Portfolio Disclosure, Portfolio Selection, and Mutual Fund Performance Evaluation
CFR Working Paper No. 04-09
Alexander Kempf and Klaus Kreuzberg
Portfolio Disclosure: Effects on Portfolio Optimality and Performance
Alexander Kempf Klaus Kreuzberg
∗ABSTRACT
We analyze the portfolio selection and performance of mutual funds in a model in which funds report their portfolio holdings to their investors. Portfolio disclosure gives a signal of the quality of managerial information to the fund investors and thus reduces their fund risk. We show that both optimal fund portfolios and fund performance depend on portfolio disclosure. More frequent disclosure allows fund managers to respond more strongly to their private information and to choose riskier positions. We introduce two new performance measures which incorporate risk reduction from portfolio disclosure.
They combine disclosed portfolio holdings with readily available fund returns.
(JEL classification: G11, G23).
∗The authors are from the University of Cologne, Germany, and from the Centre for Financial Research (CFR), Cologne, Germany.Correspondence Information: Alexander Kempf, University of Cologne, Depart- ment of Finance, Albertus-Magnus-Platz, 50923 K¨oln, Germany, www.wiso.uni-koeln.de/finanzierung. Phone:
+49-221-4702714, Fax: +49-221-4703992,mailto:kempf@wiso.uni-koeln.de. The authors are grateful to Vikas Agarwal, John Boyd, Olaf Korn, Daniel Mayston, Christoph Memmel, Eric Theissen, Pradeep Yadav and participants of the 2003 Annual Meetings of the European Finance Association (EFA), the European Fi- nancial Management Association (EFMA), the German Finance Association (DGF), the Southern Finance As-
Since the seminal work of Markowitz (1952) the portfolio selection problem of individual investors is well understood: investors choose their portfolios so that they optimally trade-off expected portfolio return and portfolio risk conditional on their information about the assets’
risky returns.
Nowadays, an increasing number of investors do not make their portfolio decisions by themselves but delegate them to professional fund managers. Therefore, two parties are in- volved in the investment process, the fund manager and the fund investor. They both differ with respect to their role and their information: the fund manager decides on the compo- sition of the fund portfolio. The fund investor decides whether to invest her money with the fund manager. While the fund manager is supposed to generate superior information, a fund investor typically neither is informed nor does she know what information (if any) the fund manager has. This informational asymmetry inherent in delegated portfolio management makes fund investors face an additional source of risk, fund management risk.
Management risk is smaller the more fund investors know about the trading strategy of the fund manager. The fund holdings reveal, to the extent to which they are observed by the fund investors, the private information underlying the trading strategy of the fund manager.
Management risk therefore relates to the portfolio disclosure of a fund. Empirically, disclosure strategies vary considerably across different types of funds. While pension funds typically provide their sponsors with detailed portfolio information, hedge funds often do not report any portfolio information at all. Mutual funds are generally situated between those extremes.
Under the U.S. Investment Company Act of 1940 they are required to publish their portfolio
holdings on a quarterly basis, but some funds voluntarily disclose additional information on their holdings to their investors.
In this paper we present a model of delegated portfolio management where funds report to their investors information on their portfolio holdings. We analyze the impact of portfolio disclosure on the optimal composition of fund portfolios and on fund performance. The main rationale of our model is as follows: the level of portfolio disclosure determines what fund investors know about the portfolio strategy of the fund. It therefore determines the amount of management risk faced by fund investors. A fund manager acting on behalf of her investors has to take management risk into account. She has to choose a portfolio which is adapted not only to her private information but also to her level of portfolio disclosure. A fund investor evaluating the performance of the fund manager also has to consider management risk. She needs a performance measure that correctly specifies her fund risk with regard to the portfolio disclosure of the fund.
Our model provides two main results: First, the optimal fund portfolio does not only depend on the private information of the fund manager, but also on the information disclosed to the fund investors. A higher level of portfolio disclosure provides more information to fund investors. It allows the fund manager to trade more aggressively on her private information and to choose a more risky portfolio.Second, performance measures have to be adapted to the level of portfolio disclosure. Otherwise they misspecify fund risk and lead to biased performance inferences (e.g., Jensen (1972), Dybvig and Ross (1985), and Grinblatt and Titman (1989)).
(Jensen (1968) and Sharpe (1966)) and internal (Grinblatt and Titman (1993)) performance evaluation. Our measures are unbiased measures of fund performance in the notion of Dybvig and Ross (1985). Based on our measures we show that informed fund managers can increase the performance for their investors by disclosing additional holdings information.
We extend the existing literature into two directions. First, we generalize the results of Dybvig and Ross (1985), Hansen and Richard (1987), and Ferson and Siegel (2001). They show that portfolios which are optimal for informed investors are suboptimal for uninformed investors. This result implies that a better informed fund manager should not choose the port- folio which is optimal given the manager’s information for her uninformed clients. However, most funds report information on their holdings so that fund investors are not uninformed.
We generalize their results by allowing the fund investors to be partly informed through the portfolio disclosure of the fund. Our analysis demonstrates how a fund manager has to adapt the fund portfolio to her disclosure policy in order to obtain a portfolio which is optimal for her less informed fund investors.
Second, we extend the literature on performance evaluation. There is a body of litera- ture suggesting various measures for external and internal performance evaluation. These measures differ with respect to their input parameters. While external measures (e.g., Jensen (1968), Sharpe (1966), and Carhart (1997)) rely solely on fund returns, internal measures (e.g., Grinblatt and Titman (1993) and Daniel, Grinblatt, Titman, and Wermers (1997)) fo- cus exclusively on fund holdings. However, both returns and holdings have to be taken into
account. Our disclosure-based measures combine both sources of information and therefore allow unbiased performance inferences.
Our paper is organized in four sections. Section I sets up the model. In Section II we show how the optimal fund portfolio depends on the level of portfolio disclosure. In Section III we first derive new performance measures and then show how the performance of a fund depends on the level of portfolio disclosure. Section IV concludes.
