Joint Ventures and Risk Sharing

Joint Ventures and Risk Sharing ∗ Lloyd P. Blenman † , and Mingxin Xu ‡ October 13, 2008 This paper is a completed version of a paper prepared for pr...
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Joint Ventures and Risk Sharing ∗ Lloyd P. Blenman † , and Mingxin Xu ‡ October 13, 2008

This paper is a completed version of a paper prepared for presentation at the 2008 Midwest Finance Association conference. The authors thank the Special Issue Editor, Larry Prather for accepting the paper for this Special Issue. All remaining errors are entirely ours. The work of Mingxin Xu is supported by National Science Foundation under the grant SES-0518869. † Department of Finance and Business Law, University of North Carolina at Charlotte. Phone: (704)-687-2823. E-mail: [email protected] ‡ Department of Mathematics, University of North Carolina at Charlotte. Phone: (704)-687-3870. E-mail: [email protected]

Abstract We analyze the problem of optimal risk sharing in a new joint venture, in a two-entity framework, where the new venture’s products are sold in a competitive market environment. The two parties to the joint venture each have pre-existing risky projects and the profit potential of the new venture is random. Risk is driven by Brownian motion processes; there is no debt and the two entities act cooperatively. In this setting, optimal risk-sharing can be determined ex-ante, with sure expectation of no ex-post negotiation, and the resulting solution will be stable. We find the square root sharing rule, can be optimal depending on, the risk preferences of the venture partners, time horizon of the joint venture and the complementarity and substitutability of the new venture cash flows with those of existing projects owned by each partner.

1. Introduction

There are several types of cooperative governance structures, production and risk sharing mechanisms. Of these, the most prominent are strategic alliances, partnerships and joint venture agreements. These types of agreements are typically drawn up for the sharing of technical information, joint research and development projects, the sharing of production costs, interlocking equity stake holdings and joint ventures, either of the horizontal or vertical type.1 Joint ventures result from the sharing of the assets of two or more cooperating entities and the resultant newly created entity, may or may not have an independent management structure. In most cases, the amount of managerial control exerted over the new entity is a function of the relative bargaining strengths of the parties as well as their respective, capital contributions, equity stakes and shares of the start up costs of the venture.2 Chan et al. (1997) find that strategic alliances, like joint ventures, generate positive mean share price reactions when announced. McConnell and Nantell (1985) find that joint ventures are wealth increasing and attribute the gains to synergies. Chen and Ross (2003) analyze the pricing strategy of the joint venture, where the venture cooperates in the production of a common input. They show that a joint venture can

1

The complexity and range of these arrangements precludes any effort to succinctly summarize them. However, what is clear is that they are becoming a preferred mode of expansion domestically and internationally. 2 The two entities may be deemed to be cooperative with respect to the jointly owned venture, but may still be rivals in other markets and products. More worryingly still, is the fact that the two entities might be cooperative with respect to the production of the joint ventures output but might still compete on the pricing of the common product.

replicate the effects of a full-scale merger of the parents. The focus is on input and output price efficiency. Chan, Kensinger, Keown and Martin (1997) study strategic alliances and find that there is a positive response to such announcements and there is no wealth transfers among the alliance partners. For horizontal alliances, which may include joint ventures, more value accrues to the partners when they are in the pooling or sharing of technical information. The primary focus is on non-equity alliances. Some of the other reasons advocated for the rise of joint ventures include the avoidance of holdup problems (Williamson 1979, 1983); Cai (2003), transaction specificity, transactions uncertainty, and their use as financing vehicles. The deterioration of the finances of an entity might propel a firm to seek a partner. Another possible alternative motive is that joint ventures allow suppliers to share risks with buyers. Joint ventures reduce the size of the investment a supplier makes to fill an order, which puts less of its resources at risk and allows it to invest in a greater number of projects for a given amount of capital. If financial distress is costly, then risk sharing can increase value. Risk sharing should be particularly important for risky suppliers and/or suppliers that are less diversified. Johnson and Houston (2000) find no evidence of the risk sharing motives as drivers of joint ventures. They were primarily investigating firms and their suppliers in a vertical joint ventures. They find synergistic gains for suppliers in vertical joint ventures and gains for joint partners in horizontal joint ventures. Reynolds and Snapp (1986) and Kwoka (1989) study the impact of joint ventures on competition. Chen and Ross (2000) as well show that joint ventures could be used as deterrence to the market entry of a rival as well as being anti-competitive. Of course some joint ventures could be pro-competitive if they reduce the dominance of a market leading firm.

