Experiments on Asset Pricing under Delegated Portfolio Management
Experiments on Asset Pricing under Delegated Portfolio Management
Elena Asparouhova
Peter Bossaerts
Jernej Copic
University of Utah
Caltech and EPFL
UCLA
Bradford Cornell
Jaksa Cvitanic
CRA and Caltech
Caltech
Debrah Meloso∗ University of Bocconi
This version November 2009 Work in progress–please, do not circulate.
∗
Corresponding author: Debrah Meloso, Department of Decision Sciences, Bocconi University, Via Guglielmo Roentgen, 20136 Milan, Italy, Tel: +39 02 5836 5931, email: [email protected] .
1
Introduction
The fundamental CAPM papers of Sharpe (1964) and Lintner (1965) that laid the groundwork for modern asset pricing began with the assumption that investors made their own investment decisions. This assumption was motivated, in part, to allow for the derivation of a closed form asset pricing model. However, at the time the CAPM was being developed, the assumption that final investors made investment decisions was not only theoretically useful, it was empirically accurate. That is no longer the case. To provide some historical perspective, as late as 1950 American households held 91% of all common stocks. By the end of 2004, the figure had fallen to 32%.1 Furthermore, this figure overstates the importance of individual investors because very wealthy individuals, such as Bill Gates, are treated as individual investors even though they delegate most of their investment decision making to financial firms or private investment advisers. In addition, most less wealthy individual investors rely on stockbrokers, financial planners, or other advisers, when making investment decisions, even when the investment account is in their name. Finally, the impact on asset prices of money managers exceeds the level of their relative holdings because they have become such active traders. By way of comparison, the average turnover for equity mutual funds during the years from 1950 to 1965 was 17%, over the period from 1990 through the first half of 2005 it was 91%. Given the dramatic increase in the importance of delegated investing, it is surprising that the impact of delegation on asset pricing has not been more thoroughly investigated. Instead of studying the effect of delegation, efforts to overcome the empirical failures of early asset pricing models have focused on two alternatives. The first is to retain the rational valuation framework, but to build models based on increasingly sophisticated assumptions regarding the stochastic processes of returns and investor preference functions. Contributions in this regard include Breeden 1
The historical data in these paragraphs is taken from Bogle (2005).
(1979), Grossman and Shiller (1981), Hansen and Singleton (1983), Campbell (1996), Constantinides and Duffie (1996) and Campbell and Cochrane (1999), among many others. The second alternative has been to abandon the rational framework and develop “behavioral models” based on various psychological theories. Lacking a specific theoretical framework, a wide variety of behavioral theories have been developed. Fama (1998) provides a critical review of this literature. Barberis and Thaler (2002) offer a more recent, and more favorable, survey. It is somewhat ironic that the behavioral approach largely ignores a central aspect of investment behavior—delegating security selection to professionals. There are strands of literature that have looked at the problem of delegated investing. The most extensive involves the application principal-agent theory to study various hypothetical manager-agent contracts. The early seminal paper in that respect is Bhattacharya and Pfleiderer (1985). Since then, important contributions include Stoughton (1993), Admati and Pfleiderer (1997), Ross (2004) and Dybvig, Farnsworth and Carpenter (2004).2 From the perspective of the current research, this literature has two deficiencies. First, it is largely formal and theoretical and does not tie the models to the actual process of delegated investing observed in the capital market. Second, and more importantly, it does not derive the implications of the delegation process for asset pricing. There is a second smaller literature that does attempt to assess the impact of delegated investing on asset prices. An early contribution is Brennan (1993). More recently, Cornell and Roll (2005) study how the CAPM is affected when the model is extended to allow for delegated investing. They show that, with no other alterations, introducing delegation has a significant impact on the form of the CAPM that can potentially explain various failings of the original model. Unfortunately, because so little is known about the actual delegation process, papers like Cornell and Roll are forced to rely on stylized assumptions regarding the manner in which delegation 2
Stracca (2005) provides a comprehensive survey of this literature.
