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Federal Reserve Bank of Chicago Gambling for Dollars: Strategic Hedge Fund Manager Investment Dan Bernhardt and Ed Nosal November 2013 WP 2013-23 Gambling for Dollars: Strategic Hedge Fund Manager Investment Dan Bernhardt University of Illinois Ed Nosal Federal Reserve Bank of Chicago Abstract Hedge fund managers di¤er in ability and investors want to distinguish good ones from bad. Via the design of their investment strategies, better fund managers want to ease this inference problem while worse fund managers want to complicate it. We impose only the minimal restrictions on the nature the investment strategies that, on average, returns re‡ect the hedge fund manager’s ability and that returns be bounded from below, and solve for the set of equilibria that emerge. We then show that under a variety of equilibrium re…nements, a unique equilibrium obtains. In this equilibrium, investors set a cuto¤ standard for providing capital to a hedge fund: and invest if and only if returns exceed this cuto¤. This induces less able hedge fund managers to adopt risky investment strategies that maximize the probability of meeting this cuto¤ by risking large losses if they fail. Over time, as investors learn about a hedge fund manager’s ability and less able hedge fund managers are stochastically weeded out, investors set less demanding re-investment standards. Our economy reconciles many facts regarding hedge fund performance. For example, in a regression with …xed hedge fund manager e¤ects, returns of more experienced hedge fund managers decline, even though the expected pro…ts of investors rise with the hedge fund manager’s experience; more experienced hedge funds deliver less volatile returns; persistence of returns is greater for better hedge funds; hedge fund failure rates are initially very high, but fall sharply with hedge fund manager experience; returns of exiting hedge funds are substantially worse than historical returns; and the longer is an investor’s horizon, the lower is the expected return of the hedge funds in which he invests. 1 Introduction When hedge fund managers di¤er in their abilities to identify pro…table investment strategies and pro…table investment opportunities, investors have strong incentives to distinguish good hedge fund managers from bad, and to allocate their resources accordingly. Unfortunately, the very way in which hedge funds generate returns make this inference problem di¢ cult for 1 November 29, 2013 investors. In particular, hedge fund investment strategies are zealously concealed, because those strategies are the source of their pro…ts. If revealed, successful strategies can be mimicked by others, thereby competing away those pro…ts. To conceal investment strategies from other institutions, hedge funds must also conceal their strategies from investors. As a result, investors must look at realized hedge fund performance, which they observe on a periodic basis, to determine how to allocate their money. If a hedge fund does well, investors will infer that the hedge fund manager is more likely to be able, and shift investments toward the fund; if a fund does badly, investors will conclude that the hedge fund manager is less likely to be able, and move investments out. The (convex) ‡ow of cash into better-performing funds and away from poorly-performing funds has been well documented (e.g., Agarwal et al. (2009), Chevalier and Ellison (1997)), and indicates that investors use hedge fund performance to update about a manager’s investment skills. In this paper, we determine how this endogenous fund ‡ow relationship a¤ects both the incentives of di¤erent fund manager types to take on risk and the nature of that risk, and how it varies over time according to historical fund performance. In turn, we derive the consequences for the evolution of hedge fund returns, as investors learn more from continued observation of a hedge fund’s returns about its manager’s skills. The central idea is that a hedge fund manager knows his level of competence and recognizes that investors will shift investments away from poorly-performing funds toward betterperforming funds. Less able fund managers can employ appropriately tailored investment strategies to try to mimic the performance of better fund managers; and good hedge fund managers can tailor investments to try to distinguish themselves from bad ones. Hedge fund managers have enormous discretion in the design of their investment strategies. To capture this discretion, we impose almost no restrictions on the possible investment strategies that a fund manager can employ, requiring only that an investment strategy have a payo¤ that is bounded from below, and that, on average, returns re‡ect a hedge fund manager’s ability. In this paper we focus on a simple investment technology, where if a …xed amount is invested, the hedge fund’s “project”will pay o¤ an amount that depends on the hedge fund manager’s ability. Hence, each period, the investor’s decision becomes whether to re-invest in the fund, rather than how much to invest. This allows us to characterize more easily the equilibrium dynamics on investor re-investment choices, the evolution of the distribution of fund performances (mean, volatility and persistence), the impact of survivorship bias 2 on measured performance, investor horizon and unobserved, post-investment, idiosyncratic shocks to fund performance. We show that under a set of di¤erent equilibrium re…nements, the equilibrium is uniquely pinned down, and investors set simple cuto¤ standards for reinvestment that decline over time as investors have longer track records on which to assess performance. In this equilibrium, the investor is indi¤erent between investing in the hedge fund conditional on the hedge fund return achieving this re-investment standard, and investing all funds in alternative assets. Better hedge fund managers do not need to distort investments to receive continued funding. Less able fund managers, in contrast, will tailor their investment strategies to maximize the probability of meeting that re-investment standard, placing residual probability on “disaster”, i.e., on the lower bound on payo¤s. This re‡ects that the hedge fund will have to exit regardless of the degree of su¢ ciently poor performance. This equilibrium, which is the unique selection of the Grossman Perry perfect sequential equilibrium re…nement (Grossman and Perry 1986), is also the one that is best for investors given the moral hazard problem they face from hedge fund managers. It follows that there will be clusters of hedge fund returns slightly “above”expected performance. Consistent with this, Dimmock and Gerken (2013) and Bollen and Pool (2009) …nd that funds are far more likely to report small positive returns than small negative returns. While these papers argue that this re‡ects mis-reporting (by about 10% of hedge funds), our analysis shows that strategic design of investment strategies could account for much of this. It also there will be far fewer hedge funds with modestly poor returns, and ‘unusually many’ extremely poor performers. Indeed, Malkiel and Saha (2005) document that failing to account for these (liquidated) hedge funds that disappear from hedge fund databases as a result, biases up estimates of hedge fund returns by four percent. We also predict that because good hedge fund managers do not need to strategically tailor investment strategies to receive continued funding, their returns will be less volatile and more persistent, consistent with the …ndings of Jagannathan et al. (2010). Moreover, because bad fund managers are stochastically weeded out over time, we predict that the reinvestment standards that investors set over time will decline. As a result, survival rates rise with the age of hedge funds— both because bad managers are di¤erentially more likely to be weeded out, and because some intermediate hedge fund manager types cease to have to adopt riskier investment strategies. Further, consistent with Boyson (2005), those senior hedge funds that do fail are those that pursue riskier strategies— these are run by less able 3 hedge fund managers who initially got lucky, but still have to have their riskier investment strategies succeed to win continued funding. In turn, if one tracks the performance of a surviving hedge fund, we predict that more experienced hedge funds will have both less volatile returns, and lower average returns, as, for example Boyson (2005) documents in her …xed (hedge fund manager) e¤ects regressions. This latter result re‡ects that hedge fund managers that had to employ risky investment strategies survived because they initially got lucky, and achieved a return that exceeded their expected return. Over time the upside of their risky investment strategies declines, so their returns will fall, regardless of whether they continue to get lucky or not. It follows that tracking a hedge fund’s performance over time, the performance of older hedge funds will, on average, decline. This decline in returns for older hedge funds is reinforced when hedge funds are subject either to pre-investment idiosyncratic shocks that a¤ect the quality of their investment opportunities at a moment in time, or to post-investment idiosyncratic shocks to returns after their investment strategies have been chosen. In both cases, ceteris paribus, surviving hedge funds will tend to be those that got positive idiosyncratic shocks, implying that future expected returns will be lower. This reduced performance of older funds would be further reinforced were our hedge fund investment technology to be partially scalable, so that taking larger positions in di¤erent assets have price impacts that are more limited in nature. In this case, an investor puts more money into better performing hedge funds, causing the expected marginal return to decline (see Berk and Green 2004, and Fung et al. 2008). Our …ndings collectively resolve some seeming paradoxes: Why is investor learning over time about which hedge fund managers are better associated with lower expected future hedge fund manager returns for a given fund manager? If this is so, why shouldn’t investors put their money into newer hedge funds? The answer is that, over time, the stochastic …ltering of worse hedge fund managers means that the average quality of hedge fund managers rises with time— in the cross-section, more experienced hedge fund managers are better. However, the expected performance of any given hedge fund manager should decline, and this decline should be sharper for (low and intermediate quality) hedge fund managers who pursued (and possibly continue to pursue) riskier investment strategies. In turn, these predictions point to the very di¤erent theoretical predictions for …xed e¤ect regressions (returns of more experienced fund managers should fall) versus cross-sectional regressions (returns of more experienced fund managers should rise). In particular, we resolve these “paradoxes”, 4 without appeal to the various biases in hedge fund databases (e.g., hedge funds only enter the databases if they succeed, in which case their initial superior returns are ‘back…lled’) or irrational investor behavior. While not dismissing the importance of these sources, we highlight how strategic behavior by fully rational agents alone delivers these predictions. Our analysis of how an investor’s investment horizon a¤ects expected hedge fund returns o¤ers and then resolves a similar paradox: the longer is the investor’s horizon, the lower is the expected return of the hedge funds in which he invests. The resolution is that investors with longer horizons are, in e¤ect, more patient, setting lower standards for continued reinvestment in the hedge fund because they value the embedded option of learning more about the hedge fund manager’s ability— and they can only learn and continue to invest if the hedge fund survives. While very good hedge managers receive continued funds regardless of an investor’s horizon, those with intermediate abilities who would receive funds in a full information setting, are more likely to obtain funding when their investors’are more long-sighted. Our theoretical model indicates that lesser hedge fund managers would like to adopt investment strategies that place as much probability mass as possible on returns just above what an investor would demand for renewed funding— which, typically will be a little above the ‘market’return. We have allowed for enormous discretion in the design of those investment strategies, but have remained silent about how they might construct those strategies in practice. One such practical vehicle are asset-backed securities (e.g., mortgage-backed securities) that pay o¤ a little above the market return unless there is a large negative common shock, as happened to the U.S. housing market at the outset of the …nancial crisis. Arguably, the explosion of mediocre institutional investors over this period who pursued this investment strategy even drove down the price of this risk, a general equilibrium e¤ect that we do not model (see Diamond and Rajan 2009 for an extensive discussion of this). The Literature In a companion paper, we analyze a two-period model where investments are partially scalable, so that an investor must now decide given a hedge fund’s track record, how much to invest, rather than just decide whether to invest. We assume that the expected average payout from a hedge fund whose manager has ability ! and capital k is !f (k), where f is strictly concave and increasing in k. Such partial scalability emerges naturally when the price impact of taking greater positions in an asset is positive. Investors put more money into better performing funds, which, with decreasing returns to scale, causes returns of better 5 past performers to fall, as Berk and Green (2004) posit, and, for example, Fung et al. (2008) …nd. Now hedge fund managers must trade o¤ investment scale versus likelihood of success in their decisions. A higher equilibrium payout means that the hedge fund is more likely to be run by an able manager, so that it will receive more funds. When investors set higher standards for substantial re-investments, they can more reliably identify good hedge fund managers, but higher standards also make it harder to unravel the quality of badly performing hedge funds, since more will be run by able managers who just happened to be unlucky. That is, higher standards reveal more information about winners, but less about losers. Our companion paper shows that when f (k) = k , and > 1=2, then in the best equilibrium that maximizes expected payo¤s of those who invest in hedge funds, bad fund managers should adopt risky investment strategies that seek to mimic good hedge fund managers (who do not employ risky investment strategies), as total surplus is maximized when investors identify as many good hedge fund managers as possible. However, when a hedge fund’s investments are less scalable, < 1=2, then in the best equilibrium, good fund man- agers employ risky investment strategies and try to separate themselves away from bad fund managers— with less scalability, it becomes more important to ensure that a hedge fund that receives signi…cant investment is run by an able hedge fund manager. Our framework adopts the premise that while a hedge fund manager can strategically design investment strategies to try to fool investors about their intrinsic investing skills, they cannot engage in fraud or misreport returns. As Stulz (2007) observes hedge funds often hold securities that are not traded on exchanges, and hence have signi…cant discretion in pricing them as they see …t. A recent literature provides evidence of such hedge fund misreporting (see e.g., Agarwal et al. (2009, 2011), Bollen and Pool (2009, 2012), Cumming and Dai (2010), Dimmock and Gerkin (2012, 2013), and Jylha (2011)). While not dismissing the evidence of such misreporting and illegal behavior designed to conceal a lack of hedge fund investment skills, our paper focuses on the legal ways in which a hedge fund can do this, and the consequences for the dynamics of hedge fund returns. A typical compensation contract for hedge fund managers— 1-2 percent of the net asset value of the fund and 15-25 percent of asset returns above a speci…ed hurdle rate— tends to invite risk taking (Stulz (2007)). Taylor (2003) examines a model where fund managers can invest in safe or risky assets, and …nds that managers switch to risky strategies if their current performance lags a benchmark that must be achieved to receive new investment funds. 