The typical assumption, in a two-agent model, is a 50% equity stake in the new firm, even though other organizational forms have become much more common. Cai (2003) notes that contrary to the predictions of standard Grossman, Hart and Moore theory, where joint ownership of assets is predicted as suboptimal, joint ventures are becoming the dominant form of expansion in China, Africa and Latin America. Reuer and Koza (2000) indicate that joint ventures are dynamic and growing component of all corporate forms of alliances. There are however wide disagreements over the causal factors behind this growth. Among the factors that have been suggested as drivers of this trend are (i) ability of having additional expertise in screening projects (ii) desire to have access to resource markets (iii) knowledge acquisition, and learning and (iv) risk and profit sharing. If risk sharing contracts are optimal and renegotiation proof this will enhance the stability of the joint venture. The available empirical evidence shows that joint ventures are subject to early terminations and have shorter life spans than those of other types of corporate structures, Kogut (1989, 1988), Nakamura (2005) and Reuer and Koza (2000). Evidence also suggests that where there is relative agreements on the assets in question, firms are more apt to select joint ventures as a mode of market entry. When there is disagreement or asymmetry of beliefs over asset valuations, growth of product markets, market direction etc., firms are more likely to choose acquisitions as a means of market entry. Hennart (1988), posits that joint ventures are more likely to occur, rather than acquisitions, when the costs of integrating assets in a merger are high. It could be that information asymmetry obscures the verification of the desirable versus non-desirable assets. In a joint venture these costs can be reduced. Under conditions of asymmetric information, a joint venture mitigates the firm’s need to engage in

costly efforts to reduce valuation uncertainties, as well as the risks of either offering too little and failing to complete the transaction or overpaying for the targeted resources (e.g., Varaiya, 1988). Joint ventures are also attractive vehicles for reducing the uncertainty and costs of valuing complementary assets ex ante. They conjecture that joint ventures are unattractive as diversification tools and are more suitable for obtaining scale economies when target firms have a dominant position. Joint ventures are also attractive if there is need to forge resource integration. In our paper we specifically focus on risk sharing as a possible driver of a joint venture. In order to focus on the pure risk -sharing motive, we abstract from moral hazard problems, (hidden action) on the part of both venture partners as well as the formal organizational structure of the joint venture. The parties act cooperatively and have symmetric information about the characteristics of the project. The returns to the project are also ex-post verifiable and the management of the joint venture acts to maximize the joint venture profits.3 The paper is organized as follows. Section I is the introduction. Section II provides the basic model set-up and assumptions. Section III derives the basic risk-sharing rule. What is necessary is that the cashflow of the common project should provide a hedge to existing cash flows for one partner. The optimal sharing rule is a square root rule. For this to be achieved, it depends on the risk preferences of the venture partners, time horizon of the joint venture and the complementarity and substitutability of the new venture cash flows with those of existing projects owned by each partner. The optimal solution once attained in this context is stable as there will be no tendency to move

3

Pratt (2000 and 1964), Pratt and Zechauser (1989) provide conditions for risk-sharing not to be sub-optimal. In the model we analyze in the paper, the optimal solution with risk sharing, is the first-best solution. Neither joint venture partner can be made better off by trying to terminate the joint venture and/or by trying to do it as a solo project.

away from it. In this context the solution does not depend on any bargaining outcome. Section IV concludes the paper.