I. The Model
This section sets up the structure of our model. The model captures two features of the mu- tual fund market: first, some fund managers may be better informed than fund investors are, but fund investors cannot identify the better informed managers. Second, fund managers pro- vide noisy signals about their trading strategies which the fund investors can use to infer the information of the fund managers.
Securities: We consider an economy with N risky assets (stocks) and one riskless asset (bond). The returns of the assets are exogenous in our model. Stock i (i=1,...,N) pays the random return ˜Ri, the return of the bond is normalized to zero. The vector of the stock returns, ˜R, consists of three parts:
R˜ = μ+s˜+γ˜ (1)
μ is a vector of constants. ˜s and ˜γ are vectors of independent, normally distributed random variables with unconditional means of zero and covariance matrices Σ(s˜) and Σ(˜γ), respec- tively. Thus, stock returns, ˜R, are normally distributed with the unconditional meanμand the covariance matrixΣ=Σ(s˜) +Σ(γ)˜ .
Agents: There are two agents in our model: a fund manager and a fund investor. The fund manager runs a mutual fund which is offered to the fund investor. The manager charges a fixed management fee which reduces the gross return of the fund byc. The fund investor is risk averse. She decides on whether to invest with the fund or directly into the stocks based on a standard mean-variance preference function.
Informational Structure: Fund manager and fund investor differ with respect to the infor- mation they have. The vector of the unconditional mean returns, μ, and the unconditional covariance matrix,Σ, are public information. The vector of signals, ˜s, is private information of the manager and ˜γis unobservable for both. Σ(s˜)determines the quality of the managerial signal, ˜s. The largerΣ(s˜)the more of the stock return variance is explained by the managerial signal and the more valuable the signal is. We assume that the managerial signal is neither useless nor perfect, i.e. Σ(s˜)is not the zero matrix andΣ−Σ(s˜) =Σ(˜γ)is positive definite.
The fund investor has no prior information whether the fund manager is informed.1 She does not know the quality of the managerial signal,Σ(s˜). What she gets from the fund manager is a noisy signal, ˜x, about the fund strategy. In our model, the signal ˜xis the only conditioning
1This prior is consistent with the view that the market share of informed fund managers is negligible. This view is supported by empirical studies showing that there is no persistent outperformance of funds (see, e.g., Brown and Goetzmann (1995), Malkiel (1995), and Carhart (1997)).
information of the fund investor.2 Since ˜x depends on the fund holdings,w(s˜), and the fund holdings reflect the private information ˜s, the portfolio signal ˜xprovides information about ˜s.
The fund return, ˜RP, is determined by the portfolio shares invested in each stock,w(s˜), the stock returns, ˜R, and thefixed management cost,c:
R˜P=w(s˜)
μ+s˜+˜γ
−c. (2)
The fund investor uses her portfolio signal to update her beliefs about the fund return. Given the portfolio signal,x, the investor expects a fund return ER˜P|x
with a variance of varR˜P|x . The preferences of the fund investor can therefore be written as
Φ = E w(s˜)
μ+s˜+γ˜x
−c−λ
2var w(s˜)
μ+s˜+γ˜x
, (3)
whereλ>0 is the parameter of risk aversion. The fund manager who invests on behalf of the fund investor has to maximize (3). She has to use her private information, ˜s, but she also has to take into account that her fund investor only observes the portfolio signal, ˜x.
2To highlight the impact of disclosure on portfolio composition and performance, we ignore the fact that the fund investor might infer information about the managerial signal by jointly observing fund and stock returns.
Research into that direction is currently being done by Iskoz and Wang (2003) analyzing the skewness of the fund returns and by Mamaysky, Spiegel, and Zhang (2003)filtering the fund returns through a Kalmanfilter.
II. Optimal Fund Portfolio
This section focuses on the portfolio selection decision of the fund manager. We derive the fund portfolio that uses the private information of the fund manager, ˜s, and is optimal for the fund investor who only receives the portfolio signal, ˜x. We begin our analysis with two examples at different extremes: the first example assumes full portfolio disclosure, i.e. the fund manager reports the exact portfolio strategy to the fund investor. The second extreme assumes that the fund manager discloses no information at all about the fund strategy. Finally, we solve our model for the general case of arbitrary portfolio disclosure.
A. Full Disclosure
This section assumes that the fund manager reports her exact portfolio strategy to the fund investor. Although funds typically do not disclose their entire portfolio strategies, this analysis is a useful starting point as it will allow us to separate between effects simply resulting from the delegation of the portfolio selection to a fund manager and those emanating from incomplete portfolio disclosure of that manager. With full portfolio, i.e. ˜x=w(s˜), the fund manager has to choose her portfolio strategywto maximize
Φ = E w(s˜)
μ+s˜+γ˜w
−c−λ
2var w(s˜)
μ+s˜+˜γw
(4)
given her private information,s, and her full disclosure policy.
Theorem 1 With full portfolio disclosure, the optimal fund portfolio is given by
w
Full(s˜) = 1
λΣ(˜γ)−1 μ+s˜
. (5)
Proof: The proof is a special case of the proof of Theorem 3 in the Appendix which holds for arbitrary portfolio disclosure.
Solving the optimal managed portfolio (5) for s, the fund investor can infer the manage- rial signal from the reported fund holdings, w
Full. Therefore, full disclosure eliminates the informational asymmetry between fund manager and fund investor. It reveals the quality of the managerial information to the fund investor. With manager and investor sharing the same information, the fund manager chooses the fund portfolio as if she were to invest her own money. With full portfolio disclosure, the decision to delegate the portfolio selection to a fund manager has no impact on the composition of optimal fund portfolios. The portfolio problem of the fund manager is equivalent to the original portfolio problem analyzed by Markowitz (1952).
B. No Disclosure
We now turn to the other extreme of the fund manager withholding all information on the portfolio strategy. While mutual funds are obliged to meet minimum disclosure requirements under the Investment Company Act, hedge funds are exempt from mandatory portfolio disclo-
sure. They typically do not report their investors any information on their portfolio strategies.