3
occurs. One approach that has not been tried is to study the impact of delegation in a laboratory setting. This is surprising because experiments could provide useful way to study delegated portfolio management and its effect on asset pricing. Unlike field data, the experimenter can control many of the variables that are crucial for understanding financial markets, such as total supplies, information, contractual agreements, enforcement of contracts, etc. Indeed, it is only under experimental conditions that CAPM pricing has ever been observed (see Bossaerts and Plott, 2004) and that it has been shown why CAPM pricing may obtain even if no-one chooses to hold the market portfolio (see Bossaerts, Plott, and Zame, 2007 ). This raises the obvious question of whether the CAPM would continue to hold if managers were introduced into the experiments. As experiments allow one to control the exact contract between “investor-subjects” and “investor-managers,” as well as the flow of information between the two groups, in principle, experiments could also provide important insights about the impact of contracts on asset prices and fund composition. The present paper provides a first step towards the experimental analysis of delegated portfolio management. It presents the results of a baseline experiment against which general equilibrium effects of incentives (contract design) on performance, prices and choices could be studied. Specifically, it reports on an experiment that was set up in exactly the same way as earlier experiments that have reliably generated CAPM pricing (see earlier references), with one exception: subjects do not trade for their own account, but for the benefit of other subjects. The “investorsubjects” (the Investors) are endowed with assets and cash and allocate those to “manager-subjects” (the Managers). The latter can then trade the assets for cash in anonymous electronic open-book markets within a pre-specified time period. The Investors are paid a share of the (liquidating) dividend of the assets as well as the final cash in the resulting portfolio, while the Managers are paid a percentage of the
4
value of the assets and cash that they received (i.e., they receive payment for order flow). Basic performance metrics are then reported, based upon which Investors may assign fresh allocations of assets and cash to the Managers. Earlier experiments (Bossaerts, Plott, and Zame, 2007) showed that when subjects trade for their own account, CAPM pricing obtains, and while portfolio holdings are quite erratic, choices reflect demands that deviate from mean-variance optimality only because of a mean-zero error term. Evidently, this error term is independent across subjects, which is why CAPM pricing still obtains – provided, of course, there are enough subjects who all have an endowment that is small relative to the market as a whole. As such, CAPM pricing obtains because of the (functional) Law of Large Numbers. We replicate some of these results here in a trading session where there are no Investors and Managers could trade for their own account. This “pre-experimental session” provides calibration for the actual delegated portfolio management experiment. Our main experimental setup is based on the simplest possible management contract design. In fact, it is plausible to argue that portfolio delegation should have no effect on asset prices or holdings. All Managers have equal information, so, barring unequal trading skill, there is no reason to believe that one Manager could outperform the others. Assuming mean-variance preferences, managers should all choose portfolios on the mean-variance frontier, and the Investors should choose to invest in managers in a way so that the Investor’s overall portfolio exactly corresponds to his optimal portfolio. The erratic nature of portfolio choices could, however, cause some Managers to outperform others by pure chance. Being indifferent as to how to allocate their assets and cash across Managers, Investors may resolve their indifference by choosing to send their endowment to the Manager who happened to have had the best performance in the past. This way, the Investors hedge against the (remote) possibility that this Manager actually outperformed because of skill. But if many investors choose to
5
do so, this Manager’s fund will be large relative to the market, and her demand will eventually influence prices, unless she happens to have exact mean-variance optimal demands, a rather unlikely event in view of the past evidence. As a result, CAPM pricing will no longer obtain. The following provides a succinct description of our findings. In early rounds, Investors allocate their shares and cash so that Managers all effectively receive the same initial endowments. Despite rather erratic holdings at the end of trading, CAPM pricing obtains. After the third round, however, most Managers did very poorly (because the market portfolio did badly) except for a few Managers who had chosen portfolios that over-weighted the assets that happened to do well. In subsequent rounds, these successful Managers received increasingly large allocations of assets and cash – at one point, four out of the 30 Managers were given 40% of the available assets and cash. We find that there was a correlation of 0.66 between a manager’s funds distributed back to investors in the previous period (after the realization of uncertainty and after fees) and the flow to that investor in the next period. Managers with large fund flows were more likely to have large fund flows in the next period as well, a finding that does not disappear when past returns are accounted for. Thus, there seems to be “stickiness” in fund flows. Moreover, investors do not appear to use mean-variance as a criterion in choosing the funds to invest in. While not given the information about the expected return and variance of managers portfolios directly, the investors are given enough information to be able to infer it. We find that next period fund flows depend positively on last period portfolio variance and negatively on last period expected return. As expected from the above described investor behavior, as the size of the largest manager grew, CAPM pricing no longer obtained. As a matter of fact, the equity premium became negative. The size of the largest manager significantly positively correlated with the mis-pricing in the market (as measured by the difference of the Sharpe ratio of the market portfolio from the Sharpe ratio of the mean-variance optimal portfolio). Our results indicate
6
that it is more the size of the largest manager more than the segmentation of the markets into large and small funds that matters for explaining the CAPM mispricing.
2
Experimental Setup
The experiment consists of one main session and one “end” session. Subsection 2.4 describes the design of the end session. Here we outline the design of the main session. This session is conducted in a series of six periods. Individuals who participate in trading during the trading periods are called managers. The 32 managers are the same individuals over all periods. A separate group of participants, called investors, are the initial owners of assets and cash. The investors need not be the same individuals each period and their number (equal to 70 on average) can possibly change. Investors receive their endowments in the beginning of each period. Their endowments consist of units of two risky assets, called A and B, and some cash. The assets are risky because their end-of-period payoffs (in US dollars) depend on the random realization of one of three states of the world, called X, Y , and Z. Investors cannot buy or sell assets directly, thus in the beginning of each period they must assign fractions of their initial resources to different managers to trade on their behalf. The allocations from different investors to one manager constitute this manager’s initial (for that period) portfolio. The fraction of this initial portfolio that is due to a single investor is this investor’s share in the manager’s fund. The managers’ initial portfolios are the sole determinant of their payoffs: the current period payoff of a manager is equal to 40% of the (expected) value of her initial portfolio. We refer to this payoff as the manager’s fee. The allocation of assets from investors to managers constitutes the first stage of a period. Managers (only) then participate in the second stage of the period, called trading stage and lasting thirty minutes. In this stage they may trade their initial portfolio to a new, final portfolio, which generates a dividend for investors according to a random realization of the state of the world (that only becomes known after the conclusion of trading). The dividend, with the 7
management fee subtracted from it, is distributed to the investors according to their shares in the fund. If this residual dividend is negative, the end-of-period payoff of all investors in the fund is equal to $0. To facilitate borrowing, in addition to trading securities A and B, managers can trade a risk-free security called a “Bond.” The Bond pays off $1 in any state of the world and is in zero net supply. All securities are traded in parallel continuous (limit-order) markets. The third and final stage of a period is the information disclosure stage. It can also be regarded as stage “zero” of the following period. In this stage a series of performance indicators for each manager are published on the experimental webpage and the university newspaper. The details of the information disclose are presented in subsection 2.3. Periods are weekly, and except for the information about past periods and the fact that managers are always the same, they are independent events. In order to avoid any last-period effects in managers’ behavior, the design includes one “last” period. In this period the managers are paid based on the realized dividend of their final portfolio, i.e., instead of being paid 40% of the expected value of the initial portfolio, they are paid 40% of the realized dividend. The remaining 60% are distributed to the investors as before, i.e., according to their shares in each fund.3 The following subsections give more detail about payoffs and trading rules, as well as about the disclosed information. Additionally, the investors’ and managers’ instructions are available on the web at http://clef.caltech.edu/exp/dp.
