Risk Sharing, Risk Shifting and the Role of Convertible Debt

Transcription

1 Risk Sharing, Risk Shifting and the Role of Convertible Debt Saltuk Ozerturk Departent of Econoics, Southern Methodist University Abstract This paper considers a financial contracting proble between a risk neutral entrepreneur and a risk averse investor. Once the venture is started, the entrepreneur chooses an action that deterines the riskiness of the venture s payoff. When action choice is contractible, the optial risk sharing consideration under liited liability calls for a pure debt contract and the low risk action is adopted. When the action choice is not contractible, due to the risk shifting proble ipleenting the low risk action requires a deviation fro the optial risk sharing. I focus on situations where despite this deviation, the risk averse investor prefers to ipleent the low risk action and show that a convertible debt contract outperfors pure debt, pure equity and any ixture of debt and equity. JEL CLASSIFICATION NUMBERS: D23, G24, G32 KEYWORDS: Convertible Debt, Second Order Stochastic Doinance, Financial Contracting. REVISED & RESUBMITTED to JOURNAL OF MATHEMATICAL ECONOMICS Assistant Professor of Econoics, Southern Methodist University, Departent of Econoics, 33 Dyer Street, Suite: 31, Dallas, TX. Tel: , Fax: , e-ail: ozerturk@ail.su.edu. I a grateful to an anonyous referee for extreely useful coents. I would also like to thank Charles A.Wilson, Alberto Bisin, Bogachan Celen, Boyan Jovanovic, Douglas Gale, Kyle Hyndan, Levent Kockesen, Santanu Roy and seinar participants at New York University and Southern Methodist University for their helpful coents. The usual disclaier applies.

2 1 Introduction Financing a young entrepreneurial fir with a risky business plan is subject to iportant inforational and incentive probles. Rather than ore coon instruents like debt or equity, investors who provide financing for entrepreneurial firs typically hold a convertible debt clai. In a recent epirical study on financial contracting by Kaplan and Stroberg (23), convertible securities account for over 9% of all financing agreeents in their saple of start-up firs. Previous work by Sahlan (199) and Gopers (1997) also report the extensive use of convertible debt in venture capital backed entrepreneurial firs. A convertible debt contract cobines the properties of debt and equity. The conversion option gives the claiholder the right to convert the debt clai into copany s equity. This paper offers an explanation on why convertible debt can be superior to pure debt, pure equity and ixed debt-equity in the venture capital context. I describe a financial contracting proble where a risk neutral entrepreneur finances his venture by funds provided by a risk averse financier/venture capitalist. The adissable sharing rules on the final payoff of the venture are constrained by a liited liability condition. Upon receiving the funds, the entrepreneur adopts a business strategy, which cannot be specified ex ante by the contract. This action choice deterines the riskiness of the venture s payoff. In particular, I consider a siple odel with two possible actions where the high risk action a H yields a payoff distribution which is a ean preserving spread of the distribution induced by the low risk action a L. When the action choice is enforceable, under liited liability a pure debt contract achieves optial risk sharing between parties. Furtherore, under enforceability of actions the financier prefers to enforce the low risk action. When the action choice is not enforceable, a debt contract induces the entrepreneur a preference for the high risk action due to its convex residual payoff. Therefore, ipleenting the low risk action requires a deviation fro the optial risk sharing arrangeent provided by the debt contract. I focus on situations where despite this deviation fro optial risk sharing, ipleenting the low risk action akes the risk averse financier better off copared to opting for the high risk action. In this setting, a convertible debt contract outperfors pure debt, pure equity and any ixture of debt and equity. This result follows because a convertible debt contract cobines two desirable properties in this proble. Its debt coponent assigns the whole payoff to the risk averse financier at the low end of payoff realizations and provides better insurance against the downside risk. The conversion into equity option, on the other hand, provides a convex payoff schedule for the financier at the upper end of payoff realizations, and corrects the entrepreneur s high risk incentives arising fro the debt portion of the 1

