研究-长江商学院.pdf
研究-长江商学院.pdf
MANAGEMENT SCIENCE Articles in Advance, pp. 1–26 ISSN 0025-1909 (print), ISSN 1526-5501 (online) https://pubsonline.informs.org/journal/mnsc The Endowment Model and Modern Portfolio Theory Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Stephen G. Dimmock,a,b Neng Wang,b,c,d,e,* Jinqiang Yangf,g a National University of Singapore, Singapore 119077; b Asian Bureau of Finance and Economic Research, Singapore 117592; c Columbia Business School, Columbia University, New York, New York 10027; d Cheung Kong Graduate School of Business, Beijing 100006, China; e National Bureau of Economic Research, Cambridge, Massachusetts 02138; f School of Finance, Shanghai University of Finance and Economics, Shanghai, China 200437; g Shanghai Institute of International Finance and Economics, Shanghai 200433, China *Corresponding author Contact: dimmock@nus.edu.sg, https://orcid.org/0000-0003-3404-9307 (SGD); neng.wang@columbia.edu, https://orcid.org/0000-0001-7895-6502 (NW); yang.jinqiang@mail.sufe.edu.cn (JY) Received: December 13, 2021 Revised: June 29, 2022 Accepted: August 18, 2022 Published Online in Articles in Advance: April 28, 2023 https://doi.org/10.1287/mnsc.2023.4759 Copyright: © 2023 INFORMS Abstract. We develop a dynamic portfolio choice model with illiquid alternative assets to analyze the “endowment model,” widely adopted by institutional investors, such as pension funds, university endowments, and sovereign wealth funds. In the model, the alternative asset has a lockup but can be liquidated at any time by paying a proportional cost. We model how investors can engage in liquidity diversification by investing in multiple illiquid alterna tive assets with staggered lockup expirations and show that doing so increases alternatives allocations and investor welfare. We show how illiquidity from lockups interacts with illi quidity from secondary market transaction costs resulting in endogenous and time-varying rebalancing boundaries. We extend the model to allow crisis states and show that increased illiquidity during crises causes holdings to deviate significantly from target allocations. History: Accepted by Bruno Biais, finance. Funding: S. G. Dimmock gratefully acknowledges financial support from the Singapore Ministry of Educa tion [Grant R-315-000-133-133]. N. Wang gratefully acknowledges support from CKGSB Research Insti tute. J. Yang gratefully acknowledges the support from the National Natural Science Foundation of China [Grants 71772112, 71972122, and 72072108], Innovative Research Team of Shanghai University of Finance and Economics [Grant 2016110241], and Shuguang Program of Shanghai Education Devel opment Foundation and Shanghai Municipal Education Commission. Supplemental Material: Data are available at https://doi.org/10.1287/mnsc.2023.4759. Keywords: endowment model • portfolio choice • liquidity • modern portfolio theory • asset allocation • alternative assets 1. Introduction Takahashi and Alexander 2002). The endowment model lacks a framework that formalizes the trade-off between the benefits of alternative assets and the costs of their illi quidity. It also lacks a framework for evaluating how investor heterogeneity affects this trade-off. For example, pension funds and wealthy individuals differ greatly in their flexibility to adjust spending across periods, yet the endowment model does not provide guidance on how such differences should affect asset allocation and spending. In this paper, we develop a tractable dynamic as set allocation model based on modern portfolio theory (MPT) to formally assess the heuristic risk-return-based endowment model championed by Swensen (2000) and widely adopted by practitioners, for example, university endowments, family offices, and other institutional in vestors. To do this, we incorporate illiquid investment opportunities and important institutional features of alter native assets, for example, private equity and hedge funds, into a generalized MPT framework (Merton 1971). We show that the illiquidity of alternative assets and incomplete markets have first order effects on the inves tor’s dynamic spending and asset allocations. Further, we Sophisticated long-term investors hold substantial frac tions of their portfolios in illiquid alternative assets. For example, public pension funds allocate 27% of their port folios to illiquid alternatives, and both university endow ments and family offices of high-net-worth individuals allocate more than half of their aggregate portfolios to illiquid alternatives.1 And, as of 2020, more than $14.2 trillion was allocated to alternative assets.2 In recent decades, an investment strategy of high allocations to illiquid assets has been adopted to varying degrees by almost all types of institutional investors. This strategy, called the “endowment model” as it was initially cham pioned by university endowments,3 advocates that longterm investors should hold high allocations of alternative assets so as to earn illiquidity premiums and exploit the inefficiencies found in illiquid markets. Despite its perceived success and growing popularity among institutional and high-net-worth investors, the endowment model has significant limitations and lacks theoretical foundations. Standard references of the en dowment model are based on static mean-variance anal ysis adjusted with ad hoc rules of thumb (Swensen 2000, 1 Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 2 show how the features of alternative assets interact with the investor’s characteristics. Our model captures the key features of alternative assets in a manner that is sufficiently realistic and analyti cally tractable. First, the alternative asset’s risk is not fully spanned by public equity. Second, the model includes a secondary market for the alternative asset, in which the investor can voluntarily transact at any time by paying a proportional transaction cost. This captures a form of illi quidity costs. Third, we allow the alternative asset to pro vide some “natural” liquidity. This can be in the form of dividends, such as rental cash flows from private real estate. Or it can be in the form of liquidity events, when the alternative asset (or a fraction of it) becomes fully liq uid, such as when a hedge fund lockup expires or a pri vate equity fund makes a distribution to investors. This is an important feature of our model as advocates of the endowment model argue that natural liquidity, such as private equity cash distributions, offsets much of the apparent illiquidity of alternative assets (see Swensen 2000, Takahashi and Alexander 2002) and, therefore, should influence an investor’s asset allocation decisions. We model how this natural liquidity can provide “liquidity diversification,” using a reduced-form ap proach.4 In our model, the investor can stagger invest ments into the alternative asset over time, resulting in distinct positions that have staggered natural liquidity events. Although we do not endogenize the investor’s liquidity diversification decisions, we show how stag gering investments in the alternative asset affects the investor’s welfare, portfolio allocations, and spending. Incorporating both secondary markets and natural liquidity events for alternative assets allows us to closely match the relevant features of alternative assets and differentiates our model from prior work. We provide an analytical characterization for the in vestor’s certainty equivalent wealth under optimality, P(Wt , Kt , t), which is the time-t total wealth that makes the investor indifferent between permanently forgoing the opportunity to invest in the illiquid asset and keeping the status quo with liquid wealth Wt and illiquid wealth Kt with the opportunity to invest in the illiquid asset. We exploit the model’s tractability to provide a quantitative yet intuitive analysis of a long-term investor’s optimal portfolio choice, spending rule, and welfare measured by P(Wt , Kt , t). Our qualitative and quantitative results significantly differ from the standard predictions of MPT. In contrast to the classic MPT prediction, we show that the alloca tion to the illiquid alternative asset follows a double bar rier policy in which the allocation can rise or fall until it reaches the endogenous rebalancing boundaries.5 This result is in sharp contrast to the classic MPT prediction that the allocation ratio between any two assets is con stant over time. Our model is among the first to show Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS how a double barrier policy arises in a setting with incomplete markets, unspanned risks, and illiquidity. We examine the endogenous rebalancing boundaries over time and show that the two types of illiquidity— arising from lockups and transaction costs—interact over time. As an automatic liquidity event approaches, the investor becomes less willing to liquidate alternative as sets. The rebalancing policies are also strongly affected by liquidity diversification; investors who stagger the maturi ties of their alternative asset investments over time can maintain more stable portfolio allocations, which results in higher ex ante allocations to alternatives. We calibrate our model and evaluate the effects of liquidity diversification by exogenously varying the num ber of alternative asset investments and then comparing the investor’s optimal portfolio allocations. We show that the investor’s ideal allocation to alternative assets is higher as the number of distinct alternative investments increases and the investor’s welfare is higher. The results show that the benefits of liquidity diversification are reached rapidly, and only a small number of distinct investments are required to realize most of the benefits. The calibrated results show that investors’ preferences for smooth intertemporal spending have first order effects on their allocations to illiquid assets. We use Epstein and Zin (1989) preferences, which separate risk aversion from the elasticity of intertemporal substitution (EIS). This sep aration is economically important as, by varying the EIS, we conveniently capture the heterogeneity in spending flexibility. For example, defined benefit pension plans have little spending flexibility and so have a low EIS. In contrast, family offices have high spending flexibility. In contrast to the full-spanning case in which the investor’s portfolio allocation is independent of the EIS, the EIS significantly affects portfolio allocations when the alternative asset is illiquid and its risk is unspanned by publicly traded assets. This is the realistic scenario. An investor with a high EIS can accept higher allocations to the illiquid asset as they are more willing to substitute spending across states and over time. This flexibility of deferring spending with little utility loss boosts the investor’s ability to make long-term illiquid investments. We show that allocation results crucially depend on the compensation the alternative asset provides for liquidity risk and skill. This is important as there is significant vari ation in allocations to alternative assets even within inves tor types (e.g., endowments or pension funds). And it is consistent with empirical findings of large and persistent heterogeneity in investors’ realized excess returns on alter native investments.6 Our quantitative results show that asset allocations are sensitive to the unspanned volatility of the alternative asset. We further show that, controlling for the level of the alternative asset’s total risk, the spanned and unspanned risks have quantitatively very different effects on asset allo cation. Whereas the investor can offset the alternative Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 3 Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS asset’s spanned risk by adjusting allocations to public equity, unspanned volatility is specific to the alternative and cannot be hedged. Alternative asset performance metrics such as internal rates of returns and public market equivalents, although useful, do not directly guide inves tors’ asset allocation as these metrics ignore the distinction between spanned and unspanned volatilities. We extend the model to include new capital contribu tions (donations) into the portfolio (Section 6) and a mini mum spending-rate constraint (Section 7). The results for contributions show that contributions increase alloca tions to alternative assets and, by providing an inflow of liquidity, decrease the variation in allocations over time. Contributions also result in higher and more stable spending rates. The results for a spending constraint show large effects on allocations to alternative assets. Even if the spending constraint rarely binds, the states in which it binds are those with high allocations to alterna tive assets, resulting in the investor choosing a signifi cantly lower allocation to illiquid securities. We also extend the model to include the possibility of crisis states. During crisis states, Robinson and Sensoy (2016) and Brown et al. (2021) document that capital calls are significantly higher in crisis and distributions to investors are much lower, and Ramadorai (2012) and Nadauld et al. (2019) document that secondary market transaction costs are much higher. To capture these im portant institutional features, we extend our model to include stochastic arrivals of crisis states, for example, in Barro (2006) and Wachter (2013), during which alternative assets become even more illiquid than usual. We find that investors’ holdings of alternative assets often significantly deviate from the optimal target allocations, and hence, the utility loss from being unable to hedge stochastic calls and distributions can be large in the crisis state. Our paper incorporates private equity and other alter native asset funds into the dynamic asset allocation liter ature. The most closely related papers are Sorensen et al. (2014) and Ang et al. (2016). Sorensen et al. (2014) focus on a single investment in isolation and do not consider the possibility of staggering alternative investments over time, nor do they realistically address alternative assets as a component of a larger portfolio. Ang et al. (2016) develop an optimal asset allocation model with both illiquid and liquid assets. In their model, illiquidity is due to the assumption that assets cannot be traded for a random duration governed by a Poisson process. The prior portfolio choice literature (with illiquid assets) can be broadly divided into two branches. One branch models illiquidity from trading restrictions in which the asset is freely tradable at certain points in time but cannot be traded at other times, for example, Longstaff (2001, 2009), Kahl et al. (2003), Gârleanu (2009), and Dai et al. (2015).7 The other branch of the literature models illiquid ity arising from transaction costs, for example, Davis and Norman (1990), Grossman and Laroque (1990), Vayanos (1998), Lo et al. (2004), Collin-Dufresne et al. (2012), and Gârleanu and Pedersen (2013, 2016).8 Motivated by the structures of private equity and hedge funds as well as the secondary markets for these illiquid alternatives, we combine the features of both types of models. In our model, the alternative asset becomes fully liquid at maturity (e.g., when a private equity fund is dissolved). But the alternative asset can also be sold prior to maturity by paying a proportional trans action cost, such as by selling a private equity fund at a discount in the secondary market (see, for example, Ramadorai 2012, Nadauld et al. 2019). We show that these two types of illiquidity interact and this interaction varies over the life cycle of the alternative asset. We further show how liquidity diversification—holding multiple dis tinct investments in the alternative asset with staggered lockup expirations—affects portfolio choice. Finally, we generalize our model to include crisis states featuring increases in both types of illiquidity and show that inves tors’ holdings deviate significantly from target allocations. Finally, our paper contributes to the literature by pro viding a rigorous foundation for analyzing the endow ment model. Although the endowment model is highly influential to practice and is used to allocate trillions of dollars, it is based on ad hoc rules of thumb and practi tioner’s lore. Our model formalizes the endowment model by developing a generalized dynamic portfolio theory with the key features of illiquid private equity. 