The Closed-End Fund Puzzle: Management Fees and Private Information

Transcription

1 The Closed-End Fund Puzzle: Management Fees and Private Information Stephen L. Lenkey September 011 Abstract We present a dynamic partial equilibrium model in a simple economy with a closed-end fund. Our model demonstrates that a combination of management fees and private information can account for several empirically observed characteristics of closed-end funds simultaneously. The model is consistent with a number of time-series and cross-sectional attributes of fund discounts, explains why funds tend to issue at a premium, and provides a rationale as to why investors purchase funds that trade at a premium even though those funds are expected to underperform funds that trade at a large discount. Tepper School of Business, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, PA 1513; slenkey@andrew.cmu.edu. I thank Rick Green and seminar participants at Carnegie Mellon University for helpful comments and suggestions. I bear sole responsibility for all errors contained herein.

2 1 Introduction The Law of One Price is one of the most basic principles in financial economics. Simply put, it states that two portfolios with identical cash flows must have the same price. Yet at first blush, closed-end funds, which are companies that hold a portfolio of financial assets, appear to violate this fundamental rule the shares of a closed-end fund typically trade at a price different from the value of the assets in its portfolio. Like other companies whose shares are publicly traded, a closed-end fund undergoes an initial public offering IPO where it sells a fixed number of shares. In contrast to other types of managed investment vehicles such as mutual funds, however, the shares of a closed-end fund generally are not redeemable. Rather, investors buy and sell shares in a closed-end fund at a price determined by the market, and this price typically does not equal the net asset value NAV of the fund. A fund that trades at a price greater than its NAV is said to trade at a premium while a fund that trades at a price less than its NAV is said to trade at a discount. Lee, Shleifer, and Thaler 1990 identify four prominent time-series features of closed-end fund prices that economists have struggled to explain. First, new funds are issued at a premium but begin to trade at a discount within a year Cherkes, Sagi, and Stanton 009, with most discounts arising between 0 and 100 days following the IPO Weiss 1989 and Peavy Furthermore, new funds often issue at times when seasoned funds are trading at a premium Lee, Shleifer, and Thaler 1991 and Higgins, Howton, and Howton 003. Second, most seasoned funds tend to trade at a substantial discount. Third, there is both time-series and cross-sectional variation in fund discounts, and discounts across funds are positively correlated Pontiff 1997 and Chan, Jain, and Xia 005. Fourth, the price converges to a fund s NAV when the closed-end fund is terminated, either through liquidation Brickley and Schallheim 1985, merger, or reorganization into an open-end mutual fund Brauer The source of divergence between the price of a closed-end fund and its NAV has proven to be elusive. Brickley and Schallheim 1985 show that closed-end fund discounts are indeed real rather than merely a byproduct of accounting. Although a closed-end fund may trade at a price different from its NAV despite a lack of frictions, as demonstrated by Spiegel 1999 in an overlapping generations model with finite-lived agents and capital supply shocks, it appears that a friction 1

3 of some sort is necessary to explain the behavior of discounts. We propose a dynamic partial equilibrium model in which a closed-end fund manager periodically acquires private information regarding the future performance of an underlying asset. The manager exploits her information to earn positive abnormal returns for the fund prior to deducting management fees, but the quantity and quality of her information fluctuate over time. Whether the fund trades at a discount or a premium depends on the value of the manager s information in relation to the fees she collects for managing the fund. The closed-end fund trades at a premium when the informational advantage outweighs the management fees, and the fund trades at a discount when the manager s compensation is greater than the value she adds via her private information. Our model explains several puzzling characteristics of closed-end fund prices, including the primary time-series attributes identified by Lee, Shleifer, and Thaler Additionally, the model explains why funds issue at a premium and provides a rationale as to why investors purchase seasoned funds at a premium even though it is well documented that premium funds tend to underperform those trading at a large discount. Our model also is consistent with other empirical observations reported in the literature, such as the statistical relations between discounts and returns and the excess volatility of fund returns. While numerous frictions have been suggested as the basis for the behavior of closed-end fund prices, we demonstrate that a model combining three fundamental elements can explain most of the salient facts about closed-end funds simultaneously: i an informational advantage for the fund manager; ii management fees; and iii a very limited relaxation of full rationality on the part of investors. This last feature is required simply to overcome a no-trade theorem that would otherwise prevent trade between an uninformed investor and a better-informed manager. Investors in our model have rational expectations in the sense of Muth 1961 and Lucas and Prescott 1971, as they correctly anticipate the distribution of future asset prices and choose utility-maximizing portfolios based on their prior information. The only limit to their rationality is that they are unable to reverse engineer or invert the price function as in Radner 1979 or Grossman and Stiglitz 1980 to infer the manager s private information. This is arguably a weaker assumption than the device traditionally used by theorists to overcome no-trade theorems in a model like ours; namely, noise traders who provide shocks to supply that are completely random and do not respond

