Mutual Fund Flows and Optimal Manager Replacement

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Mutual Fund Flows and Optimal Manager Replacement

  1. 1. Mutual Fund Flows and Optimal Manager Replacement∗ Thomas Dangl Department of Managerial Economics and Industrial Organization Vienna University of Technology Theresianumgasse 27, A-1040 Vienna Phone:0043-58801-33063 Thomas.dangl@tuwien.ac.at Youchang Wu Department of Finance, University of Vienna Bruennerstrasse 72, 1210 Vienna Phone:0043-1-4277-38211 youchang.wu@univie.ac.at Josef Zechner Department of Finance, University of Vienna Bruennerstrasse 72, 1210 Vienna Phone:0043-1-4277-38072 josef.zechner@univie.ac.at May 2004 ∗ We thank Russ Wermers, Michael Brennan, the participants of the seminars at University of Tuebingen, University of Vienna for their suggestions and comments. Youchang Wu and Josef Zechner gratefully acknowl- edge the financial support by the Austrian Science Fund (FWF) under grant SFB 010 and by the Gutmann Center for Portfolio Management at the University of Vienna.
  2. 2. Abstract This paper develops a continuous-time model in which a portfolio manager is hired by a management company. Based on past portfolio returns, all agents, including the portfolio manager, update their beliefs about the manager’s skills. In response, investors can move capital into or out of the mutual fund, the portfolio manager can alter the risk of the portfolio and the management company can replace the manager. The resulting interplay between internal governance and product market discipline generates a rich set of results in accordance with the empirical literature. In particular we show how fund inflows are related to a fund’s performance, fund size, and the uncertainty about manage- rial skills. We also derive optimal management replacement rules using the real options approach and demonstrate how manager turnover depends on the model parameters.
  3. 3. 1 Introduction The portfolio management industry has undergone dramatic growth in the last few decades, thereby also generating an increasing research interest among academics. A particular ques- tion which has attracted much attention is the governance issue associated with delegated portfolio management. Numerous authors have approached this question using the standard principal-agent paradigm, trying to characterize the optimal contracts that alleviate the agency problem between fund investors and managers, 1 Some researchers have also empirically in- vestigated the effectiveness of funds’ boards of directors in controlling agency costs 2 and the impact of incentive fees.3 This paper takes a broader view of the governance mechanisms in the portfolio manage- ment industry by explicitly taking into account the role of the product market. A key char- acteristic of the mutual fund industry is that fund investors are at the same time consumers. The open-end structure of mutual funds allows individual investors to ”fire” the manager whenever they feel dissatisfied with the management quality. This not only disciplines the manager directly, as pointed out by Fama and Jensen (1983), but also gives the management company strong incentives to fire underperforming managers in order to avoid losing market share. The interplay between these external and internal governance is the key to understand many phenomena observed in the mutual fund market. We formalize these two alternative governance forces simultaneously in a fully-dynamic framework. More specifically, we an- alyze how beliefs about managerial ability are updated over time and how investors allocate their money in response to changing beliefs; we also derive the fund management company’s optimal manager replacement rule and the equilibrium response of fund flows to an observed manager replacement. In our model, the portfolio manager may have some stock picking ability that generates an expected rate of abnormal return by taking idiosyncratic risks. However, active portfo- lio management exhibits a diseconomy of scale, as introduced by Berk and Green (2004). 1 See,for example, Stoughton (1993), Heinkel and Stoughton (1994), Admati and Pfleiderer (1997),Das and Sundaram (2002), Ou-Yang (2003). 2 See, for example, Tufano and Sevick (1997) for the case of open-end funds and Guercio, Dann, and Partch (2003) for the case of closed-end funds. 3 See, for example, Elton, Gruber, and Blake (2003). 1
  4. 4. Furthermore, the manager’s ability to manage a specific fund is not known either to the man- agement company, to the fund investors or to the manager himself. All agents in the model only know a common prior distribution of the ability and keep updating their beliefs using the observed fund performance. Since investors keep moving money into or out of a fund based on the updated beliefs about the managerial ability, there exists an equilibrium relation be- tween beliefs and fund size. The management company’s problem is to determine a threshold of the updated beliefs that triggers a manager replacement. Since the management company’s right to replace the manager is essentially an American put option written on the stochastic process of beliefs, the optimal threshold can be derived using the real options approach. We consider two polar cases. In the first case, the manager is endowed with a given ability which remains constant over time. In the second case the manager’s ability may change over time. In this case we assume that the manager’s ability follows a driftless Wiener process, so that the precision of the ability estimate remains constant. We first characterize the inference of managerial ability using the nonlinear filtering the- ory developed by Lipster and Shiryayev (1978). We then analyze two closely-related ques- tions under the equilibrium condition that the expected abnormal return to fund investors is zero: (i). What are the determinants of fund flows? We find that (1) the expected rate of net fund inflows is negatively correlated with fund size and management fee, but positively correlated with uncertainty about managerial ability; (2) the net fund inflows are pos- itively correlated with the fund’s idiosyncratic returns but negatively correlated with the market return; (3) the sensitivity of net fund inflows to fund performance increases with uncertainty about managerial ability; and (4) the volatility of net fund inflows de- creases with fund size and increases with uncertainty about managerial ability. We also find that fund size is positively related to historical performance but uncorrelated with future performance. (ii). Given the equilibrium relation between fund size and beliefs about managerial ability, what is the optimal manager replacement rule that maximizes the value of the manage- ment company? Our analysis shows that (1) in the case of constantly changing ability, 2
  5. 5. manager changes are preceded by poor risk adjusted performance and a decrease in net fund inflows; (2) in the case of constant ability, the optimal manager replacement threshold increases over the manager’s tenure; it can reach a level significantly above the expected ability of a new manager; (3) manager tenure is positively correlated with fund size; (4) the conditional probability of manager departure is non-monotonic in manager tenure, it increases at the early stage of manager tenure and then keeps de- creasing. Our results help to understand a large number of findings documented in the empirical liter- ature on mutual fund flows and portfolio manager turnover. Since the early 1990s, a considerable amount of effort has been devoted to research on the behavior of mutual fund flows. A well-documented finding is that there is a strong positive relation between a fund’s money inflows and its past performance, indicating that investors are chasing good past performers.4 Similar to Berk and Green (2004), our model generates results where rational investors behave in a way fully consistent with this empirical obser- vation. Another stylized fact that have drawn much attention is that the flow-performance relation is not only positive but also convex, meaning that out-performers receive dispro- portionately more fund inflows, while under-performers are not penalized in an offsetting manner.5 This feature is captured by the positive drift of the flow dynamics we derive. In ad- dition, it has also been documented that fund inflows, as well as their volatility and sensitivity to past performance, are negatively correlated with fund size and fund age.6 Finally, Warther (1995) reports a negative relation between the market return and subsequent aggregate fund inflows. Our model is consistent with all these results. There is also a strand of literature on the turnover of portfolio managers. Khorana (1996) documents an inverse relation between the probability of manager replacement and past fund performance, measured by asset growth rate and portfolio returns. Chevalier and Ellison (1999a) find a similar result. They also find that managerial turnover is more performance- sensitive for younger managers. Chevalier and Ellison (1999b) find that managers with longer 4 See, for example, Ippolito (1992), Gruber (1996), Chevalier and Ellison (1997), Sirri and Tufano (1998), Bergstresser and Poterba (2002),and Boudoukh, Richardson, Stanton, and Whitelaw (2003). 5 See Chevalier and Ellison (1997) and Sirri and Tufano (1998). 6 See, for example, Chevalier and Ellison (1997), Sirri and Tufano (1998), Bergstresser and Poterba (2002), Boudoukh, Richardson, Stanton, and Whitelaw (2003). 3