3 contract. This role can not be achieved by siple ixtures of debt and equity, since ixed debt-equity contracts also yield a concave payoff for the financier (and hence a convex one for the entrepreneur) and ipleent the high risk action. My priary focus is to offer an explanation on why convertible debt can be superior to ore traditional financial contracts such as pure debt, pure equity and ixed debt-equity. As entioned, this superiority stes fro the fact that (i) unlike an equity contract convertible debt gives the whole payoff to the risk averse financier below a certain payoff realization and iproves risk sharing, (ii) unlike debt and ixed debt-equity, convertible debt assigns the financier a convex increasing payoff schedule at the upside and prevents the entrepreneur fro increasing risk ex-post. For further insight, in an extension I analyze whether an optial solution can exhibit these two properties. I show that these two properties can eerge provided that (i) the adissable contracts exhibit a onotonicity property, i.e., the financier s payoff is constrained to be non-decreasing in the venture s payoff (as in Innes (199)), and (ii) the likelihood ratio f H /f L of the density functions for the high and low risk distributions is first decreasing and then increasing. This regularity condition on the likelihood ratio f H /f L is consistent with the ean preserving spread assuption, and is satisfied, for exaple, in the case of a noral distribution. Since the risk sharing consideration between the risk averse financier and the risk neutral entrepreneur is a crucial aspect of the analysis, the assuption on the risk attitutes of the two parties deserves further coent. The financial contracting proble posed in this paper can be best placed in the venture capital industry context. A typical venture capitalist perodically raises what is called a venture capital fund fro cash rich institutions such as pension funds and insurance copanies. A venture capital fund has a lifetie of five to seven years, at the end of which the returns are distributed to the fund contributors. In that sense, the venture capitalist acts as a fund anager by choosing which ventures to invest. The ability to raise new capital for future funds depends on the perforance of the earlier investents. Sahlan (199) notes that if a venture capitalist s fund suffers hugelossesorevenincases of oderate failures, the chances for raising new capital for the next fund are very liited. As pointed out by Cheanur and Fulghieri (1999), in a given fund cycle the venture capitalist can only invest in a few ventures, all of which are quite risky, and therefore the capitalist s investent portfolio reains poorly diversified. 1 They also consider a risk averse venture capitalist and argue that the copensation schee of a venture capitalist fro anaging a fund involves significant penalties for failures, thereby inducing risk averse behavior. 2 Furtherore, the success or failure of a 1 Sahlan (199) analyzes a saple of 383 venture capital investents and reports that 35% of all projects yielded a net loss and another 5% were only oderately succesful. 2 For a siilar discussion for the risk averse behavior of banks, see Diaond (1984). 2

4 particular project can significantly affect the reputation of the venture capitalist who ade the decision to invest in that venture, again leading to risk aversion. 3 As in Innes (199) and Cheanur and Fulghieri (1999), I consider a risk neutral entrepreneur with the following justification. Unlike the agent of a standard principal-agent odel (typically interpreted as a risk averse eployee of the principal), an entrepreneur who quits a well paying job to pursue a fortune by launching a new copany does not see to be exhibiting risk averse behaviour. Furtherore, what an entrepreneur loses fro a failing venture is considerably less than what a venture capitalist loses. For a typical entrepreneur, even receiving funds for the venture can work as a badge of success, and having been in charge of a copany, even if it eventually fails, is a valuable experience for a future career, especially in a growing young industry. 4 Related Literature. Previous explanations of convertible securities have focused either on the efficient allocation of control rights paradig or on an effort type oral hazard proble. Berglof (1994) provides a odel where control refers to the right to bargain with an outside party bidding for the venture and shows that convertible security allocates the control to the party who axiizes the joint surplus of the entrepreneur and the financier. Another control based explanation is Marx (1998) where a ixture of debt and equity doinates pure debt and pure equity in giving the financier the efficient liquidation incentives. However, as Gopers (1997) convincingly argues, allocation of cash flow rights and allocation of control rights can be separated by use of covenants and explicit contractual clauses. Indeed, Gopers (1997) docuents the frequent use of covenants that give investors control rights. We take the view that such control rights are soewhat independent fro the design of financial instruent and the priary purpose of convertible security is ore likely to be risk sharing and agency considerations, which is the focus of this paper. Cornelli and Yosha (23) show that conversion into equity option can be desirable, because it ay prevent the entrepreneur fro window dressing (short-teris) which does not contribute to the long ter success of the venture. Trester (1998) shows that a financier s conversion into equity option ay prevent the entrepreneur fro defaulting strategically and walking away fro the venture. However, it sees that what 3 Consistent with this view, Sahlan (199) notes that when valuing a copany, a venture capitalist coputes the present value of a copany by applying a very high discount rate, usually in the range of 4% to 6%. 4 The following rearks of Joseph Park, the founder of Kozo.co, illustrates this point: Let s say I copletely failed in 6 onths after launching the copany and lost all the oney I raised. So what? I will have an ipressive resue to apply to a business school (fro the docuentary fil e-dreas). Kozo.co was a venture capital driven online copany that proised free one-hour delivery of anything fro DVD rentals to Starbucks cofffe. After raising about $28 illion, the copany had to shut down its operations in 21. Perhaps ironically, it is now a widely studied exaple of the dot-co excess and ade Joseph Park a celebrity in business school case studies. 3