4 in any way to expectations about future prices. Although most of our results are qualitatively robust to an alternative setting with a stochastic asset supply and fully rational investors who form a posterior distribution of the manager s private information by inverting the stock price a la Grossman and Stiglitz 1980, we choose to model investors as falling short of full rationality for a number of reasons. First, since introducing noise traders is equivalent to assuming that there is a subset of investors who have no rational basis for trading or at least not one that is explicitly modeled, we find it much more palatable to assume that all investors are mostly rational, even if they are unable to reverse engineer a price function. The lack of full rationality also is reflective of reality, as the behavioral economics literature has long recognized that investors have difficulty updating their beliefs when presented with new information. 1 Our model also produces solutions that are much easier to interpret than the ones obtained in the alternative setting. We later discuss the robustness of our results in greater detail. Though somewhat unconventional, we believe that our assumption about investors inability to invert the price function is not inappropriate in the context of our model. If our objective were to study something like information flows, market efficiency, or disclosure regulations, then our tune would be quite different. In any of those cases, where the indirect revelation of private information is central to the analysis, our assumption that prices are uninformative would be completely inappropriate. However, our goal is to model at a primitive level the advantage that a fund manager has over ordinary investors and demonstrate that this advantage, in combination with management fees, can account for several time-series and cross-sectional characteristics of closed-end fund discounts. The fundamental concept of our model is unaffected if instead we take the more traditional approach and assume that investors learn from prices because the manager s informational advantage still exists in such a setting, albeit to a lesser extent. Granted, the fund manager s capacity to exploit private information is diminished in an economy with partially revealing prices, but the 1 For instance, Bruner and Potter 1964 find that individuals who form initial beliefs based on limited information tend to cling to those beliefs when later presented with better information. Even when individuals update their prior beliefs, Tversky and Kahneman 1974 find that the adjustment process is ordinarily insufficient as the updated beliefs are biased toward the original values. Tversky and Kahneman 1974 also report that individuals are generally unable to eliminate this bias through learning. See, e.g., Rabin 1998 or Barberis and Thaler 003 for a review of the behavioral economics literature. Both versions of our model account for the basic time-series features of discounts, the excess volatility of fund returns, and many statistical relations between discounts and returns. Additionally, both versions explain why funds issue at a premium rather than at NAV. However, in the version of our model with fully rational investors and noise traders, premium funds do not significantly underperform funds with large discounts because most of the manager s private information is revealed in equilibrium. We discuss this issue, as well as a possible remedy, in Section 4. 3

5 opportunity to earn excess returns still exists. We are not the first to incorporate less than fully sophisticated investors into a model of closedend fund pricing. De Long, Shleifer, Summers, and Waldmann 1990 speculate that the existence of irrational noise traders creates additional risk for rational investors with a short investment horizon and results in a lower price for closed-end funds. This theory predicts that new funds will issue at a premium when noise traders are overly optimistic about future performance and that discounts will vary with the fluctuations in noise trader opinion, or investor sentiment. Lee, Shleifer, and Thaler 1991 find empirical support for this hypothesis by conjecturing that the investor sentiment driving closed-end fund discounts also affects stock prices of small firms since individual investors, who are the source of noise-trader risk, are the predominant holders of both types of assets in the U.S. However, Dimson and Minio-Kozerski 1999 note that closed-end funds in the U.K. are predominantly held by institutions but nevertheless tend to trade at a discount. Furthermore, Chan, Jain, and Xia 005 find that noise traders are not a significant contributor to fund discounts. Other studies have produced mixed evidence in support of the investor sentiment hypothesis see, e.g., Chen, Kan, and Miller 1993, Chopra, Lee, Shleifer, and Thaler 1993 and Elton, Gruber, and Busse Our model involves a different type of unsophistication. In contrast to the investor sentiment model, we do not assume the existence of irrational noise traders who trade on correlated but unfounded beliefs. Rather, we assume that all investors are mostly rational as they choose portfolios based on economic fundamentals such as expected return and volatility, and our slight relaxation of full rationality merely enables the manager to exploit and profit from the private information by facilitating trade it does not by itself result in a discount. Furthermore, investors in our model form beliefs about the underlying assets held by a fund, which in turn affect their expectations concerning the future performance of the fund; investors do not form beliefs about funds that are independent of the underlying assets as they do in the investor sentiment model. As mentioned above, the traditional approach to generating trade among asymmetricallyinformed agents is to assume the existence of irrational noise traders. While adding noise to a model can generate trade among rational, asymmetrically-informed agents, a major drawback of this type of approach is that the added noise diminishes tractability, and so further assumptions, such as linear pricing or risk neutrality, are often invoked to solve such models. For instance, Oh 4

6 and Ross 1994 construct an equilibrium model based on an information asymmetry between a fund manager and investor. To obtain a solution, they assume a linear trading rule for the fund manager in addition to stochastic supply. They show that the precision of the manager s private information can impact a fund s discount, but since trading takes place at only a single date in their model, it is unable to explain the time-series properties of closed-end fund discounts. Another method of creating trade is to assume that rational investors and noise traders submit orders to a market maker without observing the equilibrium price as in Kyle In this vein, Arora, Ju, and Ou-Yang 003 utilize liquidity traders to generate trade in a two-period model where the fund manager has an initial informational advantage but is constrained by contractually imposed investment restrictions. They numerically show that the fund can issue at a premium and later trade at a discount, but their model does not explain either the cross-sectional or time-series variation in discounts. In contrast, our model accounts for several time-series and cross-sectional features of discounts. Several other explanations for the behavior of closed-end fund discounts have been proposed with varying degrees of success. For example, taxes may provide a partial explanation for the existence of discounts. Since investors who purchase a closed-end fund with unrealized capital appreciation face a future tax liability, the shares of such a fund should trade at a price lower than an equivalent fund with no unrealized capital appreciation. Malkiel 1977 finds some empirical support for this argument, but he demonstrates that taxes alone cannot quantitatively account for the observed discounts. Kim 1994 argues that tax-timing options also contribute to discounts. On the other hand, Brickley, Manaster, and Schallheim 1991 observe a negative correlation between unrealized capital appreciation and the discount, which is inconsistent with the taxation argument. Furthermore, taxes do not explain why funds issue at a premium or why the price converges to NAV upon termination. Management fees are another potential source of the discount that has been proposed. Ross 00a demonstrates that a closed-end fund will trade at a discount equal to the capitalized management fees if the manager receives a constant percentage of the fund s NAV in perpetuity. This simple model, however, fails to explain why funds issue at a premium or why discounts fluctuate over time. In related work, Ross 00b explores variations of the model and shows that with asymmetrically-informed investors funds may issue at a premium and that dynamic distribution 5