2. The Model

We develop the results of the paper, in various stages of complexity so that the reader can follow the implications for the relaxation of several assumptions as we progressively remove restrictions on costs, profit and cash flow rights allocations, ex-ante and ex-post bargaining and develop the necessary conditions for the existence of a renegotiation proof risk-sharing contract. We assume there are two entities (i = 1, 2) that want to share the risks associated with undertaking a new single project in a joint venture. The new project may represent either a vertical or horizontal expansion of activity for each entity. Furthermore no synergistic benefits are assumed for either party. Regardless of the nature of the individual expansion type the two entities are assumed to act cooperatively. The interest rate is deterministic r and all projects have equal lives. That is the expected life of the new project equals the remaining life of the existing projects. We assume each individual project has deterministic costs x0i but random profit potential.4 The output q of the joint venture is chosen to optimize the profit of the new entity, but since sale price p% is random, the expected excess profit is random. Expected excess revenue is posited to be a Brownian motion process.5

4

The model is sufficiently flexible to handle the case of random costs for the projects. Including this more realistic specification does not alter the analysis as the ensuing uncertainty will be encapsulated in the random profit function. 5 It can be shown that the mean rate of drift of the revenue and costs processes, if we were to assume a stochastic cost structure, would have no bearing on the optimal investment stake and the allocation share of the profits.

2.1 Case A; Linear Risk Sharing Rules

Each entity has an existing investment in a single risky project other than the common risky project. Each wants to share the profit risk associated with the common project. The cost of the common project is completely known as y0 . There is no hidden information between the two entities as far as the common project is concerned. All parties have access, ex-post, to the true realized returns of the common project. The management of the joint venture is assumed to be acting to maximize the profits of the new entity. The industry environment in which the new venture is being launched is assumed to be a competitive one and the entity takes on no debt. All profits of the new venture are to be shared between the two entities and we search for an equitable and optimal sharing rule, that would not depend on the ex-ante relative bargaining power of the two entities and would permit the creation of the venture.

We define the parameters:

γ i ∈ R+ : risk aversion parameter of the i − th entity; xi ∈ R+ : initial wealth of the i − th entity; xi + Wi (t ) : time t wealth of the i − th entity where Wi are standard Brownian

motions; W3 (t ) : profit of the risky project represented by a third Brownian motion with the

correlation structure dW1 (t )dW3 (t ) = ρ13dt and dW2 (t )dW3 (t ) = ρ 23dt . Each entity wants to maximize its own CARA utility function:

u1 (α ) = E[1 − e −γ 1 ( x1 +W1 ( T )+αW3 ( T )) ],

(1)

u2 (α ) = E[1 − e −γ 2 ( x2 +W2 ( T )+(1−α )W3 ( T )) ].

(2)

Of course, one investor’s optimizer could be far from the other’s and therefore any negotiation process may not result in a risk sharing contract. In this paper, we would like to provide necessary and sufficient conditions and the design of the risk-sharing contract such that negotiation will stop where both individuals achieve optimal expected utilities simultaneously. This will ensure that there should be no ex-post bargaining, given the above assumptions, once the contract is signed and the project is underway.6 Case A: Theorem 2.1. In the special case when ρ13 < 0 , ρ 23 < 0 and ρ13 + ρ 23 + 1 = 0 , u1 and u2 are simultaneously maximized at α = − ρ13 = ρ 23 + 1 . Otherwise, the Pareto equilibrium occurs on intervals given below •

ρ13 ≥ 0 and ρ 23 ≥ 0 : α ∈ [0, 1] ;



ρ13 ≥ 0 and ρ 23 < 0 : α ∈ [0, ρ 23 + 1] ;



ρ13 < 0 and ρ 23 ≥ 0 : α ∈ [ − ρ13 , 1] ;



ρ13 < 0 and ρ 23 < 0 : α ∈ [min{ − ρ13 , ρ 23 + 1 }, max{ − ρ13 , ρ 23 + 1 }] ;

i.e., when α is chosen as a number in the given interval, neither entity could improve its utility without hurting the other one. In this case, for one party, participation in the joint venture is a sub-optimal strategy. PROOF. To simplify the equations (1) and (2), define

6

Linear rules are not optimal in this setting as we do not permit side payments from one venture partner to the other. In some settings, especially in international joint ventures, some side payments are not permitted.

