• No results found

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

N/A
N/A
Protected

Academic year: 2021

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

Copied!
46
0
0

Loading.... (view fulltext now)

Full text

(1)

The Closed-End Fund Puzzle:

Management Fees and Private Information

Stephen L. Lenkey∗

September 2011

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 15213; slenkey@andrew.cmu.edu. I thank Rick Green and seminar participants at Carnegie Mellon University for

(2)

help-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 (2009)), with most discounts arising between 20 and 100 days following the IPO (Weiss (1989) and Peavy (1990)). 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 (2003)). 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 (2005)). Fourth, the price converges to a fund’s NAV when the closed-end fund is termi-nated, either through liquidation (Brickley and Schallheim (1985)), merger, or reorganization into an open-end mutual fund (Brauer (1984)).

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 overlap-ping generations model with finite-lived agents and capital supply shocks, it appears that a friction

(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 period-ically 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 pre-mium 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 (1990). Additionally, the model explains why funds issue at a premium and provides a rationale as to why investors pur-chase 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 re-turns and the excess volatility of fund rere-turns.

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 in-formation.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.2 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 inap-propriate. 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 dis-counts. 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 advan-tage 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 1For 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 (2003) for a review of the behavioral economics literature.

2

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.

(5)

opportunity to earn excess returns still exists.

We are not the first to incorporate less than fully sophisticated investors into a model of closed-end 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 hori-zon 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 pre-dominantly held by institutions but nevertheless tend to trade at a discount. Furthermore, Chan, Jain, and Xia (2005) 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 (1998)).

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 un-founded 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 asymmetrically-informed 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

(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 (1985). In this vein, Arora, Ju, and Ou-Yang (2003) 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 im-posed 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 ap-preciation 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 (2002a) demonstrates that a closed-end fund will trade at a discount equal to the capitalized man-agement 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 (2002b) explores variations of the model and shows that with asymmetrically-informed investors funds may issue at a premium and that dynamic distribution

(7)

policies can result in a fluctuating discount. Nevertheless, he does not simultaneously model both issue premiums and fluctuating discounts. In contrast to Ross (2002a) and Ross (2002b), 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 (1992) and Johnson, Lin, and Song (2006) 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 (2002) 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 (2009) 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 illiq-uid assets. While their model demonstrates that liqilliq-uidity 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 (2005) find that international barriers can affect discounts of funds that hold foreign assets, while Kumar and Noronha (1992) find that holding a portfolio of foreign stock does not necessarily impact the discount. Nonetheless, foreign holdings do not explain dis-count dynamics for funds that hold only domestic assets.

Agency costs also have been explored as a potential factor affecting discounts. Barclay, Hold-erness, 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. 3The 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,

(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 (2007) 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 2, 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 equilib-rium 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.

2

Basic Model

We begin by describing the framework of the model. Time is discrete and indexed by t ∈ {1,2,3,4}. Trading in the financial market occurs att= 1,2,3 while consumption occurs att= 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, ˜Y, att= 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

˜

Y ≡X˜1+ ˜X2+ ˜X3, (1)

where ˜Xt∼ N µt, σ2t

for t= 1,2,3. 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

(9)

is endogenous and denoted byPts.

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 timetand pays one unit att+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 non-zero 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 (2000)) nor changes in interest rates (Gemmill and Thomas (2002) 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 (1999)). The fund, whose shares are traded in the market, is in unit supply, and the equilibrium price at timet is endogenous and denoted by Ptf. 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 (2002), Johnson, Lin, and Song (2006), Bradley et al. (2010), and Lenkey (2011)), we assume that the fund will not be liquidated prior tot= 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), whereγi and γm denote the coefficients of risk aversion for the investor and

manager, respectively. Both the investor and fund manager behave like price-takers. Att= 1, the investor receives an exogenous endowment of wealth,Wi, and he observes the fund’s initial wealth

that is designated for investment, Wf. 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 timetthe

(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, cm, 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+bS3fY˜ +B3f −V1

, (2)

whereStf and Btf denote the quantity of stock and number of bonds held by the fund from timet

tot+ 1 and

Vt≡StfPts+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: V1 = Wf.

The fund’s time-tdiscount, Dt, is defined as the difference between the price of the fund and NAV,

Dt≡Vt−Ptf, (4)

which means that the fund’s issue premium is

ρ≡P1f −V1. (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 Dt.

5This 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.

(11)

The investor, meanwhile, receives utility solely from the consumption,ci, of the payoff from his portfolio. Hence, ˜ ci =S3iY˜ +B3i +F3 S3fY˜ +B3f −φ˜, (6)

whereStidenotes the quantity of stock, Bti denotes the number of bonds, andFtdenotes the shares

of the fund held by the investor from timet tot+ 1.

