BOSTON UNIVERSITY GRADUATE SCHOOL OF ARTS AND SCIENCES. Dissertation STOCHASTIC CONTROL PROBLEMS WITH PERFORMANCE FEES AND INCOMPLETE MARKETS GU WANG

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1 BOSON UNIVERSIY GRADUAE SCHOOL OF ARS AND SCIENCES Dissertation SOCHASIC CONROL PROBLEMS WIH PERFORMANCE FEES AND INCOMPLEE MARKES by GU WANG B.S., Peking University, 7 M.S., Boston University, Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy 3

2 Approved by First Reader Paolo Guasoni, Ph.D. Associate Professor of Mathematics Second Reader Kostas Kardaras, Ph.D. Reader in Statistics London School of Economics

3 Acknowledgements First of all, I would like to thank my father, Ansheng and mother, Ying. hey may not understand very much of Mathematics, but it is from them I learned to be positive in times of hardship and get over it with persistence. Nothing in this thesis could have been written without them. I want to express my sincere gratitude to my advisor, Professor Paolo Guasoni, for his continuous encouragement during my Ph.D. studies at Boston University. Paolo introduced the field of Mathematical Finance to me and always patiently answered my endless questions. I benefit a lot from discussions with Paolo, because he is strong in Mathematics and at the same time, familiar with Finance theories and literatures, both of which are essential for being a good researcher in Mathematical Finance. His high standards on research set a model for my future academic career. I feel lucky that I asked Paolo to be my advisor in the first place. I want to thank Professor Kostas Kardaras, Professor Murad aqqu, Professor Mark Kon and Professor Eugene Wayne, for their interest in my research and agreeing to be members of the exam committee. I would like to thank Chong Liu, Guoxin Rong and Yang Song, for friendship and support during our first six years in the United States. Finally, I want to thank Zhu Liang. Your love fills every day of my life with happiness and laughter. iii

4 SOCHASIC CONROL PROBLEMS WIH PERFORMANCE FEES AND INCOMPLEE MARKES Order No. GU WANG Boston University, Graduate School of Arts and Sciences, 3 Major Advisor: Paolo Guasoni, Associate Professor of Mathematics ABSRAC his dissertation applies stochastic control theory to two problems: i portfolio choice of hedge fund managers compensated by performance fees, and ii consumption and investment in incomplete markets. Part I. Optimal portfolios are derived in closed form for a fund manager, who is paid performance fees with a high-water mark provision, and invests both the fund s assets and private wealth in separate and potentially correlated, constant investment opportunities. he manager s goal is to maximize expected utility from private wealth in the long run, with constant relative risk aversion. At the optimum, the private portfolio depends only on the private investment opportunity, and the fund s portfolio only on fund s opportunity. he manager invests earned fees in the safe asset, allocating remaining private wealth in a constantproportion portfolio with his own risk aversion. he fund is managed as a constantproportion portfolio with risk aversion shifted towards one. he optimal welfare is the maximum between the optimal welfare of each investment opportunity alone. Part II. An agent maximizes isoelastic utility from consumption with infinite horizon in an incomplete market, in which state variables are driven by diffusions. First, a general verification theorem is provided, which links the solution of the iv

5 Hamilton-Jacobi-Bellman equation to the optimal consumption and investment policies. o tackle the analytical intractability of such problems, approximate policies are proposed, which admit an upper bound, in closed-form for their utility loss. hese policies are optimal for the same agent in an artificial and complete market, in which the safe rate and the state variable follow different dynamics, but excess returns remain the same. he approximate policies have closed form solutions in common models, and become optimal if the market is complete, or utility is logarithmic. v

6 Contents Introduction. Merton s Model Financial Intermediaries Stochastic Investment Opportunities and Incomplete Markets Optimal Portfolio with High-Water Mark Performance Fees 7. Introduction Model Investment Opportunities and Preferences he Case of ν F = Solutions and Discussions Main heorem Portfolio Separation Attention Separation Growth and Fees Heuristic Solution Verification he Fund Value and he Fees Proof of heorem vi

7 3 Optimal Consumption and Investment in Incomplete Markets Introduction Model Market Preferences Optimal Solution Finite Horizon Bounds Long Run Optimality Approximate Optimality in Incomplete Markets Example Stochastic Expected Returns and Volatilities Stochastic Interest Rates Appendix: Proofs Bibliography 4 Curriculum Vitae 8 vii

8 List of ables 3. Parameters for the model of stochastic expected returns and volatilities Parameters for the model of stochastic interest rates viii

9 List of Figures. Fund s value, high-water mark and accumulative performance fees he upper bound of the CEL against the correlation coefficient in the model of stochastic expected returns and volatilities he upper bound of the CEL against the relative risk aversion in the model of stochastic expected returns and volatilities Bounds of the optimal consumption in the model of stochastic expected returns and volatilities he upper bound of the relative error in the approximate consumption in the model of stochastic expected returns and volatilities he approximate and the Merton consumption in the model of stochastic expected returns and volatilities he approximate portfolio in the model of stochastic expected returns and volatilities he upper bound of the CEL against the correlation coefficient in the model of stochastic interest rates he upper bound of the CEL against the relative risk aversion in the model of stochastic interest rates Bounds of the optimal consumption in the model of stochastic interest rates ix

10 3. he upper bound of the relative error in the approximate consumption in the model of stochastic interest rates he approximate and the Merton consumption in the model of stochastic interest rates he approximate portfolio in the model of stochastic interest rates.. 78 x

11 List of Abbreviations HJB ESR SDE ODE CEL Hamilton-Jacob-Bellman Equivalent Safe Rate Stochastic Differential Equation Ordinary Differential Equation Certainty Equivalent Loss xi

12 Chapter Introduction Consumption and investment in continuous time models is central to Mathematical Finance. he question is how investors in financial markets allocate wealth between consumption and investment across different assets, in order to maximize utility from wealth and/or consumption. Markowitz 95 solves this problem in a static setting: investors select portfolios of risky and risk-free assets in order to minimize the variance of the portfolio for every level of expected return. he results indicate that diversification is beneficial, because for the same expected returns, it reduces the variance of portfolios of correlated assets. Merton 969, 97 studies this problem in a continuous-time, dynamic setting: investors, in the face of risky assets moving like geometric Brownian Motions, maximize power utility from terminal wealth or consumption. he answer to this problem both describes agents optimal policies in a financial market, and solves the corresponding pricing problem. Indeed, as duality theory shows, the pricing kernel is proportional to the marginal utility at optimal payoffs in Merton s case, terminal wealth or consumption. Following is a brief summary of Merton s results, which are the archetype for the problems discussed in this dissertation.