5 prevents an entrepreneur fro walking away fro the venture is the fact that her stakes are vested over tie and only becoe liquid after a certain period of tie. Furtherore, any agreeents give the investor the right to purchase a departing entrepreneur s share at a low price (see Sahlan 199). Schidt (23) shows that convertible debt gives efficient investent incentives when the entrepreneur and investor ove sequentially in a double oral hazard type proble. Another related paper is Innes (199) who considers an effort type agency odel, where higher levels of costly effort by the entrepreneur induces better payoff distributions in the sense of the onotone likelihood ratio property. Innes first allows for non-onotonic sharing rules and shows that the optial contract punishes the entrepreneur by giving the investor all the realized payoff below a threshold, whereas the entrepreneur is awarded by receiving all the payoff over this threshold. To rule out this non-onotonic optial contract, Innes then iposes a onotonicity constraint on the adissable sharing rules and establishes that under this additional constraint a siple debt contract eerges as optial. Different than Innes who assues risk neutrality for both parties, the risk sharing consideration plays an iportant role in y odel. Furtherore, the entrepreneur s action choice deterines the riskiness of the venture s payoff, but does not affect its expected value. The plan of the paper is as follows. Section 2 presents the odel. Section 3 provides the benchark case when the action choice of the entrepreneur is enforceable. Section 4 considers the case when action choice is not observable. Section 5 concludes. 2 The Model There are three dates, t =, 1, 2. There is an entrepreneur (henceforth EN) who owns a venture idea. The venture requires a fixed investent of $K at date. The ENhasnowealthofhisownandreliesonafinancier/investor (henceforth FI) to provide the investent capital. This financier can be thought as a venture capitalist focusing on young entrepreneurial firs. At date 2, the venture generates a rando payoff ỹ. The realizations of the rando variable ỹ (that I denote with y) are drawn fro a support [, ). The EN is risk neutral and axiizes expected wealth. The FI axiizes a strictly concave VNM utility function v(.) with li y v (y) =. The distribution of the venture s payoff ỹ depends on the action that EN chooses at date 1. This action can be thought as the business strategy eployed by EN upon receiving the required funds. For siplicity, I consider two utually exclusive actions denoted by a L and a H.Forally,fori {H, L}, let F i (y) denote the distribution function fro action a i with a continuously differentiable density f i.following Rotschild and Stiglitz (197), the following assuption ranks the riskiness of the two distributions under two actions. 4

6 Assuption 1. The payoff distribution F H is a ean preserving spread of the payoff distribution F L, i.e., Z x (F H F L ) dy for all x>, and (1a) yf L dy = yf H dy = µ. The above assuption says that the action a H yields a riskier payoff distribution than action a L. In the context of a start-up copany operating in an innovative industry, the riskiness of the business plan is an iportant deterinant of failure or success. Aong the any possible ways to increase risk in start-up environents, the ost coon ones are: rushing the product to the arket although further testing is warranted, changing the scope of venture s operations and drifting into uncharted territory, insisting on a very abitious design feature and thus increasing technical risk. 5 While the realization of the venture s payoff is observable and contractible, I assue that the action a i is not contractible, i.e., the two parties cannot write an enforceable contract clause at date that dictates EN to choose a particular action. A sharing rule (a financial contract) is an integrable function s(y) :R + R + which specifies the payent to FI for each payoff outcoe y. If the realized payoff is y, then the FI receives s(y) and EN as the residual claiant receives y s(y). As in Innes (199), I assue that the sharing rule s(y) ust exhibit liited liability so that s(y) y for all y. This liited liability constraint iplies that EN cannot be forced to pay FI ore than what the venture generates (s(y) y) and FI cannot be forced to ake an additional transfer once the venture s payoff is realized (s(y) ). I consider a setting with any entrepreneurs seeking funds for their ventures, but only a few financiers/venture capitalists who can provide financing for young entrepreneurial firs. The venture capitalists typically specialize in certain industries such as biotechnology and telecounications and their industry specific expertise also serves as an entry barrier for less specialized financiers (Gopers 1997). Furtherore, especially at the initial stage of a venture, the business plan and the skills of an entrepreneur are copletely untested, giving EN not uch bargaining power. Accordingly, I assue that FI has all the bargaining power and chooses the sharing rule that gives her the axial expected utility subject to a participation constraint for EN, any required incentive copatibility constraint and the liited liability constraint. The assuption that FI extracts all the surplus does not affect the qualitative results. 6 5 I do not consider the possibility of learning about the venture s prospects over tie. For a odel with this feature, see Bergeann and Hege (1998). 6 The forulation that axiizes FI s expected utility subject to giving EN his reservation utility is erely a tool to describe the Pareto optial solution for any given reservation utility level for EN. 5 (1b)