  6. 6. tenure tend to manage larger funds and have a higher probability to retain their positions.7 Hu, Hall, and Harvey (2000) classify the managerial departure into promotion and demotion, and find that the probability of a manager being fired or demoted is negatively correlated with the fund’s current and past performance whereas the promotion probability is positively related to the fund’s current and past performance. Khorana (2001) documents significant performance improvements after the replacement of an underperforming manager and per- formance deterioration after the replacement of outperforming managers. Ding and Wermers (2002) find that fund managers underperform their counterparts during the year of their re- placement, and the underperformance vanishes after the manager is replaced. These findings are in accordance with the predictions from our model. Several theoretical models have been advanced to explain subsets of the phenomena men- tioned above. Berk and Green (2004) present an interesting discrete-time learning model to explain one seemingly puzzling phenomenon mentioned above: Given that the evidence on the predictability of fund performance is rather weak, why do investors always chase past winners? There are three key elements in their answer to this question: First, there is com- petitive provision of capital by investors to mutual funds. Second, there is decreasing return to scale in active portfolio management. Finally, past returns contain information about man- agerial ability. The combination of these three elements explains both the lack of performance persistence and the flow-performance relation: Good performance leads to higher assessment of managerial ability, which in turn leads to fund inflows, however this will hurt future fund performance due to the decreasing return to scale. Since the capital provision is competitive, the fund size will be adjusted to a level where no predictability in fund performance exists. Our model on mutual fund flows incorporates all the three key elements of Berk and Green (2004). However, in contrast to Berk and Green (2004), we explicitly distinguish between fund management companies and fund managers. While Berk and Green (2004) assume an exogenous shut-down threshold, we allow the fund management company to fire the manager if it is optimal to do so. This not only enables us to explain both fund flows and managerial turnover in a unified framework, but also shed some new light on the flow- performance relation. In particular, while the Berk and Green (2004) model predicts that 7 Thepositive tenure-fund size relation is also documented by Fortin, Michelson, and Jordan-Wagner (1999) and Boyson (2003). 4
  7. 7. when the fund performs very badly, there will be a 100% cash outflow due to the shut-down of the fund, our model predicts that there will be a cash inflow due to the replacement of the underperforming manager. Therefore, in contrast to Berk and Green (2004), our model implies a globally convex flow-performance relation. Our model also differs from the Berk and Green (2004) model in other important aspects. Instead of the discrete-time setup in Berk and Green (2004), we derive a continuous-time formulation. Instead of assuming a convex cost function, we model directly the increasing difficulty of generating abnormal return when the fund becomes larger. Our formulation not only makes the analysis much easier to handle, but also yields new results. For example, while Berk and Green (2004) predict that the proportional fund inflows, after controlling for fund age, are independent of fund size, our model shows that the proportional fund inflows depend negatively on fund size, even after controlling for fund age, as documented by the empirical investigations mentioned above. Our setup also predicts a negative relation between fund inflows and past market return, as documented by Warther (1995). Lynch and Musto (2003) develop a two-period model to explain the convexity of the flow-performance relation. Their key insight is that funds that underperform will change their strategies while those outperforming will not. Therefore bad past performance con- tains less information about future performance than good past performance does. Taking this effect into account, rational investors will be less sensitive to past performance when it is poor. Our model differs from the Lynch and Musto (2003) model in several aspects. First, Lynch and Musto (2003) do not allow for diseconomies of scale. As a result, active funds will on average earn above normal returns and good performance should exhibit strong persistence. Both of these implications are in contradiction with the majority of empirical findings. Second, whereas Lynch and Musto (2003) analyze the decision to change strate- gies in a two-period model, we model the manager replacement decision in a fully dynamic framework, taking into account flow responses, replacement costs, and the option value of postponing the replacement. Third, although firing bad managers increases the convexity of the flow-performance relation, such a convexity exists in our model even without this consid- eration. Heinkel and Stoughton (1994) consider a two-period contracting problem between a port- 5
  8. 8. folio manager and a client. Their model contains both adverse selection and moral hazard components. Our model differs in several important dimensions. First, we develop a fully dynamic model in continuous time. Second, we do not model a direct contracting relation- ship between the investor and the portfolio manager. Instead, the principal-agent relationship we consider is between portfolio managers and management companies. Third, our model formalizes simultaneously the external governance of the product market and the internal governance mechanism. Another closely-related paper is Jovanovic (1979), which considers a situation where a worker’s productivity in a particular job is not known ex ante and becomes known more precisely as his job tenure increases. While we model learning in a way similar to Jovanovic (1979), the latter paper does not address the specific issues arising in the mutual fund market. In fact, it abstracts completely from product market considerations. The rest of the paper is structured as follows: Section 2 and Section 3 analyze the dynam- ics of equilibrium fund size and optimal manager replacement rules under the assumption that the true managerial ability evolves randomly over time. Section 4 reexamines these two issues under the assumption that the true managerial ability is constant over time. 2 Learning and mutual fund flows 2.1 The dynamics of NAV The net asset value per share, denoted by NAV, of an open-end fund is assumed to evolve as follows: dNAVt = [r + λσm + αt − δt − bt ]dt + σm dWmt + σit dWit (1) NAVt where r is risk free rate, λ is the market price of risk, αt is the expected rate of abnormal return generated by the manager due to his stock-picking ability, Wmt and Wit are two uncorrelated standard Wiener processes driving the market return and the idiosyncratic component of the fund’s return respectively. The subscript t denotes the incumbent manager’s tenure, σm is the fund’s constant exposure to market risk while σit is the fund’s time-varying exposure to 6
  9. 9. idiosyncratic risk, δt is the instantaneous dividend yield and bt is the management fee ratio. Our specification of the NAV dynamics explicitly accounts for active portfolio manage- ment. The term r + λσm is the fair return given the fund’s exposure to systematic risk. The expected rate of abnormal return, αt , can be interpreted as Jensen’s α. The dividend yield, δt , and the management fee ratio, bt , are subtracted from the NAV return because they represent cash paid out to the investors and the management company respectively. The exposure to market risk, σm , is assumed to be constant since we do not intend to model the manager’s market-timing ability.8 In contrast to the constant exposure to the market risk, the fund’s exposure to the idiosyncratic risk, σit , can be changed by the manager at any time, under the constraint that it must be positive. The expected abnormal rate of return is assumed to have the following functional form: αt = θσit − γAt σ2 it (2) where θt is the incumbent manager’s stock-picking ability, At is the market value of assets under management and γ is a positive constant characterizing the decreasing return to scale in active portfolio management. The manager’s ability θt is assumed to be fund specific, which implies that the manager’s track record in other funds does not matter. Thus, we assume that the abnormal return is a joint product of the manager and the management company. A good manager in one company is not necessarily a good manager in another company because every company has its own organizational structures, research networks and business culture. This assumption implies that our model is essentially a match model in the spirit of Jovanovic (1979). It allows us to abstract from the observable heterogeneity among the manager candidates available to replace the incumbent manager. Our specification of the expected abnormal return has several desirable features that we would like to capture. More specifically, the specification has the following properties. ∂αt (i). ∂θt = σit > 0. For any given level of idiosyncratic risk and fund size, managers with 8 There is little evidence showing that mutual fund managers have market-timing ability. However, recent studies using portfolio holdings data do provide support for the notion that some managers have superior stock- picking ability. See, for example, Grinblatt and Titman (1993), Daniel, Grinblatt, Titman, and Wermers (1997), Wermers (2000). 7