6 Agents in Degeorge et al. (1996, 2004) have private information regarding their quality. As in our model, low quality types may have an incentive to undertake a risky strategy or gamble so that they might be mistaken as a high type. In Degeorge et al. (2004), depending on model parameters, low quality types may also undertake less risky/sure thing strategies that perfectly reveal their type. Both Taylor (2003) and Degeorge et al. (1996, 2004) substantially restrict the possible investment alternatives (e.g., returns follow a normal distribution). But giving hedge fund managers little discretion in the design of their investment strategies, limits the insights about how they will design and shape those strategies in a world where hedge fund managers have signi…cantly more discretion. We impose minimal structure on the investment strategies that fund managers can employ, beyond bounding fund payouts from below and requiring that they be “actuarily fair”; we then derive the precise forms that equilibrium investment strategies take, and how they evolve over time as investors learn more about a hedge fund manager’s ability. In our economy, fund managers employ risky investment strategies because the (endogenous) minimum return needed to obtain continued funding gives rise to a non-concave payo¤ structure for a fund manager. Those fund managers with abilities that lie in the non-concave portion of the payo¤ structure have strict incentives to gamble. In this regard, our motivation for gambling mirrors that in Ljungqvist (1994), where …rm managers gamble on behalf of owners to exploit a non-concave payo¤ structure. Although one can, in principle, design compensation contracts to prevent excessive risk taking, Stulz (2007) points out they are not always successful. For example, many contracts stipulate that managers must recover past losses before they get a performance fee, the so-called “high-water”mark. But this constraint need not reign in excessive risky behavior if a manager can just close a fund after a large loss. DeMarzo et al. (2013) point out that optimal contracting can limit or rule out risky behavior, but in many cases the cost associated with eliminating this sort of behavior is too expensive. So in the end, principals have to live with the risky behavior that characterizes their agents. 2 The Basic Model We consider a single hedge fund run by a manager with an N zon. In each period n = 0; : : : ; N 2 period investment hori- 1, a type-! hedge fund manager has potential access to investment strategies that require a unit of capital to implement, and have expected period 7 payout !. We refer to ! as the hedge fund manager’s ability. It is common knowledge that ! is drawn from the cumulative distribution function ( ) with support [a; b], where a > 0. The manager’s ability ! to identify good investment strategies is private information to the manager. Each period, the hedge fund manager must raise the unit of capital needed to implement his strategies from risk-neutral investors. We assume that neither the hedge fund manager nor the risk-neutral investors discount future payo¤s. The assumption that a hedge fund manager’s investment strategies are not scalable— the fund manager needs a unit of capital to implement investment strategies, but additional capital is not productive— greatly facilitates analysis. It means that an investor’s problem reduces to deciding whether or not to invest in the hedge fund given its track record, and not deciding how much to invest. Our companion paper allows for partial scalability in a two-period setting, showing the ways in which our …ndings qualitatively extend when we incorporate the realistic features that price impacts of large positions and a hedge fund manager’s scarce human capital give rise to decreasing returns to scale in hedge fund investment strategies. In practice, hedge fund managers have enormous discretion in their design of investment strategies. The period-n investment strategy chosen by a type-! hedge fund manager is fully described by the distribution G! it induces over period payouts Xn . To capture the discretion that hedge fund managers have in designing investment strategies, we suppose that a type-! hedge fund manager can employ the unit capital investment in any period-n R investment strategy Gn! ( ) with the properties that EGn! (Xn ) = Xn z Xn dGn! (Xn ) ! and Xn z; where z < a. That is, the expected period hedge fund payout is bounded from above by the fund manager’s ability, and realized payouts are bounded from below by z. This lower bound z means that a fund manager does not have access to an investment strategy that does substantially better than ! in almost all states of the world, which would allow the manager to almost always succeed in mimicking the performance of a more able fund manager, o¤set by losing an arbitrarily large amount of money with a vanishingly small probability. One can motivate z from a feasibility standpoint— a hedge fund manager might not be able to lose more than 100 percent of his investment capital, in which case the lower bound on payouts is z = 0. It may also be that losing vast amounts of money is inconsistent with the strategy style that a manager purports to adopt; or that a hedge fund manager who loses too much money risks running afoul of the law, and this deters fund managers from adopting such extreme investment strategies. Our analysis will focus on the limited liability 8 bound of z = 0, but none of our qualitative characterizations hinge on this choice. We …rst suppose that the hedge fund manager must rely on capital raised from a sequence of risk-neutral short-term investors who have one-period investment horizons. Later, we characterize how outcomes are a¤ected when these investors have longer investment horizons. Investors observe a hedge fund’s historical performance. Thus, while they do not observe a hedge fund manager’s ability ! directly, a period-n investor knows the hedge fund manager’s track record of past investment strategy payouts, X n = (X0 ; X1 ; : : : ; Xn 1 ), and past capital in‡ows, k n = (k0 ; k1 ; : : : ; kn 1 ), where the non-scalability of investment strategies means that without loss of generality we can focus on ki 2 f0; 1g. A period-n investor uses this information to draw inferences about the fund manager’s ability. The investor also has the option to invest unlimited amounts in an alternative asset with known gross expected return R. For example, R could be the expected return on some broad market index. If investors provide the unit of capital to the hedge fund manager, they receive share 1 of the fund’s end-of-period payout, with the hedge fund manager retaining share payment for his services. We take this share > 0 as as exogenous in order to focus on how the inferences investors make from fund payouts interact to in‡uence the investment strategy choices of di¤erent hedge fund manager types.1 Due to the non-scalability of a hedge fund manager’s investment strategies, a period n investor’s decision reduces to deciding, given X n and k n , whether or not to invest in the hedge fund.2 We assume that (1 )E (!) > R. This is a minimum requirement for a short-horizon investor to be willing to invest in a hedge fund that does not have a track record. When )E (!) > R, a period-0 investor is willing to invest k0 = 1, as long as hedge fund man- (1 agers adopt fair investment strategies. An investment strategy Gn! is fair if EGn! (Xn ) = !. An investment strategy or investment gamble is unfair if EGn! (Xn ) < ! because, for example, the hedge fund manager needlessly wastes resources via some complicated, costly trading strategy. We use “investment strategy”, and “investment gamble” interchangeably, and when an !-type fund manager places probability one on the payout Xn = !, we say that the hedge fund manager does “not gamble”or adopts a “sure-thing”investment strategy. 1 In practice, there is limited variation in hedge fund managers’ compensation. Often, they receive two percent of the funds under management plus 20 percent of pro…ts above some hurdle. If we pose our analysis in an in…nite horizon setting, so there is no terminal period, then as long as hedge fund managers are su¢ ciently patient, our analysis qualitatively extends with this convex option-like feature to compensation. 2 The analysis extends immediately if the scale of the required (…xed) investment varies over time. 9 An investment strategy for a period-n investor maps the hedge fund’s track record and past capital in‡ows into a capital investment choice, In (X n ; k n ) ! k~n 2 [0; 1], where k~n is the probability the period-n investor invests in the hedge fund. A strategy for a hedge fund manager is a sequence of period investment functions mapping his type, track record, and past capital in‡ows into a feasible investment strategy when kn = 1, Fn (!; X n ; k n ) ! Gn! ( ). Investors form beliefs regarding the hedge fund manager’s type, !. The beliefs of a periodn investor about ! are described by a cumulative distribution function, , that depends on past capital in‡ows, k n , and the manager’s track record of past investment strategy payo¤s, X n . We omit dependence of beliefs on k n in our notation, where it does not cause confusion. The equilibrium concept is a perfect Bayesian equilibrium with re…nements. 3 Two-Period Hedge Fund Investment Horizon We …rst characterize equilibrium outcomes for hedge fund managers with a two-period investment horizon. Speci…cally, we consider a hedge fund manager whose investment horizon extends from period 0 to period 1, and a sequence of two short-term investors, one who cares about period-0 investments and one who cares about period-1 investments. It turns out that there are many perfect Bayesian equilibria. We begin by describing a simple cuto¤-rule equilibrium. We later show that this equilibrium is selected by multiple equilibrium re…nements. In this simple cuto¤-rule equilibrium, the period-0 investor always makes the unit capital investment in the hedge fund because (1 )E (!) > R means that, ex-ante, the expected return from investing in a hedge fund manager who employs a fair investment strategy exceeds that on the alternative asset. The period-1 investor employs a simple cut-o¤ rule strategy, providing capital to the hedge fund if and only if the hedge fund period-0 payout is su¢ ciently high, X0 Here, E (X0 ) [!] X c , where (1 )E (X0 ) [!] = R for X0 = X c . is the expected hedge fund manager type when the period-0 payout is X0 , and investors update to hold belief function (X0 ), over !. If X0 < X c , the period-1 investor invests solely in the alternative asset. In this equilibrium, in period 0, good hedge fund manager types ! X c adopt sure- thing investment strategies, as this guarantees that they will continue to receive capital from investors in period 1. In contrast, lesser hedge fund manager types ! < X c cannot employ 10 sure-thing investment strategies, else they would not receive investment capital in period 1. This cut-o¤ for funding induces them to adopt risky investment strategies that sometimes succeed and payo¤ X c , and sometimes fail and lose everything. To reduce notational clutter, we de…ne R R= (1 ). Proposition 1 There exists a simple cuto¤-rule equilibrium that is characterized by a critical value X c that solves R Xc RaX c a ! 2 (d!) ! (d!) = R: (1) In period 0, hedge fund manager types ! < X c adopt an investment strategy that places probability !=X c on the payout X c , and remaining probability on 0, and better types ! X c employ a sure-thing investment strategy that delivers X0 = ! with probability 1. The period-1 investor makes the unit capital investment in the hedge fund if X0 X c ; otherwise he invests only in the alternative asset. Each hedge fund manager type ! who receives continued funding in period 1 adopts the sure-thing investment strategy that delivers X1 = ! with probability 1. The period-1 investor’s beliefs satisfy 8 R Xc c (X !)! (d!) > Ra X c if X0 = 0 > c !) (d!) > > a (X > < y(X if 0 < X0 < X c 0 ) 2 [a; R) : E (X0 ) [!] = R if X0 = X c > > > > X if X0 2 (X c ; b] > : 0 y(X0 ) 2 [a; b] if X0 > b If all hedge fund managers adopted sure-thing investment strategies, then a period-1 in- vestor would provide the unit capital investment to a hedge fund manager if and only if the period payout X0 exceeded R. But then, lesser-type hedge fund managers ! < R would not receive period-1 capital investment. Such less able hedge fund managers could do better if they adopted a risky investment strategy that stochastically concealed their lower abilities, placing positive probability on payo¤s X0 R. In the simple cuto¤-rule equilibrium, less able hedge fund managers ! < X c do best to maximize the probability of receiving period-1 funding. They do this by not ‘wasting’probability mass on unnecessarily high investment strategy payo¤s, X0 > X c ; and since any ‘bad’ investment payo¤ X0 < X c results in not receiving funding, they also do not ‘waste’probability mass on X0 2 (0; X c ).3 3 More generally, they do not ‘waste’probability mass on X0 2 (z; X c ). 11 A period-1 investor understands this strategic reasoning, and he also understands that some hedge fund managers whose investment strategies pay o¤ X0 = X c will be less able fund manager types ! < R who got lucky. Less able fund managers are less likely to have a successful investment outcome, but some will. To protect against investing in these less able, but lucky, hedge fund managers, a period-1 investor raises his cuto¤ for providing capital above R, to the level X c that leaves him indi¤erent between investing in the hedge fund and investing solely in the alternative asset when all hedge fund managers with ! < X c adopt the risky investment strategy that places as much probability as possible on X c . Of course, in equilibrium, there will also be some unlucky hedge fund managers ! 2 (R; X c )— those who realized X0 = 0— in whom a period-1 investor would like to invest. But, not knowing their true types, a period-1 investor does not provide capital to failed hedge fund managers. The equilibrium identi…ed in Proposition 1 is a special case of a class of equilibria identi…ed in Proposition 2, and we prove the result there. Proposition 1 identi…es only one of a ^ c > 0 such continuum of equilibria that we broadly index by the lowest hedge fund payout X ^ c will receive that a hedge fund manager whose period-0 investment strategy pays o¤ X0 = X capital from the period 1 investor. We index equilibria in this way because a fund manager ^ c will employ a fair investment strategy that pays o¤ X ^ c with probability !=X ^c with ! < X and zero with residual probability. To describe these equilibria, it helps to de…ne a particular cuto¤, X. Cuto¤ X has the feature that if: (a) all fund managers ! 2 (a; X) adopt a period- 0 investment strategy that places as much probability mass as possible on X, and remaining probability on 0; and (b) all better fund manager types ! > X do not have X0 2 0; X in the support of their investment strategy, then E (X0 =0) [!] = R. This leaves a period-1 in- vestor indi¤erent between investing in a fund manager whose investment strategy completely ^ c > X, if all fund managers failed and investing only in the alternative asset. But then for X ^ c ) place as much probability as possible on X0 = X ^ c , and the rest on X0 = 0, ! 2 (a; X ^ c do not have X0 2 [0; X ^ c ] in the support of their investment and all fund managers ! > X strategy, a period-1 investor would want to invest even in hedge funds whose managers had unsuccessful investment outcomes X0 = 0 because E (X0 =0) [!] > R. Thus, X represents the highest possible cuto¤, or standard, that can be set on the period-0 hedge fund payout for providing capital to the hedge fund manager in period 1 such that a fund manager whose investment strategy pays out less than the standard is not funded. Since the probability 12 that a hedge fund manager ! < X places on X0 = 0 is 1 !=X, the critical value X solves RX !(X !) (d!) a = R; RX ( X !) (d!) a or RX (! 2 X = aR X (! a !R) (d!) : R) (d!) We now describe a broad class of equilibria that includes the simple cuto¤-rule equilibrium identi…ed in Proposition 1. ^ c 2 (0; X), there exists an equilibrium in which all hedge fund Proposition 2 For each X ^ c are funded in period 0. Hedge fund managers ! ^c managers with realization X0 = X X ^ c on payout X ^ c , and residual adopt period-0 investment strategies that place probability !=X ^ c . Fund managers ! > X ^ c are alprobability on 0. They are funded if and only if X0 = X ways funded, but they may adopt risky investment strategies in which they deliver realizations ^ c with probability one. In these equilibria, the expected period-1 payo¤ for a hedge X0 X fund manager of type ! takes the form !V (!), where V (!) is the piecewise linear function, V (!) = ! ^c ; 8! < X c ^ X and V (!) = 1; 8! ^ c: X ^ c > X. There does not exist an equilibrium in which the lowest standard for reinvestment is X Proof. All proofs can be found in the Appendix. ^ c 2 [X c ; X) can be supported by period-1 pessimistic out-of-equilibrium Equilibria with X ^ c ). When X ^ c 2 (X c ; X), we say that investor beliefs that E (X ) [!] 