Dry powder and short fuses: Private equity funds in emerging markets

Transcription

1 Dry powder and short fuses: Private equity funds in emerging markets Dawei Fang Department of Economics, University of Gothenburg 31 st July, 2015 Abstract: Private equity (PE) investors in emerging markets often prefer funds with a short fuse, i.e., a much shorter lifespan than their developed market counterparts. However, based on a simple agency model, we show that, unless managerial compensation is sufficiently concavified, the short fuse can exacerbate agency conflicts by encouraging the manager to burn money quickly. The money burning incentives produced by the short fuse can be countered by concavifying compensation or by lengthening the fuse. When PE funds target at projects with high profitability potential, concavifying compensation is unattractive to PE managers because it forces them to concede rents to investors. Thus, when the capital market is competitive, the equilibrium financing arrangement coincides with the long-fused convex compensation contracts used in developed markets. Thus, our model predicts that the growth in competition for PE funds in emerging markets will lead to the adoption of the long-lifespan PE contracts typical in developed PE markets. Keywords: dry powder, emerging markets, lifespan, limited partnership, PE/VC Contact dawei.fang@cff.gu.se. I thank Alexander Guembel, Ulrich Hege, Tim Jenkinson, Alan Morrison, Thomas Noe, Ludovic Phalippou, John Quah, David Robinson, Linus Siming, and Anjan Thakor for helpful comments, and seminar participants at University of Gothenburg, University of Oxford, and Queen Mary University of London. Any remaining errors are my responsibility.

2 1 Introduction Globally, private equity (PE) funds, including venture capital (VC) and buyouts (BO), manage about $3.8 trillion with more than $1.1 trillion dry powder (unspent capital) as of June 2014 (Preqin (2015)). While developed markets still account for the majority of total assets under management, emerging markets have attracted significant interest from global investors for PE investments in recent years, with China and India ranked among the top 5 regions that receive most PE investment (Metrick and Yasuda (2011)). The domestic PE funds in these emerging markets typically replicate the limited partnership structure used in the western world for PE practice. These partnerships have two distinguishing features: a finite lifespan and convex managerial compensation. The life of a PE fund is contractually partitioned into an investment period and a harvesting period. New investments are limited to the investment period, during which the management team of a PE fund, who serves as the general partner (GP), seeks new investment opportunities and invests the capital committed by limited partners (LPs) in a portfolio of projects. During the harvesting period, the GP manages and eventually exits from those investments to realize capital gains. GP compensation generally includes an option-like performance component, known as carried interest, which usually amounts to 20% of realized profit. 1 While the general structure of a PE fund in emerging markets is similar to its western counterpart (Guo and Jiang (2013)), emerging market PE funds typically have a short fuse, i.e., their investment period and lifespan are much shorter than their developed market counterparts. For example, while the life of a US PE fund is usually years with the first 5 years as the investment period (Robinson and Sensoy (2013) and Metrick and Yasuda (2010)), the life of a Chinese PE fund is only 4-7 years with the investment period often being as short as 2 years (BVCA 2011). 2 However, the short fuse appears to encourage the GP to burn dry powder quickly. As a fund manager in DT Capital Partners, a leading PE firm in China, said, for a newly established PE fund (in China), typically the total life is 7 years, with the first two or three years being the investment period, followed by a five-year harvesting period... So there is no time for the GP to hesitate. Just spend the money in time and do not worry much about making bad deals. 3 1 See Sahlman (1990), Gompers and Lerner (1999), Phalippou (2007), Cumming and Johan (2009), and Metrick and Yasuda (2011) for a discussion of the limited partnership structure and GP compensation. 2 The shorter lifespan of Chinese PE funds is a reflection of the current landscape of China s LP community, which comprises businesses and wealthy individuals as the main capital providers (Zero2IPO (2013)). These LPs are new players in the market and, compared to their foreign peers, are less accustomed to the idea of entrusting their capital to the GP for a lengthy period of time. 3 This conversation appears in a Chinese article. The original article is written in Chinese and is 1

3 In this paper, we study PE limited partnership contracting by focusing on the GP s moral hazard of burning dry powder and the length of the fuse, i.e., the length of the investment period and the lifespan of a PE fund. We base our analysis on a simple agency model in which we endogenize GP compensation, fund size, and the length of the fuse. We show that, while a long fuse combined with convex GP compensation induces the GP to burn dry powder conditionally, i.e., conditioned on low cumulative performance of the fund, a short fuse combined with convex GP compensation may induce the GP to burn dry powder unconditionally and, hence, the short fuse exacerbates agency conflicts. Minimizing agency conflicts typically requires both the use of the short fuse and a concavification of GP compensation, which might force the GP to concede rents to LPs. When rent concession is large, the equilibrium contracting under a competitive capital market typically features a long fuse and convex GP compensation, which coincides with the PE fund structure used in developed markets. To be more specific, in our model, to weed out the unskilled fly-by-night operators, the PE fund gives the GP a positive payoff only if the GP creates value for the fund. Thus, the GP bears little of the downside risk and has an inclination toward risk shifting. If the partnership has a long fuse, the GP has an incentive to undertake higher quality projects in the early phase of the investment period and thus postpones his investment decisions regarding the lower quality projects until he acquires private information on the progress of his earlier investments. This strategic timing behavior enables the GP, in the middle of the investment period, to better predict the fund s final performance and use this prediction in the late phase of the investment period to game the performancebased compensation scheme. Specifically, if the GP anticipates the success of his earlier investments, he will pass up lower quality projects, while, if he foresees the failure of his earlier investments, he will go for broke by taking those bad deals. Shortening the fuse only inhibits the GP s use of interim information for gaming but does not guarantee an improvement in efficiency. When project profitability potential is high, both a short fuse and a concavification of GP compensation are typically required to minimize agency conflicts. However, although this contract is most efficient, it may not be an equilibrium contract because it forces the GP to concede rents to LPs. When rent concession is large, a long-fused convex compensation contract, albeit less efficient, sustains an equilibrium. We further show that, when a long-fused convex compensation contract is used, restricting fund size can alleviate overinvestment, at the expense of causing underinvestment: the GP has to pass up some good investment opportunities if he encounters a sufficiently large number of good projects. Thus, restricting fund size improves efavailable at 2