2.1
Trading: Assets and Dividends
Table 1 summarizes the dividends, expressed in US Cents, of the three traded assets in each state of the world. In each period the three states, X, Y, and Z, are equally likely. This information is known to both managers and investors. 3
The initial design included eight periods followed by the “last” period. While we conducted eight periods, we report the results only of the first six due to an error that caused incorrect reporting in the information disclosure stage. Thus, in effect the “last” period’s purpose of eliminating the
8
Table 1: State-dependent asset dividends in US Cents. State X Y Z Asset A 5 80 0 Asset B 0 30 80 Bond 100 100 100 According to their initial endowment, there are two types of investors. An investor of type A holds 100 units of asset A, and $6 of cash, while investor of type B holds 70 units of asset B, and $9 of cash, respectively. Because the total number of investors as well as the fraction of investors of each type varied from one period to the other, the market portfolio also varied across periods. The exact numbers for every period are summarized in Table 2. Table 2: Number of participants by type and the corresponding market portfolio weights for each period. Participants Market Portfolio Weights A B Cash Period Type A Type B I 30 34 46.9 37.2 7.59 II 28 38 42.4 40.3 7.73 37 34 52.1 33.5 7.44 III IV 37 35 51.4 34 7.46 V 34 33 50.7 34.5 7.48 35 35 50 35 7.5 VI
Investors can only choose the number of units of the risky assets (A or B, depending on the investor’s type) in their portfolio to allocate to each manager. If a manager is allocated a fraction of an investor’s risky portfolio, the same fraction of the investor’s cash is allocated to that manager as well. Investors distribute holdings to managers using a form over the internet. Before trading starts, each manager knows her initial portfolio but not those of the other managers. During the trading stage all securities can be sold short and a bankruptcy rule is used to prevent agents from committing to trades that would imply negative cash last period effect was fulfilled by periods 7 and 8, which are now excluded from the data analysis.
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holdings in the end of the period. Whenever a manager attempts to submit an order to buy or sell an asset, her cash holdings after dividends are computed for all states of the world, given her current asset holdings and her outstanding orders (orders awaiting trade) that are likely to trade, including the order she is trying to submit. If these hypothetical cash holdings turn out negative for some state of the world, she is not allowed to submit the order.
2.2
Payoffs
Below we formalize the payoff functions for the investors and managers in the main experiment. Let wg be the initial endowment of assets for a type g investor, g ∈ {A, B}, and denote investor ig ’s let hg be his initial cash holdings. Let mig = m1ig , . . . , m32 ig distribution of his initial asset endowment among the 32 managers, ig ∈ {1, . . . , Ig }, where Ig is the total number of investors of type g. The initial portfolio of manager j, j ∈ {1, . . . , 32}, is composed of mjA units of asset A, and mjB units of asset B, where mjA =
IA X
mjiA ,
mjB =
iA =1
IB X
mjiB .
iB =1
Manager j’s initial cash holding is IA IB X X mjiA mjiB h = hA + hB . w w A B i =1 i =1 j
A
B
The expected value of manager j’s initial portfolio determines that manager’s ¯ g denote the mean dividend of asset g. The mean-value of payoff. Specifically, let D manager j’s portfolio is ¯ A mj + D ¯ B mj + hj . V¯ j mj , hj = D A B
10
(2.1)
Manager j’s payoff is 40% of the mean-value of her initial portfolio. That is, P ay j = 0.4 × V¯ j mj , hj .
(2.2)
Investor ig ’s share in fund j is the fraction j
¯ g mj + mig hg D ig w j sig = ¯ j j gj . V (m , h )
(2.3)
Given her initial portfolio of risky assets, mj = mjA , mjB , and her cash holding, hj , manager j can trade to a final portfolio, denoted m ˜j = m ˜ jA , m ˜ jB , and final ˜ j .4 These holdings and the realized state of the world, cash plus bond holdings h x ∈ {X, Y, Z} determine manager j’s earnings, Πj , as follows: ˜ j ; x = DA (x) m ˜j, Πj m ˜ j, h ˜ jA + DB (x) m ˜ jB + h where Dg (x) denotes the dividend of asset g in state of the world x. Manager j’s earnings after the manager’s fee, to which refer as manager j’s resid ual, equal Πj − 0.4 × V¯ j . Investor ig ’s earnings from manager j equal his share in this manager’s residual, i.e., sjig Πj − 0.4 × V¯ j . Investor ig ’s payoff is the sum of his earnings from all managers, as given in the expression below:
P ayig (x) =
32 X
sjig
h i j j ˜j j j j ¯ Π m ˜ , h ; x − 0.4 × V m , h .
