A Dynamic Model of Private Asset Allocation

Abstract

We build a state-of-the-art dynamic model of private asset allocation that considers five key features of private asset markets: (1) the illiquid nature of private assets, (2) timing lags between capital commitments, capital calls, and eventual distributions, (3) time-varying business cycle conditions, (4) serial correlation in observed private asset returns, and (5) regulatory constraints on certain institutional investors' portfolio choices. We use cutting-edge machine learning methods to quantify the optimal investment policies over the life cycle of a fund. Moreover, our model offers regulators a tool for precisely quantifying the trade-offs when setting risk-based capital charges.

Full text

A Dynamic Model of Private Asset Allocation∗
Hui Chen†Giovanni Gambarotta‡Simon Scheidegger§Yu Xu¶
March 4, 2025
Abstract
We build a state-of-the-art dynamic model of private asset allocation that considers
five key features of private asset markets: (1) the illiquid nature of private assets, (2)
timing lags between capital commitments, capital calls, and eventual distributions, (3)
time-varying business cycle conditions, (4) serial correlation in observed private asset
returns, and (5) regulatory constraints on certain institutional investors’ portfolio choices.
We use cutting-edge machine learning methods to quantify the optimal investment
policies over the life cycle of a fund. Moreover, our model offers regulators a tool for
precisely quantifying the trade-offs when setting risk-based capital charges.
JEL classification: C63, G11, G23.
Key words: alternative assets, business cycle, liquidity, machine learning, portfolio choice,
private equity, return smoothing, risk-based capital.
∗We thank Liberty Mutual Investments for generous support on this project.
†MIT Sloan and NBER, huichen@mit.edu
‡Liberty Mutual Investments, giovanni.gambarotta@lmi.com
§Department of Economics, HEC Lausanne, simon.scheidegger@unil.ch
¶Lerner College of Business and Economics, University of Delaware, yuxu@udel.edu
arXiv:2503.01099v1 [q-fin.PM] 3 Mar 2025

1 Introduction
Recent estimates put the size of global private markets’ assets under management at $13.1
trillion as of June 30, 2023 (McKinsey & Company,2024). A multitude of factors make
private asset allocation decisions particularly challenging. First, the illiquid nature of private
assets means that it takes considerable time and costs to establish, rebalance, and exit from
a position, making careful long-term planning an integral part of private asset management.
Second, this illiquidity can lead to mark-to-market induced serial correlation in observed
private asset returns, which further complicates dynamic allocation decisions. Third, private
market investments require investors to first commit capital and subsequently manage the
liquidity needs arising from future capital calls. Fourth, some institutional investors such as
insurance companies face additional regulatory constraints on their allocation choices. Fifth,
investors must take all of these factors into account as market and business cycle conditions
constantly change.
We develop a state-of-the-art dynamic model of private asset allocation. It captures
(1) the illiquidity inherent in private asset investing, (2) the lag between initial capital
commitments, subsequent capital calls, and eventual distributions, (3) time-varying market
and business cycle conditions, (4) serial correlation in observed alternative asset returns, and
(5) regulation-induced portfolio constraints. The interplay among these factors introduces
substantial nonlinearities and expands the state space, rendering the model numerically
intractable with traditional solution methods. To address this challenge, we use advanced
machine learning techniques—Deep Kernel Gaussian Processes (Wilson, Hu, Salakhutdinov,
and Xing,2016)—to solve the underlying dynamic programming problem. The solution
quantifies the optimal private asset allocation policies over the life cycle of a fund, and
accounts for the multitude of factors described above. To the best of our knowledge, (i) our
model is the most comprehensive private asset allocation model to date in terms of the range
of factors considered, and (ii) we are the first to use Deep Kernel Gaussian Processes in the
context of solving dynamic models in economics and finance.
We solve a dynamic portfolio optimization problem for a risk-averse limited partner (LP)
investor with a fixed (but long) horizon. The LP chooses among stocks, bonds, and a private
equity (PE) fund, subject to various constraints. To invest in the PE fund, the LP must first
make capital commitments before capital is called and distributed from the PE fund. The
illiquid nature of PE assets makes liquidity management central: the LP must finance capital
calls out of its liquid wealth (i.e., stocks and bonds). Default occurs if the LP does not have
sufficient liquid wealth to meet a capital call, in which case the LP’s PE stake is sold at a
1

discount on the secondary market and the LP loses future access to the PE fund. In addition
to such liquidity-driven defaults, we also allow for strategic defaults on capital calls.
Our model further incorporates three realistic features that make PE investing challenging.
First, a unique feature of alternative investments is the serial correlation in observed returns
(Geltner,1991;Getmansky, Lo, and Makarov,2004). As Getmansky, Lo, and Makarov
(2004) argue, sparsely-traded illiquid assets can lead to reported PE fund returns that appear
smoother than their true economic counterparts, owing to mark-to-market practices. To
capture this effect, we embed the Getmansky, Lo, and Makarov (2004) return-smoothing
model into our PE return structure, allowing observed PE returns to depend on past lags of
true economic returns. Consequently, the LP must make allocation decisions in the presence
of mark-to-market induced smoothing, as is the case in reality.
Second, key determinants of PE investing depend on business cycle conditions. For
example, PE returns, capital calls, and distributions can all vary over the business cycle (see,
e.g., Neuberger Berman 2022). We capture these time-varying conditions by modeling a
macroeconomic state that evolves according to a Markov chain. All our model parameters,
including those governing PE returns, call rates, and distributions, can depend on this
macroeconomic state. Consequently, our framework provides guidance for optimal PE
allocations over different phases of the business cycle.
Third, many PE investors are institutional investors that must comply with regulatory
constraints on their asset holdings. For example, U.S. insurers’ portfolio allocations are
governed by Risk-Based Capital (RBC) standards overseen by the National Association of
Insurance Commissioners (NAIC), while European insurers are subject to analogous RBC
requirements under Solvency II (see Eling and Holzmuller 2008 for an overview of RBC
standards). Violations of these requirements can be extremely costly because they trigger
regulatory interventions: in severe cases, an insurer may be placed under regulatory control,
which can lead to rehabilitation or even liquidation. We capture these regulatory constraints
by imposing a risk budget on the LP’s portfolio, applying separate risk charges to stocks,
bonds, and PE. As long as the LP’s total risk charge remains below a specified threshold, no
costs arise; however, costs escalate once that threshold is breached. This modeling approach
aligns with risk-charge frameworks commonly used by rating agencies to assess insurers’ RBC
adequacy (see, e.g., S&P Global 2023).
The comprehensive set of factors considered in our model makes it challenging to solve the
LP’s dynamic portfolio optimization problem for two main reasons. First, the interactions
among these factors introduce substantial nonlinearities that must be resolved to obtain
accurate allocation policies. Second, they enlarge the state space, making traditional methods
2

infeasible due to their vulnerability to the curse of dimensionality.
To overcome these challenges, we employ Deep Kernel Gaussian Processes (DKGPs) to
track the LP’s value and policy functions within our dynamic optimization framework. As
proposed by Wilson, Hu, Salakhutdinov, and Xing (2016), DKGPs combine the strengths
of Gaussian Processes (GPs) and Neural Networks (NNs). While both GPs and NNs can
address high-dimensional challenges, GPs are generally more cost-effective to train since they
involve fewer hyperparameters. This is especially relevant in our setting where each sample
point requires solving a constrained optimization problem with multi-dimensional controls
and adaptive quadrature to accurately compute expectations. However, standard GPs often
lack the flexibility of NNs in capturing complex nonlinear behavior. DKGPs resolve this
issue by embedding NNs into GPs, thereby harnessing the ability to model highly nonlinear
dynamics while reducing the need for expensive sample points. While we use DKGPs to solve
our model, this methodology may be of independent interest, as many models in economics
and finance face similar challenges.
Our calibration assumes a 10-year investment horizon for the LP, and we use private
equity (PE) data from Liberty Mutual Investments to calibrate our model. The calibrated
model yields several quantitative insights into optimal asset allocation over a fund’s life
cycle—from the initial stage, where no PE investments are held, to the transition phase,
during which the LP ramps up its PE exposure, and finally to the maintenance stage, where
the desired balance between PE and public investments is achieved.
First, the optimal allocation policy unfolds as follows. In the early years, the LP
aggressively commits new capital to build up the fund, taking into account the various factors
described above to ensure a smooth transition. At the same time, the LP adjusts its liquid
portfolio by initially allocating heavily to stocks to achieve the desired overall aggregate risk
exposure as PE investments ramp up. As PE commitments are called and the PE share
of the portfolio increases, the LP gradually reduces stock exposure. This transition phase
lasts approximately 4–5 years, after which the LP enters a maintenance phase, consistently
managing the portfolio to preserve the optimal asset mix. Notably, under the optimal policy,
the cumulative default rate is only 0.1% over the ten-year investment horizon.
Although similar outcomes are observed in practice, the complexity of the problem has
traditionally led practitioners to rely on heuristics. For instance, standard approaches often
use static mean-variance analysis combined with heuristic adjustments for capital commitment
lags (see, e.g., Takahashi and Alexander 2002). Our contribution is to rigorously quantify
the optimal policies in a realistic setting, moving beyond rules of thumb to help PE investors
achieve improved outcomes. Indeed, we demonstrate that the optimal portfolio allocations
3

obtained under our fully dynamic framework can differ dramatically from those derived using
heuristic methods.
Second, the optimal allocation accounts for business cycle conditions as follows. To first
order, the optimal allocation maintains the intended transition profile for the PE portfolio,
as if short-term business cycle fluctuations did not exist. This is because adjusting PE
allocations can be prohibitively costly given their illiquidity, commitment lags, and long
planning horizons. Instead, the LP modulates overall risk exposure over the business cycle
by adjusting its public stock holdings, lowering exposure during recessions when necessary.
Since transaction costs for public stocks are negligible in comparison, this approach offers a
cost-effective means of managing risk over the business cycle.
Failing to account for business cycles is extremely costly. To quantify these costs, we
compare the outcomes for a naive LP that ignores business cycle variation against outcomes
under the optimal policy. The naive LP adopts overly aggressive PE allocations, leading to a
dramatic increase in default risk: the cumulative default frequency is 13.6% for the naive
approach, compared to only 0.1% under the optimal policy. Ex-ante, this suboptimal strategy
translates to a 9.3% loss in initial wealth in certainty-equivalent terms.
Third, we examine whether it is necessary to “unsmooth” returns before making allocation
decisions. Prior research has argued that mark-to-market accounting induces serial correlation,
making observed alternative asset returns appear smoother than their underlying true
economic performance (Getmansky, Lo, and Makarov,2004). This smoothing can cause
private assets’ true risk to be underestimated and, consequently, lead to suboptimal portfolio
allocations. As a result, some have advocated unsmoothing returns prior to drawing portfolio
inferences.
Our model revisits this issue from the perspective of a long-term PE investor and
demonstrates that the need to unsmooth returns may be a moot point for such an investor.
The reasons are as follows. First, long-term investors care primarily about the characteristics
of long-horizon returns, which are less affected by short-term mark-to-market fluctuations.
Second, even if serial correlation is an inherent feature of true PE returns, its benefits may
be negated once realistic implementation lags and adjustment costs are considered. Indeed,
in our baseline calibration—where observed PE returns have a quarterly autocorrelation
of 0.2—we observe little difference in outcomes between a LP that optimizes its portfolio
directly using the raw returns and one that first unsmooths the return series to remove
autocorrelation while preserving the long-run moments.
Fourth, our model can be used to evaluate the impact of varying risk charges on the
LP’s portfolio outcomes by quantifying the associated risk–return trade-off. In our baseline
4