In this case, the portfolio signal ˜xis uninformative, i.e. Σ
˜ s|x
=Σ(s˜), and the informational asymmetry between fund manager and fund investor persists. The portfolio problem of the fund manager, conditional on her private information, ˜s, and her no disclosure policy, is to choose the fund portfoliowto maximize:
Φ = E w(s˜)
μ+s˜+γ˜
−c−λ
2var w(s˜)
μ+s˜+˜γ
(6)
Since the fund portfolio, w(s˜), is conditioned on the signal ˜sreceived only by the manager, the fund holdings are random for the fund investor. The portfolio problem (6) consists in determining the optimal reaction function,w, to the managerial signals,s.
Theorem 2 Without portfolio disclosure, the optimal fund portfolio is given by
w
No(s˜) = 1
λΣ(γ)˜ −1
μ+s˜ ψ(s˜) E
ψ(s˜), (7)
ψ(s˜) = 1 1+
μ+s˜
Σ(γ)˜ −1
μ+s˜. (8)
Proof: The proof is a special case of the proof of Theorem 3 in the Appendix which holds for arbitrary portfolio disclosure.
The optimal fund portfolio without disclosure differs from the one with full disclosure (5) by a scaling factorψ(s˜)
E ψ(s˜)
. The size of this factor depends on the size of the realized
signal. Compared to full disclosure, the optimal strategy (7) requires the fund manager to respond more moderately to extreme signals, but allows her to rely more on small signals.3
Figure 1 illustrates two examples of optimal managed portfolios (a) with full portfolio disclosure and (b) without portfolio disclosure for a single stock (N =1). The figures show the optimal proportion of the fund invested in the stock,w, depending on the signal of the fund manager,s.
Insert Figure 1 here.
The two disclosure strategies give rise to different optimal fund portfolios. With full disclo- sure, the optimal stock investment increases linearly with the size of the managerial signal.
Without disclosure, the relation between stock investment and size of the managerial signal is not linear, it is not even monotonic. The larger the signal, the more the optimal portfolios with full and without disclosure differ from each other. The reason for this to appear is the different fund risk associated with the two portfolios. With full disclosure, the fund investor can infer the private information of the fund manager from the reported fund holdings. She only bears the residual risk from the stock market,Σ(γ)˜ . Without portfolio disclosure, the fund investor cannot observe the portfolio strategy of the fund manager. She therefore cannot infer the private information underlying the managerial strategy. She faces an additional manage- ment risk. Analytically, this management risk results from the fund holdings being random for her. Economically, it reflects her uncertainty about the quality of the managerial information.
Choosing the portfolio which is optimal for the fund investor, the fund manager should not trade too aggressively on signals that differ heavily from their unconditional means. Other- wise, extreme signals would lead to large portfolio adjustments and, consequently, to large management risk for the fund investor.
The impact of portfolio disclosure on the composition of the optimal fund portfolio is economically important. Take for example a fund manager who reports no information on her holdings to the fund investor, but chooses the portfoliowFullfrom (5). This manager chooses too extreme positions for her client. With such a manager, the fund investor might be better off by following a simple benchmark strategy instead of giving the money to the better informed fund manager.4 Ignoring the level of portfolio disclosure when deciding on the composition of the fund portfolio thus can be disadvantageous.
C. Arbitrary Level of Disclosure
Most funds report their holdings with a time lag and only at infrequent points in time. Further- more, there is empirical evidence that funds polish their portfolios around disclosure dates in order to appear better to their fund investors.5 In this case, the portfolio disclosure represents a noisy signal, ˜x, about the fund holdings,w(s˜). The fund manager then has to maximize (3) given her private information, ˜s, and her portfolio disclosure, ˜x.
4See the Appendix for a proof of this statement.
5Lakonishok, Shleifer, Thaler, and Vishny (1991) and Musto (1999)find evidence for ”window dressing” of U.S. pension funds and money market funds, respectively, analyzing fund holdings and returns around disclosure dates.
Theorem 3 With portfolio disclosure,x, the optimal fund portfolio is given by˜
w(s˜,x˜) = 1
λΣ(γ)˜ −1
μ+s˜ ψ(s˜) E
ψ(s˜)|x˜, (9)
whereψ(s˜)is defined according to (8).
Proof: See the Appendix.
The optimal fund portfolio depends on both the private information of the fund manager and the portfolio disclosure of the fund. As before, it can be decomposed into the portfolio (5) that is optimal with full portfolio disclosure and a scaling factor,ψ(s˜)
E
ψ(s˜)|x˜
. This factor now depends not only on the size of the managerial signal,s, but also on the portfolio signal received by the fund investor,x. The fund manager chooses the fund portfolio such that it is adapted to the disclosure policy of the fund. Compared to full disclosure this, again, requires that the manager trades less in response to extreme signals.
To analyze how the portfolio strategy of the fund manager is affected by the level of portfo- lio disclosure, we measure the disclosure level by the fraction of the managerial signal that can be inferred from portfolio disclosure,R2=1−var(s˜|x)
σ2s. A largerR2 indicates a stronger portfolio signal. To eliminate the impact of the realized portfolio signal on our results, we characterize the optimal fund portfolio (9) by its mean and standard deviation.
The volatility of the fund holdings, std
w(s˜,x˜)
, is an indicator of the trading intensity of the fund manager. A higher volatility of the fund holdings indicates a higher aggressiveness
of the managerial trades. Figure 2 shows the optimal trading intensity of the fund depending on the level of portfolio disclosure,R2, for a single stock (N=1).
Insert Figure 2 here.
A higher level of portfolio disclosure makes the fund manager trade more aggressively on her private information. In our numerical example, the standard deviation of the stock holding is about 65% without portfolio disclosure (R2=0). With full portfolio disclosure (R2 =1), it rises to more than 100%.
The fund manager trades not only more aggressively with a higher level of portfolio disclo- sure but also chooses more risky positions, on average. This is made clear in Figure 3. Figure 3 shows the average stock holding of the fund for varying levels of portfolio disclosure.
Insert Figure 3 here.