13 . Merton s Model Consider a financial market with a risk-free asset S and n risky assets S = S,..., S n, dst St =rdt,.. ds i t S i t =rdt + dr i t, i n,.. dr i t =µ i dt + n j= σ ij dw j t, i n,..3 where r, µ i, σ ij are constants, for all i n and j n. Σ = σσ is positive definite and W = W,, W n is an n-dimentional multivariate Brownian Motion. An agent allocates wealth among these assets according to a portfolio π R n, where π i t represents the proportion of wealth invested in S i at time t, and the proportion invested in the safe asset is n i= πi t. In addition, the agent consumes at a continuous rate c t. hen, with initial wealth X = x, the corresponding wealth process X π,c t follows: dx t = rx π,c t dt + π µdt + σdw t X π,c t c t dt...4 From a set A = {π, c X π,c t for all t }, the agent chooses optimally the pair π, c, in order to maximize power utility from terminal wealth at a future date, [ X max E,..5 π,c A or from consumption until, [ βt c t max E e π,c A dt,..6

14 3 where is the agent s relative risk aversion and β is the subjective time-preference parameter. For objective..5, since only the terminal wealth matters, the optimal consumption is, so that every dollar is saved until, and the optimal portfolio is ˆπ t = Σ µ...7 For objective..6, the optimal policies are ˆπ t = Σ µ,..8 ĉ t = ˆX t β + r + µ Σ µ...9 under the well-posedness condition: β + r + µ Σ µ >. his result indicates that to maximize utility from either terminal wealth or consumption, the optimal portfolio is the same constant in..7 and..8, which is called the Merton portfolio. his portfolio, with consumption c t = for all t, also maximizes the exponential growth rate of the utility from terminal wealth: max π,c A [ ln E X π,c = r + µ Σ µ... he optimal consumption in..9 is a convex combination of the subjective timepreference parameter β and the maximal exponential growth rate of utility in... Merton s model is popular for its tractability, and allowing to understand the effects of risk aversion, investment opportunity and subjective time preference on a- gents optimal choice of consumption and portfolio. But its assumptions of i agents direct investment in risky assets, and ii constant investment opportunities, are ideal-

15 4 izations that can not fully describe practices in financial markets. Following are two stylized facts that deviate from Merton s model.. Financial Intermediaries With financial intermediaries, which have been growing fast in the last several years, investors do not have to trade directly in exchanges and construct portfolios of every single asset, which offers many interesting research topics, because what investors can control is only the selection among financial intermediaries with a variety of investment styles and managerial contracts, and it is not clear what choices are best for investors interest. For example, the hedge fund industry, which, as pointed out in Aragon and Nanda, has assets worth more than one trillion U.S. dollars under management in 9, compensates hedge fund managers by performance fees, provided by funds profit. Since managers want to maximize utility from their personal wealth including earned fees, not the fund s value, this compensation structure raises the question of moral hazard: Are fund managers encouraged to manage hedge funds in a way that is in the best interests of clients, or themselves? When deciding which fund to invest in, besides the fund s investment styles, one key aspect of investors decision is whether the fund s compensation scheme can prevent managers from sacrificing shareholders interests for more fees? From market regulators view point, it is also of importance what performance fee schedules can best align the manager s and investors interests. he first step to answer investors and regulators questions is to know how managers behave in the face of different compensation schemes. Chapter studies the effects of a performance fee schedule that is commonly used in practice the high-water mark performance fees, and answers the following questions: i will managers portfolio choice for the fund s investment

16 5 change with private investment opportunities for performance fees, compared to the case in which fees are restricted to the safe asset? ii will managers hedge against the risk in the fund investment by private investment of fees? iii Can managers compound the return in the private investment to the return in the fund investment, as Merton investors do with two risky assets? iv Will private investment of fees induce managers preference to higher rate of performance fees?.3 Stochastic Investment Opportunities and Incomplete Markets Merton s setting assumes that the risk-free rate, expected excess returns and volatilities of risky assets are all constants, in contrast to many empirical studies, which indicate that investment opportunities change over time. Campbell and Shiller 989 and Fama and French 989 study the predictability of stock returns; Results in Fama and Bliss 987 and Cochrane and Piazessi 5 show that bond returns are also predictable; Ball and orous find that correlations across international stock markets are stochastic. Furthermore, it is always impossible to hedge all risks that affect these changing market parameters, solely by holding a portfolio of risky assets traded in the financial market, which makes the market incomplete. Consumption and investment problems with stochastic investment opportunities and incomplete markets are more complex, because besides making decisions based on the current state of economy, investors have to anticipate changes in the distribution of asset returns in the future and try to hedge as much as possible this additional risk. Chapter 3 investigates this issue in a general framework of incomplete markets, in which all market parameters are functions of stochastic state variables. In particular, i what are the effects of state variables on the optimal consumption and portfolios? ii

17 6 What are the effects of the market s incompleteness on the optimal consumption and portfolios? Both Chapter and 3 are based on papers I co-authored with Poalo Guasoni.

