# «This Version: February 2009 ∗ Axel Buchner is at the Technical University of Munich and at the Center of Private Equity Research (CEPRES); ...»

Modeling the Cash Flow Dynamics of Private

Equity Funds –

Theory and Empirical Evidence∗

Axel Buchner, Christoph Kaserer and Niklas Wagner

This Version: February 2009

∗ Axel Buchner is at the Technical University of Munich and at the Center of Private Equity

Research (CEPRES); Christoph Kaserer is at the Technical University of Munich and at the

Center for Entrepreneurial and Financial Studies (CEFS); Niklas Wagner is at Passau University. Earlier versions of this paper have beneﬁted from comments by seminar participants at the 15th CEFS-ODEON Research Colloquium, Munich, the 6. K¨lner Finanzmarktkolo loquium ”Asset Management”, Cologne, the 14th Annual Meeting of the German Finance Association (DGF), Dresden, the XVI International Tor Vergata Conference on Banking and Finance, Rome, the Faculty Seminar in Economics and Management of the University Fribourg, the 11. Finanzwerkstatt at the University of Passau, and the Finance Seminar of the Santa Clara University, Leavey School of Business. We are also grateful to the European Venture Capital and Private Equity Association (EVCA) and Thomson Venture Economics for making the data used in this study available. All errors are our responsibility. Corresponding author: Axel Buchner; phone: +49-89-232 495616; email address: axel.buchner@wi.tum.de.

Modeling the Cash Flow Dynamics of Private Equity Funds – Theory and Empirical Evidence Abstract We present a novel continuous-time approach to modeling the typical cash ﬂow dynamics of private equity funds. Our model consists of two independent components. First is a mean-reverting square-root process is applied to model the rate at which capital is drawn over time. Second are capital distributions which are assumed to follow an arithmetic Brownian motion with a timedependent drift component that incorporates the typical time-pattern of the repayments of private equity funds. Our empirical analysis shows that the model can easily be calibrated to real world fund data by the method of conditional least squares (CLS) and nicely ﬁts historical data. We use a data set of mature European private equity funds as provided by Thomson Venture Economics (TVE). Our model explains up to 99.6 percent of the variation in average cumulated net fund cash ﬂows and provides a good approximation of the empirical distribution of private equity fund cash ﬂows over a typical fund’s lifetime. Overall, the empirical results indicate that our model is of economic relevance in an eﬀort to accurately model the cash ﬂows dynamics of private equity funds.

**Keywords:**

Venture capital, private equity, stochastic modeling JEL classiﬁcation code: G24 1 Introduction Illiquid alternative assets, such as private equity, have become an increasingly significant portion of institutional asset portfolios as investors seek diversiﬁcation beneﬁts relative to traditional stock and bond investments. Various recent studies deal with the problem of estimating the risk and return characteristics of private equity investments.1 So far, only little research has focused on the cash ﬂow dynamics of private equity investments.

The present article proposes a novel stochastic model on the typical cash ﬂow dynamics of private equity funds. The uncertain timing of capital drawdowns and proceeds poses a challenge to the management of future investment cash ﬂows.

Our stochastic model of the cash ﬂow dynamics consists of two components. First is the stochastic model of drawdowns from the committed capital. Second is the stochastic model of the distribution of dividends and proceeds. Our model diﬀers from the work of Takahashi and Alexander (2002) and Malherbe (2004, 2005) in that it solely relies on observable cash ﬂow data.2 Based on cash ﬂow data, our analysis reveals that our model is ﬂexible and can well match the various typical drawdown and distribution patterns that are observed empirically. Our model is easy to understand as well as to implement and it can be used in various directions. For example, an institutional investor may employ it for the purpose of liquidity planning.

He may also measure the sensitivity of some ex-post performance measure – such as the IRR – to changes in typical drawdown or distribution patterns. Moreover, the model may be used as a tool for many risk management applications.