The fund manager relies on her signal the more, the more portfolio information she reports to her fund investor. Without portfolio disclosure (R2=0), the average optimal stock holding of the informed fund manager is only 95% of that of an uninformed fund manager.6 With incomplete portfolio disclosure (R2=50%), this ratio increases to about 105%. It goes up to 117.5% in the case of full portfolio disclosure of the fund manager (R2=1).
6For an uninformed fund managerw=λσμ2 =100%.
The following corollary discusses the relation between two optimal portfolios using the same information signal, ˜s, but with different levels of portfolio disclosure, ˜xA and ˜xB, respec- tively.
Corollary 1 A fund portfolio, which is optimal for a fund investor B with portfolio signalx˜B, is not efficient for another fund investor A who receives a different portfolio signal,x˜A. The sole exception where it is efficient for investor A is the degenerated case where the signalx˜B
can be inferred fromx˜A.
Proof: Since w(s˜,x˜A) is the optimal fund portfolio with portfolio signalx˜A, any fund port- folio that is efficient for an investor with portfolio signal x˜A can be represented as linear combination of w(s˜,x˜A) and the riskless bond, conditional on x˜A. Replacingx in (9) by˜ x˜A andx˜B, respectively, we get the relation between the two optimal fund portfolios:
w(s˜,x˜B) = w(s˜,x˜A)E[ψ(s˜)|x˜B]
E[ψ(s˜)|x˜A] (10)
From (10), w(s˜,x˜B) is only a linear combination of w(s˜,x˜A) and the bond, conditional on
˜
xA, if E
ψ(s˜)|x˜B
can be inferred from x˜A. Assume that x˜B can be inferred from x˜A. Then E
ψ(s˜)|x˜B
can be inferred fromx˜A, too. From (10), the portfolio w(s˜,x˜B)therefore is efficient with portfolio signalx˜Ain this case. Otherwise,E
ψ(s˜)|x˜B
cannot be inferred fromx˜A. From (10), it therefore is not efficient if portfolio disclosure isx˜Ain this case.
q.e.d.
The corollary implies that the optimal fund portfolio without portfolio disclosure (7) is also efficient for all other levels of portfolio disclosure. In contrast, the optimal portfolio with full portfolio disclosure (5) is only efficient with full disclosure. In general, two fund managers sharing the same private information, ˜s, but differing with respect to the level of portfolio disclosure have to choose different fund portfolios.7
III. Mutual Fund Performance
We now turn to the implications of portfolio disclosure for performance evaluation. Investors have to decide on a fund investment without knowing the private information of the fund manager. They need performance measures that help them to overcome their informational handicap and to direct their money to a fund manager with superior information. Traditional performance measures, e.g. the Jensen (1968) alpha or the Sharpe (1966) ratio, estimate mutual fund performance based exclusively on unconditional fund and benchmark returns.
As shown in Section II, portfolio disclosure conveys information about the private signal of the fund manager. This section addresses the question of how fund investors can combine this additional information with readily available fund returns for performance evaluation.
Section A introduces two new performance measures which take the level of portfolio dis- closure into account. The first measure is based on Jensen’s alpha and the second extends
7Corollary 1 is a generalization of Hansen and Richard (1987). They show that efficient portfolios implied by an unconditional model are also conditionally efficient, whereas efficient portfolios implied by a conditional model are not unconditionally efficient.
the Sharpe ratio. Both measures combine observed fund returns with reported fund holdings.
While existing performance measures (e.g., Jensen (1968), Sharpe (1966), and Grinblatt and Titman (1993)) ignore either reported holdings or fund returns and can therefore lead to bi- ased performance inferences, our measures are unbiased in the notion of Dybvig and Ross (1985). They lead to correct performance inferences when applied to funds choosing opti- mal portfolios. Subsequently, Section B discusses the effects of portfolio disclosure on fund performance. It shows that additional disclosure can lead to a higher fund performance as documented by an unbiased performance measure.
A. Disclosure-Based Performance Evaluation
There are two sources of risk when investing the money with mutual funds: stock market risk and management risk. The amount of management risk depends on the level of portfolio disclosure of the fund. Therefore, incorporating portfolio disclosure in performance measures essentially means modifying the risk adjustment to allow for risk reduction resulting from portfolio disclosure.
Disclosure-Based Alpha: Traditional security market line analysis tracing back to Jensen (1968) defines mutual fund performance by the deviation of the expected fund return from the security market line. The expected fund and benchmark return and fund risk are determined using unconditional moments.Alpha therefore implicitly assumes that a fund investor only knows the unconditional moments of the fund return. However, when a mutual fund reports
its holdings, a rational fund investor uses this information to update her beliefs about the risk of the fund. Alpha, which ignores reported fund holdings, is then misspecified. The following performance measure extends Jensen’s alpha to include the level of portfolio disclosure:
DBJP(x) = μP−βP(x)μE (11)
μP and μE denote the expected returns of the fund and the unconditionally efficient bench- mark.8 The systematic fund risk,βP(x), is a function of the level of portfolio disclosure of the fund:
βP(x) = E
covR˜P,R˜E|x˜ varR˜E|x˜
(12)
Our disclosure-based alpha resembles the unconditional alpha of Jensen (1968) except for the systematic fund risk, βP(x).9 In Jensen (1968) the systematic fund risk corresponds to the unconditional fund beta. Here, it corresponds to the expected conditional fund beta, i.e. the updated systematic fund risk averaged across all possible realizations of the portfolio signal,
˜ x.10
8In conjunction with homogeneous beliefs of uninformed investors our model allows for the capital asset pricing model to hold approximately (see, e.g., Hirshleifer (1975) and Mayers and Rice (1979)). In this case, a natural benchmark is the unconditionally efficient market portfolio.
9By the law of iterated expectations, the expected fund and benchmark returns of an investor receiving the disclosure, ˜x, equal their unconditionally expected counterparts.
10Variations of the conditional fund beta reflect portfolio adjustments in response to the private information of the manager, which the fund investor can infer from her portfolio signal, ˜x. They therefore do not contribute to the systematic risk borne by the fund investor.
Dybvig and Ross (1985) evaluate performance measures by means of whether they lead to correct inferences when applied to fund managers choosing optimal portfolios. Without management fees and costs a fund manager with more information can make her investor better off by delivering a higher expected return for a given fund risk. Therefore, the fund investor should award the manager with a higher performance.