2. Model We analyze a long-term investor’s dynamic spending (or equivalently consumption) and asset allocation decisions by incorporating an illiquid investment opportunity into the classic modern portfolio theory developed by Merton (1969, 1971) and Samuelson (1969). We interpret the illiq uid investment opportunity in our model as the repre sentative portfolio of alternative assets including private equity, hedge funds, private real estate, etc. For technical convenience, we develop our model in continuous time. 2.1. Liquid Investment Opportunities: Bonds and Public Equity The risk-free bond pays interest at a constant (annual ized) risk-free rate r. Public equity can be interpreted as the market portfolio of publicly traded securities, and its cum-dividend market value, St, follows a geometric Brownian motion (GBM): dSt � µS dt + σS dBSt , St (1) where BSt is a standard Brownian motion and µS and σS are the constant drift and volatility parameters. The Sharpe ratio for public equity is ηS � (µS � r)=σS : The liquid investment opportunity in our model is the same as in Merton (1971). Next, we introduce the alternative Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 4 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. asset, which is the investor’s third investment opportu nity and the key building block in our model. 2.2. The Alternative Asset Adding the alternative asset expands the investment opportunity set and, thus, makes the investor better off. Additionally, provided the alternative asset is not per fectly correlated with public equity, it provides diversifica tion benefits. Unlike public equity, however, alternative assets are generally illiquid and involve some form of lockup. For example, private equity funds typically have 10-year life spans, hedge funds often have lockup periods and gate provisions, and private real estate often has lim ited liquidity for its secondary market. A key feature of alternative assets is that their illiquid ity is not constant over time. For example, private equity funds are highly illiquid for much of their lives but even tually mature and return liquid capital to their investors. We model these liquidity events as follows. Let {At ; t ≥ 0} denote the alternative asset’s fundamental value process with a given initial stock A0. The fundamental value refers to the fully realizable value of the asset if it is held to maturity. However, with illiquidity, at any time t prior to maturity, the asset’s fundamental value differs from its market value. Let {Kt ; t ≥ 0} denote the accounting value of the alternative asset holding process with a given initial stock K0. To capture the target finite duration of the lockup and holding period, we assume every mT years, where m is a positive integer, a δT fraction of the stock of illiquid alternative asset KmT automatically becomes liquid at no cost. Naturally, the investor’s liquid asset value at time mT increases by δT KmT� . Therefore, in the absence of any active acquisition or divestment of the illiquid asset at mT, we have KmT � (1 � δT )KmT� . 2.2.1. Fundamental Value Process A for the Alterna tive Asset. We assume that the fundamental value A, in the absence of a scheduled automatic liquidity event (at time mT) or any interim acquisition or divestment, ev olves via the following GBM: dAt � µA dt + σA dBA t � δA dt, At� (2) where BA t is a standard Brownian motion, µA is the cumpayout expected return (net of fees), σA is the constant volatility of returns, and δA is the alternative asset’s pay out rate. That is, the alternative asset pays dividends at the rate of δA At with an implied payout yield of δA. Intu itively, δA is one way for illiquid alternative assets to pro vide liquidity to investors. We use ρ to denote the correlation coefficient between the shocks to alternative S assets, BA t , and the shocks to public equity, Bt . Note that, in complete markets, the investor can fric tionlessly and dynamically trade the alternative asset without restrictions or costs. Therefore, the alternative asset’s market value equals its fundamental value, and the Modigliani–Miller theorem holds, meaning that whether we explicitly model the alternative asset’s pay out yield δA is irrelevant. In this ideal case, the alternative asset is conceptually no different than liquid public equity. In contrast, when the alternative asset is illiquid and not fully spanned by public equity, we must sepa rately keep track of the payout yield δA and expected capital gains µA � δA . That is, the cum-dividend return µA is no longer a sufficient measure of the total expected returns for the alternative asset as its (current) payout yield and expected capital gains influence the investor’s portfolio optimization problem differently. 2.3. Interim Acquisition and Liquidation of the Alternative Asset Holding At any time, the investor can choose to change the alter native asset holdings through acquisitions or liquida tions. Let dLt denote the amount of the alternative asset that the investor liquidates at any time t > 0 and let dXt denote the amount of the alternative asset that the inves tor purchases at time t. Then, we can incorporate the investor’s acquisition and liquidation options into the alternative asset’s fundamental value process as follows: dKt � (µA � δA )Kt� dt + σA Kt� dBA t � dLt + dXt � δT Kt� I{t�mT} : (3) Here, I{t�mT} is the indicator function, which is equal to one if and only if t is an integer multiple, m, of T. The first two terms correspond to the standard drift and vola tility terms, the third and fourth terms give the liquida tion and acquisition amounts, and the last term captures the lumpy payout to the investor at the scheduled liquid ity event dates t � mT, where m � 0, 1, : : : Although the acquisition and liquidation costs for the alternative asset do not appear in (3), they appear in the liquid wealth accumulation process. We assume that the cost of voluntary liquidation is proportional. That is, by liquidating an amount dLt > 0, the investor realizes only (1 � θL )dLt in net, and the remaining amount θL dLt is the liquidation cost. Similarly, if the investor acquires an amount dXt > 0, the transaction cost θX dXt is paid out of the liquid asset holding. Naturally, 0 ≤ θL ≤ 1 and θX ≥ 0. Higher values of θL or θX indicate that the alter native asset is less liquid. Intuitively, θL can be interpreted as the illiquidity dis count on secondary market sales of alternative assets (e.g., see Kleymenova et al. 2012, Albuquerque et al. 2018, Nadauld et al. 2019). Such discounts can arise to compensate buyers for search costs or asymmetric infor mation risks or because of market power when there are few buyers. The parameter θX can be interpreted as the transaction costs of purchasing alternative assets, such as search costs, legal fees, placement agent fees, consultant fees, etc. The costs of interim liquidation (θL) and pur chases (θX) can be asymmetric as voluntary liquidation Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 5 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS is generally more costly, particularly when there are few buyers and many sellers, such as during the recent finan cial crisis. Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 2.3.1. Alpha, Beta, and Epsilon (Unspanned Volatil ity). Suppose that the instantaneous return for the alter native asset, dAt =At� , is perfectly measurable. We can then regress dAt =At� on dSt =St and obtain the alterna tive asset’s beta with respect to public equity: ρσA : (4) βA � σS However, in reality, because investors cannot dynami cally rebalance their holdings in the illiquid asset without incurring transaction costs, investors demand compensa tion in addition to the risk premium implied by the covariance with public equity. We decompose the total volatility of the alternative asset, σA, into two orthogonal components: the part spanned by the public equity, ρσA , and the remaining unspanned volatility, ε, given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ε � σ2A � ρ2 σ2A � σ2A � β2A σ2S : (5) This volatility, ε, introduces an additional risk into the investor’s portfolio as markets are incomplete and ad justing the alternative asset holding is costly. We show that the spanned and unspanned volatilities play distinct roles in the investor’s dynamic asset allocation.9 Anticipating our subsequent risk–return trade-off an alysis in the context of dynamic portfolio construction, we next introduce the α implied by a single-index model using public equity. That is, we define α as follows: α � µA � (r + βA (µS � r)), (6) where βA is the alternative asset’s beta given by (4). In frictionless capital markets in which investors can continuously rebalance their portfolio without incurring any transaction costs, α measures the risk-adjusted ex cess return after benchmarking against public equity. However, importantly, in our framework with illiquid assets, α also includes compensation for bearing any sys tematic risk that is unspanned by public equity, which for simplicity, we refer to as illiquidity premium. 2.4. Optimization Problem 2.4.1. Liquid Wealth and Net Worth. We use W to denote the investor’s liquid wealth and Π to denote the amount allocated to public equity. The remaining liquid wealth, W � Π, is allocated to the risk-free bond. Thus, liquid wealth evolves according to dWt � (rWt� + δA Kt� � Ct� )dt + Πt� ((µS � r)dt + σS dBSt ) + (1 � θL )dLt � (1 + θX )dXt + δT Kt� I{t�mT} , (7) where the first two terms in (7) are the standard ones in Merton’s consumption/portfolio choice problem. The third and fourth terms describe the effect on liquid wealth W resulting from the investor’s interim liquida tion and purchase of the alternative asset, where θL and θX capture the proportional cost of interim liquidations and purchases of the alternative asset, respectively. Finally, the last term captures the lumpy payout to the investor at the automatic liquidity event dates t � mT. 2.4.2. Recursive Preferences and Value Functions. The investor’s preferences allow for separation of risk aver sion and EIS. Epstein and Zin (1989) and Weil (1990) develop this utility in discrete time by building on Kreps and Porteus (1978). We use the continuous-time formula tion of this nonexpected utility, introduced by Duffie and Epstein (1992). That is, the investor has a recursive preference defined as follows: �Z ∞ � Vt � Et f (Cs , Vs )ds , (8) t where f(C, V) is known as the normalized aggregator for consumption C and the investor’s utility V. Duffie and Epstein (1992) show that f(C, V) for Epstein–Zin nonex pected homothetic recursive utility is given by ζ C1�ψ � ((1 � γ)V)χ f (C, V) � , 1 � ψ�1 ((1 � γ)V)χ�1 �1 (9) where χ� 1 � ψ�1 : 1�γ (10) The parameter ψ > 0 measures the EIS, γ > 0 is the coef ficient of relative risk aversion, and ζ > 0 is the investor’s subjective discount rate. This recursive, nonexpected utility formulation allows us to separate the coefficient of relative risk aversion (γ) from the EIS (ψ), which is important for our quantitative analysis. For example, a key source of preference hetero geneity among investors is the elasticity and flexibility of their spending. The expected constant relative risk aver sion (CRRA) utility is a special case of recursive utility in which the coefficient of relative risk aversion, γ, equals the inverse of the EIS, γ � ψ�1 , implying χ � 1.10 There are three state variables for the optimization problem: liquid wealth Wt, the alternative asset’s value Kt, and calendar time t. Let V(Wt , Kt , t) denote the corre sponding value function. The investor chooses consump tion C, public equity investment Π, and the alternative asset’s cumulative (undiscounted) liquidation L and cu mulative (undiscounted) acquisition X to maximize (8). Naturally, at each automatic liquidity event date iT, if WiT � W(i�1)T � W and KmT � K(m�1)T � K, we must have V(W, K, mT) � V(W, K, (m � 1)T): (11) Hence, it is sufficient for us to characterize our model over (0, T] as the solution is stationary every T years. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 6 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 3. Model Solution We solve the model as follows. First, we analyze the investor’s problem in the region in which there is no vol untary adjustment of the alternative asset in the absence of automatic liquidity event (i.e., when t ≠ mT). Second, we characterize the investor’s voluntary liquidation and acquisition decisions for the alternative asset when t ≠ mT. Finally, we integrate the periodic liquidity event that occurs at t � mT to complete our analysis. allocation to fulfill two objectives: to obtain the desired mean-variance exposure and hedge the fraction of the alternative asset’s risk spanned by public equity. 