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Private Equity Funds: Valuation, Systematic
Risk and Illiquidity∗
Axel Buchner†
Christoph Kaserer‡
Niklas Wagner§
April 14, 2016
∗We would like to thank Stefano Bonini, George Chacko, Douglas Cumming, Sanjiv Das, JoachimGrammig, Oliver Gottschalg, Robert Hendershott, Alexander Kempf, Roman Kraussl, Josh Lerner,Glenn Pettengill, Ludovic Phalippou, Matthew Spiegel, Martin Wallmeier, Jochen Wilhelm, MikeWright, and Lixin Wu for helpful comments and discussions. Earlier versions of the paper have alsobenefited from comments by seminar participants at Cologne, Fribourg, Hong Kong, Munich, Passau,Santa Clara and Melbourne as well as at the Annual Meeting of the German Finance Association, theXVIth International Tor Vergata Conference on Banking and Finance, the European Financial Manage-ment Symposium on Entrepreneurial Finance & Venture Capital Markets, the Annual Meeting of theMidwest Finance Association, and the Private Equity Forum in Paris. We are grateful to the EuropeanVenture Capital and Private Equity Association and to Thomson Venture Economics for making thedata set available. All errors and omissions are our own responsibility.
†Passau University, Innstrasse 27, 94030 Passau, Phone: +49 851 509 3245, Fax: +49 851 509 3242,E-mail: [email protected]
‡Technical University of Munich, Arcisstrasse 21, 80333 Munich, Phone: +49 89 289 25490, Fax:+49 89 289 25491, E-mail: [email protected]
§Passau University, Innstrasse 27, 94030 Passau, Phone: +49 851 509 3241, Fax: +49 851 509 3242,E-mail: [email protected]
Private Equity Funds: Valuation, Systematic
Risk and Illiquidity
Abstract
We derive a novel model of the cash flow dynamics and equilibrium values of
private equity funds. Based on intertemporal capital asset pricing results for an
investor with logarithmic utility, the model explains the typical life cycle patterns
of systematic fund risk, expected returns and fund value. Given our model, we
also consider the effects of market illiquidity. Model calibration for a sample of
European funds illustrates that sample funds have an average risk-adjusted excess
value of 14 percent relative to committed capital, which amounts to estimated
illiquidity costs of 1.4 percent annually. We show how equilibrium expected fund
returns, systematic risk, and illiquidity discounts decrease over fund lifetime. As
compared to venture capital funds, buyout funds on average exhibit lower system-
atic risk, faster payback, lower life cycle maximum values, but higher initial excess
values.
Keywords: private equity, venture capital, buyout funds, fund life cycle, equilib-
rium fund values, illiquidity, expected returns, time-varying systematic risk
JEL Classification: G24, G12
Private equity investments amount to an increasingly significant portion of institutional
portfolios as investors seek diversification benefits relative to traditional stock and bond
holdings. Despite the increasing importance of private equity funds as an asset class, our
understanding of their pricing dynamics is quite limited. Among others, three questions
are mostly unresolved in the current literature. What is the value of a private equity fund
and how does it develop over time? How does a fund’s expected return and systematic
risk change over time? How does illiquidity affect fund values and equilibrium expected
returns? These questions ask for models that explicitly tie the variables of interest to
the cash flow dynamics of private equity funds.
In this paper, we propose a model which allows us to address the cash flow dynamics
and the equilibrium values of private equity funds in more detail. We thereby consider
the fact that private equity funds are investments which exhibit a bounded life cycle and
specific dynamics of cash drawdowns and cash distributions. A mean-reverting square-
root process represents the rate at which committed capital is drawn over time. Capital
distributions are assumed to follow a geometric Brownian motion with a time-dependent
drift that incorporates the typical repayment patterns. Given our model, we examine
the dynamics of private equity funds in three directions.
First, by applying equilibrium intertemporal asset pricing considerations, we en-
dogenously infer dynamic fund values as the difference between the present value of all
outstanding future distributions and drawdowns. Our derivation is based on the Mer-
ton (1973) intertemporal capital asset pricing model with a representative agent and
logarithmic utility. We apply Girsanov transforms to derive the dynamics of fund dis-
tributions and drawdowns under the risk-neutral measure. Our closed-form solutions
illustrate how the evolution of fund value is related to the underlying cash flow dynam-
ics, to the riskless rate, and to the correlation of cash distributions with the returns on
the market. As private equity funds are characterized by their illiquidity—no liquid or-
1
ganized secondary markets are typically available—we extend our model to demonstrate
the effect of illiquidity on fund values over time.
Second, we employ our model to explore the dynamics of the expected returns and
systematic risk of private equity funds. We start by deriving analytical expressions for
conditional expected returns and systematic risk. This allows us to evaluate how these
variables depend on the underlying economic fund characteristics and how they change
over time. In particular, we show that the beta of a fund is time-varying, which implies
that systematic fund risk and expected returns depend on the fund’s maturity level. As
such, our model demonstrates the existence and importance of a life cycle effect in the
systematic risk of private equity funds. We further show that incorporating illiquidity
into the analysis induces an additional time-varying component to expected fund returns.
Third, we calibrate our model to fund data, examine its goodness-of-fit and analyze
the model’s implications. We use data of mature European private equity funds that has
been provided by Thomson Venture Economics. Our empirical results allow for several
novel conclusions with respect to expected return, systematic risk, and the illiquidity of
private equity funds. We document that equilibrium expected returns, and systematic
risk of private equity funds decrease over fund lifetime. Capital drawdowns increase fund
beta as long as the committed capital has not been completely drawn. The economic
rationale behind this is that stepwise capital drawdowns lever up the investor’s position
in the fund. Venture funds exhibit higher betas and higher ex-ante expected returns
than buyout funds. While we are the first to highlight the existence of a life-cycle effect
in systematic fund risk, our average lifetime beta coefficients (3.30 for venture capital
and 1.08 for buyout funds) are broadly consistent with the reported values of previous
studies. Regarding valuation and illiquidity costs, we document that private equity funds
create excess value on a risk-adjusted basis. The average risk-adjusted after-fees excess
value in our sample is 14 percent relative to committed capital. Excess values result
2
for both venture and buyout funds, though, in our sample, buyout funds have a slightly
higher present value. By interpreting these excess values as compensation required by
investors for illiquidity, we derive (upper boundary) estimates of illiquidity compensation
yielded by the sample funds. Overall, our results suggest that annual illiquidity costs
amount to up to 1.4 percent of committed capital per annum. Interestingly, our results
imply that buyout funds have higher illiquidity costs than venture funds. We finally
document how fund values and illiquidity discounts of the funds evolve over time and
how illiquidity discounts increase with fund maturity.
Our paper is related to three branches of the private equity literature. First is the
growing literature on the returns and risks of private equity investments, which includes
Peng (2001), Moskowitz and Vissing-Jorgenson (2002), Das et al. (2003), Ljungquist
and Richardson (2003a,b), Cochrane (2005), Kaplan and Schoar (2005), Phalippou and
Gottschalg (2009), Cumming and Walz (2010), Korteweg and Sorensen (2010), Driessen
et al. (2012), Ang et al. (2013), Cumming and Zambelli (2013), Ewens et al. (2013),
Higson and Stucke (2013), Phalippou (2013), Buchner and Stucke (2014), Fang et al.
(2014), Harris et al. (2014), and Hochberg et al. (2014), among others. These studies
deal with risk and return characteristics either on a fund, individual deal or aggregate
industry level. Our model contributes to this literature by developing implications for the
dynamics of expected returns and systematic risk of private equity funds. It is obvious
that the aspect of time-dependence is important with respect to an evaluation of past
risk and return. As such, our model can, for example, help to explain the cross-sectional
return differences between fund types and the time-series return differences between fund
maturities. Additionally, our empirical results extend the recent evidence that private
equity fund outperform traded stocks on a risk-adjusted basis. Second is the literature
on the level of compensation for the illiquidity of private equity funds, an issue, which is
still largely unresolved. Metrick (2007), Franzoni et al. (2012), and Sorensen et al. (2013)
3
provide supporting evidence that investors are being compensated for holding illiquid
private equity funds.1 Our empirical results extend this evidence, while our model
additionally predicts that illiquidity discounts will depend on fund maturity. Third and
last is the literature on the cash flow dynamics and valuation of private equity funds.
Important empirical work in this area includes Ljungquist and Richardson (2003a,b)
and Robinson and Sensoy (2011). Similarly important theoretical contributions in the
area of private equity fund modeling and valuation include Takahashi and Alexander
(2002), Malherbe (2004), Bongaerts and Charlier (2009), Metrick and Yasuda (2010),
and Sorensen et al. (2013). None of these papers shows that there is a life cycle effect
in the systematic risk and expected returns of private equity funds. In addition, the
dynamics of our model are solely based on observable cash flow data, which appears to
be a promising approach in private equity fund modeling, as it allows to endogenously
derive fund values under equilibrium intertemporal asset pricing considerations.
The remainder of this paper is organized as follows. In the next section, we set forth
the notation, assumptions, and structure of the model. Section 2 shows how equilibrium
private equity fund values are derived. Section 3 presents our expressions for expected
fund returns and systematic risk. In Section 4 we present the results of the model
calibration and discuss its empirical implications. The paper concludes with Section 5.
1 The Model
This section develops our model for the cash flow dynamics of private equity funds. We
start with a brief description of the typical construction of private equity funds. Our
1In addition, Lerner and Schoar (2004) examine the role of transfer restrictions imposed by fundmanagers as a proxy of fund illiquidity and document that these restrictions are more likely in situa-tions where asymmetric information problems are more severe. Kleymenova et al. (2012) investigatedeterminants of liquidity of private equity fund interests sold in the secondary market.
4
choice of a continuous-time framework allows us to obtain analytical results. We assume
that all random variables introduced in the following are defined on a probability space
(Ω,F ,P), and that all random variables indexed by t are measurable with respect to the
filtration Ft, representing the information commonly available to investors.