Theorem 4 The disclosure-based alpha (11) is an unbiased measure of mutual fund perfor- mance. Before fees
(i) it is zero for uninformed fund managers,
(ii) it is positive for informed fund managers who choose the optimal fund portfolio (9), (iii) it is the larger ceteris paribus, the more private information fund managers have.
Proof: See the Appendix.
Two special cases of our new performance measure deserve further discussion: without portfolio disclosure, the disclosure-based alpha corresponds to the unconditional alpha of Jensen (1968):11
DBJP(xNo) = JP = μP−covR˜P,R˜E
σ2E μE (13)
11Without portfolio disclosure, the conditional beta of the fund corresponds to the unconditional beta of the fund return, andβP(x)is consequently equal toβP=cov(R˜P,R˜E)
σ2Ein that case.
With full portfolio disclosure, the disclosure-based alpha simplifies to the portfolio change measure suggested by Grinblatt and Titman (1993):12
DBJP(xFull) = GTP = E
w(s˜)−E
w(s˜)R˜
−c (14)
Since the unconditional alpha (13) and the portfolio change measure (14) are both special cases of our measure (11), they are both unbiased measures of mutual fund performance, but only if applied to funds with specific levels of portfolio disclosure. The Jensen measure is an unbiased measure only for funds without portfolio disclosure, whereas the portfolio change measure of Grinblatt and Titman is unbiased only for funds with full portfolio disclosure.
Typically, mutual funds disclose only a noisy signal about their portfolio strategies. Then, only the disclosure-based alpha (11) allows unbiased performance inferences.
Figure 4 illustrates that both the unconditional alpha (13) and the portfolio change measure (14) can lead to erroneous performance inferences when not adapted to the portfolio disclosure of the fund. It shows the disclosure-based alpha (solid line), Jensen’s alpha (dotted line), and the portfolio change measure (dash-dotted line) of an informed fund manager who chooses the optimal fund portfolio (9). In Figure 4 the level of portfolio disclosure, R2, varies from zero (no disclosure) to one (full disclosure).
12In our model the portfolio change measure can alternatively be written as:GTP=μP−E[w(s˜)]μ. With full disclosure, the conditioning information of the fund investors corresponds to the private information of the fund manager. Therefore, the systematic fund risk (12) simplifies toβP(x) =E[w(s˜)]β, whereβis the vector of stock betas. If we use the efficiency of the benchmark with respect to public information, it follows that μ=βμE. Together, both arguments imply (14). Grinblatt and Titman (1993) uset−kportfolio holdings as proxy for the expected holdings in (14) and consider performance before fees (c=0).
Insert Figure 4 here.
Only the disclosure-based alpha (11) correctly assigns a positive performance to the in- formed fund manager for all levels of portfolio disclosure, R2. Both Jensen’s alpha and the portfolio change measure of Grinblatt and Titman can be negative although the manager has superior information and chooses the portfolio strategy (9) which is optimal for the fund in- vestor. This example demonstrates that both measures might classify a successful fund man- ager as a poor performer. They might assign a worse performance to a better informed fund manager and might even suggest that an informed fund manager is worse than the uninformed benchmark.13
Disclosure-Based Sharpe Ratio: Sharpe (1966) defines mutual fund performance by the un- conditional expected fund return per unit of the fund return standard deviation. Using similar arguments as above, we modify the Sharpe ratio to take the portfolio disclosure of mutual funds into account:
DBSP(x) = μP E
varR˜P|x (15)
The disclosure-based Sharpe ratio (15) resembles the unconditional Sharpe ratio except for the parameter of fund risk which again is the expected conditional fund risk.
13Thefirst result follows from the fact that both measures are unbiased for some disclosure policy and therefore by Theorem 4 assign a zero performance to an uninformed fund manager who by definition has less informa- tion than the informed manager. The second result can be seen if we recall that the performance of a passive benchmark investment under both measures is equal to zero.
Theorem 5 The disclosure-based Sharpe ratio (15) is an unbiased measure of mutual fund performance. Before fees
(i) it is less or equal to the Sharpe ratio of the benchmark for uninformed fund managers, (ii) it exceeds the Sharpe ratio of the benchmark for informed fund managers who choose
the optimal fund portfolio (9),
(iii) it is the larger ceteris paribus, the more private information fund managers have.
Proof: See the Appendix.
Without portfolio disclosure a fund investor only knows the unconditional fund returns, and the disclosure-based Sharpe ratio (15) simplifies to the unconditional performance mea- sure of Sharpe (1966). Therefore, the traditional Sharpe ratio is unbiased, but only when applied to funds without disclosure. For all other funds only the disclosure-based Sharpe ratio provides unbiased performance inferences.
Figure 5 compares the disclosure-based Sharpe ratio (solid line) and the traditional Sharpe ratio (dotted line) of an informed fund manager who chooses the optimal fund portfolio (9) with the Sharpe ratio of an uninformed benchmark investment (dash-dotted line). We vary the level of portfolio disclosure,R2, from zero (no disclosure) to one (full disclosure).
Insert Figure 5 here.
Only the disclosure-based Sharpe ratio (15) correctly assigns a better performance to the fund manager than to the benchmark investment for all levels of portfolio disclosure,R2. The
traditional Sharpe ratio can be smaller than the Sharpe ratio of the uninformed benchmark if the fund discloses portfolio information. Using the traditional Sharpe ratio for funds disclosing information on their holdings might classify informed fund managers as poor performers.
Better informed fund managers might be penalized by a smaller Sharpe ratio than their less informed counterparts and the Sharpe ratio of informed fund managers might even be smaller than the Sharpe ratio of the benchmark.
