, we could assume that the intermediary s subjective probability structure differs from that of shareholders due to superior information regarding the possible outcomes of risky assets in which he invests on behalf of shareholders. Superior managerial information leads him to assign different weights than those of shareholders to the various outcomes of the risky claims. 9

10 to systematic risk. We take a partial equilibrium approach and assume that the economy is characterized by an (exogenously-given) stochastic discount factor which can price all claims dm t m t = µ m dt σ m dw m t, (1) where µ m denotes the constant drift of the pricing kernel, σ m the volatility of the pricing kernel, and {Wt m } t0 t T a P-standard Brownian motion with respect to the filtration {F t } reflecting systematic t0 t T shocks.3 It can be shown that µ m = r since a riskless asset must earn the risk free rate. Firm policies. In contrast to Adrian and Shin (2015), we explicitly model the underlying distribution of firm dynamics. We assume that firm value evolves stochastically over time with dynamics following a geometric Brownian motion, and allow the financial intermediary s choice of financing and investment policies to affect firm value through the drift term, the diffusion, and the initial value of the underlying asset. In order to model the behavior of sophisticated financial intermediaries who control firm size through their financing and investment choices, we assume that intermediaries have decision rights over both the investment policy (which entails selecting the degree of riskiness of the financial security to invest in) and the financing decision (which defines how much leverage to take in order to lever up the risky position). Formally, this amounts to assuming that the investment decision determines the diffusion σ a of the underlying process whereas the financing decision determines the notional amount of debt D. For simplicity, we assume that debt represents a discount bond with maturity date T. In the contingent claims framework, managerial discretion of firm policies can be shown to affect the pricing of corporate liabilities. Whereas the intermediary s investment decision determines the volatility of the claims underlying asset, his financing decision determines the strike price of the option embedded in equity, and by virtue of the balance sheet identity and the degree of seniority attached to corporate claims, the promised payment to the creditor. Firm dynamics are thus described by the following process under the historical measure P 4 da t A t = µ + ν (D) dt + σ a dw a t, (2) where σ a denotes the constant volatility of assets, {Wt a } t0 t T a standard Brownian motion defined on the probability space (Ω, F, P), and F {.} the associated Brownian filtration. The drift term of firm dynamics is composed of two components: (i) µ represents the firm s momentum independent of managerial dis- 3 See Bakshi and Chen Because we rule out taxes and bankruptcy costs, the total value of the firm is identical to the value of the firm s assets. Allowing for leverage-driven balance sheet growth necessarily implies that we depart from the Modigliani-Miller setting. As firm value explicitly depends on the firm s financing policy, the theorem of capital structure irrelevance no longer holds. Compared to most other papers, because we assume away taxes, bankruptcy costs, or any other frictions, the dependence of firm value on the firm s financing structure matters not through the debt-equity mix, but rather through the firm s leveraged balance sheet. 10

11 cretion (Cadenillas et al., 2003), and (ii) ν (D) captures decreasing returns to scale, ν (.) < 0 and ν (.) < 0. We assume that macroeconomic shocks and shocks to firm value are positively correlated, where W a, W m t = ρ a t with ρ a < 1. By assuming an (imperfect) correlation between idiosyncratic risk (which is discretionarily chosen by the manager) and systematic risk (which is exogenously determined), yet assuming away diversification opportunities, we allow for only the latter risk and the correlated component of the former with the latter to be priced risk factors. Since firm value is neither observable nor a tradable asset, we cannot invoke the risk-neutral valuation argument in order to price the firm s liabilities as contingent claims on assets. We thus price claims under the objective (or physical) measure. Even if the discounted firm process is not a martingale under the objective measure, the fact that trading is restricted in the model precludes agents from exploiting arbitrage opportunities. More importantly, because we assume that all agents assign the same probability to each state s occurrence, they all have the same valuation of each claim on assets. In other words, all agents price equity and corporate risky debt similarly. Another key assumption which distinguishes our paper from that of Adrian and Shin (2015) relates to banks capacity to expand their current balance sheet size through leverage. Whereas they assume that balance sheet capacity expansion exhibits constant returns to scale which implies that the borrower would want to borrow as much as possible we instead model firm dynamics to account for the fact that, empirically, value creation through the expansion of the firm s assets with leverage entails diminishing returns to scale. We thus impose that the drift of asset dynamics be composed of an exogenous component µ and an endogenous component ν which is function of the face value of debt D, and impose ν(.) < 0 and ν(.) > 0 to reflect decreasing returns to scale (of firm value) in leverage Optimal (second-best) contract in the presence of agency problems As in Adrian and Shin (2015), our objective is to modelize the behavior of financial intermediaries in the context of equity stickiness, and therefore to allow for leverage-driven balance sheet expansion. Whereas these authors do not make any explicit assumption on the distribution of the underlying asset, we rely on the contingent claim framework which assumes that the stochastic process of firm dynamics is modeled as an Itô process. Endogenizing balance sheet size in the contingent claim framework entails that the intermediary s 5 On a technical note, because the assumption that firm value follows a geometric Brownian motion necessarily implies that the firm s assets are scalable at constant returns to scale, our modeling framework would absent the assumption that the drift term be a decreasing function of the face value of debt fail to account for the fact that banks cannot arguably scale up balance sheet size through leverage while still enjoying the same expected return µ. These assumptions will be shown to have key implications on managerial incentives with respect to both leverage and asset volatility. By imposing that the drift term be decreasing in leverage, we explicitly allow for decreasing returns to scale of firm value with leverage and, therefore, that value creation through the expansion of the firm s assets with leverage entails diminishing returns to scale. 11