1.1 Institutional Framework
Investments in private equity are frequently intermediated through private equity funds,
which are pooled investment vehicles for securities of companies that are usually unlisted.
Private equity funds typically represent closed-end funds with a finite lifetime and are
organized as a limited partnership. The private equity firm serves as the general partner
(GP). The bulk of the capital invested in is typically provided by institutional investors,
who then act as limited partners (LPs). The LPs commit to provide a certain amount
of capital to the private equity fund, which is the committed capital C. The GP has an
agreed time period (usually five years) in which to invest committed capital, denoted
as commitment period Tc. 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 pre-specified commitment period. Hence, calls of the fund occur unscheduled over
the commitment period, where the exact timing does only depend on the investment
decision of the GP. Total capital calls, also called drawdowns, cannot exceed the total
committed capital C. As drawdowns occur, cash is immediately invested in managed
assets of the investment portfolio. Once an investment is liquidated, the GP distributes
the proceeds to its LPs either in marketable securities or in cash. The agreed time period
in which to return capital to the LPs (usually in the range of ten to fourteen years) is
the total legal lifetime of the fund, Tl, where Tl ≥ Tc.2
2For a more thorough description and discussion of the background of private equity funds refer toGompers and Lerner (1999), Desai et al. (2003), and to the survey of Phalippou (2007).
5
Given the above construction of private equity funds, our stochastic model of the cash
flow dynamics consists of two components that are modeled independently. First, the
stochastic model for the drawdowns of the committed capital and second, the stochastic
model of the distribution of dividends and proceeds.
1.2 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. The fund is set up at time t = 0, when D0 = 0 and U0 = C are
given by definition. At any time t ∈ [0, Tc], the identity, Dt = C −Ut, must hold. In the
following, we assume that capital is drawn at some non-negative rate from the remaining
undrawn committed capital Ut = C −Dt.
Assumption 1.1 Capital drawdowns over the commitment period Tc occur in continu-
ous time. The dynamics of the cumulated drawdowns Dt can be described by the ordinary
differential equation
dDt = δtUt10≤t≤Tcdt, (1.1)
where δt ≥ 0 denotes the fund’s drawdown rate at time t, and 10≤t≤Tc is an indicator
function.
Capital drawdowns of private equity funds tend to be 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
pattern is well reflected in the structure of equation (1.1). Cumulated capital drawdowns
6
Dt are given by
Dt = C − C exp
(−∫ t≤Tc
0
δudu
)(1.2)
and instantaneous capital drawdowns, dt = dDt/dt, are equal to
dt = δtC exp
(−∫ t≤Tc
0
δudu
)10≤t≤Tc. (1.3)
Equation (1.3) shows that initial capital drawdowns dt converge to zero for t → Tc,
where the undrawn amounts, Ut = C exp(−∫ t≤Tc
0δudu
), decay exponentially over time.
Furthermore, equation (1.2) ensures that the cumulated drawdowns Dt can never exceed
the total amount of committed capital C, i.e., Dt ≤ C for all t ∈ [0, Tc]. At the same
time the model allows for a fraction of C to remain undrawn, as the commitment period
Tc acts as a cut-off point for capital drawdowns. As investment opportunities typically
do not arise constantly over the commitment period Tc, we introduce a stochastic process
for the drawdown rate δt.
Assumption 1.2 The drawdown rate is given by a stochastic process δt, 0 ≤ t ≤ Tc,
which is adapted to (Ω,F ,P). The specification is given by
dδt = κ(θ − δt)dt+ σδ
√δtdBδ,t, (1.4)
where θ > 0 is the long-run mean of the drawdown rate, κ > 0 governs the rate of
reversion to this mean, σδ > 0 reflects the volatility of the drawdown rate, and Bδ,t is a
standard Brownian motion. It is assumed that dBδ,tdBW,t = ρδW , where BW,t is a second
Brownian motion driving aggregate stock market returns.
Assuming a non-zero correlation ρδW takes into account the important possibility
that the speed of capital drawdowns may be affected by overall stock market conditions.
7
For instance, if ρδW > 0, then fast capital drawdowns become more likely during stock
market booms.
The drawdown rate behavior implied by the above square-root diffusion ensures
that negative values of the drawdown rate are precluded3 and that the drawdown rate
randomly fluctuates around some mean level θ. Under (1.4), and given that Es[·] denotes
the expectations operator conditional on the information set available at time s, expected
cumulated drawdowns at time t ≥ s are given by
Es[Dt] = C − Us Es
[exp
(−∫ t
s
δudu
)]
= C − Us exp[A(s, t)− B(s, t)δs], (1.5)
where A(s, t) and B(s, t) are deterministic functions (see Cox et al. (1985), p. 393):
A(s, t) ≡ 2κθ
σ2δ
ln
[2fe[(κ+f)(t−s)]/2
(κ+ f)(ef(t−s) − 1) + 2f
],
B(s, t) ≡ 2(ef(t−s) − 1)
(f + κ)(ef(t−s) − 1) + 2f, (1.6)
f ≡(κ2 + 2σ2
δ
)1/2.
Expected instantaneous capital drawdowns, Es[dt] = Es[dDt/dt], are
Es[dDt/dt] =d
dtEs[Dt] =
= −Us[A′(s, t)−B′(s, t)δs] exp[A(s, t)− B(s, t)δs], (1.7)
3As we model capital distributions and capital drawdowns separately, we have to restrict capitaldrawdowns to be strictly non-negative at any time t during the period [0, Tc]. The square-root diffusionwas initially proposed by Cox et al. (1985) as a model of the short rate and is frequently denoted asCIR model. If κ, θ > 0, then δt will never be negative. If 2κθ ≥ σ2
δ , then δt remains strictly positive forall t, almost surely. See Cox et al. (1985), p. 391.
8
where A′(s, t) = ∂A(s, t)/∂t and B′(s, t) = ∂B(s, t)/∂t.
1.3 Capital Distributions
Once capital drawdowns occur, the available cash is immediately invested into managed
assets, the fund portfolio accumulates, cash or marketable securities are received and
finally returns and proceeds are distributed to the LPs. We denote cumulated capital
distributions up to time t ∈ [0, Tl] as Pt. We restrict instantaneous capital distributions,
pt = dPt/dt, to be strictly non-negative at any time t ∈ [0, Tl] and assume that pt follows
a geometric Brownian motion.
Assumption 1.3 Capital distributions over the legal lifetime Tl occur in continuous
time. Instantaneous capital distributions follow a geometric Brownian motion, such that
the dynamics of ln pt are given by
d ln pt = µtdt+ σPdBP,t, (1.8)
where µt denotes the time-dependent drift and σP is a constant volatility. BP,t is a third
standard Brownian motion, which can also be correlated with BW,t, i.e., dBP,tdBW,t =
ρPW .
Note that the correlation ρPW here takes into account that capital distributions
may also be affected by overall stock market conditions. In addition, note that if both
correlations with aggregate stock market returns (ρδW and ρPW ) are non-zero, then
capital drawdowns and distributions will also be correlated in the model.
It follows from specification (1.8) that instantaneous capital distributions, pt, have a
lognormal distribution. For an initial value ps, the solution to the stochastic differential
9
equation (1.8) is
pt = ps exp
[∫ t
s
µudu+ σP (BP,t − BP,s)
], t ≥ s. (1.9)
Taking the time-s conditional expectation of (1.9) yields
Es[pt] = ps exp
[∫ t
s
µudu+1
2σ2P (t− s))
]. (1.10)
The dynamics of (1.9) and (1.10) both depend on the specification of the time-
dependent drift µt. We propose a parsimonious yet realistic specification for µt, which
incorporates the typical time pattern of the capital distributions of private equity funds.
In the early years of a fund, capital distributions tend to small as investments have not
had the time to be “harvested”. The middle years of a fund’s life cycle tend to display
the highest distributions. Finally, later years usually are marked by decline in capital
distributions. We model this behavior by defining a fund multiple by Mt ≡ Pt/C. In
this definition, the cumulated capital distributions Pt are scaled via C. The multiple
can also be expressed as Mt =∫ t
0pudu/C with M0 = 0. As more and more investments
of the fund are exited, the multiple increases over time. We assume that its expectation
converges towards some long-run mean m.
Assumption 1.4 Let Ms
t = Es[Mt] denote the conditional expectation of the fund mul-
tiple at time t, given the available information at time s ≤ t. We assume that the
dynamics of Ms
t can be described by the ordinary differential equation
dMs
t = αt(m−Ms
t)dt, (1.11)
where m is the long-run mean of the expectation and αt = αt governs the speed of
10
reversion to this mean.
Solving for Ms
t , using the initial condition, Ms
s = Ms, yields
Ms
t = m− (m−Ms) exp
[−1
2α(t2 − s2)
]. (1.12)
As pt = (dMt/dt)C, expected instantaneous capital distributions Es[pt] = (dMs
t/dt)C
turn out to be
Es [pt] = α t(m C − Ps) exp
[−1
2α(t2 − s2)
]. (1.13)
Equations (1.10) and (1.13) both define expected instantaneous capital distributions.
Setting (1.10) equal to (1.13), we can solve for the integral∫ t
sµudu. Substituting the
result back into equation (1.9), the instantaneous capital distributions at time t ≥ s are
given by
pt = αt(mC − Ps) exp
−1
2[α(t2 − s2) + σ2
P (t− s)] + σP ǫt√t− s
, (1.14)
where ǫt√t− s ∼ (BP,t − BP,s) and ǫt ∼ N(0, 1). The above solution implies that high
capital distributions in the past decrease average future capital distributions, as the term
(mC−Ps) decreases with increasing levels of time-s cumulated capital distributions Ps.4
1.4 Model Illustration
This section illustrates our model’s ability to reproduce important features of the cash
flow patterns of private equity funds. We first examine the influence of the model
parameters on the cash flow dynamics and then turn to a numerical example.
4This assumption can be relaxed by making the long run multiple m also dependent on the availableinformation at time s.