4 ficiency only if the GP has low chance of facing many good investment opportunities, which is likely the case when economic conditions are bad. Our model has policy implications on the PE practice in emerging markets, such as China and India, whose PE partnership contracts typically feature a short fuse and convex GP compensation. While LPs in these markets, especially non-institutional LPs, believe that it is too risky to entrust their money to the GP for a lengthy period of time and thus demand a short fuse, our model implies that the short fuse, combined with convex GP compensation, might in fact exacerbate the agency risk faced by these LPs. Our model suggests that there might exist at least two ways to make such a partnership contract more efficient, either by lengthening the fuse or by concavifying GP compensation. Our model also suggests that, if economic conditions are not sufficiently good, shrinking fund size can serve as an alternative way to improve efficiency. Our model predicts that, when the PE sector in emerging markets becomes more mature, the growth in competition for PE funds in these markets will lead to the adoption of the long-lifespan PE contracts typical in developed PE markets. Our model also offers a new explanation for the existing PE fund structure used in developed markets whose efficiency has been questioned by both academics and practitioners. Arcot, Fluck, Gaspar, and Hege (2015) point out, for funds late in the investment period with substantial dry powder, the same (incentive) contract creates adverse incentives to window dress. Consistent with this point, recent empirical studies of US PE funds find evidence of GP opportunism in the late phase of the investment period. For example, Dass, Hsu, Nanda, and Wang (2012), Braun and Schmidt (2014), and Arcot, Fluck, Gaspar, and Hege (2015) all find that a PE fund s late investments significantly underperform its early investments. Degeorge, Martin, and Phalippou (2015) find that the secondary buyouts (SBOs) made in the late phase of the investment period of a PE fund not only underperform the SBOs made in the early phase but have negative NPVs on average. Our model shows that such GP opportunism relates to the GP s incentive to go for broke when his earlier investments turn bad, which receives empirical support from Barrot (2014) that the GP takes high risks if fund cumulative performance has been low while plays safe if cumulative performance has been high. The documented GP opportunism naturally raises the question of why PE funds in developed markets are structured in the existing way. Our model rationalizes the use of the existing PE contract and shows that although this contract might not be most efficient, it saves the GP from conceding rents to LPs and is more efficient than alternative rentsaving contracts. This rent-saving argument is consistent with the evidence in Phalippou (2014) and Sorensen, Wang, and Yang (2014) that LPs do not earn risk-adjusted excess returns from investing in PE funds. 3

5 Our paper relates to the theoretical literature analyzing PE fund structures. Kandel, Leshchinskii, and Yuklea (2011) argue that the finite lifespan of the partnership can induce the GP to make inefficient investments in risky projects. Their analysis is based on the exogenously specified GP compensation structure and a predetermined investment period of the fund, both of which have been endogenized in our paper. Inderst, Mueller, and Münnich (2007) argue that investment pooling, combined with capital rationing, helps the GP commit to efficient liquidation decisions. However, in our paper, capital rationing never minimizes agency conflicts. Axelson, Strömberg, and Weisbach (2009) study the GP s financing of a portfolio of projects. They show that, in the pure ex ante financing setting, it is never optimal to make the GP capital-constrained, and they find that the optimal financing structure is a combination of ex ante pooled financing and ex post deal-by-deal financing. In contrast, our paper shows that, in the pure ex ante financing setting, although pooled financing can increase efficiency, the GP may prefer having the fund capital-constrained to save rents. Fulghieri and Sevilir (2009) study the optimal size and scope of the fund s portfolio by focusing on the double-sided moral hazard problem between the GP and the entrepreneurs of the projects. They show that the GP finds it optimal to limit portfolio size when project profitability potential is high and the GP expands the portfolio size only when project fundamentals are more moderate. In contrast, our paper abstracts from the effort-incentive problem and focuses on the GP-LP relationship rather than the GP-entrepreneur relationship. Our paper finds that, even with exogenously fixed project quality, heterogeneous fund sizes can arise in equilibrium and the GP deliberately limits fund size only when project profitability potential is not too high. Our paper also relates to the theoretical literature on investment pooling and investment timing. To our knowledge, our paper is the first paper that studies the investment timing issue under the principal-agent framework when investment pooling is used. The theoretical literature on investment pooling typically focuses on how pooling helps design incentive compatible contracts, through the channels of risk diversification (Diamond (1984)) or inside wealth creation (Laux (2001)). In contrast, our paper focuses on the interaction of investment pooling and investment timing and shows that, given investment pooling, the agent has an incentive to game the compensation scheme through investment timing, which moral hazard has to be taken into account when designing optimal contract. The theoretical literature on investment timing typically abstracts from the agency problem and takes a real options approach that emphasizes the value of waiting to invest under uncertainty (see Pindyck (1991) for an excellent survey). The effects of moral hazard on investment timing is studied in Grenadier and Wang (2005), with a real options model in which the agent manages a single project. The key difference be- 4