calibration, we impose a 50% risk charge on both public and private equities, reflecting
representative industry figures (see, e.g., S&P Global 2023, Table 14). Because these charges
differ across asset classes, we also consider a scenario in which the risk charge is increased
to 100%, near the upper end of observed ranges. Under this higher risk charge scenario,
the calibrated model indicates that long-run realized returns decline from 8.4% to 7.1%,
while long-run realized volatility decreases from 2.78% to 2.35%.
1
These findings underscore
how our framework can help both institutional investors and regulators assess the cost and
benefits of alternative risk-charge levels.
Related literature. We contribute to two strands of literature. First, we advance the
research on optimal private asset allocation (see, e.g., Korteweg and Westerfield (2022) for a
recent survey). Early industry approaches to estimating exposures and cash flows include
Takahashi and Alexander (2002). More recent work has focused on dynamic models with
stochastic shocks, and highlight two key features of private asset investing: the inherent
illiquidity of these assets (Ang, Papanikolaou, and Westerfield,2014;Sorensen, Wang, and
Yang,2014;Dimmock, Wang, and Yang,2024) and the delays between capital commitments,
calls, and distributions (Giommetti and Sorensen,2024;Gourier, Phalippou, and Westerfield,
2024). We build on these prior works by incorporating additional factors that pose significant
challenges for PE investors, including time-varying business cycle conditions, serial correlation
in observed PE returns, and regulation-induced portfolio constraints. To the best of our
knowledge, our model is the most comprehensive stochastic model of private asset allocation
to date.
Second, we contribute to a growing literature that uses machine learning methods to
tackle portfolio choice problems featuring higher dimensions, realistic trading frictions,
and complex constraints. Most of this work has focused on liquid asset markets such as
stocks and government bonds (see, e.g., Gaegauf, Scheidegger, and Trojani 2023;Duarte,
Duarte, and Silva 2024), leaving alternative asset classes like private equity understudied in
comparison—despite their rising importance in institutional portfolios. Our contribution is
to extend machine learning techniques to the alternative asset allocation setting, enabling us
to characterize optimal private asset allocation under a range of realistic features. To the
best of our knowledge, we are the first to deploy Deep Kernel Gaussian Processes for solving
dynamic models in economics and finance. This methodological advance is of independent
1
Note that the seemingly low standard deviation is because we are reporting the standard deviation of
annualized long-horizon returns over the LP’s ten year investment horizon. For example, if annual returns
rt`k
are iid with a volatility of
σ1
, then the standard deviation of annualized returns over a horizon of
H
years is σ`1
Hprt`1`rt`2`... `rt`Hq˘“σ1{H.
5

interest, as similar challenges frequently arise in various settings in economics and finance.
For example, Duarte, Fonseca, Goodman, and Parker (2022) apply neural networks to a
portfolio choice problem in the household context that incorporates multiple realistic features.
While their methodology characterizes optimal portfolio choice under full commitment, our
approach instead characterizes the optimal time-consistent policy.
2 Model
Time is discrete and runs from
t“
0to the terminal date
t“T
. The limited partner (LP)
investor maximizes utility over terminal wealth WT,
upWTq “ W1´γ
T
1´γ,(1)
where
γ
is the coefficient of relative risk aversion. At each time period
t
, the LP can invest in
a risk-free bond, a public stock index, and private equity (PE). The timing of the decisions is
illustrated in Figure 1and is described below.
Time-varying macroeconomic conditions. The LP’s investment opportunities set varies
over the business cycle. We capture time-varying business cycle conditions through a Markov
process
stP t
1
,
2
u
that takes two values corresponding to recessions (
st“
1) and booms
(
st“
2). We denote by
pss1“probpst`1“s1|st“sq
the probability of transitioning from
state sto state s1.
Liquidity constraint. The LP enters period
t
with liquid wealth
Wtě
0, uncalled PE
commitments
Ktě
0, and illiquid wealth
Ptě
0. The latter corresponds to the net asset
value (NAV) of the LP’s previously called PE investments. The LP then makes three choices:
(1) new PE commitments
Ntě
0, and its allocations to (2) stocks
St
and (3) bonds
Bt
. These
choices are subject to the following liquidity constraint:
St`Bt`γNpWt`PtqˆNt
Wt`Pt´n˙2
`γSpWt`PtqˆSt
Wt`Pt´s˙2
“Wt(2)
where
γNpWt`Ptq´Nt
Wt`Pt´n¯2
and
γSpWt`Ptq´St
Wt`Pt´s¯2
are adjustment costs on
new PE commitments and stocks, respectively.
2
That is, stock and bond allocations and
2In Appendix B, we show that the problem is homogeneous in total wealth Wt`Pt. The forms of these
adjustment costs conveniently preserves this homogeneity property.
6

tt`1
Liquid wealth: Wt
Illiquid wealth: Pt
Uncalled commitments: Kt
Macroeconomic conditions: st
Expected returns on PE: µP,t
Allocation to stocks
and bonds:
Stand Bt
New commitments
to PE: Nt
PE, stock, and
bond returns:
RP,t`1, RS,t`1, Rfpstq
PE Distribution:
λDpst`1qPtRP,t`1
Capital calls:
λKpst`1qKtand λNpst`1qNt
Default on capital calls?
Figure 1: Timing of the Model
adjustment costs are paid out of liquid wealth.
Additionally, we show in Appendix B that solvency concerns rule out shorting so that
Btě0, Stě0.(3)
Risk budget. The LP’s overall portfolio is subject to internal risk controls which we model
through risk weights. Specifically, the risk weight of the LP’s overall portfolio is defined as
θt“
θBBt`θS„St`γSpWt`Ptq´St
Wt`Pt¯2ȷ`θPPt
Bt`St`γSpWt`Ptq´St
Wt`Pt¯2
`Pt
(4)
The interpretation of the risk weight
(4)
is as follows. The LP’s combined portfolio consists
of
Bt
invested in bonds,
St`γSpWt`Ptq´St
Wt`Pt¯2
in stocks (including stock adjustment
costs), and
Pt
in PE. Bonds carry a risk weight of
θB
, while stocks and PE carry risk weights
of
θS
and
θP
, respectively. The risk weight
(4)
is the value-weighted average over the risk
weights of the three classes of investments in the LP’s portfolio. We assume the risk weights
are ordered as follows:
0“θBăθSďθP.(5)
That is, risk-free bonds carry a zero risk weight as they are risk free while public stocks and
PE carry positive risk weights. The risk weight for PE is no lower than that of public stocks.
The LP incurs a cost if it violates its risk budget. Specifically, the cost of violating its
7

risk budget equals pWt`PtqˆΓpθtqwhere the proportional cost equals
Γpθtq “ κ`θt´θ˘21tθtąθu.(6)
That is, no costs are assessed if the LP’s risk weight
(4)
falls below the threshold
θ
; quadratic
costs are assessed above the threshold
θ
. Without loss of generality, we can take the threshold
to be
θ“
1.
3
The risk budget cost is paid during period
t`
1in a manner we describe below.
Our modeling of the risk budget through equations
(4)
and
(6)
is in line with regulatory
constraints faced by institutional investors. For example, U.S. insurers’ exposure to risky assets
are subject to Risk-Based Capital (RBC) standards developed by the National Association
of Insurance Commissioners (NAIC); similarly, the European Solvency II directive imposes
RBC requirements on European insurers (see Eling and Holzmuller 2008 for an overview of
Risk-Based Capital standards). Such RBC requirements are also reflected in rating agencies’
assessments of insurers’ RBC adequecy (see, e.g., S&P Global 2023). The risk-weights
θB
,
θS
, and
θP
in our setting capture risk charges used to calculate RBC adequecy. The cost
parameter
κ
can be interpreted as the shadow cost of the contribution of an insurer’s PE
investments to its overall RBC adequency.
Laws of motion. New PE commitments affect future uncalled commitments through the
law of motion
Kt`1“Kt`Nt´ rλNpst`1qNt`λKpst`1qKts
loooooooooooooooomoooooooooooooooon
capital calls
.(7)
New commitments
Nt
are added to the outstanding pool of uncalled commitments
Kt
before
capital calls are subsequently made. A fraction
λNpst`1q
and
λKpst`1q
of new and previously
uncalled commitments are called at the start of the next period
t`
1, respectively. Both
fractions depend on next period’s business cycle conditions through
st`1
. The amount of
outstanding PE commitments at the start of period
t`
1,
Kt`1
, is then given by equation
(7)
.
The next period’s illiquid wealth is determined through the law of motion
Pt`1“PtRP,t`1´λDpst`1qPtRP,t`1`λKpst`1qKt`λNpst`1qNt.(8)
The NAV of the LP’s PE investments at the start of period
t
,
Pt
, earns risky gross returns
RP,t`1
so that
PtRP,t`1
is the updated NAV. Two adjustments are subsequently made. First,
a fraction
λDpst`1q
of the updated NAV is distributed to the LP by its PE funds; these
3We can always rescale κand the risk weights θSand θPappropriately to effectively enforce θ“1.
8

distributions depend on business cycle conditions. Second, the NAV increases by the LP’s
new PE contributions which equals capital calls λNpst`1qNt`λKpst`1qKt.
The LP’s liquid wealth evolves as follows:
Wt`1“λDpst`1qRP,t`1Pt`RS,t`1St`RfpstqBt
´λKpst`1qNt´λNpst`1qNt´ pWt`PtqΓpθtq.(9)
The next period’s liquid wealth
Wt`1
consists of distributions from its PE funds totaling
λDpst`1qRP,t`1Pt
and the proceeds from the LP’s investments in public stocks
RS,t`1St
and
risk-free bonds
RfpstqBt
. Here,
RS,t`1
and
Rfpstq
denote the gross risky return on stocks
and the gross risk-free return on bonds, respectively. Additionally,
Wt`1
is decreased by
capital calls totaling λKpst`1qNt`λNpst`1qNtand risk budget costs pWt`PtqΓpθtq.
Finally, we assume that illiquid wealth can be converted into liquid wealth one-for-one at
the terminal date t“T, and that any uncalled commitments at t“Tare discarded.
Default and liquidation. If the investor does not have sufficient liquid wealth to meet
capital calls at
t`
1, it is considered a “default.” This occurs if the realized return shocks
at
t`
1are such that next period’s liquidity
(9)
is negative:
Wt`1ă
0. In case of default,
all uncalled commitments are written off, the LP’s PE holdings are liquidated at a discount
αpst`1q ă
1, and the investor loses access to PE investments in the future. That is, the LP
enters default with only liquid wealth totaling
WD,t`1“λDpst`1qRP,t`1Pt`RS,t`1St`RfpstqBt
´ pWt`PtqΓpθD,tq`αpst`1q r1´λDpst`1qsRP,t`1Pt.(10)
The differences between the post-default liquid wealth
(10)
and its counterpart in the absence
of default
(9)
are as follows. First, capital calls are not paid should the LP choose to default.
Second, the liquid wealth
(10)
includes the proceeds from liquidating the LP’s PE holdings,
αpst`1qr1´λDpst`1qsRP,t`1Pt
. Third, since PE holdings are liquidated and no longer a part
of the LP’s portfolio at t`1, the risk weight (4) is modified accordingly in default:
θD,t “
θBBt`θS„St`γSpWt`Ptq´St
Wt`Pt¯2ȷ
Bt`St`γSpWt`Ptq´St
Wt`Pt¯2
`Pt
.(11)
9