11
Considering capital drawdowns, the main model parameter governing the timing of
the drawing process is the long-run mean drawdown rate θ. Increasing θ accelerates
expected drawdowns over time. Thus, higher values of θ, on average, increase capital
drawn at the start of the fund and hence decrease capital drawn in later stages. The
influence of the mean reversion coefficient κ and the volatility σδ is relatively small and
of about the same magnitude, while their directional influence differs in sign. Increases
in σδ tend to slightly decelerate expected drawdowns, whereas increases in κ tend to
slightly accelerate them. Note that the overall volatility of the capital drawdowns is
partly influenced by the mean reversion coefficient κ. High values of κ tend to decrease
the volatility of the capital drawdowns, as a high levels of mean reversion happen to
compensate for some of the volatility of the drawdown rate, σδ.
The timing and magnitude of the capital distributions is determined by three pa-
rameters. The coefficient m is the long-run multiple of the fund. The total amount of
capital that is expected to be returned to the investors over the fund’s lifetime is deter-
mined by m times the committed capital C. The coefficient σP governs the volatility
of the capital distributions. Finally, α governs the speed at which capital is distributed
over the fund’s lifetime. The α-parameter is related to the expected payback period
of a fund, tA, i.e. the expected time needed until the cumulated capital distributions
are equal to or exceed committed capital C. It follows from equation (1.12) that α is
inversely related to a fund’s expected payback period tA, as
E0[MtA ] ≡ 1 = m
[1− exp
(−1
2· α · t2A
)](1.15)
yields
α =2 ln m
m−1
t2A. (1.16)
12
Our stochastic processes of drawdowns and distributions can be used to generate
discrete-time sample paths. Figures 1 and 2 compare the expected cash flows, i.e. draw-
downs, distributions and net fund cash flows, of two hypothetical funds. Given the sets of
model parameter values in Table 1, both funds have the same long-run multiple m = 1.5
and a committed capital C that is standardized to 1. The funds differ in the timing
and volatility of the capital drawdowns and distributions. For the first fund (Fund 1)
it is assumed that drawdowns occur rapidly in the beginning, whereas capital distri-
butions take place rather late. Conversely, for the second fund (Fund 2) it is assumed
that drawdowns occur rather progressive while distributions take place sooner. From
equation (1.16) it follows that the expected payback periods of Fund 1 and 2 are given
by 8.6 and 6.1 years, respectively. The capital drawdowns and distributions of Fund 2
exhibit higher variability as observed in Figures 1 and 2. The figures also illustrate that
the volatility of the cash flows is high when average cash flow levels are high, and vice
versa.
The cash flow patterns in Figures 1 and 2 reproduce the typical development cycle of
a fund and the characteristic J-shaped curve for the cumulated net cash flows. We may
therefore attest our model’s potential ability to generate adequate patterns of capital
drawdowns and distributions.
2 Valuation
In this section, we derive equilibrium private equity fund values. The state variables
underlying the valuation, i.e. the assumed cash flow processes, do not represent traded
assets. In such an incomplete market setting, preference-free pricing based on arbitrage
considerations alone is not feasible. For this reason, we impose additional assumptions
on the preferences of the private equity investors in our model.
13
2.1 Arbitrage-Free Fund Values
Under the assumption that the no-arbitrage condition holds, the fund value V Ft at time
t ∈ [0, Tl] is defined as the present value of all expected future cash flows, including
capital distributions and drawdowns. Thus, the arbitrage-free value is
V Ft = EQ
t
[∫ Tl
t
e−rf (τ−t)dPτ
]
︸ ︷︷ ︸≡V P
t
−EQt
[∫ Tl
t
e−rf (τ−t)dDτ
]1t≤Tc
︸ ︷︷ ︸≡V D
t
. (2.1)
where V Pt is the time-t present value of capital distributions and V D
t is the time-t present
value of capital drawdowns. Discounting drawdowns and distributions at the riskless
rate rf is appropriate in equation (2.1), as expectations are defined under the risk-
neutral measure Q. However, the risk sources underlying our model in (2.1) have to be
transformed. Applying Girsanov’s Theorem, as for example outlined in Duffie (2001),
the underlying stochastic processes for the capital drawdowns and distributions under
the Q-measure are given by
dδt = [κ (θ − δt)− λδσδ
√δt] dt+ σδ
√δt dB
Qδ,t, (2.2)
d ln pt = (µt − λPσP )dt+ σP dBQP,t, (2.3)
where BQδ,t and BQ
P,t areQ-Brownian motions and λδ and λP are the corresponding market
prices of risk, which are defined by
λδ ≡µ(δt, t)− rf
σ(δt, t), λP ≡ µ(ln pt, t)− rf
σ(ln pt, t). (2.4)
In order to determine the market price of risk λδ, note that capital commitments
are contractually binding payments, where only the exact timing is uncertain. For
14
this reason, investors typically hold undrawn commitments in a riskless asset and sell
fractions of this asset when drawdowns occur. Thus, the expected rate of return of the
undrawn capital simply equals the riskless rate of return rf . This in turn implies that
capital drawdowns can be discounted be the riskless rate rf and the market price of risk
of the drawdown rate equals zero, i.e., λδ = 0.
Determining the market price of risk λP is more involved and requires an additional
assumption on the investors’ preferences. It is assumed that investors have CRRA utility
of the logarithmic form. Under this assumption, Merton (1973) shows that equilibrium
expected returns will satisfy a version of the intertemporal capital asset pricing model,5
which requires that
µi − rf = σiW , (2.5)
where µi is the expected rate of return on some asset i and σiW is the covariance of the
returns on asset i with the return of the market portfolio W . Using (2.5), it follows that
λP = σPW/σP , with σPW = σPσWρPW .
Using these specifications gives the following theorem for the arbitrage-free value of
a private equity fund.
Theorem 2.1 The arbitrage-free equilibrium value of a private equity fund at any time
t ∈ [0, Tl] can be stated as
V Ft = α (m C − Pt)
∫ Tl
t
e−rf (τ−t)D(t, τ) dτ
+ Ut
∫ Tl
t
e−rf (τ−t)C(t, τ)dτ 1t≤Tc, (2.6)
5This simple version arises from the assumption of logarithmic utility. It permits to omit terms thatare related to stochastic shifts in the investment opportunity set, which otherwise arise. See Merton(1973) and Brennan and Schwartz (1982) for a detailed discussion.
15
where C(t, τ) and D(t, τ) are deterministic functions given by
C(t, τ) =(A′(t, τ)− B′(t, τ)δt) exp[A(t, τ)−B(t, τ)δt],
D(t, τ) = exp
[ln τ − 1
2α(τ 2 − t2)− σPW (τ − t)
],
and A(t, τ), B(t, τ) are given in (1.6).
Proof: see Appendix A.
Except for the two integrals that can be evaluated by numerical techniques, Theo-
rem 2.1 provides an analytically tractable solution for the arbitrage-free fund value.
Figure 3 illustrates the fund value dynamics for varying values of the correlation
coefficient ρPW .6 We point out that the time patterns of fund values as shown in Fig-
ure 3 conform common expectations of private equity fund behavior. The value of a
fund increases over time as the investment portfolio is build up and decreases once fewer
investments are left. In the context of our model, this characteristic behavior follows
mainly from the fact that capital drawdowns occur, on average, earlier than the capital
distributions (see also Figures 1 and 2 above). Thus, over a fund’s lifetime, the present
value of outstanding capital drawdowns decreases faster than the present value of out-
standing capital distributions. Figure 3 also illustrates the effect of different levels of
the correlation ρPW . The figure shows that increasing levels of the correlation reduce
fund values. High levels of the correlation imply a situation where capital distribution
tend to be high in states of the world in which the return of the market portfolio (and
hence aggregate wealth) is also high. This is an unfavorable relationship for investors
6Note that for all t > 0, Figure 3 displays fund values as unconditional expectations, V Ft = E0[V
Ft ].
Otherwise, fund values are stochastic as the value of a fund at time t in Theorem 2.1 depends on thecumulated capital distributions Pt and the undrawn amounts Ut up to time t.
16
and reduces fund values.
2.2 Accounting for Illiquidity
As pointed out above, private equity funds are highly illiquid investment vehicles. Both,
theory and empirical evidence suggest that investors attach a lower price to less liquid
but otherwise identical assets. Thus, in case private equity investors value liquidity,
they will discount the value of a private equity fund for illiquidity. Let C illt denote the
illiquidity discount of a fund at time t. Then, we define the fund value under illiquidity,
V F,illt , as
V F,illt = V F
t − C illt , (2.7)
where V Ft is the arbitrage-free fund value, as defined above in Theorem 2.1.
Following Amihud and Mendelson (1986), let C illt represent the time-t present value
of all illiquidity costs of the fund during its remaining lifetime (Tl − t). Based on risk-
neutral valuation arguments, we can rewrite the fund value under illiquidity as
V F,illt = V F
t − EQt
[∫ Tl
t
e−rf (τ−t)cillτ dτ
], (2.8)
with the instantaneous illiquidity costs cillt . Assuming for simplicity that instantaneous
illiquidity costs are constant over time, cillt = cill, it follows
V F,illt = V F
t − cill1− e−rf (Tl−t)
rf. (2.9)
The fund value under illiquidity, V F,illt , is generally unobservable. However, we know
that investors enter private equity funds at initiation in t = 0 without paying an explicit
17
cost.7 Therefore, the boundary condition, V F,ill0 = 0, must hold and we can implicitly
derive instantaneous illiquidity costs cill from equation (2.9) above. Solving for cill yields
cill =V F0 rf
1− e−rfTl. (2.10)
and substituting (2.10) back into (2.9) establishes the following theorem for the value of
an illiquid fund.