(2.4)
j=1
2.3
Disclosure of Information
Experimental periods were run on Tuesday every week. Indicators of managers’ performance in a given trading period were published on the following Monday. 4
The dividend on Bonds is $1. Additionally, we normalize prices so that the price of Bonds is ˜ j in all definitions we give here. $1. This allows us to merge Bonds and cash in one term, h
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Numerical values of the indicators were published both in the university newspaper (The California Tech) and on the experimental web-page. Indicators of performance were announced for all managers and for a market index, called the Dow-tech, which was composed of one unit of asset A, one unit of asset B, and $1. For each fund and the index, we reported the following four indicators: Returns, Market Share5 , Residual, and Risky Share. In what follows we describe these indicators in detail. We first define the value of a mutual fund’s initial portfolio to be the valuation of this portfolio in terms of asset prices (as opposed to its mean-value, which is defined in terms of asset mean dividends). The value, V j , of manager j’s initial portfolio is V j mj , hj ; p = pA mjA + pB mjB + hj , where p = (pA , pB , 1) is the vector of asset prices, normalized so that the price of the bond is equal to 1 (the Bond is the numeraire). In continuous markets assets are traded at many different prices. We take p to be the average price over the last five minutes of trade in a period. The (realized) Return of mutual fund j when the realized state of the world is x and the average trading price is p, is given by
rj =
˜ j ; x − V j (mj , hj ; p) ˜ j, h Πj m V j (mj , hj ; p)
.
(2.5)
The Market Share of mutual fund j is the ratio of its mean-value with respect to the total mean-value of all mutual funds. Specifically, it is given by V¯ j (mj , hj ) . msj = P32 ¯k k k k=1 V (m , h )
(2.6)
The Residual of mutual fund j is the difference between this fund’s earnings 5
The Market Share indicator was called Volume in the published reports.
12
and the manager’s fee (as defined in Subsection 2.2), if this number is non-negative. The residual is zero otherwise. That is, j j j ˜j ¯ Π = max{Π m ˜ , h ; x − 0.4 × V¯ j mj , hj , 0}.
(2.7)
The Risky Share of mutual fund j is the fraction of the total value of mutual fund j’s final portfolio in terms of prices, that is given by holdings of risky assets. It is made specific in the following equation: νj =
pA m ˜ jA + pB m ˜ jB , ˜j pA m ˜ jA + pB m ˜ jB + h
(2.8)
where p is again taken to be the average price for the last five minutes of trade in a period. Tables 3 to 6 contain the values of these indicators, as published in the university newspaper and the experimental web-page, for every experimental period.
2.4
End Session
One week after the last period of the main session, another pseudo-period was run, that looked like other periods of the main session in all except the managers’ payoff. Investors made their distributions of assets among managers and managers participated in a 30-minutes trading period, in exactly the same way as in the main session. At the end of the trading time, each manager received a fee equal to 40% of the earnings generated by her final portfolio and the draw of the state of the world. Investors were again paid their share of the manager’s earnings after the fee, which in this case equaled their share taken from the remainder 60% of earnings. In other ˜ j ; x in state of words, manager j’s payoff in the end session equaled 0.4 × Πj m ˜ j, h i P32 j h j j ˜j the world x. Investor ig ’s payoff equaled j=1 sig 0.6 × Π m ˜ ,h ;x . The reasoning underlying the addition of the end session is the prevention of an unraveling of manager’s reputation considerations, and a consequent mistrust of 13
Table 3: Return (in %) of every mutual fund and every period of the main experimental session. The number reported in the table is 100 × rj , where rj is as defined in equation (2.5). Experimental Period (State Realization) I (Z) II (Y) III (X) IV (Y) V (Y) VI (X) 20.54 82.99 -77.86 56.99 57.00 -79.70
Dow-Tech Mutual Fund: Albite -44.18 Alexandrite 119.13 10.47 Allanite Alunite 38.82 65.85 Amazonite Amblygonite -134.56 Amosite 0.72 -21.88 Andalusite Anthophyllite 33.23 Atacamite 33.52 13.60 Barite 43.71 Bassanite Beidelite 17.53 -1.39 Bementite Bentonite -2.15 Bertrandite 115.89 10.24 Biotite 28.48 Birnessite Bloedite -98.78 Boracite 22.92 Calcite 26.90 Carnallite -100.51 Celestite 13.00 Chalcopyrite 15.59 Chlorite 5.94 Colemanite -1.19 Cornadite 23.03 Cristobalite 33.17 Cryolite 67.25 Dolomite -15.93 Dumortierite 161.05 Dunite -39.27
396.59 113.11 49.25 151.01 90.56 128.56 190.00 78.60 79.59 99.66 42.12 103.08 125.26 256.98 160.26 -4.83 99.33 -8.78 220.04 134.16 143.55 255.01 38.49 123.11 100.74 194.06 197.94 124.76 51.22 2.40 98.53 70.75
-82.53 8.04 -44.77 -92.82 -65.71 -151.04 -75.85 -139.44 -119.18 -73.12 -68.75 -84.90 -88.90 -83.28 -79.58 -79.86 -65.89 -68.71 -56.92 -32.08 -36.50 -100.67 -68.88 -121.49 -90.37 42.61 -69.36 -74.29 -100.37 -100.54 -68.26 -54.42
14
124.55 95.97 183.56 37.93 140.10 169.80 46.72 133.15 45.22 11.05 -5.77 180.47 57.75 65.47 34.24 193.54 64.34 -101.90 -50.20 65.30 -7.17 206.02 58.69 -3.00 103.16 -28.34 157.00 -80.28 -22.45 69.67 49.24 57.26
66.00 -22.00 105.00 134.00 139.00 156.00 8.00 50.00 -22.00 72.00 28.00 69.00 9.00 177.00 62.00 145.00 59.00 -46.00 -99.00 66.00 -17.00 206.00 -12.00 103.00 71.00 52.00 149.00 2.00 37.00 -65.00 48.00 -3.00
-190.00 95.00 -5.00 -107.00 -125.00 -93.00 -149.00 -527.00 -145.00 -96.00 -97.00 -119.00 54.00 -13.00 -109.00 93.00 -74.00 56.00 -58.00 -120.00 -52.00 -127.00 3.00 -60.00 -125.00 -68.00 -131.00 -131.00 -121.00 -74.00 -90.00 76.00
Table 4: Market Share (in %) indicator for every mutual fund in every period. The number reported in the table is 100 × v j , where v j is as defined in equation (2.6).