That is, compared to the risk weight
(4)
, the risk weight in default
(11)
does away with the
θPPt
term in the numerator. The risk cost
(6)
is then calculated using the risk weight in
default (11).4
The LP can also choose to strategically default. Strategic default occurs when the LP has
sufficient liquidity
Wt`1
at
t`
1to meet capital calls, but chooses not to do so. The LP’s
decision to strategically default is endogenously determined as part of the LP’s optimization
problem and is characterized in Section 2.1.
Asset returns. The gross return at
t`
1for the risk-free bond
Rfpstq
depends on the
macroeconomic state at time t,st, whose dynamics follows a first-order Markov chain.
The gross returns for private equity
RP,t`1
and the public stock index
RS,t`1
follow a
log-normal distribution:
log «RP,t`1
RS,t`1ff„N˜« µP,t
µSpstqff,«σPpstq2ρpstqσPpstqσSpstq
ρpstqσPpstqσSpstqσSpstq2ff¸.(12)
Public stocks have expected return
µSpstq
, volatility
σSpstq
, and correlation
ρpstq
with private
equity returns; all of these quantities depend on macroeconomic conditions through st.
The expected return to investing in private equity follows the law of motion
µP,t “ϱP,1µP,t´1`ϱP ,2log RP,t `νPpstq,(13)
where
νPpstq
captures business cycle variation in private equity returns. The specification
(13) allows for serial correlation in private equity returns—a hallmark feature of alternative
investments (see, e.g., Getmansky, Lo, and Makarov 2004). To see this, write
log RP,t “
µP,t´1`σPpst´1qεP,t
where
εP,t ” plog RP,t ´µP,t´1q {σPpst´1q „ Np0,1q
is a standard normal
shock. Equation (13) can then be expressed as
µP,t “ pϱP,1`ϱP,2qµP ,t´1`ϱP,2σPpst´1qεP,t `νPpstq(14)
from which we see
ϱP,1`ϱP,2
is the autocorrelation coefficient for
µP,t
. In Appendix A, we
show that equation
(14)
can be motivated through the Getmansky, Lo, and Makarov (2004)
model of smoothed returns for illiquid alternative investments.
4This is necessary for utilities to be well-defined—see Appendix B for details.
10

2.1 Recursive Formulation
The problem has six state variables: liquid wealth
W
, illiquid wealth
P
, uncalled PE
commitments
K
, expected PE returns
µP
, the macroeconomic state
s
, and time
t
. In the
notation that follows, we drop time subscripts and use a prime to denote variables for the
next period (e.g., W1denotes Wt`1).
The problem after defaulting can be expressed recursively as follows:
VDpt, W, sq “ max
Sě0,Bě0
ErVDpt`1, W 1, s1q|W, ss(15)
subject to
W1“R1
SS`RfpsqB´WΓpθq,
θ“
θBB`θS”S`γSW`S
W˘2ı
B`S`γSW`S
W˘2,
W“S`B`γSWˆS
W˙2
,
with the terminal condition being
VDpT, W, sq “ W1´γ
1´γ.
Note that the problem after defaulting
(15)
does not depend on
K
,
P
, and
µP
because the
LP loses access to PE investing after defaulting.
The problem before defaulting is given by
Vpt, W, P, K, µP, sq(16)
“max
Ně0,Sě0,Bě0
Ermax tVpt`1, W 1, P 1, K 1, µ1
P, s1q, VDpt`1, W 1
D, s1qu|W, P, K, µP, s s
subject to
W1“λDps1qR1
PP`R1
SS`RfpsqB´λKps1qK´λNps1qN´ pW`PqΓpθq,
P1“ r1´λDps1qsR1
PP`λKps1qK`λNps1qN,
K1“ r1´λKps1qsK` r1´λNps1qsN,
µ1
P“ϱP,1µP`ϱP,2log R1
P`νPps1q,
11

W“S`B`γNpW`PqˆN
W`P´n˙2
`γSpW`PqˆS
W`P˙2
,
θ“
θBB`θS”S`γSpW`Pq`S
W`P˘2ı`θPP
B`S`γSpW`Pq`S
W`P˘2`P,
W1
D“ rλDps1q ` αps1qp1´λDps1qqsR1
PP`R1
SS`RfpsqB´ pW`PqΓpθDq,
θD“
θBB`θS”S`γSpW`Pq`S
W`P˘2ı
B`S`γSpW`Pq`S
W`P˘2`P,
and the terminal condition
VpT, W, P, K, µP, sq “ pW`Pq1´γ
1´γ.
We additionally impose the boundary condition
Vpt, W 1, P 1, K1, µ1
P, s1q “ ´8
if
W1ă
0
since the LP has no option but to default if it does not have sufficient liquid wealth to
meet capital calls. Finally, note that the
max
term in the objective function of problem
(16)
captures the LP’s strategic default decision—even with sufficient liquidity to meet capital
calls, the LP can choose to default if the value of defaulting next period is higher than the
value of not defaulting.
Scaling. The value functions
(15)
and
(16)
are homogeneous in total wealth
W`P
and
can be scaled as follows:
VDpt, W, sq “ rW vDpt, sqs1´γ
1´γ(17)
Vpt, W, P, K, µP, sq “ rpW`Pqvpt, w, k, µP, sqs1´γ
1´γ(18)
where
w”W{pW`Pq
is the liquid fraction of the LP’s total wealth and
k”K{pW`Pq
is
the LP’s uncalled commitments relative to its total wealth.
The scaled value function
v
represents the certainty equivalent values for terminal wealth
per unit of current total wealth, while the scaled value function
vD
provides the certainty
equivalent values conditional on the LP being in default. Both scaled value functions are
characterized by Bellman equations, as summarized in Appendix B. We solve these scaled
value functions using the numerical techniques described in Section 3; Appendix Cprovides
further details.
12

3 Solution Technique
Our goal is to compute the global solution for the scaled value functions outlined towards
the end of Section 2.1. This requires us to track the value function
vpt, w, k, µP, sq
over
time; unfortunately, the large number of state variables make it challenging to do so using
traditional methods. We instead use Machine Learning methods—Deep Kernel Gaussian
Process dynamic programming —to make the problem computationally feasible. We outline
the key steps below and refer readers to Rasmussen and Williams (2005) for a textbook
treatment of Gaussian Processes (GPs), and Scheidegger and Bilionis (2019) and Renner and
Scheidegger (2018) for an introduction to using GPs in a dynamic programming context.
To the best of our knowledge, we are the first to use Deep Kernel GPs in the context of
solving dynamic models in economics and finance.
3.1 Gaussian Process Regression
Let
f
:
RDÑR
be a multivariate function of interest. In our context,
fp¨q
is either a
value function or a policy function. We can measure
fpxq
for input
xPRD
by querying an
information source. We allow for noise in the information source: we measure
y“fpxq ` ϵ
where ϵ„Np0, σ2
nqcaptures measurement noise.
In our setting, the information source is computer code that solves an optimization problem
at point
x
in the state space. We make queries at
N
sample points
X“␣xp1q, ..., xpNq(
and observe the corresponding measurements
y“␣yp1q, ..., ypNq(
. Computational constraints
limit the number
N
of measurements that we can make as individual function evaluations are
computationally expensive. The key idea of Gaussian Process Regression (GPR) is to replace
the computationally expensive function
fp¨q
with a cheap-to-evaluate surrogate learned from
the training inputs Xand training targets y.
GPR constructs the surrogate as follows. Before making any queries, we model our prior
knowledge of fp¨q by assigning it a GP prior:
fpxq „ GP pmpx;θq, kpx,x1;θqq.
That is,
fpxq
is a GP if any finite collection of function values
tfpx1q, ..., f pxMqu
has a
joint Gaussian distribution with mean function
mpx
;
θq “ Erfpxqs
and covariance kernel
kpx,x1
;
θq “ Erpfpxq´mpx
;
θqqpfpx1q´mpx1
;
θqqs
. The vector
θ
denotes the hyperparameters
of the GP model which must be estimated.
13

A typical specification for the mean function is simply a constant,
mpx;θq “ m0,(19)
while an often-used kernel is the Matern 5/2 kernel. The automatic relevance detection
(ARD) Matern 5/2 kernel is defined by
kmatern52px,x1q “ σ2
fˆ1`?5r`5
3r2˙exp ´´?5r¯,(20)
where
r“g
f
f
eD
ÿ
d“1
pxd´x1
dq2
ℓ2
d
and
xd
is the value of
x
along dimension
d
. The parameter
σf
modulates the output
amplitude of the GP and is referred to as the signal standard deviation. The parameter
ℓd
is
the characteristic lengthscale of along dimension
d
; it captures how quickly the GP changes
as the input changes along dimension d. The resulting hyperparameters under specification
(19) and (20) are θ“ rm0, σf, ℓ1, ..., ℓDs.
We use the training data
X
and
y
to update our beliefs regarding
fpxq
. Specifically, the
posterior for fpxqis still a GP with posterior mean and variance given by
r
mpx;θq “ mpx;θq ` Kpx,X;θqT“K`σ2
nIN‰´1py´mq,(21)
and V ar pfpxq|X,y;θq “ kpx,x;θq ´ Kpx,X;θqT“K`σ2
nIN‰´1Kpx,X;θq,(22)
respectively. Here,
Kpx,X
;
θq
denotes a column vector with
i
th entry
kpx,xpiq
;
θq
,
K
denotes a
NˆN
matrix with
ij
th entry
kpxpiq,xpjq
;
θq
,
IN
denotes the
NˆN
identity
matrix, and mis a column vector with ith entry mpxpiq;θq.
The posterior mean
(21)
serves as the cheap-to-evaluate surrogate for
fpxq
that we
seek. Finally, the hyperparameters
θ
must be carefully chosen to ensure that the surrogate
accurately represents
fpxq
. This is done by maximizing the marginal likelihood for the
training data:
log ppy|Xq“´1
2yT“K`σ2
nIN‰´1y´1
2log ˇˇK`σ2
nINˇˇ´N
2log p2πq.(23)
Figure 2illustrates a GPR in a one-dimensional setting in which the true function is
fpxq “ xsinpxq
. The example is for
N“
5observations with observation noise
σn“
0
.
01.
The true function is modeled as a GP with specification
(19)
and
(20)
for the mean and
14

Figure 2: Illustration of Gaussian Process Regression. The dotted is the true function
while the solid line is the posterior mean of the fitted GP. The shaded region plots the 95% confidence
interval.
covariance kernel, respectively. The posterior mean is calculated according to equation
(21)
and is shown in the solid line. The shaded region plots the 95% confidence interval around
the posterior mean and is computed based on the posterior variance (22).
3.2 Deep Kernel Gaussian Process
We use the Deep Kernel Gaussian Process (DKGP) proposed by Wilson, Hu, Salakhutdinov,
and Xing (2016). A DKGP is a GP whose covariance kernel embeds a neural network
component within a “standard” kernel as follows:
kDKpx,x1;θ,wq “ kpN N px;wq, NN px1;wq;θq(24)
where
NN p¨
;
wq
is a neural network with hyperparameters
w
, and
kp¨,¨
;
θq
is a “regular”
kernel such as the Matern 5/2 kernel
(20)
. That is, a deep kernel
(24)
first uses a neural
network
NN p¨
;
wq
to process the input
x
before feeding the extracted feature(s)
NN px
;
wq
into a standard kernel kp¨,¨;θq.
To see why it is necessary to use a deep kernel in our setting, consider Figure 3. Panel A
illustrates a slice of the scaled value function
vpt, w, k, µP, sq
along the liquid wealth share
w
dimension (holding all other variables fixed). We see that the value function consists of two
relatively flat regions, below
w“
0
.
15 and above
w“
0
.
30, with a sharp transition in the
15