In summary, focusing exclusively on the fund return, Jensen’ alpha and the Sharpe ratio both neglect part of the relevant information. Therefore, they rely on an upwards biased estimate of fund risk and underestimate the performance of informed fund managers.14
B. Impact of Disclosure on Fund Performance
When offering a mutual fund, a fund manager has to decide how much information on her holdings she intends to report to her investors. This decision requires the fund manager to know how portfolio disclosure affects the performance of mutual funds. Frank, Poterba, Shackelford, and Shoven (2004) and Wermers (2001) highlight potential costs resulting from frequent portfolio disclosure of funds. This section demonstrates that more frequent portfo- lio disclosure can also be beneficial to fund investors. It investigates the impact of portfolio disclosure on the performance of a fund manager equipped with private information, ˜s. We compare the performance of this manager under two disclosure policies. In one case the man-
14Admati and Ross (1985) and Dybvig and Ross (1985) get similar results using the portfolio (5), which is optimal with full portfolio disclosure.
ager discloses a strong portfolio signal, ˜xA, and in the other case only a weak portfolio signal,
˜
xB. Corollary 2 states that the performance of the manager is the better, the more information she reports to her fund investor.
Corollary 2 The performance of an informed fund manager is better with a high level of portfolio disclosure than with a low level of portfolio disclosure. This result holds for the disclosure-based alpha (11) as well as for the disclosure-based Sharpe ratio (15).
Proof: See the Appendix.
Portfolio disclosure mitigates the informational asymmetry between fund manager and fund investor. Additional portfolio disclosure reduces the management risk faced by the fund investor. This leads to a reduction of the systematic fund risk,βP(x), and the overall fund risk, E
varR˜P|x
. Since by the law of iterated expectations the expected gross return of the fund and the expected benchmark return are unaffected by portfolio disclosure, additional portfolio disclosure increases the performance of mutual funds. Two fund investors with different in- formation about the fund holdings bear different amounts of risk investing in the same fund.
They rightly assign different performances to the same fund.
As shown by the examples in the Figures 4 and 5, the difference in performance due to different levels of portfolio disclosure is economically significant for reasonable parameter values. According to Figure 4, a fund manager with private information of qualityR2s =15%
can boost her alpha from about 3% to more than 8% by fully reporting her holdings instead of forgoing any portfolio disclosure. In the same example, the Sharpe ratio comes up from less
than 1.45 in the case of no disclosure to almost 1.6 with full portfolio disclosure. Evidently, the disclosure policy of funds matters. Funds can improve their performance not only by providing better research, but also by reporting more information on their portfolio holdings to their investors.15
IV. Conclusion
This paper presents a model that captures a characteristic aspect of delegated fund manage- ment. Fund investors typically do not know whether fund managers have private information which enables them to generate superior returns that outweigh the management fees and ex- penses. Fund investors therefore face additional management risk. Fund managers have to take management risk into account when deciding on the composition of the fund portfolio.
Fund investors must incorporate management risk into performance evaluation.
Our results highlight the role of portfolio disclosure for portfolio selection and perfor- mance evaluation of mutual funds. First, optimal fund portfolios depend not only on the private information of the fund manager but also on the portfolio information reported to the fund investors. More detailed portfolio disclosure allows the fund manager to respond more aggressively to her private information and to choose more risky positions. Fund managers who neglect the level of portfolio disclosure will in general choose suboptimal portfolios. With
15Whether more frequent disclosure is beneficial to the fund manager is an open question. It requires to model the tradeoff between the benefits and costs of disclosure. Afirst approach into this direction is currently being done by George and Hwang (2005).
such a manager, investors might be worse off by investing with the informed fund manager instead of holding the uninformed benchmark.
Second, the level of portfolio disclosure is an important determinant of mutual fund per- formance. Performance measures must incorporate the level of portfolio disclosure of funds in order to reflect fund risk and performance appropriately. This requires that performance measures are based on both observed fund returns and reported fund holdings. We develop performance measures which combine both sources of information and therefore allow unbi- ased performance inference. Assuming reasonable parameters values, we show that the biases of existing measures focusing exclusively on the fund returns (e.g., Jensen (1968) and Sharpe (1966)) or the fund holdings (e.g., Grinblatt and Titman (1993)) are important. Both measures might classify successful fund managers as poor performers. Based on our unbiased measures, we show that additional portfolio disclosure can be beneficial to fund investors by increasing the performance of the fund.
The level of portfolio disclosure was exogenous in our model. Our research can be gener- alized to determine the optimal level of portfolio disclosure by additionally modeling potential disclosure costs. For example, one could think of modeling the management fee as a function of the level of portfolio disclosure in order to incorporate disclosure costs. We leave this for further research.
References
Admati, Anat, and Stephen Ross, 1985, Measuring investment performance in a rational expectations equilibrium model,Journal of Business58, 1–26.
Brown, Stephen J., and William N. Goetzmann, 1995, Performance persistence,Journal of Finance50, 679–698.
Carhart, Mark M., 1997, On persistence in mutual fund performance,Journal of Finance52, 57–82.
Daniel, Kent D., Mark Grinblatt, Sheridan Titman, and Russ Wermers, 1997, Measuring mutual fund performance with characteristic based benchmarks,Journal of Finance52, 1035–1058.
Dybvig, Philip H., and Stephen A. Ross, 1985, Differential information and performance measurement using a security market line,Journal of Finance40, 383–399.
Ferson, Wayne E., and Andrew F. Siegel, 2001, The efficient use of conditioning information in port- folios,Journal of Finance56, 967–982.
Frank, Mary M., James M. Poterba, Douglas A. Shackelford, and John B. Shoven, 2004, Copycat funds: Information disclosure regulation and the returns to active management in the mutual fund industry,Journal of Law and Economics67, 515–541.
Gelfand, I. M., and S. V. Fomin, 1963,Calculus of Variations(Prentice-Hall, Englewood Cliffs, N.J.).
George, Thomas J., and Chuan-Yang Hwang, 2005, Disclosure policies of investment companies, Un- published Working Paper.
Grinblatt, Mark, and Sheridan Titman, 1989, Portfolio performance evaluation: Old issues and new
, 1993, Performance measurement without benchmarks: An examination of mutual fund re- turns,Journal of Business66, 47–68.
Hansen, Lars P., and Scott F. Richard, 1987, The role of conditioning information in deducing testable restrictions implied by dynamic asset pricing models,Econometrica55, 587–613.