2 [a; R) for X0 2 (0; X 0 the standard for continued investment in the hedge fund is “excessively high.” ^ c 2 (0; X c ) for continued capital One may wonder how equilibria with low standards X investment are supported. Such equilibria are supported by risky investment strategies taken by some hedge fund managers who are always …nanced. These hedge fund manager types !, ^ c < ! < X, ~ adopt fair investment strategies that place positive probability on both where X ^ c and X. ~ In turn, these investment strategies raise the expected quality of hedge fund manX ^ c . Hedge fund managers X ^c < ! < X ~ are indi¤erent among risky agers who deliver X0 = X strategies, as well as sure thing strategies, that guarantee that they receive capital investment at period 1. One may think that this indi¤erence can be broken by introducing arbitrarily 13 small costs to adopting riskier investment strategies. But even with such costs, equilibria in which high ability fund managers employ risky investment strategies can be supported by ^ c and (pessimistic) o¤-equilibrium investor beliefs that E (X ) [!] 2 [a; R) for both X0 < X 0 ^c ~ Hence, additional re…nements are needed to eliminate these equilibria. In X0 2 (X ; X). ^ c use risky investment strategies, investor beliefs are equilibria where fund managers ! > X non-monotone in fund payouts, X0 . The non-monotonicity in beliefs induces fund manager ^ c ; X) ~ to adopt fair investment strategies that place positive probability only on types ! 2 (X ^ c ; Xg. ~ fX Most of the equilibria described in Proposition 2 seem unreasonable; for example, those with either excessively high cuto¤s or non-monotone investor beliefs with risk taking by hedge fund managers who are sure to be funded. As well, there exist additional equilibria that are not detailed in Proposition 2 that seem odd. These additional equilibria feature “truly un^c necessary” risk-taking by hedge fund managers ! > X c when X X c or fund managers ~ when X ^ c < X c . The risk taking is truly unnecessary in the sense that there exist ! >X equivalent equilibria to these additional equilibria where the above mentioned fund managers do not gamble and all other fund managers play the same strategies. However, these equilibria with unnecessary gambling cannot be eliminated by simply assuming small costs to adopting riskier investment strategies because they can be supported by non-monotone (pessimistic) investor beliefs. There are even more implausible equilibria. For example, there exist equilibria exist in which good hedge fund managers employ unfair period 0 investment strategies, with fund ^ c placing all probability mass on the outcome X ^ c , supported by period-1 inmanagers ! > X vestor beliefs that E ^ c ) [!] (X0 6=X 2 [a; R), but E ^ c ) [!] (X0 =X R. Indeed, such unfair period-0 investment strategies can always support a “no-initial funding” equilibrium in which, even though E[!] > R, hedge fund managers do not receive capital investment in period 0, and only receive funding in period 1.4 This no-funding equilibrium is supported by beliefs of period-0 investors that too many hedge fund manager types will pursue unfair period-0 investment strategies if they receive capital investment. Hedge fund managers would undertake unfair strategies for some perverse period-1 investor beliefs. For example, if capital were provided to fund managers at date 0, a period-1 investor’s beliefs might be such that E (X0 ) [!] > R for X0 = R, but E (X0 ) [!] < R, for X0 6= R. Then, in a subgame in which cap- 4 Fund managers always pursue fair,sure thing investment strategies in the last period of their investment horizons. 14 ital is provided in period 0, hedge fund managers ! > R would burn resources by undertaking unfair investment strategies, getting the sure payo¤ of R, while hedge fund managers ! < R adopt an investment strategy that places probability !=R on R and residual probability on 0. But, since any hedge fund manager’s investment strategy pays either R or zero in period 0, the expected payo¤ to a period-0 investor is strictly less than R. So he does not provide capital investment. Since the period-1 investor sees no track record and hedge fund managers pursue a fair investment strategy in period-1, the investor invests in the hedge fund. Finally, as mentioned above, there is also an ‘always-invest’ equilibrium, in which the period-1 investor continues to invest in a hedge fund whose period-0 investment strategy lost everything. This is supported as an equilibrium when hedge fund managers pursue fair ^c > X investment strategies that place positive probability only on the (unlikely) outcome X and on 0, so that most investment strategies fail. By construction, the expected type of a hedge fund manager who pursued a failed investment strategy exceeds R. We want to eliminate such implausible equilibria. To do this, we introduce a slight aversion of hedge fund managers to riskier investment strategies: Assumption LC. There is a lexicographical cost associated with riskier investment strategies. A type ! hedge fund manager strictly prefers investment strategy G1! to investment strategy G2! if G1! has a higher expected lifetime payo¤ . If, however, G1! and G2! have the same expected payo¤, then the fund manager strictly prefers the investment strategy with the smaller support on period investment payouts, as measured by the di¤erence between the highest and lowest possible period investment payout. Assumption LC implies that a hedge fund manager …rst seeks to maximize expected lifetime pro…ts. Then, from the set of investment strategies that maximize expected lifetime pro…ts, the hedge fund manager selects an investment strategy with the least possible risk.5 We now provide two arguments for why the simple cuto¤-rule equilibrium described in Proposition 1 is the “natural”one. Proposition 3 The ex-ante expected payo¤ of each period investor is at least as high in the simple cuto¤-rule equilibrium as in any other equilibrium. Adding Assumption LC, among the set of equilibria that give period investors the highest expected payo¤, hedge fund managers 5 The particular formulation of “a slight aversion to riskier investment strategies” is unimportant for our …ndings. 15 ! > X c strictly prefer the simple cuto¤ rule equilibrium, and hedge fund managers ! Xc are indi¤erent. Furthermore, under Assumption LC, the simple cuto¤-rule equilibrium is the unique equilibrium to survive the Grossman-Perry re…nement. A period-0 investor is indi¤erent among all equilibria in which each hedge fund manager type adopts a fair investment strategy. From a period-1 investor’s perspective, he is worse o¤ in an equilibrium where the cuto¤ on investment strategy payouts X0 for providing capital ^ c > X c . This is because, relative to the simple cuto¤ rule equilibrium, the exceeds X c , i.e., X equilibrium with an excessive standard has good fund managers ! > X c adopting risky investment strategies and, hence, some of these fund managers are needlessly denied funding in period 1 when their period-0 investment strategy fails. There are, however, no o¤setting ben^ c . In particular, the composition of the pool of types e…ts associated with the higher cuto¤ X ^ c , i.e., E ^ c [!j! < X c ] = R. Therefore, ! < X c is not improved with the higher standard X (X ) i h ^ c are a period-1 investor’s payo¤ will be increased if able hedge fund managers ! 2 X c ; X funded in period 1, and this can be accomplished by adopting a lower cuto¤, X c . One might think that a period-1 investor might be better o¤ in an equilibrium in which ^ c 2 (R; X c ) berisky investment strategies employed by types ! > X c to support a cuto¤ X ^ c ; X c ) are always funded, rather than just sometimes funded. cause then some types ! 2 [X However, the reduced weeding out of low ability hedge fund managers ! < R necessarily ^ c is R, dominates. To see this, note that when the expected return conditional on X0 = X then it is a weighted average of two populations, where the expected return on the popu^ c exceeds R, implying that the expected return on lation of ! > X c who deliver payout X the population of ! ^ c is less than R. Thus, if the investor additionally X c who deliver X knew that ! X c , he would strictly prefer to invest instead in the alternative asset. (When ^ c = X c , the investor knows that ! X c when X ^ c is delivered and does not strictly prefer X to invest in the alternative asset.) From an ex-ante perspective, prior to learning his type, a hedge fund manager always prefers an equilibrium with a lower cuto¤ since it raises the probability that he will receive capital if he turns out to be less able. When there are costs associated with riskier investment strategies, each hedge fund manager type prefers less risky investment strategies. In particular, given Assumption LC, all su¢ ciently able fund managers who can ensure period 1 capital in‡ows would like to adopt sure thing strategies. If they could adopt such ^ c < X c cannot be supported, meaning that the “best” a strategy, then equilibria with X 16 equilibrium from a su¢ ciently able hedge fund manager’s perspective is the simple cuto¤ equilibrium. But, the costs associated with riskier investment strategies by themselves fail to ^ c < X c since pessimistic out-of-equilibrium eliminate (unreasonable) equilibria with cuto¤s X beliefs can be used to support them. However, since the Grossman-Perry re…nement requires that out-of-equilibrium beliefs be updated in a consistent Bayesian fashion, it excludes such pessimistic beliefs that are used to support the unreasonable equilibria. It is worth observing that the non-scalability of hedge fund investment strategies is important for the result that in the best equilibrium (from the perspective of investors) only the least able hedge fund managers adopt risky investment strategies. Our companion paper shows that when hedge fund investment strategies are partially scalable, the structure of the “best”possible equilibrium hedge fund manager investment strategies can take a more complicated form. Now the primary problem for an investor in a hedge fund becomes how much to invest, with more capital provided to hedge funds run by managers who are perceived as more able, rather than whether to invest. For example, it may be optimal to have better hedge fund managers adopt risky investment strategies to separate away from less able hedge fund managers whenever the extent of decreasing returns to scale in the investment technology are extreme enough to make it more important from a capital allocation perspective to identify which hedge fund managers are less able, rather than which ones are more able. Who gains from unobservable ability? Compared to a full information environment in which investors know a hedge fund manager’s ability, the simple cuto¤-rule equilibrium has some hedge funds managers being made better o¤, some being made worse o¤ and some being una¤ected. All less able hedge fund manager types ! 2 [a; R) bene…t since, under full information, they never receive capital. These fund managers receive a strictly positive expected payo¤ in period 0 in the simple cuto¤-rule equilibrium, and whenever they get lucky with their period-0 investment strategy, then they also receive a strictly positive payo¤ in period 1. Intermediate hedge fund manager types ! 2 [R; X c ) are hurt since, under full information, they would always receive capital in period-1 and, therefore, receive positive payo¤s in both periods. In the simple cuto¤-equilibrium, these fund managers employ a risky period-0 investment strategy. While their expected period 0 payo¤ is the same as in a full information setting, their period 0 investment strategy fails with positive probability; and if this happens, their period 1 payo¤ is zero. Finally, su¢ ciently able hedge fund mangers types ! 2 [X c ; b] receive the same payo¤s in both information environments. 17 Empirical Regularities. We will show that this very simple model generates predictions that can reconcile many empirical regularities regarding hedge fund performance. We defer this presentation until after we solve for the equilibrium to the more general N + 1-period hedge fund manager investment horizon. Robustness. The analysis extends routinely if, after period-0 investors provide their capital, a hedge fund manager’s period-0 expected payo¤ is subject to a period-0 shock 0 a, so that a type ! hedge fund manager’s investments have maximal expected period payo¤ !+ 0, as long as period-1 investors can infer fund. Period-1 investors can obviously infer also be able to infer 0 0 0 before deciding whether to invest in the if it is directly observable. Investors may if it is a common shock that hits many hedge fund managers, so that investors can extract 0 from the cross-section of hedge fund manager performances via the law of large numbers. While we focus on a single hedge fund for simplicity, in practice, investors draw inferences about a fund manager’s ability from his relative performance, and our model should be interpreted in that light. In Section 6 we examine how outcomes are a¤ected when hedge fund managers abilities or payo¤s are subject to idiosyncratic shocks. We examine two cases: 1. A type ! fund manager is hit with a mean-zero, idiosyncratic shock n that is indepen- dently distributed over time and privately observed prior to undertaking his investment strategy. This shock re‡ects the stochastic arrival of investment opportunities of different quality. Hence, a fund manager of type ! who is hit with shock access to investment strategies with expected payo¤ EGn! (X) !+ n now has n. 2. Each fund manager is hit with an ex-post idiosyncratic (manager speci…c) shock to his payo¤ Xn after adopting an investment strategy. That is, the fund manager does not know the shock when selecting his investment strategy. In each case, we characterize how the shocks a¤ect equilibrium designs of a hedge fund manager’s investment strategy. One should caution that an investor’s ability to infer n from the cross-section presumes that the distribution of hedge fund manager abilities is common knowledge. One could imagine a setting in which investors also learn about the distribution of hedge fund manager 18 abilities,6 in e¤ect, learning about the expected over-all worth of investing in hedge funds rather than more standard assets. In this case, some of a positive common n shock (or ex-post shock) might be attributed to a better over-all distribution of hedge funds, so that only “some”of it is accounted for by investors in their decision making. Our analysis also presumes that a given hedge fund’s prospects do not systematically improve or worsen over time from the ex-ante perspective of an investor contemplating providing capital to a hedge fund. Such perturbations would not dramatically alter our qualitative predictions,7 but they can substantially complicate analysis, especially when a hedge fund has a longer investment horizon. We next analyze precisely this case. 4 Multiperiod hedge fund investment horizon We now consider a hedge fund manager who has an N -period investment horizon, where 3. The hedge fund manager must rely on capital raised from a sequence of risk neutral N investors who have one-period investment horizons. Thus, a period-n investor only cares about investment payo¤s in period n. Because these investors have only one-period investment horizons, there are no dynamic learning considerations. In particular, short-horizon investors have no incentive to set lower cuto¤s on hedge fund period payouts for providing continued capital in‡ows because they do not attach a positive option value to being able to switch subsequently to the alternative asset. As a result, a hedge fund manager will receive capital in period n if and only if E n (X n 1) R, where X n [!] the fund manager’s period investment strategies, i.e., X n 1 1 is the history of payo¤s from = fX0 ; :::; Xn 1 g. Once again, there are many perfect Bayesian equilibria. However, we focus on the multiperiod analogue of the simple cuto¤-rule equilibrium, where the minimum payout that the hedge fund must achieve in period n 1 in order to receive capital in period n, Xnc 1 , satis- …es E n (X n 1 =X (n 1)c ) [!] = R, where X (n 1)c = fX0c ; : : : ; Xnc 1 g is the history where, in each period, the hedge fund pays out the minimum amount required to receive continued capital 6 The explosion in the total dollars invested in hedge funds and other institutional investors is consistent with this. For example, in the 1990s and early 2000s, many universities (e.g., Harvard) began to allocate substantial portions of their portfolios to hedge funds and other institutional investors. The subsequently relatively poor performance of hedge funds (see The Economist Dec 22, 2012) suggests that they may have over-estimated the average ability of these institutional investors to provide superior returns. 