Theorem 2.2 The value of an illiquid private equity fund at any time t ∈ [0, Tl] can be
stated as
V F,illt = V F
t − willt V F
0 , (2.11)
where
willt =
1− e−rf (Tl−t)
1− e−rfTl
and V Ft denotes the arbitrage-free fund value as given in Theorem 2.1.8
The weighting factor willt in Theorem 2.2 is an increasing function of the fund’s
remaining lifetime (Tl − t). The longer the remaining lifetime, the larger willt and the
more illiquidity affects value. When the fund is liquidated, t = Tl, it follows that willt = 0
and hence, V F,illTl
= V FTl
= 0, where the last equation holds by definition.
Figure 4 illustrates our results by comparing the values of an illiquid fund with
the values of the corresponding liquid fund over time. Excluding fund liquidation at
Tl, the value of the illiquid fund is below the value of the liquid fund. Over time,
7This holds, at least, when we ignore all direct and indirect transaction costs, such as search costsfor the investor.
8Under this specification V Ft may become slightly negative at the end of fund lifetime in case the
present value of the constant illiquidity costs cill at some time t is higher than the value of the liquidfund, V F
t . A rational investor will never sell a fund given that the costs of doing so are higher than its
current value. Therefore, we may alternatively define: V F,illt = max
[V Ft − will
t V F0; 0].
18
the difference between the values decreases as the illiquidity discount of the fund is a
decreasing function of the fund’s remaining lifetime.
Regarding our treatment of illiquidity costs, three additional points are notable.
First, instantaneous illiquidity costs cill are expected liquidation costs of a fund per unit
time and equal the product of the probability that an investor wants to liquidate his
fund investment during the time interval (t, t + dt] and of the costs of selling the fund;
see Amihud and Mendelson (1986) for a similar definition of illiquidity costs. Second, in
a narrow interpretation, the costs of selling represent transaction costs. More broadly,
however, the costs of selling would also include other costs, as for instance costs arising
from delay or search. Lastly, we assume that illiquidity costs are constant over time.
While a more general framework would account for time-varying, possibly stochastic,
illiquidity costs, our approach offers a straightforward method to explicitly calculate the
average illiquidity costs over a fund’s lifetime by equation (2.10).
3 Expected Return and Systematic Risk
This section derives expressions for expected fund returns and systematic risk and shows
how returns are related to underlying economic characteristics of a fund’s cash flows. As
in the preceding section, we assume that the no-arbitrage condition holds and initially
ignore illiquidity. We then relax this assumption to show how illiquidity affects expected
fund returns in equilibrium.
19
3.1 Valuation Framework
Fund returns are defined by fund cash flows plus changes in value, divided by current
value. Appendix B shows that the conditional expectation of fund returns is
Et
[RF
t
]=
Et
[dV F
t
dt
]+ Et
[dPt
dt
]−Et
[dDt
dt
]
V Ft
. (3.1)
To evaluate condition (3.1), we obtain expressions for the conditional expectation of
a fund’s instantaneous price change, i.e. for Et[dVFt /dt]. This quantity, in turn, is deter-
mined by the difference of the expected instantaneous changes in present values of the
capital distributions and capital drawdowns, as Et[dVFt /dt] = Et[dV
Pt /dt]−Et[dV
Dt /dt]
holds. Appendix B shows that these two quantities can be represented as
Et
[dV D
t
dt
]= −Et
[dDt
dt
]+ rf V D
t (3.2)
and
Et
[dV P
t
dt
]= −Et
[dPt
dt
]+ (rf + σPW )V P
t . (3.3)
Substituting (3.2) and (3.3) into (3.1), the expected instantaneous fund return is
Et
[RF
t
]= rf + σPW
V Pt
V Pt − V D
t
. (3.4)
That is, the expected return of a private equity fund is given by the riskless rate of
return rf plus the risk premium σPWV Pt /(V P
t − V Dt ). This risk premium depends on
two components. First, this risk premium is determined by the covariance σPW . The
economic intuition behind this follows from standard asset pricing arguments. Second is
that the risk premium additionally depends on the ratio V Pt /(V P
t − V Dt ), which implies
20
that equilibrium expected fund returns will vary as the systematic fund risk changes
over time. This result becomes more obvious once we consider expected fund returns in
(3.4) from the traditional beta perspective. It follows
Et
[RF
t
]= rf + βF,t(µW − rf), (3.5)
where βF,t is the beta coefficient of the fund returns at time t and µW is the expected
return of the market portfolio. Setting (3.4) equal to (3.5), the fund beta turns out to
be
βF,t = βPV Pt
V Pt − V D
t
, (3.6)
where βP = σPW/σ2W is the constant beta coefficient of the capital distributions of the
fund and (2.5) requires that µW − rf = σ2W . From specification (3.6), it is obvious that
the fund beta varies over time as the present values V Pt and V D
t are stochastic.
3.2 Accounting for Illiquidity
In order to account for illiquidity, we start by noting that the conditional expectation
of a fund’s net return in an economy with instantaneous illiquidity costs, cillt , can be
written as
Eillt
[RF
t − cilltV Ft
]=
Et
[dV F
t
dt
]+ Et
[dPt
dt
]− Et
[dDt
dt
]
V Ft
. (3.7)
Then, expected fund returns in beta representation form are given by
Eillt
[RF
t
]= rf +
cilltV Ft
+ βF,t(µW − rf ), (3.8)
where the fund’s beta, βF,t, is given by (3.6). The required excess return is the sum of
the relative illiquidity costs, cillt /V Ft , and the fund’s beta times the risk premium. The
21
relative illiquidity costs reflect the compensation required by investors for the lack of an
organized and liquid market. It is important to acknowledge that illiquidity costs imply
a second time-varying component in expected fund returns. Relative illiquidity costs,
cillt /V Ft , will vary over time as the fund value V F
t follows a distinct life cycle pattern.
More importantly, cillt /V Ft will also vary with with instantaneous illiquidity costs, cillt .
When illiquidity costs are high, expected returns are also high. This may help to ex-
plain why several studies document that private equity returns are highly cyclical. For
example, Gompers and Lerner (2000) and Kaplan and Schoar (2005) provide supporting
evidence in favor of a of boom and bust cycle in private equity fund returns. In the light
of equation (3.8), this cyclicality could be a result from time-varying illiquidity costs.
4 Empirical Evidence
In this section, we show how our model can be calibrated to data and discuss its empirical
implications. We start by introducing the private equity fund data set and outline our
estimation methodology. Then, the empirical results are presented.
4.1 Data Set and Sample Selection
This section makes use of a dataset of European private equity funds that has been
provided by Thomson Venture Economics (TVE).9 The unique advantage of the data
is that is contains the exact timing and size of cash flows and residual net-asset-values
(NAVs) on a quarterly basis. All cash flows and reported NAVs are net of management
9Note that TVE uses the term private equity to describe the universe of all venture capital, buyout,and mezzanine investing. Fund of fund investing and secondaries are also included in this broadestterm. TVE is not using the term to include angel investors or business angels, real estate investments,or other investing scenarios outside of the public market. For a detailed overview on the TVE datasetand a discussion of its potential biases see Kaplan and Schoar (2005) and Stucke (2011).
22
fees and carried interest payments. The dataset contains a total of 777 funds and covers
the period ranging from January 1, 1980 through June 30, 2003. Unfortunately, updated
versions of the data are no longer available for research purposes.
Based on our data set above, we derive a subsample of mature private equity funds
during the period January 1, 1980 to June 30, 2003. As we focus on core private
equity funds, we immediately exclude 14 funds of funds. Next, as our study in principle
requires the knowledge of the entire cash flow history of the analyzed funds, our sample
selection procedure has to deal with the problem of the limited number of liquidated
funds available. Once we restrict ourselves to funds which are fully liquidated at the end
of the observation period, Table 2 shows that this reduces our data set to a total number
of 95 funds only. We therefore increase our sample by adding funds that have small net
asset values relative to their realized cash flows at the end of the observation period. We
thereby add non-liquidated funds to our sample given that their reported June 30, 2003
residual net asset value is less or equal to 10 percent of the undiscounted sum of the
absolute values of all previously accrued fund cash flows. Treating the net asset value
at the end of the observation period as a final cash flow, will only have a minor impact
on our model estimation results. Table 2 reports that the extended sample consists of a
total of 203 funds of which 102 are venture capital funds and 101 are buyout funds. Our
subsequent analysis is based on this extended sample of mature private equity funds.
4.2 Estimation Methodology and Results
Our procedure for model calibration is an explicit parameter estimation based on his-
torical fund cash flow data. We use the concept of Conditional Least Squares (CLS),
which is a general approach for estimating the parameters involved in a continuous-time
stochastic process (see Klimko and Nelson (1978)). Details of this estimation procedure
23
are outlined in Appendix C.
Table 3 shows the estimated model parameters for the sample of all (Total), venture
capital (VC) as well as buyout (BO) funds. For capital drawdowns, the estimated
annualized long-run mean drawdown rate θ of all sample funds amounts to 0.47. This
implies that in the long-run approximately 11.75 percent of the remaining committed
capital is drawn on average in each quarter of a fund’s lifetime. The high reported value
for the volatility σδ indicates that drawdowns are fluctuating heavily over time. When
comparing venture as opposed to buyout funds, it appears that venture and buyout
funds on average draw down capital at a similar pace as the coefficients θ are almost
equal among the two sub-samples. However, venture funds draw down capital with
somewhat higher uncertainty than their buyout counterparts, which is pointed out by
a higher value of the volatility σδ for venture funds, while the higher mean reversion
coefficient κ tends to absorb some of the higher volatility. For capital distributions,
the long-run multiple m for all sample funds is estimated to equal 1.85. That is, on
average, funds distribute 1.85 times their committed capital over the total lifetime. The
reported α coefficient further implies via equation (1.16) that the sample funds have an
average payback period of 7.41 years (i.e. around 89 months). The sub-sample of venture
funds returned substantially more capital to the investors than the corresponding sample
buyout funds. A comparison of the α-coefficients reveals that the average buyout fund
tends to pay back its capital much faster than the average venture fund, an observation
that is statistically significant. This result corresponds to the common notion that
venture funds invest in young and technology-oriented start-ups, whereas buyout funds
invest in mature and established companies. Growth companies typically do not generate
significant cash flows during their first years in business and it usually takes longer until
these investments can successfully be exited (for example, by an IPO or a trade-sale to
a strategic investor). These conceptual differences help to explain the differences in the
24
standard deviations σP as given in Table 3. Buyout funds distribute their capital with
less uncertainty as measured by the lower estimated volatility σP in our sample.