I Mutual Fund: Albite Alexandrite Allanite Alunite Amazonite Amblygonite Amosite Andalusite Anthophyllite Atacamite Barite Bassanite Beidellite Bementite Bentonite Bertrandite Biotite Birnessite Bloedite Boracite Calcite Carnallite Celestite Chalcopyrite Chlorite Colemanite Cornadite Cristobalite Cryolite Dolomite Dumortierite Dunite
2.02 2.38 2.42 2.34 4.62 5.31 2.33 3.36 2.78 3.38 2.93 2.84 3.17 2.96 2.31 2.01 2.48 2.47 3.44 3.06 3.35 3.29 2.78 4.41 4.42 3.52 2.74 3.12 3.42 4.02 2.84 3.48
Experimental Period II III IV V 0.34 11.61 1.96 2.27 6.68 4.38 0.98 0.84 1.70 4.39 2.19 3.18 1.83 0.64 0.69 10.57 1.09 7.71 0.28 1.44 1.80 0.62 1.82 1.58 2.21 0.97 3.62 1.87 6.15 12.36 1.53 0.70
8.14 10.36 1.16 5.69 6.19 2.58 3.89 1.22 0.89 2.72 1.12 2.11 2.06 4.66 1.62 2.98 1.56 0.99 3.72 1.67 2.90 3.90 1.64 1.54 2.96 2.99 6.65 3.07 2.82 2.28 1.52 2.38
15
4.28 19.95 2.03 1.88 3.71 6.60 1.81 1.08 2.03 1.25 0.46 1.09 1.37 3.20 1.12 0.90 1.19 1.04 1.86 3.30 3.80 2.32 1.23 2.94 2.67 15.66 4.42 1.55 0.91 1.20 1.82 1.34
2.42 20.34 2.91 1.78 5.65 5.48 0.77 2.27 0.86 1.74 0.84 5.66 0.89 2.23 0.78 4.54 1.36 0.58 0.79 3.77 1.41 6.53 0.89 0.49 2.09 4.82 9.08 2.22 1.13 1.97 0.91 2.80
VI 2.53 8.67 2.74 2.21 6.80 11.69 1.29 0.85 0.63 0.74 0.57 2.68 0.74 5.27 0.76 6.27 1.07 0.51 0.46 2.44 0.71 10.80 0.50 2.65 1.33 6.26 13.42 0.69 0.70 1.18 1.07 1.77
Table 5: Residual (in US Dollars) for every mutual fund in every experimental period. ¯ j , as defined in equation (2.7). The number reported in the table is Π
Albite Alexandrite Allanite Alunite Amazonite Amblygonite Amosite Andalusite Anthophyllite Atacamite Barite Bassanite Beidellite Bementite Bentonite Bertrandite Biotite Birnessite Bloedite Boracite Calcite Carnallite Celestite Chalcopyrite Chlorite Colemanite Cornadite Cristobalite Cryolite Dolomite Dumortierite Dunite
I 7.41 95.01 37.85 51.37 131.63 0.00 31.70 28.41 58.54 71.76 48.50 66.23 53.88 38.20 29.67 77.38 38.99 47.79 0.00 55.80 65.03 0.00 45.34 76.05 64.75 46.27 51.21 66.14 98.90 139.73 39.79 16.50
Experimental Period II III IV V 27.75 0.00 198.71 71.12 398.53 145.59 787.06 177.48 42.66 2.83 124.41 110.61 92.75 0.00 46.76 78.94 197.72 0.00 187.20 258.46 178.10 0.00 386.10 276.77 45.57 0.00 49.10 11.98 23.16 0.00 52.91 57.78 44.82 0.00 54.55 7.64 142.55 0.00 22.37 52.83 45.34 0.00 6.37 16.88 100.13 0.00 65.96 169.05 65.45 0.00 40.82 14.38 36.56 0.00 102.01 122.24 27.67 0.00 27.04 22.03 109.82 0.00 57.66 215.50 33.03 0.00 37.33 37.52 73.34 0.00 0.00 1.84 15.66 0.00 4.84 -7.15 51.65 8.82 104.29 109.79 67.26 12.62 51.06 14.14 38.52 0.00 155.34 403.01 35.75 0.00 37.25 9.66 54.24 0.00 42.34 18.37 67.77 0.00 109.17 63.70 49.33 66.15 126.92 125.06 172.12 0.00 241.04 435.21 67.02 0.00 0.00 31.73 134.87 0.00 8.88 25.18 134.47 0.00 39.79 -2.42 46.54 0.00 50.08 22.95 15.99 1.12 39.85 37.17
16
VI 0.00 319.36 35.66 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 20.00 58.24 0.00 227.22 0.00 13.95 0.17 0.00 1.27 0.00 7.37 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 57.17
Table 6: Risky Share (in %) for every mutual fund in every experimental period. The number reported in the table is 100 × ν j , where ν j is as defined in equation (2.8).