7 policies can result in a fluctuating discount. Nevertheless, he does not simultaneously model both issue premiums and fluctuating discounts. In contrast to Ross 00a and Ross 00b, our model can explain the time-series attributes of closed-end funds without relying on a dynamic distribution policy or information asymmetry among investors at the time of issuance. 3 The empirical evidence regarding the impact of management fees on discounts is mixed. Malkiel 1977 and Barclay, Holderness, and Pontiff 1993 find that management fees are an insignificant contributor to the discount, but Kumar and Noronha 199 and Johnson, Lin, and Song 006 report that discounts are significantly and positively related to fund expenses. A few studies have offered explanations for these conflicting empirical observations. Gemmill and Thomas 00 find that fees might be significant but are highly collinear with other explanatory variables, while Deaves and Krinsky 1994 argue that higher fees increase the probability of an open-ending attempt, which in turn results in lower discounts. Cherkes, Sagi, and Stanton 009 develop an equilibrium model based on the trade-off between management fees and the liquidity benefits provided by a closed-end fund that holds primarily illiquid assets. While their model demonstrates that liquidity concerns can lead to new funds issuing at a premium during times when seasoned funds are trading at a premium and then subsequently falling into discount, their model is unable to explain the behavior of discounts for funds that hold liquid assets. Nevertheless, Malkiel 1977 provides some empirical evidence that funds investing in restricted stocks experience deeper discounts. Somewhat related to the liquidity argument is the notion that investors may be willing to pay a premium for funds which hold stocks that trade in an otherwise inaccessible market, such as foreign countries. Bonser-Neal, Brauer, Neal, and Wheatley 1990 and Chan, Jain, and Xia 005 find that international barriers can affect discounts of funds that hold foreign assets, while Kumar and Noronha 199 find that holding a portfolio of foreign stock does not necessarily impact the discount. Nonetheless, foreign holdings do not explain discount dynamics for funds that hold only domestic assets. Agency costs also have been explored as a potential factor affecting discounts. Barclay, Holderness, and Pontiff 1993 find that funds with concentrated block ownership tend to have larger discounts, which they attribute to managers diverting fund resources for their own private benefit. 3 The information asymmetry in our model arises after the IPO and is between the fund manager and the investor. Although this may seem like a minor distinction, our model illustrates that contemporaneous asymmetric information among investors is not necessary to produce a premium at the IPO or a subsequent discount because only investors, who always have identical information sets, trade shares in the fund. 6

8 However, agency costs do not explain the time-series attributes of fund discounts. Perhaps the most robust prior research of the closed-end fund discount is a model by Berk and Stanton 007 in which there exists a tradeoff between a reduced-form managerial ability and fees. By allowing a manager with high ability to extract the surplus she creates via a pay raise, their model is able to explain most of the time-series characteristics of closed-end funds. While our model is similar in concept to theirs, we develop some important new insights into the closed-end fund puzzle by specifically modeling managerial ability as the ability to acquire private information. To wit, our model explains why funds issue at a premium and rationalizes the underperformance of funds that trade at a premium. We also obtain a closed-form expression for the discount. The remainder of this article is organized as follows. In Section, we outline the basic features of the model and solve for the equilibrium over a short time horizon using symbolic computational methods. The techniques discussed in that section are utilized in Section 3 to solve for the equilibrium over a longer time horizon. We then simulate data and assess the model s ability to account for several empirical observations reported in the literature. We discusses the robustness of our results to an alternative setting with fully rational investors and a stochastic asset supply in Section 4, and Section 5 concludes. Basic Model We begin by describing the framework of the model. Time is discrete and indexed by t {1,, 3, 4}. Trading in the financial market occurs at t = 1,, 3 while consumption occurs at t = 4. The economy consists of three types of financial assets a stock, a sequence of one-period bonds, and a closed-end fund. The stock, which is in unit supply, pays a random amount, Ỹ, at t = 4 but does not pay any dividends prior to that time. The payoff on the stock consists of the sum of three independent and normally distributed random variables, 4 Ỹ X 1 + X + X 3, 1 where X t N µ t, σ t for t = 1,, 3. As discussed in greater detail below, the value of each Xt is initially unknown but is observed as time progresses. The equilibrium price of the stock at time t 4 Throughout this article, a tilde denotes a random variable whose value has yet to be realized or observed. 7

9 is endogenous and denoted by Pt s. A couple of simplifying assumptions are made regarding the bonds. Each one-period bond has a constant interest rate that is normalized to zero; accordingly, a bond costs one unit at time t and pays one unit at t+1. Additionally, the supply of each one-period bond is elastic. These assumptions dramatically improve the tractability and computational efficiency of the model. Although a nonzero interest rate would impact the prices of the stock and closed-end fund, empirical studies have found that neither the short-term interest rate Coles, Suay, and Woodbury 000 nor changes in interest rates Gemmill and Thomas 00 and Lee, Shleifer, and Thaler 1991 significantly affects the discount. The closed-end fund is an endogenous, time-varying portfolio comprised of the stock and bond. This relatively simple setup highlights the effect of asymmetric information on the discount, though in reality closed-end funds typically specialize in a diversified portfolio of either stocks or bonds see, e.g., Dimson and Minio-Kozerski The fund, whose shares are traded in the market, is in unit supply, and the equilibrium price at time t is endogenous and denoted by P f t. At t = 1, the fund undergoes an IPO. The fund is liquidated at t = 4, and its assets are distributed to the fund s shareholders at that time after deducting management fees. Although the potential for early liquidation or open-ending can impact the discount see Brauer 1988, Deaves and Krinsky 1994, Gemmill and Thomas 00, Johnson, Lin, and Song 006, Bradley et al. 010, and Lenkey 011, we assume that the fund will not be liquidated prior to t = 4 with certainty. A single fund manager she and single representative investor he are present in the market. Both actors exhibit preferences, which are common knowledge, characterized by constant absolute risk aversion CARA, where γ i and γ m denote the coefficients of risk aversion for the investor and manager, respectively. Both the investor and fund manager behave like price-takers. At t = 1, the investor receives an exogenous endowment of wealth, W i, and he observes the fund s initial wealth that is designated for investment, W f. At each trading date, the manager chooses the composition of the closed-end fund according to her preferences by allocating the fund s financial resources among the bond and stock while the investor optimally allocates his wealth across the bond, stock, and fund. The fund is prohibited from issuing new shares or repurchasing existing shares. As is typical in practice, the investor is unable to observe the contemporaneous composition of the fund, but he acquires knowledge of the prior period composition as time progresses; that is, at time t the 8