Information regarding the final stock payoff, ˜Y, 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 Ii

t and I f

t denote the information set at time t for the investor and

fund manager, respectively. Initially, the value of each Xt is unknown to both the investor and

manager: Ii

1 = I

f

1 = ∅. At t = 2, the manager observes X1 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: Ii

2 =∅andI f

2 ={X1}. Att= 3, the investor acquires knowledge

of X1 and both actors observe X2, so the information sets are once again symmetric: I3i =I f

3 =

{X1, X2}. Finally, all information is available at the terminal date: I4i =I f

4 ={X1, X2, X3}. 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 Ii

1 = I

f

1, the investor can

infer the fund’s portfolio composition from the equilibrium stock price. Att= 2, 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 observeX1, 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

(12)

previous date, which is announced prior to trading. Finally, the management fees are paid, the portfolios are liquidated, and consumption occurs att= 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 thatbΓ> γi, where Γ≡γi+γm. The second assumption is thatγi2γm2σ14+θσ21σ22+b2γm2Γ2σ24 >0

whereθ≡ −γi

(Γ +γm) b2Γ2+γi2

−2bΓ3. Both of these assumptions are entirely reasonable if

γm γi, 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 2.1 characterizes the equilibrium att= 3. Those results are then drawn on in Section 2.2 to derive the equilibrium at t = 2, which in turn is relied upon to characterize the equilibrium at t = 1 in Section 2.3. Some implications of the basic model are discussed in Section 2.4.

2.1 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

P3s=X1+X2+µ3−γiΓγmσ23 (7)

S3f =γi(bΓ)−1 (8)

(13)

and the closed-end fund discount is given by

D3 =a+b(V3−V1). (10)

The remaining portion of this subsection describes the equilibrium derivation, beginning with the fund manager’s objective. The fund manager’s goal att= 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 S3f −√1 2πσ2 3 Z exp [−γm˜cm] exp h −( ˜X3−µ3)2 2σ2 3 i dX˜3 (11) subject to ˜ cm= ˜φ+ρ (12) Bf3 =S2fP3s+B2f −S3fP3s (13)

and also (1), (2), and (5). Since, conditional on X1 and X2, the manager’s consumption is

log-normally distributed, her expected utility can be rewritten in closed form as

E3(um|X1, X2) =−exp h −γm a+b S3f(X1+X2+µ3−P3s) +S2fP3s+B2f −V1 +ρ−12γmb2 S3f 2 σ32 i , (14)

where Et is the expectation operator conditional on information available at time t, after

substi-tuting (1), (2), (12), and (13) into (11) and integrating. The manager’s demand function is then derived by differentiating (14) with respect toS3f and solving the corresponding first-order condition to obtain S3f = X1+X2+µ3−P s 3 bγmσ23 . (15)

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

(14)

constraint, taking into account the portfolio held by the fund:6 max Si 3, F3 −√1 2πσ2 3 Z exp [−γi˜ci] exp h −( ˜X3−µ3)2 2σ2 3 i dX˜3 (16) subject to B3i =S2iP3s+B2i +F2P3f − S3iP3s+F3P3f (17)

as well as (1), (2), (6), (13), and (15). Since the investor’s consumption is also conditionally log-normally distributed, his expected utility can be rewritten as

E3(ui|X1, X2) =−exp h −γi S3i(X1+X2+µ3−P3s) +S2iP3s+B2i +F2P3f +F3 Sf3(X1+X2+µ3−P3s) +S f 2P3s+B f 2 (1−b) +bV1−P3f −a −12γi S3i +F3S3f(1−b) 2 σ23 i (18)

after substituting (1), (2), (6), (13), and (17) into (16) and integrating. Differentiating (18) with respect toS3i and solving the first-order condition provides the investor’s stock demand function,

S3i = X1+X2+µ3−P

s 3 γiσ32

−F3(1−b)S3f. (19)

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 F3 = 1 and S3i +S f

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.

6Recall that the investor can infer the fund’s portfolio att= 3 for a given stock price since he has knowledge of

(15)

Finally, the fund price is obtained by differentiating the investor’s expected utility, (18), with respect toF3, substitutingF3 = 1 into the first-order condition, and solving for the price,

P3f =V3−a−b(V3−V1). (20)

Thus, the fund price is equal to the NAV of the fund minus an adjustment for the management fees. It follows immediately from (20) 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) whenevera+bV3> bV1; 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 att= 4, net of management fees, is

S3fY˜ +B3f −a−b

S3fY˜ +B3f −V1

, (21)

an appropriate hedging portfolio would consist of−B3f(1−b)−a+bV1

bonds and−S3f(1−b) shares of stock, but because the cost of this hedging portfolio equals −P3f, 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.

2.2 Equilibrium at t= 2

The equilibrium prices and allocations derived in Section 2.1 are used to determine the equilib-rium prices and allocations at t = 2. Recall that information is asymmetric at t= 2 as the fund manager observes the value ofX1but the investor does not. The following proposition characterizes

the equilibrium in the presence of asymmetric information.

