ARTICLE IN PRESS. Journal of Mathematical Economics xxx (2008) xxx xxx. Contents lists available at ScienceDirect. Journal of Mathematical Economics

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1 Journal of Matheatical Econoics xxx (28) xxx xxx Contents lists available at ScienceDirect Journal of Matheatical Econoics journal hoepage: Risk sharing, risk shifting and the role of convertible debt Saltuk Ozerturk Southern Methodist University, Departent of Econoics, 33 Dyer Street, Suite: 31, Dallas, TX, United States article info Article history: Received 16 Deceber 25 Received in revised for 8 April 28 Accepted 12 April 28 Available online xxx JEL classification: D23 G24 G32 Keywords: Convertible debt Second order stochastic doinance Financial contracting 1. Introduction 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 is superior to pure debt, pure equity and any ixture of debt and equity. 28 Published by Elsevier B.V. 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 adissible 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 second order stochastically doinated by 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 the low risk action. When the action choice is not Tel.: ; fax: E-ail address: ozerturk@ail.su.edu /$ see front atter 28 Published by Elsevier B.V. doi:1.116/j.jateco Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

2 2 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx 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 can be superior to 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 contract. This role cannot 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. 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 attitudes 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 periodically 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 5 7 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 huge losses or even in cases 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 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 behavior. 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 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. 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 successful. 2 For a siilar discussion for the risk averse behavior of banks, see Diaond (1984). 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 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). Q1 Kozo.co was a venture capital driven online copany that proised free 1-h delivery of anything fro DVD rentals to Starbucks coffee. 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. Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

3 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx 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 adissible 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 odel 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 EN has no wealth of his own and relies on a financier/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) aredrawnfro 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, for i {H, L}, letf i (y) denote the distribution function fro action a i with a continuously differentiable density f i. Following Rothschild and Stiglitz (197), the following assuption ranks the riskiness of the two distributions under two actions. Assuption 1. The payoff distribution F L second order stochastically doinates F H, i.e., (F H F L )dy for all x>, and (1a) yf L dy = yf H dy =. (1b) 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 : 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. Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

4 4 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx 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 action a i under a sharing rule s. An optial contract with action a i is then the solution to a prograing proble in which the sharing rule s 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, U i (s) w>, (4) and the liited liability constraint 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 the action choice is enforceable, the optial risk sharing contract with action a i for i {H, L} is a pure debt contract s i Min(y, i ), (6a) where the face value i is uniquely deterined by EN s participation constraint and solves i i = w + F i dy. (6b) Proof. In an optial solution to (P1), EN s participation constraint (4) ust hold as an equality, and hence we ust have E[s] = s(y)f i dy = w. (7) Consider the fixed payent contract s i w for all y. For every y< w, the contract s i violates the liited liability constraint (5), and gives EN a negative payoff. Replacing EN s negative payoff by zero at such low values of y and adjusting FI s payoff accordingly so that (7) is still satisfied, one obtains the debt contract s i Min(y, i ) where the repayent obligation i is deterined by EN s participation constraint as 7 i (y i )f i dy = w i = w + F i dy. i I now show that the debt contract s i Min(y, i ) second order stochastically doinates any arbitrary contract ŝ i that satisfies (7) and the liited liability constraint (5). First, note that the distribution function ˆF i (.) ofs i is given by ˆF i (y) = F i (y)fory< i and ˆF i (y) = 1fory i. Let us denote the distribution function of our arbitrary contract ŝ i with Ĝ(y). Since ŝ i (y) s i (y) = y for all y< i, we have Ĝ(y) ˆF i (y) for all y< i. Therefore, it follows that T(x) (ˆF i Ĝ)dy for all x< i. (8a) What reains to be shown is that T(x) < for all x [ i, ) as well. Now note that since both contracts ŝ i and s i satisfy (7), we have E[ŝ i ] = E[s i ] (ˆF i Ĝ)dy =, (8b) 7 i In this expression, the ter F i dy reflects the adjustent in FI s payoff over w. Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