12 choice of face value of debt affects the endogenous component of the drift term whereas his choice of degree of volatility of assets determines the diffusion of the process, and that both the intermediary s debt and volatility decisions determine balance sheet size. As stated in the introduction, allowing for leverage-driven expansion of the firm s balance sheet requires that we relax Merton s assumption of exogenous asset size in order to account for the endogenous feature of both assets and leverage empirically documented by Adrian and Shin (2010). As we show below, a relaxation of this assumption results in the levered balance sheet being dependent on the value of corporate risky debt as a contingent claim on assets. Technically, the latter implies in conjunction with the assumption that firm value follows an Itô process that the market value of debt is a fixed point equation. We design a common agency framework that endogenizes the financial intermediary s compensation contract, firm policies and the balance sheet size. Whereas Adrian and Shin (2015) only consider the agency relationship between financial intermediaries and lenders, our framework extends their analysis to the inclusion of a second principal, i.e. wealthy unskilled investors. Financial intermediaries have no wealth of their own, but manage investors money (see also Allen and Gorton 2003) and are granted an equity stake in firm equity. Because they enjoy limited liability upon issuing risky debt, financial intermediaries own a call option on the assets they manage. Limited liability provides the option to default at T, i.e. value creation in excess of the face value of debt accrues to them whereas downside risk is limited to zero. Since portfolio management requires sophisticated skills, investors can only earn a low expected return if they directly invest in financial securities (direct portfolio management). By contrast, they can earn a higher expected return by delegating the management of their funds to financial intermediaries (delegated portfolio management). However, because investors cannot costlessly and easily monitor intermediaries actions, contracts are incomplete, paving the way to potential agency problems. Following Dow and Hang (2014), we model contractual incompleteness by assuming that the investor (i.e. the principal) cannot ex ante determine whether the financial intermediary (i.e. the agent) is a good one or a bad one. Good intermediaries take financing and investment decisions that maximize investors (equity) value by allocating the entire funds raised through the issuance of risky debt to investment in the financial security. By contrast, bad financial intermediaries because they derive private benefits at the expense of investors from consumption (i.e. nonproductive) goods have incentives to divert a fraction λ of funds to private goods, and therefore only allocate a fraction 1 λ to investment in the financial security. Compared to investing in the financial security, the nonproductive good provides the financial intermediary with a nonstochastic and immediate payoff. Since financial intermediaries are granted an equity stake of investors capital and become residual claimants upon issuing risky debt, only good intermediaries incentives are aligned with those of investors. The financial intermediary s optimization problem. The common agency 12