7 This would lead to correlated entry and closure of hedge funds found in the data (Grecu et al. 2007) See also http://online.wsj.com/article/SB10001424127887324640104578163251346728208.html. 19 investment. One can show that the Grossman-Perry re…nement together with Assumption LC eliminate all other equilibria. The next proposition characterizes the evolution of the sequence of cuto¤s set by investors for continued infusion of capital into the hedge fund. Proposition 4 The multi-period simple cuto¤-rule equilibrium is characterized by a sequence of cuto¤s for re-investment X0c ; R Xnc RaXnc a ! n+2 (d!) ! n+1 (d!) ; XNc 1 implicitly de…ned by = R; for n 2 f0; : : : ; N (2) 1g : A period 0 investor makes the unit capital investment in the hedge fund. Subsequent period n investors make the unit capital hedge fund investment if and only if Xn n = 0; : : : N X0c > X1c > Xnc for 1. The cuto¤s for re-investment strictly decline for older hedge funds, i.e., > XNc 1. In each period n < N , fund manager types ! 2 (0; Xnc ) that receive capital employ fair investment strategies that place probability !=Xnc on Xnc and residual Xnc adopt sure-thing investment strategies that probability on 0, and fund manager types ! deliver Xn = ! with probability 1. In period N all hedge fund managers who receive funding employ sure-thing investment strategies. Intuitively, over time, investors become more con…dent that a hedge fund manager with the successful track record X nc is “good”: Less able fund managers are stochastically weeded out. As a consequence, investors do not require such a high payout by more established hedge funds in order to be willing to provide capital. That is, the cuto¤s for period hedge fund payouts for continued capital investment fall over time. As a result, over time, some hedge fund manager types ! 2 [Xnc ; Xnc 1 ) who initially had to employ risky investment strategies in periods 0; : : : n 1 in order to have a chance of receiving continued capital in‡ows can switch to sure-thing investment strategies in period n. Of course, such fund managers only reach this point if their initially risky period investment strategies realized lucky outcomes. To gain insights into (a) the extent to which period cuto¤s for reinvestment exceed R (which measures how many good hedge fund manager types must adopt risky investment strategies), and (b) the rate at which these cuto¤s decline over time as investors learn more about a fund manager’s likely ability, it is useful to consider an explicit parameterization. Accordingly, suppose that the initial distribution over fund manager types and write the lower support as a = R, where 20 ( ) is uniform, < 1 measures the percent by which the worst fund manager provides a lower expected period return than the alternative asset. Then, from (2), the cuto¤s for reinvestment solve: (Xnc )n+3 (Xnc )n+2 ( R)n+3 1 )R: = (1 + n+2 ( R) n+2 Rearranging yields (Xnc )n+2 [(1 + 1 )R n+2 Xnc ] = ( R)n+2 R(1 + 1 n+2 0, then Xnc If the worst hedge fund manager is completely inept, ): (1 + 1 )R. n+2 This implies that investors set a very high standard for a hedge fund manager who is just starting out, X0c X1c 1:5R. However, the standards for continued re-investment then drop sharply to c = 1:01R. Conversely, when 1:33R and X2c = 1:25R, then slowly tailing o¤ so that X98 almost no hedge fund managers are bad, so that Quite generally, one can solve for X0c = R period cuto¤s for various levels of 1, then Xnc p 2 2 3 2 + (3+2 ) 4 (4 ) R. . Table 1 details di¤erent when R = 1. The table reveals that the re-investment Table 1: Period cuto¤s 0.0 0.5 0.8 1.0 X0c X1c X2c X3c X4c 1.50 1.333 1.25 1.20 1.167 1.366 1.284 1.229 1.191 1.162 1.176 1.157 1.141 1.128 1.117 1.00 1.00 1.00 1.00 1.00 cuto¤ for more experienced hedge fund managers becomes relatively insensitive to the quality of the worst fund manager type, as long as a non-trivial fraction lack ability, and declines slowly for more experienced hedge fund managers. Table 2 presents the corresponding hazards into liquidation for the hedge fund when the best fund manager type has ability ! = 2. For plausible lower bounds on fund manager abilities ( = 0:5; 0:8), hazards are initially high, initially decline quickly both due to the reduced standard for re-investment, and the stochastic weeding out of less able hedge fund managers, but then slowly tail o¤.8 8 We will show that when investors have longer investment horizons, the hazards into liquidation have lower peaks, but remain elevated for longer. 21 Table 2: Hazard Rates 0.0 0.5 0.8 1.0 H0 0.375 0.137 0.050 0.000 H1 0.158 0.103 0.038 0.000 H2 0.078 0.062 0.029 0.000 H3 0.043 0.038 0.022 0.000 H4 0.012 0.025 0.017 0.000 Empirical Implications and Discussion The simple cuto¤ equilibrium that we describe has only three sources of uncertainty: uncertainty due to the unobserved heterogeneity in fund manager abilities, uncertainty due to commonly-observed shocks that investors can identify and “subtract out” in their calculations of hedge fund performance, and endogenous uncertainty introduced by less able fund managers who seek to mimic the performance of moderately-skilled hedge fund managers. Nonetheless, the dynamics of hedge fund returns in our simple model can reconcile many features regarding hedge fund returns and survival such as: Hedge funds must initially do “quite well” to receive continued capital in‡ows— most hedge funds that succeed initially required some luck in the sense that their initial returns must exceed their long-run returns. More established hedge funds receive capital in‡ows even if their returns slightly decline over time. This re‡ects that the rate of learning about hedge fund manager ability is very high initially, but drops o¤ sharply once hedge funds have su¢ cient track records (roughly at rate 1=N ). Failure rates of hedge funds are initially very high, but fall sharply for more established (older) hedge funds. Howell (2001) …nds that the probability a hedge fund fails in its …rst year is 7.4%. The more relevant failure rate is the second year failure rate of 20.3% in the second year— in practice, hedge funds do not fail “right away” due to lock-in and restriction periods.9 As a result, hazards are initially ‡atter than our simple model predicts. Moreover, failure rates of younger hedge funds are substantially undermeasured, because many do not survive long enough to enter hedge fund data bases, and there is selective back…lling of past (earlier) returns of hedge funds in the data bases 9 Agarwal et al. (2009) document a median lock-in of one year, and there is often a restriction period, typically of four months, over which, after giving notice, one cannot withdraw funds. 22 that initially did well (Malkiel and Saha, 2005). Liquidation rates in 2008-09 were 15 percent per year and in 20 percent in 2010.10 http://www.bloomberg.com/news/201009-21/hedge-fund-closure-rate-may-rise-to-20-on-lack-of-capital-merrill-says.html Identifying failure from the data is tricky as hedge funds self report, and some of the best hedge fund managers may cease reporting because they may not be actively seek new money, or choose to opt out of the data bases for other reporting reasons, or they may have merged with another hedge fund, and so on, (Gregoriou (2002)). Malkiel and Saha (2005) report a survivorship bias in hedge fund returns of 4.5% due to failing to account for exit; Brown, Goetzman and Ibbotson, 1999 report 3 percent bias (exit was higher post 2000). Nonetheless, even after accounting for these potential upward biases, average returns of funds that exit database are very poor in the few months preceding their exit, with highly negative Sharpe ratios. More generally their returns and Sharpe ratios are far worse at the end of their reporting lives. For example, the average monthly return for exiting …rms in their …nal 3 months is -0.61% compared to 0.49% during their earlier lives, while the average Sharpe index is -1.859 in their …nal 3 months compared to 0.102 during their earlier lives (Grecu et al. 2007). The poor performance of ‘true’exiting …rms would be even worse to the extent that some …rms that cease reporting do so because they merge or cease to advertise because they do not want additional investment. Many established hedge funds deliver returns that marginally exceed the return on alternative assets. This re‡ects that as investors observe longer histories of hedge fund success, they update to conclude that the survivors are likely better, making the investors more willing to provide them capital, even if their subsequent performance is slightly less good. A hedge fund’s returns tend to fall over time. This re‡ects: (1) many hedge fund managers must initially take risky gambles, and succeed, generating payo¤s that exceed their long-run average expected payo¤s, and; (2) over time, as investors set lower cuto¤s for continued reinvestment, these hedge fund managers employ less risky gambles, generating slightly lower returns when they succeed (and better hedge fund managers 10 See http://www.bloomberg.com/news/2010-09-21/hedge-fund-closure-rate-may-rise-to-20-on-lack-ofcapital-merrill-says.html 23 who initially had to employ risky gambles may be able to adopt sure thing investment strategies). In hedge fund databases, both of these e¤ects are reinforced to the extent that there is back…ll bias, so that the sample is di¤erentially comprised of hedge funds that were initially successful. We show in Section 6.1 that the average decline over time in a hedge fund’s return is reinforced if hedge fund investment opportunities have an idiosyncratic component, so that in some periods a hedge fund can identify better investment opportunities than in other periods— i.e., they are subject to pre-investment shocks to their investment opportunities. This means that successful hedge funds initially tend to be those that received positive idiosyncratic shocks, and hence had better investment opportunities than they will typically be able to deliver in the future. Better-performing funds have more persistence in their returns (Jagannathan et al. 2010, Jame 2013). In our simple model, this re‡ects that good hedge fund managers can adopt sure thing strategies, while less able hedge fund managers must employ risky strategies, and their expected returns decline conditional on their risky strategies succeeding. Returns of more established hedge funds are less volatile, and more predictable. Even though returns for a (surviving) hedge fund tend to decline with time, in the cross-section, the returns for older funds exceed those on less established funds if one accounts for the survivorship bias.11 Some of these predictions have important implications for interpretation of empirical …ndings. In particular, they indicate that regressions of hedge fund returns on a laundry list of regressors that include …xed hedge fund manager e¤ects (to identify the manager’s individual alpha) must be interpreted carefully. One can contemplate a researcher running such a regression, designed to isolate the impact of the hedge fund manager’s experience on his performance, and concluding that more established hedge funds o¤er inferior returns on average. Rather than conclude that the hedge fund manager grows senile over time, one could imagine the researcher making conclusions regarding back…ll bias, or strategic choices of safer, lower mean investment strategies, or appeal to a Berk and Green-style explanation that revolves around successful hedge funds receiving more capital investment, which 11 This cross-sectional result can be ‘reversed’in the data if there is su¢ ciently greater under-reporting of failure by younger hedge funds (back…ll bias). 24 reduces their marginal returns. While recognizing that these features can contribute to the observed empirical relationship, appeal to such factors is unnecessary. In fact, this predicted relationship emerges in our extremely stripped-down, stylized model of strategic investment by hedge fund managers. One can also imagine an investment advisor counseling investors on the basis of this empirical relationship to put their money in a pool of less-established hedge funds in order to generate superior returns. In fact, in our model, the relevant measure is the pooled cross-section of hedge fund returns. Further, such regressions, which include manager experience as a regressor, would reveal a positive coe¢ cient on experience (absent back…ll biases). This re‡ects that, over time, less able hedge fund managers are stochastically weeded out over time when their period investment strategies fail— on average, the expected ability of surviving hedge fund managers rises with experience in the cross-section. While our model predicts that many institutional investors will adopt risky investment strategies that place signi…cant weight on returns that slightly exceed those on alternative assets, and residual weight on very poor returns, it is silent on how institutional investors should implement those risky investment strategies in practice. In fact, in the early 2000s, one easy way to implement such a strategy in practice was to invest in tranches of mortgage-backed securities, where there was a risk of correlated default that we eventually experienced in the second half of the decade. Such assets realized slightly better returns than alternative assets, in those early years. Most of the focus on the sources of the crash of the housing market has been on the moral hazard of lenders extending loans to households that were unlikely to repay, or the small down payments that put many households under water on their mortgages, once housing prices started to fall. To the extent that securitization of these mortgages has received attention, it has only been because lenders needed to repackage and sell most of their mortgages in order to have enough capital to lend at such high levels. What has received far less attention is the question: Why were institutional traders willing to buy these repackaged loans, and why were they willing to do so at such low prices, driving down the implicit price of the risk of correlated default? One might posit that these assets o¤ered simple ways in which to implement the risky investment strategies that we describe, and the explosive growth in the numbers of institutional investors may have sharply bid down the prices, as the institutional investors who lacked ability sought to deliver returns that slightly beat those on alternative assets. Our simple model, of course abstracts from these general equilibrium e¤ects. 25 5 Longer Investor Horizons To ease presentation, we have focused on investors with one-period horizons. It is, however, important to understand how longer investment horizons for potential investors a¤ect equilibrium outcomes. Investors with longer horizons value the learning associated with observing a hedge fund’s performance for longer since they have the option to switch to the alternative asset in the future. To illustrate the qualitative impacts associated with longer investor horizons, we suppose that both the investor and hedge fund manager have three-period horizons, caring about payo¤s in periods 0, 1 and 2. In each period, the investor chooses to make the unit capital investment in the hedge fund or to invest solely in the alternative asset. The investor sets ^ 1c , where X ^ nc represents the minimum period ^ 0c and X two cuto¤s for continued investment, X n payout that results in the investor providing capital to the hedge fund in period n + 1. We distinguish these critical cuto¤s from their analogues, (X0c ; X1c ), for investors with oneperiod investment horizons described in Proposition 4. We focus on the simple cuto¤-rule equilibrium— the analogue to that in Proposition 4— where the investor earns an expected future return equal to that on the alternative asset from a hedge fund whose payout just meets the cuto¤ for continued capital investment. Now, however, the relevant return for the investor is his expected future lifetime return, where, after observing X0 , the investor takes into account that if he provides capital to the hedge fund, then he can switch to the alternative asset in period 2 if he is dissatis…ed with the period 1 payout. We now show that investors with longer investment horizons initially set lower cuto¤s for providing continued capital investments in order to have the opportunity to learn more about the hedge fund manager’s ability. This reduces the probability that they prematurely cease to provide capital to a fund run by an able manager. In turn, some hedge fund managers with intermediate abilities adopt less risky investment strategies. Proposition 5 In equilibrium, a long-horizon investor sets a lower initial cuto¤ on period-0 ^ c < X c . However, payouts for continued capital investments than a short-horizon investor, X 0 0 ^ c = X c , where X ^c < X ^ c. they set the same …nal period cuto¤, X 1 1 1 0 n o c c ^ min X0 ; X0 , the period-1 distributions of hedge fund manager ^ 0c . This re‡ects types conditional on X1 = X1c are the same for period-0 cuto¤s X0c and X ^ 1c = X1c Because X 26 that a higher cuto¤ on period 0 payouts reduces the fraction of all types of hedge fund managers that are below the period-1 cuto¤ by the same proportion. To see this, note that if the period-0 cuto¤ is X c , then the fraction of all types ! < X c with X0 = X c is !