0 0.2 0.4 0.6 0.8 1
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
0 0.2 0.4 0.6 0.8 1
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
0 0.2 0.4 0.6 0.8 1
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
Figure 3: Illustration of necessity of Deep Kernel Gaussian Process. Panel A illustrates
the value function when the model is solved using DKGP at every single step. Panel B illustrates
the solution from a single backward induction step when the value function is fitted using a standard
kernel. Panel C illustrates the accumulation of error over successive steps of backward induction
when the value function is fitted using a standard kernel.
intermediate region between w“0.15 and w“0.30.
The sharp transition in the value function is a generic feature of our problem and is due to
the possibility of default. The logic is as follows. Defaulting next period is likely when current
liquidity
w
is low; in this case, the current period’s value function primarily depends on the
value of default
vDpt`
1
, s1q
next period which is low. Conversely, defaulting next period
becomes unlikely when current liquidity
w
is high and the current period’s value function
primarily depends on the much larger non-default value next period
vpt`
1
, w1, k1, µ1
P, s1q
.
The sharp transition for intermediate liquidity
w
captures the transition between defaulting
and not defaulting next period.
Sharp transitions are challenging for standard kernels to capture. The reason is that
standard kernels such as the Matern kernel
(20)
feature a constant lengthscale along each
dimension. In contrast, the value function in our setting features several lengthscales—long
lengthscales are necessary to capture the slowly-varying value function when
w
is low or high,
while a short lengthscale is needed to capture the fast-varying value function for intermediate
values of
w
. In addition, these lengthscales vary over the entire state space as both the
location and speed of the sharp transition depend on all state variables.
Panels B and C of Figure 3illustrates the pitfalls of using a single lengthscale to fit a
multi-lengthscale value function. Maximum likelihood estimation fits a short lengthscale to
capture the sharp transition. The resulting fitted GP is fast-varying and contains wiggles
(see panel B). These wiggles accumulate over successive steps of backward induction and
ultimately results in a very noisy value function after sufficiently many steps (see panel C of
Figure 3).
16

The deep kernel
(24)
captures state-dependant lengthscales through its neural network
component. This can be seen in Panel A of Figure 3which plots the fitted value function
when a DKGP is used.
4 Quantitative Analysis
We begin by calibrating the model in Section 4.1 and describing the model’s solution in
Section 4.2. We then report our model’s implications for optimal private asset allocation.
Section 4.3 and Section 4.4 describe the optimal allocations over the LP’s life cycle and how
business cycle conditions alter those allocations, respectively. Section 4.5 shows that, for
long-term PE investors, whether or not PE returns must first be “unsmoothed” prior to
making inferences and allocation decisions may be a moot point. Section 4.6 examines the
impact of the risk weights on the LP’s portfolio allocation. Appendix C provides details on
the numerical implementation of the model.
4.1 Calibration
We use the parameters listed in Table 1. We calibrated the model at a quarterly frequency
and take the investment horizon to be T“40 quarters or 10 years.
We take state
s“
1to be a recessionary state and state
s“
2to be an expansionary state.
We set the transition probabilities for the macroeconomic state to
p12 “
0
.
25 and
p21 “
0
.
05
based on the duration of NBER recessions in the post-WWII sample. This implies recessions
and expansions last for 4 and 20 quarters on average, respectively. In what follows, we use
NBER recession dates when estimating parameter values that depend on the macroeconomic
state.
We use quarterly data provided by Liberty Mutual Investments (LMI) on its PE holdings
to calibrate various PE-related parameters. This data contains LMI’s total PE exposure
broken down by asset class—buyout, growth, and venture capital. For each asset class, the
data details total exposure over time (
Kt`Pt
in the model) and breaks this exposure down
into outstanding commitments (
Kt
) and the NAV of current PE investments (
Pt
). The
data also include information on LMI’s contributions, distributions received from its PE
investments, and the number of investees LMI invests in. In estimating parameters, we
restrict attention to buyout funds from the moment when the number of buyout investees
first reach 10. The resulting sample starts in 1996Q4 and ends in 2023Q4; all estimates are
based on this sample period.
17

Description Symbol Value
s“1s“2
Investment horizon, quarters T40
Macro transition probability pss10.25 0.05
PE call rate, new commitments λNpsq0.18 0.047
PE call rate, existing commitments λKpsq0.050 0.078
PE distribution rate λDpsq0.028 0.071
PE liquidation discount αpsq0.66 0.90
Risk-free rate log Rfpsq0.0028 0.0051
Stock expected return µSpsq0.0079 0.0238
Stock return volatility σSpsq0.1493 0.0829
Stock-PE return correlation ρpsq0.9527 0.4575
PE return volatility σPpsq0.0768 0.0424
PE expected return autocorrelation ϱP,1`ϱP,20.201
PE expected return state-dependance νPpsq0.0024 0.0317
Risk aversion γ2
PE adjustment costs γN0.1
PE adjustment costs n0
Stock adjustment costs γS0.01
Risk budget threshold θ1
Risk budget cost κ1
Risk weight, bonds θB0
Risk weight, stocks θS1.5
Risk weight, PE θP1.5
Table 1: Parameters. We simulate the model at a quarterly frequency using the parameters in
this table.
Contributiont
in the LMI data corresponds to
λKpstqKt´1`λNpstqNt´1
in the model. We
estimate the call rates using the following regression:
Contributiont
Kt´1“apstq ` bpstqNt´1
Kt´1`ϵt,(25)
where the constant
apstq
maps to
λKpstq
and the slope coefficient
bpstq
maps
λNpstq
. We
estimate regression
(25)
separately for recessionary (
s“
1) and expansionary (
s“
2) regimes
to obtain state-dependant estimates for the call rates. The resulting point estimates give
λKp
1
q “
0
.
050 and
λNp
1
q “
0
.
18 during recessions, and
λKp
2
q “
0
.
078 and
λNp
2
q “
0
.
047
during expansions.
We set the PE payout rate
λDpsq
to be the average payout rate in the LMI dataset. This
results in λDp1q “ 0.028 and λDp2q “ 0.071 during recessions and expansions, respectively.
18

We calibrate the discount from liquidating PE investments
αpsq
based on estimates of
transaction costs in the secondary market for selling PE stakes from Nadauld, Sensoy, Vorkink,
and Weisbach (2019, Table 1). They find that the purchase price, expressed as a pecent of
NAV, is 86% on average and drops to 66% during 2008-09 (the only recession in their sample).
We set
αp
1
q “
0
.
66 and
αp
2
q “
0
.
90 during recessions and expansions, respectively, based on
these estimates.
We calibrate the returns processes as follows. We take the series for the risk-free rate
and aggregate stock returns from Kenneth French’s website. We set the risk-free rate and
expected stock returns to their sample averages during recessions and expansions; this results
in
log Rfp
1
q “
0
.
0028 and
log Rfp
2
q “
0
.
0051 for the risk-free rate and
µSp
1
q “
0
.
0079 and
µSp2q “ 0.0238 for expected stock returns.
LMI data includes quarterly mark-to-market information for the NAV of its PE investments;
this allows us to compute the realized PE return
RP,t
as the ratio of the mark-to-market and
the lagged NAV value. We use moments of the resulting time series for
RP,t
to estimate the
PE return process. We set the stock-PE return correlation
ρpsq
to its sample correlation; this
results in
ρp
1
q “
0
.
9527 and
ρp
2
q “
0
.
4575 during recessions and expansions, respectively.
We restrict
ϱP,1“ϱP,2
and choose the remaining parameters of the PE expected return
process (13) and PE return volatility σPpsqby matching the following data moments: (1) a
PE expected return
Erlog RP,t`1|sts
of 0.0052 and 0.0392 during recessions and expansions,
respectively, (2) a PE return volatility
σplog RP,t`1|stq
of 0.0772 and 0.0427 during recessions
and expansions, respectively, and (3) a PE return autocorrelation of
ρpRP,t, RP ,t`1q “
0
.
1425.
This results in
ϱP,1“ϱP,2“
0
.
1006,
νPp
1
q “
0
.
0024 and
σPp
1
q “
0
.
0768 during recessions,
and νPp2q “ 0.0317 and σPp2q “ 0.0424 during expansions.
We set the risk weights for stocks and private equity to
θS“
1
.
5and
θP“
1
.
5, respectively.
Since the overall risk budget
θ
is normalized to 1, these risk weights imply a 50% risk charge
for stocks and private equity. These values are in line with equity risk charges used by ratings
agencies for assessing insurers’ RBC adequency (see S&P Global 2023, Table 14). The risk
weight for risk-free bonds is
θB“
0. We set the cost parameter for the proportional risk cost
(6) to κ“1.
We set risk aversion to
γ“
2which is standard. The remaining parameters relate to
adjustment costs. We set
γN“
0
.
1and
n
for PE adjustment costs, and
γS“
0
.
01 for stock
adjustment costs.
19

0 5 10 15 20 25 30 35 40
1
1.1
1.2
1.3
1.4
1.5
1.6
Recession Expansion
0
0.2
0.4
0.6
0.8
Figure 4: Solution after defaulting. Panel A plots the scaled value function after defaulting
vDpt, sq
; it is related to the unscaled value function through equation
(17)
. Panel B plots the optimal
allocation to stocks after defaulting.
4.2 Value and policy functions
We begin by illustrating the solution for the scaled value function and the policy functions;
recall that the scaled and unscaled value functions are related through equations
(17)
and
(18).
Value functions. Figure 4plots the solution after defaulting. Panel A plots the scaled
value function
vDpt, sq
. The value function is declining over time since the investment horizon
over which to accumulate wealth shrinks over time. The value function is also higher during
expansions than recessions. Panel B plots the portfolio choice after defaulting. The investor’s
allocation depends only on the macroeconomic state and does not vary over time—the
optimal allocation to stocks is 25% and 66% during recessions and expansions, respectively;
all remaining wealth are allocated to bonds.
Next, we illustrate the solution before defaulting during which the scaled value function
vpt, w, k, µP, sq
depends on time
t
, the liquid fraction of total wealth
w“W{pW`Pq
,
uncalled commitments relative to total wealth
k“K{pW`Pq
, the expected return for PE
µP, and the macroeconomic state s.
Figure 5plots the distribution of the expected PE return
µP
. Panel A plots the uncondi-
tional distribution while panel B plots the conditional distribution given the macroeconomic
state. Expected returns for PE are higher during expansions under our baseline calibration
with expected returns averaging
ErµP,t|st“
1
s “
0
.
0052 and
ErµP,t|st“
2
s “
0
.
0392 per
quarter during recessions and expansions, respectively. From panel B, we also see that the
20

