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I examine a simple corporate finance q-theory incentive modeling framework of financial intermediation with financial constraints to investigate the key economic drivers of optimal PEF contracts in private investment markets. Following Holstrom and Tirole \(\\cite{Holmstrom:QJE:1997}\), the model setup and notation is developed below.\\
At an initial time \(\ \tau \) , an experienced PEF firm has sourced an attractive private markets investment project with a fixed cost of $I^{PEF}_t$. The investment project is inherently risky and will require full GP effort to maximize the probability of the project’s success $p$. If successful, the project is expected to produce a future profitable total cumulative cash flow of $R_{T}$ $(R_{T}>I^{PEF}_{t})$ at time $T$ $(T>t)$. Yet, there is also a probability of $1-p$ that the project could be unsuccessful and ultimately yield a future total cumulative cash flow of $R_{T}=0$, irrespective of GP effort \footnote{Although it is extremely unlikely for most PEF firms to produce an $R_{T}=0$, this simplification can be relaxed in more realistic models.}.\\
The investment project is large and risky, but overall, it is expected to generate a positive NPV. Financially constrained by its limited net worth of only $A_{t}$, the PEF firm is unable to fully fund the total project cost of \(\$I^{PEF}_t$ ($I^{PEF}_{t}>A_{t}$)\) independently. As such, the GP will seek to secure an external capital source to fund the total cost of the investment project. The PEF firm will form a PEF investment vehicle to secure an LP investor to specifically provide the additional funding required to acquire and complete the investment project.\\
Total project cost $I^{PEF}_{t}$ and future cumulative project cash flow $R_{T}$ are to be shared by LP and GP per contractually agreed terms. Typically, the LP will incur the full range of investment project costs (which includes direct investment costs and PEF related expenses in the form of fees), whereas the GP will only incur direct investment costs associated with a GP ownership stake in the proposed investment project (GP as manager of the PEF pays no PEF expenses). Total investment project cost is allocated between LP and GP
\begin{equation}
I^{PEF}_{t}=I^{LP}_{t}+I^{GP}_{t} .
\label{E1}
\end{equation}
The future total PEF cumulative cash flow payoff $R_{T}$ will be split between LP and GP per terms of the PEF contract agreement (known as the waterfall provision of the contract). The PEF waterfall provision governs the rules, priorities and preferences of how PEF distributions are shared between the parties.
\begin{equation}
R_{T}=R^{LP}_{T}+R^{GP}_{T}
\label{E2}
\end{equation}
where $R_{T}\in [0,\infty)$.\\
The PEF firm has sourced a profitable fixed cost investment project
\begin{equation}
R_{T}>^{PEF}I_{t}
\label{E3}
\end{equation}
and is financially constrained by insufficient firm net worth of only $A_{t}$ to pursue the investment project independently, that is
\begin{equation}
^{PEF}I_{t}>A_{t}.
\label{E4}
\end{equation}
To move the investment project forward, the firm will set up a PEF investment vehicle to obtain the additional equity capital necessary to fully fund the total investment project cost. The additional required LP equity contribution for the investment project is given by
\begin{equation}
I^{LP}_{t}=I^{PEF}_{t}-\alpha_{t}A_{t}
\label{E5}
\end{equation}
where $\alpha_{t}\in (0,1]$ represents the fraction of the PEF firm’s net worth that the GP will contribute as equity to fund the investment project. The LP’s equity contribution will fund most of the direct investment cost and all of the PEF fees and expenses. The GP’s equity contribution will only fund the remainder of the direct investment cost (GP pays no PEF expenses).\\
In this simple two state model economy, the risk-free rate $r_{f}$ is set to zero (minimize algebraic overhead), the expected market return on equity $r_{e}$ is assumed to be positive ($r_{e}>0$), and both LP and GP are assumed to be risk neutral agents.\\
As an initial assessment, the LP will compare the expected payoff of the investment project to alternative investment opportunities available in the market. At a minimum, the LP investor would require the expected future payoff of the investment project to exceed the expected payoffs of the alternative equity investment opportunities readily available
\begin{equation}
pR^{LP}_{T}> I^{LP}_{t}(1+r_{e})=(I^{PEF}_{t}-\alpha_{t}A_{t})(1+r_{e})
\label{E6}
\end{equation}
where $R^{LP}_{T}\in [0,\infty)$.\\
The GP is aware of the LP’s alternative investment opportunities and suggests to provide a minimum preferred equity return $r_{pref}$ (that exceeds the prevailing market return on equity $r_{e}$) to the LP on the investment project as an inducement
\begin{equation}
pR^{LP}_{T}\geq{pI^{LP}_{t}(1+r_{pref})}>I^{LP}_{t}(1+r_{e}) .
\label{E7}
\end{equation}
If the investment project is successful, the LP would (on average) be better off with investing in the PEF with GP than otherwise. The LP would earn at least the preferred return $r_{pref}$ ($r_{pref}>r_{e}$) offered by the PEF firm. \\
Although the preferred equity return $r_{pref}$ on the project is more attractive to the LP, the investment project’s future payoff is inherently risky ($R_{T}=0$\footnote{Although it is extremely unlikely for most PEF firms to produce $R_{T}=0$, this simplification can be relaxed in more realistic models.} with probability $1-p>0$) and will require full GP focus and effort to maximize the project’s probability success $p$. The LP also recognizes the potential agency risk or cost associated with the investment project and the uncertainty of future GP behavior (which may undermine the potential success of the project). Properly incentivised, the GP can choose to behave and to expend his full effort (not “\textit{shirk}”) to increase the probability of the success of the investment project to $p_H$. On the other hand, the GP can also choose to misbehave or to “\textit{shirk}” on the project to derive a private benefit $B$ $(B>0)$. Effectively, GP misbehavior reduces the probability of success of the investment project to $p_L$ $(p_{L}<p_{H})$. The LP will only want to invest projects that are expected to be profitable with GP good behavior
\begin{equation}
p_{H}R_{T}-I^{PEF}_{t}>0>p_{L}R_{T}-I^{PEF}_{t}+B .
\label{E8}
\end{equation}