10 investor has knowledge of all fund portfolios through t 1. In some cases, the investor can infer the fund s portfolio at the current date based on the fund s prior portfolios in addition to the other parameters and state variables. The fund manager obtains utility solely from the consumption, c m, of fees, φ, earned from managing the closed-end fund plus any issue premium, ρ. The management contract is exogenous and pays the manager a fixed amount, a, plus a fraction, b, of the return on the fund, 5 φ = a + b S f 3 Ỹ + Bf 3 V 1, where S f t and B f t denote the quantity of stock and number of bonds held by the fund from time t to t + 1 and V t S f t P s t + B f t 3 denotes the fund s time-t NAV, which is equal to the market value of the assets in the fund s portfolio. The initial NAV equals the fund s initial wealth designated for investment: V 1 = W f. The fund s time-t discount, D t, is defined as the difference between the price of the fund and NAV, D t V t P f t, 4 which means that the fund s issue premium is ρ P f 1 V 1. 5 Defining the discount as the absolute difference, as opposed to the more conventional definition of percentage or log difference, between the price of the fund and its NAV results in simpler expressions for the discount. Percentage and log discounts can easily be obtained from D t. 5 This contractual form differs from most compensation contracts in the industry which pay the fund manager a fraction of the total assets under management. While the results of the basic model developed in this section are robust to these more prevalent contractual forms, the two-part contract produces more realistic solutions to the extended model presented in Section 3. In the extended model, the manager is compensated with a sequence of fees over a longer time horizon. If the compensation contract paid the manager a fraction of the total assets under management so that each fee depended on the NAV at a particular date, then portfolio choices would affect not only the contemporaneous fee but also all future fees. Hence, such a contract effectively makes holding stock riskier for the manager than when she is compensated via the two-part contract. To offset the increased risk, the manager would tend to allocate a small amount of the fund s wealth to the stock at early dates and gradually increase the allocation over time. In contrast, the two-part contract leads to stock allocations that are stationary, which is more realistic. 9

11 The investor, meanwhile, receives utility solely from the consumption, c i, of the payoff from his portfolio. Hence, c i = S3Ỹ i + Bi 3 + F 3 S f 3 Ỹ + Bf 3 φ, 6 where S i t denotes the quantity of stock, B i t denotes the number of bonds, and F t denotes the shares of the fund held by the investor from time t to t + 1. Information regarding the final stock payoff, Ỹ, evolves over time. As time progresses, the manager obtains an informational advantage over the investor which she exploits to earn an excess return for the fund. Let I i t and I f t denote the information set at time t for the investor and fund manager, respectively. Initially, the value of each X t is unknown to both the investor and manager: I i 1 = If 1 =. At t =, the manager observes X 1 whereas the investor does not. Since the investor does not infer the manager s private information from the equilibrium price, the information sets are asymmetric: I i = and If = {X 1}. At t = 3, the investor acquires knowledge of X 1 and both actors observe X, so the information sets are once again symmetric: I i 3 = If 3 = {X 1, X }. Finally, all information is available at the terminal date: I i 4 = If 4 = {X 1, X, X 3 }. This information structure enables the study of equilibrium dynamics and, in particular, the impact of an informational advantage on the closed-end fund price. All acquisition of information is costless; consequently, potential moral hazard issues relating to information acquisition do not arise. The sequence of events is as follows. The fund undergoes an IPO at t = 1, and the investor and manager subsequently choose portfolios at market-clearing prices. The investor allocates his wealth among the bond, stock, and fund, while the manager allocates the fund s financial resources among the bond and stock. Because preferences are common knowledge and I1 i = If 1, the investor can infer the fund s portfolio composition from the equilibrium stock price. At t =, the fund discloses its portfolio holdings from the previous date, the manager acquires private information regarding the terminal payoff of the stock, and both the investor and manager rebalance their respective portfolios. The investor cannot infer the precise composition of the fund s current portfolio since he does not observe X 1, although he does form beliefs about a distribution of the fund s portfolio based on the manager s preferences. At t = 3, the fund manager s informational advantage disappears, both actors rebalance their respective portfolios, and the investor can once again infer the fund s portfolio from the equilibrium stock price and the composition of the fund s portfolio from the 10