(16)

are given by P2s=µ2+µ3− γiγm Γ σ 2 3 + γi σ12+σ22 X1+ (bΓ−γi)µ1−bγiγm σ12+σ22 σ22 γiσ12+bΓσ22 (22) Sf2 = (bΓ−γi) (X1−µ1) +bγiγm σ 2 1 +σ22 bγm γiσ21+bΓσ22 (23) Si2= (bΓ−γi) µ1−X1+bγmσ 2 2 bγm γiσ21+bΓσ22 (24)

and the closed-end fund discount is given by

D2 =b(V2−V1) +a−λ (25) where λ≡ (1−b) (bΓ−γi)σ 2 1 γi(1−b) (bΓ−γi+bγm)σ21+b2γm2σ22 . (26)

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= 2, 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 S2f −1 2πσ2 2 Z exp −γm a+b h S3f X1+ ˜X2+µ3−P˜3s +S2fP˜3s+B2f −V1 i +ρ− 1 2γmb 2Sf 3 2 σ23 exp −(X˜2−µ2) 2 2σ2 2 dX˜2 (27) subject to Bf2 =S1fP2s+B1f −S2fP2s (28)

(17)

Substi-tuting (3), (7), (8), and (28) into (27), the manager’s objective can be rewritten as max S2f −1 2πσ2 2 Z exp −γm a+bhS2fX1+ ˜X2+µ3−γiΓγmσ32−P2s +S1f(P2s−P1s)i +γ2iγm 2Γ2 σ23+ρ exp −(X˜2−µ2) 2 2σ2 2 dX˜2. (29)

Since her utility conditional onX1 is log-normally distributed, integrating (29) results in a

closed-form expression for the manager’s expected utility att= 2,

E2(um|X1) =−exp h −γm a+b h S2f X1+µ2+µ3−γiΓγmσ23−P2s +S1f(P2s−P1s) i +γi2γm 2Γ2 σ32+ρ−12γmb 2Sf 2 2 σ22i. (30)

Differentiating (30) with respect toS2f and solving the first-order condition provides the manager’s demand function, S2f = X1+µ2+µ3−P s 2 − γiγm Γ σ32 bγmσ22 . (31)

This expression is similar to the manager’s demand att= 3 but contains an extra term to account for future uncertainty. Furthermore, the presence of X1 in the manager’s demand function

repre-sents an additional source of risk for the investor.

Since the investor does not observe X1 att= 2, 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= 2 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 Si 2, F2 − 1 2π√σ2 1σ22 Z Z −exp −γi S3iX˜1+ ˜X2+µ3−P˜3s +S2iP˜3s+Bi2+F2P˜3f +F3 h S3fX˜1+ ˜X2+µ3−P˜3s + ˜S2fP˜3s+ ˜B2f(1−b) +bV1−P˜3f−a i −12γ S3i +F3S3f(1−b) 2 σ23 exp −(X˜1−µ1) 2 2σ2 1 exp −(X˜2−µ2) 2 2σ2 2 dX˜2dX˜1 (32) subject to B2i =S1iP2s+B1i +F1P2f − S2iP2s+F2P2f (33)

(18)

plus (3), (7), (8), (9), (20), (28), (31), and F3 = 1, where the investor’s objective function follows

from (18). Since the fund’s stock holdings, (31), do not depend on X2, conditional on X1 the

investor’s utility is log-normally distributed. Therefore, integration with respect toX2 is relatively

straightforward. Substituting the aforementioned equations (except (31)) into (32) and integrating with respect toX2, the investor’s objective can be rewritten as

max Si 2, F2 −√1 2πσ2 1 Z exp −γi S2i +F2(1−b) ˜S2f X˜1+µ2+µ3−P2s−γiΓγmσ 2 3 +B1i +F2 B1f −a+S1iP2s+F2S1f(bP s 1 + (1−b)P2s)−P f 2 (F2−F1) +γiγ2m 2Γ2 σ32−12γi S2i +F2(1−b) ˜S2f 2 σ22 exp −(X˜1−µ1) 2 2σ2 1 dX˜1. (34)

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 F1 = 1 into (34), the

investor’s objective can again be rewritten as

max Si 2 −1 2πσ2 1 Z exp h Gi2+H2iX1+J2iX˜12 i exp −(X˜1−µ1) 2 2σ2 1 dX˜1, (35) where Gi2 ≡Gi2S2i, F2, P2s, S1i, S f 1, B1i, B f 1, P1s;γi, γm, µ2, µ3, σ22, σ32, a, b H2i ≡H2i S2i, F2, P2s;γi, γm, µ2, µ3, σ22, σ32, b J2i ≡J2i F2;γi, γm, σ22, 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

E2(ui) =− 1 q 1−2σ12J2i exp " Gi2+2µ1H i 2+σ21 H2i 2 + 2µ21J2i 2 1−2σ2 1J2i # . (36)

Then, differentiating this expression with respect toSi

2, setting F1 = 1, and solving the first-order

7

The expressions forGi2,H2i, andJ2i, as well as the analogous expressions for the constant terms in (45) and (49),

are not reported but are available upon request.