The application of our model to fund valuation and to the calculation of expected
fund returns requires four additional parameter estimates. These are the riskless rate
rf , the expected return of the market portfolio µW , the standard deviation of market
portfolio returns σW and finally the covariance σPW between changes in log capital dis-
tributions and market returns. Table 4 summarizes our choices of parameter values for
these variables. We set the riskless rate equal to the sample mean of the annualized
monthly money market rates for three-month funds as reported by Frankfurt banks
(data are available at http://www.bundesbank.de) over the period January 1, 1980 to
June 30, 2003, which results in rf = 0.0587. The parameters µW and σW are estimated
based on continuously compounded monthly returns of the MSCI World Index over the
same observation period. Estimation of the covariance σPW is more involved. Using
the full sample of 777 funds, we calculate monthly rolling annual differences of aggre-
gate log capital distributions. We then estimate σPW by the covariance between these
rolling changes in log capital distributions and the corresponding rolling yearly MSCI
World continuously compounded returns. This results in estimated covariances of 0.0296
(Total), 0.0433 (VC) and 0.0157 (BO). These and the respective correlations ρPW are
reported in Table 4. The higher correlation of venture compared to buyout illustrates
that capital distributions of venture capital funds react more to the overall stock market
development than buyout funds. This finding is consistent with a greater importance of
stock markets as exit channels for venture capital compared to buyout funds.
25
4.3 Model Validation and Application
A. Goodness-of-Fit
An examination of the calibrated model by assessing its goodness-of-fit is essential. A
simple way to evaluate our model specification is to examine whether the model’s implied
cash flow patterns are consistent with the time series data of our sample. The present
test of model expectations addresses our model’s ability to match the first moment of
the fund cash flows over time. This ability is a major characteristic of the valuation
model, as values are discounted sums of expected cash flows.
Figure 5 compares the historical average capital drawdowns, capital distributions and
net fund cash flows of all sample funds to the corresponding expectations that can be
constructed from our model by using the parameters reported in Table 3. Overall, the
results from Figure 5 indicate an excellent fit of the model. In particular, as measured by
the coefficient of determination, R2, our model can explain a very high degree of 97.73
percent of the variation in average yearly net fund cash flows. In addition, the mean
absolute error (MAE) of the approximation is only 1.56 percent annually (as measured
in percent of committed capital). Splitting the overall sample for venture and buyout
funds, we find that the quality of the approximation is slightly lower for venture funds
(R2 with 94.69 percent and MAE with 2.74 percent) than the for buyout funds (R2 with
97.02 percent and MAE with 1.58 percent).
B. Valuation Results
All reported valuation results are derived based on our model with the calibrated pa-
rameter values as shown in Table 3 and Table 4. We first employ the model to calculate
risk-adjusted excess values and implicit illiquidity costs for our sample of funds. We
then show how fund values and illiquidity discounts typically evolve over the fund life
cycle.
26
B.1 Excess Value and Illiquidity Costs
Table 5 presents the valuation results for the overall sample as well as for venture and
buyout funds separately. The first panel of Table 5 reports the present values V P0 and
V D0 of the capital distributions and capital drawdowns, respectively, and the resulting
fund values V F0 at the starting date t = 0. Fund values V F
0 are equivalent to the ex-ante
risk-adjusted net present values of investing in a fund and represent an ex-ante present-
valued return on committed capital. Thus, each Dollar or Euro committed is worth one
plus V F0 in present value terms, where V F
0 represents the excess value.
The second panel of Table 5 reports instantaneous illiquidity costs cill. We thereby
Ljungquist and Richardson (2003a) and interpret overall excess values as a compensation
required by investors for illiquidity. We point out that this approach yields an upper
boundary for the costs of fund illiquidity, as excess values may include other premiums.
Private equity investors frequently lack sufficient diversification and hence may require
a premium for bearing idiosyncratic risk. Additional compensation may be required for
costs of the investors that arise from asymmetric information. Such additional sources
would reduce the estimated illiquidity costs. Hence, based on an average fund lifetime
of 15 years, equation (2.10) implies an upper boundary on annual illiquidity costs. In-
stantaneous upper boundary illiquidity costs in Table 5 are annualized and stated as a
percentage of committed capital.
Our valuation results have several implications. First, our results confirm that private
equity funds create excess value on a risk-adjusted basis. The results in Table 5 show
that the risk-adjusted net-of-fees excess value of an average fund is 14.41 percent relative
to committed capital, i.e., 100 currency units committed are on average worth 114 in
present value terms. Second, excess values result for both venture and buyout funds. In
our sample, buyout funds create higher excess values. Third, based on our assumptions
above, equation (2.10) implies an upper boundary on annual illiquidity costs of about
27
1.44 percent of committed capital. As buyout funds create higher excess values, our
approach implies that buyout funds offer a higher compensation for illiquidity than
venture funds. This result is consistent with the observation that investors of buyout
funds require a higher compensation for illiquidity due to the larger size of the individual
investments of these funds. In line with this, Franzoni et al. (2012) show that investment
size is a positive and significant determinant of compensation for illiquidity. It appears
that larger investments are more sensitive to exit conditions and, via this channel, are
more heavily exposed to liquidity risk.
B.1 Value Dynamics and Illiquidity Discounts
We now illustrate the dynamics of fund values over time. Table 6 provides the dynamics
of fund values under liquidity, V Ft , under illiquidity, V F,ill
t , and the illiquidity discounts,
C illt , for all three samples of funds. All reported values are unconditional, i.e. time-zero,
expectations of the variables.
The dynamics of the fund values shown in Table 6 conform to common perceptions
of private equity fund behavior. In particular, fund values first increase over time as
the investment portfolio expands and then gradually decrease once fewer investments
are left for exit. Three differences between venture and buyout funds are highlighted in
the table. First, buyout funds reach their value maximum earlier than venture funds.
Second, buyout fund values decrease faster towards the end of the fund life cycle. This
follows as venture funds have a slightly slower drawdown schedule and also distribute
capital slower than buyout funds. Finally, venture funds potentially reach higher max-
imum values as they distribute more capital, on average. Table 6 also reports upper
boundary illiquidity compensations over the fund cycle. In line with economic intuition,
the results show that illiquidity discounts increase non-linearly with the expected re-
maining fund lifetime. That is, a higher expected remaining lifetime implies a higher
illiquidity discount. Thereby, the non-linearity of the relationship stems from the fact
28
that capital distributions are not spread equally over a fund’s lifetime.
C. Systematic Risk and Expected Return
We next turn to the implications of our model with respect to systematic fund risk and
expected returns. Table 7 illustrates the model dynamics of the systematic fund risk, as
measured by beta coefficients, and of the corresponding expected fund returns. The beta
coefficients and expected returns in Table 7 are calculated based on the unconditional
expectations of the fund values as shown in Table 6.
Our results document that the beta coefficients, βFt , and expected fund returns under
liquidity, Et[RFt ], follow distinct time-patterns during the fund cycle. The highest values
of these variables can be observed at the start of the funds. Over time, both variables
decrease towards some constant level. Mathematically, the pattern is given by equation
(3.6) for the fund beta. Over time, the fund draws down capital from its investors, builds
up its investment portfolio, and the present value of the remaining capital drawdowns
V Dt decreases. On average, drawdowns occur earlier than capital distributions over the
fund’s lifetime, i.e. the value V Dt decreases faster than the value V P
t . The present value
of the capital drawdowns V Dt eventually decreases to zero, the fund beta converges to
the beta coefficient of the capital distributions, βP = σPW/σ2W , and expected returns
converge to rf + βP (µW − rf). From an economic standpoint, the fact that committed
capital is not instantly invested at the start of a fund acts like a leverage for the investor’s
position in the fund. Therefore, stepwise capital drawdowns increase a fund’s systematic
risk as long as the committed capital has not been entirely drawn. The results also show
that venture funds have higher beta coefficients and therefore generate higher ex-ante
expected returns than buyout funds. For all times during the observed fund lifetime,
the systematic risk of venture funds is higher than that of buyout funds. Average
lifetime betas in Table 7 amount to 3.30 for venture capital and 1.08 for buyout funds.
The higher systematic risk of venture capital investments has also been documented
29
in previous studies. Table 8 summarizes the empirical evidence on the systematic risk
of private equity investments. A comparison underlines that while we are the first to
highlight the existence of a life-cycle effect in systematic fund risk, our average beta
coefficients are broadly consistent with the reported values of previous studies.
For the buyout segment, our lifetime average of 1.08 does closely match with the
results of previous studies. Ljungquist and Richardson (2003a) even report an identical
beta of 1.1 for buyout funds. Jones and Rhodes-Kropf (2003), Woodward (2009), and
Franzoni et al. (2012) find slightly lower beta coefficients, while Driessen et al. (2012),
Ang et al. (2013), and Buchner and Stucke (2014) report higher betas.
For the venture segment, estimated beta coefficients of previous studies range from
1.1 to 2.8. While this range implies a remarkable variation, the average over the re-
ported values of all studies of around 2.0 shows that there is a consistent view that
venture capital investments exhibit high levels of systematic risk with beta coefficients
considerably above one. Our estimated lifetime average of 3.30 is consistent with this
result. In addition, it is important to note that the results of the more recent studies
cited in Table 8 are even better in line with our lifetime average. For example, Ewens
(2009) and Korteweg and Sorensen (2010) estimate beta coefficients of 2.4 and 2.8 on
the individual deal level. On the fund level, Driessen et al. (2012) and Buchner and
Stucke (2014) report beta coefficients of venture capital of 2.7 and 2.8, respectively.