13 setting involves two embedded optimization problems wherein investors first contract with financial intermediaries, who subsequently contract with creditors. We assume that the intermediary cares for his job and is thus averse to job loss. In the event of default, he derives a disutility which is given by the product of his coefficient of aversion to job loss γ 0 and the probability of default. Since he is given an equity ownership, the intermediary selects firm policies that maximize the value of his (residual) claims on assets, net of his disutility in the event of default. The agent s decisions entails the amount of leverage and the degree of riskiness of the asset. We assume that the agent issues zero-coupon discount bond risky debt with face value D and maturity T. Proceeds of the debt issuance are given by the pricing of risky debt as a senior contingent claim on assets A, where ω {D, σ a } ( ) b 0 (A, ω) = e rt + E P min (A (ω), D) F 0 = De rt e rt F0 E P D A (ω), (3) where the second equality simply expresses the creditor s claim as a fixed amount equal to the face value less a put option on assets with strike price D and maturity T (Merton, 1974). Underlying firm value follows the diffusion process with controlled diffusion given in (2). 6 Prior to firm decisions by the financial intermediary, the balance sheet of the all-equity financed firm is equal to the investor s initial capital i 0 A 0 = i 0. (4) Following the issuance of risky debt, the levered firm s balance sheet is increased by the amount of proceeds of the debt issue. Depending on whether the financial intermediary is good or bad, the balance sheet is increased by the total amount of funds raised with probability (1 λ) if the intermediary is good and by a zero amount with probability λ if the intermediary is bad, i.e. A 0 (A, ω, λ, b 0 ) = i 0 + (1 λ) b 0 (A, ω) + λ.0, (5) where b 0 (A, ω) is defined in (3). 7 By the law of large numbers, the levered balance sheet is equal to the investor s initial capital i 0 and a fraction (1 λ) of proceeds of the debt issue invested in the financial security with the remaining a fraction λ of proceeds being diverted, i.e. A 0 (A, ω, λ, b 0 ) = i 0 + (1 λ) b 0 (A, ω). (6) Since initial firm value A 0 is function of the market value of debt b 0, which is itself a contingent claim on assets A, the market value of debt at issuance is 6 The dependence of firm value on asset volatility arises through the coefficient diffusion of firm dynamics. In addition, the market value of risky debt b 0 at issuance date also depends on both debt and asset volatility. 7 Because b0 is the value of the contingent claim on assets A, the initial balance sheet is function of the underlying asset A. 13

14 the solution to a fixed point equation of type u = f(u) (see Appendix) ( ) + b 0 (A, ω, λ) = De rt e rt F0 E P D A T (ω, λ). (7) The payoff to equityholders at T who enjoy limited liability is given by S T (A, ω, λ, b 0 ) = ( ) + (A T (1 λ) b 0 (ω, λ) D), }{{} (8) fraction 1 λ of b 0 where A T = i 0 + (1 λ) b 0 (A, ω) e µt +ν(d)+σau T. Formally, taking the investor s initial capital i 0 0, his fixed salary ϕ 0 and fractional equity stake n 0 as given, the financial intermediary solves for the notional value of debt D and the degree of riskiness of assets σ a that maximize his expected wealth net of his disutility in default. His optimization problem at date 0 can thus be formulated as follows {ω} = arg max ω { mt ( ) } F0 ϕ + h (λb 0 (ω, λ)) + E P ns T (A, ω, λ, b 0 ) γ1 {AT (ω,λ)<d}, (9) m 0 subject to D 0, 0 σ a 1, as well as to the dynamics of both the stochastic discount factor (1) and firm value (2). We assume h (.) 0 and h (.) 0 to capture the fact that managerial private benefits are increasing in consumption goods (see also Stulz (1990)). His financing D and investment σ a decisions are, however, constrained by two conditions, i.e. the creditor s participation constraint which defines the relation between the value of risky debt issued by the intermediary and the creditor s expected discounted (senior) claim on assets mt ( ) F0 b 0 (ω, A) E P min A T (ω), D 0, (10) m 0 and a Value-at-Risk (VaR) constraint as in Adrian and Shin (2015) E P 1 {AT (ω) A 0(ω) L} F0 α, (11) where α > 0 and L > 0 are assumed to be exogenously imposed by the regulator. 8 The creditor s participation constraint (10) entails that the expected discounted value of the creditor s defaultable claim on assets less the amount invested in the firm has to be positive. Otherwise, the creditor would never agree to the contractual terms if the amount invested in the firm at the initial date exceeded the expected discounted value of his claim. Through the issuance of debt, the intermediary commits to repaying the notional amount D at maturity. Failure to do so prompts the firm to default if the underlying firm value falls short of the promised payment amount D. 8 Note that no discounting is applied to the Value-at-Risk constraint. 14