  10. 10. higher ability can generate higher expected rate of abnormal return. ∂αt (ii). ∂At = −γσ2 < 0. For any given level of managerial ability and idiosyncratic risk, larger it funds will have a lower expected rate of abnormal return. This is the diseconomy of scale effect in active portfolio management that we want to capture. An important rea- son of this effect is the price impact of large portfolio transactions. Consider a manager who is able to identify a small number of undervalued stocks. If he is managing a small fund, he can invest all his funds in these stocks and earn a high abnormal rate of return. However, if he is managing a large fund, doing so would move the purchase prices of those stocks up and erode his performance.9 10 ∂2 αt (iii). ∂σ2 = −2γAt < 0. The marginal return of taking idiosyncratic risk is decreasing, and it it decreases faster for funds with bigger size. The marginal return of taking idiosyncratic risk must be decreasing in order to rule out unlimited expected profit opportunities. Consider a manager who can identify a small number of undervalued securities, the idiosyncratic risk of his fund will increase as he puts more and more money in those stocks.11 This will increase the fund’s expected rate of abnormal return. However, the marginal return must be decreasing because the more money he invests in those stocks, the more likely he will move the price to his disadvantage. This effect is much stronger when the fund is big. Therefore, the marginal return decreases faster when the fund has a larger size. ∂2 αt (iv). ∂At ∂σit = −2σit < 0. This has two implications: (1) The larger the fund size, the smaller the marginal return of taking idiosyncratic risk. This should be the case due to the larger price impact associated with larger fund size. (2) the more idiosyncratic risk a fund takes, the larger the diseconomy of scale effect it has to face. A fund with high idiosyncratic risk may either concentrate on a small number of securities, or hold illiquid stocks. Such funds would suffer the most from being large. However, a well- 9 Many authors have investigated this issue empirically. See for example, Perold and Salomon (1991), Indro, Jiang, Hu, and Lee (1999), Chen, Hong, Huang, and Kubik (2003). 10 Note that this condition does not imply that for a given level of managerial ability, a larger fund will always have a lower expected rate of abnormal return than a smaller fund. This is true only if both funds take the same level of idiosyncratic risk. 11 The systematic risk can be kept constant by shorting an index. 8
  11. 11. diversified fund, such as an index fund will not be hurt as much by its large size. Our formulation of the abnormal return can be interpreted in another way.12 Suppose the manager divides his assets under management into two parts: one inactive part with system- atic risk σm and zero idiosyncratic risk, and one actively managed part with systematic risk σm and idiosyncratic risk εi . The weights of these two components are 1 − wt and wt respec- tively. The inactively managed part delivers no abnormal return and does not suffer from any diseconomy of scale. The actively managed part produces an expected rate of abnormal return due to the managerial ability θ. But the abnormal return per unit of idiosyncratic risk decreases with the size of the actively managed part. It decreases faster when the idiosyn- cratic risk is higher, since the portfolio with higher idiosyncratic risk is less liquid. Therefore, the abnormal return of the fund can be written as αt = wt εi (θt − wt εi γAt ). (3) This specification is equivalent to Equation (2) since the idiosyncratic risk of the whole port- folio is σit = wt εi . 2.2 Inference about managerial ability We assume that neither the fund management company, nor the fund investors, nor the man- ager himself, can observe the managerial ability θt directly. Therefore, the Wit process is not observable either. In other words, when agents in our model observe a high (low) market adjusted NAV return, they cannot be sure whether this is due to good (bad) luck or due to the manager’s ability. All the other terms in Equation (1) are assumed to be observable. The agents have a common normally distributed prior belief about θ0 , with a mean a0 and vari- ance v0 , and they all use the Bayesian rule to update their beliefs as they observe the realized NAV process. For the true ability θt , we consider two polar cases. In one case, which is considered in Section 4, the manager has a constant ability. In another case, which is considered in this and the next section, the true ability θt is assumed to be changing randomly over time and can be 12 Berk and Green (2004) motivate their model in a similar way. 9
  12. 12. described by a driftless Wiener process: dθt = v0 dWθt (4) where t denotes the manager tenure and Wθt denotes a standard Wiener process driving θt , which is assumed to be uncorrelated with the Wiener processes driving the market return and idiosyncratic return, i.e., Wmt and Wit . We think both scenarios capture some features of reality. The constantly-changing ability assumption can be motivated as follows. A manager may improve his investment management skill by learning from his experience, this might lead to an upward trend in θt . However, it is well-recognized that the business environment is changing rapidly over time, implying that old strategies and procedures can be outdated very fast. Sometimes past experience might even be an obstacle to future success. When these two effects offset each other on average, we end up with a driftless process for managerial ability. From a technical point of view, we will see shortly that the assumption of a randomly evolving true ability simplifies our analysis significantly. To characterize the learning process, we substitute Equation (2) into Equation (1), and move all the directly observable terms to the left-hand side and denote the resulting expression by dπt : dNAVt dπt := − [(r + λσm − γAt σ2 − δt − bt )dt + σm dWmt ]. it (5) NAVt Note that the realized dWmt can be directly observed ex post as long as the agents know both the true expected and realized market return, which we assume they do. Using the new notation, we can rewrite Equation (1) as dπt = θt σit dt + σit dWit . (6) The learning in our model consists of updating the beliefs about a manager’s time-varying θt from the observed history of dπ since the start of his tenure. Under our model specifications, it follows directly from nonlinear filtering theory devel- oped by Lipster and Shiryayev (1978) that the conditional distribution of θt is normal at every 10
  13. 13. point of time, with a constant conditional variance v0 . The conditional mean of θt , denoted by at , evolves according to the following differential equation13 : v0 dat = [dπt − at σit dt]. (7) σit The fact that the conditional variance is constant over the manager tenure is due to our specification of the θt process. Since the true managerial ability keeps changing over time, the precision of the agents’ estimate of managerial ability does not increase over manager tenure, although the mean estimate is continuously updated based on the realized dπt process. Under more general specifications of the θt process, the conditional variance will usually be a deterministic function of the manager tenure. Define the innovation process, Zt as the normalized deviation of dπt from its conditional mean, at σit dt: dZt := [dπt − at σit dt]/σit , Z0 = 0. It follows that Zt is a standard Wiener process conditional on the common information set of all agents in the model. Unlike the unobservable Wit process, the Zt process is derived from an observable process and is thus observable. Rewriting the dynamics of NAVt and at in terms of dZt , we have: dNAVt = [r + λσm + (at σit − γAt σ2 ) − δt − bt ]dt + σm dWmt + σit dZt it (8) NAVt dat = v0 dZt . (9) 2.3 Equilibrium fund size We assume that after paying a one-time setup cost, the management company charges a pro- portional fee bt for its services. The operating cost of managing the fund, including the compensation to the fund manager, is assumed to be a fixed fraction s of the total fee in- 13 See theorem 12.1 of Lipster and Shiryayev (1978). For an intuitive explanation of the non-linear filter- ing theory and its application in finance, see Gennotte (1986). Recent financial research using this technique includes Brennan (1998) and Xia (2001). 11