18 Chapter Optimal Portfolio with High-Water Mark Performance Fees. Introduction A line of research that emerges in recent years focuses on the effects of convex compensation schemes, such as the high-water mark provision, on portfolio choice of hedge fund managers. Performance fees are a hedge fund manager s main source of income, and are typically % of a fund s profit after reaching a new historical maximum also called the high-water mark, with the high-water mark provision. his compensation scheme requires that previous losses must be recovered before more fees are paid. As an illustration, Figure. shows a fund s balance which follows a geometric Brownian Motion, with the corresponding high-water mark and accumulative performance fees. A manager s large exposure to fund s performance is a powerful incentive to deliver superior returns, but it is also a potential source of moral hazard, as the manager may use private wealth to hedge such exposure, and thus induces more risk taking in the fund than what is best in investors interest. From a regulator s view-

19 Figure.: One path of the fund balance following a geometric Brownian Motion vertical axis, with drift.8 and volatility.6, with the corresponding high-water mark dashed line and cumulative performance fees dotted line, which are 5% of the increase in the running maximum, against time horizontal axis. point, understanding managers behavior in the face of performance fees helps design compensation schemes and regulation that efficiently align investors and managers interests. Carpenter considers a risk averse manager compensated by a call option on the fund at a finite horizon, and finds that he takes excessive risk if the option is deeply out of the money. In Panageas and Westerfield 9, a risk neutral manager maximizes discounted fees with a long horizon, and optimally takes moderate risk because higher risk in the fund may generate fees in a short term, but reduces the long run growth rate of the fund. he risk averse manager with power utility in Guasoni and Ob lój behaves like a Merton investor with a different relative risk aversion. All these models acknowledge that reinvestment of fees makes hedging the exposure to the fund s risk possible, and is a source of potential moral hazard, but avoid it rather than modeling it, by assuming that private wealth, including earned

20 9 fees, are invested at the risk-free rate. As a result, the literature is effectively silent on the interplay between a manager s personal and professional investments. his chapter begins to fill this gap. Section.. sets up a model with two investment opportunities that are constant over time, separate and potentially correlated, one accessible to the fund, the other accessible to the manager s private account. o make the model tractable, and consistently with the literature, we consider a fund manager with constant relative risk aversion and a long horizon, who maximizes u- tility from private wealth. he assumption of a long horizon means, in particular, that the model s conclusions are driven by a stationary risk-return tradeoff, not by the short-term incentives created by finite horizons. Section.. discusses the case when the Sharpe ratio of the private investment is zero, and gives an alternative interpretation of results in Guasoni and Ob lój. he main results and their implications are summarized in Section.3. We find the manager s optimal investment policies explicitly. For the fund, the optimal portfolio entails a constant risky proportion, which corresponds to the effective risk aversion identified by Guasoni and Ob lój in the absence of private investments. he optimal policy for private wealth is more complex. he manager leaves earned fees in the safe asset, investing remaining wealth according to an optimal constant-proportion portfolio, which corresponds to the manager s own relative risk aversion. he result of these policies combined is that the manager obtains the maximum welfare between the fund and private wealth. he significance of this result is threefold. First, the model predicts that the fund composition does not affect the manager s private investments, and that such investments also do not affect the fund composition portfolio separation holds. In As an exception, Aragon and Qian restrict earned fees to reinvestment in the fund, which also excludes hedging attempts.

21 particular, even if investment opportunities are highly correlated, the manager does not attempt to hedge exposure to the fund s risk with a position in the private account. he intuition is that, for a long horizon, the benefits from hedging are surpassed by the costs of holding a short position in an asset with positive returns. Second, the manager does not rebalance all private wealth. Indeed, the optimal policy is to leave earned fees in the safe asset, and to rebalance only excess wealth. his policy effectively replicates a pocket of private wealth that grows like the highwater mark of the fund, while leaving the other pocket to grow at the optimal rate for private investments. Over time, the pocket with the larger growth rate will dominate the private portfolio, delivering the maximum welfare of the two strategies. In contrast to usual portfolio allocation with multiple assets, private investments can outperform the fund, but cannot compound its return, regardless of correlation. hird, since the manager s welfare is the maximum between the fund s and the private wealth s, our policy is always optimal, but never unique. Indeed, if the fund delivers the optimal welfare, it does not matter how the manager invests private capital in excess of earned fees. By contrast, if private investments deliver optimal welfare, it does not matter how earned fees or even the fund are invested. While lack of uniqueness is an extreme effect of the long-horizon approximation, it highlights that in this model either the fund, or private investments, become the main focus of the manager, without interactions. Further, the model yields the conditions under which the manager focuses on the fund rather than private wealth. In summary, we find that high-water marks in performance fees reduce in the long run, eliminate a manager s incentive to use private investments either to hedge fund exposure, or to augment the fund s returns. his conclusion remains valid if the manager has private access to the fund s investment opportunities, a situation that is nested in our model if correlation between investment opportunities is perfect.

22 he results can inform the decisions of investors and regulators alike. For investors, the main message is that moral hazard is likely to be higher for managers who face shorter horizons, either because the holding period of their average investor is shorter, or because redemptions are allowed more frequently. Also, arrangements that increase a manager s horizon, such as longer lock-up periods, reduce the potential for moral-hazard, as do high-water mark provisions. hese observations are broadly consistent with those of Aragon and Qian, which find that these contract features help alleviate asymmetric-information issues for hedge funds. From a regulatory viewpoint, our results suggest that restrictions on managers private investments may be redundant, if funds contracts lead managers to act with a long horizon perspective, because high-water marks, combined with long horizons, reduce managers incentives to privately trade against investors interests. Section.4 offers a heuristic derivation of the main result, using informal arguments of stochastic control in the case of logarithmic utility case, and Section.5 contains the formal verification of the main theorem.. Model.. Investment Opportunities and Preferences A fund manager aims at maximizing utility from private wealth at a long horizon. For brevity, henceforth private wealth is simply wealth, unless ambiguity arises. o achieve this goal, the manager has two tools: allocating the fund s assets X between a safe asset and a risky asset S X, and allocating wealth F, including performance fees earned from the fund, between the safe asset and another risky asset S F. he interpretation is that the fund has access to investment opportunities that, because of scale, regulation, or technology, are restricted to institutional investors. Examples