5 Conclusion
This paper presents a novel and convenient structural model of the life cycle dynamics of
private equity funds, which can be calibrated based on observable variables only. Based
on an economic specification of the underlying cash flow processes, we endogenously
30
derive equilibrium fund values from intertemporal asset pricing considerations. Our
results underline the importance of the life cycle dynamics of private equity funds and
of the distinct fund patterns which arise thereof. To the best of our knowledge, we are
first to explore the nature of the—so far relatively obscure—dynamics of equilibrium
expected fund returns, systematic fund risk and fund illiquidity premiums. Systematic
fund risk in general varies predictably through time, which is central to the literature on
the risk and return characteristics of private equity funds. Further research may deepen
our understanding of risk, return and illiquidity of private equity investments.
31
A Derivation of Fund Values
In this appendix, we derive the arbitrage-free fund values given in Theorem 2.1. The
fund value V Ft at time t ∈ [0, Tl] is defined by the difference between the present value
of capital distribution, V Pt , and capital drawdowns, V D
t .
A. Capital Drawdowns
We start with the derivation of the present value of the capital drawdowns, V Dt , that is
defined by
V Dt = EQ
t
[∫ Tl
t
e−rf (τ−t)dτdτ
]1t≤Tc. (A.1)
Note that we can first reverse the order of the expectation and the time integral in
(A.1) due to Fubini’s Theorem (see e.g. Duffie (2001)). That is,
EQt
[∫ Tl
t
e−rf (τ−t)dτdτ
]=
∫ Tl
t
e−rf (τ−t)EQt [dτ ]dτ (A.2)
holds, as the riskless rate rf is assumed to be constant. In addition, we have implicitly
assumed the drawdown rate to carry zero systematic risk. Therefore, the expectation
on the right hand side of (A.2) is the same under the risk-neutral measure Q and the
objective probability measure P, i.e., EQt [dτ ] = EP
t [dτ ]. Thus, inserting (1.7) directly
yields
V Dt = −Ut
∫ Tl
t
e−rf (τ−t)C(t, τ)dτ1t≤Tc, (A.3)
where
C(t, τ) = (A′(t, τ)−B′(t, τ)δt) exp[A(t, τ)− B(t, τ)δt],
and A(t, τ), B(t, τ) are as given in (1.6).
B. Capital Distributions
We now turn to the present value of the capital distributions, V Pt . Applying Fubini’s
32
Theorem again, this present value is defined by
V Pt =
[∫ Tl
t
e−rf (τ−t)EQt [pτ ]dτ
]. (A.4)
This reduces the problem to finding EQt [pτ ]. Solving the risk-neutralized process (2.3)
with λP = σPW/σP yields
pτ = ατ(mC − Pt) exp −1
2[α(τ 2 − t2) + σ2
P (τ − t)]
+ σP ǫt√τ − t− σPW (τ − t), (A.5)
for τ ≥ t. Taking the conditional expectations of (A.5) results in
EQt [pτ ] = ατ(mC − Pt) exp
−1
2α(τ 2 − t2)− σPW (τ − t)
. (A.6)
Inserting this into (A.4), the present value of the capital distributions can be represented
as
V Pt = α (m C − Pt)
∫ Tl
t
e−rf (τ−t)D(t, τ)dτ, (A.7)
where
D(t, τ) = exp
[ln τ − 1
2α(τ 2 − t2)− σPW (τ − t)
].
Finally, substituting (A.3) and (A.7) into the valuation identity, V Ft = V P
t − V Dt ,
provides the result as stated in Theorem 2.1.
33
B Derivation of Expected Fund Returns
The purpose of this appendix is to derive the expected return of a private equity fund
as stated in equation (3.4). The instantaneous time-t return RFt is defined by
RFt dt =
dV Ft + dPt − dDt
V Ft
. (B.1)
From an economic perspective, RFt is the return that can be earned by investing
in the fund over an infinitesimally short time interval (t, t + dt]. Dividing by the time
increment dt on both sides of equation (B.1) yields
RFt =
dV Ft
dt+ dPt
dt− dDt
dt
V Ft
. (B.2)
Substituting the conditions, V Ft = V P
t − V Dt and dV F
t /dt = dV Pt /dt − dV D
t /dt,
equation (B.2) can be rewritten as
RFt =
dV Pt
dt− dV D
t
dt+ dPt
dt− dDt
dt
V Pt − V D
t
. (B.3)
Taking conditional expectations EPt [·] of (B.3), the expected instantaneous fund return
is
Et
[RF
t
]=
Et
[dV P
t
dt
]−Et
[dV D
t
dt
]+ Et
[dPt
dt
]− Et
[dDt
dt
]
Et [V Pt ]−Et[V D
t ], (B.4)
where Et[dPt/dt] and Et[dDt/dt] denote expected instantaneous capital distributions
and capital drawdowns, respectively.
Under the specifications of our model, the expected instantaneous change of the
34
present value of the capital drawdowns Et[dVDt /dt] can be represented as
Et
[dV D
t
dt
]= −Et
[dDt
dt
]+ rf V D
t . (B.5)
This result can be derived using two different ways. The first way is to directly dif-
ferentiate the value V Dt given by equation (2.6) with respect to t and then taking the
conditional expectation of the result. After some algebraic transformations, it follows
that (B.5) holds. The second and much faster way is to directly derive (B.5) by using
the general equilibrium model given by equation (2.5). From this, it must hold that
Et
[dV D
t + dDt
V Dt
]= rfdt, (B.6)
where equality with the riskless rate of return rf follows from the fact that we have
implicitly assumed capital drawdown to carry zero systematic risk. Multiplying by V Dt
and rearranging directly leads to (B.5).
Following a similar line of argument, it can be inferred that the expected instanta-
neous change of the present value of the capital distributions Et[dVPt /dt] can be repre-
sented as
Et
[dV P
t
dt
]= −Et
[dPt
dt
]+ (rf + σPW )V P
t , (B.7)
where now, compared to equation (B.5), the additional term σPWV Pt enters as we have
assumed logarithmic capital distributions and the return on the market portfolio to be
correlated with constant covariance σPW .
Finally, substituting (B.5) and (B.7) into (B.4), expected instantaneous fund returns
turn out to be
Et
[RF
t
]= rf + σPM
V Pt
V Pt − V D
t
. (B.8)
35
C Estimation Methodology
In this appendix, we present our estimation methodology for the parameters involved in
the processes of capital drawdowns and distributions.
A. Capital Drawdowns
The modeling of the drawdown dynamics requires the estimation of the following pa-
rameters: the long-run mean of the fund’s drawdown rate θ, the mean reversion speed
κ, the volatility σδ, and the initial drawdown rate δ0.
The objective is to estimate the model parameters θ, κ, σδ, and δ0 from observable
capital drawdowns of the sample funds at equidistant time points tk = k∆t, where
k = 1, . . . , K and K = T/∆t holds. To make the funds of different sizes comparable,
capital drawdowns of all j = 1, . . . , N sample funds are first standardized on the basis of
each fund’s total invested capital. LetD∆tk,j denote the standardized capital drawdowns of
fund j in the time interval (tk−1, tk]. Using this definition, cumulated capital drawdowns
Dk,j of fund j up some time tk are given by Dk,j =∑k
i=1D∆ti,j and undrawn committed
amounts Uk,j at time tk are given by Uk,j = 1 − Dk,j. Using these definition, the
(annualized) arithmetic drawdown rate δ∆tk,j of fund j in the interval (tk−1, tk] can be
defined by
δ∆tk,j =
D∆tk,j
Uk−1,j ·∆t. (C.1)
To estimate the model parameters θ and κ, we use the concept of conditional least
squares (CLS). The concept of conditional least squares, which is a general approach
for estimation of the parameters involved in the conditional mean function E[Xk|Xk−1]
of a stochastic process, is given a thorough treatment by Klimko and Nelson (1978).
The idea behind the CLS method is to estimate model parameters from discrete-time
36
observations Xk of a stochastic process, such that the sum of squares
K∑
k=1
(Xk − E[Xk|Fk−1])2 (C.2)
is minimized, where Fk−1 is the σ-field generated by X1, . . . , Xk−1. This idea can be
slightly adapted to our particular estimation problem. As we have time-series as well as
cross-sectional data of the capital drawdowns of our sample funds, a natural approach
is to replace the Xk in relation (C.2) by the sample average Xk.
Let Uk denote the sample average of the remaining committed capital at time tk, i.e.,
Uk = 1N
∑Nj=1Uk,j. An appropriate goal function to estimate the parameters θ and κ is
then given byK∑
k=1
(Uk −E[Uk|Fk−1])2, (C.3)
where Fk−1 is the σ-field generated by U1, . . . , Uk−1. The required conditional expecta-
tion E[Uk|Fk−1] can be derived from the continuous-time specification dUt = −δtUtdt.
The dynamics of the undrawn amounts are given in discrete-time by
Uk = Uk−1(1− δ∆tk ∆t). (C.4)
Taking the conditional expectation E[·|Fk−1] and replacing Uk−1 by the sample average
Uk−1, it follows that
E[Uk|Fk−1] = Uk−1(1−E[δ∆tk |Fk−1]∆t). (C.5)
Under the process defined by (1.4), the conditional expectation of the drawdown rate
37
E[δ∆tk |Fk−1] is given by (see Cox et al. (1985), p. 392):
E[δ∆tk |Fk−1] = θ(1− e−κ∆t) + e−κ∆tδ∆t
k−1, (C.6)
where δ∆tk denotes the average (annualized) drawdown rate of the sample funds that is
defined by
δ∆tk =
1N
N∑j=1
D∆tk,j
1N
N∑j=1
Uk−1,j∆t
. (C.7)
Substituting (C.6) and (C.5) into (C.3), the goal function to be minimized turns out
to beK∑
k=1
Uk − Uk−1[1− (θ(1− e−κ∆t) + e−κ∆tδ∆tk−1)∆t]2. (C.8)
Appropriate estimates of θ and κ can then be found by a numerical minimization of
(C.8). This also requires the knowledge of the initial value of the drawdown rate. For
simplicity, this value is set to zero. That is, we assume δ0 = δ∆t0 = 0 in the following.