6 tween our model and the real options models is that the NPV of a project is fixed in our model whereas in real options models, it generally follows a stochastic process. Thus, in contrast to real options models, in our model, the manager s incentive to wait is not driven by project quality fluctuation but by agency conflicts and investment pooling. The remainder of the paper is structured as follows. Section 2 describes the model. Section 3 analyzes the socially optimal financing arrangement and shows that this arrangement may force the GP to concede rents to LPs. Section 4 analyzes alternative financing arrangements, which are less efficient but rent-saving. Section 5 determines the equilibrium financing arrangement. Section 6 provides the implications of the model. Section 7 concludes. All the proofs are relegated to the Appendix. 2 The model 2.1 The players There are three types of players in the model: a GP, investors who are perfectly competitive, 4 and fly-by-night operators. All players are risk-neutral, with no discounting of the future, and have access to a storage technology yielding the risk-free interest rate, which is normalized to zero. 2.2 The projects The model has five dates (time 0, 1, 2, 3, and 4). At time 0, all the players anticipate that the GP will be endowed with two projects at time 1 and they have the common prior that the quality of each project will be independently drawn from the same distribution, with probability λ (0,1) of being good (G) and probability 1 λ of being bad (B). 5 At time 1, two projects arrive. The quality of each project is privately observable to the GP, who possesses certain expertise that normal investors lack. Each project requires an investment of I to be undertaken. To undertake a project, the GP must invest I in the project at either time 1 or time 2; otherwise, the investment opportunity disappears. We assume that it takes two periods for an investment to produce cash flows, i.e., time 1 s investments produce cash flows at time 3 while time 2 s investments produce cash flows at time 4. To focus on the investment timing effect on investment efficiency through the GP s opportunistic behavior but not through the change in project quality, we assume no change in a project s quality no matter whether it is undertaken at time 1 4 We use investors and LPs interchangeably throughout the paper. 5 There is no change in the qualitative nature of the result if we assume positive project quality correlation. This extension is available from the author upon request. 5

7 or 2. 6 Particularly, we assume that a good project, if undertaken at time 1 (time 2), will succeed, producing cash flow R > I, with certainty at time 3 (time 4). A bad project, if undertaken at time 1 (time 2), will succeed with some probability, i.e., it will produce, at time 3 (time 4), R with probability p (0,1) and 0 with probability 1 p. We assume that bad projects are negative NPV investments, i.e., pr < I. To simply the analysis, we further assume that R > 2I. (1) Thus, the costs of investing in both projects can be covered when at least one investment succeeds. 7 Given that pr < I, condition (1) further implies that p < 1 2. To give the GP incentives to postpone part of his investment decisions, we assume that the progress of time 1 s investments is privately observable to the GP at time 2. To deliver the key insight without getting into unnecessary complications, we assume that, at time 2, the GP knows whether time 1 s investments will eventually succeed or fail. Note that, if the GP aims to maximize efficiency, then such interim information is useless, because a project s cash flow distribution is independent of the outcome of the other project. However, if there is misalignment of interests, this interim information can be useful for the GP s investment decision at time 2, as we will see later. 2.3 The GP s problem The GP has no money of his own and has to raise money from investors. 8 We focus on ex post information asymmetry regarding project quality by assuming that the GP has to raise funds by establishing a partnership with investors at time 0 (when the GP has no superior information on project quality). This assumption can be justified on the grounds that PE fundraising is time consuming. At time 0, to raise money, the GP offers investors a contract that specifies fund size, the length of the investment period, and GP compensation. Fund size is determined by investors committed capital, denoted by K. Because there are at most two good projects and because, in our model, having 6 Introducing project quality deterioration caused by investment delay will not change the qualitative nature of our result so long as the reduction in the probability of success for a bad project caused by delay is sufficiently small. 7 There is no change in the qualitative nature of the result if we extend the analysis to I < R 2I. This extension is available from the author upon request. 8 Although PE funds may have capital contribution provisions that require the GP to contribute 1% of the committed capital, the GP is frequently permitted to borrow money from the fund to meet his capital contribution requirements (see Harris (2010)). 6