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
0
10
20
30
40
50
60
70
80
-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0.06
0
10
20
30
40
50
60
70
80
90
Figure 5: Distribution for
µP
.Panel A plots the stationary distribution for
µP
whose law of
motion is given by equation
(13)
. Panel B plots the distribution for
µP
conditional on the current
macroeconomic state.
distribution of expected PE returns is wider during expansions than recessions; this is on
account of higher PE return volatilities during recessions.
Figure 6a illustrates the scaled value function
vpt, w, k, µP, sq
at
t“
0. Panel A plots
the value function in the expansionary state (
s“
2) when the expected return on PE is
equal to its mean conditional on
s“
2. We see that the value function
v
is a non-monotonic
function of the liquid fraction of total wealth w. Specifically, vis first increasing in wwhen
w
is sufficiently small (e.g., for
wă
0
.
30 when
k“
0) and then decreasing in
w
when
w
is
sufficiently large (e.g., for wą0.30 when k“0).
The reason for this non-monotonicty is as follows. Defaulting becomes likely when
w
is
small; this is illustrated in Figure 7which plots the one quarter ahead default probabilities
at
t“
0. When the current period’s liquid wealth is low, the LP becomes less likely to have
enough liquid wealth next period to meet its capital commitments and to pay its risk budget
cost (see equation
(9)
). Hence, the value function becomes increasing in
w
when
w
is small,
as having more liquidity helps the LP avoid default. Instead, when
w
is sufficiently large so
that there is no risk of default, the LP’s value function becomes decreasing in
w
because the
LP would be able to achieve better investment outcomes had it allocated more of its wealth
to PE.
Similarly, the value function
v
is non-monotonic in uncalled commitments relative to
total wealth
k
. For example, in panel A of Figure 6a we see that
v
is decreasing in
k
for
small
w
(e.g., when
w“
0
.
2) while
v
is increasing in
k
for large
w
(e.g., when
w“
0
.
8). The
reason is as follows. When the LP has little liquidity, default concerns become first order and
21

0 0.2 0.4 0.6 0.8 1
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
1
1.5
2
2.5
0 0.2 0.4 0.6 0.8 1
1
1.5
2
2.5
(a) Slices of the value function along the wdimension.
(b) Surface plots of the value function
Figure 6: Scaled value function at
t“
0.This figure illustrates the scaled value function
before default
vpt, w, k, µP, sq
at
t“
0. In both subfigures, the first (second) row displays the value
function in the expansionary (recessionary) state. The first (second) column sets
µP
to its mean
conditional on the expansionary (recessionary) state.
22

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Figure 7: One quarter ahead default probabilites at
t“
0.This figure illustrates the one
period ahead default probability at
t“
0. The first (second) row displays the default probabilities
in the expansionary (recessionary) state. The first (second) column sets
µP
to its mean conditional
on the expansionary (recessionary) state.
having more uncalled commitments increases the likelihood of default, thereby decreasing
v
.
In contrast, when the LP has sufficient liquidity and default is no longer a concern, higher
uncalled commitments
k
increases the LP’s PE exposure which increases
v
. This is because
the presence of adjustment costs means that it takes time for the LP to build up its PE
exposure through capital commitments.
Panel B of Figure 6a plots the value function in the expansionary state when the PE
expected return is low while panels C and D illustrate the value function in the recessionary
state. The value function is monotonically increasing in the PE expected return
µP
, as higher
expected PE returns unambiguously translate into higher expected utility. Moreover, the
value function is lower during recessions (panels C and D) compared to expansions (panels A
and B), reflecting the less favorable investment opportunities during economic downturns.
Despite these level differences, we see that the shape of the value function remains similar
across different macroeconomic states and PE expected returns. Figure 6b illustrates further
details of the value function using three-dimensional surface plots.
New commitments. Figure 8illustrate the optimal policy for new PE commitments, scaled
with respect to total wealth, at
t“
0. We see that new PE commitments are increasing in the
liquid fraction of total wealth
w
and decreasing in the uncalled commitments relative to total
23

0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.2 0.4 0.6 0.8 1
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Figure 8: New commitments at
t“
0.This figure illustrates new commitments relative to
total wealth,
N{pW`Pq
, at
t“
0. The first (second) row displays the results for the expansionary
(recessionary) state. The first (second) column sets
µP
to its mean conditional on the expansionary
(recessionary) state.
wealth
k
. The LP becomes less willing to commit new capital to PE when default concerns
are high as defaulting leads to costly PE liquidations. This is why new PE commitments
are low when
w
is low or
k
is high. Instead, when default is unlikely, the LP commits more
capital to PE in order to increase its PE exposure and thereby lower its future liquid wealth
share (as discussed above, the value function becomes decreasing in
w
in the absence of
default concerns).
Comparing panels A and B of Figure 8, and panels C and D of Figure 8, we see that current
PE expected returns
µP
have surprisingly little impact on new commitments across both
expansionary and recessionary states. This muted response stems from the inherent delay
between making commitments and their eventual conversion into PE investments. Because PE
expected returns exhibit low quarterly autocorrelation in the baseline calibration (
ϱP“
0
.
20),
initial differences in
µP
largely dissipate before committed capital is called. Consequently,
optimal commitment levels depend primarily on the expected conditional return
ErµP,t|sts
rather than the current level of
µP
, since only invested capital (NAV) ultimately generates
24

0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.2
0.4
0.6
0.8
1
Figure 9: Allocation to stocks at
t“
0.This figure illustrates the optimal allocation to public
stocks, expressed as a fraction of total wealth, at
t“
0. The first (second) row displays the optimal
allocation in the expansionary (recessionary) state. The first (second) column sets
µP
to its mean
conditional on the expansionary (recessionary) state.
PE returns.
Finally, comparing across macroeconomic states, we see from Figure 8that new com-
mitments are higher during expansions than recessions. Figure C.1a in Appendix C uses
three-dimensional surface plots to illustrate further details regarding new commitments.
Allocation to stocks. Figure 9illustrates the optimal allocation to public stocks as a
fraction of total wealth at
t“
0(see Figure C.1b in Appendix C for surface plots that
display further details). Panel A depicts this allocation during the expansionary state, with
PE expected returns at their mean conditional on
s“
2. When liquid wealth
w
exceeds
0.3, making default risk negligible, two patterns emerge. First, stock allocation increases
with
w
, reflecting a rebalancing mechanism: as liquid wealth rises and PE exposure (1
´w
)
mechanically falls, the LP maintains its desired risk exposure by increasing its allocation to
public stocks. For example, the portfolio shifts from 50% PE, 17% stocks, and 33% bonds
at
w“
0
.
5to 20% PE, 46% stocks, and 34% bonds at
w“
0
.
8. Second, stock allocation
25

remains independent of uncalled commitments
k
. This is because, absent default risk, the
value function depends primarily on next period’s total wealth growth rate. Since uncalled
commitments affect total wealth only after their conversion to PE investments (NAV), which
occurs with a lag, they have no immediate impact on portfolio growth (see equation
(B.9)
)
and therefore do not interact with the allocation decision for public stocks.
When default risk is non-negligible, a different mechanism is at play. For example, default
occurs with certainty when
w“
0
.
1for the levels of
k
considered here (see Figure 7). In
this case, the LP makes its stock allocation decision knowing that it will default on its PE
commitments and incur a PE liquidation loss next period. The LP chooses to maximize its
stock allocation to compensate for this loss when macroeconomic conditions are favorable
(see panels A and B of Figure 9). Conversely, it takes a conservative approach and does
not allocate anything to public stocks when macroeconomic conditions are unfavorable (see
panels C and D of Figure 9).
However, this default mechanism is rarely at work along the optimal path when we
simulate the model. The LP tries to avoid default since defaulting is costly. The cumulative
default rate over the LP’s 10 year investment horizon is only 0.1%.
4.2.1 Marginal value of liquidity
The illiquid nature of PE assets makes liquidity management central. Indeed, the LP must
finance its capital calls out of its liquid wealth or suffer the consequences of default. The
marginal value of liquidity at time
t
, measured in units of certainty equivalent terminal
wealth, is given by
B
BWpW`Pqvpt, w, k, µP, sq “ v` p1´wqvw´kvk.(26)
On the left-hand side of equation
(26)
,
pW`Pqvpt, w, k, µP, sq
expresses the value function
Vpt, w, k, µP, sq
in certainty equivalent units of terminal wealth (see equation
(18)
). Here,
W`P
is current total wealth and
vpt, w, k, µP, sq
is the certainty equivalent value per unit
of wealth. On the right-hand side, the
v
term captures the direct effect of an additional
unit of wealth. The
p
1
´wqvw
and
´kvk
account for compositional effects due to changes in
the liquid fraction of wealth
w
and uncalled commitments relative to total wealth
k
. The
formulation of the marginal value of liquidity
(26)
is analogous to the marginal value of cash
in the liquidity management model of Bolton, Chen, and Wang (2011,2013).
Figure 10a illustrates the marginal value of liquid wealth
(26)
at
t“
0; Figure 10b uses
surface plots to provide additional details. We see that the marginal value of liquid wealth
26

0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
30
0 0.2 0.4 0.6 0.8 1
0
5
10
15
20
25
30
(a) Slices of the marginal value of liquid wealth along the wdimension.
(b) Surface plots of the marginal value of liquid wealth
Figure 10: Marginal value of liquid wealth at
t“
0.This figure illustrates the marginal
value of liquid wealth
(26)
at
t“
0. In both Figure 10a and Figure 10b, the first (second) row
displays the value function in the expansionary (recessionary) state; the first (second) column sets
µPto its mean conditional on the expansionary (recessionary) state.
27

is highly sensitive to default risk. As the liquid fraction of wealth
w
decreases, default
probabilities increase (see Figure 7), leading to a spike in the marginal value of liquidity.
For instance, panel A of Figure 10a shows that when
k“
0
.
4, the macroeconomic state is
expansionary, and
µP
equals its conditional mean in the expansionary state, the marginal
value of liquidity equals 2.28 when
w“
1and default is not a concern. However, as
w
drops
to 0
.
22, the marginal value of liquidity rises to 30.21; at this point, the one-quarter-ahead
default probability is 0.21 and rapidly approaches 1 as
w
declines further. Eventually, as
w
continues to decrease, the marginal value of liquidity levels off because default becomes
inevitable, so that marginal increases in liquid wealth no longer helps to avert default. When
w“
0and default is certain, the marginal value of liquidity falls to 1.62—this lower certainty
equivalent value reflects the lower growth rate of wealth after the LP defaults and loses access
to PE investing.
The plots further demonstrate how the marginal value of liquidity varies over the state
space. For example, the location of the peak in the marginal value of liquidity depends
strongly on the ratio of uncalled capital to wealth
k
; higher values of
k
result in the peak
occurring at larger values of w.
4.3 Life cycle dynamics
We now turn to the life cycle dynamics of the LP’s portfolio. We assume that the LP starts
out at
t“
0without any PE investments (i.e.,
P0“K0“
0) and a units of liquid wealth
W0“
1so that
k0“
0and
w0“
1. The initial macroeconomic state
s0
and PE expected
return
µP,0
are drawn from their stationary distributions. We then simulate the LP’s portfolio
allocation over its investment horizon of 10 years. Figure 11 shows the outcome—the solid
line plots the average outcome at each point in time while the shaded region plots the 95%
confidence interval around the average.
Starting out, the LP aggressively makes new capital commitments in the first two years to
rapidly increase its PE exposure (see panel A of Figure 11). For example, new commitments
average 100% of total wealth during the first year before dropping to 47% and 10% of total
wealth on average in the second and third years, respectively. As a result, uncalled capital
increases and peaks at an average of 90% and 85% of total wealth during the second and
third years (see panel B). This is accompanied by a drop in the liquid portion of wealth (see
panel C) or, equivalently, an increase in the LP’s PE exposure. Specifically, the LP’s NAV
in PE investments rises to 26% and 47% of total wealth on average in the second and third
years, respectively, before reaching 62% of total wealth by year 5. From year 5 onwards, the
28