23 of such investments are institutional funds, restricted shares, such as Rule 44a securities, or high-frequency trading strategies. By contrast, the manager s wealth is invested in securities available to individual investors. he fund s S X and private S F risky assets follow two correlated geometric Brownian Motions, with expected returns, volatilities and the Sharpe ratio µ X, σ X, ν X = µx σ X and µ F, σ F, ν F = µf σ F respectively. Formally, consider a filtered probability space Ω, F, F t, P equipped with the Brownian Motions Wt X t and Wt F t, with correlation ρ i.e., W X, W F t = ρt, and define the risky assets as dst X St X ds F t S F t =µ X dt + σ X dw X t,.. =µ F dt + σ F dw F t... he manager chooses the proportion of the fund π X to invest in the asset S X, and the proportion of wealth π F to invest in the asset S F. he strategies π X and π F are square-integrable processes, adapted to F t, defined as the augmentation of the filtration generated by W X and W F. he high-water mark X t is the running maximum X t = max s t X s of the net value of the fund. o ease notation, we assume a zero safe rate. hen, the net fund return equals the gross return on the amount invested Xt πx πt X, minus performance fees, which are a fraction of the increase in the high-water mark. hus, dx πx t = Xt πx πt X µ X dt + σ X dwt X α α dxπx t...3 In this SDE, the last term reflects the fact that each dollar of gross profit is split into α as performance fees, plus α as net return, whence performance fees are Guasoni and Ob lój consider a constant safe rate, and find that its value does not affect the optimal fund s policy, suggesting that the assumption of a zero safe rate is inconsequential.

24 3 α/ α times the net return. Similarly, the return on the manager s wealth equals the return on the risky wealth F πx,π F t πt F, plus the fees earned from the fund, i.e. df πx,π F t = F πx,π F t πt F µ F dt + σ F dwt F α + α dxπx t...4 Note, in particular, that while the fund evolution depends only on its policy π X, the evolution of wealth F πx,π F t depends both on π F and on π X, as the latter drives earned fees. he fund manager chooses ˆπ t X and ˆπ t F as to maximize expected utility from fees in the long run, that is, the equivalent safe rate ESR of wealth cf. Grossman and Zhou 993; Dumas and Luciano 99; Cvitanic and Karatzas 995: ESR π X, π F = lim lim [ ln E F πx,π F [ E ln F πx,π F, <,, =...5 his equivalent safe rate measures the manager s welfare, and has the dimension of an interest rate. It corresponds to the hypothetical safe rate which would make the manager indifferent between i actively managing fund and wealth, and ii retiring from the fund, investing all wealth at this riskless rate... he Case of ν F = When ν F =, the private investment opportunity is not attractive at all, and all wealth should be invested the safe asset, which is confirmed in heorem. below. hus, the case of ν F = is effectively equivalent to the model where wealth is restricted to the safe asset and the fund s optimal portfolio should be the same.

25 4 he latter model is studied in Guasoni and Ob lój, and the following gives an alternative interpretation of the model and results in that paper. Since the risk free rate is, from.5.4 below, for any π X, F πx, t = α X πx t X = α α α X e αrx t,..6 where R X is defined in.5. below. he fund manager invests the fund optimally, in order to maximize the ESR of wealth: lim sup = lim sup ln E [ ln E [ F πx, e α RX = lim sup [ ln E X e αrx = α lim sup [ ln E e R X..7 where = α + α. Note that e RX is the running maximum of wealth with π X invested in S X. hus, the model in which performance fees are restricted in the safe asset and managers maximize ESR of wealth can be alternatively interpreted as: for an agent with relative risk aversion, who faces constant investment opportunities, what is the optimal investment strategy that maximizes α of the ESR of wealth s running maximum? Here the objective is not the terminal wealth, but its running maximum. Hence this is like a path dependent derivative, where the buyer has certain degrees of freedom, not to choose the time to exercise optimally, as for American type options, but to choose the investment strategy optimally. he results in Guasoni and Ob lój imply that the optimal strategy is ˆπ X = µ X and the optimal welfare is ανx σ X, which is the Merton portfolio and α of the optimal welfare, respectively, for investors with relative risk aversion

26 5 and constant investment opportunity ν X. he authors interpretation is that highwater mark performance fee schedules shift the manager s risk aversion, and the manager acts like a Merton investor for the fund investment. From the above new interpretation of the model, this result also shows that in the long run, maximizing utility from running maximum of wealth is equivalent to maximizing utility from terminal wealth..3 Solutions and Discussions.3. Main heorem he main result below identifies the manager s optimal policies, and the corresponding welfare. It is proved in the case of logarithmic risk aversion or lower, hence it allows to understand the risk-neutral limit. heorem.. Let,, and set = α + α. he investment policies ˆπ t X = µx σ X,.3. ˆπ t F = α X ˆπX t X µ F α F t σ F,.3. attain the manager s maximum equivalent safe rate of wealth, which equals to ESR ˆπ X, ˆπ F = max ν X + α α, ν F..3.3 he optimal fund policy in.3. shows that the manager invests in the constantproportion portfolio with relative risk aversion between one and the manager s own relative risk aversion. his policy, even with ν F, coincides with the one obtained

27 6 by Guasoni and Ob lój in the absence of private investment opportunities, which corresponds in our model to ν F =. In this case, the private risky opportunity has zero return, hence it is never used. he private policy in.3. is best understood in terms of the total risky and safe positions, which amount respectively to ˆF tˆπ t F = ˆF t ˆF t ˆπ t F = ˆF t α ˆX t X α α α µ F σ F, α ˆX t X. α ˆX t X µf σ F.3.5 hese formulas show that the manager divides wealth into earned fees α ˆX α t X, which are set aside in the safe asset, and the rest, which is invested in the constantproportion portfolio with the manager s own relative risk aversion. he manager s welfare in.3.3 equals the maximum between the welfare of fees and the welfare of private investments..3. Portfolio Separation A salient feature of this result is that the fund policy is independent of private positions, and vice versa. In other words, ˆπ X does not depend on µ F, σ F, and ˆπ F does not depend on µ X, σ X. Further, neither ˆπ X nor ˆπ F depend on the correlation coefficient ρ between investment opportunities. We call this property portfolio separation. Portfolio separation entails that a manager with long horizon has no incentive to hedge the exposure of future performance fees to the fund s investments with private risky assets, regardless of their correlation. In fact, hedging does not take place even in the limit case µ X = µ F, σ X = σ F and ρ =, which corresponds to a manager who