3.2. Certainty Equivalent Wealth P(W , K, t) We express the investor’s value function V(W, K, t) dur ing the time period t ∈ ((m � 1)T, mT) as V(W, K, t) � (b1 P(W , K, t))1�γ , 1�γ (15) 3.1. Dynamic Programming and First Order Conditions (FOCs) Fix time t within the time interval ((m � 1)T, mT), where m is a positive integer. Using the standard dynamic pro gramming approach, we have the following standard Hamilton–Jacobi–Bellman (HJB) equation for the inves tor’s value function V(Wt , Kt , t) in the interior region: where b1 is a constant given by 0 � max f (C, V) + (rW + δA K + (µS � r)Π � C)VW C, Π Guided by MPT, we can interpret P(W, K, t) as the inves tor’s certainty equivalent wealth, which is the minimal amount of total wealth required for the investor to per manently give up the opportunity to invest in the alter native asset. Thus, imagine that, at any time t, the investor has two options: either (1) adhere to the optimal portfolio and spending plan prescribed by the model or (2) surrender both the liquid asset W and illiquid asset holdings K in exchange for immediately and perma nently giving up the opportunity to invest in the alterna tive asset but with a liquid wealth level of Ω from which the investor can continuously spend and rebalance bet ween public equity and bonds. Liquid wealth Ω � P(W, K, t) makes the investor indifferent between these two options. Mathematically, in the interim period in which (m � 1)T < t < mT, the following equation holds: (ΠσS )2 + VWW + Vt + (µA � δA )KVK 2 σ2 K2 + A VKK + ρΠKσS σA VWK : 2 (12) The first three terms on the right side of (12) capture the standard effects of consumption and asset allocation (both drift and volatility effects) on the investor’s value function, V(Wt , Kt , t) as in Merton (1971). The investor’s opportunity to invest in the illiquid alternative asset gen erates three additional effects on asset allocation: (1) the effect of target holding horizon T captured by Vt, (2) the risk–return and volatility effects of changes in the value of the alternative asset K, and (3) the additional diversification/hedging benefits resulting from the corre lation between public equity and the alternative asset. By optimally choosing C and Π, the investor equates the right side of (12) to zero in the interior region in which there is no interim liquidation nor acquisition. The optimal consumption C is characterized by the fol lowing standard FOC: fC (C, V) � VW (W, K, t), (13) which equates the marginal benefit of consumption with the marginal value of savings VW. The optimal invest ment in public equity is given by Π�� ηS VW ρσA KVWK � : σS VWW σS VWW (14) The first term gives the classical Merton’s mean-variance demand, and the second term captures the investor’s hedging demand with respect to the illiquid alternative asset. Note that the hedging demand depends on the cross-partial VWK and is proportional to ρσA =σS (which is equal to βA as shown in (4)). Both results are intuitive and follow from the standard hedging arguments in Merton (1971); the investor chooses the public equity ψ 1 b1 � ζψ�1 φ11�ψ , and φ1 is the constant given by � � η2S : φ1 � ζ + (1 � ψ) r � ζ + 2γ V(W, K, t) � J(P(W, K, t)): (16) (17) (18) Here, J(·) is the value function for an investor who can invest only in liquid public equity and risk-free bonds. We show that J(·) is given by J(W) � (b1 W)1�γ , 1�γ (19) where b1 given in (16) is the same constant appearing in the value function for the classic Merton’s problem. We emphasize that certainty-equivalent wealth P(W, K, t) is more natural and intuitive than the investor’s value function V(W, K, t) to measure welfare. This is because the unit for P(W, K, t) is dollars, whereas the unit for V(W, K, t) is utils. Appendix B contains a proof for the characterization of the certainty-equivalent wealth P(W, K, t). 3.3. Homogeneity Property In our model, the certainty equivalent wealth P(W, K, t) has the homogeneity property in W and K, and hence, it is convenient to work with the liquidity ratio wt � Wt =Kt Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 7 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS and the scaled certainty equivalent wealth function p(wt , t) defined as follows: Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. P(Wt , Kt , t) � p(wt , t) · Kt : (20) This homogeneity property is due to the Duffie– Epstein–Zin utility and the value processes for public equity and the alternative asset. Importantly, this homo geneity property allows us to conveniently interpret the optimal portfolio rule and target asset allocation. 3.4. Endogenous Effective Risk Aversion gi To better interpret our solution, it is helpful to introduce the following measure of endogenous relative risk aver sion for the investor, denoted by γi (w, t) and defined as follows: VWW γi (w, t) ≡ � × P(W, K, t) VW p(w, t)pww (w, t) � γpw (w, t) � : (21) pw (w, t) In (21), the first identity sign gives the definition of γi , and the second equality follows from the homogeneity property. What economic insights does γi (w, t) capture, and what is the motivation for introducing it? First, recall that the standard definition of the investor’s coefficient of absolute risk aversion is �VWW =VW . To convert this to a measure of relative risk aversion, we need to multi ply absolute risk aversion �VWW =VW with an appropri ate economic measure for the investor’s total wealth. Under incomplete markets, although there is no marketbased measure of the investor’s economic well-being, the investor’s certainty equivalent wealth P(W, K, t) is a nat ural measure of the investor’s welfare. This motivates our definition of γi in (21).11 We show that the illiquidity of alternative assets causes the investor to be effectively more risk-averse, meaning pw (w, t) > 1 and pww (w, t) < 0 so that γi (w, t) > γ. In contrast, if the alternative asset is publicly traded (and markets are complete), γi (w, t) � γ as pw (w, t) � 1 and pww (w, t) � 0. 3.5. Optimal Policy Rules Again, by using the homogeneity property, we may ex press the scaled consumption rule c(wt , t) � C(Wt , Kt , t)= Kt as follows: c(w, t) � φ1 p(w, t) pw (w, t)�ψ : (22) Because illiquidity makes markets incomplete, the inves tor’s optimal consumption policy is no longer linear and depends on both the certainty equivalent wealth p(w, t) and the marginal certainty equivalent value of liquid wealth pw (w, t). The allocation to public equity is Πt � π(wt , t)Kt , where π(w, t) is given by � � ηS p(w, t) ρσA γp(w, t) π(w, t) � � �w , (23) σS γi (w, t) σS γi (w, t) where γi (·) is the investor’s effective risk aversion given by (21). Intuitively, the first term in (23) reflects the mean-variance demand for the market portfolio, which differs from the standard Merton model in two ways: (1) risk aversion γ is replaced by the effective risk aversion γi (w, t), and (2) net worth is replaced by certainty equiv alent wealth p(w, t). The second term in (23) captures the dynamic hedging demand, which also depends on γi (w, t) and p(w, t). 3.6. Partial Differential Equation (PDE) for p(w, t) Substituting the value function (15) and the policy rules for c and π into the HJB Equation (12) and using the homogeneity property and the definition of the in vestor’s effective risk aversion, γi (w, t) given by (21), we obtain the following PDE for p(w, t) at time t for the liquidity ratio wt in the interior region and when (m � 1)T < t < mT: ! φ1 (pw (w, t))1�ψ � ψζ γσ2A + µA � δ A � p(w, t) 0� ψ�1 2 + pt (w, t) + ε2 w 2 pww (w, t) 2 + [(δA � α + γε2 )w + δA ]pw (w, t) � (ηS � γρσA )2 pw (w, t)p(w, t) + : 2γi (w, t) γε2 w2 (pw (w, t))2 2 p(w, t) (24) Because of incomplete spanning (e.g., ε ≠ 0), unlike Black–Scholes, (24) is a nonlinear PDE, and moreover, pw (w, t) > 1 as we show. The numerical solution for p(w, t) involves the standard procedure. Next, we analyze how the investor actively rebalances the allocation to the illiquid alternative asset. 3.7. Rebalancing the Illiquid Alternative Asset During the Interim Period Although, under normal circumstances, the investor plans to hold the alternative asset until an automatic liquidity event occurs at date mT, under certain circum stances, the investor may find it optimal to actively reba lance even at time t ≠ mT. As acquisition and voluntary liquidation are costly, we have an inaction region at all times, including t � mT. Let W t and W t denote the lower liquidation and upper acquisition boundaries for liquid wealth Wt at time t, respectively. Next, we sketch out the key steps and relegate technical details to Appendix B. To help understand how we obtain our key results, we first state the following two key conditions for the unscaled certainty-equivalent wealth P(W, K, t) implied by the standard value-matching and smooth-pasting conditions for the investor’s value function V(W, K, t) at Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 8 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. the lower liquidation boundary Wt : PK (Wt , Kt , t) � (1 � θL )PW (Wt , Kt , t), (25) PKW (Wt , Kt , t) � (1 � θL )PWW (Wt , Kt , t): (26) Equation (25) is the smooth-pasting condition, which states that the marginal certainty equivalent wealth of illiquid wealth, PK , must equal (1 � θL )PW , the marginal certainty equivalent wealth of liquid wealth when the investor sells a unit of the illiquid asset. This corresponds to the investor’s indifference condition when liquidating the alternative asset. Because the investor is optimally choosing the liquidation boundary, the derivatives of the two sides of (25) with respect to W must equal. This gives the supercontact condition (26) for the illiquid asset liquidation decision (Dumas 1991). Using the homogeneity property to simplify (25) and (26), we obtain the following smooth-pasting and super contact conditions at the lower liquidation boundary w t :12 p(w t , t) � (1 � θL + w t )pw (w t , t), pww (w t , t) � 0: (27) (28) Using essentially the same analysis, we obtain the fol lowing smooth-pasting and supercontact conditions at the upper acquisition boundary w t : p(w t , t) � (1 + θX + w t )pw (w t , t), pww (w t , t) � 0: (29) (30) In sum, it is optimal for the investor to keep the liquidity ratio wt within the (w t , w t ) region by voluntarily liquidat ing a portion of the alternative asset if wt is too high and acquiring the alternative asset if wt is too low. Next, we analyze the investor’s decision at t � mT when the portfolio’s liquidity changes discretely because of the automatic liquidity event, that is, when an alterna tive investment in the investor’s portfolio pays a lumpy liquidating dividend. course, the investor optimally decides considering both the “marginal analysis” for the liquidity ratio and the automatic liquidity event at t � mT. As a result, we have b mT ≤ two cases to consider at t � mT: case (i) in which w b mT > w mT . As the automatic w mT and case (ii) in which w liquidity event always increases liquid asset holdings, b mT is always larger than w mT . Hence, we need only w consider these two cases. b mT ≤ w mT , the optimal liquidity ratio In case (i) when w b mT as it is optimal for the investor not to volun at mT is w tarily rebalance the illiquid alternative asset holding. The intuition is that, even with the automatic increase in liquidity at mT, the liquidity ratio still lies within the inaction range limt→mT (wt , w t :). Therefore, the continuity cmT , of the value function implies P(WmT , KmT ) � P(W bmT ), which can be simplified as K b mT , t)(1 � δT ), lim p(w, t) � p(w t→mT b mT is given in (31). where w b mT > w mT , the optimal liquidity In case (ii) when w ratio at mT is w mT as it is optimal for the investor to vol untarily acquire the illiquid alternative asset. In this case, b mT > the automatic liquidity events result in wmT � w w mT , which means the investor holds too much of the liquid asset. To bring the portfolio liquidity ratio back into the inaction region, the investor must acquire more of the alternative asset so that wmT � w mT . In Appendix B, we show lim p(w, t) � p(w mT , mT)(1 � δT + λ), t→mT b mT ≡ w cmT W wt + δT � lim : b t→mT 1 � δT K mT (31) By now, we have outlined the procedures for calculating both w mT (ignoring the automatic liquidity event) and b mT (focusing only the automatic liquidity event). Of w (33) where λ reflects the effect of rebalancing and is given by wt + δT � w mT (1 � δT ) : t→mT 1 + θX + w mT λ � lim (34) Finally, the homogeneity property allows us to express the value-matching condition (11) in terms of p(w, t) at t � mT: p(w, mT) � p(w, (m � 1)T): 3.8. Value and Decisions When There is an Automatic Liquidity Event at t 5 mT At time t � mT, a fraction δT of the alternative asset auto matically becomes fully liquid without any voluntary cmT and K bmT to denote the corre liquidation. We use W sponding levels of liquid wealth and the alternative asset at t � mT if the investor chooses not to do any voluntary cmT � limt→mT (Wt + rebalancing. It is immediate to see W b b mT denote δT Kt ) and K mT � limt→mT (Kt � δT Kt ). Let w the corresponding liquidity ratio: (32) (35) Next, we summarize the main results of our model. Proposition 1. The scaled certainty equivalent wealth p(w, t) in the interim period when (m � 1)T < t ≤ mT solves the PDE (24) subject to the boundary conditions (27), (28), (29), (30), b mT ≤ and (35). Additionally, p(wmT� , mT�) satisfies (32) if w b mT > w mT , where w b mT is given by (31). w mT and (33) if w 4. Data and Calibration 4.1. Data and Summary Statistics As a guide to the calibration parameters and a bench mark for interpreting our findings, we use university endowment fund data from the National Association of College and University Business Officers and Common fund Endowment Fund Survey (NCES). See Brown et al. (2010, 2014) and Dimmock (2012) for more details. We Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 9 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS focus on the cross-section of 774 university endowment funds as of the 2014–2015 academic year end. Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 4.1.1. Asset Allocation. NCES provides annual snap shots of endowment funds’ portfolio allocations. To link the NCES data to the model, we aggregate endowment allocations in the NCES data into the three asset classes in our model: (1) the risk-free asset, which aggregates cash and fixed income; (2) public equity, which aggre gates public equity and real estate investment trusts (REITs); and (3) the alternative asset, which aggregates hedge funds, private equity, venture capital, private real estate, and natural resources. For determining some of the calibration parameters, we use the disaggregated subasset classes (e.g., venture capital), which are reported in Appendix D. Table 1 shows the summary statistics as of the end of the 2014–2015 academic year. The first and second col umns show the equal- and value-weighted averages, respectively. The remaining columns show averages with in various size categories of endowment funds (e.g., “0%–10%” summarizes the variables for the smallest dec ile of funds). The average endowment size is $677 million, but the distribution is highly positively skewed, and the median decile size is $116 million. On an equal-weighted basis, public equity has the largest average allocation at 50.7%. On a value-weighted basis, alternative allocations has the largest average allocation at 57.1% compared with 32.0% for public equity and 10.8% for cash and fixed income. The average spending rate is 4.2%. 