8R

e−ξx2−2νxdx=qπξe

ν2

(19)

condition provides the investor’s stock demand att= 2, S2i = (bγm−(1−b)γiF2) µ1+µ2+µ3−P s 2 − γiγm Γ σ23 bγiγm σ12+σ22 , (37)

which is a function of the stock price and underlying parameters.

The stock price, which is given by (22), is obtained by imposing the market-clearing conditions and solving for price. A cursory examination of (22) 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 (23) and (24). These expressions reduce to the same stock allocations as att= 3 if there is no uncertainty regarding ˜X1. Furthermore, the closed-end fund holds a larger amount of stock when the realization

of ˜X1 is higher,9 and a comparison with thet= 3 allocations indicates that, on average, the fund

holds a greater quantity of stock and the investor holds less in his personal portfolio att= 2. Finally, the closed-end fund price is obtained by substituting (37) into (36), differentiating the resulting expression with respect to F2, substituting F2 = 1 into the first-order condition, and

solving for price,

P2f =V2−b(V2−V1)−a+λ, (38)

whereλis defined in Proposition 2 and represents the expected benefit from the manager’s private information, i.e., the expected value of the manager’s private information beforeX1 is realized. The

discount, which is given by (25), follows immediately from (38). The fund will trade at a discount (as opposed to a premium) wheneverb(V2−V1)> λ−a. In contrast to t= 3 where any amount

of NAV appreciation leads to a discount, at t= 2 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.

9Under the current setup, extreme realizations of ˜X

1 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.

(20)

2.3 Equilibrium at t= 1

The results from t = 2,3 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 P1s=µ1+µ2+µ3−γiΓγm σ12+σ22+σ32 (39) S1f =γi(bΓ)−1 (40) S1i = (bΓ−γi) (bΓ)−1 (41)

and the closed-end fund discount is given by

D1 =a−λ. (42)

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= 2 fund portfolio and stock price, the manager’s goal at t= 1 is to maximize her expected utility subject to a budget constraint:

max S1f −1 2πσ2 1 Z exp −γm a+b h ˜ S2f ˜ X1+µ2+µ3−γiΓγmσ32−P˜2s +S1f ˜ P2s−P1s i +γi2γm 2Γ2 σ 2 3+ρ− 12γmb 2S˜f 2 2 σ22 exp −(X˜1−µ1) 2 2σ2 1 dX˜1 (43) subject to B1f =Wf −S1fP1s (44)

(21)

rewritten in closed form as E1(um) =− 1 q 1−2σ2 1J f 1 exp   G f 1+ 2µ1H1f +σ12 H1f2+ 2µ21J1f 21−2σ2 1J f 1    (45)

after substituting (22) and (23) into (43) and integrating, where

Gf1 ≡Gf1S1f, Ps1;γi, γm, µ1, µ2, µ3, σ12, σ22, σ23, a, b, ρ H1f ≡H1fS1f;γi, γm, µ1, σ21, σ22, b J1f ≡J1f γi, γm, σ21, σ22, b .

Differentiating (45) with respect to S1f and solving the first-order condition gives the manager’s demand function at t= 1, S1f = µ1+µ2+µ3−P1s− γiγm Γ σ 2 3 γi2σ21+b2Γ2σ22 bγi2γmσ21 σ12+σ22 − bΓσ2 2 γiσ21 , (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 fromt= 2 as well as the closed-end fund’s current composition:10 max Si 1, F1 −√1 2πσ2 1 Z exp −γi ˜ S2i +F2(1−b) ˜S2f X˜1+µ2+µ3−P˜2s− γiγm Γ σ 2 3 +B1i +F2 B1f−a +S1iP˜2s+F2S1f bP1s+ (1−b) ˜P2s −P˜2f(F2−F1) +γiγ2m 2Γ2 σ 2 3 −12γi ˜ S2i +F2(1−b) ˜S2f 2 σ22 exp −(X˜1−µ1) 2 2σ2 1 dX˜1. (47) subject to B1i =Wi−S1iP1s−F1P1f (48)

in addition to (22), (23), (24), (44), (46), andF2= 1. Substituting these constraints into (47) and

10

(22)

integrating provides a closed-form expression for the investor’s expected utility att= 1, E1(ui) =− 1 q 1−2σ12J1i exp " Gi1+2µ1H i 1+σ21 H1i 2 + 2µ21J1i 2 1−2σ12J1i # , (49) where Gi1 ≡Gi1S1i, F1, P1s, P f 1, Wi, Wf;γi, γm, µ1, µ2, µ3, σ12, σ22, σ23, a, b H1i ≡H1i S1i, F1;γi, γm, µ1, σ12, σ22, b J1i ≡J1i γi, γm, σ12, σ22, b .