8 a financial slack does not help contracting, without loss of generality, we restrict the choice of K to K {I,2I}. When K = I (K = 2I), the fund size is small (big) and the GP can undertake at most one project (two projects). We denote the length of the investment period by l {1,2}, where l = 1 (l = 2) refers to a short (long) investment period, including time 1 only (both time 1 and 2). Note that new investments are limited to the investment period, so the GP is allowed to make new investments at time 2 only when the investment period is long, i.e., when l = 2. Since it always takes a project the GP invests in two periods to produce cash flows, the total lifespan of the partnership is automatically determined by the length of the investment period, i.e., if the contract specifies a short (long) investment period, the partnership will be dissolved by time 3 (time 4). Thus, a short (long) investment period corresponds to a short (long) lifespan of the partnership. Let s(x) denote the GP security, which maps the final cash flow x to the GP s payoff and satisfies the following limited liability and monotonicity conditions: Limited Liability (LL): 0 s(x) x for all x 0; Monotonicity (M): s(x) and x s(x) are nondecreasing in x. The monotonicity condition is quite common in the financial literature on security design, e.g., Harris and Raviv (1989) and Nachman and Noe (1994). This condition contracts off the GP s moral hazard of increasing his own payoff by secretly borrowing money from a third party or by burning money. Given s, the investor security is automatically determined by x s(x). To introduce agency conflicts in a simple way (otherwise the two parties interests can be perfectly aligned by using an equity contract), we follow Axelson, Strömberg, and Weisbach (2009) by introducing an ex ante adverse selection problem, where flyby-night operators play a role. We assume that there is an infinite supply of fly-by-night operators who cannot bring success to any project but can store money at the riskfree interest rate. At time 0, without contracts as a signalling device, investors cannot distinguish capable GPs from unskilled fly-by-night operators. The ex ante contracting must resolve this adverse selection problem, because if fly-by-night operators have an incentive to mimic capable GPs by offering the same contract, given the infinite supply of fly-by-night operators, the probability that investors meet a fly-by-night operator would be one, in which case investors must lose money. Because fly-by-night operators can store money at the risk-free interest rate, to prevent them from mimicking, the contract has to ensure that they earn nothing by storing money. Thus, the GP security must satisfy the following condition: No-Profit-No-GP-Earning (NP): for raised capital K, s(x) = 0 whenever x K. Condition (NP) makes the GP bear little of the downside risk and thus induces the 7

9 GP s risk-shifting tendency. This condition is also consistent with GP compensation in practice: the GP earns no performance-based compensation before LPs are paid in full (see Sahlman (1990)). The GP s problem is to maximize his expected payoff by offering investors, at time 0, a contract (K, l, s) that specifies fund size, investment duration, and GP compensation. Investors accept the contract and provide the GP with K if their expected payoff is nonnegative. Figure 1 illustrates the timeline of the model. Time 0 Time 1 Time 2 Time 3 Time 4 GP establishes partnership Partnership contracting: fund size investment period GP security Projects arrive GP observes quality GP makes investment decisions GP observes progress of time 1 s investments If GP has dry powder and investment period is long, GP decides whether to use dry powder Cash flows from time 1 s investments are realized Cash flows from time 2 s investments are realized Figure 1: The timeline of the model 3 The socially optimal financing arrangement 3.1 Conditions Since all the ex post efficiency loss will be borne by the GP ex ante, it is natural to start our analysis by investigating the optimal contract that minimizes inefficiency. At time 1, there are three states of the world, distinguished by the combination of two projects quality: (1) state GG, in which both projects are good, (2) state GB, in which one project is good while the other is bad, 9 and (3) state BB, in which both projects are bad. Because of condition (NP), the first-best world, in which the GP invests in all good but no bad projects in all the three states, is unattainable: in state BB, the GP always has an incentive to invest in at least one bad project, since otherwise he earns nothing. Thus, it is impossible to contract off the efficiency loss from undertaking one bad project in state BB. It is thus natural to examine the second best, i.e., the financing arrangement that induces the GP to i. invest in two good projects in state GG, ii. invest in only the good project in state GB, iii. and invest in only one bad project in state BB. 9 Given that the two projects are identical, it is irrelevant which of the two projects is good. We therefore treat these two separate but symmetric cases effectively as a single case. 8

10 The investment surplus from the above prescribed investment behavior equals π s = λ 2 (2R 2I) + 2λ(1 λ)(r I) + (1 λ) 2 (pr I) (2) = ( 2λ + (1 λ) 2 p ) R (1 + λ 2 )I. In equation (2), 2R 2I is the surplus from investing in two good projects in state GG, which occurs with probability λ 2 ; R I is the surplus from investing in only the good project in state GB, which occurs with probability 2λ(1 λ); pr I is the surplus from investing in only one bad project in state BB, which occurs with probability (1 λ) 2. Financing is feasible only if the surplus expressed in (2) is nonnegative. This condition is equivalent to R/I 1 + λ 2 2λ + (1 λ) 2 p = r 0(p,λ). (3) We think of R/I as representing a project s profitability potential. Thus, financing is feasible only if project profitability potential is greater than the threshold, r 0, defined in (3). In what follows, we assume that condition (3) holds. We call the financing arrangements that induce the GP to follow the above prescribed investment behavior socially optimal arrangements. Note that, socially optimal arrangements must satisfy the following three conditions: Unconstrained Capital: sufficient funds raised for investing in two projects; Investment Pooling: putting the two projects together under one roof; Short Investment Period: new investments are limited to time 1. The first condition unconstrained capital is necessary to contract off underinvestment; it requires that the GP raises 2I at time 0. The second condition investment pooling is necessary to induce the GP to abandon a bad project, since if the GP instead establishes two funds, each used to finance one project, given that each fund s contract has to satisfy condition (NP), the GP will never abandon any bad project. Given unconstrained capital and the use of investment pooling, the third condition short investment period is necessary to minimize overinvestment. To see this, note that, if the contract specifies a long investment period, in state BB, the GP has an incentive to invest in one bad project at time 1 and, if he foresees the failure of this investment at time 2, he will burn the dry powder by undertaking the other bad project at time 2, since otherwise he earns nothing. The above analysis implies that a socially optimal arrangement must specify large fund size and a short investment period, i.e., K = 2I and l = 1. Conditioned on K = 2I and l = 1, to minimize agency conflicts, the GP security, s, must satisfy the following 9