Hirshleifer, J., 1975, Speculation and equilibrium: Information, risk, and markets, Quarterly Journal of Economics89, 519–542.
Iskoz, Sergey, and Jiang Wang, 2003, How to tell if a money manager knows more,Working Paper.
Jensen, Michael C., 1968, The performance of mutual funds in the period 1945-1964, Journal of Fi- nance23, 389–416.
, 1972, Optimal utilization of market forecasts and the evaluation of investment performance, in Giorgio P. Szeg¨o and Karl Shell, ed.: Mathematical Methods in Investment and Finance(North Holland, Amsterdam).
Lakonishok, Josef, Andrei Shleifer, Richard Thaler, and Robert Vishny, 1991, Window dressing by pension fund managers,American Economic Review81, 227–231.
Malkiel, Burton, 1995, Returns from investing in mutual funds 1971 to 1991, Journal of Finance50, 549–572.
Mamaysky, Harry, Matthew Spiegel, and Hong Zhang, 2003, Estimating the dynamics of mutual fund alphas and betas,Working Paper.
Markowitz, Harry M., 1952, Portfolio selection,Journal of Finance7, 77–91.
Mayers, David, and Edward M. Rice, 1979, Measuring portfolio performance and the empirical content of asset pricing models,Journal of Financial Economics7, 3–28.
Musto, David, 1999, Investment decisions depend on portfolio disclosures, Journal of Finance 54, 935–952.
Schott, James R., 1997,Matrix Analysis for Statistics(John Wiley and Sons, New York).
Seierstad, Atle, and Knut Sydsaeter, 1987,Optimal Control Theory with Economic Applications(North Holland, Amsterdam).
Sharpe, William F., 1966, Mutual fund performance,Journal of Business39, 119–138.
Wermers, Russ, 2001, The potential effects of more frequent portfolio disclosure on mutual fund per- formance,Investment Company Institute Perspective7.
Appendix A. Proofs
Proof of Theorem 3:In integral notation, the problem of the fund manager (3) is:
maxw Φ = ∞
−∞...
∞
−∞w(s,x) μ+s
f(s|x)ds−c (A1)
−λ 2
∞
−∞...
∞
−∞w(s,x) μ+s μ+s
+Σ(˜γ)
w(s,x)f(s|x)ds
− ∞
−∞...
∞
−∞w(s,x) μ+s
f(s|x)ds 2
Here, f(s|x) = f
s1,...,sN|x1,...,xN
denotes the joint density of the signals s1,...,sN conditional on portfolio disclosure x1,...,xN. The problem (A1) is an optimal control problem. It can be solved using multidimensional calculus of variations (e.g. Seierstad and Sydsaeter (1987), chapter 1, and for the multivariate extension Gelfand and Fomin (1963), chapter 7). The Euler-Lagrangefirst order conditions to the problem (A1) are
0 =
μ+s˜
−λ μ+s˜ μ+s˜
+Σ(γ)˜
w(s˜,x˜) (A2) +λ
μ+s˜ ∞
−∞...
∞
−∞w(s,x) μ+s
f(s|x)ds,
wherew(s,˜x)˜ denotes the optimal reaction function of the manager. Solving (A2) forw(s,˜ x), we get:˜
w s,˜x˜
= 1
λ μ+s˜ μ+s˜
+Σ(γ)˜ −1 μ+s˜
1+λER˜
P|x˜
= 1 λ
Σ(˜γ)−1−Σ(γ)˜ −1 μ+s˜
μ+s˜ Σ(˜γ)−1 1+
μ+s˜
Σ(γ)˜ −1 μ+s˜
μ+s˜
1+λER˜
P|x˜
(A3)
= 1
λΣ(γ)˜ −1
μ+s˜ 1+λER˜
P|x˜ 1+
μ+s˜
Σ(γ)˜ −1 μ+s˜,
where we use that
ER˜
P|˜x
= ∞
−∞...
∞
−∞w(s,x) μ+s
f(s|x)ds (A4)
defines the conditional expectation of ˜RPgiven disclosurex. The second equality in (A3) follows from the use of matrix algebra (e.g. Schott (1997), p. 9f.).
Tofind an explicit solution for w(s,˜ x), we˜ first calculate ER˜
P|x˜
. Inserting (A3) into (A4), we get:
ER˜
P|˜x
= 1 λE
μ+s˜
Σ(˜γ)−1
μ+s˜ 1+λER˜
P|x˜ 1+
μ+s˜
Σ(γ)˜ −1
μ+s˜x˜
(A5)
Solving (A5) for ER˜
P|x yields:
ER˜
P|x˜
= 1 λ
E μ+s˜
Σ(γ)˜ −1 μ+s˜
ψ(s˜)x˜ E
ψ(s˜)|x˜ (A6)
= 1 λ
1 E
ψ(s)|˜ x˜−1
,
whereψ(s˜)is defined according to (8). Replacing ER˜
P|x˜
in (A3) by (A6), we get the explicit solution (9) for the optimal fund portfolio. This proves Theorem 3.
With full portfolio disclosure, ˜x=w
Full(s˜). Therefore, E ψ(s˜)|x˜
=E
ψ(s˜)|w
Full(s˜)
=ψ(s˜)pro- vided that the optimal fund portfolio with full disclosurew
Full(s)is strictly monotonic ins. Replacing E
ψ(s)|˜ x˜
by ψ(s)˜ in (9) yields the solution (5) for the optimal fund portfolio with full disclosure.
Inspecting (5) we verify that w
Full(s) indeed is strictly monotonically increasing in s. This proves
Without disclosure the expectations of the fund investors in (9) correspond to the unconditional expectations. Replacing E
ψ(s˜)|x˜ by E
ψ(s˜)
in (9) we get the solutionw
No(s˜)according to (7). This proves Theorem 2.
q.e.d.