12 previous date, which is announced prior to trading. Finally, the management fees are paid, the portfolios are liquidated, and consumption occurs at t = 4. We also make a couple of technical assumptions regarding the relative magnitudes of the actors risk aversion coefficients to ensure well-defined and meaningful solutions. The first assumption is that bγ > γ i, where Γ γ i + γ m. The second assumption is that γi γ mσ1 4 + θσ 1 σ + b γmγ σ 4 > 0 [ where θ γ i Γ + γm b Γ + γi bγ 3 ]. Both of these assumptions are entirely reasonable if γ m γ i, which is not unrealistic since in actuality the mass of investors is far larger than that of fund managers and the coefficients of risk aversion are equivalent to the inverses of the actors risk tolerances. The equilibrium is solved recursively with the aid of symbolic computational methods. Section.1 characterizes the equilibrium at t = 3. Those results are then drawn on in Section. to derive the equilibrium at t =, which in turn is relied upon to characterize the equilibrium at t = 1 in Section.3. Some implications of the basic model are discussed in Section.4..1 Equilibrium at t = 3 Since information is symmetric at t = 3, the third period in this equilibrium model is similar to standard single-period equilibrium pricing models with symmetrically-informed investors who have CARA preferences. Consequently, many of the results contained in this section are typical, or slight variations, of standard outcomes found in the literature. Nevertheless, the results of this section are needed to solve for the non-standard equilibrium at prior dates. Equilibrium prices and allocations are derived from the utility-maximizing objectives of the manager and investor. The following proposition characterizes the equilibrium. Proposition 1. At t = 3, there exists a unique equilibrium in which the stock price and allocations are given by P s 3 = X 1 + X + µ 3 γ iγ m Γ σ 3 7 S f 3 = γ i bγ 1 8 S i 3 = bγ γ i bγ

13 and the closed-end fund discount is given by D 3 = a + b V 3 V The remaining portion of this subsection describes the equilibrium derivation, beginning with the fund manager s objective. The fund manager s goal at t = 3 is to maximize her expected utility from consumption of the management fees and issue premium by choosing the composition of the closed-end fund subject to a budget constraint: max S f 3 1 πσ 3 exp [ γ m c m ] exp [ X ] 3 µ 3 d X 3 11 σ 3 subject to c m = φ + ρ 1 B f 3 = Sf P s 3 + B f Sf 3 P s 3 13 and also 1,, and 5. Since, conditional on X 1 and X, the manager s consumption is lognormally distributed, her expected utility can be rewritten in closed form as E 3 u m X 1, X = exp [ γ m a + b [ S f 3 X 1 + X + µ 3 P3 s + S f P 3 s + B f V 1] ] + ρ 1 γ mb S f 3 σ 3, 14 where E t is the expectation operator conditional on information available at time t, after substituting 1,, 1, and 13 into 11 and integrating. The manager s demand function is then derived by differentiating 14 with respect to S f 3 and solving the corresponding first-order condition to obtain S f 3 = X 1 + X + µ 3 P s 3 bγ m σ The investor faces a problem similar to that of the manager. The investor s objective is to maximize his expected utility from consumption of the assets in his portfolio subject to a budget 1

14 constraint, taking into account the portfolio held by the fund: 6 max S i 3, F 3 1 πσ 3 exp [ γ i c i ] exp [ X ] 3 µ 3 d X 3 16 σ 3 subject to B i 3 = S i P s 3 + B i + F P f 3 S i 3P s 3 + F 3 P f 3 17 as well as 1,, 6, 13, and 15. Since the investor s consumption is also conditionally log-normally distributed, his expected utility can be rewritten as E 3 u i X 1, X = exp [ γ i S3 i X 1 + X + µ 3 P3 s + SP i 3 s + B i + F P f 3 [ + F 3 S f 3 X 1 + X + µ 3 P3 s + S f P 3 s + B f 1 b + bv1 P f 3 a] 1 γ ] i S i 3 + F 3 S f 3 1 b σ 3 18 after substituting 1,, 6, 13, and 17 into 16 and integrating. Differentiating 18 with respect to S i 3 and solving the first-order condition provides the investor s stock demand function, S i 3 = X 1 + X + µ 3 P s 3 γ i σ 3 F 3 1 bs f The first term of this equation represents the investor s optimal stock holdings in the absence of a closed-end fund while the second term adjusts for the investor s indirect holdings of the stock through the fund. Imposing the market-clearing conditions, which require that F 3 = 1 and S i 3 + Sf 3 = 1, and solving for the stock price results in 7. Since the stock price at t = 3 is typical of models with CARA preferences, it appears as though the presence of a closed-end fund does not distort the price of the underlying asset provided that information is symmetric. Substituting the stock price, 7, back into the demand functions, 15 and 19, reveals that optimal risk-sharing occurs whereby the fund and investor each hold a positive constant fraction of the stock given by 8 and 9, respectively. Again, this result is characteristic of models with CARA investors who have identical information sets. 6 Recall that the investor can infer the fund s portfolio at t = 3 for a given stock price since he has knowledge of the prior composition of the fund in addition to the other state variables and parameters. 13

15 Finally, the fund price is obtained by differentiating the investor s expected utility, 18, with respect to F 3, substituting F 3 = 1 into the first-order condition, and solving for the price, P f 3 = V 3 a b V 3 V 1. 0 Thus, the fund price is equal to the NAV of the fund minus an adjustment for the management fees. It follows immediately from 0 that the discount, which is given by 10, stems from the management fees when the investor and manager have identical information sets and there is no possibility of a future information asymmetry. The fund will trade at a discount as opposed to a premium whenever a + bv 3 > bv 1 ; that is to say, appreciation of the NAV is a sufficient condition for the fund to trade at a discount. Since the closed-end fund generally trades at a price different from its NAV, it is conceivable that an arbitrage opportunity exists. We show here, however, that the discount does not present an arbitrage opportunity. If the fund is trading at a discount relative to NAV, then a potential arbitrage strategy would entail purchasing shares in the fund and simultaneously taking an offsetting position in a hedging portfolio. Since the fund payoff at t = 4, net of management fees, is S f 3 Ỹ + Bf 3 a b S f 3 Ỹ + Bf 3 V 1, 1 an appropriate hedging portfolio would consist of B f 3 1 b a + bv 1 bonds and S f 3 1 b shares of stock, but because the cost of this hedging portfolio equals P f 3, there is no arbitrage opportunity. In other words, arbitrage does not exist because the discount at t = 3 arises solely from the future management fees, which also reduce the fund payoff.. Equilibrium at t = The equilibrium prices and allocations derived in Section.1 are used to determine the equilibrium prices and allocations at t =. Recall that information is asymmetric at t = as the fund manager observes the value of X 1 but the investor does not. The following proposition characterizes the equilibrium in the presence of asymmetric information. Proposition. At t =, there exists a unique equilibrium in which the stock price and allocations 14