  14. 14. come. Therefore, the instantaneous net profit of the management company is bt (1 − s)At . This specification can be motivated as a linear sharing rule between the fund manager and the management company. In the open-end fund market, investors will allocate their money to funds whose expected rate of abnormal return is higher than the management fee and withdraw money from funds whose expected rate of abnormal return is lower than the management fee. For simplicity, we assume that such fund flows are free of charge, i.e., there is perfect capital mobility. Since there is no asymmetric information in our model, the investors will update their beliefs about the managerial ability exactly as the manager and the management company do, and they will also watch the size and idiosyncratic risk of the fund to judge whether it is worthwhile to invest in it. Due to the diseconomy of scale, the performance of funds attracting more money tends to decrease and the performance of funds experiencing money outflows tends to improve. In equilibrium, the size of every fund will be adjusted to such a level that the expected abnormal return is equal to the management fee:14 at σit − γAt σ2 = bt , it (10) Given an updated belief about managerial ability, the management company and the manager will choose an optimal fee bt and an optimal level of idiosyncratic risk σit to maximize the value of the management company.15 We assume that both bt and σit can be changed without any cost. Since learning of managerial ability is independent of the bt and σit processes, at any point of time, bt and σit will be set at the level such that the instantaneous net fee income is maximized, under the investor’s free movement constraint (Equation 10) and the constraints that bt , σit and At must all be positive. Formally, the manager (or the management 14 Ippolito (1992),Edelen (1999) and Wermers (2000) show that this condition holds approximately in reality. 15 Sincethere is no asymmetric information, it makes no difference whether the management company or the manager sets the fee ratio and the level of idiosyncratic risk. Therefore from now on we will adopt the convention that the fee ratio is set by the management company while the level of risk is determined by the manager. 12
  15. 15. company) solves the following maximization problem, maxbt ,σit bt (1 − s)At s.t. at σit − γAt σ2 = bt it bt > 0, σit > 0, At > 0. Solving for At from the free movement constraint and substituting it into the objective function, i.e., the net fee income, we see immediately that the net fee income depends only on the ratio bt σit , but not on bt or σit per se. Since the net fee income is a quadratic function bt bt at of σit , one can easily see that it is maximized at σit = 2. When at ≤ 0, the problem has no solution, since the constraints cannot be met jointly. This means that managers whose mean estimated ability is non-positive will be driven out of business: they get no assets to manage even if they are not fired by the management company. However, when at > 0, the management company has the flexibility of choosing any combination of bt and σit such that at the ratio equals 2. Therefore, our model does not determine a unique optimal value of bt or σit , but it does predict a negative one-to-one correspondence between bt and σit for a given at . This prediction is consistent with empirical findings (see for example, (Golec 1996)). bt Since the profits of the management company are only determined by the ratio σit , we assume for the moment that bt is constant over time and denote it by b. This allows us to focus on the influence of beliefs about managerial ability.16 Clearly, when bt is fixed at b, σit will be adjusted as the beliefs about managerial ability are updated over time, so does the fund size At . More specifically, we have the following proposition: Proposition 1. Under the conditions of perfect capital mobility and constant fee ratio b, the equilibrium size At∗ of an open-end fund and the optimal level of idiosyncratic risk σ∗ are it 16 In reality, management fees are usually very stable. They serve as a contract between investors and fund management company. It makes no sense to change them very frequently. So our assumption is not very unrealistic. 13
  16. 16. given respectively by   at2 if at > 0 At∗ = 4bγ (11)  0/ if at ≤ 0   2b if a > 0 ∗ t σit = at (12)  0 if a ≤ 0 / t / where 0 denotes the empty set. Proof. See the discussion preceding the proposition. Several features of the equilibrium in the open-end fund market are worth mentioning. First, the equilibrium fund size At is a convex function of the conditional mean of man- agerial ability at . It is independent of a fund’s dividend policy δt , its systematic risk σm , the market price of risk λ, and the risk free rate r. Since the management fee is proportional to fund size, this implies that these factors have no influence on the fee income of the manage- ment company. Second, the expected abnormal return to the open-end fund investors is zero (see Equa- tion (10)). This does not imply that managerial ability does not exist, it just means that the investors cannot benefit from it because the provision of capital is competitive. All the eco- nomic rents generated by managerial ability are captured by the manager and the management company. Therefore manager ability is better measured by the fund size and management fee than by the return to fund investors. Third, the level of idiosyncratic risk σit is negatively related to the conditional mean of managerial ability at . This is counter-intuitive because one might expect managers with higher estimated ability will take higher idiosyncratic risk. However it is a natural result of the convex relation between fund size and managerial ability and the negative relation between fund size and idiosyncratic risk. From Equation (11) and (12) we can easily get at σit = , (13) 2γAt which shows that controlling for fund size, idiosyncratic risk is positively correlated with managerial ability while controlling for managerial ability, idiosyncratic risk is negatively 14
  17. 17. correlated with fund size.17 2.4 The dynamics of fund size and fund flows From Proposition 1 we can derive the dynamics of the fund size in a straightforward way. By Ito’s Lemma, we know from Equation (11) that the percentage change of fund size is given by dAt 1 at = (dat )2 + dat (14) At 4bγAt 2bγAt v20 v0 = dt + dZt (15) 4bγAt bγAt v2 v0 = 0 dt + σit dZt , (16) 4bγAt b where the second equality is derived using Equation (9) and Equation (11), and the third equality is derived using Equation (13). Equation (15) shows that the fund has a positive expected growth rate, which is positively related to uncertainty about the managerial ability, and negatively related to the fee ratio and fund size. The positive expected growth rate is due to the convex relation between the equilibrium fund size and the estimated managerial ability. The convexity in this relation implies that the fund size is more responsive to the upward movement of the conditional mean of the managerial ability than to the downward movement. Since the conditional mean follows a driftless Wiener process, this asymmetric response leads to a positive drift of the fund size. Since dZt follows a standard Wiener process, the exposure of the fund growth rate to dZt in Equation (15) , i.e, √v0 , is the volatility of fund growth rate. It increases with v0 and bγAt decreases with b and At . Equation (16) tells us the sensitivity of fund growth to fund performance. To see this, note that σit dZt can be interpreted as a performance measure. Substituting Equation (10) into (8), we get dNAVt = (r + λσm − δt )dt + σm dWmt + σit dZt . (17) NAVt 17 A negative relation between idiosyncratic risk and fund size has been documented by Golec (1996), Cheva- lier and Ellison (1999b) and Boyson (2003) 15
  18. 18. This shows that σit dZt is the market adjusted abnormal NAV return. It equals the unexpected NAV return, adjusted by the unexpected market return: dNAVt σit dZt = ( + δt dt) − (r + λσm )dt − σm dWmt . (18) NAVt Therefore, the coefficient of σit dZt , i.e., v0 b, is the sensitivity of fund growth to fund perfor- mance. It increases with v0 and decreases with b. Another implication of Equation (16) is that a fund’s size is determined by its history of performance. Today’s largest funds are usually yesterday’s best performing funds. However, as Equation (17) shows, since the current fund size has fully reflected the current assessment of managerial ability, no expected abnormal return exists at any fund. Therefore, fund size is uncorrelated with future performance. This is exactly what Ciccotello and Grant (1996) find in their investigation of the size-performance relation. We summarize our results in the following corollary: Corollary 1. The dynamics of the fund size is characterized by the following features: (i). Both the expected rate and the volatility of fund growth are negatively correlated with the management fee ratio and the fund size, and positively correlated with the uncer- tainty about managerial ability. (ii). The fund growth rate is positively related to fund performance; the sensitivity of the fund growth rate to fund performance decreases with the management fee and increases with the uncertainty about managerial ability. (iii). Fund size is positively correlated with past performance but uncorrelated with future performance. In the empirical literature, fund flows are normally defined as the fund’s asset growth rate minus its NAV return. Since the NAV return equals the sum of capital gain and dividend yield, this suggests the following definition for fund flows: dAt dNAVt FLOW := −( + δt dt) (19) At NAVt 16