7 3 Action choice enforceable As a benchark, this section considers the case when the action choice a i is enforceable. The optial contract in this case is deterined only by risk sharing considerations. To state the optial contracting proble forally, define U i (s(.)) (y s(y)) f i dy for i {H, L}. (2) as the expected payoff for EN fro adopting action a i under a sharing rule s(y). An optial contract that prescribes EN to undertake action a i is then the solution to a prograing proble in which the sharing rule s(y) is chosen to axiize the expected utility of FI, V i (s(.)) v(s(y))f i dy (3) subject to a participation constraint for EN, and the liited liability constraint U i (s(.)) w>, (4) s(y) y for all y. (5) I refer to the above proble as (P1). The following proposition establishes that the optial risk sharing consideration under liited liability calls for a pure debt contract. Proposition 1a. If action choice a i is enforceable, the optial risk sharing contract that prescribes EN to undertake action a i is a pure debt contract s i (y) =Min(y, i ), (6a) where the face value i of debt is uniquely deterined by EN s participation constraint and solves Z i i F i dy = µ w. (6b) Proof: In an optial solution to (P1), EN s participation constraint (4) holds as an equality. Under a pure debt contract s i (y) =Min(y, i ) that prescribes a i, the participation constraint becoes Z i U i = (y i ) f i dy = µ i + F i dy = w. i and yields (6b). I now show that s i (y) =Min(y, i ) is the optial solution to (P1). Suppose, contrary to our clai, that there is an alternative sharing rule ŝ i (y),different than the debt contract s i (y) in (6a-6b), which is a solution to (P1). This would iply v(ŝ i (y))f i dy 6 v(s i (y))f i dy. (7a)

8 Let Ĝ and ˆF i denote the distribution functions of ŝ i (y) and s i (y), respectively. Since v(.) is a strictly concave increasing function, the inequality in (7a) iplies that Ĝ second order stochastically doinates ˆF i (see Hadar and Russell (1969)), and therefore we have Z x ³ Ĝ ˆFi dy for all x>. (7b) Note that the distribution function ˆF i (.) of s i (y) is given by ˆF i (y) =F i (y) for y< i and ˆF i (y) =1for y i. To obtain a contradiction, suppose that for realizations y [, i ], our alternative optial sharing rule ŝ i (y) is different than the debt contract s i (y) in (6a-6b). With the liited liability restriction in (5), this iplies that ŝ i (y) y for all y [, i ] and ŝ(y) <yfor soe y [, i ]. But then we have which further iplies that Ĝ(y) ˆF i (y) =F i (y) for all y [, i ] and Ĝ(y) > ˆF i (y) =F i (y) for soe y [, i ], Z i ³ ˆFi Ĝ dy <, and contradicts (7b). Accordingly, the proposed alternative optial sharing rule ŝ i (y) cannotbedifferent than the debt contract s i (y) in (6a-6b) for y [, i ],as otherwise it would be second order stochastically doinated by s i (y). This arguent establishes that an optial solution to (P1) ust have s(y) =y for y [, i ]. To prove that ŝ i (y) cannotbedifferent than the debt contract s i (y) in (6a-6b) for y> i either, note that both ŝ i (y) and s i (y) ust give FI an expected payoff of µ w and therefore we have ³ Ĝ E[ŝ i (y)] = E[s i (y)] ˆFi dy =. (8a) It is already shown that ŝ(y) =s i (y) =y for y [, i ], and hence the equality in (8a) becoes ³ Ĝ ˆFi dy =. (8b) i Since ˆF i (y) =1for y i, the equality in (8b) iplies that Ĝ(y) =1for y i. This proves that ŝ i (y) = i for y i as well. Accordingly, the unique solution to (P1) is given by s i (y) in (6a-6b). Q.E.D. (7c) 7

9 One can write the expected utility of FI fro holding a debt clai s i (y) = Min(y, i ) as V i (s i (y)) = Z i = v( i ) v(y)f i dy + v( i )(1 F i ( i )) (9) Z i v (y)f i dy, where the second equality follows fro integration by parts. In the above expression, the second ter R i v (y)f i dy accounts for the reduction in FI s expected utility due to riskiness of the debt clai. For a given i, this ter is increasing in the riskiness of the distribution F i and the risk aversion of FI. This is because under the high risk action a H default is ore likely and a ore risk averse FI suffers ore at the low end of payoff realizations. As a result, to copensate for this reduction one would expect the face value to be higher when the high risk choice a H is adopted. I now prove that this is indeed the case and we have H > L. Recall that the face value i is deterined by the action a i undertaken and EN s reservation wage w. Fro (6b), we have Z L Z H L H = F L dy F H dy. (1a) For a contradiction, suppose that L H and rewrite (1a) as L H = Z H (F L F H ) dy + Z L H F L dy. (1b) By Assuption 1, the first ter R H (F L F H ) dy. Therefore, if L H fro (1b) we would have L H Z L H F L dy. (1c) But this last inequality in (1c) is a contradiction, since F L (y) < 1 for y [ H, L ] and hence R L H F L dy < L H. This arguent establishes that L < H. Using this observation, one can show that when the action choice is enforceable, FI strictly prefers the low risk action a L. Proposition 1b. If the action choice is enforceable, FI strictly prefers the low risk action a L to be undertaken by offering s L (y) =Min(y, L ). Proof. Given L < H, I show that s L second order stochastically doinates s H. The desired result then follows fro the strict concavity of v(.). Since L < H, one can write equation (1a) as, Z L (F H F L ) dy = Z H L (1 F H ) dy. (11) 8