11 incentive-compatibility conditions: 10 s(r + I) ps(2r) + (1 p)s(r), (IC GB ) ps(r + I) p 2 s(2r) + 2p(1 p)s(r). (IC BB ) The left-hand side of (IC GB ) is the GP s expected payoff in state GB if he only undertakes the good project. The right-hand side of (IC GB ) is the GP s expected payoff in state GB if he undertakes both projects. When (IC GB ) holds, the GP only undertakes the good project in state GB. The left-hand side of (IC BB ) is the GP s expected payoff in state BB if he only undertakes one bad project. The right-hand side of (IC BB ) is the GP s expected payoff in state BB if he undertakes both projects. When (IC BB ) holds, conditioned on l = 1, the GP only undertakes one bad project in state BB. It is noticeable that (IC BB ) implies (IC GB ). Thus, a contract is socially optimal if and only if it is in the form of (K = 2I,l = 1,s) with s satisfying (IC BB ) and conditions (LL), (M), and (NP) specified in Section 2.3. Since our aim is to derive the equilibrium financing arrangement, that is, given a perfectly competitive capital market, the contract that maximizes the GP s payoff, it is natural for us to examine, among all the socially optimal contracts, the contract that maximizes the GP s payoff. Denote this contract by Cs and denote by s s the associated GP security. The above analysis implies that Cs = (K = 2I,l = 1,s s). In the next part of this section, we solve for s s. 3.2 Security design With investment pooling, the final cash flow x can potentially take on one of the six different values, i.e., x {0,I,2I,R,I + R,2R}. The GP security, s, must specify the GP s payoff for each of these six cash flow realizations, among which, by condition (NP), s(0) = s(i) = s(2i) = 0. Thus, s s must solve the following program: maxe[s(x)] = λ 2 s(2r) + [ 2λ(1 λ) + (1 λ) 2 p ] s(r + I), s (P) subject to (LL), (M), (NP), (IC BB ), and the following investors break-even condition E[s(x)] π s, (BE) where π s is given by equation (2). 10 Since we have assumed that good projects will succeed for sure, the incentive-compatibility conditions that induce the GP to undertake good projects are implied by the monotonicity condition (M). 10

12 There are two possible payoffs to the GP in the maximand. The first payoff, s(2r), occurs only in state GG. The second payoff, s(r + I), occurs both in state BB when the bad project the GP invested in at time 1 succeeds and in state GB. 11 For investors to break even, the GP s payoff must not exceed the investment surplus, given by (2). To solve problem (P), we first solve the following relaxed problem: maxe[s(x)] = λ 2 s(2r) + [ 2λ(1 λ) + (1 λ) 2 p ] s(r + I), s (P ) subject to (LL), (M), (NP), and (IC BB ). Denote the solution to the above relaxed problem by s o. The only difference between the relaxed problem and the primal problem is the exclusion/inclusion of condition (BE). Note that (BE) is investors break-even condition which imposes the investment surplus, π s, as an upper bound on the objective function. Thus, if the maximal value of the relaxed problem falls below the investment surplus, the solution to the primal problem equals the solution to the relaxed problem, while if otherwise, the solution to the primal problem equals a scale transformation of the solution to the relaxed problem. This result is formally presented in the following lemma. Lemma 1. Suppose s s and s o represent the solutions to the primal problem (P) and to the relaxed problem (P ), respectively. Then [ s s(x) = s o π ] s (x)min 1, E[s o (x)] x 0, (4) where E[s o (x)] represents the maximal value of the relaxed problem (P ) and π s is the investment surplus given by equation (2). Our first proposition presents the expressions for s o and E[s o (x)]. Proposition 1. The solution to the relaxed problem (P), s o, and the associated maximal value E[s o (x)] are as follows: s o (x) = 0 for all x 2I, and i. if R/I 3 2p, then s o (R) = R 2I, s o (R + I) = R I, and s o (2R) = 2R 2I, in which case E[s o (x)] = ( 2λ + (1 λ) 2 p ) (R I); (5) ii. if 3 2p < R/I 3, then a. if λ λ, where λ = p 2 2p + p p 2 1 p 2, (6) 11 If in state BB, the bad project the GP invested in at time 1 fails, the final cash flow is I. By condition (NP), s(i) equals zero and is hence missing in the maximand. 11