=X c . A higher period-0 cuto¤ reduces the fraction of all types below the period-1 cuto¤ by a factor ^ c =X c , compared to the lower period-0 cuto¤. As a result, the short-horizon and longof X 0 0 horizon investors set the same period 1 cuto¤ for providing capital in the terminal period 2. In contrast, a long-horizon investor sets a lower cuto¤ on period 0 payouts for providing capital in period 1. A long-horizon investor places a positive value on basing period 2 capital investment decisions on information revealed in period 1, whereas a short-term investor does not. Most obviously, because the cuto¤ for continuing to provide capital falls over time, a ^ c; X ^ c ] are long-term investor attaches a positive value to learning in period 1 that all ! 2 [X 1 0 valuable hedge fund managers, and should receive capital. This leads a long-horizon investor to set a lower period-0 cuto¤ in order to avoid prematurely ceasing to provide capital to good hedge fund managers. These qualitative insights extend: The longer is an investor’s horizon, the lower are the cuto¤s that the investor sets early in the investment horizon for providing capital to the hedge fund. In turn, this implies that the hazard into liquidation does not have as high a peak (less able hedge fund managers are weeded out more slowly), but also that the hazard remains high for longer (because more less able hedge fund managers remain). Setting a lower period-0 cuto¤ means that more hedge fund managers ! 2 (X1c ; X0c ) receive capital in all three periods, which is socially bene…cial since X1c > R, and only a fraction !=X1c of remaining !-type managers with ! < X1c , are funded in the …nal period. The lower cuto¤s for continued capital provision set by longer-horizon investors have implications for hedge-fund performance that may initially seem paradoxical: Proposition 6 Expected hedge fund payouts in periods 1 and 2 are lower with long-horizon investors than short-horizon investors. One could imagine a researcher isolating two groups of investors, one with long horizons, and the other with short horizons, calculating their hedge fund returns, and concluding from the lower average measured hedge fund returns in the portfolio of the long-horizon investors that these investors are “worse” than the short-horizon investors, or that short-horizon in27 vestors are better o¤ than their long-horizon counterparts. Obviously, such conclusions are misplaced. The expected performance of a hedge fund is lower in both periods 1 and 2 with longhorizon investors than with short-horizon investors precisely because the former has more “lower ability”hedge fund managers than the latter. That is, they both fund all able hedge fund manager types ! > X0c , but a short-horizon investor is less likely to fund types ! < X0c . But some of these lower ability hedge fund managers are able to o¤er an expected payo¤ that ^ c < X c and R < X ^ c = X c . The correct exceeds that on the alternative asset since R < X 0 0 1 1 comparison is not of these lower ability hedge fund managers with higher ability ones, but rather the comparison of their expected return with the alternative asset. Thus, it would be incorrect to compare the lower expected payouts from surviving hedge funds that receive capital from long-horizon investor with those associated with a short-horizon investor. The correct conclusion to draw is that too many short-horizon investors prematurely switch to investing in the alternative asset in period 1 because of their high period-0 cuto¤— too few short-horizon investors provide capital to hedge funds. 6 Pre- and Post-Investment Shocks So far we have characterized equilibrium outcomes in settings where any common shocks that hit the payo¤s of all hedge fund managers are …ltered out via comparisons of relative performances. Hence, the only remaining uncertainty that a hedge fund manager faces is the endogenous uncertainty created by his choice of investment strategy. In reality, hedge funds receive exogenous idiosyncratic shocks outside of their control that a¤ect their payouts. Because investors in hedge funds do not observe these shocks, they complicate their e¤orts to extract information about a hedge fund manager’s ability from observed hedge fund payouts. One must distinguish the impacts of idiosyncratic shocks that occur prior to a hedge fund manager’s investment choices from those that occur afterwards. Pre-investment, a hedge fund manager may receive an idiosyncratic shock that a¤ects the quality of his investment strategy opportunity— in some periods a fund manager may be able to uncover better investment possibilities than in others. From a forecasting perspective, a potential investor wants to isolate the permanent component that describes a fund manager’s ability, as it is this long-term ability that drives expected future hedge fund performance. However, since the hedge fund 28 manager observes the pre-investment shock prior to selecting an investment strategy, he can directly condition his investment strategy choice on both his ability and the shock realization. After a hedge fund manager chooses an investment strategy, his portfolio may be hit with a (post-investment) shock that a¤ects the ultimate hedge fund payo¤. While a hedge fund manager will take account of the possibility of post-investment shocks when choosing his investment strategy, he is unable to condition his investment strategy choice on a particular shock realization since he does not know it at that time. To highlight the qualitative e¤ects of such shocks we return to our two-period economy with short-horizon investors. We focus on the simple cut-o¤ equilibrium, where a period 1 investor provides capital to a hedge fund if and only if its period 0 total payout meets or exceeds the cuto¤ X c . In this cuto¤ equilibrium, the expected ability of the hedge fund manager who generates a payout of X c is equal to R. We now explore how these two sources of uncertainty, which are outside the control of a hedge fund manager, a¤ect investor cuto¤s for continued capital investment, and the design of a fund manager’s investment strategies. 6.1 Pre-Investment Shocks To ease presentation, we assume that a hedge fund manager’s ability ! is drawn from a uniform distribution on [a; b], with associated pdf, ( ). With pre-investment shocks, In both periods 0 and 1, the hedge fund manager receives an idiosyncratic shock n prior to selecting an investment strategy that determines the quality of the investment opportunities to which he has access in the period. Thus, a fund manager with ability ! who is hit with a shock n can implement any investment strategy that delivers a non-negative payo¤ with an expected payo¤ that does not exceed ! + n. We assume that n is identically and independently distributed over time on [ y; y] according to a pdf h( n ). We suppose that h( n ) is symmetric, so that h( n ) = h( n ): We also assume that y is small enough that the solutions characterized below are interior, i.e., we assume that y < a and b > X c + y. In period 0, if !+ 0 < X c , then it is optimal for the hedge fund manager to undertake the risky investment strategy f0; X c g, i.e., the hedge fund manager adopts an investment strategy that pays out X c with probability (! + 0 0 ) =X c , and pays out zero otherwise. If, instead ! + > X c , then it is optimal for the hedge fund manager to pursue the sure-thing investment strategy. In e¤ect, in period 0 the hedge fund manager behaves as if his permanent type is !+ 29 0. Interestingly, even though the expected value of the pre-investment shock is zero, we have, Proposition 7 Pre-investment idiosyncratic shocks reduce the equilibrium cuto¤ X c set by investors for continued capital investment in the hedge fund. The intuition underlying Proposition 7 is that achieving a given cut-o¤ provides the investor with better news when there are non-trivial idiosyncratic shocks 0 compared to when there are not. The news is better because some high ! hedge fund manager types happen to get unlucky (negative) draws of — and must pursue a risky investment strategy— and some low ! types happen to get lucky draws of — and pursue a sure-thing investment strategy. With the symmetric density, h( ), lucky low ! types are “replaced”by unlucky high ! types, and this raises the expected ability of the hedge fund manager whose payo¤ hits the cuto¤. Thus, pre-investment shocks have three important qualitative empirical implications: (1) they reduce the spread between the return that less able hedge fund managers seek to achieve and the return on standard assets; (2) hazards into liquidation have a lower peak, but remain high for longer; and (3) because surviving hedge funds are more likely to have received positive idiosyncratic shocks, their future expected performance will fall to re‡ect that their period-expectation is always zero. 6.2 Post-Investment Shocks With post-investment shocks, after each hedge fund manager type ! implements his period0 investment strategy G0! (X0 ), the fund’s payout is hit by a (post-investment) shock 0. We examine a multiplicative shock structure, where the “…nal” total payout in period 0 is ~ 0 = 0 X0 , where E[ 0 ] = 1, and 0 > 0. The period 1 investor only observes X ~ 0 and not X its separate components, X0 and 0. We focus on a multiplicative shock structure because it delivers a non-negative fund payout. One can formulate additive shock structures in a ~ 0 = 0 + X0 < 0 or imposing a high variety of ways (e.g., with E[ 0 ] = 0 and allowing for X “enough”bound on X0 ) that yield qualitatively identical results. To ease presentation, we suppose that the post-investment shock 0 is drawn from a uniform distribution on [1 ; 1 + ], where 0 < < 1. The associated cdf, F ( 0 ), is 8 for 0 1 < 0 1 1 + 2 0 for 0 2 [1 ;1 + ] : F ( 0) = 2 : 1 for 0 1 + 30 (3) Our more general presentation is designed to highlight how the results extend. Denote the date 0 cuto¤ as X c . In equilibrium, the period 1 investor provides capital to ~ 0 = X0 0 the hedge fund if and only if X X c . Given a cuto¤ X c , a hedge fund manager ~0 with ability ! seeks to maximize the probability that X X c by choosing an appropriate investment strategy, G0! (X0 ). The hedge fund manager must receive a shock of 0 X c =X0 to obtain investment capital in period 1. This occurs with probability 1 F (X c =X0 ), where 8 for X0 X c = (1 + ) < 0 c 1+ 1 X for X0 2 [X c = (1 + ) ; X c (1 )] : (4) 1 F (X c =X0 ) = 2 X0 : 2 c ) 1 for X0 X = (1 Note that the probability of success, 1 [X c = (1 + ) ; X c (1 )]. F (X c =X0 ), is strictly concave in X0 for X0 2 The hedge fund manager’s period 0 optimization problem is, Z [1 F (X c =X0 )] dG0! (X0 ) max G0! X0 0 Z Z 0 subject to dG! (X0 ) = 1 and X0 dG0! (X0 ) = !; X0 0 (5) X0 0 where the constraints re‡ect that the manager chooses a fair investment strategy. To understand the nature of the period 0 strategy G0! (X) that hedge fund manager ! employs, consider the concave hull of 1 X 2 (X u ; X c = (1 F (X c =X). It is either strictly concave for some ))— as 0AB in Figure 1— or piecewise linear for all X > 0— as 0AB in Figure 2. In either case, this implies that all su¢ ciently weak hedge fund managers pursue a risky strategy that places probability !=X u on X0 = X u , and residual probability on X0 = 0, while hedge fund managers with ! > X u employ sure-thing investment strategies. Thus, for hedge fund manager types, ! < X u , the period-0 optimization problem (5) simpli…es to max [1 u X If X u is less than X c = (1 F (X c =X u )] =X u : )— as in Figure 1— then X u is given by the implicit solution to the …rst-order condition,12 f (X c =X u ) X c (X u )3 12 (6) [1 F (X c =X u )] = 0: (X u )2 With uniformly-distributed shocks, second-order conditions hold. 31 Figure 1: “Large” Figure 2: “Small” 32 We can re-arrange this …rst-order condition to solve for X u in terms of the hazard, Xu = Xc Adding the assumption that f (X c =X u ) : 1 F (X c =X u ) (7) has a uniform distribution, one can solve explicitly for Xu = 2X c ; 1+ (8) when X u is interior. When there is little “ex post” uncertainty, i.e., when is su¢ ciently small, then— as in Figure 2— the hedge fund manager resolves all funding uncertainty whenever his risky strategy succeeds by choosing X u = X c = (1 ). When X u = X c =(1 ) and the hedge fund man- ager’s risky strategy succeeds, he is always funded in period 1 since his lowest possible …nal payout is (1 )X c =(1 u ) = X c . Thus, the solution to (6) is given by X u = min c so that X = 2X = (1 + ) for c u 1=3 and X = X = (1 ) for Xc 1 ; 2X 1+ c ; 1=3. Consider now a cut-o¤ equilibrium characterized by: (1) a cuto¤, X c ; (2) the fair risky investment strategy f0; X u g adopted by hedge fund managers ! 2 [a; X u ), with ! > X u pursuing the sure-thing investment strategy; and (3) investor beliefs Xc , which are consistent with the distribution of hedge fund manager abilities conditional on the hedge fund period ~ 0 and E (X c ) [!] = R. Suppose further that the lowest type ^b < b that pursues the 0 payo¤ X sure-thing investment strategy is always funded. This implies that are small. Thus, only types ! < X u who are hit by E (X c ) [!] = R Xu !2 Xu a R Xu ! Xu a (d!) (d!) 0 =1 R Xu = RaX u a 1 , 3 i.e., ex-post shocks realize X c implying that ! 2 (d!) ! (d!) (9) = R: It follows that when possible ex-post shocks are small, X u equals the cuto¤ X c for continued investment when there are no ex-post shocks (compare (9) with (1) in Section 3). Thus, small ex-post shocks do not alter the investment strategies chosen by hedge fund managers. However, such ex-post shocks do reduce the cuto¤ for continued funding to X c = (1 )X c . Further, while there is clustering of hedge fund payouts, the cluster is now spread over [(1 )X c ; (1 + )X c ], and, in fact, peaks with the uniform distribution at (1 + )X c , as more types ! > X c who do not adopt risky investment strategies still receive unlucky 0 draws and hence lower ex-post payouts. Importantly, when the extent of ex-post uncertainty is small, i.e., when 1=3, the probability with which hedge fund managers secure continued 33 re-investment is una¤ected, as only the probability with which the risky investment strategies of bad hedge fund managers ! < X c succeed at delivering X u = X c determines whether they are funded. Suppose now that the lowest type ^b < b that pursues a sure-thing investment strategy is not always funded. This implies that 2X c this is so, X u = 1+ 1 , 3 > i.e., ex-post shocks can be large. When . Consequently, some hedge fund managers whose ‘initial’investment strategies succeed still receive unlucky draws that cause them not to be funded. We can solve for the critical Xu 0 = 2X c 1+ 0 0 that a hedge fund manager ! < X u requires to receive period 1 funding: = Xc ) 0 1+ 2 = c c . Furthermore, all types ! 2 ( 2X ; 1X ) sometimes fail to 1+ receive funding. Equilibrium demands that R=E (X c ) [!] = R Xu a R Xu a Adding the assumption that R=E 0 !2 f (X c =X u ) Xu (d!) + ! f (X c =X u ) Xu u (1+ ) R X2(1 ) Xu (d!) + !f (X c =!) (d!) u (1+ ) R X2(1 ) Xu : c f (X =!) (d!) has a uniform distribution yields: (X c ) [!] = R Xu a R Xu a !2 Xu (d!) + ! Xu u (1+ ) R X2(1 ) (d!) + Xu R ! (d!) X u (1+ ) 2(1 ) Xu : (10) (d!) It follows directly that at X u = X c , the right-hand side of (10) exceeds R, and indeed for a …xed X u , the right-hand side is increasing in . To retrieve equality, it must be that as ex-post shocks grow larger, i.e., as increases, then X u falls. That is, greater ex-post uncertainty causes hedge fund managers to introduce less endogenous strategic investment uncertainty. These ex-post shocks impair an investor’s ability to learn about a hedge fund manager’s ability— some relatively poor performers may be skilled but unlucky hedge fund managers. In turn, this causes the investor to be more willing to extend funding. Gathering the observations from these extensions of our basic model reveals that enriching our base stylized model to allow for investors with longer investment horizons or preor post-investment shocks to hedge fund payo¤s all qualitatively serve to induce increased patience in hedge fund investors. In equilibrium, investors become more willing to tolerate lower initial investment returns, which causes less able hedge fund managers to pursue less risky investment strategies, thereby slowing the exit rate of hedge funds. One can also show that ex-post shocks serve as a re…nement of sorts, reducing the set of equilibria that can be supported by perverse investor beliefs in the absence of those shocks 34 (i.e., relative to the basic no-shock model in Section 3). While such shocks do not eliminate the no fund equilibrium, they help reduce the set of equilibria with excessively high minimum standards for continued investment. For example, with large ex-post shocks, i.e., > 1=3, and lexicographic costs associated with riskier investment strategies, a unique funding equilibrium obtains without appeal to re…nements. This is because the lexicographic costs ensure that good hedge fund manager types ! > 2X c =(1 + ) employ sure thing investment strategies;13 and the shocks are large enough that, in equilibrium, some hedge funds realize “successful initial”payout X u , but bad post-investment shocks result in a ‘…nal period’pay~ 0 = X u 0 < X c , so that they do not receive continued capital investment. Therefore, out X we must have, R = E (X c ) [!]: ~ 0 2 (0; X c ) cannot pessimistic o¤-equilibrium beliefs given X be used to support other equilibria, making additional re…nements unnecessary. 