  19. 19. This definition represents the percentage asset growth rate in excess of the growth that would have occurred if no new funds had flowed in and if all dividend had been reinvested in the fund. Substituting (16) and (17) into (19), we see that the fund flow is given by: v2 v0 FLOW = ( 0 − r − λσm )dt − σm dWmt + ( − 1)σit dZt . (20) 4bγAt b Equation (20) shows that the net fund flows can be decomposed into three parts: the expected inflows, the response to unexpected market return, and the response to the market adjusted abnormal NAV return. As we will discuss later, under reasonable parameterization, v2 both the expected inflows, ( 4bγAt − r − λσm ), and the sensitivity of fund flows to the market 0 adjusted abnormal NAV return,( vb0 − 1), are positive (see footnote 23). However, the relation between fund inflows and market return is always negative. Since σit = √ b , we also have bγAt v2 v0 − b FLOW = ( 0 − r − λσm )dt − σm dWmt + dZt , (21) 4bγAt bγAt This equation shows that the volatility of fund flows is positively related to the fund’s expo- sure to systematic risk and uncertainty about the managerial ability, and negatively correlated with fund size and management fee. We summarize our results in Corollary 2. Corollary 2. The net fund flows into a specific fund are characterized by the following fea- tures: (i). The expected fund inflows decrease with the management fee ratio and fund size, but increase with the uncertainty about managerial ability; (ii). The net fund inflows are positively correlated with the fund’s idiosyncratic return but negatively correlated with the market return; (iii). The sensitivity of net fund inflows to market adjusted abnormal NAV return decreases with the management fee ratio and increases with the uncertainty about managerial 17
  20. 20. ability. (iv). The volatility of net fund inflows decreases with management fee and fund size, and increases with the exposure to systematic risk and the uncertainty about managerial ability. The predictions of Corollary 2 are generally consistent with the empirical literature on mutual fund flows, as discussed in the introduction.18 In particular, our model rationalize a puzzling phenomenon documented in Warther (1995): The negative short-run relation be- tween aggregate fund inflows to the mutual fund industry and past market return. Within our framework this is quite intuitive. A fund will attract net inflows if and only if its inter- nal growth rate, namely its NAV return, is less than its equilibrium growth rate described by Equation (16). While the positive unexpected market return dWmt has a positive impact on a fund’s internal growth rate, it has no influence on its equilibrium growth rate, because it does not affect the beliefs about managerial ability. Therefore, its influence on fund size must be offset by corresponding fund outflows. Things are different when the market adjusted abnor- mal return σit dZt is high. Higher market adjusted return not only increases the asset value but also results in a higher evaluation of managerial ability, and the latter effect generally dominates the former. Therefore fund flows will respond positively to it. A direct implication of our analysis is that funds delivering abnormal returns in a downward market will attract money inflows most dramatically. Testing this prediction is an interesting topic for further empirical examinations of the performance-flow relation. Although we have not yet introduced manager change into our model formally, the im- pact of a given manager change on fund flows can be easily derived from the Proposition 1. By assumption, all new managers have the same unconditional mean ability a0 . There- fore, when a manager with ability at is replaced by a new manager with ability a0 , the fund at2 a2 19 0 size will adjust from 4bγ to 4bγ . Whether this change is positive or negative depends on whether the conditional mean of the departing manager’s ability, at , is higher or lower than the unconditional mean a0 . We summarize this result in Corollary 3: 18 Note that the convexity in the flow-performance relation is reflected in the positive drift of the flow dynamics in our continuous-time model. 19 Note that investors have no incentive to respond earlier in our simplified world without transaction costs, even if the manager change is fully anticipated. 18
  21. 21. Corollary 3. Manager changes in the open-end fund will be accompanied by a discrete a2 −at2 0 change in the fund size. The magnitude of this change is given by 4bγ , where at denotes the outgoing manager’s expected ability. The prediction that fund size will adjust immediately to reflect the new manager’s ability is admittedly quite strong, because in reality, managerial ability is not the only determinant of fund performance. However some anecdotic evidence do show that investors are quite sensitive to fund manager change. For example, it was reported that when William von Mueffling, a star manager running the hedge-fund business of Lazard Asset Management company, resigned in January 2003, Lazards 4 billion hedge-fund business dwindled to less than 1 billion in just a few weeks (The Economist (2003)). Corollary 3 also has important implications on the flow-performance relation. Since man- agers with extremely bad performance will be replaced, thus inducing money inflows, bad performance is not necessarily associated with money outflows. This introduces additional convexity in the flow-performance relation. It is in sharp contrast with the prediction of the Berk and Green (2004) model, which assumes that funds will shut down when its perfor- mance toughes a lower boundary. 3 The optimal manager replacement rule 3.1 The optimal replacement threshold We now examine the optimal manager replacement rule of the management company, taking the relation between the fund size and the beliefs about managerial ability as given. To elim- inate the horizon effect and simplify our analysis, we assume for the moment that manager tenure can in principle be infinitely long. However, a manager can be fired by the manage- ment company or quit at his own will. Whenever the incumbent manager departs, either because he is fired or he leaves voluntarily, the management company has to incur a cost when hiring a new manager. This cost can either be interpreted as a search cost or a cost for training the new manager. The ability of the new manager is again believed to be normally distributed with mean a0 and variance v0 . The cost is assumed to be a fixed proportion k of 19
  22. 22. the initial value of the management company. Since the focus of this paper is the optimal firing decision of the management company, we do not endogenize the job quitting decision of the manager. Instead we assume that the manager’s job quitting can be described by a Poisson process qt with a constant mean arrival rate µ:   0 with probability 1 − µdt dqt = (22)  1 with probability µdt The qt process is assumed to be uncorrelated with either the Wmt or the Zt process. The management company’s problem is to determine an optimal threshold of manager replacement. This is similar to optimizing the exercise decision for the owner of an American option and can be analyzed using dynamic programming techniques. In the setup we consider so far, the Bellman equation can be written down as follows: F(at ) = max{(1 − k)F(a0 ), bt (1 − s)At dt + e−rdt E[F(at ) + dF(at )]}. (23) where the value function F(at ) is the market value of the management company, which is essentially the present value of all future net profits under the optimal replacement rule. On the right-hand side, the first term is the value of the management company when the manager is replaced. We call it the replacement value. The second term is the continuation value, which consists of the immediate net profit in the period from t to t + dt and the expected company value at t + dt, discounted back at the risk free rate. The risk free rate is used because the change in the market value of management company is driven by dZ and dq, both of which are idiosyncratic. Since the owners of the management company bear no systematic risk, the fair expected return is the risk free rate. Note that there is only one state variable, namely at , in the Bellman equation (23). This is due to two special features of the model we consider so far: (1) The conditional variance of the managerial ability is constant so it can be omitted; (2) There is no upper limit on manager tenure, consequently there is no horizon effect. We will consider the case of a tenure-dependent value function in Section 4. 20
  23. 23. Since for at ∈ (0, +∞), which is the relevant range for our model, bt (1 − s)At − r[(1 − k)F(a0 )] is monotonically increasing with at (note that At is a quadratic function of at and F(a0 ) − K is a constant), there will be a single cutoff a, with continuation optimal if at is above a and replacement optimal otherwise.20 This cutoff value is exactly the optimal threshold that we wish to identify. In the continuation region, the Bellman equation is rF(at )dt = bt (1 − s)At dt + E[dF(at )], (24) which is essentially a no-arbitrage condition. Suppose that the function F(a) is continuous and twice-differentiable. Using Ito’s Lemma and taking into account the possibility of job quitting, we have   1 F (a )v2 dt + F (a )v dZ if dq = 0 2 aa t 0 a t 0 dF(at ) = (25)  (1 − k)F(a ) − F(a ) if dq = 1 0 t where Fa (at ) and Faa (at ) denote the first and second order derivative of F(at ) with respect to at . From Equation (25) and (22), we can see that the expected change of F(at ) is 1 E[dF(at )] = Faa (at )v2 dt + µ[(1 − k)F(a0 ) − F(at )]dt. 0 (26) 2 Substituting Equation (11) and (26) into (24), we get the following ordinary differential equation for F(at ): 1 (1 − s)a2 (r + µ)F(at ) = Faa (at )v2 + µ(1 − k)F(a0 ) + 0 . (27) 2 4γ The general solution of this ordinary differential equation is √ √ F(at ) = F0 +C(1)e− 2(r+µ)at /v0 +C(2)e 2(r+µ)at /v0 , (28) 20 See Dixit and Pindyck (1994), pp. 128-130. 21