28 7 has unfettered access to the fund with private capital, and hence faces a dynamically complete market. he biggest concern about high-water mark performance fees combined with fees private investment opportunities is that when the fund and private investment opportunities are highly correlated, managers may, like a Merton investor, short in private investments and take a larger risk in the fund investment, which may be against clients interests. o understand why such hedging is ineffective, consider a manager whose fund trades well below its high-water mark, as it is the case for the most of the time. In this case, there is nothing to hedge against the short position in private investments, because the high-water mark and hence future income is insensitive to small variations in the fund value, and therefore reduces the growth of wealth in the long run. Mathematically, though the two investment opportunities are correlated, the highwater mark is increasing, hence of bounded variation, and has zero local covariance with the private investment. Since high-water mark performance fees are the manager s source of income and thus the only thing he cares about from the fund side, correlation does not matter. Conversely, portfolio separation implies that the manager has no incentive to take more or less risk in the fund, in view of private investment opportunities outside the fund. A priori, it may seem plausible that a manager takes more risk in the fund if outside opportunities are attractive, because more risk is likely to lead to earlier performance fees, which could then be invested in outside opportunities. However, this tactic can only generate a one-time transfer of wealth, but not a lasting increase in the growth rate of the manager s wealth, hence it is long-run irrelevant. In summary, the message of portfolio separation is largely positive: if horizons are long, then moral hazard concerns are limited, because high-water marks essentially

29 8 defeat any hedging incentives between the fund and wealth. Yet, portfolio separation has a downside attention separation..3.3 Attention Separation As a consequence of portfolio separation, the manager s welfare in.3.3 is the maximum between the welfare from performance fees, and the welfare of remaining wealth. hus, while the joint policy in.3. and.3. is optimal in all cases, it is never unique. Indeed, if the manager s welfare in.3.3 is due to the fund i.e. performance fees, then private investment opportunities becomes irrelevant, and can be replaced, for example, with the policy π F =. Vice versa, if remaining wealth drives welfare, then the fund policy is irrelevant, and utter negligence π X = will deliver the same result. his rather extreme implication is clearly driven by the assumption of a longhorizon, which focuses on the risk-adjusted long-term growth rate, neglecting all other welfare effects. Still, it makes it clear that a manager s commitment to the fund will easily wane, unless its investments are superior to outside opportunities. he manager s attention inevitably shifts to either the fund, or wealth, whichever is more profitable. Indeed, equation.3.3 shows that the manager focuses on the fund if and only if the fund s Sharpe ratio ν X exceeds the private Sharpe ratio ν F by a multiple, which depends on the fund s fees and on the manager s risk aversion: ν X ν F α α For example, in the case of a logarithmic manager =, and performance fees of %, the manager focuses on the fund, provided that its Sharpe ratio is.8% higher

30 9 than private investments. Such a condition is likely to hold in practice: Getmansky, Lo and Makarov 4 find high Sharpe ratios in the hedge fund industry, even after controlling for return smoothing and illiquidity. he right-hand side in.3.6, which represents the manager s attention threshold, grows as risk aversion declines. he explanation is as follows: as risk aversion declines to zero, the effective risk aversion = α + α induced by the high-water mark converges to α, which entails finite leverage in the fund. On the other hand, the private portfolio is driven by the true risk aversion, which declines to zero, leading to increasingly high leverage. Because leverage can arbitrarily magnify expected returns, for sufficiently low risk aversion the private portfolio is always more attractive. Overall, attention separation brings both some bad news, as the manager may grossly neglect a fund if it does not offer sufficiently attractive returns, and some good news, since the conditions for attention to the fund seem mild, and a manager with very low risk aversion is likely to leverage wealth rather than the fund..3.4 Growth and Fees A puzzling feature of extant models of performance fees is that a manager prefers lower performance fees, i.e. welfare is decreasing in α. he explanation of this finding, common to the models of Panageas and Westerfield 9 with risk-neutrality, and of Guasoni and Ob lój with risk aversion, is that higher fees today reduce the growth rate of the fund, leading to lower fees tomorrow. Both models assume that fees are invested at the safe rate in the manager s account, and raise the question of whether reinvestment can induce preference for higher fees. Equation.3.3 offers a qualified negative answer. If private risky investments are available, there will be a threshold α, below which the manager prefers lower fees, as

31 in the absence of private investments, and above which the manager is indifferent to changes in fees, because the fund becomes irrelevant, as welfare is entirely driven by wealth. his threshold is in fact the value of alpha for which.3.6 holds as equality. his result is essentially a consequence of portfolio separation. Because the manager is unable to compound fund growth with wealth growth, either private investments make fees negligible, or are negligible themselves. Overall, the model shows that the reinvestment value of fees is not sufficient to obtain a manager s preference for higher payout rates, which in turn is likely to involve intertemporal preference for consumption, or fund flows..4 Heuristic Solution his section derives a candidate optimal solution with heuristic stochastic control arguments. For brevity, this heuristic argument is presented only for the case of logarithmic utility, while the rigorous proof for all cases < is in the next section. o ease notation, in the argument below we drop the superscripts π X and π F from X t and F t. Denoting by Z t = Xt X, the manager s value function is V t, x, f, z = sup π X,π F E t [ln F X t = x, F t = f, Z t = z..4. By Iˆto s formula, dv t, x, f, z = V t + xµ X π Xt V x + fµ F π Ft V f + σx xπt X V xx + σf fπt F V ff + ρσ X σ F xfπt X πt F V xf dt + σ X xπt X V x dwt X + σ F fπt F V f dwt F + V z + α α V f V x dxt.