7 Concluding Remarks Our paper begins with the observation that hedge fund managers zealously conceal investment strategies. This complicates the inference problem of investors who only see the returns of those investment strategies on a periodic basis and must forecast what future returns will be. Hedge fund managers know their “true abilities”and understand that investors will shift investments away from poorly-performing funds toward better performers. We consider a simple setting in which the problem of a potential investor in a hedge fund is whether to provide capital and not how much capital to provide. This allows us to model the enormous discretion hedge fund managers have in the design of their investment strategies. We allow hedge fund managers to tailor their investments however they see …t, requiring only that the investment strategy have a payo¤ that is bounded from below and that, in expectation, returns re‡ect the hedge fund manager’s ability. Given these modest restrictions, we characterize the di¤erent investment strategies that can emerge in equilibrium, together with the criteria set by investors for providing continued capital to the hedge funds. 13 That is, b > X u . To see this, suppose instead that b X u . Then all hedge fund managers puru sue the risky strategy f0; X g, and the expected quality of a hedge fund manager with payout X c is Rb Rb Rb 2 ! (d!)= a ! (d!). But, since a ! (d!) > R, a contradiction obtains: a R=E (X c ) [!] = Rb ! 2 (d!) a Rb a ! (d!) 35 > Z a b ! (d!) > R: We then show that equilibrium is uniquely pinned down under a set of di¤erent equilibrium re…nements. In this equilibrium, investors set simple cuto¤ standards for re-investment that slightly exceed the expected return on alternative investments. Facing such re-investment standards, less able fund managers employ risky investment strategies that maximize the probability of meeting the re-investment standards at the cost of a positive probability of producing disastrously low returns, while more able fund managers choose not to introduce extra risk to their investment strategies. Re-investment standards decline for more experienced hedge fund managers. This re‡ects that investors have longer track records on which to assess performance, and that less able fund managers are stochastically weeded out over time. We are not the …rst to recognize a manager’s incentive to gamble. The notion that the unobservability of investment strategies can induce hedge fund managers to “employ” “unnecessarily” risky strategies dates back at least to Degeorge et al. (1996). Related incentives show up in Ljungqvist (1994). The key contributions of our paper are …rst to derive the structure of those risky strategies in an environment with minimal restrictions on their structure. Unlike most of the literature, we do not require that managers choose from a set of narrowly speci…ed investment alternatives. Instead, we allow hedge fund managers to freely design their own investment payo¤ structure. We do this not for the sake of generality, but rather because this is what we observe in practice, and because this minimalist approach to investment selection allows us to match a set of empirical regularities regarding hedge fund performance and to resolve a set of seeming paradoxes. We derive a host of implications for the equilibrium dynamics of hedge fund returns and survival. For example, we predict that in a regression with …xed hedge fund manager e¤ects the returns of more experienced hedge fund managers should decline, even though the average pro…ts of investors in hedge funds rise with hedge fund manager experience due to learning. We show how idiosyncratic shocks to a hedge fund manager’s investment opportunities or idiosyncratic ex post shocks to fund payouts strengthen this relationship. So, too, we show that the longer is an investor’s horizon, the lower is the expected return of the hedge funds in which he invests. We predict that more experienced hedge funds deliver less volatile returns; that persistence of returns is greater for more able hedge fund managers; hedge fund failure rates are initially very high, but fall sharply with hedge fund manager experience due both to the improved selection and the declining reinvestment standards; that returns of exiting hedge funds will be far worse than historical returns; and so on. 36 Although our model provides new insights and helps us understand many qualitative empirical features of hedge fund performance, it is restricted in an important regard: the size of a hedge fund is either zero or one. This means that we cannot analyze the relationship between incremental ‡ows into and out of hedge funds and hedge fund performance. We address this in our companion paper. 8 References Agarwal, Vikas, Naveen D. Daniel, and Narayan Y. Naik (2009) “Role of Managerial Incentives and Discretion in Hedge Fund Performance,”Journal of Finance, 2221-2256. 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Taylor, Jonathan (2003) “Risk-taking behavior in mutual fund tournaments,”Journal of Economic Behavior & Organization 50, 373–383. 9 Proofs Proof to Proposition 2 ^ c 2 (X c ; X), where X c is de…ned in (1), hedge fund managers ! 2 [a; X ^ c ) use the When X ^ c g and hedge fund managers ! 2 [X ^ c ; b] use the sure-thing risky fair investment strategy f0; X investment strategy. Period-1 investor beliefs are summarized by: 8 R X^ c !(X c !) (d!) > a > < R if X0 = 0 RX > ^c > (X c !) (d!) > a > > ^ c) > if X0 2 (0; X < y(X0 ) 2 [a; R) RX ^c 2 ! (d!) E (X0 ) [!] = . a ^c R if X0 = X RX ^c > > ! (d!) > a > > ^ c ; b] > X if X0 2 [X 0 > > : ^ cg y(X0 ) 2 [a; b] X0 > maxfb; X The beliefs of investors are consistent with hedge fund manager behavior. Given the strategy of hedge fund managers, investors optimally provide a unit capital investment to hedge fund ^ c in period 1; and given the beliefs and strategies of investors, the managers with X0 X best response for hedge fund managers is the strategy described above. ^ c 2 (0; X c ), hedge fund managers ! 2 [a; X ^ c ) use the risky fair investment When X ^ c g, hedge fund managers ! 2 (X ^ c ; X) ~ use the risky fair investment strategy strategy f0; X ^ c ; Xg ~ (where the notation fx; yg denotes the fair investment strategy in which the hedge fX fund manager places as much probability as possible on the higher fund payout, y and the ~ then hedge fund managers remaining probability on the lower fund payout x), and if b > X, 39 ~ b] use the sure-thing investment strategy. Period-1 investor beliefs are given by: ! > [X; 8 R X^ c !(X c !) (d!) > a > <R if X0 = 0 RX > ^c c !) (d!) > > a (X > > > ^c > y(X0 ) 2 [a; R) if X0 2 0; X > > > c R R ~ ^ X > ~ ~ ^c ^ (d!)+ X > !(!=X) ^ c !(X !)=(X X ) (d!) a X > ^c R if X0 = X > R R c ^ ~ > ^ (d!)+ X ~ ~ ^c < aX (!=X) ^ c (X !)=(X X ) (d!) X ^ c ; X) ~ ; E (X0 ) [!] = y(X0 ) 2 [a; R) if X0 2 (X > R ~ X > c ^ > > RX^ c~ !(! X ) (d!) > R ~ > if X0 = X X > ^c > ^ c (! X ) (d!) X > > > ~ < X0 b > X0 if X > > > ~ > if X b < X0 or > > : y(X0 ) 2 [a; b] ~ if b X < X0 ~ su¢ ciently large. Such an X ~ exists given the assumption that E [!] > R. These for X beliefs are consistent with hedge fund manager behavior, and it is optimal for the period1 investor to provide a unit capital investment to the hedge fund manager if and only if ^ c g [ fXg ~ [ (X; ~ b), where (X; ~ b) = ; if b < X. ~ In turn, it is a best response for hedge X0 2 fX ^ c g if ! 2 (a; X ^ c ], to use the fund managers to use the risky far investment strategy f0; X ^ c ; Xg ~ if ! 2 (X ^ c ; X), ~ and to use the sure-thing investment risky fair investment strategy fX ~ b] when b > X. ~ strategy if ! > [X; The form of V (!) is immediate from the construction. ^ c > X because the date-1 investor would invest A cuto¤ equilibrium does not exist if X in a manager that has a date-0 realization of X0 = 0 since E (0) [!] > R. Proof to Proposition 3 Any equilibrium in which hedge fund managers receive the unit capital investment at date 0 and adopt fair investment strategies are optimal from the perspective of the period 0 investor since E[!] > R. We …rst focus on equilibria in which the probability the period 1 investor funds a period 1 project is either zero or one, i.e., k(X0 ) 2 f0; 1g, for all X0 . We then solve for the set of ! that the period 1 investor should always fund in equilibrium given that lesser hedge fund manager types choose their investment strategies optimally. We then extend the analysis to deal with k(X0 ) 2 (0; 1). Let ! ^ be the worst type that is always funded, i.e., ! ^ = minf!jk(X0 = !) = 1, where the 40 assumption that a < R implies that it is not optimal to fund all hedge fund manager types. Then the optimal investment strategies of hedge fund managers ! < ! ^ place probability !=^ ! on X0 = ! ^ , and residual probability on 0. Since k(X0 ) for X0 > ! ^ does not a¤ect the investment strategies of hedge fund managers ! < ! ^ , it follows that it is (weakly) optimal for the investor to set k(X0 ) = 1 for all X0 > ! ^ . Thus, ! ^ solves Z !^ Z b !2 ! max (1 ) !+ + 1 (1 R (d!) + ! ^ ! ^ ! ^ ^ a X )2! (d!): The Leibnitz terms drop out of the associated …rst-order condition for the above maximization problem yielding, Z a Substituting for R = (1 which implies that ! ^ = X c. ! ^ (1 !2 ! ) 2 + 2R ! ^ ! ^ )R, multiplying by ! ^ 2 =(1 R !^ 2 ! (d!) = R; Ra!^ 1 ! (d!) a (d!) = 0: ), and rearranging yields We now extend the analysis to deal with the possibility of equilibria with k(X) 2 (0; 1). In this analysis, rather than exhaustively delineate all possible equilibria, we work with the concave hull of the equilibrium funding probabilities, k( ). Thus, k(X0 ) is the probability with which someone with type ! = X0 expects to be funded in equilibrium. Optimization by hedge fund managers together with the fact that an investor who mixes between funding and not must be indi¤erent implies that the concave hull of the equilibrium funding probabilities must be a concave piecewise linear function of X0 with at most two kinks. Importantly, both hedge fund managers and period investors are indi¤erent between the equilibrium and a setting in which a hedge fund manager with type ! = X0 does not adopt a risky investment strategy and is funded with probability k(X0 ). Obviously, the latter is not typically an equilibrium (investors would update, and want to either fund for sure or never fund), but both deliver the same funding probabilities for any given hedge fund type (e.g., in the simple cuto¤ equilibrium, a type ! = X c =2 delivers X c with probability one-half, and zero with the remaining probability one-half, and hence is funded with probability one-half, and k(X c =2) = 1=2), and hence are payo¤ equivalent for all parties. To see that the concave hull is linear with at most two kinks, note that k(X0 ) 2 (0; 1) implies E (X0 ) [!] = R. Denote the smaller kink (when it exists) by XA and the larger (when 41 it exists) by XB . Then E (X0 ) [!] = R for any X0 2 [XA ; XB ) in the support of an equilib- rium investment strategy, and k(X0 ) = 1 for X0 XB is in the support of an equilibrium investment strategy (presuming there is a larger kink)), and, if there is a kink, then all ! < XA adopt the investment strategy f0; XA g. Note that types ! > XB place probability 0 on X0 < XB . There are 3 cases to analyze: Case 1: No kinks on [0; b]. Suppose there is an equilibrium in which k(X) is linear for all X 2 [0; b], where k (0) = 0 and k (b) < 1. The equilibrium is payo¤ equivalent to one where all hedge fund managers ! 2 [a; b] employ fair investment strategy f0; bg and the investor provides period 1 funding when X = b with probability k(b) < 1. For these strategies, E (X0 ) [!] > R, which means that the expected payo¤ to the investor will increase if he pro- vides period 1 funding when X = b with probability one. Hence, an equilibrium in which all hedge fund manager types employ fair investment strategy f0; bg and the investor provides period 1 funding with probability one if and only if X b generates a higher expected payo¤ to the investor than any equilibrium in which k(X) is linear for all X 2 [0; b] and k(b) < 1. Therefore, an equilibrium in which k(X) that is linear for all X 2 [0; b], where k (0) = 0 and k (b) < 1 cannot maximize the investor’s payo¤. Case 2: One kink on [0; b], no kinks on X > b. Suppose there is an equilibrium in which ^ where k (0) = 0, X ^ b and k (X) = 1 for all X X. ^ The k(X) is linear for all X 2 [0; X], ^ equilibrium payo¤s are equivalent to strategies that have hedge fund managers ! 2 [a; X) ^ hedge fund managers ! 2 [X; ^ b] play sure thing employing fair investment strategies f0; Xg, ^ and with X, ^ = X c. zero probability otherwise. From above, the investors’payo¤s are maximized by X strategies and investors providing period 1 funding with probability 1 for X Therefore, the simple cuto¤ equilibrium provides the highest payo¤ among all equilibria in ^ where k (0) = 0, X ^ b and k (X) = 1 for all X X. ^ which k(X) is linear for all X 2 [0; X], Case 3: Two kinks. Suppose there is an equilibrium in which k(X) is piecewise linear over X > 0 with kinks at XA and XB , where a < XA < b, XA XB R b, and k (X) = 1 for all X XB , k (0) = 0, 0 < k(XA ) 1, XB . The equilibrium payo¤s are equivalent one in which hedge fund managers ! 2 [a; XA ) employ fair investment strategies f0; XA g, hedge fund managers ! 2 [XA ; XB ] employ fair investment strategies fXA ; XB g, hedge fund man- agers ! 2 [Xb ; b] play sure thing strategies and investors providing period 1 funding with probability k(XA ) for X = XA , with probability one for X 42 XB , and with zero probability otherwise. For these (equilibrium) strategies, consider the choices of XA , XB and k (XA ) that maximize the expected payo¤ of the investor. In such an equilibrium, the critical values, XA and XB , that maximize the investor’s expected payo¤ solve Z XA ! ! max (1 ) k (XA ) (XA + !) + (1 k (X1 )) (XA + R) XA ;XB a XA XA ! + 1 R d (!) XA Z maxfXB ;bg ! XA XB ! (XB + !) + k (X1 ) (XA + !) + + (1 ) X B XA XB XA XA + (1 k (X1 )) (XA + R)] d (!) Z b 2!d (!) ; + (1 ) minfXB ;bg where k(XA ) is the probability that the period-1 investor provides a unit capital investment to a hedge fund manager that has a period-0 realization equal to XA . Note that by construction, the period-1 investor will provide funds to a hedge fund manager whose period-0 realization is XB with probability one. Since 0 < k(XA ) < 1, investor indi¤erence implies that "Z # Z XA 2 Z maxfXB ;bg Z maxfXB ;bg XA ! XB ! ! XB ! d (!)+ ! d (!) = R d (!) + d (!) : XA X B XA XA X B XA a XA a XA Substituting this into the above maximization problem and Z XA ! ! max R (1 ) (XA + !) + 1 XA ;XB a XA XA Z maxfXB ;bg ! XA XB + (1 ) (XB + !) + XB X A XB XA Z b + (1 ) 2!d (!) = max XA Z a simplifying, we get14 d (!) ! (XA + !) d (!) XA minfXB ;bg XA (1 !2 ) !+ XA + 1 ! XA R (d!) + Z b (1 )2! (d!): XA Note that XB does not enter the objective function. The associated …rst-order condition for the above maximization problem is, Z XA (1 a ) !2 ! + R XA X A 14 (d!) = 0; It is important to note that the …nal expression that follows is also valid if k (XA ) = 1. This is because, as we will see, the solution to the maximization problem has XA = X c and for the simple cuto¤ equilibrium we know that E (X c ) [!] = R. 43 and can be simpli…ed to read R XA RaXA a c ! 