The investor’s stock demand function is found by differentiating (49) with respect toS1i and solving the first-order condition to obtain

S1i = µ1+µ2+µ3−P1s− γiγm Γ σ 2 3 γ 2 iγm2σ14+θσ12σ22+b2γm2Γ2σ24 γi3γ2 mσ21 σ12+σ22 −Γσ 2 2 (1−b)γi[2bγm−(1−b)γi]σ21−b2γm2σ22 γ2 iγmσ21 σ12+σ22 −F1(1−b)S 1 f. (50)

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 F1,

substituting (28), (39), (40), (41), and F1 = 1 into the derivative, and solving for price,

P1f =V1−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 (42). Whether the fund

(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 thatP2f can be replicated by forming a portfolio consisting of (1−b)S1f

shares of stock and bS1fPs

1 +B

f

1 −a+λbonds at t = 1. Since the cost of this portfolio, S f 1P1s+ B1f −a+λ, equalsP1f, however, there is no arbitrage.

2.4 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 (1990). 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 (1995). Furthermore, the fund will issue at a premium if

λ > a, and a discount will necessarily emerge byt= 3 provided thatV3 > V1, although a discount

could arise at t = 2 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 att= 4.11

While it is apparent from (42) 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 ˜X1 and ˜X2 are components of another

variable ˜Z ≡ X˜1 + ˜X2, where ˜Z ∼ N µz, σ2z

, and that the respective distributions of ˜X1 and

11

(24)

Table I: Parameter Values.

Variable Symbol Value

Investor’s coefficient of risk aversion γi 1

Manager’s coefficient of risk aversion γm 40

Fixed component of management fee a 0.018 Investor’s initial wealth Wi 1

Variance of ˜Z σz2 0.0167 Variance of ˜X3 σ32 0.0083 ˜ X2 are ˜X1 ∼ N αµz, ασz2 and ˜X2 ∼ N (1−α)µz,(1−α)σ2z

. Under this specification, the distribution of the stock payoff, ˜Y, 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 σ21 =ασz2 and σ22 = (1−α)σ2z into (26), the expression for λcan be rewritten in terms of ability as

¯

λ≡ α(1−b) (bΓ−γi)

α(1−b)γi(bΓ−γi+bγm) + (1−α)b2γm2

. (52)

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 man-ager, 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 pre-mium rather than at NAV.12 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 −D1), 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 graph-ically. In order to plot these relationships, we assume numerical values for various parameters in 12Pursuant to the Investment Company Act of 1940, a closed-end fund may sell its common stock at a price less

(25)

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 0.2 0.4 α b ρ

(a) Issue Premium

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 −1 −0.8 −0.6 −0.4 −0.2 0 α b E(u m )

(b) Manager’s Expected Utility

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0.3 0.4 0.5 0.6 α b −E(u i )

(c) Negative of Investor’s Expected Utility

0 0.25 0.5 0.75 1 0 0.25 0.5 0.75 1 0 10 20 30 40 50 α b ∂ S2 f / ∂ X 1

(d) Degree of Portfolio Adjustment

Figure 1: Issue Premium, Expected Utility, and Response to Private Information.

the model. The investor’s coefficient of risk aversion is normalized to one, and the manager’s coef-ficient of risk aversion of 40 is chosen to satisfy the assumption thatbΓ> γi and still accommodate

relatively low values for b. The fixed component of the management fee is set to 0.018, and the investor’s initial wealth is normalized to one, but neither affect either the shape of the issue pre-mium or the actors’ expected utilities. Finally, the variance of the stock payoff is chosen to match the return precision from Berk and Stanton (2007), with two-thirds of the total variance allocated to the portion of the payoff that can potentially comprise the manager’s private information, ˜Z, and one-third allocated to the remaining portion, ˜X3. These parameter values are summarized in

Table I.

Figure 1(a) plots the issue premium, Figure 1(b) plots the manager’s expected utility, and Fig-ure 1(c) plots the negative of the investor’s expected utility as a function of managerial ability, α,

(26)

and the variable component of the management fee, b. Comparing these figures, it is evident that the choice ofbthat maximizes the premium for a given level of ability also maximizes the manager’s expected utility and minimizes the investor’s expected utility.13 These relationships explain why closed-end funds tend to issue at a premium. Provided thatais not too large, the fund manager’s utility-maximizing choice for b gives rise to an issue premium, and the investor is willing to pay a premium for the fund because he expects to receive the risk-adjusted market rate of return.

The aforementioned figures also highlight the significance of the variable component of the man-agement fee, but the reason why the choice ofbis important is because it affects the degree to which the manager adjusts the fund’s portfolio in response to realizations of her private information, as demonstrated by Figure 1(d), which plots the partial derivative of the fund’s stock holdings with respect to the realization of the manager’s private information, ∂S2f/∂X1, for various values ofα

and b. This derivative measures how aggressively the manager trades on her private information and is a function of γi, γm, α, b, and σz2. By comparing Figures 1(a) and 1(d), it is clear that

regardless of the manager’s ability level, the extent to which the manager reacts to her private information, which depends on her stake in the equity owned by the fund, directly impacts the issue premium.