13 then s o (R) = R 2I, s o (R+I) = R I, and s o (2R) = R I +(3I R)(1 p)/p, in which case E[s o (x)] = (2λ + (1 λ) 2 p λ 2 ) R p (2λ + 2λ 2 + (1 λ) 2 p 3λ 2 p ) I, b. while if λ > λ, then s o (R) = (I pr)/(1 p), s o (R+I) = (I pr)/(1 p)+i, and s o (2R) = (I pr)/(1 p) + R, in which case ( E[s o (x)] = 1 + λ 2 + (1 λ) 2 p 1 ) ( R+ (1 λ)(3λ 1) + (1 λ) 2 p + 1 ) I; 1 p 1 p (8) iii. if R/I > 3, then a. if λ λ, then s o (R) = I and s o (R + I) = s o (2R) = 2I, in which case E[s o (x)] = ( 4λ 2λ 2 + 2(1 λ) 2 p ) I, (9) (7) b. while if λ > λ, the expressions for s o and E[s o (x)] are the same as those presented in (iib). Proposition 1 has implications on socially optimal security design. When project profitability potential, R/I, is low (below the threshold 3 2p), part (i) in Proposition 1 implies that s o has all the marginal cash flows in the region [2I,2R] pledged to the GP. In this case, s o is a call option with strike price equal to LPs committed capital. Since, by Lemma 1, s s is a scale transformation of s o, s s is a call-option-like security with strike price equal to LPs committed capital. This case is illustrated in Panel A of Figure 2. When project profitability potential is not too low (above the threshold 3 2p), an incentive-compatible security must not have all the marginal cash flows in the region [2I,2R] pledged to the GP. In this case, if the projects have high probability of being good (λ > λ), parts (iib) and (iiib) in Proposition 1 imply that s o is a call option with strike price between 2I and R, which further implies, by Lemma 1, that s s is a calloption-like security with strike price between 2I and R. This case is illustrated in Panel B of Figure 2. If projects have low probability of being good (λ λ), part (iia) in Proposition 1 implies that s o must have some marginal cash flows in the region [R + I,2R] pledged to LPs when projects have medium profitability potential, i.e., 3 2p < R/I 3, while part (iiia) implies that s o must have all the marginal cash flows in the region [R + I,2R] and some marginal cash flows in the region [2I,R] pledged to LPs when projects have high profitability potential, i.e., R/I > 3. This implies, by Lemma 1, that in both cases, s s must exhibit concavity in the high cash flow region. These two cases are illustrated in Panels C and D in Figure 2. 12

14 Low Profitability Potential Medium or High Profitability Potential (in the case where λ>λ) s s (x) s s (x) G P G P P A Y O F F 0 2I R R+I 2R x P A Y O F F 0 2I R R+I 2R x Cash Flow Cash Flow Panel A. R/I 3 2p Panel B. R/I > 3 2p and λ > λ s s (x) Medium Profitability Potential (in the case where λ λ) s s (x) High Profitability Potential (in the case where λ λ) G P G P P A Y O F F 0 2I R R+I 2R x P A Y O F F 0 2I R R+I 2R x Cash Flow Cash Flow Panel C. 3 2p < R/I 3 and λ λ Panel D. R/I > 3 and λ λ Figure 2: The GP security, s s, gives the GP the highest payoff among all the socially optimal securities. The graphs illustrate the shapes of s s in four cases, distinguished by parametric assumptions. Panel A illustrates the case where project profitability potential, R/I, is low (below 3 2p). Panel B illustrates the case where project profitability potential is medium or high (above 3 2p) while a project s probability of being good is high, i.e., λ > λ. Panel C illustrates the case where project profitability potential is medium (between 3 2p and 3) while a project s probability of being good is low, i.e., λ λ. Panel D illustrates the case where project profitability potential is high (above 3) while a project s probability of being good is low, i.e., λ λ. Proposition 1 also presents the expression for E[s o (x)], which is the GP s maximal payoff by using a socially optimal contract with LPs break-even condition omitted. Including LPs break-even condition just imposes the investment surplus, πs, as an upper bound on the GP s expected payoff. Thus, with LPs break-even condition included, the GP s expected payoff by using the socially optimal contract equals the minimum of E[s o (x)] and πs. This result is presented in the next proposition. Proposition 2. The GP s expected payoff by using the socially optimal contract, Cs = 13

15 (K = 2I,l = 1,s s), where s s solves problem (P), equals E[s s(x)] = min[e[s o (x)],π s ], (10) where E[s o (x)] is presented in Proposition 1 and π s is given by equation (2). 3.3 LPs rents By part (i) in Proposition 1, when project profitability potential is low, s o pledges all the marginal cash flows in the region [2I,2R] to the GP and gives the GP a payoff expressed in equation (5). Since this payoff is greater than the investment surplus, π s, by Proposition 2, when projects have low profitability potential, s s, which is a calloption-like security with strike price equal to LPs committed capital, gives the GP all the investment surplus and thus leaves LPs with no rents. However, when project profitability potential is medium or high, to satisfy the incentive-compatibility condition (IC BB ), s o must leave some marginal cash flows in the region [2I,2R] to LPs. Note that these marginal cash flows are pledged to LPs not because of LPs break-even condition but because of (IC BB ). Thus, it is possible that, at the solution to the primal security design problem (P), condition (IC BB ) is binding while LPs break-even condition is not. In this case, LPs earn positive rents. We use the case stated in part (iiia) in Proposition 1 as an example to show that LPs may earn rents when the project profitability potential is not too low. In part (iiia), projects have high profitability potential while low probability of having good quality. In this case, as shown in (iiia), regardless of the final cash flow, the GP s payoff cannot exceed 2I. Since the GP s expected payoff equals the minimum of E[s o (x)] and the investment surplus, π s, and since, by equation (9), E[s o (x)] does not depend on R, an increase in R only increases π s but not E[s o (x)]. When R is sufficiently large, π s exceeds E[s o (x)], in which case, by using a socially optimal contract, the GP earns at most E[s o (x)], leaving π s E[s o (x)] as rents conceded to LPs. Note that, in practice, good projects are typically a minority of a PE fund s project candidates while PE-backed projects, conditioned on success, often produce stellar profits; examples include Apple, Federal Express, Genentech, Intel, Microsoft, etc. Thus, the parametric conditions in (iiia) seem to be consistent with the PE practice. Thus, the result in (iiia) suggests that the socially optimal contract can be costly for the GP to use because of rent concession. The rents paid to LPs can be considerably high if project profitability potential is sufficiently high. For instance, suppose p = 0.1, λ = 0.2, R = 8, and I = 1. These parameters satisfy the conditions in (iiia). Given 14