5 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx i.e., we have T(x = ) =. Since ˆF i (y) = 1 for all y [ i, ), we have ˆF i (y) Ĝ(y) = 1 Ĝ(y) for all y [ i, ), (8c) and therefore T(x) is increasing in x for every x [ i, ). But if T(x = ) = and T(x) is increasing for every x [ i, ), then we ust have T(x) < for all x [ i, ). This arguent establishes, together with (8a), that T(x) (ˆF ) i Ĝ dy for all x>. (9) Accordingly, the debt contract s i Min(y, i ) second order stochastically doinates all other contracts that satisfy (7) and the liited liability constraint (5). The optiality of the debt contract s i in (P1) then follows fro the strict concavity of v(.). The next result provides a coparison of FI s expected utility under the optial debt contracts s L and s H. Proposition 1b. We have V L (s L ) >V H (s H ), i.e., the optial debt contract s H with the high risk action yields a strictly lower expected utility to the risk averse FI than the optial debt contract s L with the low risk action. Proof. First I establish that H > L.Fro(6b), wehave L H = L F L dy H F H dy. For a contradiction, suppose that L H and rewrite (1a) as L H = H (F L F H )dy + L (1a) H F L dy. (1b) By Assuption 1, the first ter ( H (F L F H )dy). Therefore, if L H fro (1b) we would have L H L H F L dy. But this last inequality in (1c) is a contradiction, since F L (y) < 1fory [ H, L ] and hence L F L dy< L H. Therefore, H we ust have L < H. Now, given L < H, I show that s L second order stochastically doinates s H. The desired result V L (s L ) >V H (s H ) then follows fro the strict concavity of v(.). Let ˆF i denote the distribution function for s i. We need to establish that K(x) (ˆF L ˆF H )dy for all x>. First, note that ˆF i (y) = F i (y) fory< i and ˆF i (y) = 1fory i. Therefore, (ˆF L ˆF H )dy = (F L F H )dy for all x< L, and hence K(x) for all x< L. To see that K(x) for all x L as well, recall that the expected payoff is the sae under both s L and s H, and hence we have (ˆF L ˆF H )dy =. But since ˆF i (y) = 1fory i, it follows that K(x) = for all x H. Since ˆF i (y) = 1 for all y i,wehaveˆf L (y) ˆF H (y) for all y [ L, H ), which iplies that K(x) is increasing in x for all x [ L, H ). Given K(x) = for all x H and K(x) is increasing for all x [ L, H ), we ust have K(x) < for all x [ L, H ) which copletes our clai that K(x) for all x>. An iediate corollary of Proposition 1b is that when the action choice is enforceable, FI strictly prefers the low risk action a L. After characterizing this benchark case under enforceability, I now analyze the case when the 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 well known result in the financial 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 (1c) Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