σ 1 (α ) = 1 + 2αρ13 + α 2 , σ 2 (α ) = 1 + 2(1 − α ) ρ 23 + (1 − α ) 2 . Then

W1 ( T ) σ1 (α )

+ ασW13(α( T) ) and

W2 ( T ) σ 2 (α )

+ (1−σα2)(Wα3)( T ) are new Brownian motions and therefore

normally distributed and u1 (α ) = 1 − e −γ 1x1 E[e

W (T )

− γ 1σ1 (α )( σ1 ( α ) +

u2 (α ) = 1 − e −γ 2 x2 E[e

1

αW3 ( T ) ) σ1 ( α )

] = 1− e

W ( T ) (1−α ) W ( T ) − γ 2σ 2 (α )( σ 2 ( α ) + σ ( α3 ) ) 2 2

− γ 1 x1 + 12 γ 12σ1 (α )2 T

] = 1− e

,

− γ 2 x2 + 12 γ 22σ 2 (α )2 T

.

Next we find the derivatives of the utility with respect to parameter α : ∂u1 (α ) ∂σ (α ) = −(1 − u1 (α ))σ 1 (α )γ 12T 1 = −(1 − u1 (α ))γ 12T ( ρ13 + α ), ∂α ∂α ∂u2 (α ) ∂ σ 2 (α ) = −(1 − u2 (α ))σ 2 (α )γ 22T = (1 − u2 (α ))γ 22T ( ρ 23 + 1 − α ). ∂α ∂α

When ρ13 > 0 and ρ 23 > 0 ,

∂u1 (α ) ∂α

< 0 and

∂u2 (α ) ∂α

> 0 for α ∈ [0, 1] . u1 is maximized

when α = 0 and u2 is maximized when α = 1 . The Pareto equilibrium is the interval [0, 1] and too wide to be meaningful as a result of any negotiation process. The rest of

the cases can be similarly analyzed noting that: when ρ13 < 0 , u1 first increases, then decreases; when ρ13 > 0 , u1 always decreases; and when ρ 23 < 0 , u2 first increases, then decreases; when ρ 23 > 0 , u2 always increases. Now suppose ρ13 < 0 and ρ 23 > 0 . If we choose α < − ρ13 , then both expected utility have positive derivative at α and can be simultaneously increased if α increases. However, this situation does not happen if α ∈ [ − ρ13 , 1] . In this case the posited mechanism to share the profits and allocate costs does not result in a solution to proceed with a joint venture that is optimal for both parties, except in the limited situation where ρ13 < 0 , ρ 23 < 0 and ρ13 + ρ 23 + 1 = 0 . If this condition holds, u1 and u2 are simultaneously maximized at α = − ρ13 = ρ 23 + 1, and both parties will commit to the joint venture. In all the other cases listed, optimal joint

ownership is not feasible with the given set of rules. So a linear risk sharing rule is generally not optimal for this simple case. It simply does not permit both parties to simultaneously participate in the project, even if a decision is made to proceed with the project.

2.2 Case B: Non-linear risk sharing rules

In the first case discussed above, a linear sharing rule does not give the flexibility of assigning differential proportions in investment and reward. No expertise is required for designing the particular risk-sharing contract and we have seen that such a simple method does not result in a non-trivial risk-sharing plan. In this case, we relax the previous set of assumptions and we assign the entities α and 1 − α proportions of the cost of the project, while getting λ (α ) and 1 − λ (α ) proportions of the payoff. With the same basic parameters as in Case A, we will further define the discounting rate, costs of the projects and extend the risk aversion parameters to be any real numbers. For i = 1, 2 , define •

r ∈ R+ : interest rate;