Figure 11: Life cycle dynamics. This figure illustrates the life cycle dynamics of the LP’s
portfolio. The solid line plots the average outcome in each year while the shaded region provides
the 95% confidence interval. Panels A through D plot the outcomes for new commitments, uncalled
commitments, liquid wealth, and the stock allocation, respectively. All outcomes are shown relative
to total wealth.
LP is in maintenance mode, making new commitments to maintain its PE exposure at an
average of 62% of total wealth until the investment horizon is reached at year 10.
Stock allocation initially starts high (see panel D); for example, stock allocation averages
55% of total wealth in the first year. This is because the LP can only achieve its target
aggregate risk exposure through public stocks before its PE exposure is built up. The wide
confidence interval for stock allocation during this initial phase is due to differences in stock
allocation across the business cycle which we discuss in Section 4.4. The stock allocation
subsequently drops to 5% of total wealth on average during the LP’s maintenance phase.
Figure 12 shows the distribution of outcomes at the terminal date
T
. Panel A plots the
distribution of for total wealth
WT`PT
. Starting from a unit initial total wealth, the LP’s ter-
minal wealth equals 2.41 on average with a standard deviation of 0.66. This wealth distribution
translates into a certainty equivalent wealth of
Ervpt“
0
, w0, k0, µP, sq|w0“1, k0“0s “
2
.
25
on the initial date. Panel B illustrates the distribution of final outcomes in terms of the
annualized realized total returns
1
10 logppWT`PTq{pW0`P0qq
. The annualized realized total
return over the LP’s investment horizon has a mean of 8.41% and a standard deviation of
29

Figure 12: Total wealth outcome distribution. This figure illustrates the distribution of
total wealth outcomes at the end of the investment horizon
t“T
. The initial total wealth is
W0`P0“
1. Panel A plots the distribution for total wealth
WT`PT
. Panel B plots the distribution
for the realized growth in total wealth per annum, logpWT`PTq{10.
2.78%. Note that this seemingly low standard deviation is because we are reporting the
standard deviation of annualized long-horizon returns (see Footnote 1).
4.3.1 Heuristic versus dynamic private asset allocation
In practice, investors often use heuristics to simplify private asset allocation decisions. One
common heuristic is to split the allocation problem into two steps. In the first step, investors
determine a long-run target portfolio by solving a static portfolio choice problem that abstracts
from liquidity risk and the timing lags inherent in PE investing (e.g., a traditional Markowitz
approach). In the second step, they adjust their allocations to reach the targets from the
first step by accounting for the delays between capital commitments, capital calls, and
eventual distributions (see, for example, Takahashi and Alexander 2002). We demonstrate
that portfolio allocations derived from our dynamic model can differ dramatically from these
heuristic approaches. Specifically, we show that portfolio outcomes from our dynamic model,
starting from year 5 when the LP has reached the maintenance phase, substantially differ
from the long-run target portfolio obtained under the heuristic approach. We provide details
below.
Within our framework, the first step of the heuristic approach corresponds to solving
Vheuristicpt, Wtotal , sq “ max
S,B,P ě0
E“Vheuristicpt`1, W 1
total, s1q |Wtotal , s ‰(27)
30

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Figure 13: Portfolio choice: heuristic vs. dynamic private asset allocation. This figure
compares the portfolio outcomes under the heuristic problem
(27)
(shown by the diamonds) against
the outcomes under the full dynamic allocation problem
(16)
(shown in the circles). Outcomes
for the latter correspond are from year 5 onwards during which the maintenance phase has been
reached.
subject to
W1
total “R1
PP R1
SS`RfpsqB´WtotalΓpθq,
θ“θBB`θSS`θPP
B`S`P,
Wtotal “S`B`P,
and the terminal condition
VheuristicpT , W, sq “ W1´γ{p
1
´γq.
The heuristic problem
(27)
resembles the full problem
(16)
but reduces the total number of state variables by abstracting
away from the illiquidity of PE investments. In implementing the heuristic problem
(27)
,
we further set
µP“ErµP,t|st“ss
to its mean conditional on the macroeconomic state. In
Appendix B, we show that the solution to the heuristic problem
(27)
amounts to optimizing
the certainty-equivalent growth rate of total wealth at each point in time. The solution is
static in the sense that the optimal portfolio weights depend solely on the macroeconomic
state sand do not otherwise change over time.
Figure 13 compares the portfolio outcomes under the heuristic approach
(27)
with those
under our full dynamic model
(16)
. The circles represent the portfolio allocations derived
from our dynamic model for year 5 onwards—when the limited partner (LP) has entered
its maintenance phase—while the diamonds show the static portfolio choices obtained from
solving (27) for each macroeconomic state.
31

Under our baseline calibration (Table 1), the heuristic approach yields an optimal portfolio
allocation of 67.3% in private equity (PE) and 0% in stocks (with the remainder in bonds)
in the expansionary state, and 0% in PE and 36.4% in stocks in the recessionary state. In
contrast, the optimal dynamic allocation maintains a PE allocation of at least 45% during
the maintenance phase, while the stock allocation does not exceed 10%. This difference in
outcomes is especially stark during the recessionary state, where the heuristic outcome falls
entirely outside the range of values generated by the optimal dynamic allocation.
4.4 Business cycle dynamics
Figure 14 illustrates how business cycle conditions affect the life cycle dynamics of the
LP’s portfolio. The solid and dashed lines plot the average outcomes conditional on the
macroeconomic state being in a expansion and a recession, respectively; the shaded 95%
confidence intervals are also conditional on the macroeconomic state. Overall, we see that
the allocation patterns across the business cycle are similar compared to the unconditional
case discussed in Section 4.3.
The most notable difference across the business cycle is the difference in the LP’s stock
allocation, particularly during the ramping-up phase (see panel D of Figure 14). For instance,
in year 1, stock allocations average 62% of total wealth during expansions compared to 38%
during recessions. This difference narrows to about 2% of total wealth by the maintenance
phase from year 5 onwards.
There is a smaller difference in new commitments across the business cycle (see panel
A). For example, During the ramping-up phase in the first three years, new commitments
are lower during recessions compared to expansions by about 5% of total wealth on average.
Similarly, panels B and C show that the difference in uncalled commitments and liquid wealth
over the business cycle is smaller on average compared to the differences in stock allocation.
For example, the difference in PE allocation across the business cycle averages 2% of total
wealth over the lifetime of the fund.
The main reason for the pronounced difference in stock allocation across the business cycle,
as opposed to PE allocation, is due to adjustment costs. PE allocation involves significant
adjustment costs, whereas adjustment costs for stocks are negligible. Therefore, the LP keeps
the PE allocation relatively stable across the business cycle and primarily adjusts its exposure
to aggregate risks through stock allocation.
Figure 15 plots the distribution of outcomes at the terminal date
T
conditional on business
cycle conditions encountered over the LP’s investment horizon. The transition probabilities
32

Figure 14: Life cycle dynamics over the business cycle. This figure illustrates the life
cycle dynamics of the LP’s portfolio after conditioning on business cycle conditions. Panels A
through D plot the outcomes for newcommitments, uncalled commitments, liquid wealth, and the
stock allocation, respectively, where all outcomes are shown relative to total wealth. The solid
(dashed) line shows the average outcome conditional on a expansionary (recessionary) state. Shaded
regions indicated 95% confidence intervals, conditional on the macroeconomic state.
for the macroeconomic state from Table 1imply that recessions occur 1/6 of the time. The
solid blue (dashed red) bars plot outcome distributions conditional on the LP encountering
recessions for less (more) than 1/6 of the time over its investment horizon. Panel A plots
the outcomes for wealth. On average, the LP ends up with a terminal wealth of 2.60 and
2.17 conditional on experiencing a lower- and higher-than expected number of recessions,
respectively. The corresponding standard deviations of terminal wealth are similar: 0.63 and
0.61 conditional on experiencing a lower- and higher-than expected number of recessions,
respectively. The outcomes in terms of annualized realized total returns are shown in panel
B. Conditional on experiencing a lower than expected number of recessions, the realized
annualized returns average 9.29% with a standard deviation of 2.40%. The annualized return
averages 7.35% with a standard deviation of 2.83% if a higher than expected number of
recessions is encountered instead.
33

Figure 15: Business cycles and total wealth outcomes. This figure illustrates the
distribution of total wealth outcomes at the end of the investment horizon
t“T
. The initial total
wealth is
W0`P0“
1. Panel A plots the distribution for total wealth
WT`PT
. Panel B plots
the distribution for the realized growth in total wealth per annum,
logpWT`PTq{
10. The solid
(dashed) bars plot the outcomes conditional on the LP experiencing a lower (higher) than expected
number of recessions over its investment horizon.
4.4.1 Cost of ignoring business cycle when making PE allocations
Next, we illustrate the importance of accounting for business cycle conditions through the
following exercise. We consider a naive LP whose decisions are based on a model that ignores
business cycles. Specifically, this naive LP solves the model under the assumption that the
economy is always in an expansionary state (i.e., this LP sets
s0“
2,
p21 “
0, and uses
the remaining parameters from Table 1). The naive LP then applies the resulting policy
functions, which ignore business cycle fluctuations, in a setting where such fluctuations are
present. We compare the life cycle dynamics of this naive LP to the baseline case, where the
LP optimally accounts for business cycle conditions. The results are shown in Figure 16.
We observe that without considering the possibility of recessions, the LP’s PE allocation
is significantly more aggressive compared to the optimal policy that accounts for business
cycle fluctuations. For instance, in year 1, new commitments average 117% of total wealth
compared to 100% in the baseline case. During the maintenance phase, the PE allocation
averages 67% of total wealth, as opposed to 62% in the baseline case. Consequently, the
naive LP defaults substantially more frequently—the naive LP’s 10-year cumulative default
probability is 13.6% compared to only 0.1% under the optimal policy.
This large increase in default frequency has a significant negative impact on the naive
LP’s final portfolio outcomes. Figure 17 compares the outcome distributions for the naive
investor against the baseline; panels A and B compare outcomes for terminal total wealth
34

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0
0.2
0.4
0.6
0.8
1
1.2
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0.4
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1
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0.8
1
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0
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Figure 16: Life cycle outcomes without accounting for business cycles. The solid and
dashed lines plot the average outcomes when business cycles are accounted for and not accounted
for, respectively.
and annualized realized returns, respectively. Compared to the baseline investor that takes
business cycles into account, the wealth total wealth distribution for the naive LP has a
lower mean (2.32 vs. 2.41) and a higher volatility (0.75 vs 0.66). Similarly, the naive LP’s
annualized realized returns have a lower mean (7.83% vs 8.41%) and a higher volatility (3.58%
vs 2.78%). The biggest difference in outcomes, however, is heavier left-tail encountered by
the naive investor. For example, the first and fifth percentiles for the naive LP’s realized
annualized returns equal -3.06% and 1.14%, as opposed to 1.34% and 3.68% for the baseline
case. The naive LP’s suboptimal policies translate into a 9.3% loss in initial total wealth—the
naive LP’s certainty equivalent value is 2.04 compared to 2.25 for the baseline case.
These results underscore the importance of considering business cycle conditions when
making optimal PE allocation decisions.
35