6 6 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx Q2 illustrate the risk shifting incentives in our fraework, consider any pure debt contract with a repayent obligation. The change in EN s payoff fro switching to the high risk action a H under the debt contract is given by 27 (F H F L )dy for all >. (11) 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 EN s preference for high risk is the convexity of the residual payoff for EN under a pure debt contract. To see this ore transparently, consider any concave sharing rule s c (including the pure debt contract) that yields a convex residual payoff y s c for EN. By Assuption 1, wehave (y s c (y))(f H f L )dy = s c (y)(f L f H )dy. (12a) and hence EN s preference for high risk extends to any concave sharing rule s c. For exaple, a ixed debt equity contract of the for s Min(y, + (y )) with >and, which gives 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 (1 ) (F H F L )dy. (12b) 4.2. Ipleenting the low risk action 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: MaxV L (s) = v(s(y))f L dy, (P2) s s.t. (y s(y))f L dy w>, (13a) (y s(y))(f L f H )dy, 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. 8 Note that ipleenting a L by satisfying (13b) involves an agency cost relative to the first best achieved by s L = 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 = Min(y, H ) to ipleent a H and ensure an expected utility V H (s H ) = v( H ) H v (y)f H dy. In other words, the fully optial incentive copatible contract in the absence of enforceability is the better of the two contracts: the optial solution to (P2) that ipleents a L and the optial debt contract s H discussed in Section 3 that ipleents a H.IfwehaveV L (s ) <V H (s H ) at an optial solution s to (P2), then the second best fully optial contract is s H and the high risk action a H is to be ipleented. 9 One question is whether FI is better off fro satisfying (13b) and solving (P2) copared to ipleenting a H and receiving V H (s H ). Given that the other contract has already been discussed in Section 3, in what follows I consider the case where despite the deviation fro optial risk sharing and associated agency cost the second best involves ipleenting a L. 1 8 I a grateful to an anonyous referee for suggesting this discussion. 9 This outcoe 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. 1 The only restriction iposed on F H and F L is a stochastic doinance condition, which is too weak to atheatically characterize a siple condition that ensures FI is better off fro ipleenting a L by solving (P2). However, it should be noted that a sharing rule that satisfies (13b) and ipleents a L can yield the risk averse FI a higher expected utility than V H(s H), 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 which is not necessarily an optial solution to (P2). In Appendix A, I construct and solve an exaple to illustrate this point. (13b) (14) Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

7 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx Convertible debt contract My ain objective is to provide an explanation on why convertible debt can be preferred to ore standard financial contracts such as pure debt, pure equity and ixed debt equity. The convertible debt contract can be described as a sharing rule s, ax{in{y, },y} for and [, 1). (15) The above contract specifies a payoff realization below which FI receives all the realized payoff. Therefore, the paraeter 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 faily. The contract s =,> corresponds to a pure equity contract, whereas the contract s >,= corresponds to a pure debt contract. Let us now consider a variation of (P2) by restricting the set of feasible contracts to convertible debt contracts as described in (15). As noted, the class of contracts in (15) also includes all pure debt and pure equity contracts. Forally, within this restricted class the proble becoes choosing and [, 1) to axiize / V L (, ) = v(y)f L dy + v()f L dy + v(y)f L dy. (P3) / / s.t. (y )f L dy + (1 )yf L dy w>, (16) / / (y )(f L f H )dy + (1 ) y(f L f H )dy. (17) / Let us refer to the above proble as (P3). In the above forulation, the constraint in (16) is EN s participation constraint, and (17) stands for the incentive copatibility constraint that ust be satisfied to ipleent the low risk action. The proposition below establishes that the solution to (P3) is neither debt nor equity, but it is a non-degenerate convertible debt contract. Proposition 2. The optial solution to (P3) is a non-degenerate convertible debt contract s, with > and >. Proof. The preceding analysis ruled out the optiality of a pure debt contract, since it ipleents the high risk action, and violates (17). I now show that a non-degenerate convertible debt contract s, with > and > is superior to an equity contract as a solution to (P3). Let us define / U L (, ) (y )f L dy + (1 )yf L dy = w, (18a) / / W(, ) (y )(f L f H )dy + (1 ) y(f L f H )dy. Let be such that (y )f L dy = w. / Since U L (, ) is strictly decreasing in and, one can define a function :[, ] [, 1] by U L (, ()) = w. Forally, the function () satisfies /() (y )f L dy + (1 ())yf L dy = w. (19) /() Iplicit differentiation of (19) yields /() 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. To show that a solution with >is superior to pure equity, it suffices to show that as, we have dv L (, ()) d > and dw(, ()) d. (18b) Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