10 Let ˆF i denote the distribution function for s i. We have ˆF i (y) =F i (y) for y< i and ˆF i (y) =1for y i. Since the expected payoff is the sae under both s L (y) and s H (y), equation (11) iplies Z x ³ ˆFL H ˆF dy < for x< H and Z x ³ ˆFL H ˆF dy = for x H, which establishes that s L second order stochastically doinates s H. Q.E.D. The analysis in this section established that when the action choice is enforceable, for a given action a i (and hence a given distribution F i )apuredebtcontractprovides the best possible insurance (as uch as liited liability allows) for the risk averse FI by allocating her all the realized payoff at the downside. Furtherore, under enforceability FI strictly prefers to enforce the low risk action a L. After characterizing this benchark case under enforceability, I now analyze the case when action choice is not enforceable. 4 Action choice not enforceable 4.1 Risk Shifting Suppose now the action choice of EN is not enforceable, and EN chooses the action that axiizes his expected payoff given the sharing rule specified at date. A wellknownresultinthefinancial contracting literature is that under a risky debt contract the risk neutral EN has a preference for high risk (see, for exaple, Jensen and Meckling (1976), Green (1984)). To illustrate the risk shifting incentives in our fraework, consider any pure debt contract with a repayent obligation. Fro (6c), EN s expected payoff fro choosing action a i under such a debt contract is given by Z U i = µ + F i dy for i {H, L}. (12) The change in EN s payoff fro switching to the high risk action a H under the debt contract can then be written as U H U L = Z (F H F L ) dy for all >. (13a) Therefore, a pure debt contract, which provides optial risk sharing under enforceability, ipleents the high risk action a H when action choice is not enforceable. The reason behind the entrepreneur s preference for high risk is the convexity of the residual payoff for EN under a pure debt contract. To see this ore transparently, 9

11 consider any concave sharing rule s c (y) (including the pure debt contract) that yields a convex residual payoff y s c (y) for EN. By Assuption 1, we have U H U L = (y s c (y)) (f H f L ) dy = s c (y)(f L f H ) dy. (13b) and hence EN s preference for high risk extends to any concave sharing rule s c (y). Accordingly, a ixed debt-equity contract of the for s(y) Min(y, + (y )) with > and >, which gives the FI a share of the upside of the venture, is also concave, and ipleents the high risk action a H. In a ixed-debt equity contract, EN s payoff fro switching to a H is given by U H U L =(1 ) Z 4.2 Ipleenting the Low Risk Action (F H F L ) dy. (13c) Due to the risk shifting proble, ipleenting a L requires FI to deviate fro optial risk sharing achieved by the debt contract and satisfy an additional incentive copatibility constraint. Forally, the proble that FI needs to solve to ipleent a L optially can be stated as follows. Max s(.) V L (s(y)) = v(s(y))f L dy (P2) s.t. (y s(y)) (f L f H ) dy = (y s(y)) f L dy w>, (14a) s(y)(f H f L ) dy, (14b) and the liited liability constraint s(y) y for all y. I refer to the above proble as (P2). Before proceeding, a discussion is in order. Note again that ipleenting a L by satisfying (14b) involves an agency cost relative to the utility level achieved by s L (y) =Min(y, L ), since it requires a deviation fro the optial risk sharing rule. On the other hand, FI can always opt for offering the debt contract s H (y) = Min(y, H ) to ipleent a H and ensure an expected utility V H (s H (y)) = v( H ) Z H v (y)f H dy. One question is whether FI is better off fro satisfying (14b) and solving (P2) copared to ipleenting a H and receiving V H (s H (y)). While the answer depends on the relative riskiness of F H and F L, the only restriction I have is a ean preserving spread condition, which is too weak to atheatically characterize a siple condition that ensures FI is better off fro ipleenting a L. I focus on situations where 1