In our empirical analysis, we use a dataset of 203 mature European private equity funds of which 95 were fully liquidated during the January 1980 to June 2003 sample period. Our analysis shows that the model can easily be calibrated to real world fund data. The results of two consistency tests underline the economic relevance of our model, which explains up to 99.6 percent of the variation in average cumulated net cash ﬂows of the liquidated funds. Overall, our model provides a good approximation of the empirical distribution of typical cash ﬂows over the most relevant periods of fund lifetime.

The remainder of this article is organized as follows. In Section 2, we present our model. Section 3 illustrates the model dynamics and analyzes the impact of the various parameters on the timing and magnitude of fund cash ﬂows. Section 4 presents our calibration results and shows a risk management application of the model. Finally, Section 5 gives a conclusion and identiﬁes areas for future research.

1 Recent studies include, among others, Cochrane (2005), Diller and Kaserer (2009), Kaplan and Schoar (2005), Ljungquist and Richardson (2003a,b), Moskowitz and Vissing-Jorgenson (2002), Peng (2001a,b) and Phalippou and Gottschalg (2009).

2 Takahashi and Alexander (2002) propose modeling the cash ﬂow dynamics of a private equity fund. However, their model is deterministic and thus fails to reproduce the erratic nature of real world private equity cash ﬂows. Malherbe (2004, 2005) develops a continuoustime stochastic version of the model of Takahashi and Alexander (2002). While his model considers randomness, it relies on the speciﬁcation of the dynamics of an unobservable fund value and therefore has to account for an inaccurate fund valuation by incorporating an error term.

2 Modeling the Cash Flow Dynamics of Private Equity Funds

2.1 Institutional Framework Private equity investments are typically intermediated through private equity funds.

Thereby, a private equity fund denotes a pooled investment vehicle whose purpose is to negotiate purchases of common and preferred stocks, subordinated debt, convertible securities, warrants, futures and other securities of companies that are usually unlisted. The vast majority of private equity funds are organized as limited partnerships in which the private equity ﬁrm serves as the general partner (GP).

The bulk of the capital invested in private equity funds is typically provided by institutional investors, such as endowments, pension funds, insurance companies, and banks. These investors, called limited partners (LPs), commit to provide a certain amount of capital to the private equity fund – the committed capital denoted as C.

The GP then has an agreed time period in which to invest this committed capital – usually on the order of ﬁve years. This time period is referred to as the commitment period of the fund and will be denoted by Tc in the following. In general, when a GP identiﬁes an investment opportunity, it “calls” money from its LPs up to the amount committed, and it can do so at any time during the prespeciﬁed commitment period. That is, we assume that capital calls of the fund occur unscheduled over the commitment period Tc, where the exact timing does only depend on the investment decisions of the GPs. However, total capital calls over the commitment period Tc can never exceed the total committed capital C. As those drawdowns occur, the available cash is immediately invested in managed assets and the portfolio begins to accumulate. When an investment is liquidated, the GP distributes the proceeds to its LPs either in marketable securities or in cash. The GP also has an agreed time period in which to return capital to the LPs – usually on the order of ten to fourteen years. This time period is also called the total legal lifetime of the fund and will be referred to by Tl in the following, where obviously Tl ≥ Tc must hold. In total, the private equity fund to be modeled is essentially a typical closed-end fund with a ﬁnite lifetime.3

2.2 Modeling Capital Drawdowns We begin by assuming that the fund to be modeled has a total initial committed capital given by C as deﬁned above. Cumulated capital drawdowns from the LPs up to some time t during the commitment period Tc are denoted by Dt, undrawn committed capital up to time t by Ut. When the fund is set up, at time t = 0, D0 = 0 and U0 = C are given by deﬁnition. Furthermore, at any time t ∈ [0, Tc ], the simple identity Dt = C − Ut (2.1) must hold. In the following, we assume that the dynamics of the cumulated capital drawdowns, Dt, can be described by the ordinary diﬀerential equation

refer to Gompers and Lerner (1999), Lerner (2001) or to the recent survey article of Phalippou (2007).

where δt ≥ 0 denotes the rate of contribution or simply the fund’s drawdown rate at time t and 1{0≤t≤Tc } is an indicator function. This modeling approach is similar to the assumption that capital is drawn over time at some non-negative rate δt from the remaining undrawn committed capital Ut = C − Dt.