4.1.2. Portfolio Illiquidity and Target Horizons. Table 1 reports the average number of alternative asset funds held by the endowments, which is an important compo nent of liquidity management. Suppose two endowment funds have the same allocation to alternative assets, but endowment A holds a single private equity fund with a 10-year lockup and endowment B staggers its holdings across 120 different private equity funds such that one lockup expires every month. Although both endow ments have the same allocations, their liquidity expo sures are very different. Endowment A can only adjust its exposure through the secondary market, whereas endowment B can costlessly adjust its exposure as lock ups expire each month. Thus, by holding multiple in vestments with staggered maturities, the endowment can enhance the liquidity of its portfolio, which we refer to as liquidity diversification. In our quantitative analy sis, we explore the relation between liquidity diversifica tion and investor welfare. Table 1 shows there is a strong positive relation bet ween endowment size and the number of alternative asset funds. On average, endowments in the largest dec ile hold 86.5 alternative asset funds; endowments in the smallest decile own only a single fund. Thus, liquidity diversification is more effective for larger endowments, lowering the unspanned risk. We also estimate the average target holding period for alternative assets based on investors’ portfolio allocations and the horizons of each subasset class within alterna tives.13 Table 1 reports the average alt target horizon, the period when the alternative investment is locked up, is 4.2 years for the full sample and 5.9 years for the largest decile of funds. 4.1.3. Parameter Choices and Calibration. Table 2 sum marizes the baseline parameter values. Following the lit erature, we choose the following standard parameter values. The investor’s coefficient of relative risk aversion is set to γ � 2. We set the EIS to be ψ � 0:5 so that it corresponds to expected utility with γ � 1=ψ � 2. We set the annual risk-free rate r � 4%, and we also set the Table 1. Summary of Endowment Fund Asset Allocation Endow. size ($M) Cash & fixed inc. Public equity Alternatives Spending rate No. alt. funds Alt. target horizon Average Value-weighted average 0%–10% 45%–55% 90%–100% 677 21.0% 50.7% 28.3% 4.2% 16.9 4.2 12.7 35.6 51.7 4.4 56.0 5.5 17 33.1 60.1 6.3 4.5% 1.1 3.6 116 22.4 54.7 22.9 3.9 7.2 4.0 13,409 10.8 32.0 57.1 4.5 86.5 5.9 Notes. This table summarizes endowment fund portfolios as of the end of the 2014–2015 academic year for 774 endowments. The first two columns show the equal- and value-weighted averages, respectively. The columns 0%–10% to 90%–100% show averages within size-segmented groups of endowment funds. For example, the column 0%–10% shows the value-weighted average portfolio allocation for the smallest decile of endowment funds. The table shows summary statistics for endowment fund size (reported in millions of dollars), asset class allocations and spending rates (reported in percentages), the number of alternative asset funds that the endowment holds, and the average target horizon for the alternative assets. Cash & fixed income includes cash, cash equivalents, and fixed income securities (except for distressed securities). Public equity includes domestic and foreign equity as well as REITs. Alternatives includes hedge funds, private equity, venture capital, private real estate, and illiquid natural resources. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 10 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Table 2. Summary of Key Parameters Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Parameter Coefficient of relative risk aversion Elasticity of intertemporal substitution Subjective discount rate Risk-free rate Public equity expected return Volatility of market portfolio Beta of the alternative asset Alternative asset alpha Alternative asset expected return Volatility of alternative asset Alternative asset target holding horizon Proportional cost of liquidation Proportional cost of acquisition Implied parameters Correlation between risky assets Unspanned volatility Payout rate Symbol Value γ ψ ζ r µS σS βA α µA σA H θL θX 2 0.5 4% 4% 10% 20% 0.6 2% 9.6% 19.2% 6 0.1 0.02 ρ ε δA 0.625 15% 4.00% investor’s discount rate equal to the risk-free rate, ζ � r. For public equity, we use an annual volatility of σS � 20% and an aggregate equity risk premium of µS � r � 6%. We calibrate the properties of the alternative asset by building up from the university endowments’ allocations and the characteristics found in the literature. Appendix D provides the details and additional discussion. We set the alternative asset βA � 0.6 and the unspanned volatil ity of the alternative asset to ε � 15%. We set the horizon of the representative alternative asset H � 6 years. In our model, the alpha of the illiquid alternative investment includes compensation for skill, value added from governance, liquidity risk, and other risks un spanned by public equities. We set α � 2%, which we view as reasonable given the empirical findings in the lit erature. For example, Franzoni et al. (2012) find that private equity earns a net-of-fees liquidity risk premium of 3% annually. Aragon (2007) and Sadka (2010) find similar net-of-fees liquidity risk premia for hedge funds. Given this assumed alpha, the expected overall return on the alternative asset is µA � 0:02 + 0:04 + 0:6 × (0:10� 0:04) � 0:096 � 9:6%. For voluntary liquidations, we assume that the pro portional transaction cost is θL � 10%. Appendix D pro vides details showing how we reach this cost by building up from asset allocations and empirical evi dence on secondary market discounts (see Kleymenova et al. 2012, Ramadorai 2012, Nadauld et al. 2019). For acquisitions, we assume that the proportional acquisi tion cost is θX � 2%, which is equal to the average placement agent fee reported by Rikato and Berk (2015) and Cain et al. (2020). Calibrating the model also requires a payout para meter, which determines the liquidity generated by automatic liquidity events (e.g., from intertemporally staggered investments in the alternative asset maturing and paying out capital). The payout rate depends on the number of distinct investments into the alternative asset. For example, given the target horizon of H � 6 years, an investor with a single alternative asset investment re ceives a large payout once every six years. In contrast, an investor with a large number of distinct investments receives smaller but more frequent payouts. For any given number of investments, denoted by i, Appendix E shows how it is possible to impute the payout rate using the previously described parameter values. For our base line calibration, we use i → ∞, which implies a continu ous payout rate of δA � 4:0%. For comparison, we also consider cases with i � 1 and i � 6. 5. Quantitative Results In this section, we analyze the model using the parame ter values from Table 2. As a benchmark, we also analyze the case when the alternative asset is fully liquid.14 5.1. Certainty Equivalent Wealth and Net Worth We introduce the widely used net worth as the account ing value of the investor’s portfolio: Nt ≡ Wt + Kt : (36) In general, because of illiquidity, net worth is not an eco nomic measure of the investor’s true welfare. Figure 1 plots P(Wt , Kt , t)=Nt , the ratio of the certainty equivalent wealth to the portfolio’s book value (net worth) Nt, as a function of zt � Kt Kt 1 � � , N t Wt + K t wt + 1 (37) the proportion of the portfolio allocated to alternative assets. Recall that we use the liquidity ratio, wt, as the effective state variable when analyzing the model and its solution in Sections 2 and 3. Here, we use zt to exposit our quantitative results as practitioners typically work with portfolio allocations. Also, note that zt is typically between zero and one, making the results easier to interpret. As the optimal w is a range (w, w) and z decreases with w, the corresponding range for the optimal z is (z, z), where 1 1 z� and z � : (38) w+1 w+1 Therefore, the lower liquidation boundary w maps to the upper liquidation boundary z, and the upper acquisition boundary w maps to the lower acquisition boundary z: Let z∗t denote the “desired” target of zt � Kt =(Wt + Kt ). Mathematically, z∗t is the value of zt at which the investor attains the highest certainty-equivalent wealth for a given level of Nt, P(Wt , Kt , t)=Nt , where P(Wt , Kt , t) p(wt , t) � � zt p((1 � zt )=zt , t): Nt wt + 1 (39) Figure 1 includes the case of i → ∞, and for comparison, it also includes the cases of i � 1 and i � 6. For i → ∞, we see that z lies between (z, z) � (27:5%, 64:9%): That Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Figure 1. (Color online) Plot of P=N � p(w, t)=(w + 1), the Ratio of the Certainty Equivalent Wealth P(W, K, t) � p(w, t)K and Net Worth N � W + K as a Function of the Portfolio’s Per centage Allocation to Alternative Assets z � K=N Notes. The lines show results for the inaction region in which the left (right) ends of the lines are the lower (upper) rebalancing boundaries. For the i � ∞ case with δA � 4%, the no-trade region is (z, z) � (0:275, 0:649) and the desired target is z∗ � 0:345, which corresponds to the maximal P=N value of 1.0785 (the dashed). Other parameter values are given in Table 2. For the i � 1 and i � 6 cases, the figure shows results at t � mT. is, if the allocation to alternatives z falls to the endoge nous acquisition boundary, z � 27:5%, the investor im mediately sells just enough units of the liquid assets and invests the proceeds in the illiquid alternative asset to keep z ≥ 27:5%. If the allocation to alternatives rises to the endogenous liquidation boundary, z � 64:9%, the investor sells just enough units of the illiquid asset so that z falls back to 64.9%. Hypothetically, if investors could costlessly choose z, they would choose the desired target z∗ � 34:5%. At this point, certainty equivalent wealth is 7.85% higher than net worth. The curve is noticeably asymmetric and declines more rapidly to the right of the maximum as the investor approaches the voluntary liquidation boundary because liquidating alternative assets is more costly than acquiring them (i.e., θL � 10% > θX � 2%). In sharp contrast, when the alternative asset is per fectly liquid, as in the case of full spanning, the admissi ble illiquid alternative asset holding is not a range, but instead is a singleton with the value of z∗ � 44:4%. For the case of i � 1, the rebalancing boundaries are fur ther to the left, indicating that the investor holds less of the alternative asset when there is less liquidity diversifi cation from staggering maturities across time. The curve for i � 6 is similar to that for i → ∞, indicating that even a moderate number of distinct investments in the alterna tive asset provides benefits from liquidity diversification. 11 5.2. Rebalancing Boundaries Figure 2 shows the rebalancing boundaries over a period (m � 1)T < t < mT for any positive integer m ≥ 1. We plot the optimal liquidation and acquisition boundaries in panels (a) and (b), respectively. First recall that, for the i � ∞ case, the rebalancing boundaries are constant over time, which corresponds to two horizontal lines at z � 27:5% and z � 64:9%. Next, we turn to the cases that fea ture time-varying no-trade regions. The figure shows the cases of i � 1, 3, and 6. In our discussion, we focus on the case of i � 1 (the investor has made only a single investment into the alternative asset) because the i � 1 case provides the greatest contrast with the i � ∞ case. In the case of i � 1, there is an automatic liquidity event every six years at which time the alternative asset becomes fully liquid (see Appendix E for details). The differences between the cases of i → ∞ and i � 1 high light one of the unique features of our model: that it can accommodate the liquidity diversification from investing in illiquid assets with staggered lockup expirations. The initial rebalancing boundaries at time t � 0 are lower for the i � 1 case than the i → ∞ case because the effective cost of illiquidity resulting from trading restric tions is greater, resulting in lower demand for the illiquid asset. The comparison between the cases of i � 1 and i → ∞ illustrates the interconnection of illiquidity from transactions costs and trading restrictions. For the case of i � 1, both boundaries increase as t → mT. This means that the investor becomes less willing to liqui date alternative assets and more willing to voluntarily acquire alternative assets as the automatic liquidity event at t � mT approaches. This is intuitive as the investor’s liquid holdings increase significantly at t � mT. Anticipating the natural liquidity event, the inves tor is more willing to accept large allocations to the illiquid asset. As a result, as t → mT, the liquidation boundary z t becomes exceedingly large, and the acqui sition boundary, z t also increases. Note that the quanti tative effects for the acquisition boundary are smaller than for the liquidation boundary. Figure 2 shows that investors with fewer distinct investments in the alternative assets experience less sta ble portfolio allocations and deviate from their desired target to a larger extent. Given that smaller investors are more likely to hold fewer distinct investments, this figure highlights a potential reason for the empirical correlation between investor size and alternative asset allocations: that small investors are less able to engage in liquidity diversification. 5.3. Comparative Statics In this section, we conduct comparative static analysis for the i → ∞ case. As shown earlier, because of transac tion costs, the model generates an illiquid asset no-trade region (z, z). To ease interpretation, we report asset allo cations and the spending rate at the desired target, z∗ (at Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 12 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Figure 2. (Color online) Rebalancing Boundaries for the Portfolio’s Percentage Allocation to Alternative Assets over Time: z � K=N, Where N � W + K Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. (a) (b) Notes. In panels (a) and (b), we plot the (upper) time-dependent liquidation boundary z t and the (lower) acquisition boundary, z t , respectively. All parameter values (other than the number of distinct investments i) are given in Table 2. For the stationary case (i → ∞), the annual payout rate is set at δA � 4%, and the optimal z lies within (z t , z t ) � (0:275, 0:649) for all t (see the two dash-dotted horizontal lines). For the cases with finite numbers of investments, for example, i � 1, i � 3, and i � 6, the payout rates are, respectively, δT � 21:34% every six years (H � 6), δT � 7:69% every two years (H � 2), and δT � 3:92% every year (H � 1). The payout rates across the four cases are set so that they have effectively the same total payouts over long time periods. which the investor’s value function and certainty equiva lent wealth are maximized) defined in the text preceding (39) and shown in Figure 1. The columns “Region” and “Deviation” show, respectively, the rebalancing bound aries for the allocation to the alternative asset and the average deviation from the desired target for the given parameter values. The tables also show the average devi ation of the spending rate from the desired target. For each table, the row in bold font shows results for the baseline case in which the parameter values are given in Table 2. 