Figure 17: Total wealth outcome without accounting for business cycles. This figure
illustrates the distribution of total wealth outcomes at the end of the investment horizon
t“T
. The
initial total wealth is
W0`P0“
1. Panel A plots the distribution for total wealth
WT`PT
. Panel
B plots the distribution for the realized growth in total wealth per annum,
logpWT`PTq{
10. The
solid and dashed bars plot the outcomes for the baseline and a naive investor that ignores business
cycles, respectively.
4.5
Does serial correlation in PE returns matter for long-term
investors?
Positive serial correlation is a hallmark feature of the observed returns of illiquid alternative
asset, PE investments included. A large literature investigates the sources of this serial
correlation and its implications for investors (see, e.g., Geltner 1991 and Getmansky, Lo,
and Makarov 2004). A likely explanation is that serial correlation can arise from having
to mark-to-market illiquid assets, which can lead to observed returns that appear smooth
even when the underlying true returns do not have serial correlation (Getmansky, Lo, and
Makarov,2004). As a result, the literature has argued for “unsmoothing” such returns to
obtain a more accurate picture of the underlying asset’s risk and return characteristics; for
example, naively using smoothed returns can lead to overestimation of Sharpe ratios and
faulty portfolio allocations.
In this section, we revisit the question of whether unsmoothing is necessary for optimal
portfolio choice. We differ from the existing literature in that we take the perspective of a
long-term investor. Our headline result is that for long-term investors, whether or not returns
must be first unsmoothed may be a moot point. The intuition is as follows. First, long-term
investors care about properties of long-horizon returns, which are much less affected by short-
term mark-to-market fluctuations. Second, even if serial correlation is a true feature of short
run PE returns, it may not be exploitable by investors after taking realistic implementation
36

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0.4
0.6
0.8
1
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0.4
0.6
0.8
1
123456789
0
0.2
0.4
0.6
0.8
1
1.2
123456789
0
0.2
0.4
0.6
Figure 18: PE return autocorrelation and life cycle dynamics. The solid and dashed
lines plot the average lifecycle outcomes under (1) our baseline model, and (2) a model where the
PE returns process has zero autocorrelation but otherwise have the same return moments.
lags and adjustment costs into account. We provide details below.
Consider the following thought experiment. Suppose, for argument’s sake, that the
quarterly autocorrelation of PE expected returns of
ϱP,1`ϱP,2“
0
.
20 in our baseline
calibration is entirely due to smoothing. The LP may then want to base its portfolio choice
on unsmoothed returns by considering a model without serial correlation:
ϱP,1“ϱP,2“
0. To
keep the overall properties of PE returns unchanged, this LP simultaneously recalibrates the
other parameters of the PE returns process so that the expected PE return
Erlog RP,t`1|sts
and volatility
σplog RP,t`1|stq
remain identical to that of the baseline calibration in both of the
macroeconomic states. This involves setting
νPp
1
q “
0
.
0051,
νPp
2
q “
0
.
0392,
σPp
1
q “
0
.
0772,
and
σPp
2
q “
0
.
0427 while keeping the remaining parameters unchanged from their baseline
values in Table 1.
Figure 18 compares the resulting difference in life cycle dynamics between the baseline
case and the case where the LP ignores serial correlation in PE returns. We see that the
LP’s portfolio allocations appear near-identical across the two cases. Furthermore, the two
37

outcomes remain near-identical when we further condition outcomes on the macroeconomic
state (see Figure D.1 in Appendix D).
The results in this section show that for long-term investors, the presence of serial
correlation in PE returns may not be a critical factor in determining optimal portfolio
allocations.
4.6 Risk budget
The risk weights in our baseline calibration
θS“θP“
1
.
5correspond to 50% risk charges for
PE and stocks (the risk budget threshold
θ
is normalized to 1). In this section, we conduct
comparative static exercises to investigate how the LP’s outcomes depend on these risk
charges. This is a useful exercise in at least two contexts. First, while risk charges of 50% are
representative numbers on average, risk charges can vary widely depending on asset quality.
For example, S&P Global assesses anywhere between 35% to 99% market risk charges for
equities depending on asset quality (S&P Global,2023, Table 14). Second, individual LPs
may assess different risk charges based on their own individual circumstances. For example,
an insurer may assess higher risk charges for its PE investments depending on properties of
its insurance portfolio; the risk weights
θP
and
θS
can then be interpreted as effective risk
charges whose values depend on the riskiness of other components of the insurer’s balance
sheet.
We demonstrate the effects of risk charges through two scenarios. In the first scenario, we
set
θS“θP“
2, so that risk charges are uniformly higher. In the second scenario, we increase
the risk charge for PE to
θP“
2while keeping
θS“
1
.
5unchanged. For both scenarios, we
take the remaining parameters from the baseline calibration in Table 1.
Figure 19 illustrates the impact of risk charges on average life cycle outcomes. Compared
to the baseline outcome (solid line), we see that overall allocations to PE and stocks are
lower under the first scenario (dotted line) in which the risk charges for both PE and stocks
are increased to
θP“θS“
2. For instance, during the initial ramp-up phase, uncalled
commitments peak at 66% of total wealth in year 2, as opposed to 90% in the baseline
calibration. Similarly, stock allocation in year 1 is reduced to 42% of total wealth on average
compared to 55% in the baseline. In the maintenance phase from year 5 onwards, the PE
allocation is 42% of total wealth, down from 62% in the baseline. The LP does, however,
compensates with a slightly higher stock allocation during the maintenance phase: 7% of total
wealth on average compared to 5% on average for the baseline. Figure D.2 in Appendix D
additionally displays outcomes for the first scenario after further conditioning on the business
38

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0.6
0.8
1
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0.2
0.4
0.6
0.8
1
12345678910
0
0.2
0.4
0.6
0.8
1
12345678910
0
0.2
0.4
0.6
Figure 19: Life cycle dynamics under different risk charges. This figure displays average
life cycle outcomes under alternative risk charges. For reference, the solid line plots outcomes under
the baseline calibration in which
θP“θS“
1
.
5. The dotted line plots outcomes when both
θS
and
θPequal 2, and the dashed line plots outcomes when only θPis increased to 2.
cycle.
The second scenario isolates the effect of risk charges by only increasing
θP
. In this case,
we see from Figure 19 that only PE allocations are affected to first order. Specifically, the
average outcome for stock allocation remains roughly unchanged from the baseline (compare
the solid and dashed lines in panel D) while the outcomes for PE allocation are similar to
that under the first scenario (compare the dotted and dashed lines in panels A, B, and C).
Figure D.3 in Appendix Dadditionally displays outcomes for the second scenario after further
conditioning on the business cycle.
Figure 20 compares the distribution of final outcomes under the different risk charge
scenarios. We see that higher risk charges do indeed decrease the riskiness of the LP’s overall
portfolio. For example, the standard deviation of annualized realized returns decreases to
2.35% (from the baseline value of 2.78%) under the second scenario when
θP
increases to
2; it further decreases to 2.14% under the first scenario when both
θP
and
θS
increase to
2. This decrease in risk does come at the expense of lower returns. The average annualized
return decreases to 7.08% under the second scenario (down from 8.41% for the baseline), and
further decreases to 6.79% under the first scenario.
39

Figure 20: Total wealth outcomes under different risk charges. This figure illustrates
the distribution of total wealth outcomes at the end of the investment horizon
t“T
. The initial
total wealth is
W0`P0“
1. Panel A plots the distribution for total wealth
WT`PT
. Panel B plots
the distribution for the realized growth in total wealth per annum,
logpWT`PTq{
10. Both panels
compare outcomes under (1) the baseline in which
θP“θS“
1
.
5, (2) scenario 1 in which both
θP
and θSare increased to 2, and (3) scenario 2 in which only θPis increased to 2.
The costs of imposing higher risk charges in certainty equivalent terms are as follows.
Increasing
θP
alone to 2 (i.e., scenario 2) reduces the certainty equivalent wealth from the
baseline value of 2.25 to 2.00 or, equivalent, a 11% reduction in certainty equivalent wealth.
When we additionally increase
θS
to 2 (i.e., scenario 1), the certainty equivalent wealth
further decreases to 1.92 for a 15% reduction in certainty equivalent wealth compared to the
baseline.
5 Conclusion
Exciting recent developments in machine learning have enabled the solution of portfolio
choice problems featuring higher dimensionality, more realistic trading frictions, and more
complex constraints. In this paper, we employ Deep Kernel Gaussian Processes to accurately
characterize the optimal policies in a private asset allocation model that incorporates a
host of real-world complexities, including illiquidity, commitment lags, serial correlation in
returns, business cycle fluctuations, and regulation-induced constraints. Our approach not
only provides novel insights into private equity investing but also offers a flexible blueprint
for tackling similarly challenging problems in economics and finance that involve substantial
nonlinearities and a large number of state variables.
40

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42

Appendix
A Specification for expected PE returns
In this section, we use the Getmansky, Lo, and Makarov (2004) model of smoothed returns for
illiquid alternative investments to motivate our specification (14) for PE expected returns.
Applied to our context, the Getmansky, Lo, and Makarov (2004, equation 21) model posits that
the observed return on PE investments logpRP,t`1qis a weighted average of contemporaneous and
lagged values of the true but unobserved return on PE investments:
logpRP,t`1q “ ω0logpr
RP,t`1q ` ω1logpr
RP,tq ` ... `ωklogpr
RP,t`1´kq,(A.1)
In terms of notation for this section, we use a tilde to denote quantities associated with the true but
unobserved returns. For example,
logpRP,t`1q
is the observed PE return at
t`
1while
logpr
RP,t`1q
is the unobserved contemporaneous true return. The coefficients
ωj
in equation
(A.1)
are positive
weights that sum to 1. We use the geometric weighting scheme considered in Getmansky, Lo, and
Makarov (2004, equation 43):
ω0“ p1´ϕq{p1´ϕk`1q, ωj“ω0ϕjfor j“1, ..., k, (A.2)
where
ϕP p
0
,
1
q
. This weighting scheme leads to the convenient AR(1) process for the expected
returns on observed PE returns (14).
Specifications
(A.1)
and
(A.2)
imply that the expected return for observed PE returns
µP,t ”
EtrlogpRP,t`1qs evolves as follows:
µP,t “ω0Etrlogpr
RP,t`1qs ` ϕEtrω0logpr
RP,tq ` ... `ω0ϕk´1logpr
RP,t`1´kqs
“ω0Etrlogpr
RP,t`1qs ` ϕEtrlogpRP,t q ´ ω0ϕklogpr
RP,t´kqs
“ω0r
µP,t `ϕlogpRP,t q ´ ω0ϕk`1Etrlogpr
RP,t´kqs (A.3)
«ω0rµP,t `ϕlogpRP,t q(A.4)
“ω0rµP,t `ϕµP,t´1`ϕσP,t´1εP ,t.(A.5)
Equation
(A.3)
is a direct consequence of specifications
(A.1)
and
(A.2)
;
r
µP,t ”Etrlogpr
RP,t`1qs
is the
expected return for the true return process. The approximation
(A.4)
holds if
k
is large enough or if
ϕ
is small enough. Equation
(A.5)
writes the observed return as
logpRP,tq “ µP,t´1`σP,t´1εP,t
where
µP,t´1”Et´1rlogpRP,t qs
,
σP,t´1”aV art´1rlogpRP,t qs
, and
εP,t ” plogpRP,t q ´ µP,t´1q {σP,t´1
.
Comparing equation
(A.5)
to equation
(14)
, we see that the AR(1) process for expected PE returns
(14)
is a more flexible version of equation
(A.5)
in which the loading on the
σP,t´1εP,t
term is allowed
to differ from the autoregressive coefficient.
We make further functional form assumptions in our implementation. Specifically, the spec-
ification in equations
(12)
and
(14)
assumes (1) the expected true return
r
µP,t “rµPpstq
depends
only on the macroeconomic state (hence
νPpstq
in equation
(14)
corresponds to the
ω0rµP,t
term
from equation
(A.5)
), (2) the volatility of observed PE returns
σP,t´1“σPpst´1q
depends only
on the macroeconomic state, and (3) the shock to observed PE returns
εP,t „Np
0
,
1
q
is normally
distributed.
43