16 are given by P s = µ + µ 3 γ iγ m Γ σ 3 + γ i σ 1 + σ X1 + [ bγ γ i µ 1 bγ i γ m σ 1 + σ ] σ γ i σ1 + bγσ S f = bγ γ i X 1 µ 1 + bγ i γ m σ 1 + σ bγ m γi σ1 + bγσ 3 S i = bγ γ i µ 1 X 1 + bγ m σ bγ m γi σ1 + bγσ 4 and the closed-end fund discount is given by D = b V V 1 + a λ 5 where λ 1 b bγ γ i σ1 γ i 1 b bγ γ i + bγ m σ1 + b γmσ. 6 The derivation of the equilibrium is described in the remaining portion of this subsection. Since the manager s expected utility is independent of the investor s portfolio, the manager s problem is relatively straightforward and is analogous to her problem at t = 3. On the other hand, because the investor does not observe the manager s private information, his situation is more complicated and involves additional uncertainty. At t =, the manager chooses the fund allocation to maximize her expected utility subject to a budget constraint, bearing in mind the future stock price and fund portfolio: max S f 1 πσ [ [ exp γ m a + b S f 3 X 1 + X + µ 3 P 3 s + S f P ] 3 s + B f V 1 ] + ρ 1 γ mb S f 3 σ 3 exp [ ] X µ d X 7 σ subject to B f = Sf 1 P s + B f 1 Sf P s 8 in addition to 3, 7, and 8, where the manager s objective function follows from 14. Substi- 15

17 tuting 3, 7, 8, and 8 into 7, the manager s objective can be rewritten as max S f 1 πσ [ [ exp γ m a + b S f X 1 + X ] + µ 3 γ iγ m Γ σ3 P s + S f 1 P s P1 s + γ i γm σ Γ 3 + ρ ] exp [ ] X µ d X. 9 σ Since her utility conditional on X 1 is log-normally distributed, integrating 9 results in a closedform expression for the manager s expected utility at t =, [ E u m X 1 = exp [ γ m a + b S f X1 + µ + µ 3 γ iγ m Γ σ3 P s + S f ] 1 P s P1 s + γ i γm Γ σ 3 + ρ 1 γ mb S f σ ]. 30 Differentiating 30 with respect to S f and solving the first-order condition provides the manager s demand function, Γ σ 3 S f = X 1 + µ + µ 3 P s γ iγ m bγ m σ. 31 This expression is similar to the manager s demand at t = 3 but contains an extra term to account for future uncertainty. Furthermore, the presence of X 1 in the manager s demand function represents an additional source of risk for the investor. Since the investor does not observe X 1 at t =, he cannot infer the precise composition of the fund s portfolio. Given a price, however, he can infer a distribution of the quantity of stock held by the fund. The investor s problem at t = is to maximize his expected utility subject to a budget constraint, taking into consideration the results from t = 3 and the uncertainty surrounding the current composition of the fund: max S i, F 1 π σ1 σ [ exp γ i S3 i X1 + X + µ 3 P 3 s + S i P 3 s + B i f + F P 3 + F 3 [S f 3 X1 + X + µ 3 P 3 s + S f P 3 s + B f 1 b + bv 1 P ] f 3 a ] 1 γ S3 i + F 3 S f 3 1 b σ 3 exp [ ] X 1 µ 1 exp [ ] X µ d X d X 1 3 σ 1 σ subject to B i = S i 1P s + B i 1 + F 1 P f S i P s + F P f 33 16

18 plus 3, 7, 8, 9, 0, 8, 31, and F 3 = 1, where the investor s objective function follows from 18. Since the fund s stock holdings, 31, do not depend on X, conditional on X 1 the investor s utility is log-normally distributed. Therefore, integration with respect to X is relatively straightforward. Substituting the aforementioned equations except 31 into 3 and integrating with respect to X, the investor s objective can be rewritten as max S i, F 1 πσ 1 [ exp γ i S i + F 1 b S f X1 + µ + µ 3 P s γ iγ m Γ σ3 + B1 i + F B f 1 a + S1P i s + F S f 1 bp 1 s + 1 b P s P f F F 1 + γ iγm σ Γ 3 1 γ i S i + F 1 b S ] f σ exp [ ] X 1 µ 1 d X σ 1 The investor must also consider his uncertainty regarding the fund s portfolio when selecting his own portfolio. Substituting the fund s stock holdings, 31, as well as F 1 = 1 into 34, the investor s objective can again be rewritten as max S i 1 πσ 1 [ exp G i + HX i 1 + J i X ] 1 exp [ ] X 1 µ 1 d X 1, 35 σ 1 where G i G i S, i F, P s, S1, i S f 1, Bi 1, B f 1, P 1 s ; γ i, γ m, µ, µ 3, σ, σ3, a, b H i H i S i, F, P s ; γ i, γ m, µ, µ 3, σ, σ3, b J i J i F ; γ i, γ m, σ, b are functions of the underlying parameters and state variables. 7 Integrating 35 through symbolic computation gives a closed-form expression for the investor s expected utility, 8 [ 1 E u i = exp G i 1 σ1 J i + µ 1H i + σ 1 H i ] + µ 1 J i 1 σ1 J i. 36 Then, differentiating this expression with respect to S i, setting F 1 = 1, and solving the first-order 7 The expressions for G i, H, i and J, i as well as the analogous expressions for the constant terms in 45 and 49, are not reported but are available upon request. 8 e ξx νx dx = π e ν ξ ξ if ξ > 0. The assumption that bγ > γ i ensures that the restriction on ξ is satisfied. 17