  24. 24. where µ(1 − k)F(a0 ) (1 − s)[v2 + at2 (r + µ)] 0 F0 = + . r+µ 4γ(r + µ)2 and C(1) and C(2) are two constants that need to be determined by boundary conditions. Note that F0 is the value of the management company with managerial ability a if it had √ no option to fire the manager. The rest of the company value, i.e., C(1)e− 2(r+µ)at /v0 + √ C(2)e 2(r+µ)at /v0 , arises from the option to fire managers with low ability. When a goes to plus infinity, the manager will never be replaced so the option becomes worthless. This asymptotic consideration implies that the undetermined constant associated with the positive square root, namely, C(2), must be zero. Furthermore, the function F(at ) has to satisfy the following boundary conditions: (i). Value matching condition. At the replacement threshold a, we must have F(a) = (1 − k)F(a0 ) (29) (ii). Optimality condition. As a first order condition of optimality, we require 21 Fa (a) ≥ 0, Fa (a)a = 0. (30) (iii). Starting value F(a0 ). The starting value F(a0 ) should satisfy the general solution (28). 21 This is a generalized version of the smooth-pasting condition frequently used in option pricing literature, i.e., Fa (a) = 0. We use the Kuhn-Tucker optimality condition instead of the standard smooth-pasting condition because a can- not be negative. When the replacement cost is very high and uncertainty about the managerial ability is also very high, it might be optimal for the management company not to replace the manager even if the conditional mean at is zero, because there is still some chance that he will turn out to be or become a good manager later. However, since the fund size would be zero when at = 0, the manager has to be replaced as long as at = 0, otherwise the fund will cease to exist. In reality, funds may face some additional constraints, for example, the minimum size or maximum risk con- straint, which may potentially result in non-optimal manager replacement. However, these types of constraints will never be binding in our model, since funds can always satisfy such constraints by lowering their fee ratio. Note, however, that although fee adjustment can help to get round those constraints, it cannot be a substitute for the manager replacement in our model, since it has no influence on the net profit of the management company. 22
  25. 25. Since C(2) = 0, we have µ(1 − k)F(a0 ) (1 − s)[v2 + a2 (r + µ)] √ 0 0 − 2(r+µ)a0 /v0 F(a0 ) = + +C(1)e (31) r+µ 4γ(r + µ)2 These conditions allow us to prove the following proposition: Proposition 2. The optimal manager replacement threshold a and the initial value of the management company F(a0 ), are the solution to the following system of equations22 : v0 4rγ(1 − k)F(a0 ) v2 0 a = Max [ 0, − + − ], (32) 2(r + µ) 1−s 2(r + µ)  √  (1 − s)[v2 + a2 (r + µ) + 2(r + µ)v0 ae− 2(r+µ)(a0 −a)/v0 ]    0 0 if a > 0,   4γ(r + µ)(r + µk)  F(a0 ) = √ (33)   (1 − s)[v2 + a2 (r + µ) − v2 e− 2(r+µ)a0 /v0 ]     0 0 0 √ if a = 0.  4γ(r + µ)[(r + µk) − r(1 − k)e− 2(r+µ)a0 /v0 ] Proof. See Appendix A. 3.2 The distribution of manager tenure Given the optimal manager replacement threshold and the job quitting process, we can derive the following proposition about the manager tenure. Proposition 3. Under the optimal replacement threshold a and the constant job quitting density µ, the distribution of the manager’s tenure has the following properties: (i). The expected tenure of the manager is given by √ 1 − 2µ v0 (a0 −a) T = [1 − e ]; (34) µ 22 Generally, there are two roots for this equation system, but only one of them makes economic sense, because the larger root of a is bigger than a0 . This root can be excluded because it implies that no manager will be employed at all. 23
  26. 26. (ii). The cumulative distribution function of the manager’s tenure is given by a0 − a P(t) = 1 − e−µt [2Φ( √ ) − 1]; (35) v0 t Proof. See Appendix B. We can also derive the conditional density of manager departure and the expected man- agerial ability given the manager’s tenure. These results are summarized in Proposition 4: Proposition 4. Under the optimal replacement threshold a and the constant job quitting density µ, we have: (i). For a manager who has survived until t, the probability density of departure is given by (a0 −a)2 − 2v2 t (a0 − a)e 0 f (t) = µ + √ . (36) 2πv0t 2 [2Φ( a0 −a ) − 1] 3 √ v t 0 (ii). For a manager who has survived until t, the expected ability is given by a0 − 2a[1 − Φ( a0 −a )] v t √ E(at |t) = 0 . (37) 2Φ( a0 −a ) − 1 √ v0 t Proof. See Appendix C 3.3 Results and Comparative statics Table 1 summarizes the base case parameter values. Suppose that the idiosyncratic risk σit is fixed at 0.10, then under this set of parameter values, a newly hired manager can generate on average 2 percent abnormal rate of return for the first dollar he manages. However, the expected rate of abnormal return goes to zero if the fund size increases to 200 million dollars. A manager whose ability is one standard deviation above the mean can generate on average 24
  27. 27. 5.5 percent abnormal rate of return for the first dollar he manages. He can be expected to outperform the market as long as his fund size is less than 550 million dollars.23 Panel A of Table 2 summarizes the initial value of the management company, the replace- ment threshold, and the expected tenure of the base case under the current model specifica- tion, i.e., the true ability θt follows a driftless Wiener process and manager tenure can be infinitely long. Note that the replacement threshold, a = 0.0732, is significantly below the expected ability of a new manager, a0 = 0.20. Since the downward movement of at before it hits the replacement threshold is associated with bad performance and decrease in fund in- flows, our model is consistent with the empirical findings of Khorana (1996), and (Chevalier and Ellison 1999a), who document that managerial turnover is usually preceded by bad per- formance and a lower asset growth rate. Figure 1 plots the cumulative distribution function of manager tenure, P(t); the condi- tional density of manager departure as a function of tenure f (t); and the expected ability of a manager with tenure t, E(at |t). The P(t) curve shows that about 70% of the managers will have a tenure less than 5 years. The conditional density of manager departure, f (t), increases rapidly when the manager starts his tenure, and then keeps going down. The non- monotonicity of the conditional departure density reflects the effect of learning and firing. Without learning and firing, the conditional departure density will always be constant due to the assumption of constant quitting density. Since it takes some time for the manage- ment company to learn about the manager’s ability, the initial density of firing is low. But it goes up very quickly because many new managers may be not good enough to survive. As more and more incompetent managers are fired over time, managers who survive longer will generally have a higher ability, as shown in the E(at |t) curve. Therefore, consistent with the empirical evidence discussed in the introduction, they tend to manage larger funds, have lower probability of being fired, and their departures are more likely due to non-performance reasons. As to the comparative static analysis, we first state some analytical results, which are 23 Under this base case parameter set, consider a fund with a median size of 100 million dollars, and a typ- ical fee ratio of 1.5% (taking into account the front- and back-load fee), as well as a typical expected an- nual NAV return of 14%. According to Equation (20), the expected net fund inflows of this fund will be 0.122 10% (= 4∗0.015∗10−8 ∗108 − 0.14), and the sensitivity of fund flows to market adjusted abnormal return will be 0.12 7(= 0.015 − 1). Both terms are strongly positive. 25