3

Extended Model

The basic model described in Section 2 can be extended to an economy that includes a longer time horizon and random ability. The aims of this section are to illustrate the evolution of the discount over time and test whether the combination of managerial ability and management fees can explain some empirical features of closed-end funds. We first describe the framework of the extended model in Section 3.1 and present the equilibrium in Section 3.2. We then simulate data in Section 3.3 and compare the model’s predictions to some empirical characteristics of closed-end funds reported in the literature.

13

This can be verified numerically. The relationships also hold if instead the management contract pays the manager a fraction of the total assets under management, although the shape of the issue premium is slightly different.

(27)

3.1 Assumptions

This subsection outlines the assumptions of the extended model. Most of these assumptions parallel those of basic model described in Section 2. Unless otherwise noted, the assumptions of the basic model continue to hold in the extended setting.

Time is now indexed by t= 1,2, . . . , T + 1, where T is a multiple of 3. As described in more detail below, the information and trading sequences of the basic model repeat for N = T3 cycles, which are indexed by n. Each cycle is comprised of three dates. Throughout this section, for all

n= 1,2, . . . , N, we refer tot= 3n−2 as the “beginning” of a cycle,t= 3n−1 as the “middle” of a cycle, andt= 3n as the “end” of a cycle, which are analogous tot= 1,2,3, respectively, in the basic model. Consumption occurs at T+ 1.

The stock pays a random amount, ˜Y, at T+ 1. As in the basic model, the payoff of the stock consists of the sum of independent and normally distributed random variables; accordingly, ˜Y is redefined as ˜ Y ≡ T X t=1 ˜ Xt, (53) where ˜Xt∼ N µt, σ2t

for allt≤T. The assumptions regarding the bonds remain unchanged. The closed-end fund undergoes an IPO att= 1 and is liquidated atT+ 1, but it does not make any distributions prior to liquidation. The fund manager collects a sequence of fees, ˜φn, at the

end of each cycle and consumes cm =ρ+PNn=1φ˜n atT + 1. The parameters of the management

contract, aand b, are constant over time, and then-th cycle fee is given by

˜ φn=a+b Sf3n3sn+1+B3fn−V3n−2 . (54)

The fees are deducted from the fund’s NAV at the time they are earned; hence, between cycles the fund’s NAV evolves according to

V3n+1=S3fnP3sn+1+B f

3n−φn (55)

for all n, and within a cycle the fund’s NAV evolves according to

Vt+1 =StfPts+1+B f

(28)

for all t 6= 3n. Additionally, no arbitrage requires that the terminal stock price be equal to its payoff: ˜PTs+1 = ˜Y.

Like in the basic model, the investor obtains utility solely by consuming the payoff from his portfolio, ˜ ci=STiY˜ +BTi +FT STfY˜ +BTf −φ˜N . (57)

The investor’s budget constraint satisfies

StiPts+Bit+FtPtf =Sti−1Pts+Bit−1+Ft−1Ptf (58)

for all t≤T.

The information structure follows a pattern similar to that of the basic model. At the beginning of every cycle, the investor and fund manager have knowledge of all prior realizations ofXt: I3in−2=

I3fn2 = {X1, X2, . . . , X3n−3} for n = 1,2, . . . , N. The manager acquires private information at

the middle of every cycle through observation of X3n−2, which is unobservable to the investor

at that time. Consequently, the information sets are asymmetric: Ii

3n−1 = I3in−2 and I f 3n−1 =

{X1, X2, . . . , X3n−2}. At the end of every cycle, the information sets are once again symmetric:

Ii 3n = I

f

3n = {X1, X2, . . . , X3n−1}. For the sake of completeness, note that I0i = I f 0 = ∅ and Ii T+1 =I f T+1={X1, X2, . . . , XT}.

An important feature of the extended model is that the manager’s ability to acquire information at the middle of every cycle is stochastic. For tractability, we assume that the manager’s ability for a given cycle can be either low, α`, or high, αh, with probability Υ` and Υh, respectively, and

0 < α` < αh < 1. Both the manager and investor observe the manager’s ability for cycle n at

the beginning of the cycle, i.e., t = 3n−2, but the ability is unknown to both of them prior to that time. Modeling ability in this fashion reflects the notion that the value of the manager’s skills idiosyncratically depends on (unmodeled) economic conditions which evolve over time. Similar to the basic model, for each cycle we assume that ˜X3n−2 and ˜X3n−1 are components of another

variable ˜Zn ≡X˜3n−2+ ˜X3n−1, where ˜Zn ∼ N µz, σ2z

for all n. The distributions of ˜X3n−2 and

˜

X3n−1, which depend on the manager’s ability for a particular cycle, are ˜X3n−2∼ N αkµz, αkσ2z

and ˜X3n−1 ∼ N (1−αk)µz,(1−αk)σz2

for k ∈ {`, h}. Managerial ability is independent of Zn

(29)

The sequence of events is as follows. Att= 1, the fund undergoes an IPO. At the beginning of every cycle, both the investor and manager observe the manager’s ability for that cycle and choose portfolios at market-clearing prices subject to their respective budget constraints. The management fee earned during the immediately preceding cycle is deducted from the NAV of the fund and placed into escrow until T + 1 prior to selecting the fund’s new portfolio. At the middle of every cycle, the manager acquires private information regarding the payoff of the stock. The investor and manager then proceed to rebalance their respective portfolios. At the end of every cycle, the fund manager’s informational advantage disappears, and both actors select new portfolios. AtT+ 1, all management fees are paid, the fund is liquidated, and consumption occurs.