19 condition provides the investor s stock demand at t =, S i = bγ m 1 b γ i F µ 1 + µ + µ 3 P s γ iγ m Γ σ3 bγ i γ m σ 1 + σ, 37 which is a function of the stock price and underlying parameters. The stock price, which is given by, is obtained by imposing the market-clearing conditions and solving for price. A cursory examination of reveals that the stock price equals the expected payoff less an adjustment for future risk, plus a term that incorporates the risk associated with the information asymmetry. Then, substituting the price back into the demand functions, 31 and 37, provides the stock allocations for the fund and investor, which are given by 3 and 4. These expressions reduce to the same stock allocations as at t = 3 if there is no uncertainty regarding X 1. Furthermore, the closed-end fund holds a larger amount of stock when the realization of X1 is higher, 9 and a comparison with the t = 3 allocations indicates that, on average, the fund holds a greater quantity of stock and the investor holds less in his personal portfolio at t =. Finally, the closed-end fund price is obtained by substituting 37 into 36, differentiating the resulting expression with respect to F, substituting F = 1 into the first-order condition, and solving for price, P f = V b V V 1 a + λ, 38 where λ is defined in Proposition and represents the expected benefit from the manager s private information, i.e., the expected value of the manager s private information before X 1 is realized. The discount, which is given by 5, follows immediately from 38. The fund will trade at a discount as opposed to a premium whenever b V V 1 > λ a. In contrast to t = 3 where any amount of NAV appreciation leads to a discount, at t = the NAV must appreciate beyond a particular level in order for a discount to emerge. Furthermore, the investor cannot arbitrage the discount by taking a position in the fund along with an offsetting position in a hedging portfolio because he cannot infer the exact composition of the fund. 9 Under the current setup, extreme realizations of X1 may result in either the fund or investor taking a short position in the stock, which may seem somewhat unrealistic given that there are only two actors in the model. Conceptually, however, the fund manager could acquire a small amount of private information about several assets in a portfolio, which she could then exploit to earn an excess return for the fund that is comparable to the current setup without taking an extreme position in one of the underlying assets. 18

20 .3 Equilibrium at t = 1 The results from t =, 3 are utilized in deriving the equilibrium at t = 1. Like at t = 3, information is symmetric at t = 1. Accordingly, many of the results parallel those derived earlier. The following proposition characterizes the equilibrium. Proposition 3. At t = 1, there exists a unique equilibrium in which the stock price and allocations are given by P1 s = µ 1 + µ + µ 3 γ iγ m Γ σ 1 + σ + σ3 39 S f 1 = γ i bγ 1 40 S i 1 = bγ γ i bγ 1 41 and the closed-end fund discount is given by D 1 = a λ. 4 Although the results are similar to those obtained at t = 3, the derivation here is much more complicated due to the presence of a future information asymmetry. We describe the derivation in the remaining portion of this subsection, starting with the fund manager s objective. Taking into account the t = fund portfolio and stock price, the manager s goal at t = 1 is to maximize her expected utility subject to a budget constraint: max S f 1 1 πσ 1 [ [ exp γ m a + b Sf X1 + µ + µ 3 γ iγ m Γ σ3 P ] s + S f 1 P s P1 s ] + γ i γm σ Γ 3 + ρ 1 γ mb Sf σ exp [ ] X 1 µ 1 d X 1 43 σ 1 subject to B f 1 = W f S f 1 P s 1 44 as well as and 3, where her objective function follows from 30. The objective can be 19

21 rewritten in closed form as 1 µ 1 H f E 1 u m = exp G f 1 σ1 J f σ 1 H f 1 + µ 1 J f 1 1 σ 1 1 J 45 f 1 after substituting and 3 into 43 and integrating, where G f 1 Gf 1 S f 1, P s 1 ; γ i, γ m, µ 1, µ, µ 3, σ1, σ, σ3, a, b, ρ H f 1 Hf 1 S f 1 ; γ i, γ m, µ 1, σ1, σ, b J f 1 J f 1 γi, γ m, σ 1, σ, b. Differentiating 45 with respect to S f 1 and solving the first-order condition gives the manager s demand function at t = 1, S f 1 = µ 1 + µ + µ 3 P s 1 γ iγ m Γ σ 3 γ i σ 1 + b Γ σ bγ i γ mσ 1 σ 1 + σ bγσ γ i σ 1, 46 which is equal to the expected payoff on the stock minus the price and a risk adjustment, scaled by a term that incorporates the future information asymmetry. Turning to the investor, his problem at t = 1 is to maximize his expected utility subject to a budget constraint while considering the results from t = as well as the closed-end fund s current composition: 10 max S i 1, F 1 1 πσ 1 [ exp γ i Si + F 1 b S f X1 + µ + µ 3 P s γ iγ m Γ σ3 + B1 i + F B f 1 a + S1 i P s + F S f 1 bp1 s + 1 b P s P f F F 1 + γ iγm σ Γ 3 1 γ i Si + F 1 b S ] f σ exp [ ] X 1 µ 1 d X σ 1 subject to B i 1 = W i S i 1P s 1 F 1 P f 1 48 in addition to, 3, 4, 44, 46, and F = 1. Substituting these constraints into 47 and 10 As at t = 3, the investor can infer the fund s current portfolio composition. 0