32 hus, the Hamilton-Jacobi-Bellman HJB equation for V t, x, f, z is, for < x < z + X, V t = sup π X,π F σ xµ X πt X V x + fµ F πt F X xπt X V f + σ F fπt F V xx + V ff + ρσ X σ F xfπt X πt F V xf, with the boundary condition: V z + α α V f V x = when x = z + X. By the usual scaling property of logarithmic utility, and in the long-horizon limit, we can rewrite the value function as V t, x, f, z = βt+ln z +vξ, ϕ, where ξ = ln x z and ϕ = ln f, and the HJB equation becomes z β = sup π X,π F σ µ X πt X v ξ + µ F πt F X πt X v ϕ + σ F πt F v ξξ v ξ + v ϕϕ v ϕ + ρσ X σ F πt X πt F v ξϕ, for < x < z + X, while the boundary condition reduces to α v ξ α exp ξ + α + v ϕ α exp ϕ α =, when x = z + X. Since the manager s aim is to maximize the ESR of wealth and X becomes large at optimum in the long run, the initial fund s value X should not matter in this optimization problem. hus, Xt Z t = Xt X, and we can approximate the HJB equation and the boundary condition with β = sup π X,π F σ µ X πt X v ξ + µ F πt F X πt X v ϕ + σ F πt F v ξξ v ξ + v ϕϕ v ϕ + ρσ X σ F πt X πt F v ξϕ,.4. when < ξ <, and α v ξ + v ϕ α exp ϕ α =,.4.3

33 when ξ =. he first order conditions for π X and π F in.4. are πt X = σf µ X v ξ v ϕϕ v ϕ ρσ X µ F v ξϕ v ϕ σ X σ,.4.4 F v ξξ v ξ v ϕϕ v ϕ ρ vξϕ πt F = σx µ F v ϕ v ξξ v ξ ρσ F µ X v ξϕ v ξ σ X σ F..4.5 v ξξ v ξ v ϕϕ v ϕ ρ vξϕ Plugging.4.4 and.4.5 in.4. yields: when < ξ <, β = µ X σ F v ξ v ϕϕ v ϕ ρσ X σ F µ F µ X v ξ v ϕ v ξϕ + σ X µ F v ϕ v ξξ v ξ σ X σ F =. v ξξ v ξ v ϕϕ v ϕ ρ vξϕ.4.6 Since v also solves.4.3, v ξ v ϕ α exp ϕ α is a constant for any ϕ R when ξ =. hus, we guess the solution to.4.3 as vξ, ϕ = δξ + b ln α α expϕ, where β, b and δ are three parameters to be found. Plugging this guess into.4.3 gives b = δ. Finally, plugging vt, ξ, ϕ = δξ+ δ ln α α expϕ α into.4.4,.4.5 and.4.6 yields: ˆπ t X = µx σ X,.4.7 ˆπ t F αz µ F = αf σ F,.4.8 βδ = δ ν X + δ ν F..4.9 α Notice that ˆπ X t and ˆπ F t do not depend on δ and taking δ to be when νf αν X and α when ν F < α ν X yields and.4.8 help us conjecture the optimal policies for the fund and private investments: the manager puts performance fees, which are αz αf of wealth in the safe asset, and invests the rest in the Merton portfolio. For the fund, the same strategy as in Guasoni and Ob lój

34 3 is adopted, as to ensure the maximum ESR from performance fees..5 Verification We now show that the policies.3. and.3. are optimal, and lead to the maximum ESR from wealth in he Fund Value and he Fees We start by defining the following processes, which represent cumulative log returns, before fees, on the fund and wealth, respectively, R X t = R F t = t t R X t, = R X R X t, R F t, = R F R F t. [ µ X πs X σ X πs X ds + σ X πs X dws X [ µ F πs F σ F πs F ds + σ F πs F dws F,.5.,.5. R X and R F depend on π X and π F, respectively, and should be denoted as R X,πX and R F,πF. We drop these superscripts for ease of notation, unless it causes ambiguity. SDEs..3,..4, Proposition 7 and Lemma 8 in Guasoni and Ob lój imply that X t = X e RX t αrx t,.5.3 Xt = X e αrx t,.5.4 F t = F e RF t + α α t e RF s,t dx s..5.5

35 4.5. Proof of heorem. he discussion begins with two simple lemmas that are used often in the proof of the main theorem. Lemma.. Let X t and Y t be two continuous processes, and define X t and Y t = min s t Y s. hen X t + Y t X + Y t. = max s t X s Proof. Since X + Y t X s + Y s X s + Y t for all s t, it follows that X + Y t X t + Y t..5.6 Lemma.3. Let G t t be a continuous filtration and let F G be a σ-algebra, and denote by E F and E Gt the conditional expectation with respect to F and G t, respectively. If A t is an increasing process adapted to G t for t, and X t is a positive, continuous stochastic process such that E Gt [X t, C, for all t, and some constant C, where X t, = X X t, then [ E F X t, da t CE F [A A. Proof. Since A t is an increasing process, for a partition of [, : = t n < t n < < t n n =, t n k = k for i n, n X t, da t = lim n n k= X t n k, A t nk A t nk..5.7 hus, [ [ E F X t, da t = E F lim n n k= X t n k, A t n k A t n k,.5.8