We now turn to the estimation of the volatility σδ. The conditional variance of the
capital drawdowns of some fund j in the interval (tk−1, tk] can be stated by
E[D∆tk,j − E[D∆t
k,j|Fk−1]|Fk−1]2 = V ar[δ∆t
k,jUk−1,j∆t|Fk−1]
= (Uk−1,j∆t)2V ar[δ∆tk,j|Fk−1]. (C.9)
Under the specification of the mean reverting square root process defined by (1.4), the
conditional variance V ar[δ∆tk,j|Fk−1] of the drawdown rate is given by (see Cox et al.
(1985), p. 392):
V ar[δ∆tk,j|Fk−1] = σ2
k,j(η0 + η1δ∆tk−1,j), (C.10)
38
where
η0 =θ
2κ
(1− e−κ∆t
)2,
η1 =1
κ
(e−κ∆t − e−2κ∆t
).
The conditional expectation E[D∆tk,j|Fk−1] can in discrete-time be written as
E[D∆tk,j|Fk−1] = E[δ∆t
k,j|Fk−1]Uk−1,j∆t
= (γ0 + γ1δ∆tk,j)Uk−1,j∆t, (C.11)
with
γ0 = θ(1− e−κ∆t
),
γ1 = e−κ∆t.
Substituting equation (C.10) and (C.11) into (C.9), an appropriate estimator of the
variance σ2j of the drawdown rate of fund j turns out to be
σ2j =
1
K
K∑
k=1
[D∆tk,j − (γ0 + γ1δ
∆tk−1,j)Uk−1,j∆t]2
(Uk−1,j∆t)2(η0 + η1δk−1,j), (C.12)
where γ0, γ1 and η0, η1 are evaluated at (θ, κ). In the following, the sample variance
is then defined to be the simple average of the individual fund variances, i.e., σ2δ =
1N
∑Nj=1 σ
2j .
39
B. Capital Distributions
The modeling of the distribution dynamics requires the estimation of the following pa-
rameters: the long-run mean of the fund’s multiple m, the coefficient α, and the volatility
σP .
We estimate these model parameters from observable capital distributions of the
sample funds at equidistant time points tk = k∆t, where k = 1, . . . , K and M = T/∆t
holds. To make the funds of different sizes comparable, capital distributions of all
j = 1, . . . , N sample funds are also standardized on the basis of each fund’s total invested
capital. P∆tk,j denotes the standardized capital distributions of fund j in the time interval
(tk−1, tk]. Cumulated capital distributions Pk,j of fund j up some time tk are given by
Pk,j =∑k
i=1 P∆ti,j .
From these definitions, the multiple Mj of fund j at the end of the lifespan T is given
by
Mj =K∑
i=1
P∆ti,j . (C.13)
An unbiased and consistent estimator for the long-run mean m is given by the sample
average, i.e.,
m =1
N
N∑
j=1
Mj . (C.14)
We now turn to the estimation of the coefficient α. This model parameter cannot
directly be observed from the capital distributions of the sample funds. However, it can
be estimated by using the CLS method introduced above. In this case the conditional
least squares estimator α minimizes the sum of squares
K∑
k=1
(Pk −E[Pk|Fk−1])2, (C.15)
40
where Pk =1N
∑Nj=1 Pk,j is the sample average of the cumulated distributions at time tk
and Fk−1 is the σ-field generated by P1, . . . , Pk−1.
By definition, the conditional expectation E[Pk|Fk−1] of the cumulated capital dis-
tributions is given by (see equation (1.12)):
E[Pk|Fk−1] = mC − (mC − Pk−1) exp[−1
2α(t2k − t2k−1)]. (C.16)
Substituting this condition into equation (C.15), the corresponding sum of squares to
be minimized is
K∑
k=1
Pk − mC + (mC − Pk−1) exp[−
1
2α(t2k − t2k−1)]
2
, (C.17)
where the conditional expectation E[Pk|Fk−1] is evaluated with m and tk = k∆t. An
estimate for the parameter α can then be found by a numerical minimization of (C.17).
In order to estimate the volatility of the capital distributions, σP , we first calculate
the variances of the log capital distributions in each time interval (tk−1, tk] by
σ2k = ln
[1
N
N∑
j=1
(P∆tk,j )
2
]− 2 ln
[1
N
N∑
j=1
P∆tk,j
]. (C.18)
The variance of the capital distributions, σ2P , can then be defined by the average of the
individual period variances, σ2k (k = 1, . . . , K), where weighting is done with the average
distributions that occur during the given time period:
σ2P =
K∑
k=1
1N
N∑j=1
P∆tk,j
mσ2k
. (C.19)
41
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46
Tables and Figures
Table 1: Model Parameters
This table gives the model parameters for the capital drawdowns and distributions of two differenthypothetical funds. The committed capital C of both funds is standardized to 1. The initial drawdownrates δ0 are assumed to be equal to their corresponding long-run means θ.
Model Drawdowns Distributions
κ θ σδ m α σP
Fund 1 2.00 1.00 0.50 1.50 0.03 0.20
Fund 2 0.50 0.50 0.70 1.50 0.06 0.30
47
Table 2: Descriptive Statistics
This table provides a descriptive overview on the fund data provided by Thomson Venture Economics(TVE). The complete data set includes 777 European private equity funds. In accordance with TVE,we use the following stage definitions: Venture capital funds (VC) represent the universe of ventureinvesting. It does not include buyout investing, mezzanine investing, fund of fund investing or sec-ondaries. Angel investors or business angels are also not included. Buyout funds (BO) represent theuniverse of buyout investing and mezzanine investing.
All Liquidated Funds Extended Sample
Number of FundsVC
absolute 456 47 102relative 58.69% 49.47% 50.25%
BOabsolute 321 48 101relative 41.31% 50.53% 49.75%
Totalabsolute 777 95 203relative 100.00% 100.00% 100.00%
48
Table 3: Drawdown and Distribution Process Parameters
This table shows the estimated (annualized) model parameters for the capital drawdowns and distri-butions of the 203 sample funds. Standard errors of the estimates are given in parentheses. Standarderrors of the estimated θ, κ and α coefficients are derived by a bootstrap simulation. Note that we setδ0 = 0 for all (sub-)samples.
Drawdowns Distributions
κ θ σδ m α σP
Total 7.3259 0.4691 4.7015 1.8462 0.0284 1.4152(5.8762) (0.1043) - (0.1355) (0.0007) -
VC 13.3111 0.4641 5.2591 2.0768 0.0230 1.4667(6.4396) (0.0869) - (0.2254) (0.0010) -
BO 4.9806 0.4797 4.5696 1.6133 0.0379 1.1966(2.9514) (0.1142) - (0.0833) (0.0006) -
49
Table 4: Market Derived Parameters
This table shows the estimated (annualized) model parameters for the riskless rate of return (rf ), theexpected return (µW ) and standard deviation (σW ) of the market portfolio, and the covariance (σPW )(correlation (ρPW )) between changes in log capital distributions and market returns.
Covariance/(Correlation)
Interest Rate Market Returns Total VC BO
rf µW σW σPW σPW σPW
Parameter 0.0587 0.1072 0.1507 0.0296 0.0433 0.0157(0.3082) (0.4411) (0.1739)
50
Table 5: Fund Value and Illiquidity Costs
The first panel of this table gives the present values V P0
and V D0
and the resulting fund values V F0
for the sample funds overall and broken down by venture capital versus buyout funds. The secondpanel gives the instantaneous illiquidity costs cill. These are derived implicitly by assuming that theex-ante excess values V F
0are compensation for holding an illiquid fund. Instantaneous illiquidity costs
are annualized and are given as a percentage of the committed capital of the funds.
Model Value Illiquidity Costs
V P0 V D
0 V F0 cill
All 1.0104 0.8663 0.1441 1.44%
VC 0.9744 0.8770 0.0974 0.98%
BO 1.0293 0.8541 0.1752 1.76%
51
Table 6: Value Dynamics over Time for Liquid and Illiquid Funds
This table illustrates the dynamics of fund values under liquidity (V Ft ), the corresponding fund values under illiquidity (V F,ill
t ) and the illiquiditydiscounts (Cill
t ) for all three (sub-)samples of funds. Note that unconditional expectations of the fund values and illiquidity discounts are shownfor all t > 0 for illustrative purposes.