Dow-Tech Albite Alexandrite Allanite Alunite Amazonite Amblygonite Amosite Andalusite Anthophyllite Atacamite Barite Bassanite Beidellite Bementite Bentonite Bertrandite Biotite Birnessite Bloedite Boracite Calcite Carnallite Celestite Chalcopyrite Chlorite Colemanite Cornadite Cristobalite Cryolite Dolomite Dumortierite Dunite
I 86.61 78.88 100.49 110.26 99.75 101.77 99.77 65.11 78.9 101.64 75.97 76.8 92.39 99.87 77.55 83 99.84 77.81 13.33 57.99 99.62 70.9 117.61 45.65 80.66 37.15 0 108.77 87.53 99.97 71.69 76.44 20.89
II 84.75 242.13 100 74.32 100 96.11 99.99 112.09 120.55 99.97 106.65 62.05 99.84 111.98 106.61 98.58 69.59 75.01 11.6 88.08 99.41 61.12 122.34 22.71 99.04 99.55 101.46 120.29 82.17 63.35 31.31 96.19 65.04
Experimental Period III IV V 85.24 86.92 86.92 93.6 47.98 113.42 -10.92 109.86 82.65 57.29 99.36 26.77 100.76 99.98 99.33 72.49 114.23 76.35 172.01 89.47 82.67 86.3 90.28 113.63 150.07 104.64 120.57 127.92 101.73 104.15 80.78 8.47 60.86 85.09 29.38 39.22 99.75 99.93 96.06 99.68 55.49 11.08 92.29 11.92 54.4 90.21 77.15 90.66 98.77 115.29 70.92 77.77 99.6 93.41 75.28 -4.27 7.45 65.02 -73.78 -174.07 34.69 89.15 66.86 40.1 4.79 -42.74 123.86 119.12 119.08 75.19 98.78 56.31 135.73 -23.95 -0.49 99.56 90.74 96.46 -39.2 99.4 77.81 74.92 71.86 114.19 81.65 30.96 83.43 109.82 101.35 99.74 109.33 -17.45 35.92 74.98 61.67 93.61 58.14 99.38 -2.33
17
VI 86.6 197.28 -117.53 -7.2 119.57 121.52 102.85 145.98 475.63 159.8 101.55 92.17 121.64 -63.61 12.15 119.3 -152.87 79.08 -91.59 67.24 128.35 30.56 155.1 -18.67 75.45 120.65 97.51 145.84 132.77 122.84 78.45 95.43 -110.85
investors. In the absence of the end session, managers have no reputation reason to perform in the investors’ best interest in the last period. Investors’ knowledge of this could have lead to an equilibrium where investors do not use any performance indicators to allocate their assets and managers do not exert any effort to satisfy investors. Thus, by design, the incentives of the managers were fully aligned with those of investors. Moreover, conditional on having identical relative holdings of assets and cash, a manager endowed with a larger initial portfolio at the beginning of the end session was better off than one receiving a smaller allocation from investors. This means that the possibility of reputation considerations on the side of the managers was left intact for the main session.