3.2 Equilibrium

The equilibrium in the extended setting is complicated by the fact that the manager’s ability is now stochastic. However, since the manager’s ability remains unchanged between dates within a cycle and there are no wealth effects associated with CARA preferences, the same techniques utilized to solve for the equilibrium in the basic setting can be used to solve for the equilibrium prices and allocations for both the stock and closed-end fund at the beginning and middle of each cycle in the extended setting. Solving for the equilibrium at the end of a cycle is slightly more complicated due to the uncertainty surrounding the manager’s ability for the immediately ensuing cycle.

When selecting their respective portfolios at the end of each cycle (except cycleN), the investor and manager are exposed to risk associated with both a component of the stock payoff and the manager’s ability for the immediately ensuing cycle.14 Since the manager’s ability is independent of

the stock payoff and the actors’ preferences do not exhibit wealth effects, however, the uncertainty surrounding the manager’s ability does not impact the stock price. Therefore, the stock price at the end of a cycle is obtained using the same techniques as in the basic model and consists of the expected payoff less an adjustment for risk. The stock price at the end of cycle nis

P3sn= 3n−1 X τ=1 Xτ+ T X τ=3n µτ −γiΓγmσ2τ . (59) 14

(30)

Because the manager and investor possess symmetric information, the respective equilibrium stock allocations at the end of cycle nare the same constant fractions as in the basic setting:

S3fn=γi(bΓ)−1 (60)

and

S3in= (bΓ−γi) (bΓ)−1. (61)

While the investor is able to infer the fund’s portfolio at the end of each cycle since it contains a constant fraction of the stock, owning the fund exposes the investor to risk associated with the manager’s ability for the next cycle. If the manager’s ability happens to be high during the next cycle then she will obtain a greater informational advantage and the fund price will be higher at the beginning of the next cycle than if the manager’s ability turns out to be low. This uncertainty is incorporated into the fund price as a weighted average of the investor’s expected benefit from the manager’s private information,

δ ≡ ¯ λ`Υ` r 1−(1−α`)α`γi(Γ+γm)(bΓ−γi)2 (γm(α`γi+(1−α`)bΓ))2 + ¯λhΥh r 1− (1−αh)αhγi(Γ+γm)(bΓ−γi)2 (γm(αhγi+(1−αh)bΓ))2 Υ` r 1−(1−α`)α`γi(Γ+γm)(bΓ−γi)2 (γm(α`γi+(1−α`)bΓ))2 + Υh r 1−(1−αh)αhγi(Γ+γm)(bΓ−γi)2 (γm(αhγi+(1−αh)bΓ))2 (62) where ¯ λk≡ αk(1−b) (bΓ−γi) αk(1−b)γi(bΓ−γi+bγm) + (1−αk)b2γm2 (63)

fork∈ {`, h}. The fund price at the end of cyclen is

P3fn=V3n−a−b(V3n−V3n−2) + (N−n) (δ−a), (64)

and the corresponding discount,

D3n=a+b(V3n−V3n−2)−(N −n) (δ−a), (65)

is a combination of the future management fees and benefits from private information.

(31)

Like in the basic setting, the information asymmetry is incorporated into the stock price. The equilibrium stock price at the middle of cyclenis

P3sn1 = γiX3n−2+ (1−αk) αk(bΓ−γi)µz−bγiγmσ 2 z αkγi+ (1−αk)bΓ + 3(n−1) X τ=1 Xτ + T X τ=3n−1 µτ− T X τ=3n γiγm Γ σ 2 τ, (66)

which is analogous to P2s in the basic setting. The expressions for the stock allocations at the middle of cyclenalso reflect the information asymmetry and are analogous to the stock allocations in the basic setting:

S3fn1 = (bΓ−γi) (X3n−2−αkµz) +bγiγmσ 2 z bγm(αkγi+ (1−αk)bΓ)σz2 (67) and S3in−1= (bΓ−γi) αkµz−X3n−2+ (1−αk)bγmσ2z bγm(αkγi+ (1−αk)bΓ)σz2 . (68)

The fund price is given by

P3fn1 =V3n−1−b(V3n−1−V3n−2)−a+ ¯λk+ (N −n) (δ−a) (69)

fork∈ {`, h}and depends on whether the manager currently possesses a low or high level of ability. Accordingly, the discount at the middle of cyclen,

D3n−1 =b(V3n−1−V3n−2) +a−λ¯k−(N −n) (δ−a), (70)

also depends on the current level of managerial ability.