8 8 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx Note that as, we have ( / dv L d = f L dy 1 ) / (v (y)/v ())yf L dy / yf > L dy ( ) / / yf L dy f L dy 1 / yf =, L dy where the inequality follows fro the fact that li y v (y) =, and hence the ratio v (y)/v () approaches to zero as. Siilarly, straightforward algebra yields / [ ] / dw(.) d = / yf H dy / f H dy f L dy / yf > (f H f L )dy> L dy }{{} <1 by Assuption 1 as. This follows, since as Assuption 1 iplies that f H (y) >f L (y) for all y (, /). A convertible debt contract has two desirable properties copared to pure debt and pure equity. Unlike a pure equity contract, the debt coponent of convertible debt assigns the whole payoff to the risk averse FI at the low 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. 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 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. Acknowledgeents 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. Appendix A Appendix A presents an exaple to illustrate that FI can be better off fro ipleenting a L even with an equity contract s e = e y (which satisfies (13b) but not necessarily an optial solution to (P2)) rather than opting for a H by offering s H = Min(y, H ). Let us specify FI s utility function by v(x) = ln x and set EN s reservation wage as w = 3. Suppose that for action a i with i {H, L}, the probability density function f i is log-noral and it is given by [ ] 1 f i (y; i, i ) = exp (ln y i) 2 y i i To ensure that F H is a ean preserving spread of F L when both F H and F L are log-noral, one needs the following two conditions to be satisfied 315 L + 2 L 2 = H + 2 H 2 and L < H, Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco

9 S. Ozerturk / Journal of Matheatical Econoics xxx (28) xxx xxx which we will ensure with the following paraeter specification: L = 1, H = 2, L = 4.5, H = With the above paraeter specification, we have [ ] yf i dy = exp i + 2 i = for i {H, L}. 2 One can copute the face value H of the debt contract under action a H by solving (6b) which takes the for H H = yf H dy 3 + F H dy, and yields H = The debt contract s H Min(,y) yields FI an expected utility H V H (s H ) = H ln(y)f H dy + H ln( H )f H dy = Let us now copute the equity contract s e e y by solving e = 1 w yf L dy = =.79. This equity contract is not necessarily an optial solution to (P2), but it ipleents the low risk action a L and yields FI an expected utility V L (s e ) = References ln( e y)f L dy = >V H (s H ). Bergeann, D., Hege, U., Venture capital financing, oral hazard and learning. Journal of Banking and Finance 22, Berglof, E., A control theory of venture capital finance. Journal of Law Econoics and Organization 62, Cheanur, T., Fulghieri, P., A theory of going public decision. Review of Financial Studies 12, Cornelli, F., Yosha, O., 23. Stage financing and role of convertible securities. Review of Econoic Studies 7, Diaond, D., Financial interediation and delegated onitoring. Review of Financial Studies 51, Gopers, P., Ownership and control in entrepreneurial firs: an exaination of convertible securities in venture capital investents. Harvard Business School Working Paper. Green, R., Investent incentives, debt and warrants. Journal of Financial Econoics 13, Innes, R., 199. Liited liability and incentive contracting with ex ante action choices. Journal of Econoic Theory 52, Jensen, M., Meckling, W., Theory of the fir: anagerial behavior, agency costs and ownership structure. Journal of Financial Econoics 3, Kaplan, S., Stroberg, P., 23. Financial contracting theory eets the real world: an epirical analysis of the venture capital contracts. Review of Econoic Studies 7, Marx, L., Efficient venture capital financing cobining debt and equity. Review of Econoic Design 3, Rothschild, M., Stiglitz, J., 197. Increasing risk I: a definition. Journal of Econoic Theory 2, Sahlan, W., 199. The structure and governance of venture capital organizations. Journal of Financial Econoics 27, Schidt, K., 23. Convertible securities and venture capital finance. Journal of Finance 58, Trester, J., Venture capital contracting under asyetric inforation. Journal of Banking and Finance 22, Please cite this article in press as: Ozerturk, S., Risk sharing, risk shifting and the role of convertible debt, J Math Econ (28), doi:1.116/j.jateco