B Scaled value functions
Scaled choice variables. We work with the following scaled versions of the choice variables:
n”N
W`P,(B.1)
r
ϕS”
S`γSpW`Pq´S
W`P¯2
W´γNpW`Pq pn´nq2.(B.2)
Here,
n
is the new PE commitments scaled by total wealth. The interpretation for
r
ϕS
is as follows.
The denominator in equation
(B.2)
is the amount of liquid wealth that is available to be invested in
stocks and bonds after deducting adjustment costs for new PE commitments from current liquid
wealth. Of this amount,
r
ϕS
is the fraction invested in stocks inclusive of stock adjustment costs.
We also denote by
ϕS”S
W´γNpW`Pq pn´nq2.(B.3)
the fraction invested in stocks excluding stock adjustment costs. The relation between
r
ϕS
and
ϕS
is
ϕS“´1`b1`4γSrw´γNpn´nq2sr
ϕS
2γSrw´γNpn´nq2s.(B.4)
Scaled value function in default. The recursive formulation for the scaled value function in
default is
vDpt, sq “ max
r
ϕSPr0,r
ϕS,maxpsqs
E”`g1
D˘1´γ|sı1
1´γE“vDpt`1, s1q1´γ|s‰1
1´γ(B.5)
where
g1
D”W1
W“R1
SϕS`Rfpsq´1´r
ϕS¯´ΓpθSr
ϕSq(B.6)
is the growth rate of total wealth in default and
vDpT, sq “ 1
is the terminal condition. Note that when computing the scaled value function
(B.5)
, equations
(B.2)
,
(B.3)
, and
(B.4)
for
r
ϕS
,
ϕS
, and their relation, respectively, are computed with
P“
0and
n“
0. In addition, note that while the
E“uDpt`1, s1q1´γ|s‰1
1´γ
on the right-hand side of equation
(B.5)
affects the scaled value function, it does not affect the optimal portfolio choice. This is because
the growth rate of total wealth
(B.6)
does not depend on
s1
. As a result, the optimal policies in
default only depend on the macroeconomic state s.
The upper bound
r
ϕS,maxpsq ” sup !r
ϕSP r0,1s: infR1
Sg1
D“Rfpsq´1´r
ϕS¯´ΓpθSr
ϕSq)
appear-
ing in problem
(B.5)
ensures that the growth in wealth
g1
D
is strictly positive so that utilities are
well-defined. The upper bound equals
r
ϕS,maxpsq “ $
’
&
’
%
1if θSďθ,
min #θ
θS`´Rfpsq`bRfpsq2`4κθSRfpsqpθS´θq
2κθ2
S
,1+if θSąθ. (B.7)
44

Similarly, the lower bound
r
ϕSě
0rules out shorting which is also necessary to ensure that utilities
are well-defined.
Scaled value function before default. The recursive formulation for the scaled value
function before default is
vpt, w, k, µP, sq “ max
n, r
ϕS
E”max ␣g1
DvD`t`1, s1˘, g1v`t`1, w1, k1, µ1
P, s1˘(1´γ|w, k, µP, s ı1
1´γ
(B.8)
subject to
g1
D“p1´wq“λDps1q ` αps1qp1´λDps1qq‰R1
P´ΓpθDq
`“w´γNpn´nq2‰”ϕSR1
S` p1´r
ϕSqRfpsqı,
θD“θS“w´γNpn´nq2‰r
ϕS
1´γNpn´nq2,
g1“ p1´wqR1
P`“w´γNpn´nq2‰”ϕSR1
S` p1´r
ϕSqRfpsqı´Γpθq,(B.9)
θ“θPp1´wq ` θS“w´γNpn´nq2‰r
ϕS
1´γNpn´nq2,
k1“r1´λKps1qsk` r1´λNps1qsn
g1,
w1“#λDps1qR1
Pp1´wq ´ λKps1qk´λNps1qn´Γpθq
`“w´γNpn´nq2‰”ϕSR1
S` p1´r
ϕSqRfpsqı+
g1,
µ1
P“ϱP,1µP`ϱP,2log R1
P`νPps1q,
nP rnminpwq, nmax pwqs,r
ϕSP r0,r
ϕS,maxpw, n, sqs,(B.10)
with terminal condition
vpT, w, k, µP, sq “ 1.
Here,
g1
D“W1
D{pW`Pq
and
g1“ pW1`P1q{pW`Pq
denote the growth rates of total wealth
conditional on defaulting and not defaulting, respectively.
The limits on the choice variables
(B.10)
ensure that utility is well-defined. To see why, note
that the log returns on PE and the stock are normally distributed and are therefore unbounded.
Hence, this would require
maxtg1
D, g1u ą
0for all realization of return shocks. This condition implies
inf g1
D“ rw´γNpn´nq2sp
1
´r
ϕSqRfpsq ´
Γ
pθDq ě
0where the infimum is taken over both
R1
P
and
R1
S. From this, we obtain
nminpwq “ max "n´cw
γN
,0*, nmaxpwq “ n`cw
γN
,
45

and
r
ϕS,maxpw, n, sq
“$
’
&
’
%
1if r
θSpw, nq ď θ,
min #θ
r
θSpw,nq`´r
Rfpw,n,sq`br
Rfpw,n,sq2`4κr
θSpw,nqr
Rfpw,n,sqrr
θSpw,nq´θs
2κr
θSpw,nq2,1+if r
θSpw, nq ą θ,
where
r
θSpw, nq ” θSrw´γNpn´nq2s
1´γNpn´nq2,and r
Rfpw, n, sq”rw´γNpn´nq2sRfpsq.
Scaled value function for the heuristic problem
(27)
.The value function for the
heuristic problem
(27)
can be scaled as
Vheuristicpt, Wtotal , sq “ ”Wtotalvheuristicpt,sqı1´γ{p
1
´γq
where vheuristic solves
vheuristicpt, sq “ max
ϕS,ϕP
E”`g1
total˘1´γ|sı1
1´γE“vheuristicpt`1, s1q1´γ|s‰1
1´γ(B.11)
subject to
g1
total ”W1
total
Wtotal
“R1
SϕS`R1
PϕP`Rfpsq p1´ϕS´ϕPq ´ ΓpθSϕS`θPϕPq,
ϕS, ϕPP r0,1s,
ϕS`ϕP“1
and the terminal condition vheuristicpT , sq “ 1.
C Numerical Implementation
We first solve the scaled problem in default
(B.5)
. In doing so, we note that the the value function
can be split as
vDpt, sq “ E”`uDpt`1, s1q˘1´γ|sı1
1´γmax
r
ϕSPr0,r
ϕS,maxpsqs
E”`g1
D˘1´γ|sı1
1´γ.(C.1)
This is because
g1
D
does not depend on
s1
(see equation
(B.6)
). Hence, the optimal portfolio choice
problem depends salone and is obtained by solving
max
r
ϕSPr0,r
ϕS,maxpsqs
E”`g1
D˘1´γ|sı1
1´γ.(C.2)
To ensure numerical accuracy, we use adaptive quadrature in to compute expectations.
After obtaining
vDpt, sq
, we solve the scaled problem before default (see equation
(B.8)
) using
backward induction. The procedure is as follows:
1. Initiate the value function at t“Tusing the terminal condition vpT, w, k, µP, sq “ 1.
46

2.
Obtain sample points for the state variables
pw, k, µPq
. In doing so, we sample 800 points via
a Halton sequence over the hypercube
tpw, k, µPq
:
wP r
0
,
1
s, k P r
01
.
5
s, µPP rµP,lb, µP,ubsu
where the bounds
µP,lb
and
µP,ub
are set to be the 0.1st and 99.9th percentiles of the stationary
distribution for µP, respectively.
3.
Given a value function
vGP pt`1, w, k, µP, sq
that has been previously fitted via Deep Kernel
Gaussian Process Regression, do the following:
(a)
For each
s
at time
t
, solve problem
(B.8)
for the sample points for
pw, k, µPq
obtained in
step 2. When doing so, we use the fitted value function
vGP pt`1,¨q
as the continuation
value when not defaulting next period. As in the problem after default, we use adaptive
quadrature to compute expectations to ensure numerical accuracy. To ensure that we
find a global maximum, we first consider a coarse grid of policy points. We then run an
interior-point optimization algorithm starting from the best policy based on the coarse
grid. The end result is a set of observations for the value and policy functions at the
sample points.
(b)
We use the observations from step (a) to fit deep kernel Gaussian processes for the
value and policy functions at time
t
for each state
s
. Our choice for the deep kernel
(24)
is as follows. The neural network component
NN
takes the three-dimensional input
pw, k, µPq
and passes it through 3 hidden layers with 64, 32, and 16 units, respectively,
before outputting 2 features (the shape of the hidden layers follow a so-called “decoder
structure”); we use Gaussian error linear unit (GELU) activation. These 2 output
features are then passed through a Matern 5/2 kernel
(20)
. We train the parameters of
the deep kernel Gaussian processes by maximizing the marginal likelihood
(23)
using
the Adam optimizer (Kingma and Ba,2015) with a learning rate of 0.001. To ensure
that the training converges to a global optimum, we run the Adam optimizer 10 times
from different initial points. We always use the trained parameters from step
t`
1as
one of the initial points; the remaining initial points are randomly drawn with Xavier
initialization for the neural network weights. We standardize both input and output
variables when training.
4. Repeat step 3 until t“0.
Figure 6b and Figure C.1 illustrate the value function and policy functions at
t“
0, respectively.
Heuristic portfolio choice problem.
D Additional Figures
47

(a) New commitments.
(b) Stock allocation.
Figure C.1: Policies at t“0.
48

12345678910
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Figure D.1: Life cycle comparison (conditional), PE return autocorrelation. The
solid and dashed lines plot the average lifecycle outcomes under (1) our baseline model, and (2) a
model where the PE returns process has zero autocorrelation but otherwise have the same return
moments.
49

1 2 3 4 5 6 7 8 9 10
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Figure D.2: Life cycle comparison (conditional), risk budget 1.
50

1 2 3 4 5 6 7 8 9 10
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0
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Figure D.3: Life cycle comparison (conditional), risk budget 2.
51