12 ipleenting the low risk action brings a higher expected utility to the risk averse FI than the axiu she can get under the high risk action. This would be the case if F H is risky enough and/or FI is risk averse enough so that V H (s H (y)) is low enough. As is coon in the agency literature, in what follows I consider the case that despite the deviation fro optial risk sharing and associated agency cost, the second best involves ipleenting a L rather than opting for a H. 7 It should be noted that a sharing rule that satisfies (14b) and ipleents a L can yield the risk averse FI a higher expected utility than V H (s H (y)), even if it is not an optial solution to (P2). In particular, FI ay be better off fro ipleenting a L even with a pure equity contract s E (y) =y which satisfies (14b), but is not necessarily an optial solution to (P2). The following exaple, developed and analyzed ore copletely in the Appendix, illustrates this point. Let v(y) =2y 1/2 and specify the two distributions as F L (y) = y 2 for y [, 2] and F H(y) = y 4 for y [, 2) and f H (2) = 1 4. Set EN s reservation wage to w = 1 4. One can copute that (see the Appendix for a general derivation), the optial debt contract that ipleents a H is given by s H (y) =Min(y, H =1.26) which yields FI an expected utility V H (s H (y)) = On the other hand, an equity contract s E (y) =.75y gives EN a reservation wage w = 1 4, ipleents a L and yields FI an expected utility V L (s E (y)) = Superiority of Convertible Debt over Pure Debt and Equity My ain objective is to provide an explanation on why convertible debt can be superior to ore coonly observed financial contracts such as pure debt, pure equity and ixed-debt equity. The convertible debt contract can be described as a sharing rule s(y;, ) ax {in {y, },y} for and [, 1) (15) The above contract specifies a payoff realization below which FI receives all the realized payoff. Therefore, again serves as the face value of FI s debt clai. The conversion into equity option is described by the share of venture s equity. Upon realization of the venture s payoff, FI has the option to exchange the debt clai for a share of the venture s equity. This conversion into equity option is exercised for payoff realizations y /, whereas for y</fi retains the debt clai. It should be noted that the pure equity and the pure debt contracts are special cases in this 7 I a grateful to an anonyous referee for raising this question. If we have V L (s (y)) <V H (s H (y)) at an optial solution s (y) to (P2), then the second best contract is s H (y) and the high risk action a H should be ipleented. This would be siilar to an agency odel with costly hidden effort where the agency cost of inducing the agent the high effort level is higher than the efficiency benefits on expected output and as a result the principal has to settle for the low effort. 11

13 faily. The contract s(y; =,) corresponds to a pure equity contract, whereas s(y;, =) corresponds to a pure debt contract. The figure below illustrates FI s payoff schedule fro a convertible debt contract s(y;, ). FI s payoff / y Figure 1. The convertible debt contract A convertible debt contract has two desirable properties copared to pure debt and pure equity in the context of (P2). Unlike a pure equity contract, the debt coponent of convertible debt assigns the whole payoff totheriskaversefiatthelow end of payoff realizations and provides better insurance at the downside. Furtherore, unlike a pure debt contract (or ixed debt-equity), the conversion into equity option of convertible debt creates a convex payoff schedule for FI at the upper end of payoff realizations and corrects EN s high risk incentives arising fro FI s debt clai. Accordingly, the convertible debt contract provides better insurance by its debt coponent, while eliinating EN s preference for high risk with its conversion into equity coponent. I now foralize the above arguent and show that convertible debt outperfors pure equity and pure debt in proble (P2). Let e rewrite (P2) by restricting attention to the class of sharing rules s(y;, ) for and [, 1). The FI s expected utility fro s(y;, ) is given by Z Z V L (s(y;, )) = v(y)f L dy + v()f L dy + The participation constraint for EN takes the for v(y)f L dy. (16) U L (s(y;, )) = Z (y ) f L dy + (1 )yf L dy w. (17) Finally, define W (s(y;, )) U L (s(y;, )) U H (s(y;, )). Then the incentive copatibility constraint in (14b) becoes W (s(y;, )) Z (y )(f L f H )dy +(1 ) y(f L f H )dy. (18) 12

14 Within the faily of sharing rules s(y;, ), FI chooses and [, 1) to axiize V L (s(y;, )) subject to (17) and (18). The proposition below foralizes that a non-degenerate convertible debt contract s(y; >, >) outperfors pure debt and pure equity in (P2). Proposition 2. equity in (P2). A convertible debt contract outperfors pure debt and pure Proof. The preceding analysis ruled out a pure debt contract, since it ipleents the high risk action and violates (18). I now show that s(y; >, >) outperfors the equity contract s(y; =, > ). Define by U(, ) = w. Since U is strictly decreasing in and, one can define a function :[, ] [, 1] by U(, ()) w. Forally, the function () satisfies Z U(, ()) () and iplicit differentiation yields (y ) f L dy + (1 ()) yf L dy = w () () = R () R () f L dy yf L dy <, which iplies that for a given reservation payoff w for EN, increasing FI s debt clai requires decreasing the equity clai, so that the participation constraint continues to hold as an equality. Note that a pure equity contract also ipleents the low risk action and therefore we have W (,)=. To show that a solution with > is superior, one needs to show that for > sufficiently sall, we have dv (, ()) d > and dw (, ()) d The idea is to start with =which satisfies (18) and then to find a sufficiently sall >that akes FI better off yet still ipleenting the low risk action. The equity share (.) defined by the binding participation constraint above adjusts accordingly. Note that Z dv (.) d = f L dy 1 v (y) v () yf Ldy > yf L dy Z. f L dy 1 yf L dy = yf L dy where the inequality follows fro the fact that v () converges to a large nuber for sufficiently sall and therefore the ratio v (y)/v () approaches to zero. 13