15 The creditor s participation constraint therefore entails that the value of the market value of corporate risky debt b 0 (ω, A) reflects risk inherent to the creditor s defaultable claim on assets, i.e. min (A T (ω), D) where the creditor is paid off the promised payment D at maturity date T only if firm value is sufficiently high. Otherwise, the creditor seizes the firm s assets. Condition (10) not only requires that the amount of cash raised by the intermediary through the issuance of debt satisfies the creditor s participation constraint, but perhaps more importantly, that when solving his own maximization problem, the intermediary takes into account the impact of firm policy choices on the market value of debt (see section 3). By contrast, the risk-based capital constraint (11) requires the financial intermediary to comply with the regulator s condition on the maximum loss possible by satisfying the constraint implied by the Value-at-Risk. For a confidence level 1 α imposed by the regulator, the Value-at-Risk L defines the maximum loss allowed over the contracting period such that the probability that the loss exceeds the regulatory level defined ex-ante L is below the critical level α. 9 Investor s optimization problem. Given the financial intermediary s optimization problem, and taking the dynamics of the stochastic discount factor and firm value as given, shareholders problem consists in solving for the optimal managerial compensation schedule that maximizes their net expected payoff max ϕ,n mt (1 n) E P max (A T (ω) D, 0) F 0 ϕ, (12) m 0 subject to (i) providing the intermediary with at least his reservation utility, i.e. mt ( ) ϕ + E P nmax AT (ω) D, 0 γ1 {AT (ω)<d} F0 v, (13) m 0 (ii) satisfying the incentive-compatibility requirement of second-best contracts given by (9), and (iii) ensuring the feasibility condition on the managerial fractional equity ownership, i.e. 0 n 1. The optimal second-best contract therefore involves a sequential agency framework. In the first optimization, shareholders determine the optimal compensation scheme {ϕ, n} that provides the intermediary with his reservation utility and ensures that the latter s decisions are incentive compatible. In the second optimization, the intermediary determines firm policies {D, σ a } that maximize his net expected payoff subject to the creditor s participation constraint and the Value-at-Risk constraint As detailed in the following sections, the VaR constraint is actually a constraint on both the level of riskiness of assets and leverage. 10 Hence, the optimal contract involves in a sequential fashion (1) shareholders determining the optimal compensation scheme subject to the intermediary s participation and incentive compatibility constraints, and (2) the intermediary determining the optimal financing and investment policies subject to the creditor s participation constraint and the regulator s Value-at-Risk constraint. Using the fact that the intermediary s participation constraint binds in the optimum (and assuming without loss of generality that the intermediary s reservation utility v is zero) implies that the optimal 15

16 3 Managerial incentives in the case of endogenous balance sheet size 3.1 Financial intermediaries leverage and asset-risk taking incentives Endogenizing balance sheet size within the contingent claim analysis can be shown to significantly affect the incentive structure stemming from intermediaries contingent claims on assets. The impact on managerial incentives essentially arises through the creditor s participation constraint (to the intermediary s contract) which determines the value of the latter s claim within the contractual agreement. It is straightforward to show that the creditor s participation constraint must necessarily be satisfied with strict equality in order for both parties to agree to the contract. 11 A binding creditor participation constraint, contract solves the following sequential agency frameworks max n mt (1 n)e P max (A T (ω) D, 0) F 0 ϕ (14) m 0 mt s. t. ϕ + E P (nmax (A T (ω) D, 0) γ1 m {AT 0 (ω)<d} { mt ϕ + λb(d) + E P {ω} = arg max ω ) F0 v (15) ) F0 } (nmax (A T (ω) D, 0) γ1 m {AT 0 (ω)<d} mt ( ) F0 s. t. b 0 (ω, A) E P min A T (ω), D 0, (16) m 0 E P 1 {AT (ω) A 0 (ω) L} F0 α. (17) Because the intermediary s participation constraint binds at the optimum, the optimal contract rewrites as follows mt } F0 max E P {max (A T (ω) D, 0) γ1 n m {AT 0 (ω)<d} (18) { mt ) } F0 s. t. {ω} = arg max λb(d) + E P (nmax (A T (ω) D, 0) γ1 ω m {AT 0 (ω)<d} (19) mt ( ) F0 s. t. b 0 (ω, A) E P min A T (ω), D 0, (20) m 0 E P 1 {AT (ω) A 0 (ω) L} F 0 α. (21) for 0 α 1, and L > 0, as well as nonnegativity and feasibility constraints on all endogenous variables. In addition, as the combination of a binding participation constraint and a zero reservation utility implies that the intermediary s net expected payoff is zero, the incentive compatibility constraint no longer includes the intermediary s fixed compensation, the latter being determined residually ex-post from the binding participation constraint, i.e. mt ( ϕ = E P γ1 m {AT (b 0 0 ( ω), σa)< (A D} nmax T ( ω) D, )) F Indeed, creditors who are in net positive supply have no bargaining power when establishing a contract with the intermediary. In addition, the financial intermediary himself will never agree to issuing debt if proceeds from the debt issuance are lower than the value of the creditor s expected discounted claim on assets. Participation of both the creditor and the intermediary thus requires that debt issued be priced at market value, i.e. that the constraint holds with equality. See also Adrian and Shin, Furthermore, the requirement that the constraint holds with equality implies that contrary to the shareholder-intermediary agency relationship, there is no incentive design involved in the intermediary-creditor relationship thus implying that the interests of both principals are not strictly speaking at odds. 16