5.3.1. EIS c. Table 3 shows comparative static effects of changing the EIS ψ. Panel A reports results for four cases ranging from very low to high values of the EIS: ψ � 0:1, 0:5, 1, 2. Panel B shows results for the case of full spanning. We see that varying the EIS has very large quantitative effects on the spending rate. An investor who is unwilling to substitute spending over time (e.g., ψ � 0:1) has a spending rate of 6.36%, which is relatively high (in light of the permanent income argument). In contrast, an investor who is willing to substitute con sumption over time, (e.g., ψ � 2 as in the long-run risk literature following Bansal and Yaron 2004), has a spend ing rate of only 1.33%. The intuition is that an investor with a high EIS defers spending to exploit the investment opportunity. Not only does the EIS have a first order effect on spending, it also influences asset allocation. As the EIS increases, the investor allocates more to the illiquid alter native asset and less to public equity and bonds. For example, an investor with ψ � 0:1 allocates 56.20% to public equity and 30.77% to alternatives compared with an investor with ψ � 2 who allocates 44.39% to public equity and 50.25% to alternatives. The large effect of the EIS on asset allocation in our incomplete markets model is due to the interactive effect between asset allocation and optimal spending policies. An investor with a higher EIS is more willing to defer consumption in response to better investment opportunities rather than engage in costly liquidation of alternative assets. This allows for both a higher target allocation to alternative assets and a wider illiquid asset no-trade region. This contrasts with the case of full spanning, in which the EIS has no effect on asset allocation (see Equations (C.1) and (C.2)). Our model-implied results for the relation between spending flexibility and portfolio liquidity are consistent with empirical facts. Pension plans, which have low spend ing flexibility, have relatively low allocations to alternative assets as compared with investors with greater flexibility, such as endowments and family offices. Over a medium or long horizon, the combined effect of a high EIS—reducing spending and tilting toward investments that earn an illi quidity premium—have a significant impact on the accu mulation of net worth. 5.3.2. Risk Aversion g. Table 4 shows that the coeffi cient of relative risk aversion has a very large effect on Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 13 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Table 3. The Effect of the EIS ψ on Asset Allocation and Spending Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Panel A: Illiquidity case ψ � 0:1 c � 0:5 ψ �1 ψ �2 Public equity Bonds Alternatives Region Dev. Spending Dev. 56.20 53.93 50.57 44.39 13.03 11.59 9.43 5.36 30.77 34.48 40.00 50.25 (24.57, 62.50) (27.47, 64.94) (31.45, 69.44) (37.04, 82.64) 6.9 9.4 10.9 5.5 6.36 5.32 3.99 1.33 0.04 0.07 0.10 0.01 Panel B: Full-spanning case ψ � 0:5 48.33 7.22 44.44 (44.44, 44.44) 0 5.35 0 Notes. This table reports the comparative static effect of ψ on asset allocation and spending for the i � ∞ case. The three columns public equity, bond, and alternatives (alternative assets) report Π=N, (W � Π)=N, and K/N, respectively, evaluated at the desired target highlighted in Figure 1. Region (illiquid asset no-trade region) and Dev. (standard deviation) report the range between rebalancing boundaries and the average deviation from the desired target for the alternative asset, respectively. The spending column reports the corresponding desired target spending rate, C/N, and Dev. reports the average deviation from the spending rate at the desired target. All columns are presented in percentages. Panel A reports results for the case with illiquidity. Panel B reports results for the case of full spanning. The baseline parameter values are given in Table 2. For the results in this table, we fix risk aversion at γ � 2. asset allocation. For a fixed EIS of ψ � 0:5, if risk aversion decreases from γ � 2 to γ � 1, the investor increases the portfolio allocation to alternative assets from 34.48% to 53.76%. Even more strikingly, the investor changes the portfolio allocation to the risk-free asset from a long posi tion of 11.59% to a short position (borrowing 66.51% of net worth). As a result, the investor increases the portfo lio allocation to public equity from 53.93% to a levered position (112.75% of net worth). As risk aversion in creases from γ � 2 to γ � 4, allocations to bonds signifi cantly increase from 11.59% to 55.43%, allocations to alternative assets decrease by about half from 34.48% to 17.36%, and allocations to public equity decrease from 53.93% to 27.21%. Table 4 shows that risk aversion has a large effect on the illiquid asset no-trade region and the average devia tion of the desired target. Risk aversion affects not only the level of the desired allocation to illiquid assets, but also the tolerance for deviations from the desired alloca tion. As risk aversion rises, the no-trade region becomes narrower, and the average deviation becomes smaller. Comparing the results for risk aversion in Table 4 with those for the EIS in Table 3, we show that it is important to use Epstein–Zin utility as risk aversion and the reciprocal of the EIS have opposite effects on alloca tions to public equity and spending. Whereas increasing the coefficient of relative risk aversion causes allocations to public equity to fall, decreasing the EIS causes alloca tions to public equity to increase. Similarly, whereas increasing the coefficient of relative risk aversion causes spending to fall, decreasing the EIS causes spending to increase. 5.3.3. Proportional Liquidation Cost θL. Table 5 reports comparative static effects of changing the proportional liquidation cost θL. Changing the liquidation cost θL from a low value of 1% to a large cost of 50% results in a moderately large decrease in the allocation to the alterna tive asset from 43.20% to 31.65%. The investor alters the portfolio allocation to keep the overall portfolio β nearly constant, offsetting the reduced risk exposures of the alternative asset with higher allocations to public equity. We also see that progressively larger increases in θL result in progressively smaller changes in asset allocation; as the liquidation cost rises, the no-trade region becomes wider and the investor is increasingly unlikely to engage in port folio rebalancing, and so further increases in rebalancing costs become quantitatively much less important. Table 4. The Effect of γ on Asset Allocation and Spending Rates Γ �1 g�2 γ �4 Public equity Bonds Alternatives Region Dev. Spending Dev. 112.75 53.93 27.21 �66.51 11.59 55.43 53.76 34.48 17.36 (38.91, 135.14) (27.47, 64.94) (13.16, 37.04) 27.3 9.4 6.2 6.57 5.32 4.66 0.21 0.07 0.04 Notes. This table reports the comparative static effect of γ on asset allocation and spending for the i � ∞ case. The three columns public equity, bond, and alternatives (alternative assets) report Π=N, (W � Π)=N, and K/N, respectively, evaluated at the desired target highlighted in Figure 1. Region (illiquid asset no-trade region) and Dev. (standard deviation) report the range between rebalancing boundaries and the average deviation from the desired target for the alternative asset, respectively. The spending column reports the corresponding desired target spending rate, C/N, and Dev. reports the average deviation from the spending rate at the desired target. All columns are presented in percentages. The baseline parameter values are given in Table 2. For the results in this table, we fix the EIS at ψ � 0:5. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 14 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Table 5. The Effect of the Proportional Liquidation Cost θL on Asset Allocation and Spending θL � 0:01 θL � 0:05 uL � 0:1 θL � 0:25 θL � 0:5 Public equity Bonds Alternatives Region Dev. Spending Dev. 48.80 52.20 53.93 55.51 55.63 8.01 10.42 11.59 12.64 12.72 43.20 37.38 34.48 31.85 31.65 (32.10, 52.08) (29.28, 57.80) (27.47, 64.94) (25.58, 86.21) (25.45, 123.46) 5.7 7.7 9.4 11.3 11.7 5.34 5.33 5.32 5.31 5.31 0.01 0.03 0.07 0.20 0.32 Notes. This table reports the comparative static effect of the proportional liquidation cost θL on asset allocation and spending for the i � ∞ case. The three columns public equity, bond, and alternatives (alternative assets) report Π=N, (W � Π)=N, and K/N, respectively, evaluated at the desired target highlighted in Figure 1. Region (illiquid asset no-trade region) and Dev. (standard deviation) report the range between rebalancing boundaries and the average deviation from the desired target for the alternative asset, respectively. The spending column reports the corresponding desired target spending rate, C/N, and Dev. reports the average deviation from the spending rate at the desired target. All columns are presented in percentages. The baseline parameter values are given in Table 2. 5.3.4. Excess Return a. Table 6 reports comparative static results for excess return α defined with respect to the single-index model with public equity. For this rea son, α includes the risk premium and illiquidity premium of the alternative asset. That is, α is not purely a measure of the alternative asset manager’s skills. The results show that asset allocations are quite sensitive to changes in α. For example, increasing α from 2% to 3% increases the alternative asset allocation from 34.48% to 60.24%. As the allocation to the alternative asset increases, the allocation to public equity falls from 53.93% to 37.91% to manage the overall portfolio β and because of the additional liquidity risk. The sensitivity of the implied portfolio allocations to changes in α is consistent with the large cross-sectional dispersion in endowment funds’ allocations to alterna tive assets. An α of 0% can explain nonparticipation, whereas an α of 3% implies allocations that are broadly consistent with those of large endowments, such as Yale and Stanford. Thus, with reasonable parameter values, our model is consistent with both the average allocation and also the cross-sectional dispersion of allocations to alternative assets. is also consistent The sensitivity of allocations to α with the empirically observed strong relation between endowment fund size and allocations to alternative assets. Lerner et al. (2008), Brown et al. (2010), Barber and Wang (2013), and Ang et al. (2018) find that large endowment funds persistently earn significant alphas, which they attribute to superior alternative asset invest ments, whereas small endowments do not earn signifi cant alphas. Lerner et al. (2008) and Brown et al. (2011) discuss how large endowments typically have better investment committees and access to elite managers and can exploit economies of scale in selecting alternative assets. 5.3.5. Unspanned Volatility «. Table 7 shows that the unspanned volatility of the alternative asset, ε, has a quantitatively large effect on asset allocation. We use two panels to demonstrate how both the level of the unspanned volatility and the composition of total volatil ity affect asset allocation. In panel A of Table 7, we fix βA � 0:6 for all rows, which implies that the part of the alternative asset’s return volatility spanned by the public market equals ρσA � 0:12 for all four cases. The total variance, σ2A � (ρσA )2 + ε2 � :122 + ε2 , varies one-to-one with ε2 . We show that changes in the volatility unspanned by public equity have large effects on asset allocation. For example, if we decrease ε from 15% (the baseline case) to 10%, the investor more than doubles the allocation to alternative assets from 34.48% to 76.34%. Table 6. The Effect of α on Asset Allocation and Spending Rates α � 0% α � 1% a � 2% α � 3% Public equity Bonds Alternatives Region Dev. Spending Dev. 75.00 67.29 53.93 37.91 25.00 20.02 11.59 1.85 0.00 12.69 34.48 60.24 (0, 0) (9.09, 38.02) (27.47, 64.94) (51.02, 88.50) 0 7.8 9.4 9.1 5.13 5.16 5.32 5.60 0 0.05 0.07 0.10 Notes. This table reports the comparative static effect of α on asset allocation and spending for the i � ∞ case. The three columns public equity, bond, and alternatives (alternative assets) report Π=N, (W � Π)=N, and K/N, respectively, evaluated at the desired target highlighted in Figure 1. Region (illiquid asset no-trade region) and Dev. (standard deviation) report the range between rebalancing boundaries and the average deviation from the desired target for the alternative asset, respectively. The spending column reports the corresponding desired target spending rate, C/N, and Dev. reports the average deviation from the spending rate at the desired target. All columns are presented in percentages. The baseline parameter values are given in Table 2. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 15 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Table 7. The Effect of ε on Asset Allocation and Spending Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Panel A: Fixing βA � 0:6 ε � 10% « � 15% ε � 17:5% ε � 19:2% Implied σA ε2 =σ2A Public equity Bonds Alternatives Region Dev. Spending Dev. 15.6 19.2% 21.2 22.6 41.0 61.0% 68.0 71.9 27.79 53.93 60.17 63.07 �4.13 11.59 15.50 17.33 76.34 34.48 24.33 19.61 (68.03, 125.00) (27.47, 64.94) (18.55, 51.55) (14.53, 44.64) 6.3 9.4 9.6 9.0 5.59 5.32 5.26 5.24 0.05 0.07 0.06 0.06 Panel B: Fixing σA � 19:2% ε � 10% « � 15% ε � 17:5% ε � 19:2% Implied βA ε2 =σ2A Public equity Bonds Alternatives Region Dev. Spending Dev. 0.82 0.60 0.40 0 27.1 61.0 83.0 100.0 �2.12 53.93 66.41 74.57 6.88 11.59 13.01 13.61 95.24 34.48 20.58 11.82 (86.21, 135.14) (27.47, 64.94) (15.63, 48.54) (8.45, 38.17) 7.7 9.4 9.5 9.0 5.62 5.32 5.25 5.20 0.10 0.07 0.06 0.05 Notes. This table reports the comparative static effect of ε on asset allocation and spending for the i � ∞ case. The three columns public equity, bond, and alternatives (alternative assets) report Π=N, (W � Π)=N, and K/N, respectively, evaluated at the desired target highlighted in Figure 1. Region (illiquid asset no-trade region) and Dev. (standard deviation) report the range between rebalancing boundaries and the average deviation from the desired target for the alternative asset, respectively. The spending column reports the corresponding desired target spending rate, C/N, and Dev. reports the average deviation from the spending rate at the desired target. All columns are presented in percentages. Panel A shows the effect of changing ε when βA is fixed at βA � 0:6, and the column “Implied σA” shows the total volatility of the alternative asset. Panel B shows the effect of changing ε when the total volatility of the alternative asset is fixed at σA � 19:2% and the column “Implied βA” shows the implied beta of the alternative asset. In both panels, the column “ε2 =σ2A ” shows the alternative asset’s unspanned variance as a percentage of its total variance. The baseline parameter values are given in Table 2. In panel B of Table 7, we fix the total volatility of the alternative asset σA at 19.2%. Then, as we increase the unspanned volatility ε, the spanned volatility must de crease to keep the total volatility unchanged. Consider again decreasing ε from 15% (the baseline case) to 10%. The investor reacts to this decrease in unspanned volatil ity by almost tripling the allocation to alternative assets from 34.48% to 95.24%. In sum, both the amount of unspanned risk and the composition of total risk have quantitatively large effects on asset allocation. The sensitivity of allocations is strik ing given the empirical uncertainty associated with these parameter values. Our quantitative results suggest it is worth devoting much more work to improve the empiri cal estimates of unspanned volatility. 