All Funds VC Funds BO Funds
Year V Ft V
F,illt Cill
t V Ft V
F,illt Cill
t V Ft V
F,illt Cill
t
0 0.1441 0.0000 0.1441 0.0974 0.0000 0.0974 0.1752 0.0000 0.17521 0.4671 0.3270 0.1402 0.4668 0.3721 0.0947 0.4505 0.2801 0.17042 0.6886 0.5525 0.1360 0.7138 0.6218 0.0919 0.6351 0.4697 0.16543 0.8001 0.6684 0.1317 0.8491 0.7601 0.0890 0.7111 0.5510 0.16014 0.8293 0.7023 0.1270 0.9016 0.8157 0.0858 0.7069 0.5524 0.15445 0.7991 0.6771 0.1221 0.8933 0.8108 0.0825 0.6483 0.4999 0.14846 0.7293 0.6125 0.1169 0.8423 0.7633 0.0790 0.5582 0.4161 0.14217 0.6364 0.5251 0.1113 0.7630 0.6878 0.0752 0.4553 0.3200 0.13538 0.5338 0.4284 0.1054 0.6678 0.5965 0.0713 0.3536 0.2254 0.12829 0.4318 0.3326 0.0992 0.5664 0.4993 0.0670 0.2624 0.1418 0.120610 0.3375 0.2449 0.0926 0.4664 0.4038 0.0626 0.1865 0.0739 0.112611 0.2552 0.1696 0.0856 0.3732 0.3154 0.0578 0.1271 0.0231 0.104112 0.1868 0.1087 0.0782 0.2902 0.2374 0.0528 0.0833 0.0000 0.083313 0.1324 0.0622 0.0703 0.2192 0.1717 0.0475 0.0528 0.0000 0.052814 0.0908 0.0289 0.0619 0.1604 0.1186 0.0418 0.0325 0.0000 0.032515 0.0602 0.0072 0.0530 0.1133 0.0775 0.0358 0.0197 0.0000 0.019716 0.0375 0.0000 0.0375 0.0760 0.0466 0.0295 0.0108 0.0000 0.010817 0.0219 0.0000 0.0219 0.0478 0.0250 0.0228 0.0055 0.0000 0.005518 0.0114 0.0000 0.0114 0.0267 0.0111 0.0156 0.0025 0.0000 0.002519 0.0045 0.0000 0.0045 0.0113 0.0032 0.0080 0.0009 0.0000 0.000920 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
52
Table 7: Expected Return and Systematic Risk over Time
This table illustrates the dynamics of the beta coefficients (βF,t) and expected returns (Et[RFt ]) for all three (sub-)samples of funds. Expected
returns are stated annualized
All Funds VC Funds BO Funds
Year βF,t Et[RFt ] βF,t Et[R
Ft ] βF,t Et[R
Ft ]
0 9.14 50.18% 19.08 98.37% 4.06 25.57%1 3.00 20.44% 4.31 26.75% 1.65 13.89%2 2.08 15.94% 2.92 20.05% 1.17 11.52%3 1.75 14.35% 2.46 17.80% 0.98 10.65%4 1.59 13.59% 2.25 16.75% 0.90 10.21%5 1.50 13.16% 2.13 16.18% 0.85 9.97%6 1.45 12.90% 2.06 15.85% 0.81 9.82%7 1.41 12.73% 2.01 15.64% 0.80 9.73%8 1.39 12.61% 1.99 15.50% 0.78 9.67%9 1.37 12.53% 1.97 15.40% 0.77 9.63%10 1.36 12.47% 1.95 15.33% 0.77 9.60%11 1.35 12.42% 1.94 15.28% 0.76 9.58%12 1.34 12.38% 1.93 15.24% 0.76 9.55%13 1.33 12.33% 1.92 15.20% 0.75 9.51%14 1.32 12.28% 1.92 15.16% 0.73 9.42%15 1.30 12.19% 1.91 15.11% 0.69 9.22%
Average 2.04 15.78% 3.30 21.85% 1.08 11.10%
53
Table 8: Summary of Empirical Evidence on Systematic Risk
This table summarizes important empirical evidence on systematic risk (as measured by the beta coef-ficients) of venture and buyout investments. For studies that provide a range for the beta coefficients,average values are reported here.
Beta
Study Sample VC BO
Gompers and Lerner(1997)
The study examines a sample of 96 venture cap-ital investments
1.2 -
Jones and Rhodes-Kropf (2003)
Data set from 866 venture and 379 buyout fundsbetween 1980-1999
1.1 0.8
Ljungquist andRichardson (2003a)
The paper analyzes 19 venture and 54 buyoutfunds by one large LP raised from 1981 to 1993
1.1 1.1
Cochrane (2005) The paper analyzes 16,613 observations on 7,765startup firms over the period of 1987-2000
1.7 -
Ewens (2009) The data sample covering 1987-2007 with over55,000 financing events and 10,000 returns
2.4 -
Jegadeesh et al. (2009) Data samples of 24 publicly traded funds offunds (FoF) and 155 listed private equity funds(LPE) over 1994-2008
1.0(LPE)
0.7(FoF)
Woodward (2009) The data sample includes 51 observations afteradjustments, period 1996Q1 -2008Q3
2.2 1.0
Korteweg and Sorensen(2010)
Data sample of 61,356 investment rounds for18,237 companies over 1987- 2005
2.8 -
Driessen et al. (2012) Data sample of 686 mature VC and of 272 buy-out funds over 1980-2003
2.7 1.3
Franzoni et al. (2012) Data sample of 4,403 buyout investments withinvestments years 1975 to 2006
- 0.9
Ang et al. (2013) Data sample of 630 quasi-liquidated funds withvintage years 1992 to 2008
1.7 1.3
Buchner and Stucke(2014)
Data sample of 1,109 quasi-liquidated fundswith vintage years 1980 to 2001
2.8 2.7
54
0 20 40 60 800
0.05
0.1
0.15
0.2
0.25
Lifetime of the Fund (in Quarters)
Qua
rter
ly C
apita
l Dra
wdo
wns
0 20 40 60 800
0.2
0.4
0.6
0.8
1
Lifetime of the Fund (in Quarters)
Cum
ulat
ed C
apita
l Dra
wdo
wns
(a) Expected Quarterly Capital Drawdowns (Left) and Cumulated Capital Drawdowns(Right)
0 20 40 60 800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
Lifetime of the Fund (in Quarters)
Qua
rter
ly C
apita
l Dis
trib
utio
ns
0 20 40 60 800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Lifetime of the Fund (in Quarters)
Cum
ulat
ed C
apita
l Dis
trib
utio
ns
(b) Expected Quarterly Capital Distributions (Left) and Cumulated Capital Distribu-tions (Right)
0 20 40 60 80
−0.25
−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
Lifetime of the Fund (in Quarters)
Cum
ulat
ed N
et F
und
Cas
h F
low
s
0 20 40 60 80−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Lifetime of the Fund (in Quarters)
Cum
ulat
ed N
et F
und
Cas
h F
low
s
(c) Expected Quarterly Net Fund Cash Flows (Left) and Cumulated Net Fund CashFlows (Right)
Figure 1: Dynamics of Fund 1 Model expectations are obtained from equations(1.5) and (1.7). Unconditional expectations are plotted. Solid lines representexpectations, dotted lines represent expectations ± one standard deviation.
55
0 20 40 60 800
0.05
0.1
0.15
0.2
0.25
Lifetime of the Fund (in Quarters)
Qua
rter
ly C
apita
l Dis
trib
utio
ns
0 20 40 60 800
0.2
0.4
0.6
0.8
1
Lifetime of the Fund (in Quarters)
Cum
ulat
ed C
apita
l Dis
trib
utio
ns
(a) Expected Quarterly Capital Drawdowns (Left) and Cumulated Capital Drawdowns(Right)
0 20 40 60 800
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
Lifetime of the Fund (in Quarters)
Qua
rter
ly C
apita
l Dis
trib
utio
ns
0 20 40 60 800
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Lifetime of the Fund (in Quarters)
Cum
ulat
ed C
apita
l Dis
trib
utio
ns
(b) Expected Quarterly Capital Distributions (Left) and Cumulated Capital Distribu-tions (Right)
0 20 40 60 80−0.2
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
Lifetime of the Fund (in Quarters)
Qua
rter
ly N
et F
und
Cas
h F
low
s
0 20 40 60 80−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Lifetime of the Fund (in Quarters)
Cum
ulat
ed N
et F
und
Cas
h F
low
s
(c) Expected Quarterly Net Fund Cash Flows (Left) and Cumulated Net Fund CashFlows (Right)
Figure 2: Dynamics of Fund 2 Model expectations are obtained from equations(1.12) and (1.13). Unconditional expectations are plotted. Solid lines repre-sent expectations, dotted lines represent expectations ± one standard devi-ation.
56
0
0.2
0.4
0.6
0.8
1.0
0
5
10
15
20−0.5
0
0.5
1
Correlation ρLifetime of the Fund (in Years)
Val
ue
−0.2
0
0.2
0.4
0.6
0.8
Figure 3: Unconditional Market Values Unconditional expectations of market val-ues over fund lifetime for varying values of the correlation coefficient ρPW .The model parameters are: C = 1, Tc = Tl = 20, rf = 0.05, κ = 0.5, θ = 0.5,σδ = 0.1, δ0 = 0.01, m = 1.5, α = 0.025, σP = 0.5 and σW = 0.2.
57
0 10 20 30 40 50 60 70 80−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Lifetime of the Fund (in Quarters)
Val
ue
Value of the Liquid FundValue of the Illiquid Fund
Figure 4: Liquidity versus Illiquidity Unconditional expectations of liquid fundand corresponding illiquid fund values. The model parameters are: C = 1,Tc = Tl = 20, rf = 0.05, κ = 0.5, θ = 0.5, σδ = 0.1, δ0 = 0.01, m = 1.5,α = 0.025, σP = 0.5 and σPW = 0.
58
0 5 10 15 200
0.1
0.2
0.3
0.4
0.5
Lifetime of the Fund (in Years)
Yea
rly C
apita
l Dra
wdo
wns
0 5 10 15 200
0.2
0.4
0.6
0.8
1
Lifetime of the Fund (in Years)
Cum
ulat
ed C
apita
l Dra
wdo
wns
(a) Yearly Capital Drawdowns (Left) and Cumulated Capital Drawdowns (Right)
0 5 10 15 200
0.05
0.1
0.15
0.2
0.25
Lifetime of the Fund (in Years)
Yea
rly C
apita
l Dis
trib
utio
ns
0 5 10 15 200
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Lifetime of the Fund (in Years)
Cum
ulat
ed C
apita
l Dis
trib
utio
ns
(b) Yearly Capital Distributions (Left) and Cumulated Capital Distributions (Right)
0 5 10 15 20−0.5
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
Lifetime of the Fund (in Years)
Yea
rly N
et C
ash
Flo
ws
0 5 10 15 20−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Lifetime of the Fund (in Years)
Cum
ulat
ed N
et C
ash
Flo
ws
(c) Yearly Net Fund Cash Flows (Left) and Cumulated Net Fund Cash Flows (Right)
Figure 5: Model Expectations and Observations Model expectations are plottedas compared to historical observations for all N = 203 sample funds. Solidlines represent model expectations, dotted lines represent historical observa-tions.
59