16 these parameters, by equation (9), E[s o (x)] = 0.848, while by equation (2), πs = Thus, the GP s expected payoff, min[πs,e[s o (x)]], equals 0.848, while LPs rents equal = In this case, although the capital market is perfectly competitive, by using the socially optimal contract, LPs expected payoff is more than twice as much as the GP s expected payoff and LPs net expected rate of return, which equals 1.824/2, is more than 90%! It is a general result that, under the socially optimal contract, Cs, an increase in R (and hence, with I fixed, an increase in project profitability potential) weakly increases LPs rents. We enclose this result in the next proposition. Proposition 3. Under the socially optimal contract, C s = (K = 2I,l = 1,s s), where s s solves problem (P), with p, λ, and I fixed, increasing R weakly increases LPs rents. Given that the GP may not earn all the investment surplus by using the socially optimal contract, we still need to determine whether the socially optimal contract offers the GP the highest expected payoff compared to less efficient but possibly rent-saving contracts. We tackle this problem in the next section by examining those alternative financing arrangements. 4 Alternative financing arrangements Recall from the discussion in Section 3 that there are four conditions for socially optimal contracting: (i) unconstrained capital, i.e., K = 2I, (ii) investment pooling, (iii) short investment period, i.e., l = 1, and (iv) the incentive-compatibility condition (IC BB ). By relaxing one or more conditions, we eventually obtain four mutually exclusive alternative financing arrangements, listed as follows: 12 a. the capital-constrained arrangement, under which the GP establishes a small fund, i.e., K = I; 13 b. the long duration arrangement, under which the GP establishes a large fund which specifies a long investment period, i.e., K = 2I and l = 2; c. the stand-alone arrangement, under which the GP establishes two small funds, each with K = I; 12 Alleviating the overinvestment problem via ex post renegotiation is not feasible. This is because, at time 2, LPs cannot observe the progress of time 1 s investments and thus the ex post renegotiation is still subject to the problem of fly-by-night operators. As a result, any ex post contract that gives the GP a positive payoff without bringing success to any project will not be accepted by LPs. This also coincides with the PE practice that partnership agreements are rarely renegotiated. 13 With a small fund, the choice of the length of the investment period has no effect on investment efficiency. 15

17 d. the socially suboptimal short duration arrangement, under which the GP establishes a large fund which specifies a short investment period, i.e., K = 2I and l = 1, but the GP security violates (IC BB ). In what follows, we analyze each of these alternative financing arrangements. 4.1 The capital-constrained arrangement By using the capital-constrained arrangement, the GP can at most undertake one project. Because of condition (NP), the GP can only obtain a positive payoff if he creates value for the fund. Thus, the GP always has an incentive to undertake one project. In addition, in state GB, given that the good project has a higher probability of success than the bad one, the GP always picks the good project. Therefore, the GP s investment behavior by using the capital-constrained arrangement is as follows: (i) in state GG, undertake one good project, (ii) in state GB, undertake one good project, and (iii) in state BB, undertake one bad project. It is thus clear that the capital-constrained arrangement minimizes overinvestment but causes underinvestment. The next proposition shows that the capital-constrained arrangement is feasible so long as the investment surplus from the prescribed investment behavior is nonnegative, and by using a call-option-like GP security under the capital-constrained arrangement, the GP concedes no rents to LPs. Proposition 4. The investment surplus by using the capital-constrained arrangement is π c = ( 2λ λ 2 + (1 λ) 2 p ) R I. (11) The capital-constrained arrangement is feasible if and only if the surplus in (11) is nonnegative. Under the capital-constrained arrangement, the optimal GP security is given by s 0 if x I c(x) =, (12) α c (x I) if x > I ( ) where α c = R I 1 I R. By using this security under the capital-constrained 2λ λ 2 +(1 λ) 2 p arrangement, the GP concedes no rents to LPs and captures all the investment surplus, π c. 4.2 The long duration arrangement By using the long duration arrangement, the GP can at most undertake two projects. As discussed in Section 3.1, with a long investment period, in state BB, the GP has an incentive to invest in one bad project at time 1 and, conditioned on anticipating the failure 16