6. Contributions (Inflows) to the Fund In this section, we extend our baseline model of Section 2 to include contributions (inflows) into the investor’s portfolio. Inflows provide new capital, which natu rally should influence the fund’s portfolio alloca tion decisions. We assume that new contributions flow into the port folio at a rate of τ(Wt + Kt ), where τ > 0. Incorporating τ(Wt + Kt ) into (7) gives the following liquid wealth pro cess: dWt � (rWt� + δA Kt� + τ(Wt� + Kt� ) � Ct� )dt + Πt� ((µS � r)dt + σS dBSt ) + (1 � θL )dLt � (1 + θX )dXt + δT Kt� I{t�mT} : (40) The HJB equation for the value function V(Wt , Kt , t) in the interior region is given by 0 � max f (C,V) + (rW + δA K + τ(W + K) C, Π (ΠσS )2 VWW 2 σ2 K2 + Vt + (µA � δA )KVK + A VKK + ρΠKσS σA VWK : (41) 2 + (µS � r)Π� C)VW + By using essentially the same procedure and the homo geneity property as in Section 2, we obtain the follow ing nonlinear PDE for the scaled certainty-equivalent wealth p(w, t) in the interior region and when (m � 1)T < t < mT: ! φ1 (pw (w, t))1�ψ � ψζ γσ2A 0� + µA � δA � p(w, t) ψ�1 2 ε 2 w2 pww (w, t) 2 + ((δA � α + γε2 )w + δA + τ(w + 1))pw (w, t) + pt (w, t) + � γε2 w2 (pw (w, t))2 2 p(w, t) + (ηS � γρσA )2 pw (w, t)p(w, t) , 2γ(w,t) (42) where φ1 is given by (17). As in the baseline model of Section 2, the boundary conditions are given in (27), (28), (29), (30), and (35). The scaled certainty equivalent wealth p(wmT� , mT�) is given Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 16 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS b mT ≤ w mT and w b mT > w mT cases, by (32) and (33) for the w b mT is given by (31). Finally, the respectively, in which w optimal consumption rule is given by (22). Table 8 shows the effect of the contribution rate, τ, on portfolio allocations and spending. The first row shows the baseline case with τ � 0, and the additional rows report results for τ � 1%, 2%, and 5%.15 As the contribu tion rate τ increases, allocations to the alternative asset increase. Intuitively, current and future capital inflows from contributions effectively make the portfolio more titled toward liquid assets. Anticipating this, the investor reduces allocations to both public equity and bonds. Also consistent with this intuition, the range between the rebalancing boundaries widens, indicating that the in vestor is willing to tolerate greater deviations from the desired target as τ increases. But deviations from the desired target decrease sharply as τ increases as the in flows from new contributions allow the investor to main tain the alternative allocation closer to the desired target. Table 8 also shows that spending increases as the con tribute rate τ increases. This is expected as the investor is wealthier and the current spending rate C/N does not take into account future capital inflows. The standard deviation of spending rates also decreases because of the investor’s stronger ability to smooth spending over time and across states. In sum, new capital contributions make the investor wealthier in present value and also provide additional flexibility to manage illiquidity risk, resulting in not just higher allocations to alternative assets and higher spend ing, but also lower volatilities for both allocations and spending. 7. Spending Constraint In this section, we extend our baseline model of Section 2 to include a lower bound on the spending rate. In the United States, most private foundations are required to pay out at least 5% of assets every year to maintain taxexempt status (university endowments are an exception to this rule). A minimum spending requirement reduces flexibility and interacts with illiquidity in potentially important ways. We assume that the investor’s spending Ct as a frac tion of total net worth Nt cannot fall below c at any t ≥ 0: Ct ≥ cNt : (43) The optimal consumption c(wt , t) � C(Wt , Kt , t)=Kt is then given by c(w, t) � max{φ1 p(w, t) pw (w, t)�ψ , c · (w + 1)}: The PDE for the scaled certainty equivalent wealth p(w, t) is given by 0 1 pw (w, t) 2 ψc(w, t) p(w � ψζ γσ , t) 0�@ + µA � δA � A Ap(w, t) ψ�1 2 ε2 w2 pww (w, t) 2 + ((δA � α + γε2 )w + δA � c(w, t))pw (w, t) + pt (w, t) + � γε2 w2 (pw (w, t))2 2 p(w, t) (ηS � γρσA )2 pw (w, t)p(w, t) + , 2γi (w, t) (45) where c(w, t) is given by (44). All the boundary condi tions and other policy functions, for example, the alloca tion to public equity, are the same as in our baseline model of Section 2. In Table 9, we report the quantitative effect of in troducing a lower spending requirement of c � 5:2%. Although the desired spending target falls only slightly from 5.32% to 5.30%, the allocation to the alternative asset decreases from 34.5% to 27.8%, a significant 6.7% fall in levels and 20% fall in percentage terms. Addition ally, the illiquid asset no-trade region widens. Panel (a) of Figure 3 compares the investor’s certainty equivalent wealth for the spending constraint case with that of the baseline case. The expected welfare loss from the spend ing constraint is large: approximately 1% of the inves tor’s certainty equivalent wealth. Table 8. The Effect of Inflow τ on Asset Allocation and Spending Rates t�0 τ � 1% τ � 2% τ � 5% (44) Public equity Bonds Alternatives Region Dev. Spending Dev. 53.93 52.39 50.79 47.22 11.59 10.58 9.53 7.12 34.48 37.04 39.68 45.66 (27.47, 64.94) (29.24, 68.03) (30.77, 71.94) (33.56, 83.33) 9.4 8.0 6.3 2.3 5.32 5.83 6.33 7.84 0.07 0.07 0.06 0.03 Notes. This table reports the comparative static effect of τ, the contribution rate, on asset allocation and spending for the i � ∞ case. The three columns public equity, bond, and alternatives (alternative assets) report Π=N, (W � Π)=N, and K/N, respectively, evaluated at the desired target highlighted in Figure 1. Region (illiquid asset no-trade region) and Dev. (standard deviation) report the range between rebalancing boundaries and the average deviation from the desired target for the alternative asset, respectively. The spending column reports the corresponding desired target spending rate, C/N, and the Dev. column reports the average deviation from the spending rate at the desired target. All columns are presented in percentages. The baseline parameter values are given in Table 2. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 17 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Table 9. The Effect of a Spending Constraint Public equity Bonds Alternatives Region Dev. Spend Dev. 53.93 57.97 11.59 14.25 34.48 27.78 (27.47, 64.94) (22.52, 43.67) 9.4 6.0 5.32 5.30 0.07 0.04 Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. No constraint c � 5:2% Notes. This table reports the comparative static effect of a spending constraint on asset allocation and spending for the i � ∞ case. The three columns public equity, bond, and alternatives (alternative assets) report Π=N, (W � Π)=N, and K/N, respectively, evaluated at the desired target highlighted in Figure 1. Region (illiquid asset no-trade region) and Dev. (standard deviation) report the range between rebalancing boundaries and the average deviation from the desired target for the alternative asset, respectively. The spending column reports the corresponding desired target spending rate, C/N, and Dev. reports the average deviation from the spending rate at the desired target. All columns are presented in percentages. The baseline parameter values are given in Table 2. Spending constraints reduce investor welfare and sig nificantly alter asset allocation. These results are impor tant given that some politicians, such as Senator Chuck Grassley, advocate such spending requirements. 8. Financial Crisis In this section, we extend the model to include the possi bility of crisis states. This is motivated by empirical find ings that alternative asset illiquidity is time-varying and increases in crisis states.16 8.1. Model and Solution We assume that there are two states: normal and crisis. The transitions between these two states follow a continuous-time Markov chain. Let st denote the state at time t, where st � g is the normal state and st � b is the crisis state. Over a short time interval, ∆, the state switches from g to b (or from b to g) with a constant prob g g ability ξg ∆ (or ξb ∆). We denote θL and θX (θbL and θbX ) as the proportional costs of liquidation and acquisition in g the normal (crisis) state, respectively. We assume θL < θbL , which reflects a higher secondary market liquidation cost (e.g., illiquidity) during the crisis state. We also assume that, in the crisis state, the value of the alternative asset is subject to an additional downward jump shock, modeled as in the rare disaster literature, for example, Barro (2006), Pindyck and Wang (2013), and Wachter (2013). Let J denote a pure jump process with a constant arrival rate λ > 0, which is present only in the crisis state. If a jump does not occur at t (dJt � 0), the alternative asset’s fundamental value is continuous: At � At� , where At� ≡ lims↑t As denotes the left limit of the fundamental value. If a jump occurs at t (dJt � 1), the alternative asset’s fundamental value falls from At� to At � ZAt� , reflecting the proportional decline in the value of the alternative asset when the economy transi tions into the crisis state. We now write the dynamics of the fundamental value A in the crisis state b as dAt � µA dt + σA dBA t � δA dt � (1 � Z)dJt : At� (46) Figure 3. (Color online) Effects of Spending Constraints on Certainty Equivalent Wealth and Consumption for the i � ∞ Case (a) (b) Notes. Panels (a) and (b) plot P=N � p(w)=(w + 1) and C=N � c(w)=(w + 1) as functions of the percentage allocation to alternative assets z � K=N, respectively. The solid and dashed lines are for the no-constraint and the case in which the spending is constrained to be at least c � 5:2%. All other parameter values are given in Table 2. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 18 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS The dynamics of A in the normal state has no jumps and, hence, is the same as (2). Finally, to capture the empirical pattern that capital calls increase and distributions (to investors) decrease during the crisis state,17 we assume that there is a sto chastic call of the alternative asset in the crisis state at the moment the economy is hit by a downward jump shock (dJt � 1). To be precise, when dJt � 1, the value of the alternative asset that the investor owns drops from the prejump level of Kt� to the postjump level of ZKt� (and, for simplicity, we assume that the value of the investor’s liquid asset holdings also drops by (1 � Z) fraction).18 But, importantly, at this moment, the investor receives a capital call proportional to ZKt� , where the proportional ity constant is call > 0. This means that the investor must increase the position in the alternative asset by providing the amount call · (ZKt� ) to meet this capital call.19 To fund the capital call, the investor’s liquid wealth decreases from the prejump level Wt� to Wt � ZWt� � call · ZKt� and the alternative asset position changes from the prejump level of Kt� to Kt � Z(1 + call)Kt� , a combi nation of losses on their original positions and increased allocation (because of the capital call). Therefore, the alternative asset position in the crisis state b evolves as dKt � (µA � δA )Kt� dt + σA Kt� dBA t � dLt + dXt � δT Kt� I{t�mT} � (1 � Z)Kt� dJt + call · ZKt� dJt : (47) In the crisis state, the investor’s liquid wealth evolves as dWt � (rWt� + δA Kt� � Ct� )dt + Πt� ((µS � r)dt + σS dBSt ) + (1 � θbL )dLt � (1 + θbX )dXt + δT Kt� I{t�mT} � (1 � Z)Wt� dJt � call · ZKt� dJt , (48) where the last two terms capture the value loss from the prejump position and the outflow of capital from the liq uid asset holdings to meet the stochastic capital call, respectively. For brevity, we do not write down the cor responding equations for the normal state. Appendix F summarizes the solution. Next, we cali brate this generalized model and analyze the results. 8.2. Quantitative Analysis We calibrate the cost of liquidation in the crisis state to θbL � 0:25.20 We set the capital call parameter to call � 0.2, based on Robinson and Sensoy (2016) and Brown et al. (2021). The state transition probabilities are set to ξg � 0:1 and ξb � 0:5 as in Bolton et al. (2013). To focus on the effect of stochastic capital call in the crisis state, we ignore the downward jump losses (for both public equity and the alternative asset) in the crisis state (by setting Z � 1). Panel (a) of Figure 4 plots P=N � p(w)=(w + 1), the ratio of the certainty equivalent wealth P(W, K) � p(w)K and net worth N � W + K at time t � 0 on the y-axis as a function of the percentage allocation to alternative assets (z � K=N) on the x-axis. Figure 4. (Color online) Effect of a Crisis State on the Investor’s Certainty Equivalent Wealth and the Marginal Value of Liquidity (a) (b) Notes. We plot P=N � p(w, t)=(w + 1) (panel (a)) and pw (w, t) (panel (b)) at t � 0 as functions of the alternative assets z � K=N for the i � 1 case. g g The parameter values are θX � θbX � 0:02, θL � 0:1, θbL � 0:25, ξg � 0:1, ξb � 0:5, call � 0:2, and λ � 0:1 with an implied payout of δT � 21:34% every six years (H � 6). Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 19 Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS The solid curve plots results for the normal state, and the dashed line plots results for the crisis state. The most striking aspect is the extremely wide range of the alterna tive asset allocation in the crisis state, indicating high reluctance to rebalance in this state. This occurs not only because of the much higher proportional cost, but also because of the option value of waiting for a possible regime switch back to the good state when the transac tion cost is lower. Figure 4 shows that, for a fixed allocation to alternative assets (between 0.33 and 0.65), the investor’s utility is approximately the same in both states (panel (a)) and so is the marginal (certainty equivalent) value of liquidity pw (w, t � 0) (panel (b)).21 However, the distributions of z and marginal value of liquidity are significantly different in the two states, which we discuss next using Figure 5. In Figure 5, we plot the cumulative distribution func tions in the two states for the allocation to the illiquids, z, (panel (a)) and the marginal value of liquidity, p(wt , t � 0) (panel (b)). Panel (a) shows that the distribution of the allocations to alternatives z in the crisis state first order stochastically dominates the distribution of z in the nor mal state. That is, the probability to draw a large value of z is higher in the crisis state than in the normal state. Panel (b) shows that the distribution of the marginal value pw (w, t) in the crisis state first order stochastically dominates that the distribution of pw (w, t) in the normal state. That is, in the sense of first order stochastic domi nance, the marginal value of liquidity pw (w, t) is higher in the crisis state than in the normal state, again indicat ing that the cost of illiquidity is larger in crisis. Because the distributions of z in the two states are very different, the marginal value of liquidity that measures the cost of investing in alternatives also differs signifi cantly. For example, the marginal value of liquidity is pw (w, t � 0) � 1:28 at the 75th percentile of the distribu tion of z (z � 0.97) in the crisis state, which is economically much larger than pw (w, t � 0) � 1:14 at the 75th percentile of the distribution of z (z � 0.71) in the normal state. This result indicates the high value of liquidity in a crisis. 9. Conclusion The endowment model, an investment strategy of high allocations to illiquid alternative assets, is widely used by many institutional and high-net-worth investors. We build on the framework of modern portfolio theory to develop a dynamic portfolio choice model with illiquid alternative assets to analyze conditions under which the endowment model does and does not work. We capture the illiquidity of the alternative asset as follows. First, a fraction of the alternative asset periodically matures and becomes fully liquid, and the investor can benefit from liquidity diversification by holding alternative assets maturing at different dates. Second, the investor can vol untarily buy and sell the illiquid asset at any time by paying a transaction cost. Third, the alternative asset’s risk is not fully spanned by publicly traded assets. We model how investors can engage in liquidity diversification by investing in multiple illiquid alterna tive assets with staggered lockup expirations. We show that such liquidity diversification results in higher alloca tions to alternative assets and higher investor welfare. Figure 5. (Color online) Stationary Cumulative Distributions of the Percentage Allocation to Alternative Assets z � K=(W + K) (Panel (a)) and the Marginal Value of Liquidity pw (w, t) (Panel (b)) in Both the Normal and Crisis States for the i � 1 Case (a) g (b) g Note. The parameter values are θX � θbX � 0:02, θL � 0:1, θbL � 0:25, ξg � 0:1, ξb � 0:5, call � 0:2, and λ � 0:1 with an implied payout of δT � 21:34% every six years (H � 6). Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 20 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS We also show how illiquidity from lockups interacts with transactions costs in the secondary market to create endogenous and time-varying rebalancing boundaries. Our extended model with crisis states captures sto chastic capital calls and much higher secondary market transaction costs. We find that investors’ holdings of alternative assets in crisis states often significantly devi ate from the optimal target allocations, and hence, the utility loss from not being able to hedge stochastic calls and distributions can be large. Acknowledgments For helpful comments, the authors thank an anonymous associate editor, two anonymous referees, Bruno Biais (edi tor), Patrick Bolton, Winston Dou, Thomas Gilbert, Harrison Hong, Steve Kaplan, Monika Piazzesi, Jim Poterba, Tom Sar gent, Mark Schroder, and Luis Viceira as well as seminar participants at Columbia University, the European Finance Association, and National Bureau of Economic Research New Developments in Long-Term Asset Management con ferences. The authors thank Matt Hamill and Ken Redd of the National Association of College and University Business Officers and John Griswold and Bill Jarvis of Commonfund for assistance with data. Appendix A. Public Equity and Bonds with No Alternatives First, we summarize the solution for the complete markets special case of our model in which an investor with Duffie–Epstein–Zin recursive preferences has the standard investment opportunities defined by the public equity’s risky return process given by (1) and a risk-free bond that pays a constant rate of interest r. The investor dynamically adjusts consumption/spending and frictionlessly rebalances the portfolio to maximize the recursive preferences given in (8) and (9). Note that the investor only has liquid wealth W. The following proposition summarizes the solution for this frictionless benchmark. Proposition A.1. The investor allocates a constant fraction, denoted by π, of wealth Wt to public equity, that is, the total investment amount in public equity is Π � πW, where µ �r η π� S � S 2 : (A.1) γσS γσS Note that the optimal asset allocation rule is the same as that in Merton (1969, 1971). Specifically, the EIS has no effect on π in this frictionless benchmark. The optimal spending Ct is pro portional to wealth Wt: Ct � φ1 Wt , where φ1 is given in (17). Note that the optimal spending rule depends on both risk aver sion γ and the EIS ψ, which is different from Merton (1969, 1971). The investor’s value function J(W) is given by (b1 W)1�γ J(W) � , (A.2) 1�γ where b1 is a constant given by b1 � ζ ψ ψ�1 1 1�ψ φ1 : (A.3) Next, we analyze the general case in which the investor can also invest in illiquid alternative assets in addition to public equity and bonds. Appendix B. Proof for Proposition 1 B.1. Optimal Policy Functions and PDE for p(w, t) We conjecture that the value function V(W, K, t) takes the following form: V(W, K, t) � (b1 P(W, K, t))1�γ (b1 p(w, t)K)1�γ � , 1�γ 1�γ (B.1) where b1 is given in (A.3). Substituting (B.1) into the consump tion FOC given in (13) and the asset-allocation FOC given in (14), we obtain (22) for the scaled consumption rule c(w, t) and (23) for the scaled asset allocation in public equity π(w, t), respectively. Finally, substituting the conjectured value function given in (B.1) and the consumption and asset-allocation policy rules given in (22) and (23) into the HJB equation (12), we obtain the PDE (24) for the certainty equivalent wealth p(w, t). B.2. Lower Liquidation Boundary Wt and Upper Acqui sition Boundary Wt Let (Wt , Kt ) denote the investor’s time-t holdings in public equity and the alternative asset, respectively. We use ∆ to denote the amount of the illiquid alternative asset that the investor is considering liquidating. The investor’s postli quidation holdings in public equity and the alternative asset are equal to Kt � ∆ and Wt + (1 � θL )∆, respectively. Because the investor’s value function is continuous before and after liquidation, we have V(Wt + (1 � θL )∆, Kt � ∆, t) � V(Wt , Kt , t) � 0: (B.2) Dividing (B.2) by ∆ and letting ∆ → 0, we obtain under differentiability 0 � lim 1 ∆→0 ∆ + lim [V(Wt + (1 � θL )∆, Kt � ∆, t) � V(Wt + (1 � θL )∆, Kt , t)] 1 � θL ∆→0 ∆(1 � θL ) [V(Wt + (1 � θL )∆, Kt , t) � V(Wt , Kt , t)] � �VK (Wt , Kt , t) + (1 � θL )VW (Wt , Kt , t): (B.3) The preceding equation implicitly defines the boundary Wt in that VK (Wt , Kt , t) � (1 � θL )VW (Wt , Kt , t): (B.4) The optimality of Wt implies that the derivatives on both sides of (25) are equal. Therefore, VKW (Wt , Kt , t) � (1 � θL )VWW (Wt , Kt , t): (B.5) Substituting the value function given by (B.1) into (B.4), we obtain (25). Similarly, substituting the value function given by (B.1) into (B.5), we obtain (26). By using the homogeneity property, we obtain the following: PW (Wt , Kt , t) � pw (wt , t), PWW (Wt , Kt , t) � pww (wt , t)=Kt , PK (Wt , Kt , t) � p(wt , t) � pw (wt , t)wt , and PWK (Wt , Kt , t) � �Wt pww (wt , t)=K2t . Substituting these expressions into (25) and (26), we obtain p(w t , t) � pw (w t , t)w t � (1 � θL )pw (w t , t) �pww (w t , t)w t =Kt � (1 � θL )pww (w t , t)=Kt : (B.6) (B.7) Simplifying these two equations, we obtain (27) and (28). We can derive the boundary conditions for Wt and w t by using essentially the same procedure as the preceding. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 21 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. The preceding proof is applicable to the upper and lower barriers for all t such that t ≠ mT. To complete our analysis for t � mT, we need to incorporate the automatic liquidity event that takes place t � mT. B.3. Value and Decisions at t 5 mT When there is an automatic liquidity event at t � mT, it is possible that, without active rebalancing, the automatic liquidity can cause the portfolio to be overly exposed to b mT > w mT , the liquid assets. In this case, that is, when w investor may choose to reduce the liquid asset holding. units Suppose that the investor optimally purchases Λ cmT � (1 + θX )Λ � W mT , of the alternative asset such that W and the liquidity ratio is then equal to cmT � (1 + θX )Λ W Wt + δT Kt � (1 + θX )Λ w mT � � lim , (B.8) bmT + Λ t→mT Kt � δT Kt + Λ K then solving the equation gives the following expression, yielding the number of units for the alternative asset, Λ � λKmT� , that the investor plans to purchase at t � mT, where λ is given by wt + δT � w mT (1 � δT ) : λ � lim t→mT 1 + θX + w mT (B.9) Appendix C. Full Spanning with Liquid Alternative Asset In this appendix, we summarize the full-spanning case in which the alternative asset is fully liquid. An investor with Duffie–Epstein–Zin recursive preferences has three invest ment opportunities: (a) the public equity whose return pro cess is given by (1), (b) a risk-free bond that pays a constant rate of interest r, and (c) the risky liquid alternative asset. The investor dynamically adjusts consumption/spending and frictionlessly rebalances the portfolio to maximize the recursive preferences given in (8) and (9). Note that the investor’s wealth is fully liquid. The following proposition summarizes the solution for this frictionless benchmark. Proposition C.1. The investor continuously rebalances the portfolio so the investment in public equity, Π, and in the alter native asset, K, are proportional to net worth N, that is, η � ρηA Π� S N, (C.1) σS γ(1 � ρ2 ) α K � 2 N: (C.2) γε The remaining wealth, N � (Π + K), is allocated to the risk-free bond. The optimal consumption C is proportional to the net worth, N: C � φ2 N, where � � η2 � 2ρηS ηA + η2A φ2 � ζ + (1 � ψ) r � ζ + S : (C.3) 2γ(1 � ρ2 ) The investor’s value function V(N) is given by V(N) � (b2 N)1�γ � J((b2 =b1 )N), 1�γ (C.4) where b2 is a constant given by ψ 1 b2 � ζψ�1 φ21�ψ , (C.5) and J(·) is the value function given in (A.2) for an investor who only has access to public equity and bonds. By comparing φ2 given in (C.3) and φ1 given in (17), we see that diversification (| ρ| < 1) and an additional risk premium ηA > 0 both make the investor better. By intro ducing a new risky (alternative) asset into the investment opportunity set, the investor is better off because b2 > b1 . The second equality in (C.4) implies that b2 =b1 � 1 is the fraction of wealth that the investor needs as compensation to permanently give up the opportunity to invest in the liquid alternative asset and instead invest only in public equity and the risk-free asset. C.1. Proof for the Case of Full Spanning with the Liquid Alternative Asset Using the standard dynamic programming method, we have 0 � max f (C, V) + [rN + (µS � r)Π + (µA � r)K � C]V N C , Π, K + (ΠσS )2 + 2ρΠσS KσA + (KσA )2 V NN , 2 (C.6) and using the FOCs for Π, K, and C, we have fC (C, V) � VN , η VN ρσA Π�� S � K, σS VNN σS η VN ρσS K�� A � Π: σA VNN σA (C.7) (C.8) (C.9) We conjecture and verify that the value function takes the following form: V(N) � (b2 N)1�γ : 1�γ (C.10) 1�ψ Substituting (C.10) into the FOCs, we obtain C � ζψ b2 N, (C.1), and (C.2). Finally, substituting them into the HJB equa tion (C.6) and simplifying the expression, we obtain (C.3). Appendix D. Additional Details of Data and Calibration Inputs This appendix provides details on the inputs and calcula tions for some of the calibration parameters used in the paper. D.1. Subasset Classes Calibrating the model requires the standard deviation, beta, and unspanned volatility of the representative alterna tive asset. To obtain these parameters, we build up from the standard deviations and correlations of the subasset classes composing the representative alternative asset. For each sub asset class a, we combine its βa and R2a with the standard deviation of the market σS � 20% to obtain the implied stan qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dard deviation for the asset class: σa � β2a σ2S =R2a . Table D.1 shows the summary statistics for the more detailed subasset categories. Within alternative allocations, hedge funds have the largest allocation with an equalweighted average allocation of 16.7%. For all of the subas set classes, the allocations increase with endowment size, Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 22 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Table D.1. Summary of Endowment Fund Asset Allocation Subcategories Average, % Value-weighted average 0%–10% 20%–30% 45%–55% 70%–80% 90%–100% 5.1 15.9 50.7 16.7 4.6 1.7 2.7 2.7 4.0 8.7 35.6 23.4 10.9 5.7 6.1 5.9 7.2 25.8 60.1 4.6 0.2 0.3 0.4 0.8 3.5 18.2 57.9 13.0 3.1 0.5 1.8 1.9 5.6 16.8 54.7 14.3 3.1 0.4 2.7 2.0 3.8 11.4 45.9 22.3 7.1 2.2 3.1 4.2 3.5 7.4 32.0 23.8 12.3 6.9 7.0 6.7 Cash & equivalents Fixed income Public equity Hedge funds Private equity Venture capital Private real estate Natural resources Notes. This table summarizes endowment fund portfolios as of the end of the 2014–2015 academic year for 774 endowments. The first two columns show the equal- and value-weighted average, respectively. The columns 0%–10% to 90%–100% show averages within size-segmented groups of endowment funds. For example, the column 0%–10% shows the value-weighted average portfolio allocation for the smallest decile of endowment funds. Hedge funds includes managed futures. Natural resources includes illiquid natural resources, such as timberland and oil & gas partnerships. particularly for the least liquid categories: private equity, ven ture capital, private real estate, and illiquid natural resources. Panel A of Table D.2 shows the βa, R2a , and σa for each of the alternative subasset classes. For hedge funds, the β and R2 are taken from Getmansky et al. (2004) and account for return smoothing. For private equity and venture capital, the β and R2 are taken from Ewens et al. (2013). For private real estate and illiquid natural resources, the variables are based on Pedersen et al. (2014) and account for return smoothing. Panel B of Table D.2 shows the pairwise corre lations between the asset classes, which are calculated using index returns over the period 1994–2015.22 We com bine the asset allocations from Table D.1 with the data from Table D.2 to impute portfolio β, σ, and unspanned volatility (ε). Panel C of Table D.2 shows the imputed vari ables for the cross-section of endowment funds. Table D.2. Summary of Asset Class Risk and Correlations Panel A Hedge funds (HF) Private equity (PrivEqu) Venture capital (VC) Private real estate (PrivRE) Natural resources (NatRes) βa R2a σ a, % 0.54 0.72 1.23 0.50 0.20 0.32 0.32 0.30 0.49 0.07 19.1 25.4 45.1 16.0 17.0 Panel B FixedInc PubEqu HF PrivEqu VC PrivRE NatRes FixedInc PubEqu HF PrivEqu VC PrivRE NatRes 1 0.02 0.16 �0.23 �0.18 �0.13 0.04 1 0.64 0.78 0.46 0.35 0.87 1 0.73 0.52 0.31 0.67 1 0.66 0.51 0.70 1 0.17 0.46 1 0.44 1 Panel C βA σA ε Average Value-weighted average 0%–10% 20%–30% 45%–55% 70%–80% 90%–100% 0.58 18.1% 13.9% 0.61 18.7 14.2 0.53 17.7% 14.1 0.55 17.7 13.9 0.54 17.3 13.5 0.57 18.2 14.2 0.62 18.9 14.3 Notes. Panel A shows βa, R2a , and σa for each alternative asset class a. Panel B shows the pairwise correlations between these subasset classes. Panel C shows the implied parameters of the representative alternative asset: βA is the beta, σA is the standard deviation, and ε is the unspanned volatility. The first two columns show results for the equal- and value-weighted average portfolios. The remaining columns show allocations for size-segmented groups of endowments. For example, the column 0%–10% shows the value-weighted statistics for the smallest decile of endowment funds. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 23 Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS D.2. Secondary Market Costs For voluntary liquidations, we assume that the propor tional transaction cost is θL � 10% based on empirical findings and the following back-of-the-envelope calcula tion: For secondary market liquidations of private equity, Kleymenova et al. (2012) and Nadauld et al. (2019) find average discounts of 25.2% and 13.8%, respectively. For sec ondary market liquidations of hedge funds, Ramadorai (2012) finds an average discount of 0.9%, which rises to 7.8% during financial crisis. Therefore, we combine the aggregate endowment fund portfolio weights with liquidation costs of 20% for private equity and venture capital, 1% for hedge funds, and 10% for private real estate and timberland, to obtain a proportional liquidation cost of 9.3% for the repre sentative alternative asset. For acquisitions, we assume that the proportional acquisition cost is θX � 2%, which is equal to the average placement agent fee reported by Rikato and Berk (2015) and Cain et al. (2020). investment. As the payout occurs once every T years, the annualized net payout rate is then 1 egA H ISt � egI H ISt 1 e(gA �gI )H � 1 1 � Pi (g �g )jT � (1 � e�(gA �gI )T ): P i g H (g �g )jT I I I A A T e ISt × j�1 e T j�1 e T (E.1) Next, we use this annualized net payout rate to calibrate δA and δT. Although, for the sake of generality, the model includes both δA and δT, in any single calibration, we use only one of either δA or δT. Next, we provide three examples. First, consider the case when i → ∞ with fixed finite holding period H for each investment, T ≡ H=i → 0. There fore, the investor continuously receives payout at a con stant rate. This maps to the parameter δA in our model. By applying L’Hopital’s rule to (E.1), we obtain, as one may expect, the following simple expression for the net payout rate: δA � gA � gI , Appendix E. Calibrating the Payout Rates: dA and dT We focus on the steady state in which the investor always has i distinct investments in the alternative asset at any time t. This is feasible provided the investor immediately replaces each investment that exits. To simplify exposition, assume that each investment’s payoff structure involves only one contribution at its incep tion and one distribution upon its exit and the horizon (or, equivalently, the lockup period) of each investment is H. At the steady state, i/H investments mature each year, which means that there is one liquidity event every T � H=i years. For example, if the lockup period for each investment is H � 6 and there are three investments in the steady state (i � 3), then every two years (T � 6=3 � 2), an automatic liquidity event occurs. To ensure that the investor has three invest ments in the steady state, the investor immediately replaces the exited investment by making a new investment with a six-year lockup. To ensure growth stationarity, we assume that both the growth rate of each investment, gA, and the growth rate of the inception size for each investment (vintage), gI, are constant. Consider a vintage-t investment, which refers to the investment that enters the portfolio at time t. Let ISt denote the investment’s initial size (IS) at inception. Its size at (t + jT) is then (t + jT), where j � 1, 2, : : : i, and hence, the investment’s size when exiting at time t + H is egA H ISt . At time (t + H), the investor holds a total of i illiquid alternative investments ranging from vintage-t to vintage(t + (i � 1)T). Note that the value of the vintage-(t + j) investment is egA (i�j+1)T × (ISt egI (j�1)T ) as its inception size is ISt egI (j�1)T and has grown at the rate of gA per year for (i � j + 1)T years. Summing across all vintages, we obtain i X j�1 egA (i�j+1)T × (ISt egI (j�1)T ) � egI H ISt × i X e(gA �gI )jT : j�1 The net payout at time (t + H) is given by the difference between egA H ISt , the size of the exiting vintage-t invest ment, and egI H ISt , the size of the new vintage-(t + H) (E.2) which is simply the difference between the incumbent invest ment growth rate gA and the growth of the new investment’s initial size gI. For our calibration, we set µA � gA � 9:6% and gI � 5:6% (approximately equal to the average endowment fund growth rate over the past 20 years), resulting in δA � 4%. Second, consider the case when the investor has only one investment outstanding at each point in time. Then, T � H, the payout occurs once every H years, and we use δT to capture the payout rate for this case. That is, when T is relatively large, δT is given by δT � 1 � e�(gA �gI )T : (E.3) Note that δT as defined in the model is not annualized. Thus, with gA � 9:6% and gI � 5:6%, for a single invest ment (i � 1) in the portfolio and H � 6, δT � 21:34%, which is equivalent to an annualized payout of 3.28%. Third, consider an intermediate case when the investor has six distinct investments at each point in time. Then, we have T � H=i � 6=6 � 1, and we can use δT � δ1 � 1 � e�0:04 � 3:92% to capture the payout. Alternatively, we can approxi mate with a continuous constant dividend yield by annual izing δT and using this annualized value as δA in the calibration. In this case, we have δA ≈ (1 + δT )1=T � 1 � δT � δ1 � 3:92% when T � 1. As one can see, the difference bet ween the two approximations is not noticeable. Appendix F. Proofs for the Model with Financial Crisis and Stochastic Calls Let Vg (Wt , Kt , t) and Vb (Wt , Kt , t) denote the value func tions in the normal and crisis states, respectively. The HJB equation for the value function in the normal state, Vg (Wt , Kt , t), is g 0 � max f (C, Vg ) + (rW + δA K + (µS � r)Π � C)VW + C, Π (ΠσS )2 g VWW 2 σ2A K2 g g VKK + ρΠKσS σA VWK 2 + ξg (Vb (W, K, t) � Vg (W, K, t)): g g + Vt + (µA � δA )KVK + (F.1) Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory 24 Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS Similarly, the HJB equation for the value function in the crisis state, Vb (Wt , Kt , t), is given by b 0 � max f (C, Vb ) + (rW + δA K + (µS � r)Π � C)VW C, Π (ΠσS )2 b VWW + Vtb 2 σ2 K2 b b + (µA � δA )KVKb + A VKK + ρΠKσS σA VWK 2 + ξb (Vg (W, K, t) � Vb (W, K, t)) Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. + 2 See https://www.bcg.com/publications/2020/global-asset-manage ment-protect-adapt-innovate. 3 + λ(E(Vb (Z(W � call)K), Z(1 + call)K, t)) � Vb (W, K, t)): (F.2) Using the FOCs for Π and C, we obtain that same portfo lio choice and consumption rules, given by (13) and (14), respectively, for our baseline model. Using the homogeneity property, we obtain the follow ing two interconnected ordinary differential equations for the investor’s scaled certainty equivalent wealth: ! g φ1 (pw (w, t))1�ψ � ψζ γσ2A g 0� + µA � δA � p (w, t) ψ�1 2 2 2 ε w g + pt (w, t) + pg (w, t) 2 ww + ((δA � α + γε2 )w + δA )pgw (w, t) 2 g g γε2 w2 (pw (w, t))2 (ηS � γρσA ) pw (w, t)pg (w, t) + g 2 pg (w, t) 2γi ! � b �1�γ ξg p (w, t) + � 1 pg (w, t), 1�γ pg (w, t) pensions/. For university endowments see the 2020 National Asso ciation of College and University Business Officers Endowment Study at https://www.nacubo.org/Research/2020/Public-NTSE-Tables. See also Brown et al. (2010, 2014) and Dimmock (2012). For family offices, see the UBS/Campden survey http://www.globalfamilyofficereport.com/ investments/. � (F.3) David Swensen of the Yale University endowment is generally credited with originating the endowment model. See Swensen (2000), Takahashi and Alexander (2002), and Lerner et al. (2008). 4 For an empirical examination of liquidity diversification, see Rob inson and Sensoy (2016). 5 The double barrier policy is a standard feature in models with transaction costs. See Davis and Norman (1990) as an early example in the portfolio choice literature. 6 See Lerner et al. (2007, 2008) and Brown et al. (2010). 7 Benzoni et al. (2007) show how nontradable human capital affects portfolio choice. 8 Gallmeyer et al. (2006) model how transaction frictions from the taxation of realized capital gains affect portfolio choice. 9 Our approach follows the common industry practice of defining β relative to the portfolio of publicly traded equity. Although ε is unspanned by public equity, this does not imply it is purely idio syncratic volatility. For example, private equity constitutes a sub stantial fraction of total wealth in the world and is not perfectly correlated with public equity. Theoretically, Cochrane et al. (2008) and Eberly and Wang (2009) show that, in segmented markets, both segments command risk premia. Empirically, Aragon (2007), Sadka (2010), and Franzoni et al. (2012) show that alternative assets earn significant liquidity premia. 10 For the special case of CRRA, f (C, V) � U(C) � ζV, where U(C)R � ∞ ζC1�γ =(1 � γ): By integrating Equation (8), we obtain Vt � Et [ t �ζ(s�t) e U(Cs )ds]. and 0 � 1 �1�ψ b 2 p φ (w , t) � ψζ w 1 γσ B C 0�@ + µA � δA � A Apb (w, t) ψ�1 2 + pbt (w, t) + 11 See Wang et al. (2012) and Bolton et al. (2019) for similar defini tions involving endogenous risk aversion but for very different eco nomic applications. ε2 w2 b p (w, t) 2 ww + ((δA � α + γε2 )w + δA )pbw (w, t) � 12 γε2 w2 (pbw (w, t))2 2 pb (w, t) (ηS � γρσA )2 pbw (w, t)pb (w, t) 2γbi ! � g � ξb p (w, t) 1�γ + � 1 pb (w, t) 1�γ pb (w, t) 0 1 !1�γ , t) λ @ Z(1 + call)pb (w�call 1+call E + � 1Apb (w, t): 1�γ pb (w, t) + (F.4) Finally, the boundary conditions for (F.3) and (F.4) are the same as those in our baseline case. (We index the two g parameters, θL and θX, with the states g and b, i.e., θL (θbL ) g b and θX (θX ).) As p ≥ 0 and pw ≥ 0, Equation (27) implies w t ≥ �(1 � θL ), mean ing that the investor can borrow only a fraction of the alternative asset’s fundamental value. As a result, the investor can repay the liability with probability one by liquidating the alternative asset. Thus, the investor’s debt capacity is endogenously determined by the liquidation value of the alternative asset. Although the investor can borrow, in our numerical exercise, as in reality, borrowing is rare. 13 For hedge funds, we assume a horizon of six months, which approximately equals the sum of the average redemption, advance notice, and lockup periods reported in Getmansky et al. (2015). For private equity and venture capital, we assume a horizon of 10 years based on the average commitment period reported in Metrick and Yasuda (2010). For private real estate and illiquid natural resources, we also assume horizons of 10 years based on the holding periods reported in Collett et al. (2003) and Newell and Eves (2009). 14 In this case, the alternative asset simply expands the investment opportunity set. Thus, as Appendix B.1 shows, the value function is clearly higher than when the alternative asset is illiquid. 15 Endnotes During the 2009–2015 period, the value-weighted average and median annual contribution rates to university endowments were 2.8% and 2.0%, respectively. 1 16 For public pension plans, see the American Investment Council 2021 Public Pension Study at https://www.investmentcouncil.org/ See Franzoni et al. (2012), Kleymenova et al. (2012), Ramadorai (2012), and Nadauld et al. (2019), among others. Dimmock, Wang, and Yang: Endowment Model and Modern Portfolio Theory Management Science, Articles in Advance, pp. 1–26, © 2023 INFORMS 17 Robinson and Sensoy (2016) and Nadauld et al. (2019) show that the net cash flows from private equity are countercyclical. 18 We can relax this assumption at the cost of more involved nota tions and analysis. Downloaded from informs.org by [137.132.123.69] on 28 April 2023, at 18:23 . For personal use only, all rights reserved. 19 In practice, investors prefer not to be called in a crisis precisely for the reason that they do not want their portfolio allocations to be distorted in crisis times. We ignore the negotiation and bargaining between the asset manager and owners. 20 We obtain this cost by combining the average portfolio weights of endowment funds with the estimated secondary market costs in the financial crisis for hedge funds from Ramadorai (2012) and for private equity from Nadauld et al. (2019). 21 Note that, if the allocation to alternatives exceeds 98%, the ratio P/N falls below one, indicating that the investor would be willing to permanently give up the opportunity to invest in the alternative asset if it were feasible for the investor to costlessly liquidate the alternative asset holdings. 22 The indexes are Bloomberg/Barclays U.S. Aggregate Bond Index, Center for Research in Security Prices value-weighted index, Credit Suisse/Tremont Aggregate Hedge Fund Index, Cambridge Associ ates U.S. Private Equity Index, Cambridge Associates U.S. Venture Capital Index, National Council of Real Estate Investment Fiducia ries Property Index (unsmoothed), and the Standard and Poor’s Global Timber and Forestry Index. For private equity, venture capi tal, private real estate, and illiquid natural resources the returns are quarterly; the other returns are monthly. 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