2 (d!) ! (d!) = R; which implies that XA = X , where X c is de…ned in equation (1). The solution to the above problem delivers a concave hull k (X) that is identical to that of the simple cuto¤ equilibrium, i.e., XA = X c , XA XB , and k (XA ) = k (XB ) = 1. Hence, the simple cuto¤ equilibrium provides the investor with the highest possible expected payo¤. The proof to the Grossman-Perry part of the proposition is in 8 steps. Step 1 : We …rst prove that there cannot exist an equilibrium where the period-0 investor provides investment capital to the hedge fund manager and the period-1 investor never provides investment to the hedge fund manager. Suppose that such an equilibrium exists. Then, because of assumption LC, all hedge fund managers ! must be employing sure-thing investment strategies in period 0. But then for X0 > R, must re‡ect ! > R. But then it is strictly optimal for the period-1 investor to provide the hedge fund manager a unit of investment capital at date 1 (as they employ sure thing strategies at date 2), a contradiction. Step 2 : We can rule out an equilibrium where continued capital investment requires that ^ i.e., if X0 6= X, ^ then the probabilthe period 0 realization of the hedge fund be exactly X, ity of continued capital investment is zero. In such an equilibrium, a hedge fund managers ^ use the risky fair investment strategy f0; Xg ^ and hedge fund managers ! 2 (X; ^ b] ! 2 [a; X] use unfair strategies that e¤ectively destroy ! ^ units period-0 output. Such an equilibX rium does not survive the Grossman-Perry re…nement. To see this, suppose that a period-0 ~ >X ^ is observed. If outcome X ~ receives continued investment with realization equal to X ^ X) ~ have an incentive to use the risky probability one, then hedge fund managers ! 2 (X; ^ Xg— ~ ^ fair investment strategy fX; has this i increases their period-0 expected payo¤ from X to ~ b have an incentive to provide the payout X— ~ as this !— and hedge fund managers ! 2 X; ~ X. ^ Further, X ~ would not be in the support of would increase their period-0 payo¤ by X ^ as this would simply lower the the investment strategies of hedge fund managers ! 2 [a; X) probability of receiving capital investment in period 1. Therefore, E (X~ ) [!] > R, and the period-1 investor will provide a unit of investment capital to a hedge fund manager that has ~ a contradiction. (Note that this step rules out the equilibrium a period-0 realization of X, described in the text where period-0 investors do not provide capital investment to the hedge 44 fund manager. Recall that perverse period-1 investor beliefs were required to support such an equilibrium. If, however, the period-0 investor provided investment capital to the hedge fund manager, then the period-1 investor would only provide investment capital to the hedge ^ = R. The above argument demonstrates that the beliefs required fund manager if X0 = X to support such an outcome do not survive the Grossman-Perry re…nement.) Step 3 : We now show that there can be at most one value of X, say XA with k1 (XA ) 2 (0; 1) that is observed along the equilibrium path, where k1 (XA ) 2 (0; 1) should be inter- preted as a mixed capital investment strategy by the investor. If k1 (XA ) 2 (0; 1), then E (XA ) [!] = R. Suppose there exists an equilibrium characterized by XA , XB and XC such that 0 < k1 (XA ) ; k1 (XB ) < 1 and k1 (XC ) = 1, where XA < XB < XC . Since assumption LC implies that any given hedge fund manager will employ a risky investment strategy with most two points of support, in the equilibrium, hedge fund managers ! 2 (a; XA ) gamble on f0; XA g, hedge fund managers ! 2 (XA ; XB ) gamble on f0; XB g and hedge fund managers ! 2 (XB ; XC ) employ fair investment strategy f0; XC g. Since hedge fund managers with XB in the support of their investment strategy are uniformly better than those with XA , it must be that E (XA ) [!] <E (XB ) [!], a contradiction. Hence, in any equilibrium there can be at most one value of X, say XA , such that k1 (XA ) 2 (0; 1). Step 4 : Step 3 does not rule out the possibility that there exists XA and XB such that k1 (XA ) 2 (0; 1) and k1 (XB ) = 1, as this is consistent with (XA ) = R and (XB ) > R for XB > XA . We now show that the Grossman-Perry re…nement precludes this possibility. Suppose there is an equilibrium characterized by hedge fund managers ! 2 (a; XA ) using the risky fair investment strategy f0; XA g and ! 2 (XA ; XB ) using the risky fair investment strategy fXA ; XB g, where E (XA ) [!] = R and k1 (XA ) 2 (0; 1). Clearly, E (XB ) [!] > R and k1 (XB ) = 1. Suppose …rst that hedge fund managers ! 2 (a; XA ) strictly prefer the risky fair investment strategy f0; XA g to f0; XB g, i.e., k1 (XA ) =XA > 1=XB . Now suppose that ^ < XB is observed, where XB X ^ is arbitrarily small. If k1 (X) ^ = 1, then hedge fund manX ^ prefer the risky fair strategy fXA ; Xg, ^ ! 2 (X; ^ XB ) prefer the risky fair agers ! 2 (XA ; X) ^ XB g; and ! = X ^ prefers the sure-thing investment strategy to the equilibrium strategy fX; strategy because, in each case, the expect payo¤ to the hedge fund manager exceeds that from ^ su¢ ciently small, no hedge fund manager equilibrium risky strategy fXA ; XB g. For XB X ^ even if k~1 (X) ^ = ! 2 (0; XA ) has an incentive to use the risky fair investment strategy f0; Xg ^ then, using the Grossman-Perry logic, he will 1. Hence, if the period 1 investor observes X, 45 conclude that E ^ [!] (X) ^ > R, and provide the hedge fund manager with period-0 payout X a unit of capital in period 1, a contradiction. Hence, there cannot exist an equilibrium such that there exists XA and XB with k~1 (XA ) 2 (0; 1), k1 (XB ) = 1, and k1 (XA ) =XA > 1=XB . Now consider k1 (XA ) =XA = 1=XB (if k1 (XA ) =XA < 1=XB then XA is not in the support of any hedge fund manager’s investment strategy). First consider XB > X c , and suppose that ^ is observed, where X c < X ^ < XB . If k1 (X) ^ = 1, then hedge fund managers ! 2 (a; XA ) X ^ hedge fund managers ! 2 (XA ; X) ^ prefer the risky fair prefer the risky fair strategy f0; Xg, ^ hedge fund managers ! 2 (X; ^ XB ) prefer the risky fair strategy fX; ^ XB g; strategy f0; Xg, ^ prefers the sure-thing investment strategy to the equilibrium strategy because, and ! = X in all cases, the associated expected payo¤ exceeds that associated with the proposed equi^ > X c , E ^ [!] > R, and the period 1 investor will provide capital to a librium. Since X (X ) ^ in period 0, a contradiction. hedge fund manager who generates a payout of X Finally, posit an equilibrium in which there exists XA and XB such that k1 (XA ) 2 (0; 1), k1 (XB ) = 1, k1 (XA ) =XA = k1 (XB ) =XB and XB X c . Then, R XA R XB !(!=X ) (d!) + !(XB !)=(XB XA ) (d!) A a X E (XA ) [!] = R XA R XAB (!=XA ) (d!) + XA (XB !)=(XB XA ) (d!) a N : D We now sign of the derivative of E (XA ) [!] with respect to XA . Note that Z XB Z XA @E (XA ) [!] 2 !(XB !)=(XB XA )2 (d!)]D sign !(!=XA ) (d!) + = sign[ @XA XA a Z XA Z XB [ (!=XA2 ) (d!) + (XB !)=(XB XA )2 (d!)]N a Z )[ XA XA 1 1 = sign( + X A XB XA a Z XA Z XB 2 ! (d!) (XB a R XA a ! = sign[ R XA a R XB ! XA RX (d!) XAB !(XB RX (d!) XAB (XB ! (d!) Z XB !(XB !) (d!) XA !) (d!)] !) (d!) !) (d!) !(XB !)=(XB XA ) (d!) X = sign[ R AXB (XB !)=(XB XA ) (d!) XA R XA a ! 2 (d!) R XA a R XA ! (d!) R XB X (XB !) (d!)] XA (XB !) (d!) R XAB ! (!=XA ) (d!) ] > 0; R XA (!=XA ) (d!) a a since this …rst term represents the expected fund manager type who gets payout XA from the population of hedge fund managers ! 2 (XA ; XB ) with fair investment strategy fXA ; XB g and the second term represents the expected fund manager type who gets payout XA from 46 ] the population of hedge fund managers ! 2 [a; XA ) who employ fair investment strategy f0; XA g. Since XA < XB X c , @E (XA ) [!]=@XA to XB raises the expectation, but at XA = XB ; E > 0 means increasing XA all the way (XA ) [!] R, a contradiction (i.e., there cannot be an equilibrium where k1 (XA ) > 0). Step 5 : The …rst four steps allows us to focus on cuto¤ equilibria, i.e., equilibria characterized by a hedge fund manager below some critical value using risky investment strategies and, as a result, receiving continued capital investment with probability less than one, and by a hedge fund manager above some critical value that receives continued capital investment with probability one. Consider now equilibria that are characterized by an excessively high stan^ c > X c , where X c is given by equation (1). In such an equilibrium, E ^ c [!] > R. In dard X (X ) c ^ ^ cg the equilibrium, hedge fund managers ! < X use the risky fair investment strategy f0; X ^ c use the sure-thing investment strategy. Such an equilibX ^ c ). rium is supported by period-1 investor beliefs E (X ) [!] = y(X0 ) 2 [a; R) for all X0 2 (0; X and hedge fund managers ! 0 ^ c ) as a support in Given these beliefs, hedge fund managers will use any outcome in (0; X the risky investment strategy. Such beliefs, however, do not survive the Grossman-Perry ~ 0 2 [X c ; X ^ c ) is observed and the period-1 investor equilibrium re…nement. Suppose that X ~ 0 . Then, hedge fund manwill invest a unit of capital with the hedge fund manager if X0 = X n o ~ 0 will prefer to use the risky fair investment strategy 0; X ~ 0 to f0; X ^ c g; agers ! 2 0; X ~ 0 will prefer to use the sure-thing investment strategy to the risky hedge fund manager ! = X ^ c g; and hedge fund managers ! 2 (X ~0; X ^ c ) will prefer to use fair investment strategy f0; X ~0; X ^ c g to f0; X ^ c g.Therefore, any hedge fund manager the risky fair investment strategy fX ^ c ) has an incentive to defect from proposed play. Using the Grossman-Perry logic, ! 2 (0; X ~ 0 2 [X c ; X ^ c ) is the period-1 investor beliefs associated with outcome X R X~0 R^ ~ 0 ) (d!) + X ~ 0 )=(X ^ X ~ 0 ) (d!) !(!=X X ~ 0 !(! a X E (X~0 ) [!] = R ~ > R: R^ X0 ~ 0 ) (d!) + X ~ ^ ~ (!= X (! X )=( X X ) (d!) 0 0 ~0 a X Hence, the period-1 investor will provide a unit capital of investment to a hedge fund man~ 0 2 [X c ; X ^ c ) in period 0. Therefore, there does not exist ager that produces a payout of X an equilibrium characterized by an excessively high standard. ^ c < X c , note Step 6 : To rule out equilibria that have a non-monotone standard, i.e., X ^ c ; X) ~ that such a standard can be only be supported by having hedge fund managers ! 2 (X ^ c ; Xg, ~ which raises E ^ c [!] to R. Suppose that use the risky fair investment strategy fX (X ) ^ ^ c ; X) ~ and invests a unit the period-1 investor observes a period-0 payout equal to X 2 (X 47 of capital with the hedge fund manager with probability one. Then, n hedge fund o managers ^ c; X ^ have an incentive use the risky fair investment strategy X ^ c; X ^ ; hedge fund !2 X ^ has an incentive to use the sure-thing investment strategy; and hedge fund manager ! = X ^ X) ~ have an incentive to use the risky fair investment strategy fX; ^ Xg ~ managers ! 2 (X; since, in all cases, the “defecting”risky strategies have smaller payout supports compared to the equilibrium strategy, and provide the same expected payo¤. Clearly, hedge fund man^ c ) have no incentive to use X ^ as a support for their risky strategies. The agers ! 2 (0; X Grossman-Perry re…nement implies that the investor will, in fact, invest a unit of capital with ^ since E ^ [!] > R. Therefore, a hedge fund manager that has a date-0 realization X0 = X (X ) there cannot be an equilibrium that has a non-monotone standard. Step 7 : We can rule out equilibria where there is a single cuto¤ equal to X c , hedge fund managers ! 2 (0; X c ) use the risky fair investment strategy f0; X c g, hedge fund managers ! X c use the sure-thing investment strategy, the investor invests a unit of capital with probability one if X0 > X c , and invests a unit of capital with probability less than one if X0 = X c , i.e., k~ (X c ) < 1. To see this, note that fund manager ! 2 (0; X c ) would prefer to use the risky fair investment strategy f0; X c + "g, where " > 0 is arbitrarily small, instead of f0; X c g because the probability of achieving X c +" is approximately equal to that of achieving X c but the former receives a unit of capital investment in period 1 with probably 1. Step 8 : Finally, it is straight forward to rule out equilibria that are characterized by “unnecessary gambling,”since smaller gambles are preferred to larger ones. Steps 1-8 imply that the equilibrium identi…ed in Proposition 1 is only equilibrium to survive the Grossman-Perry re…nement and assumption LC. Proof to Proposition 4 We …rst show that the set of cuto¤ values is unique in the sense that there do not exist equilibria where the cuto¤ for continued capital investment in the hedge fund depends nontrivially on the entire track record of fund performance. That is, one might contemplate ~n > X ^ n at period n, distinct hedge fund payout paths, where if a hedge fund has a payout X ~ <X ^ at period then it might require a payout of only X > n to continue to receive a unit ^ n at of capital investment from the investor, while another hedge fund that had a payout of X ^ >X ~ at period period n might require a payout X 48 to continue to receive a unit of capital investment from the investor. Let Xnc 1 (X n 1 ) be the minimum payout in period n 1 such that an investor will make a unit capital investment in the hedge fund in period n, given that the hedge fund’s payout history is X n 1 . For concreteness, assume that N = 3. The min fX1c (X0 ) ; X0c g in period 0 is proportional to expected payo¤ to hedge fund manager ! ! X0c X0c + X1c ! [X c (X c ) + !] : (X0c ) 1 0 ^ 0c in period 0, where ~ 0c and X Suppose there are nontrivial cuto¤s at periods 0 and 1: X ^ 0c ) < X1c (X ~ c ). If some hedge ~ 0c ) in period 1, where X1c (X ^ 0c ) and X1c (X ^ 0c , and X1c (X ~ 0c < X X 0 fund managers prefer the lower period 0 cuto¤ and others prefer the higher period cuto¤, then there will exist a fund manager, say ! ~ , that is indi¤erent between using the risky fair c c ~ g and f0; X ^ g. This indi¤erence implies that investment strategy f0; X 0 0 9 9 8 8 = = < < h h i i ! ~ ! ~ ! ~ ! ~ c c c c c c ~ ^ ^ X X X X = 0: (11) X + ! ~ X + ! ~ + + 1 1 0 0 ~ 0c : 0 X c X ^ 0c : 0 X c X ; X ; c c ~ ^ X 0 0 1 1 If we view the left-hand side of (11) as a function of ! ~ and di¤erentiate with respect to ! ~, we get 1+ " 2 1 3~ ! 2~ !+ c c X0 X1 X0c 8 < # : 1+ 2 1 4 3~ ! 2~ !+ c ^0 ^ 0c X X1c X 39 = 5 : ; To sign this derivative, note that (11) can be rewritten as, 2 39 # 8 " < = 2 2 1 1 4 ! ~ ! ~ 5 1+ c ! 1+ c ! ~+ ~+ c = 0: ^0 : X0 X1 X0c ^c ; X Xc X 1 (12) (13) 0 Subtracting (13) from (12) yields " 2 2~ ! 1 ! ~+ c c X0 X1 X0c # 2 2 3 1 4 2~ ! 5: ! ~ + ^ 0c ^ 0c X X1c X (14) ~ 0c < X ^ 0c and X1c (X ^ 0c ) < X1c (X ~ 0c ), reveals that (14) is Comparing (13) to (14), and noting that X negative, which, in turn, implies that (12) is negative. Hence, all hedge fund managers ! > ! ~ c ^ 0 g and all fund managers ! < ! prefer the risky fair gamble strategy f0; X ~ prefer the risky fair ~ 0c g. This, however, implies that E investment strategy f0; X equilibria where E ^ c ) [!] (X 0 ^ c ) [!] (X 0 >E ~ c ) [!] (X 0 R. But > R are precluded by the Grossman-Perry equilibrium re…nement and assumption LC. This reasoning extends to N > 3. 49 We now characterize the unique equilibrium that survives the Grossman-Perry re…nement. The equilibrium is characterized by a declining set of cuto¤s for continued capital investment. Hedge fund managers ! Xnc use the sure-thing investment strategy in period n and in future periods. So over time, some “intermediate-type fund manager,”for example c Xn+1 < ! < Xnc , use the risky fair investment strategy f0; Xnc g, for the …rst n periods and then switch to the sure-thing investment strategy thereafter. Let n( ) be the endogenous cumulative distribution function of fund manager types remaining at the beginning of period n, given the equilibrium outcome that fund managers ! 2 (0; Xnc ) use the risky fair investment strategy f0; Xnc g. Then Xnc solves R Xnc !(!=Xnc ) n (d!) a = R: R Xnc (!=Xnc ) n (d!) a Note that 1 (!) = Z ! a ! X0c (d!) ; and an induction argument establishes that, Z ! !n n (!) = c c Xnc a X0 X1 (d!): 1 Hence, the cuto¤ Xnc is given implicitly by the solution to R Xnc R Xnc n+2 c ) (d!) !(!=X ! (d!) n n = RaX c = R: Ra Xnc n c) n+1 (d!) (!=X (d!) ! n n a a (15) Up to this point we have only asserted that Xnc is strictly decreasing in n. We now prove that this is, in fact, the case. De…ne E observing X0c c n (X0 ) [!] as the expected value of ! conditional on 1; : : : ; 0 when all hedge fund managers ! < X0c use the risky each period n; n fair investment strategy f0; X0c g in every period n 0 and hedge fund managers ! X0c always use the sure-thing investment strategy. Hence, R X0c n+2 ! (d!) E n (X0c ) [!] = RaX c : 0 n+1 (d!) ! a Consider now what happens to the value of E decreasing in n if and only if E for n 1, then c n (X0 ) R X0c RaX0c a c n (X0 ) [!] as n increases. Note that Xnc is strictly [!] is strictly increasing in n. If E ! n+2 (d!) ! n+1 (d!) R X0c a ! n+1 (d!) > R Xc 0 a 50 ! n (d!) ; c n (X0 ) [!] > E c n 1 (X0 ) [!] or Z Xc ! n+2 (d!) a Z Z Xc ! n (d!) > a 2 Xc ! n+1 (d!) : a :5 If we de…ne F (d!) = [! n+2 (d!)] and G(d!) = [! n (d!)]:5 , we can rewrite the above inequality as Z Xc 2 F (d!) Z Xc Z 2 G (d!) > a a 2 Xc F (d!) G (d!) which is simply a statement of Schwartz’s inequality. Therefore, E which implies that Xnc ; a c n (X0 ) [!] > E c n 1 (X0 ) [!], is strictly decreasing in n. Proof to Proposition 5 ^ c represent the long-horizon investor’s expected payo¤ in period 2 given the ^ c; X Let V3 X 1 0 ^ 0c in period 0 and X ^ 1c in period 1. The critical values, X ^ 0c and X ^ 1c , hedge fund’s payout was X ^ 0c ; X ^ 1c = R or V3 X ^ 0c ; X ^ 1c = (1 solve V3 X ^c ^ c; X V3 X 1 0 1 = ) = R. Therefore, R X^1c a ! n R X^1c a R X^1c = Ra ^ c X1 a ! ^c X 1 ! ^c X 1 h h ! ^c X 0 ! ^c X 0 ! 