36 5 and by Fatou s Lemma and the tower property, the right-hand side is less than or equal to lim inf n E F [ n k= X t n k, [ n A t n k A t n k = lim inf E [ k F E Gt n Xt n n k k, A t n A k t n. k=.5.9 Since E Gt [X t, C for all t,.5.9 is less than or equal to C lim inf n E F [ n k= A t n k A t n k = C lim inf n E F [A A = CE F [A A..5. he proof of heorem. is divided into the following two steps. First, any investment policies π X and π F satisfy the following: ESR π X, π F max ν X + α α, ν F α = max ν X, ν F,.5. which is proved in Lemma.4. Second, this upper bound is achieved by the candidate optimal policies in.3. and.3., as proved in Lemma.9. Lemma.4. For any investment strategies π X and π F, α ESR π X, π F λ = max ν X, ν F. We prove this lemma for logarithmic utility and power utility, respectively. Proof of Lemma.4 for logarithmic utility:

37 6 For convenience of notation, define X t = for t <, α α F + Xt X for t. hen X t is an increasing process, which has a jump at t =, and then grows with Xt. From.5.5, F t = α α erf t, d Xt, and..5 can be rewritten as, lim sup =λ + lim sup [ E ln E Since d W X, W F t α α [ ln e RF t, d Xt = lim sup [ E ln e RF t, d Xt.5. e λ +RF t, d Xt..5.3 = ρdt, W X t = ρw F t + ρ W, where W t is a Brownian Motion independent to W F t. Denote E W as the expectation conditional on W s s the whole trajectory of W until. By Lemma.5 below, lim sup lim sup = lim sup lim sup E E [ ln [ ln [ [E E W ln [ [ln E E W e λ +RF t, d Xt e RF t, t ν F ds+ν F dws F e α ν X t αν X ρw F t d X t.5.4 e RF t, t ν F ds+ν F dws F e α ν X t αν X ρw F t d X t.5.5 e RF t, t ν F ds+ν F dws F e α ν X t αν X ρw F t d X t,.5.6 where.5.5 follows from the tower property of conditional expectation, and.5.6 from Jensen s inequality.

38 7 Next, Lemma.6 below implies that e RF,t t ν F ds+ν F dws F with respect the filtration generated by W F s W F and W, plus the future of W. hus, s t and W s is a supermartingale s the present of E W,W F t [ e RF t, t ν F ds+ν F dw F s, t..5.7 In addition, A t = t α e ν X t αν X W F t d X t is an increasing process. hus,.5.7 and Lemma.3 imply that E W E W [ [ e RF t, t ν F ds+ν F dws F e α ν X t αν X ρw F t d X t e α ν X t αν X ρw F t d X t..5.8 hus, from.5.3 and.5.8, it follows that lim sup E [ln F λ + lim sup hen, Lemma.7 below proves that [ [ln E E W e α ν X t αν X ρw F t d X t..5.9 lim sup [ [ln E E W e α ν X t αν X ρw F t d X t,.5. whence lim sup E [ln F λ,.5. which concludes the proof for logarithmic utility.

39 Lemma.5. If λ = max lim sup lim sup E E [ ln [ ln 8 ν F, α ν X, e λ e RF t, d Xt e RF t, t ν F ds+ν F dws F e α ν X t αν X ρw F t d X t. Proof. Define the stochastic process N s = W F W F s, for s, and note that N s has the same distribution as W F s. It follows that lim E [ ν F N = lim = lim E [ ν F N E [ ν F W F αν X ρw F + lim + lim E [ αν X ρw F E [ αν X ρ W F.5. =,.5.3 where the last equality uses the fact that, for a, b, lim b E [aw = lim b ax π e x dx =..5.4 hus, lim sup = lim sup lim sup E E [ ln [ ln e λ +RF t, d Xt e λ +RF t, d Xt + lim E [ ν F N ανx ρw F.5.5 [ E ln e λ νf N ανx ρw F +RF t, d X t..5.6 Now, note that N W F W t F, λ ν F t + α ν X t and

40 9 αν X ρw F ανx ρw F, for all t. hus, it follows that t λ ν F N t αν X ρw F + RF t, ν F α t =Rt, F ν F ds + ν F dws F ν X t ν F W F Wt F α αν X ρw F t + RF t, ν X t αν X ρw F t. Plugging this inequality into.5.6 yields: lim sup lim sup E E [ ln [ ln e λ e RF t, d Xt e RF t, t ν F ds+ν F dws F α ν X t αν X ρw F t d Xt..5.7 Lemma.6. Let π t be adapted to the filtration {F t } t generated by Ws X W F. Define {G s s t t} t as the filtration generated by Ws F and Ws X hen M t = e t π s dw F s π sds is a supermartingale with respect to {Gt } t. s t s t and s. Proof. Suppose π is a simple process, i.e. π t = Σ n i=π i ti,t i for a partition of [,, = t < t < t < t n = and π i is F ti -measurable, for i =,, n. hen for any s < t, E W X,W F s [e t π udw F u π u du = e s π udw F u π u du E W X,W F s [e t s π udw F u π u du..5.8 Since there exists k s k t n such that t ks s t ks and t kt t t kt, thus s, t and all the division points in between forms a partition of [s, t, denoted by s = u < u < u < u m = t. hen, since W X and W F are two independent

41 3 Brownian Motions, E W X,W F s [e t sπ udw F u π udu = E W X,W F s [ m i= e π iwu F i Wu F i π i u i u i = m i= E W X,W F s.5.9 [ e π i Wui F Wui F π i u i u i =,.5.3 and thus, E W X,W F s [e t π u dw F u π udu = e s π u dw F u π udu..5.3 For a general π, from the definition of stochastic integral, there exists a sequence of simple processes {π n t } n=, such that t π n s dw F s πn s n ds t π s dw F s π s ds a.s.,.5.3 hence, for s t, E W X,W F s [e t π udw F u π udu = e s π udw F u π udu EW X,W F s [e t s π udw F u π udu =e s π udw F u π udu EW X,W F s [ lim inf n e s π udwu F π udu lim inf E W n X,W s F.5.33 t e s πudw n u F πn u du.5.34 [e t sπudw n u F πn u du.5.35 =e s π udw F u π udu,.5.36 which confirms that M t is a supermartingale with respect to {G t } t. Lemma.7. lim sup [ [ln E E W e α ν X t αν X ρw F t d X t.