3
Results in the Main Experiment
3.1
Prices and Final Holdings
Table 7 presents the market portfolio and the corresponding state-dependent aggregate wealth for each of the six periods. As can be seen the market portfolio changed from one period to the next, with changes primarily due to changes in the aggregate supply of security A. Table 7: Market portfolio and state-dependent aggregate endowments (in US dollars) in each period. Asset / Period I II III IV V VI Asset A 3000 2800 3700 3700 3400 3500 Asset B 2380 2660 2380 2450 2310 2450 State State State State
/ Period X Y Z
I 150 3114 1904
II 140 3038 2128
III 185 3674 1904
IV 185 3695 1960
V 170 3413 1848
VI 175 3535 1960
Table 8 presents the trading volume and the turnover for each asset in the six trading periods. As apparent from the table, the trading activity in each of the secu18
rities increased as the periods advanced, with the increase being more pronounced in security B, which was the security with more stable aggregate supply across periods. Table 8: Trading volume and asset turnover by period. Trading Volumea Asset / Period I II III IV V VI Asset A 1114 1484 1652 1739 1648 1988 Asset B 942 928 1257 1383 1601 1919 Bond 212 135 217 279 349 446
Asset / Period Asset A Asset B
Asset Turnoverb I II III IV 0.37 0.53 0.45 0.47 0.40 0.35 0.53 0.56
V 0.48 0.69
VI 0.57 0.78
a
Trading volume is the number of assets that traded during a period. Asset turnover is calculated by dividing the trading volume over a period by the total number of securities outstanding (as presented in Table 7). b
We next turn to evaluating the pricing “quality” in the markets across periods. The Arrow-Debreu securities pricing model makes predictions about state prices. Because we have complete markets, we can invert the prices of the traded securities for the state prices, and compute the ratio of the state prices and the state probabilities. According to the Arrow-Debreu model, these state price probability ratios should be inversely related to aggregate wealth in the corresponding state. The state when aggregate wealth was lowest should be most expensive relative to its probability. In this experiment (as evident from Table 7), the state with the lowest aggregate payoff was X. The state where aggregate wealth was highest should carry the lowest state price probability ratio. In all periods the highest payoff state was Y . Thus, the Arrow-Debreu asset pricing model predicts that in all periods the state price probability ratio for state X should be the highest, followed by that for state Z, and the ratio should be the lowest for state Y . The ranking of state-price probabilities evolved in time in each session. Figure 1 presents the evolution of state price probabilities in each of the periods. The state prices are computed every time a transaction occurs. 19
Figure 1: State-price probabilities in time for each session. SPP at every trading time, session 061030. The endowments in each state are ranked: X
Elena Asparouhova
Peter Bossaerts
Jernej Copic
University of Utah
Caltech and EPFL
UCLA
Bradford Cornell
Jaksa Cvitanic
CRA and Caltech
Caltech
Debrah Meloso∗ University of Bocconi
This version November 2009 Work in progress–please, do not circulate.
∗
Corresponding author: Debrah Meloso, Department of Decision Sciences, Bocconi University, Via Guglielmo Roentgen, 20136 Milan, Italy, Tel: +39 02 5836 5931, email: [email protected] .
1
Introduction
The fundamental CAPM papers of Sharpe (1964) and Lintner (1965) that laid the groundwork for modern asset pricing began with the assumption that investors made their own investment decisions. This assumption was motivated, in part, to allow for the derivation of a closed form asset pricing model. However, at the time the CAPM was being developed, the assumption that final investors made investment decisions was not only theoretically useful, it was empirically accurate. That is no longer the case. To provide some historical perspective, as late as 1950 American households held 91% of all common stocks. By the end of 2004, the figure had fallen to 32%.1 Furthermore, this figure overstates the importance of individual investors because very wealthy individuals, such as Bill Gates, are treated as individual investors even though they delegate most of their investment decision making to financial firms or private investment advisers. In addition, most less wealthy individual investors rely on stockbrokers, financial planners, or other advisers, when making investment decisions, even when the investment account is in their name. Finally, the impact on asset prices of money managers exceeds the level of their relative holdings because they have become such active traders. By way of comparison, the average turnover for equity mutual funds during the years from 1950 to 1965 was 17%, over the period from 1990 through the first half of 2005 it was 91%. Given the dramatic increase in the importance of delegated investing, it is surprising that the impact of delegation on asset pricing has not been more thoroughly investigated. Instead of studying the effect of delegation, efforts to overcome the empirical failures of early asset pricing models have focused on two alternatives. The first is to retain the rational valuation framework, but to build models based on increasingly sophisticated assumptions regarding the stochastic processes of returns and investor preference functions. Contributions in this regard include Breeden 1
The historical data in these paragraphs is taken from Bogle (2005).
(1979), Grossman and Shiller (1981), Hansen and Singleton (1983), Campbell (1996), Constantinides and Duffie (1996) and Campbell and Cochrane (1999), among many others. The second alternative has been to abandon the rational framework and develop “behavioral models” based on various psychological theories. Lacking a specific theoretical framework, a wide variety of behavioral theories have been developed. Fama (1998) provides a critical review of this literature. Barberis and Thaler (2002) offer a more recent, and more favorable, survey. It is somewhat ironic that the behavioral approach largely ignores a central aspect of investment behavior—delegating security selection to professionals. There are strands of literature that have looked at the problem of delegated investing. The most extensive involves the application principal-agent theory to study various hypothetical manager-agent contracts. The early seminal paper in that respect is Bhattacharya and Pfleiderer (1985). Since then, important contributions include Stoughton (1993), Admati and Pfleiderer (1997), Ross (2004) and Dybvig, Farnsworth and Carpenter (2004).2 From the perspective of the current research, this literature has two deficiencies. First, it is largely formal and theoretical and does not tie the models to the actual process of delegated investing observed in the capital market. Second, and more importantly, it does not derive the implications of the delegation process for asset pricing. There is a second smaller literature that does attempt to assess the impact of delegated investing on asset prices. An early contribution is Brennan (1993). More recently, Cornell and Roll (2005) study how the CAPM is affected when the model is extended to allow for delegated investing. They show that, with no other alterations, introducing delegation has a significant impact on the form of the CAPM that can potentially explain various failings of the original model. Unfortunately, because so little is known about the actual delegation process, papers like Cornell and Roll are forced to rely on stylized assumptions regarding the manner in which delegation 2
Stracca (2005) provides a comprehensive survey of this literature.