  28. 28. derived from Proposition 2 and summarized in Proposition 5: Proposition 5. When a > 0,24 the optimal replacement threshold a, the initial value of the management company F(a0 ), and the expected manager tenure T , have the following prop- erties: ∂a ∂a ∂a ∂a >0, <0, = =0; (38) ∂a0 ∂k ∂γ ∂s ∂F(a0 ) ∂F(a0 ) ∂F(a0 ) ∂F(a0 ) >0, <0, <0, < 0; (39) ∂a0 ∂k ∂γ ∂s ∂T ∂T ∂T >0, = =0. (40) ∂k ∂γ ∂s Proof. See Appendix D. All the results stated in Proposition 5 are very intuitive and self-explanatory. In particular, the optimal replacement threshold a is independent of the diseconomy of scale parameter γ and the variable cost s. This is because both the continuation value and the replacement value are proportional to these two parameters, so the replacement decision is not affected by them. The influence of v0 and µ are more difficult to sign analytically. Therefore, we resort to numerical methods. We examine the influence of one parameter at a time by fixing the value of all other parameters at the base case level. We then plot F(a0 ), a, T as a function of the parameter under consideration. The parameters under consideration include v0 , µ as well as a0 and k, although analytical results are available for the last two parameters. Figure 2 plots the initial value of the fund management company, F(a0 ), as a function of a0 , k, v0 , and µ respectively. Consistent with the analytical results, F(a0 ) increases with a0 and decreases with k. The figure also shows that the relation between F(a0 ) and µ is negative, this is so because high frequency of manager quitting not only results in high replacement cost, but also means the loss of good managers. The relation between F(a0 ) and v0 is more subtle. On the one hand, higher uncertainty about managerial ability makes the replacement decision more difficult to make; on the other hand,the management company essentially holds 24 We skip the less-realistic case where a = 0 for brevity, although many similar results can be derived for that case as well. 26
  29. 29. an option to fire the manager, and the value of this option increases with the heterogeneity and volatility of the managerial ability. The positive relation plotted in Figure 2 shows that the latter effect is dominating. Figure 3 plots the optimal replacement threshold, a, as a function of a0 , k, v0 , and µ. As expected, a increases with the a0 and decreases with k. As a0 increases, a increases almost linearly, but the slope is slightly less than one. This is because higher a0 is associated with higher value of the management company, which in turn implies higher replacement cost since the replacement cost is proportional to the company value. The figure also shows that a is negatively related to v0 . There are several reasons why this is the case: When v0 is high, (1) the company is less sure whether a manager is good or not; (2) there is a high probability that a bad manager can become a good manager later; (3) the replacement cost is higher because the company value increases with v0 . Another message conveyed by the figure is that a is relatively insensitive to the change of the manager quitting density µ. The reason is as follows. Higher frequency of manager quitting reduces the value of a management company, thus lowering the replacement cost; however, it also lowers the incentive to fire the incumbent manager, because even if the new manager is very good, his high quitting probability makes him less valuable to the company. These two effects tend to offset each other, leaving the replacement threshold largely unchanged. Figure 4 plots the expected tenure T as a function a0 , v0 , k, and µ respectively. It shows that T increases significantly with k and decreases with µ. It also shows that T increases slightly with a0 . This should be expected because as we have seen, the increase of a0 does not lead to a one-to-one increase in a. The slightly negative relation between v0 and T suggests that although the management company tends to adopt a less stringent threshold when the uncertainty is higher, the higher volatility of the mean ability process still leads to higher probability of manager firing. 27
  30. 30. 4 Constant managerial ability In this section we consider the case where the true managerial ability remains constant. In other words, we replace Equation (4) with: dθt = 0. (41) An immediate consequence of this change in assumption is that both the mean and vari- ance of the conditional beliefs about the managerial ability will change over time. More specifically, according to Theorem 12.1 in Liptser and Shiryayev (1978), the conditional dis- tribution of θ is normal at every point of time, and the conditional mean at and variance vt follow the differential equations: vt dat = [dπt − at dt] = vt dZt , (42) σi dvt = −vt2 dt. (43) The conditional variance is a deterministic function of the manager tenure, and can be solved explicitly as: v0 vt = . (44) v0 t + 1 It can be easily seen that vt shrinks over time. It goes to zero as t goes to infinity. This implies that all agents in this model will eventually learn the true value of θ if the manager’s tenure is infinitely long. 25 In a world of no transaction cost, the manager’s portfolio choice and the equilibrium fund size are independent of the conditional variance vt and the manager tenure. As a re- sult, proposition 1 continues to hold under our new specification. However, the dynamics of the equilibrium fund size and fund flows do depend on the learning process. More specif- ically, under the new assumption, we have to replace the constant conditional variance v0 in Equations (15),(16), (20), and (21) by the time-varying conditional variance vt . Since vt 25 However,the decrease of vt is slow, if v0 = 0.12 as in our base case parameterization, then after 10 years, vt will be 0.0545, which still implies a strongly positive flow-performance relation. 28
  31. 31. decreases over the tenure, our model predicts that the fund growth and fund inflows will tend to slow down, and become less volatile, less responsive to past performance as the tenure of the incumbent manager increases over time. This version of the model does capture some important empirical features that are not captured by the model with constant conditional variance. For example, both Chevalier and Ellison (1997) and Boudoukh, Richardson, Stanton, and Whitelaw (2003) find that fund flows are more sensitive to the past performance for young funds than for old funds. Boudoukh, Richardson, Stanton, and Whitelaw (2003) also find that young funds have higher expected percentage inflows than the old funds. If we interpret young funds as funds managed by man- agers with shorter tenure, then these findings are consistent with our model with decreasing conditional variance. The determination of the optimal manager replacement rule is now more complicated, since both conditional mean and variance are varying over time, i.e., we have two state vari- ables. Since the conditional variance is a deterministic function of tenure, the Bellman equa- tion for the value of the management company can be written as F(at ,t) = max{(1 − k)F(a0 , 0), bt (1 − s)At dt + e−rdt E[F(at ,t) + dF(at ,t)]}. (45) Since for at ∈ (0, +∞), b(1 − s)At − r(1 − k)F(a0 , 0) is monotonically increasing with at at any t, there exists a tenure-dependent threshold a(t), with continuation optimal when at > a(t) and replacement optimal when at < a(t). In the continuation region, we must have rF(at ,t)dt = bt (1 − s)At dt + E[dF(at ,t)]. (46) This leads to the following partial differential equation: 1 (r + µ)F(at ,t) = bt (1 − s)At + Ft (at ,t) + Faa (at ,t)vt2 + µ(1 − k)F(a0 , 0), (47) 2 where Ft denotes the partial derivative of F with respect to t, Faa denotes the second order partial derivative of F with respective to a. 29
  32. 32. Substituting Equation (11) and (44) in (47), we get (1 − s)at2 v20 (r + µ)F(at ,t) = + Ft (at ,t) + Faa (at ,t) + µ(1 − k)F(a0 , 0), (48) 4γ 2(v0t + 1)2 The partial differential equation (47) has to satisfy the following boundary conditions: (i). Value matching condition. At the replacement boundary, we must have F(a(t),t) = (1 − k)F(a0 , 0) (49) (ii). Optimality condition. At the replacement boundary, we should also have Fa (a(t),t) ≥ 0, Fa (a(t),t)a(t) = 0 (50) Similarly, we can get the partial differential equation for the expected tenure T (at ,t): v20 µT (at ,t) = 1 + Tt (at ,t) + Taa (at ,t) , (51) 2(v0t + 1)2 with the boundary conditions T (a(Tmax ), Tmax ) = 0 and T (a(t),t) = 0. Unfortunately, the partial differential equations (47) and (51) have no closed-form solu- tion. We need to resort to numerical procedures. We perform the valuation on a binomial tree. Since the state variable at has a nonconstant volatility, we use the approach developed by Nelson and Ramaswamy (1990) to construct a recombining tree. We set the maximum tenure that a single manager can have to be 30 years, after which he has to be replaced by a new manager. We find the value of the management company and the replacement threshold recursively. Using the same parameter values as in Table 1, we get the initial value of the management company and the expected tenure as reported in panel B of Table 2. We see that the value of the management company is substantially lower than in the case of randomly evolving managerial ability. This is because the value of the option to replace the manager is lower due to decreasing volatility of the conditional mean. The expected manager tenure is also shorter. 30