42 3 Proof. By integration by parts, e α ν X t αν X ρw F t d X t = αf + e α ν X t αν X ρw F t dxt α = αf α + α ν X + αν X + e α ν X αν X ρw F X X e α ν X t αν X ρw F t X t dt.5.37 e α ν X t αν X ρw F t X t d ρw F t Since X t = e αrx t, from Lemma., e α ν X t αν X ρw F t X t X e α R X νx ν X ρw F t hus, from.5.38, e α ν X t αν X W F t d X t αf + X e α R X α + α ν X X + αν X X ν X ν X ρw F e α R ν X ν X ρw F e α R X t dt ν X ν X ρw F t d ρw F t..5.4

43 3 Since e α R X α ν X ν X ρw F t e α R X ν X X + αν X X e α R ν X ν X ρw F e α R X ν X X e α R X ν X ν X ρw F α + αν X X e α R X α = ν X ν X ρw F for all t, t dt ν X ν X ρw F t d ρw F ν X ν X ρw F ρw F X e α ν X + αν X ρ W F hus, from.5.4 and.5.4, E W [ E W [F + e α ν X t αν X ρw F t d X t + α =F + E W [ + α t.5.4 R X ν X ν X ρw F..5.4 ν X + αν X ρ W F X e α R X ν X αν X ρw F ν X + αν X ρ W F X e α R X ν X αν X ρw F [ δ [ F + E W L δ δ E W K δ δ δ,.5.45 for any δ >, by Hölder s inequality, where K = + α L =e α R X ν X + αν X ρ W F X,.5.46 ν X αν X ρw F.5.47 Since δ > and δ >, by Minkowski inequality E [f + δ gp p E [f p p +

44 33 E [f p p, it follows that [ δ E W K δ δ δ + α ν X + αν X ρ E W = + α ν X + αν X ρ [ W δ δ δ F δ X.5.48 δ + δ Γ π δ δ X hus from.5.45 and.5.49, setting C = + α ν X + αν X ρ δ + δ Γ π δ δ X,.5.5 for any δ >, E W [ αf α e α ν X t αν X ρw F t d X t + C E W [ e α R X ν X αν δ X ρw F δ..5.5 hen lim sup lim sup lim sup + lim sup [ [ln E E W [ E ln e α ν X t αν X ρw F t d X t [ αf + C E α W e αδ R X ν X αν X ρw F δ.5.5 ln C.5.53 [ E αf ln + E αc W [e αδ R X ν X αν X ρw F δ..5.54

45 34 Note that the limit in.5.53 is. Since F C a.s. as, for large enough, αf αc < E W [e αδ R X ν X αν X ρw F δ hus,.5.54 is less than or equal to lim sup = lim sup = lim sup + lim inf lim sup [ E ln E W e α R X [ln E W [e αδ R X δ E δ E [ δ E δ E [ln E W [e αδ R X ν X ν X ρw F ν X ν X ρw F ν X ν X ρw F ρ αδν X W [ln E W [e αδ R X ν X ν X ρw F δ δ.5.56 ρ αδν X W [ Note that.5.58 holds because lim inf δ E ρ αδν X W =, which follows from.5.4. hen, again by Lemma., the running maximum and running minimum can be combined, and.5.59 is less than or equal to lim sup = lim sup lim sup δ E δ E δ ln E [ln E W [e αδ R X [ln E W [e αδ R X [e αδ R X ν X ν X ρw F ν X ν X W X ρ ν X W.5.6 ν X ν X W X,.5.6 where.5.6 follows from Jensen s inequality and tower property of conditional expectation. M t = e RX t ν X t ν X Wt X generated by Ws F and W s t s is a local martingale with respect to the filtration. hen, since M s t t M, which in turn is

46 35 dominated by a random variable X, and X is uniformly distributed on [, cf. 54 in Guasoni and Ob lój, for < δ <,.5.6 is less than or equal to α lim sup = lim sup [ δ ln E δ ln M αδ lim sup δ ln x αδ dx.5.6 =,.5.63 αδ which concludes the proof. Proof of Lemma.4 for power utility: For the rest of this chapter, let p =. Suppose now that the fund manager has an additional source of income, such that whenever performance fees are paid, ϵ of fees are matched like a bonus, with a restriction that this extra income must be invested in the safe asset. Let the manager s wealth under this schedule be F t, with strategy π F for private investments, the dynamics of F t is d F α π t = F t ϵ α X F t X t µ F dt + πt F σ F dwt F ++ϵ α α dx t Solving this SDE and comparing the result to.5.5 implies that, with the same π F, F t = F e RF,πF t + α α α = F t + ϵ t e RF,πF s,t dxs α + ϵ α X t X.5.65 α X t X hus, F t F t and F t ϵ α α X t X for all t, and ESR of F is less than or equal to ESR of F. Lemma.8 below shows that this upper bound is also less than or equal to λ.

47 36 Lemma.8. lim sup Proof. Let π F t [ ln E F p p λ for all < p <. = F t ϵ α α X t X F t π F t. Investing π F t of F t ϵ α α X t X in the risky asset is equivalent to investing π F t of F t. hus π F t can be regarded as an investment strategy for F t, and d F p t = p + p F t pp α π F t ϵ α X F t X t µ F dt + πt F σ F dw F F p α t F t ϵ α X t X π F t σ F dt + + ϵp p F t = p F p t π t F µ F p F + π t σ F dt + π Ft σ F dw F + + ϵp F p t α α dx t.5.67 α α dx t Solving this SDE, F p = F p e prf, πf α + p + ϵ α e prf, πf t, F p t dx t hus, lim sup = lim sup [ ln E F p p ln E [ F p e prf, πf α + p + ϵ α e prf, πf t, p F t dxt p..5.7 Since < p <, from Dembo and Zeitouni 998, Lemma..5, for any positive processes f t and g t, lim sup ln f + g p = max lim sup ln f p, lim sup ln g p..5.7