Proof that neglecting the portfolio disclosure can render the fund investor worse off:Assume for simplicity that there is only a single stock (N =1). Take for example a fund manager who chooses w
Full(s˜)according to (5) without disclosure. The optimal fraction,y, of money for the fund investor to invest in the fund is the solution to the problem
maxy Φ(y) = E yw
Full(s˜) μ+s˜
−λ
2var yw
Full(s˜)
μ+s˜+γ˜
= yE μ+s˜
λσ2γ
μ+s˜
−λ 2y2var
μ+s˜ λσ2γ
μ+s˜+˜γ
(A7)
= y
μ2+σ2s λσ2γ
−λ 2y2
σ2γ
μ2+σ2s +2σ2s
2μ2+σ2s λ2σ4γ
,
where we have inserted a multipleyofw
Full(s)˜ according to (5) into (6) and used the fact that E
˜ s4
= 3σ4s from the normality of ˜s( ˜s∼N
0,σ2s
). Solving thefirst order condition foryyields:
y = σ2γ
μ2+σ2s σ2γ
μ2+σ2s +2σ2s
2μ2+σ2s (A8)
Inserting (A8) into (A7), the maximum utility the investor can achieve investing with the informed fund manager is:
Φ y
=
μ2+σ2s2 2λ
σ2γ
μ2+σ2s +2σ2s
2μ2+σ2s (A9)
The expected utility from a passive benchmark investment (α=λ(σ2μ
γ+σ2s)) is given by:
Φ
B = αμ−λ 2var α
μ+s˜+γ˜
= μ
λ
σ2s+σ2γ−λ 2
μ2 λ2
σ2s+σ2γ2
σ2s+σ2γ
(A10)
= μ2
2λ
σ2s+σ2γ
If we subtract (A10) from (A9), the expected difference in utility is:
Φ(y)−Φ
B = σ2s
σ2−μ2
μ2+σ2s
−2μ4 2λσ2
σ2γ
μ2+σ2s +2σ2s
2μ2+σ2s (A11)
This difference is negative provided thatσ2<μ2. It is also negative for some parameter constellations whereσ2>μ2, depending on the size ofσ2s. Only ifσ2≥3μ2,Φ(y)>Φ
B for everyσ2s>0.
q.e.d.
Proof of Theorem 4:
Statement (i): If the manager is uninformed both ˜wand ˜xare independent of ˜R. If we insert (12) into
(11) and use the efficiency of the benchmark based on only unconditional information (α= 1λΣ−1μ), DBJP(x)simplifies to:
DBJP(x) = E
˜ wR˜
−E E
˜
wR˜R˜α|x˜
−E
˜ wR|˜ x˜
μα var
αR|˜ x˜
αμ
= E
˜ wR˜
−E E
w|˜ x˜
ER˜R˜α|˜x
−E
˜ wR|˜ x˜
μα αΣα
αμ
= E
˜ w
μ−E E
˜ w|x˜
μμ+Σ
Σ−1μ1λ−E
˜ w|x˜
μμΣ−1μλ1
λ12μΣ−1ΣΣ−1μ
μΣ−1μ1
λ (A12)
= E
˜ w
μ−μΣ−1μ μΣ−1μE
˜ w
μ
= 0
This proves statement (i).
Statement (ii): Inserting (12) in (11) yields forDBJP(x):
DBJP(x) = E
ER˜P|x˜
−covR˜P,R˜E|x˜
varR˜E|x˜ ER˜E|˜x +cov
covR˜P,R˜E|˜x
varR˜E|x˜ ,ER˜E|x˜
(A13)
In the remainder of the proof we need covR˜P,R˜E|x˜
and varR˜P|x˜
in optimum. Using iterated expec- tations andw(s,˜x)˜ according to (9), we get:
covR˜
P,R˜E|x˜
= E
w(s,˜ x)˜ R˜R˜αx˜
−ER˜
P|x
ER˜E|x˜
= E
w(s,˜ x)˜ ER˜R˜|˜s,ε˜ αx˜
−ER˜
P|˜x
ER˜E|x˜
(A14)
= 1 λE
ψ(s)˜ E
ψ(s)|˜ x˜ μ+s˜
Σ(˜γ)−1 μ+s˜
μ+s˜
+Σ(˜γ) αx˜
−ER˜
P|x˜
ER˜E|˜x
= 1 λ
E
ψ(s)ψ(˜ s)˜ −1 μ+s˜
αx˜ E
ψ(s˜)|x˜ −ER˜
P|x˜
ER˜E|˜x ,
where ˜ε=x˜−w(s,˜x)˜ denotes the vector of disturbances of the fund holdingsw(s,˜ x). If we substitute˜ ER˜
P|x˜ 1−E
ψ(s˜)|x˜ for 1
λE ψ(s˜)|x˜
from (A6), (A14) simplifies to:
covR˜
P,R˜E|˜x
= ER˜
P|x˜
ER˜E|˜x E ψ(s)|˜ x˜ 1−E
ψ(s)|˜˜ x
= 1
λER˜E|˜x
(A15)
varR˜
P|x˜
= E
w(s˜)R˜R˜w(s˜)x˜
−ER˜
P|x˜2
= E
w(s˜,x˜)ER˜R˜|s˜,ε˜
w(s˜)x˜
−ER˜
P|x˜2
(A16)
= 1 λ2E
ψ(s˜)2 E
ψ(s)|˜˜ x2 μ+s˜
Σ(˜γ)−1 μ+s˜
μ+s˜
+Σ(˜γ)
Σ(˜γ)−1
μ+s˜x˜
−ER˜
P|x˜2
= 1 λ2
E
ψ(s˜)2ψ(s˜)−1 μ+s˜
Σ(γ)˜ −1 μ+s˜
|x˜ E
ψ(s)|˜˜ x2 −ER˜
P|x˜2
If we substitute ER˜
P|x˜2 1−E
ψ(s)|˜˜ x2 for 1
λE
ψ(s)|˜ x˜2
according to (A6), (A16) simplifies to:
varR˜
P|x˜
= ER˜
P|x˜2E ψ(s˜)
ψ(s˜)−1−1
|x˜ 1−E
ψ(s)|˜ x˜2 −ER˜
P|˜x2
= ER˜
P|x˜2 E ψ(s)|˜ x˜ 1−E
ψ(s)|˜˜ x, (A17)