18 of this investment at time 2, the GP has an incentive to invest in the other bad project at time 2. Thus, to minimize agency conflicts conditioned on using the long duration arrangement, the GP security must induce the GP to (i) undertake two good projects in state GG, (ii) undertake only the good project in state GB, and (iii) in state BB, conditioned on foreseeing at time 2 the success of his earlier investment, keep the dry powder in the safe asset. Among the three requirements, (i) is implied by the monotonicity condition (M), while (ii) and (iii) are satisfied if and only if the GP security satisfies the weaker incentive-compatibility condition, (IC GB ), presented in Section 3.1. The next proposition shows that the long duration arrangement is feasible so long as the investment surplus from the prescribed investment behavior is nonnegative, and by using a call-option-like GP security under the long duration arrangement, the GP concedes no rents to LPs. Proposition 5. The investment surplus by using the long duration arrangement is π l = ( 2λ + (1 λ) 2 (2 p)p ) R ( 1 + λ 2 + (1 λ) 2 (1 p) ) I. (13) The long duration arrangement is feasible if and only if the surplus in (13) is nonnegative. Under the long duration arrangement, the optimal GP security is given by s l (x) = 0 if x 2I, (14) α l (x 2I) if x > 2I where α l = (2λ+(1 λ)2 (2 p)p)r (1+λ 2 +(1 λ) 2 (1 p))i. By using this security under the (2λ+(1 λ) 2 (2 p)p)r (2λ+(1 λ) 2 (3 2p)p)I long duration arrangement, the GP concedes no rents to LPs and captures all the investment surplus, π l. 4.3 The stand-alone and the socially suboptimal short duration arrangement Under the stand-alone arrangement, the GP finances two projects by establishing two small funds, with each fund financing one project. Since the GP security in each fund must satisfy condition (NP), the GP has an incentive to undertake both projects in every state, which implies that the GP undertakes two bad projects in state BB. Under the socially suboptimal short duration arrangement, the GP security violates condition (IC BB ), in which case, given the short investment period, the GP also undertakes two bad projects in state BB. Since the long duration arrangement only causes inefficiency in state BB, in which case the GP invests in 2 p bad projects on average, it is 17

19 clear that the long duration arrangement is strictly more efficient than both the standalone and the socially suboptimal short duration arrangement. Moreover, by Proposition 5, by using the long duration arrangement, the GP concedes no rents to LPs. The next proposition is thus straightforward. Proposition 6. The GP s expected payoff is higher under the long duration arrangement than under the stand-alone and the socially suboptimal short duration arrangement. 5 The equilibrium financing arrangement Proposition 6 implies that there are three financing arrangement candidates for the equilibrium: the socially optimal arrangement, the capital-constrained arrangement, and the long duration arrangement. Each has its own pros and cons. The socially optimal arrangement maximizes efficiency but may force the GP to concede rents to LPs. As discussed in Section 3.3, rent concession occurs only if project profitability potential is not too low. Thus, the socially optimal arrangement will be used if project profitability potential is sufficiently low. When project profitability potential is not too low, the socially optimal arrangement causes rent concession, in which case the GP may turn to the capital-constrained or the long duration arrangement. Both these two alternative arrangements are rent-saving, but the capital-constrained arrangement causes underinvestment while the long duration arrangement causes more overinvestment. Everything else being equal, the higher the project profitability potential, the larger the efficiency loss induced by underinvestment while the smaller the efficiency loss induced by overinvestment. Thus, the capital-constrained arrangement will be used only if project profitability potential is neither too low nor too high, whereas the long duration arrangement will be used only if project profitability potential is sufficiently high. We present these results formally in the following theorem. Theorem 1. Suppose condition (3) is satisfied so that financing is feasible. The equilibrium financing arrangement has the following features: i. if R/I 3 2p, the GP chooses the socially optimal arrangement; ii. fixing p and λ, as R/I 1/p, the GP either chooses the socially optimal arrangement or the long duration arrangement; iii. fixing p and λ, if the GP chooses the socially optimal arrangement as R/I 1/p, then he chooses the socially optimal arrangement for all R/I < 1/p; iv. fixing p and λ, if the GP chooses the long duration arrangement as R/I 1/p, then there must either (a) exist a unique threshold r sl (3 2p,1/p) such that the 18

20 GP chooses the socially optimal arrangement when R/I r sl and chooses the long duration arrangement when r sl < R/I < 1/p or (b) exist two thresholds r sc and r cl, where 3 2p < r sc < r cl < 1/p, such that the GP chooses the socially optimal arrangement when R/I r sc, chooses the capital-constrained arrangement when r sc < R/I r cl, and chooses the long duration arrangement when r cl < R/I < 1/p. Theorem 1 implies that the lower (higher) the project profitability potential, the more likely the use of the socially optimal arrangement (the long duration arrangement) and the capital-constrained arrangement is used only if project profitability potential is moderate. The result thus suggests a positive relation between project profitability potential and the length of the investment period and a U-shaped relation between project profitability potential and fund size. We use an example to show how financing arrangement changes when project profitability potential changes. In this example, p = 0.1 and λ = 0.2. We change project profitability potential by changing R/I. By assumption, 2 < R/I < 1/p. Thus, given that p = 0.1, we must have 2 < R/I < 10. Given these parametric assumptions, there exist two thresholds π sc and π cl such that the socially optimal arrangement is used if R/I falls below π sc, the capital-constrained arrangement is used if R/I is between π sc and π cl, while the long duration arrangement is used if R/I is above π cl. 6 Implications of the model In this section, we discuss the implications of the model, starting from the implications on emerging markets, followed by the implications on developed markets. 6.1 Implications on emerging markets As mentioned in the introduction, LPs, especially non-institutional LPs, in emerging markets prefer PE funds with a short fuse, i.e., a much shorter investment period and lifespan than their developed market counterparts. Although these LPs believe that it is too risky to entrust their money to the GP for a lengthy period of time and thus demand a short fuse, our model implies that the short fuse, combined with convex GP compensation, might in fact exacerbate the agency risk faced by these LPs. When the fuse is short, the GP is under big pressure to spend the money within a short period of time. In such a case, when GP compensation is convex and when the fund targets at projects with high profitability potential, the GP, aiming to maximize the value of his carried 19