γ i ∈ R : risk aversion parameter of the i -th entity;



xi ∈ R+ : initial wealth of the i -th entity;



xi + Wi (t ) : time t wealth of the i -th entity where Wi are standard

Brownian motions; • motion

W3 (t ) : profit of the risky project represented by a third Brownian

with

the

dW2 (t )dW3 (t ) = ρ 23dt .

correlation

structure

dW1 (t )dW3 (t ) = ρ13dt

and



x0i ∈ R+ : cost for project Wi ;



y0 ∈ R+ : cost for project W3 .

Equations of terminal wealth. Entity 1, ( x1 − x01 − α y0 )e rT + W1 (T ) + λ (α )W3 (T )

Entity 2, ( x2 − x02 − (1 − α ) y0 )e rT + W1 (T ) + (1 − λ (α ))W3 (T )

Each entity wants to maximize expected terminal wealth as described by the respective expected CARA utility functions : u1 (α ) = 1 − E[e −γ 1 ( β1 (α )+W1 ( T )+λ (α )W3 ( T )) ],

(3)

where β1 (α ) = ( x1 − x01 − α y0 )e rT ; u2 (α ) = 1 − E[e −γ 2 ( β2 (α )+W2 ( T )+(1−λ (α ))W3 ( T )) ], where β 2 (α ) = ( x2 − x02 − (1 − α ) y0 )e rT .

The solutions that follow below are unconstrained, in the sense that each entity maximizes its expected terminal wealth (a) without regard to the optimal attainable wealth by the joint venture partner, (b) without an added boundary condition that expected terminal wealth from doing the joint venture exceeds, expected terminal wealth without the joint venture.

(4)

3. Optimality of the sharing rule Definition 3.1. An acceptable risk-sharing contract must have a payoff function λ such that λ : [0, 1] → [0, 1] has to satisfy the following conditions: •

λ (0) = 0 , λ (1) = 1 ;



λ (α ) is an increasing and continuously differentiable function of α

on [0, 1] ; •

α and 1 − α are respectively the percentage of the project costs

contributed by the first and second party; •

λ (α ) and 1 − λ (α ) are respectively the percentage of the profit of the

risky project shared by the first and second party. Theorem 3.1. If there exist an α ∗ ∈ [0, 1] and an acceptable contract λ (α ) where

γ 1 y0e rT + γ 12T ( ρ13 + λ (α ∗ ))λ ′ (α ∗ ) = 0,

(5)

γ 2 y0e rT + γ 22T ( ρ 23 + 1 − λ (α ∗ ))λ ′ (α ∗ ) = 0,

(6)

and for all α ∈ [0, 1] the following inequalities hold: { γ 1 y0e rT + γ 12T ( ρ13 + λ (α ))λ ′ (α ) }2 + γ 12T{ ( ρ13 + λ (α ))λ ′′ (α ) + (λ ′ (α )) 2 } < 0,

(7)

{ γ 2 y0e rT + γ 22T ( ρ 23 + 1 − λ (α ))λ ′ (α ) }2 − γ 22T{ ( ρ 23 + 1 − λ (α ))λ ′′ (α ) − (λ ′ (α )) 2 } < 0,

(8)

where we assumed the second derivative λ ′′ exists, then λ (α ) is an optimal risk-sharing contract and both investors obtain maximal expected utility at a unique point α = α ∗ . Remark 3.1. If the second order conditions (7) and (8) are too complicated to check or the second derivative λ ′′ does not exist, we can replace them with first order conditions:

γ 1 y0e rT + γ 12T ( ρ13 + λ (α ))λ ′ (α ) > 0, γ 2 y0e rT + γ 22T ( ρ 23 + 1 − λ (α ))λ ′ (α ) < 0 on α ∈ [0, α ∗ ),

(9)

γ 1 y0e rT + γ 12T ( ρ13 + λ (α ))λ ′ (α ) < 0, γ 2 y0e rT + γ 22T ( ρ 23 + 1 − λ (α ))λ ′ (α ) > 0 on α ∈ (α ∗ , 1].