15 Siilarly, straightforward algebra yields Z Z dw (.) yf H dy d = f H dy f L dy Z yf L dy > {z } <1 by Assuption 1 Z (f H f L )dy > for sall enough. Note that we use the fact that for sufficiently sall, Assuption 1 iplies that f H (y) >f L (y) for all y (, ). Q.E.D. 4.4 Characterizing an Optial Solution to (P2) The preceding analysis established that a convertible debt contract is superior to pure debt and pure equity in (P2) due to its two properties: (i) at the lower end of payoff realizations it assigns the whole payoff totheriskaversefiandiproves risk sharing, (ii) at the upper end of payoff realizations, it creates convexity in FI s payoff schedule and corrects EN s high risk incentives. For further insight, in this section I provide an analysis of the optial solution to (P2) to see whether these two properties can eerge in a general solution as well. As I show shortly, the behaviour of an optial solution to (P2) depends on the likelihood ratio of the density functions f H /f L. The ean preserving spread assuption (Assuption 1) alone does not preclude the possibility of considerable local variation in the relative agnitudes of f L and f H. Consequently, to obtain a reasonably well-behaved solution, one needs to ipose additional regularity conditions on f H /f L. I ipose the following regularity condition on f H /f L, which is consistent with Assuption 1. Assuption 2. There exists a y > such that f H /f L is strictly decreasing for y<y and strictly increasing for y>y. The above assuption on f H /f L holds, for exaple, in the case of two noral distributions with the sae ean and different variances. 8 Along with the liited liability restriction in (5), I follow Innes (199) and also ipose the following onotonicity restriction on the adissable sharing rules. Assuption 3. FI spayoff s(y) ust be non-decreasing in the venture s payoff, i.e., s(y + ε) s(y) for all (y, ε) R 2 +. As in Innes (199), the above assuption can be justified as follows (for a further discussion, see also Bolton and Dewatripont (25, page 164)). Suppose that Assuption 3 is violated and there is a segent of payoff realizations such that s(y) is 8 Consider two noral distributions with the sae ean µ and variances σ 2 H and σ 2 L with σ H >σ L. Then the likelihood ratio f H/f L is strictly decreasing for y<µand strictly increasing for y>µ. 14

16 strictly decreasing in y. In that segent, EN would strictly gain by borrowing oney at par fro another source and thus arginally boost the venture s perforance. If this kind of costless borrowing goes undetected, EN would have an incentive to engage in an arbitrage activity by borrowing at any decreasing segent of s(y), artificially boosting the venture s perforance and reducing FI s payoff. 9 I now proceed with the characterization of an optial solution to (P2) under these two additional assuptions. Let λ and δ denote the Lagrange ultipliers in (P2) associated with EN s participation constraint (14a) and the incentive copatibility constraint (14b), respectively. To save notation, I do not introduce additional Lagrange ultipliers for the liited liability constraint (5) and the onotonicity constraint (Assuption 3), but check separately whether they are binding. The Lagrangean for (P2) takes the for µ µ = v(s(y))f L dy+λ (y s(y)) f L dy w +δ (y s(y)) (f L f H ) dy. An optial solution to (P2) ust give EN an expected payoff no ore than the reservation wage w and therefore λ>. Also note that δ =is not possible, as otherwise pure debt would be optial, which we know violates (14b). Therefore, both constraints are binding, and λ>, δ>. Pointwise differentiation of with respect to s(y) and using the liited liability constraint as a boundary condition, one obtains the following first order condition: v (s (y)) λ + δ 1 f H(y) if s (y) <y, f L (y) v (s (y)) λ + δ 1 f H(y) if s (y) >. f L (y) The binding participation constraint (14a) with w>requires that s (y) <yfor soe y. Therefore, one can define ½ y 1 inf y>:v (s(y)) λ + δ 1 f ¾ H(y), (19) f L (y) ½ y 2 sup y>:v (s(y)) λ + δ 1 f ¾ H(y). f L (y) The liited liability constraint s(y) y and li y v (y) = iply that there exists a value y 1 >, such that v (s(y)) >λ+ δ 1 f H(y) for y<y 1. = s (y) =y for y<y 1, f L (y) 9 As noted by Innes(199), since EN cannot be forced to borrow, the liited liability constraint s(y) y still applies here. 15