In most cases, capital drawdowns of private equity funds are heavily concentrated in the ﬁrst few years or even quarters of a fund’s life. After high initial investment activity, drawdowns of private equity funds are carried out at a declining rate, as fewer new investments are made, and follow-on investments are spread out over a number of years. This typical time-pattern of the capital drawdowns is well reﬂected in the structure of equation (2.2). Under the speciﬁcation (2.2), cumulated capital drawdowns Dt are given by t≤T c

Equation (2.4) shows that the initially very high capital drawdowns dt at the start of the fund converge to zero over the commitment period Tc. This condition leads to the realistic feature that capital drawdowns are highly concentrated in the early years of a fund’s lifetime under this speciﬁcation. Furthermore, equation (2.3) shows that the cumulated drawdowns Dt can never exceed the total amount of capital C that was initially committed to the fund under this model setup, i.e., Dt ≤ C for all t ∈ [0, Tc ].

Usually, the capital drawdowns of real world private equity funds show an erratic feature as investment opportunities do not arise constantly over the commitment period Tc. As it stands, the model for the capital drawdowns is purely deterministic.

A stochastic component can easily be introduced into the model by deﬁning a continuous-time stochastic process for the drawdown rate δt. We assume that the drawdown rate δt follows a mean-reverting square-root process given by the stochastic diﬀerential equation

** dδt = κ(θ − δt )dt + σδ δt dBδ,t, (2.5)**

where θ 0 is the long-run mean of the drawdown rate, κ 0 governs the rate of reversion to this mean and σδ 0 reﬂects the volatility of the drawdown rate; Bδ,t is a standard Brownian motion.4 The drawdown rate behavior implied by the structure of this process has the following two relevant properties: (i) Negative values of the drawdown rate are precluded under this speciﬁcation.5 This is a necessary 4 This process is known in the ﬁnancial literature as a square-root diﬀusion and was initially

condition, as we model capital distributions and capital drawdowns separately and must, therefore, restrict capital drawdowns to be strictly non-negative at any time t during the commitment period Tc. (ii) Furthermore, the mean-reverting structure of the process reﬂects the fact that we assume the drawdown rate to ﬂuctuate randomly around some mean level θ over time.

Under the speciﬁcation of the square-root diﬀusion (2.5), the conditional expected cumulated and instantaneous capital drawdowns can be inferred. Given that Es [·] denotes the expectations operator conditional on the information set available at time s, expected cumulated drawdowns at some time t ≥ s are given by

where A(s, t) and B(s, t) are deterministic functions depending on the model parameters and the time subscripts s and t. In addition, A′ (s, t) = ∂A(s, t)/∂t and B ′ (s, t) = ∂B(s, t)/∂t.6

2.3 Modeling Capital Distributions As capital drawdowns occur, the available capital is immediately invested in managed assets and the portfolio of the fund begins to accumulate. As the underlying investments of the fund are gradually exited, cash or marketable securities are received and ﬁnally returns and proceeds are distributed to the LPs of the fund. We assume that cumulated capital distributions up to some time t ∈ [0, Tl ] during the legal lifetime Tl of the fund are denoted by Pt and pt = dPt /dt denotes the instantaneous capital distributions, i.e., the (annualized) capital distributions that occur over inﬁnitesimally short time interval from t to t + dt.

We model distributions and drawdowns separately and, therefore, must also restrict instantaneous capital distributions pt to be strictly non-negative at any time t ∈ [0, Tl ]. The second constraint that needs to be imposed on the distributions model is the addition of a stochastic component that allows a certain degree of irregularity in the cash outﬂows of private equity funds. An appropriate assumption that meets both requirements is that the logarithm of instantaneous capital distributions, ln pt, follows an arithmetic Brownian motion of the form

** d ln pt = µt dt + σP dBP,t, (2.8)**