(10)

PROOF. To simplify the equations (3) and (4), define

σ 1 (α ) = 1 + 2λ (α ) ρ13 + λ (α ) 2 , σ 2 (α ) = 1 + 2(1 − λ (α )) ρ 23 + (1 − λ (α )) 2 . Then

W1 ( T ) σ1 (α )

+ λ (ασ1)W(α3 )( T ) and

W2 ( T ) σ 2 (α )

+ (1−λσ(α2 ())αW)3 ( T ) are new Brownian motions and

therefore normally distributed and the moment generating function can be computed as u1 (α ) = 1 − e −γ 1β1 (α ) E[e

W (T )

− γ 1σ1 (α )( σ1 ( α ) +

u2 (α ) = 1 − e −γ 2 β2 (α ) E[e

1

λ ( α ) W3 ( T ) ) σ1 ( α )

] = 1− e

W ( T ) (1− λ ( α )) W3 ( T ) − γ 2σ 2 (α )( σ 2 ( α ) + ) σ2 (α ) 2

− γ 1β1 (α ) + 12 γ 12σ1 (α )2 T

] = 1− e

,

− γ 2 β 2 (α ) + 12 γ 22σ 2 (α )2 T

.

The sufficient conditions results from the first and second derivatives of the expected utilities with respect to parameter α : ∂u1 (α ) ∂β (α ) ∂σ (α ) = −(1 − u1 (α )){ −γ 1 1 + γ 12T σ 1 (α ) 1 } ∂α ∂α ∂α = −(1 − u1 (α )){ γ 1 y0e rT + γ 12T ( ρ13λ (α )′ + λ (α )λ (α )′ ) },

(11)

∂u2 (α ) ∂β (α ) ∂σ (α ) = −(1 − u2 (α )){ −γ 2 2 + γ 22T σ 2 (α ) 2 } ∂α ∂α ∂α = −(1 − u2 (α )){ −γ 2 y0e rT − γ 22T ( ρ 23λ (α )′ + (1 − λ (α ))λ (α )′ ) }.

(12)

∂ 2u1 (α ) = −(1 − u1 (α )){ γ 1 y0e rT + γ 12T ( ρ13λ (α )′ + λ (α )λ (α )′ ) }2 2 ∂α −(1 − u1 (α ))γ 12T{ ( ρ13 + λ (α ))λ (α )′′ + (λ (α )′ ) 2 },

(13)

∂ 2u2 (α ) = −(1 − u2 (α )){ −γ 2 y0e rT − γ 22T ( ρ 23λ (α )′ + (1 − λ (α ))λ (α )′ ) }2 ∂α 2 +(1 − u2 (α ))γ 22T{ ( ρ 23 + 1 − λ (α ))λ (α )′′ − (λ (α )′ ) 2 }.

(14)

α ∗ is a critical point if it is a solution to equations (7)=0 and (8)=0 which yield the first order conditions (5) and (6). Strict concavity guarantees the maximum and its uniqueness.

Proposition 3.1. If the following conditions are satisfied:

γ 1 = −γ 2 ,

ρ13 = 0,

ρ 23 = −1,

1 y0e rT = − γ 1T , 2

then we can find the acceptable risk sharing plan

λ (α ) = α such that either investor will agree on any sharing point α ∈ [0, 1] . The converse is also true. PROOF. If both investors agree at any possible proportion α , then their expected utilities are constants for varying α . Setting (11)=0 and (12)=0, we get

where c1 =

y0e rT γ 1T

and c2 =

y0e rT γ 2T

c1 + ( ρ13 + λ (α ))λ (α )′ = 0,

(15)

c2 + ( ρ 23 + 1 − λ (α ))λ (α )′ = 0,

(16)