  33. 33. Figure 5 plots the tenure-dependent manager replacement threshold. The threshold starts at approximately 0.08 and keeps going up with manager tenure. It exceeds a0 , the uncondi- tional mean of the managerial ability, after about 6 years. This could never be the case in the specification with constantly changing ability, because in that case, the incumbent manager is always better than the new manager as long as at > a0 . However, in the current version of the model, the management company is not satisfied with a manager with average ability. It will fire a manager when it becomes increasing likely that he is just an average manager. Since the replacement threshold is increasing with tenure, this version of the model predicts a stronger size-tenure relation than our previous version. Figure 6 plots the cumulative distribution of manager tenure. Due to the higher replace- ment threshold, the fraction of managers whose tenure is shorter than 5 years is higher than in the case of continuously changing ability. 5 Conclusion This paper has developed a continuous-time model in which the portfolio manager’s ability is not known precisely. All parties involved, including the manager, learn about the manager’s ability from past portfolio returns. In response, investors can move capital into or out of a mu- tual fund, portfolio managers can alter the risk of the portfolio and the management company can replace incumbent managers with new managers. Thus, the model formalizes simultane- ously the external governance of the product market and the internal governance mechanism in a fully-dynamic framework, thereby generating a rich set of empirical predictions on both mutual fund flows and manager turnover. The model shows that product market forces introduce strong incentives to monitor and replace managers of open-end funds. At the same time, this internal governance also influ- ences how investors respond to past performance. Our analysis rationalizes many empirically documented findings (e.g., convex flow-performance relation; negative relation between fund risk and fund size; positive relation between fund risk and management fees; negative relation between fund inflows and market return; positive relation between manager tenure and fund size, etc.) At the same time, our analysis also generates new empirical hypotheses. For ex- 31
  34. 34. ample, our model predicts that fund inflows respond most positively to good performance in downward markets; that the convexity of the flow-performance relation and the sensitivity of flows to performance decrease with manager tenure; and that the probability of management replacement first increases and then decreases with manager tenure. The framework derived in this paper can be extended in a number of ways. In particular, we have assumed that capital can be freely moved in an out of the fund. In practice this is associated with costs. In the limit, capital cannot be withdrawn or injected in a fund at all. This would be true for the case of closed-end funds. Applying this analysis to the case of closed-end funds and analyzing the resulting discounts or premia is our current subject of research. 6 Appendix A. Proof of Proposition 2 Proof. Since C(2) must be zero by the usual asymptotic argument, from Equation (28) and the value-matching condition (29) we have µ(1 − k)F(a0 ) (1 − s)[v2 + a2 (r + µ)] √ 0 − 2(r+µ)a/v0 + +C(1)e = (1 − k)F(a0 ). (52) r+µ 4γ(r + µ)2 If a > 0, then by the optimality condition (30) we will have Fa (a) = 0, which implies (1 − s)a − 2(r + µ) −√2(r+µ)a/v0 +C(1) e = 0. (53) 2γ(r + µ) v0 Therefore we have (1 − s)v0 a √2(r+µ)a/v0 C(1) = 3 e . (54) (2(r + µ)) 2 γ Substituting Equation (54) into (52) and collecting terms, we get v0 4rγ(1 − k)F(a0 ) v2 0 a=− + − . 2(r + µ) 1−s 2(r + µ) 32
  35. 35. Substituting Equation (54) into (31), we get √ (1 − s)[v2 + a2 (r + µ) + 2(r + µ)v0 ae− 0 0 2(r+µ)(a0 −a)/v0 ] F(a0 ) = . 4γ(r + µ)(r + µk) If a = 0, then by Equation (52) we have r(1 − k)F(a0 ) (1 − s)v20 C(1) = − r+µ 4γ(r + µ)2 Substituting this equation into (31) and noting that a = 0, we have √ (1 − s)[v2 + a2 (r + µ) − v2 e− 0 0 0 2(r+µ)a0 /v0 ] F(a0 ) = √ . 4γ(r + µ)[(r + µk) − r(1 − k)e− 2(r+µ)a0 /v0 ] This completes our proof of Proposition 2. B. Proof of Proposition 3 Proof. We first derive the unconditional mean of the manager’s tenure. Since the only state variable in our model is the conditional mean of managerial ability at , we can write the expected tenure T as a function of at . Starting from any at greater than a, the expected remaining tenure T (at ) must satisfy the following Bellman Equation: T (at ) = dt + E[T (at ) + dT (at )]. (55) Since with probability µdt the manager will quit at t + dt, it follows that T (at ) = dt + (µdt ∗ 0) + (1 − µdt) ∗ [T (at ) + E(dT (at )|dqt = 0)]. (56) Using Ito’s Lemma and Equation (9), we have 1 1 µT (at ) = 1 + E[dT (at )|dqt = 0] = 1 + Taa (at )v2 . 0 (57) dt 2 One can easily see that the differential equation for T (at ) is the same as that for F(at ) 33
  36. 36. with only two adaptations: the flow term is 1 (because the flow is the time itself) and the discount rate is µ. Solving this second order ordinary differential equation, using the usual asymptotic argument and the boundary condition T (a) = 0, we get 1 √ T (at ) = [1 − e− 2µ(at −a)/v0 ] (58) µ Setting at to be a0 yields the expected tenure of a new manager, i.e., Equation (34). To calculate the cumulative distribution function of the manager’s tenure, we first note that for the manager’s tenure to be longer than t, it must be true that: (1) he has not quit before time t; (2) he has not been fired, i.e., the conditional mean of his ability has not hit the replacement threshold a before time t. Since the job quitting follows a Poisson process, the probability that the manager did not quit before time t is simply e−µt . And since the conditional mean at follows a Wiener Process with volatility v0 , the probability that at stays above a until t is given by a0 − a Pa0 ( in f as ≥ a) = 2Φ( √ )−1 (59) 0<s<t v0 t where Φ(·) denotes the cumulative standard normal probability. Due to the assumption that the quitting process qt is uncorrelated with the conditional mean process at , the joint probability that neither quitting nor firing happens before t is simply the product of two marginal probabilities. Therefore, the probability that the manager survives until t is a0 − a Psurvive (t) = e−µt [2Φ( √ ) − 1], (60) v0 t and the cumulative distribution function of the manager’s tenure is a0 − a P(t) = 1 − Psurvive (t) = 1 − e−µt [2Φ( √ ) − 1], v0 t where P(t) denotes the probability that the manager’s tenure is shorter than t. This completes our proof of Proposition 3. 34
  37. 37. C. Proof of proposition 4 Proof. From the cumulative distribution function (35) we can derive the conditional proba- bility that the manager departs in the time interval [t,t + dt], given that he has survived until t. This conditional probability is given by [P(t + dt) − P(t)]/[1 − P(t)]. Dividing the condi- tional probability by dt, we get the conditional density of manager departure f (t), which is given by Equation (36). For the expected ability of a manager with tenure t, first note that based on the reflec- tion principle of the Wiener process, the unconditional density of mean managerial ability at tenure t, given the optimal replacement threshold a and the constant quitting density µ, is given by  −(at −a0 )2 (a +a −2a)2  e−µt  − t 02  √ [e 2v2t − e  2v0 t ] if at > a,  v 2πt 0 0 g(at ) = (61)     0 if at ≤ a. From the unconditional density, we can calculate the unconditional mean of at , which equals a0 − a e−µt [a0 − 2a(1 − Φ √ )], v0 t Dividing the unconditional mean obtained above by the survival probability given by Equa- tion (60), we get the conditional expectation of at for a manager with tenure t, which is given by Equation (37). This completes our proof of proposition 4. D. Proof of Proposition 5 Proof. Substituting the first equality of Equation (33) into (32), we see immediately that the optimal replacement threshold a is independent of the diseconomy of scale parameter γ and the variable cost s. Using the implicit function theorem, it is not difficult to prove that ∂a ∂a > 0, < 0. ∂a0 ∂k 35
  38. 38. Since by Equation (34), the expected tenure T is negatively related to a, it follows from the above results that ∂T ∂T ∂T > 0, = = 0. ∂k ∂γ ∂s Noting that a is independent of γ and s, we can easily see from Equation (33) that ∂F(a0 ) ∂F(a0 ) < 0, < 0. ∂γ ∂s 4rγ(1−k)F(a0 ) v2 Substituting a = − √ v0 + 1−s − 2(r+µ) into the first equality of Equation 0 2(r+µ) (33), and applying the implicit function theorem, it is not difficult to show that ∂F(a0 ) ∂F(a0 ) > 0, < 0. ∂a0 ∂k This completes our proof of Proposition 5. 36
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  43. 43. Table 1: base case parameters value Notation Definition Base Case Value r risk free rate 0.05 s variable cost as a percentage of fee income 0.5 k percentage replacement cost 0.05 γ measure of diseconomy of scale 10−8 a0 unconditional mean of managerial ability 0.2 v0 unconditional variance of managerial ability 0.12 µ mean arrival rate of job quitting 0.05 Table 2: Base case results Panel A: Results when θt evolves randomly F(a0 ) 4.96 ∗ 107 a 0.0732 T 5.68 (years) Panel B: Results when θt is constant F(a0 ) 3.20 ∗ 107 a tenure-dependent T 4.16 (years) 41
  44. 44. P (t) 0.8 0.6 E(at |t) 0.4 0.2 f (t) 5 10 15 20 t Figure 1. The base case results with constant conditional variance. The figure plots the cumulative distribution of manager tenure P(t), the conditional density of manager departure f (t), and the expected ability of the manager for a given tenure, E(at |t). The parameter values are given in Table 1. 42
  45. 45. F (a0) F (a0) 7 7 8·10 8·10 7 7 7·10 7·10 7 7 6·10 6·10 7 7 5·10 5·10 7 7 4·10 4·10 7 7 3·10 3·10 7 7 2·10 2·10 a0 v0 0.18 0.21 0.24 0.27 0.3 0.1 0.12 0.14 0.16 F (a0) F (a0) 7 7 8·10 8·10 7 7 7·10 7·10 7 7 6·10 6·10 7 7 5·10 5·10 7 7 4·10 4·10 7 7 3·10 3·10 7 7 2·10 2·10 k µ 0.02 0.04 0.06 0.08 0.1 0.02 0.04 0.06 0.08 0.1 Figure 2. Comparative statics: The value of the management company 43

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