17 which establishes that there is a payoff realization y 1 below which all the venture s payoff accrues to the risk averse FI. For y > y 2, we have s (y) =y as well, but it should be noted that without any further assuptions on li y (f H /f L ), it is possible to have y 2 =. For y (y 1,y 2 ), the liited liability constraint is not binding ( <s (y) <y) and the optial solution s (y) is deterined by v (s (y)) = λ + δ 1 f H(y). (2) f L (y) Assuption 2 iplies that y (y 1,y 2 ) and for y (y 1,y ) the onotonicity constraint in Assuption 3 is binding. This follows, since f H /f L is strictly decreasing for y<y by Assuption 2, and hence s (y) that solves (2) is strictly decreasing in that interval. Consequently, under Assuption 3 we have s (y) =y 1 for y (y 1,y ]. Equation (2) also iplies that s (y) is strictly increasing for y (y,y 2 ), since f H /f L is strictly increasing for y>y by Assuption 2. The figure below illustrates the optial solution to (P2) under Assuptions (1-3) when y 2 =. s * (y) y 1 y 1 y * y Figure 2. An optial solution to (P2) under Assuptions (1-3) While the shape of the optial solution in the above figure resebles that of a convertible debt contract, one should be cautious with such an interpretation for three reasons. First, unlike an equity contract the portion for y>y is not linear. Second, we ay have s (y) =y at the upper end again, unless y 2 =. Third, the behavior of s (y) for y>y 1 is driven by the regularity condition on f H /f L iposed by Assuption 2 and the flat portion arises due to the onotonicity restriction iposed by Assuption 3. At the sae tie, the analysis in this section does illustrate that under the above two additional assuptions, an optial solution s (y) can exhibit the two properties of convertible debt that ake it superior to pure debt, pure equity and ixed debt-equity in (P2): at the lower end, the solution s (y) assigns the whole payoff to FI for better risk sharing; and at the upper end it has a convex part that eliinates the entrepreneur s preference for the high risk action. 16

18 5 Conclusion This paper considers a financial contracting proble between a risk neutral entrepreneur who seeks funds for his venture and a risk averse financier who can provide financing. The riskiness of the venture s payoff depends on the action that the entrepreneur takes after the financing is agreeent. When the action choice is enforceable, a pure debt contract achieves optial risk sharing between parties under liited liability and the financier prefers to enforce the low risk action. When the action choice is not enforceable, due to the well known risk shifting proble a debt contract induces the entrepreneur a preference for the high risk action due to its convex residual payoff. Accordingly, ipleenting the low risk action requires a deviation fro the optial risk sharing arrangeent provided by the debt contract. I focus on situations where despite this deviation fro optial risk sharing and associated agency cost, ipleenting the low risk action akes the risk averse financier better off copared to opting for the high risk action. In this setting, I show that a convertible debt contract outperfors pure debt, pure equity and any ixture of debt and equity. This result follows because a convertible debt contract has two desirable properties: (i) at the lower end of payoff realizations it assigns the whole payoff to the risk averse FI and iproves risk sharing, (ii) at the upper end of payoff realizations, it creates convexity in FI s payoff schedule and corrects EN s high risk incentives. This role can not be achieved by siple ixtures of debt and equity, since ixed debtequity contracts also yield a concave payoff for the financier (and hence a convex one for the entrepreneur) and ipleent the high risk action. For further insight, in an extension I also analyze whether an optial solution can exhibit these two properties of convertible debt that ake it superior to pure debt, pure equity and ixed debtequity. I show that these two properties can eerge provided that (i) the adissable contracts exhibit a onotonicity property, i.e.,thefinancier s payoff is constrained to be non-decreasing in the venture s payoff, and (ii) the likelihood ratio of the density functions for the high and low risk actions is first decreasing and then increasing. 17

19 Appendix This Appendix presents an exaple to illustrate that FI can be better off fro ipleenting a L even with an equity contract s E (y) =y (which satisfies (14b) but not necessarily an optial solution to (P2)) rather than opting for a H by offering s H (y) =Min(y, H ). Specify FI s utility function by v(y) =y 1 γ / (1 γ) where γ (, 1) and let F L (y) = y for y [, 2], 2 F H (y) = ε 2 +(1 ε 2 )y for y [, 2) and f H(2) = ε 2. The paraeter ε (, 1] easures the extent that a H shifts probability ass to the tails. One can verify that F H is a ean preserving spread of F L and both distributions yield an expected value µ =1. Let us set EN s reservation wage as w (, 1), so that there is an expected surplus to be shared. First consider (6b) that describes the face value of the debt clai H when FI offers s H (y) =Min(y, H ) and ipleents a H. Solving (6b) under the above specification yields ( 2 ε ε 2 +4w(1 ε) H = 1 ε if ε (, 1), 2(1 w) if ε =1. Consider now an equity contract s E (y) y that solves EN s participation constraint (14a) as an equality. Under the above specification of F L (y), we have =1 w. This equity contract satisfies (14b) and ipleents a L. Denote FI s expected utility fro s E (y) by V L (s E (y)) and her expected utility fro s H (y) by V H (s H (y)). When ε =1, for any γ (, 1) and w (, 1), wehave V L (s E (y)) = 21 γ (1 w) 1 γ (2 γ)(1 γ) >V H(s H (y)) = 21 γ (1 w) 1 γ. 2(1 γ) To obtain further exaples, let us set w =.25 and γ =.5. The table below reports V L (s E (y)) and V H (s H (y)) for ε { 1 8, 1 4, 1 2 }. Note that in each case, the expected utility V L (s E (y)) under the equity contract s E (y) y is higher than the expected utility V H (s H (y)) under the debt contract s H (y) that ipleents a H. ε H V H (s H (y)) V L (s E (y))

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