At the beginning of each cycle, the investor and manager bear the risk associated with a component of the stock payoff but are not exposed to risk associated with managerial ability. Since the information sets are symmetric, the equilibrium here is analogous to the equilibrium at t= 1 in the basic setting. At the beginning of cyclen, the equilibrium price of the stock is

P3sn2 = 3n−3 X τ=1 Xτ+ T X τ=3n−2 µτ−γiΓγmσ2τ , (71)

(32)

and the stock allocations are S3fn2 = γi bΓ (72) and S3in−2 = bΓ−γi bΓ . (73)

Although there is no current uncertainty regarding managerial ability, the future uncertainty is incorporated into the fund price. The price of the fund at the beginning of cyclenis

P3fn2 =V3n−2−a+ ¯λk+ (N−n) (δ−a), (74)

and the corresponding discount is

D3n−2 =a−¯λk−(N −n) (δ−a). (75)

Similar to the end of each cycle, the discount at the beginning of each cycle is a combination of the future management fees and benefits from private information.

A few basic themes emerge from the extended model. First, the closed-end fund will trade at a discount when the management fees outweigh the benefits from the manager’s private information, and vice versa. Second, the expected benefit from the manager’s private information, which is captured by ¯λkandδ, evolves over time. Specifically, at the turn of a cycle (fromt= 3ntot= 3n+

1), the expected benefit from all future private information changes by an amount equal to ¯λk−δ.

Since this adjustment is positive whenever a high ability level is realized and negative whenever a low ability level is realized, the expected benefit from the manager’s private information (and hence the discount) changes even when her ability level remains unaltered. Third, the equilibrium stock prices, fund prices, and portfolio holdings are analogous to those in the basic model, but there is additional uncertainty which stems from the time-varying ability of the manager and causes the discount to fluctuate. We next evaluate the model’s ability to account for several empirical observations documented in the literature.

(33)

3.3 Simulation

Since the focus of our analysis is private information and management fees, the only parameters that may vary across simulations are managerial ability and the realizations of the components of the stock payoff, and we assume that the distributions of these parameters are independent across simulations. The parameter values used in the simulations are listed in Tables I and II. Although there are a total of 13 free parameters in our model, only a few of these are truly discretionary. Here we discuss our rationale for selecting the parameter values, beginning with those listed in Table I. As previously mentioned, γi is normalized to one, and γm is conservatively chosen to satisfy our

assumption that bΓ > γi while still accommodating relatively low values for b. Additionally, we

choose the stock volatility to match the return precision from Berk and Stanton (2007) and set the investor’s initial wealth to one, although it does not affect the discount.

Next, we explain our reasons for choosing the parameter values listed in Table II. The variable component of the management fee, b, is set at 20% to be consistent with contracts in the money management industry that charge a fixed percentage of returns. Like the stock volatility, the time horizon of 50 cycles matches the horizon in Berk and Stanton (2007), and the fund’s initial wealth designated for investment is set to 1.5, which ensures that the fund price does not turn negative in any of the simulations. The mean stock payoff for a cycle is fixed at 0.15 so that negative payoffs occur only when the payoff realization is more than roughly one standard deviation below the mean. Additionally, we assume that two-thirds of the total payoff for a cycle can potentially comprise the manager’s private information, ˜Zn, and that the remaining one-third of the total

payoff is determined by ˜X3nfor alln. After selecting these parameter values, only a handful of

pa-rameters remain that can be used to match the empirically observed discount dynamics—these are the fixed component of the management contract, the manager’s ability levels, and the distribution of ability. To be consistent with empirical observations, the values for the ability levels and their accompanying probability distribution are chosen so that funds issue at a premium and are likely to begin trading at a discount very quickly. The probability of the manager having a high ability level for a cycle, Υh, is set at 10%, but in each simulation the initial level of ability is set atαh so

that the fund issues at a premium. This results in issue premiums and the emergence of discounts within a single cycle, but it also reflects the fact that funds rarely move form trading at a discount

References

Related documents

Attitude towards technology Routine/ frequency Usage of financial services Trust and Efficacy Efficiency Personal contact Efficacy Efficiency Performance Evaluation of BB

Part II: Integrating local ecological services into intergovernmental fiscal transfers ...69 2 Ecological public functions and fiscal equalisation at the local level in Germany

Figure 14: Projected wholesale prices by demand scenario and renewable energy target under a carbon price

After establishing the system’s reading speed, you should run a more challenging read rate test to determine the impact of factors such as line vibration, variable

Thus, HR can play a vital role in shaping organizations to have a positive environment and culture where employees feel at ease and relaxed so that they can share and learn

• Green – Not as Important, or likely high cost/extended implementation time – Fiscal notes. –

One of the most recent theorist on women’s voluntary engagement in the informal economy, Kinyanjui demands a change in the discourse, where the pull factors that lead

At the most, cost-effective energy efficiency is stated as a principle for energy strategy, such as in the Energy Strategy of 2007, but as separately worthwhile rather than