. The corresponding solutions for (15) and (16) are

λ1 (α ) = − ρ13 ± ρ132 − 2c1α + d1 , λ2 (α ) = 1 + ρ 23 ± (1 + ρ 23 ) 2 + 2c2α + d 2 . For the risk sharing plan to be mutually acceptable, the functions

λ1 (α ) = λ2 (α ) = λ (α ) have to be the same, we need − ρ13 = ρ 23 + 1 and −c1 = c2 . Therefore, γ 1 = −γ 2 and

λ (α ) = − ρ13 ± ρ132 − 2c1α + d1 . λ (0) = 0 implies d1 = 0 and ρ13 ≥ 0 . Since − ρ13 = ρ 23 + 1 , we have ρ13 = 0 and

ρ 23 = −1 . λ (1) = 1 implies ρ13 = −c1 − 12 and the negative solution is not feasible

unless ρ13 = 0 . In conclusion,

λ (α ) = α . The above arguments are equivalent and the conditions we have are both sufficient and necessary.

We note that the square root risk sharing rule is optimal for both partners. By itself this is not new but we have derived the result endogenously by looking at the risk profiles of the partners, individual project distributions and own project correlation factors of each joint venture partner. What is significant is that we show that when

γ 1 = −γ 2 , there are simply valid reasons for two joint venture partners to proceed with the common project. For the first entity, the cash flows from the common project must be uncorrelated with his existing project. For the second entity, the cash flows from the common project must act as a hedge for cash flows from existing project. The literature is replete with discussions of private benefits as the reason for joint ventures, but here we have assumed full disclosure and the parties make the decision with each having knowledge of the project characteristics and the impact that it would have on their individual non-joint projects.7

7

Many authors, including Bolton and Harris (2006), Olsen and Osmussen (2005), and Marinucci (2008) generate a simple sharing rule in the two firm, joint venture problem, where the optimal sharing rule is the ratio of the principal’s risk aversion to the sum of the principal and agent’s risk aversion parameters. In most of these cases, the analyses were conducted with respect to the flows from the joint venture alone, ignoring the possibility of the partner firms having other existing projects. However,

We can also see that even in this simple case that the 50-50 ownership stakes, based on 50-50 investment costs rule fails. As we remarked earlier in Section III where the properties of optimal rule are outlined, the normalized risk of the new venture is T, where T is also the horizon length of the venture partners. Hence the sharing rule is independent of the risk of the new venture, but depends on the risk preferences of the joint partners and the complementarity and substitutability of the new venture cash flows with those of the existing projects of the partners. Hence

λ (α ) T , measures the share of the risk of the cash flows, that accrues to a joint venture partner who invests an equity stake, α ∗ in the project. λ ∗ measures his share of the cash flow, profits from the venture. The residual risks accrue to the second partner as well as all residual cash flows.

this rule can also be derived in contexts where existing projects are not ignored.

4. Conclusion We analyze the problem of optimal risk sharing in a continuous time setting where two joint venture partners consider investing in a new project. The partners operate cooperatively in a setting of homogeneity of beliefs about the distribution of the cash flows of the new project. The partners do not compete in the product market for the revenues from the common project, so the management structure that optimizes the revenue from the common project is optimal for both parties. The optimal sharing rule from the profits and revenues, is determined endogenously and there is no incentive to deviate from the optimum. We show that the optimal sharing rule depends not on the risk of the new venture,but on the risk preferences of the venture partners, the time horizon of the joint venture and the complementarity and substitutability of the new venture cash flows with those of existing projects owned by each partner. In this simple setting, the optimal equity stakes is a function of the optimal sharing rule for the cashflows. In future work we will try to develop a more general model of the joint venture case where we permit non-cooperative behavior in product markets as well as moral hazard issues as joint venture partners try to improve their welfare at the expense of their partners. Our contribution is that we show that the joint venture problem can potentially provide more stability than is usually considered, once the partners are operating in a cooperative framework. In this setting it is optimal to share project risk. Any attempt to deviate from the optimum allocation would leave both joint venture partners worse off.

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