3 (d!) io (d!) i (d!) (16) = R; ! 2 (d!) ^ 1c = X1c , see (15). Hence, the cuto¤ set on the period 1 hedge fund which implies that X payout for continued period 2 capital investment does not vary with the investor’s horizon. Now consider the cuto¤ set by the long-horizon investor on the period 0 hedge fund payout for continued period 1 capital investment. The expected lifetime payo¤ to this investor ^ 0c in period 0 is when it achieves a payout X V3 ^c X 0 = R X^1c n a R X0c ^c X 1 ! ^c X 1 h ! (1 (1 ^ 1c R + ! (1 )X R X^0c ! a ^c X 0 ) R + ! (1 ) R X^0c ! (d!) ^c a X 0 51 i ) + 1 ! ^c X 0 (d!) (d!) : ! ^c X 1 R o ! ^c X 0 (d!) + Recognizing that V3 ^ 0c X = R X^1c a R X^1c a ! 3 (d!) = R (1 R X^1c a ) !2R + !R ! 2 (d!) from (16), this expression simpli…es to R X^ c (d!) + X^ c0 (1 1 R X^0c ! (d!) a ) !2 1 + R (d!) : (17) ^ 0c = R2 + R, i.e., the critical value at date 0 is such Note that e¢ ciency requires that V3 X that the investor is indi¤erent between investing in the fund and the alternative asset. Consider now the payo¤s to short-horizon investors. In particular, we are interested in the expected payo¤ to an investor in date 1, given that the date 0 fund payout was X0c . Denote this expected payo¤ by V1 (X0c ). To facilitate comparison between the long-horizon and short-horizon investors, note that V1 (X0c )R + R = R2 + R, since V1 (X0c ) = R. Therefore, i R X0c h ! R Xc ! c (1 ) X R (d!) + a 0 R X!c (d!) c c 1 X X a 1 0 0 V1 (X0c ) R + R = R X0c ! (d!) X0c a R Xc R X1c (d!) (d!) + X c0 (1 ) ! 2 R + R! (1 ) ! 2 R + R! a 1 (18) : = R X0c ! (d!) a ^ c = X c and subtracting the numerator of (18) ^ c = X c . Now, recalling that X Suppose that X 1 1 0 0 from that of (17), we get Z X0c (1 )! 2 = X0c R! (d!) X1c X1c Z Z X0c (1 )! R ! (d!) : (19) X1c R > 0 for all ! 2 [X1c ; X0c ], expression (19) is strictly positive, a ^ c = X c , it must be the case contradiction. Since (17) exceeds (18) when it is assumed that X 0 0 c c ^ that X < X . But since, (1 0 )! 0 Proof to Proposition 6 The expected performance of the hedge fund in period 1 is, R X !2 Rb (d!) + X ! (d!) a X ; RX ! Rb (d!) + (d!) a X X 52 (20) ^ 0 ; X0 ). The sign of the derivative of (20) where X represents the period-0 cuto¤ and X 2 [X with respect to X is Z sign X a Z X Z b Z b !2 ! (d!) + ! (d!) : (d!) (d!) X2 X2 X a X " RX # Rb ! 2 (d!) ! (d!) a X = sign + Rb RX ! (d!) (d!) a X The absolute value of the …rst term is less than X and the value of the second term exceeds X. Hence, this expression is positive, which means that an increase in the period-0 standard increases the period-1 expected fund performance. The expected performance of the hedge fund in period 2 is R X1c !3 R X !2 Rb ! (d!) (d!) + (d!) + c XX1 a X X X : R X1c !2 R 1X ! Rb (d!) + X c X (d!) + X (d!) XX1 a (21) 1 The sign of the derivative of (21) can be written as " Z Z b Z X 2 sign ! (d!) (d!) + X1c Z a X1c Z b (d!) + = sign 4 Z X1c a X The …rst line can be rearranged as 2 R X Xc R X1 X1c ! ! (d!) X1c X !3 (d!) X1 X 2 (d!) ! (d!) + RXb b ! (d!) X !2 (d!) X1 Rb Z Z X # ! (d!) : X ! (d!) (d!) b (22) 3 5: The …rst term is less than X and the second term exceeds X, meaning that the sign is positive. The second line of (22) can be rearranged as " R Xc # Rb 3 1 ! (d!) ! (d!) sign + RXb : RaX1c 2 ! (d!) (d!) a X The …rst term is less than X1c and the second term exceeds X1c . All of this implies that (22) is positive, which means that an increase in the period-0 standard increases the period-2 expected hedge fund performance. 53 Proof to Proposition 7 Since the expected ability of a fund manager whose fund pays out X0 = X c is R, X c solves R [(! + 0 )!=X c ] (!) h( 0 )d 0 d! !+ 0 <X c R = E (X c ) [!] = R [(! + 0 )=X c ] (!) h( 0 )d 0 d! !+ 0 <X c R (! + 0 )!h( 0 )d 0 d! !+ <X c ; = R 0 (! + 0 )h( 0 )d 0 d! !+ <X c 0 where the simpli…cation re‡ects the uniform uncertainty of fund managers’ type !. The conditional expectation can be expanded to read, R Xc y 2 R Xc R Xc ! Ry ! d! + X c y [ y !(! + 0 )h( 0 )d 0 + X c ! A (!; h) h( 0 )d 0 ]d! a ; Ry R Xc y R Xc R Xc ! 0 (!; h) h( )d ]d! !d! + [ (! + )h( )d + A c c 0 0 0 0 0 a X y y X ! where A (!; 0) = [! + 2(X c !)][! + 2(X c !) 0] and A0 (!; 0) = (! + 2(X c (23) !) 0 ). To understand the construction of (23), consider each of the three terms in the numerator: 1. The …rst integral is over hedge fund manager types ! < X c such that ! + y < X c . Even if they receive the highest possible idiosyncratic shock, these hedge fund managers must use a risky investment strategy in order to achieve X c . 2. The second integral is over hedge fund manager types ! < X c who do not receive a su¢ ciently large 0 shock, and hence must adopt a risky investment to reach X c . It does not include those hedge fund manager types who receive big positive 0 >X c 0 shocks, !, and hence adopt sure thing investment strategies. 3. The last integral is over hedge fund manager types ! > X c who receive su¢ ciently large negative 0 shocks that push them below X c , and hence must choose a risky in- vestment strategy in order to achieve X c . This integral term exploits the symmetry of 0, h( 0 ) = h( 0 ), to use the bounds of integration that apply to hedge fund manager types ! < X c . This facilitates comparisons of (23) with the base-case where equals zero. Thus, for any hedge fund manager type ! = X c for = X c 0 always ! 2 (0; y) the corresponding actual hedge fund manager type is ! ^ = X c + , which we write as Xc + 2(X c lucky receiving a (X c 0 )) = X c + 2 = X c + ; and the counterpart got un- < 0 shock rather than a positive 54 0 shock. Thus, the true type, ! ^ = ! + 2(X c !), enters the expectation (of period 1 ability) in the …rst brack- eted term in A(!; 0 ), and the hedge fund manager’s period 0 resources ! ^ 0, which c determine the probability of achieving X , are the second bracketed term in A(!; 0 ). The denominator can be interpreted analogously. Now, substitute A (!; 0) = (2X c ! 0 )(2X c !) = ! + 0 + 2X c (2(X c !) 0) in the numerator of (23), and A0 (!; = 2X c 0) ! 0 =!+ 0 + 2(X c (! + 0 )) in the denominator, to rewrite the conditional expectation as R Xc 2 R Xc R y ! d! + X c y X c ! 2X c (2(X c !) 0 )h( 0 )d 0 d! a R Xc R Xc R y !d! + X c y X c ! 2(X c (! + 0 ))h( 0 )d 0 d! a Z Xc Z y Z Xc 2 2X c ((X c !)h( 0 )d 0 d! + ! d! + = [ Z [ a Xc Xc y Z Xc Z Xc y 2X c ((X c Xc ! !d! + = where = R Xc R y Xc y Xc Xc ! 2 a 0 ))h( 0 )d 0 d!] y 2(X c Xc R Xc R y ! d! + 2(X c (! + multiplying the inequality) since, Z Xc Z Xc Z 2 c !d!( ! d! + X ) a y Z (! + (! + a 0 ))h( 0 )d 0 d!. Xc 0 ))h( 0 )d 0 d!] 1 ! 2X c ((X c Xc y Xc ! R Xc R Xc R y !d! + X c y X c ! 2(X c a R Xc 2 R Xc 2 c ! d! ! d! + X a > RaX c ; R Xc !d! + !d! a a a > Z Xc a R Xc Xc ! y Z ! d!( 2 a (! + (! + 0 ))h( 0 )d 0 d! 0 ))h( 0 )d 0 d! The …nal inequality follows (by cross- Xc !d! + ) = Z Xc (X c !)!d! > 0: a Thus, for any given cuto¤ X c , pre-investment shocks raise the expected hedge fund type that pays out X c in the following sense. Conditional on achieving the cuto¤ (1) in Section 3 (where there were no pre-investment shocks), the expected ability of a hedge fund manager exceeds R. To retrieve equality with R, it must be that the cuto¤ X c for continued investment in the hedge fund is reduced. 55 Working Paper Series A series of research studies on regional economic issues relating to the Seventh Federal Reserve District, and on financial and economic topics. Comment on “Letting Different Views about Business Cycles Compete” Jonas D.M. Fisher WP-10-01 Macroeconomic Implications of Agglomeration Morris A. Davis, Jonas D.M. Fisher and Toni M. Whited WP-10-02 Accounting for non-annuitization Svetlana Pashchenko WP-10-03 Robustness and Macroeconomic Policy Gadi Barlevy WP-10-04 Benefits of Relationship Banking: Evidence from Consumer Credit Markets Sumit Agarwal, Souphala Chomsisengphet, Chunlin Liu, and Nicholas S. Souleles WP-10-05 The Effect of Sales Tax Holidays on Household Consumption Patterns Nathan Marwell and Leslie McGranahan WP-10-06 Gathering Insights on the Forest from the Trees: A New Metric for Financial Conditions Scott Brave and R. Andrew Butters WP-10-07 Identification of Models of the Labor Market Eric French and Christopher Taber WP-10-08 Public Pensions and Labor Supply Over the Life Cycle Eric French and John Jones WP-10-09 Explaining Asset Pricing Puzzles Associated with the 1987 Market Crash Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein WP-10-10 Prenatal Sex Selection and Girls’ Well‐Being: Evidence from India Luojia Hu and Analía Schlosser WP-10-11 Mortgage Choices and Housing Speculation Gadi Barlevy and Jonas D.M. Fisher WP-10-12 Did Adhering to the Gold Standard Reduce the Cost of Capital? Ron Alquist and Benjamin Chabot WP-10-13 Introduction to the Macroeconomic Dynamics: Special issues on money, credit, and liquidity Ed Nosal, Christopher Waller, and Randall Wright WP-10-14 Summer Workshop on Money, Banking, Payments and Finance: An Overview Ed Nosal and Randall Wright WP-10-15 Cognitive Abilities and Household Financial Decision Making Sumit Agarwal and Bhashkar Mazumder WP-10-16 1 Working Paper Series (continued) Complex Mortgages Gene Amromin, Jennifer Huang, Clemens Sialm, and Edward Zhong WP-10-17 The Role of Housing in Labor Reallocation Morris Davis, Jonas Fisher, and Marcelo Veracierto WP-10-18 Why Do Banks Reward their Customers to Use their Credit Cards? Sumit Agarwal, Sujit Chakravorti, and Anna Lunn WP-10-19 The impact of the originate-to-distribute model on banks before and during the financial crisis Richard J. Rosen WP-10-20 Simple Markov-Perfect Industry Dynamics Jaap H. Abbring, Jeffrey R. Campbell, and Nan Yang WP-10-21 Commodity Money with Frequent Search Ezra Oberfield and Nicholas Trachter WP-10-22 Corporate Average Fuel Economy Standards and the Market for New Vehicles Thomas Klier and Joshua Linn WP-11-01 The Role of Securitization in Mortgage Renegotiation Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet, and Douglas D. Evanoff WP-11-02 Market-Based Loss Mitigation Practices for Troubled Mortgages Following the Financial Crisis Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet, and Douglas D. Evanoff WP-11-03 Federal Reserve Policies and Financial Market Conditions During the Crisis Scott A. Brave and Hesna Genay WP-11-04 The Financial Labor Supply Accelerator Jeffrey R. Campbell and Zvi Hercowitz WP-11-05 Survival and long-run dynamics with heterogeneous beliefs under recursive preferences Jaroslav Borovička WP-11-06 A Leverage-based Model of Speculative Bubbles (Revised) Gadi Barlevy WP-11-07 Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen WP-11-08 Fertility Transitions Along the Extensive and Intensive Margins Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder WP-11-09 Black-White Differences in Intergenerational Economic Mobility in the US Bhashkar Mazumder WP-11-10 2 Working Paper Series (continued) Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options? Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein Business Networks, Production Chains, and Productivity: A Theory of Input-Output Architecture Ezra Oberfield WP-11-11 WP-11-12 Equilibrium Bank Runs Revisited Ed Nosal WP-11-13 Are Covered Bonds a Substitute for Mortgage-Backed Securities? Santiago Carbó-Valverde, Richard J. Rosen, and Francisco Rodríguez-Fernández WP-11-14 The Cost of Banking Panics in an Age before “Too Big to Fail” Benjamin Chabot WP-11-15 Import Protection, Business Cycles, and Exchange Rates: Evidence from the Great Recession Chad P. Bown and Meredith A. Crowley WP-11-16 Examining Macroeconomic Models through the Lens of Asset Pricing Jaroslav Borovička and Lars Peter Hansen WP-12-01 The Chicago Fed DSGE Model Scott A. Brave, Jeffrey R. Campbell, Jonas D.M. Fisher, and Alejandro Justiniano WP-12-02 Macroeconomic Effects of Federal Reserve Forward Guidance Jeffrey R. Campbell, Charles L. Evans, Jonas D.M. Fisher, and Alejandro Justiniano WP-12-03 Modeling Credit Contagion via the Updating of Fragile Beliefs Luca Benzoni, Pierre Collin-Dufresne, Robert S. Goldstein, and Jean Helwege WP-12-04 Signaling Effects of Monetary Policy Leonardo Melosi WP-12-05 Empirical Research on Sovereign Debt and Default Michael Tomz and Mark L. J. Wright WP-12-06 Credit Risk and Disaster Risk François Gourio WP-12-07 From the Horse’s Mouth: How do Investor Expectations of Risk and Return Vary with Economic Conditions? Gene Amromin and Steven A. Sharpe WP-12-08 Using Vehicle Taxes To Reduce Carbon Dioxide Emissions Rates of New Passenger Vehicles: Evidence from France, Germany, and Sweden Thomas Klier and Joshua Linn WP-12-09 Spending Responses to State Sales Tax Holidays Sumit Agarwal and Leslie McGranahan WP-12-10 3 Working Paper Series (continued) Micro Data and Macro Technology Ezra Oberfield and Devesh Raval WP-12-11 The Effect of Disability Insurance Receipt on Labor Supply: A Dynamic Analysis Eric French and Jae Song WP-12-12 Medicaid Insurance in Old Age Mariacristina De Nardi, Eric French, and John Bailey Jones WP-12-13 Fetal Origins and Parental Responses Douglas Almond and Bhashkar Mazumder WP-12-14 Repos, Fire Sales, and Bankruptcy Policy Gaetano Antinolfi, Francesca Carapella, Charles Kahn, Antoine Martin, David Mills, and Ed Nosal WP-12-15 Speculative Runs on Interest Rate Pegs The Frictionless Case Marco Bassetto and Christopher Phelan WP-12-16 Institutions, the Cost of Capital, and Long-Run Economic Growth: Evidence from the 19th Century Capital Market Ron Alquist and Ben Chabot WP-12-17 Emerging Economies, Trade Policy, and Macroeconomic Shocks Chad P. Bown and Meredith A. Crowley WP-12-18 The Urban Density Premium across Establishments R. Jason Faberman and Matthew Freedman WP-13-01 Why Do Borrowers Make Mortgage Refinancing Mistakes? Sumit Agarwal, Richard J. Rosen, and Vincent Yao WP-13-02 Bank Panics, Government Guarantees, and the Long-Run Size of the Financial Sector: Evidence from Free-Banking America Benjamin Chabot and Charles C. Moul WP-13-03 Fiscal Consequences of Paying Interest on Reserves Marco Bassetto and Todd Messer WP-13-04 Properties of the Vacancy Statistic in the Discrete Circle Covering Problem Gadi Barlevy and H. N. Nagaraja WP-13-05 Credit Crunches and Credit Allocation in a Model of Entrepreneurship Marco Bassetto, Marco Cagetti, and Mariacristina De Nardi WP-13-06 4 Working Paper Series (continued) Financial Incentives and Educational Investment: The Impact of Performance-Based Scholarships on Student Time Use Lisa Barrow and Cecilia Elena Rouse WP-13-07 The Global Welfare Impact of China: Trade Integration and Technological Change Julian di Giovanni, Andrei A. Levchenko, and Jing Zhang WP-13-08 Structural Change in an Open Economy Timothy Uy, Kei-Mu Yi, and Jing Zhang WP-13-09 The Global Labor Market Impact of Emerging Giants: a Quantitative Assessment Andrei A. Levchenko and Jing Zhang WP-13-10 Size-Dependent Regulations, Firm Size Distribution, and Reallocation François Gourio and Nicolas Roys WP-13-11 Modeling the Evolution of Expectations and Uncertainty in General Equilibrium Francesco Bianchi and Leonardo Melosi WP-13-12 Rushing into American Dream? House Prices, Timing of Homeownership, and Adjustment of Consumer Credit Sumit Agarwal, Luojia Hu, and Xing Huang WP-13-13 The Earned Income Tax Credit and Food Consumption Patterns Leslie McGranahan and Diane W. Schanzenbach WP-13-14 Agglomeration in the European automobile supplier industry Thomas Klier and Dan McMillen WP-13-15 Human Capital and Long-Run Labor Income Risk Luca Benzoni and Olena Chyruk WP-13-16 The Effects of the Saving and Banking Glut on the U.S. Economy Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti WP-13-17 A Portfolio-Balance Approach to the Nominal Term Structure Thomas B. King WP-13-18 Gross Migration, Housing and Urban Population Dynamics Morris A. Davis, Jonas D.M. Fisher, and Marcelo Veracierto WP-13-19 Very Simple Markov-Perfect Industry Dynamics Jaap H. Abbring, Jeffrey R. Campbell, Jan Tilly, and Nan Yang WP-13-20 Bubbles and Leverage: A Simple and Unified Approach Robert Barsky and Theodore Bogusz WP-13-21 5 Working Paper Series (continued) The scarcity value of Treasury collateral: Repo market effects of security-specific supply and demand factors Stefania D'Amico, Roger Fan, and Yuriy Kitsul Gambling for Dollars: Strategic Hedge Fund Manager Investment Dan Bernhardt and Ed Nosal WP-13-22 WP-13-23 6