22 integrating provides a closed-form expression for the investor s expected utility at t = 1, [ 1 E 1 u i = exp G i 1 σ1 J 1 i 1 + µ 1H1 i + σ 1 H i ] 1 + µ 1 J1 i 1 σ1 J 1 i, 49 where G i 1 G i 1 S1, i F 1, P1 s, P f 1, W i, W f ; γ i, γ m, µ 1, µ, µ 3, σ1, σ, σ3, a, b H1 i H1 i S i 1, F 1 ; γ i, γ m, µ 1, σ1, σ, b J i 1 J i 1 γi, γ m, σ 1, σ, b. The investor s stock demand function is found by differentiating 49 with respect to S1 i and solving the first-order condition to obtain S1 i = µ 1 + µ + µ 3 P1 s γ iγ m Γ σ3 γi γ mσ1 4 + θσ 1 σ + bγmγ σ 4 γi 3γ mσ1 σ 1 + σ Γσ 1 b γi [bγ m 1 b γ i ] σ1 b γmσ γi γ mσ1 σ 1 + σ F 1 1 b Sf The stock price at t = 1, which is found by enforcing the market-clearing conditions, is given by 39. This price is analogous to the t = 3 stock price and is typical of models with CARA preferences. Thus, again it appears as though the existence of a closed-end fund does not distort the price of the underlying asset if information is symmetric, even in the presence of a future information asymmetry. Substituting the stock price, 39, back into the demand functions, 46 and 50, gives the optimal stock holdings for the manager and investor, which are given by 40 and 41 and are the same constant fractions as at t = 3. Finally, the price of the closed-end fund is found by differentiating 49 with respect to F 1, substituting 8, 39, 40, 41, and F 1 = 1 into the derivative, and solving for price, P f 1 = V 1 a + λ. 51 Thus, the fund price is equal to NAV plus an adjustment for the management fees and the manager s future informational advantage. The fund s discount at t = 1 is given by 4. Whether the fund 1

23 trades at a premium or discount depends on the volatility of the stock in the first two periods, the risk preferences of the investor and manager, and the parameters of the management contract. The discount is independent of the stock s expected return. Like at t = 3, it is conceivable that an arbitrage opportunity exists because the fund price generally does not equal NAV. Though as we now show, the discount does not present an arbitrage opportunity. If the fund is issued at a premium, as is typical in practice, then an arbitrage strategy would involve taking a short position in the fund along with an offsetting position in a hedging portfolio. It is easy to verify that P f can be replicated by forming a portfolio consisting of 1 b Sf 1 shares of stock and bs f 1 P 1 s + Bf 1 a + λ bonds at t = 1. Since the cost of this portfolio, Sf 1 P 1 s + B f 1 a + λ, equals P f 1, however, there is no arbitrage..4 Implications of the Basic Model Though relatively simple, the basic model described in the previous subsections can account for some of the puzzling behavior exhibited by closed-end funds. First, the basic model shows that a combination of private information and management fees can explain the empirical time-series attributes of discounts outlined by Lee, Shleifer, and Thaler A simple comparison of the discounts reveals that the size of the closed-end fund discount fluctuates over time. In particular, appreciation of the NAV leads to an increase in the discount, which is consistent with the empirical findings of Malkiel 1977 and Pontiff Furthermore, the fund will issue at a premium if λ > a, and a discount will necessarily emerge by t = 3 provided that V 3 > V 1, although a discount could arise at t = if the NAV appreciation is large enough. Additionally, the principle of no arbitrage along with 10 suggest that the discount will disappear once the management fees are paid just prior to liquidation at t = While it is apparent from 4 that the discount at t = 1 is a function of the management fees and the manager s future informational advantage, the model thus far has made no assumptions regarding the origin of the manager s private information. Suppose that the quantity of information obtained by the fund manager depends on her ability, α 0, 1, to acquire information, with larger values of α representing a greater ability. Assume that X 1 and X are components of another variable Z X 1 + X, where Z N µ z, σ z, and that the respective distributions of X1 and 11 These qualitative results are robust to percentage and log discounts.

24 Table I: Parameter Values. Variable Symbol Value Investor s coefficient of risk aversion γ i 1 Manager s coefficient of risk aversion γ m 40 Fixed component of management fee a Investor s initial wealth W i 1 Variance of Z σz Variance of X3 σ X are X 1 N αµ z, ασz and X N 1 α µ z, 1 α σz. Under this specification, the distribution of the stock payoff, Ỹ, is independent of ability, yet the private information acquired by a manager with high ability is superior to the information obtained by a manager with low ability. Substituting σ 1 = ασ z and σ = 1 α σ z into 6, the expression for λ can be rewritten in terms of ability as λ α 1 b bγ γ i α 1 b γ i bγ γ i + bγ m + 1 α b γm. 5 Notice that the stock volatility is replaced by managerial ability, so that the size of the closed-end fund discount depends on the ability of the manager, the risk preferences of the investor and manager, and the parameters of the management contract. The discount does not depend on either the expected return or volatility of the stock. Rather, the discount is a function of managerial ability and future management fees. Given an ability level, the contract parameters a and b can be chosen so that the fund issues at a premium. There is no inherent reason, though, why a closed-end fund should issue at a premium rather than at NAV. 1 After all, the investor will receive the equilibrium rate of return over the life of the fund regardless of whether it issues at a premium or at NAV. However, examining the relationships between the issue premium, ρ which is equal to D 1, and the actors ex ante expected utilities reveals that for any level of ability the issue premium is positively related to the manager s expected utility and negatively related to the investor s expected utility. The relationships between the issue premium and expected utilities are best understood graphically. In order to plot these relationships, we assume numerical values for various parameters in 1 Pursuant to the Investment Company Act of 1940, a closed-end fund may sell its common stock at a price less than NAV only in certain limited circumstances. 3