3
occurs. One approach that has not been tried is to study the impact of delegation in a laboratory setting. This is surprising because experiments could provide useful way to study delegated portfolio management and its effect on asset pricing. Unlike field data, the experimenter can control many of the variables that are crucial for understanding financial markets, such as total supplies, information, contractual agreements, enforcement of contracts, etc. Indeed, it is only under experimental conditions that CAPM pricing has ever been observed (see Bossaerts and Plott, 2004) and that it has been shown why CAPM pricing may obtain even if no-one chooses to hold the market portfolio (see Bossaerts, Plott, and Zame, 2007 ). This raises the obvious question of whether the CAPM would continue to hold if managers were introduced into the experiments. As experiments allow one to control the exact contract between “investor-subjects” and “investor-managers,” as well as the flow of information between the two groups, in principle, experiments could also provide important insights about the impact of contracts on asset prices and fund composition. The present paper provides a first step towards the experimental analysis of delegated portfolio management. It presents the results of a baseline experiment against which general equilibrium effects of incentives (contract design) on performance, prices and choices could be studied. Specifically, it reports on an experiment that was set up in exactly the same way as earlier experiments that have reliably generated CAPM pricing (see earlier references), with one exception: subjects do not trade for their own account, but for the benefit of other subjects. The “investorsubjects” (the Investors) are endowed with assets and cash and allocate those to “manager-subjects” (the Managers). The latter can then trade the assets for cash in anonymous electronic open-book markets within a pre-specified time period. The Investors are paid a share of the (liquidating) dividend of the assets as well as the final cash in the resulting portfolio, while the Managers are paid a percentage of the
4
value of the assets and cash that they received (i.e., they receive payment for order flow). Basic performance metrics are then reported, based upon which Investors may assign fresh allocations of assets and cash to the Managers. Earlier experiments (Bossaerts, Plott, and Zame, 2007) showed that when subjects trade for their own account, CAPM pricing obtains, and while portfolio holdings are quite erratic, choices reflect demands that deviate from mean-variance optimality only because of a mean-zero error term. Evidently, this error term is independent across subjects, which is why CAPM pricing still obtains – provided, of course, there are enough subjects who all have an endowment that is small relative to the market as a whole. As such, CAPM pricing obtains because of the (functional) Law of Large Numbers. We replicate some of these results here in a trading session where there are no Investors and Managers could trade for their own account. This “pre-experimental session” provides calibration for the actual delegated portfolio management experiment. Our main experimental setup is based on the simplest possible management contract design. In fact, it is plausible to argue that portfolio delegation should have no effect on asset prices or holdings. All Managers have equal information, so, barring unequal trading skill, there is no reason to believe that one Manager could outperform the others. Assuming mean-variance preferences, managers should all choose portfolios on the mean-variance frontier, and the Investors should choose to invest in managers in a way so that the Investor’s overall portfolio exactly corresponds to his optimal portfolio. The erratic nature of portfolio choices could, however, cause some Managers to outperform others by pure chance. Being indifferent as to how to allocate their assets and cash across Managers, Investors may resolve their indifference by choosing to send their endowment to the Manager who happened to have had the best performance in the past. This way, the Investors hedge against the (remote) possibility that this Manager actually outperformed because of skill. But if many investors choose to
5
do so, this Manager’s fund will be large relative to the market, and her demand will eventually influence prices, unless she happens to have exact mean-variance optimal demands, a rather unlikely event in view of the past evidence. As a result, CAPM pricing will no longer obtain. The following provides a succinct description of our findings. In early rounds, Investors allocate their shares and cash so that Managers all effectively receive the same initial endowments. Despite rather erratic holdings at the end of trading, CAPM pricing obtains. After the third round, however, most Managers did very poorly (because the market portfolio did badly) except for a few Managers who had chosen portfolios that over-weighted the assets that happened to do well. In subsequent rounds, these successful Managers received increasingly large allocations of assets and cash – at one point, four out of the 30 Managers were given 40% of the available assets and cash. We find that there was a correlation of 0.66 between a manager’s funds distributed back to investors in the previous period (after the realization of uncertainty and after fees) and the flow to that investor in the next period. Managers with large fund flows were more likely to have large fund flows in the next period as well, a finding that does not disappear when past returns are accounted for. Thus, there seems to be “stickiness” in fund flows. Moreover, investors do not appear to use mean-variance as a criterion in choosing the funds to invest in. While not given the information about the expected return and variance of managers portfolios directly, the investors are given enough information to be able to infer it. We find that next period fund flows depend positively on last period portfolio variance and negatively on last period expected return. As expected from the above described investor behavior, as the size of the largest manager grew, CAPM pricing no longer obtained. As a matter of fact, the equity premium became negative. The size of the largest manager significantly positively correlated with the mis-pricing in the market (as measured by the difference of the Sharpe ratio of the market portfolio from the Sharpe ratio of the mean-variance optimal portfolio). Our results indicate
6
that it is more the size of the largest manager more than the segmentation of the markets into large and small funds that matters for explaining the CAPM mispricing.
2
Experimental Setup
The experiment consists of one main session and one “end” session. Subsection 2.4 describes the design of the end session. Here we outline the design of the main session. This session is conducted in a series of six periods. Individuals who participate in trading during the trading periods are called managers. The 32 managers are the same individuals over all periods. A separate group of participants, called investors, are the initial owners of assets and cash. The investors need not be the same individuals each period and their number (equal to 70 on average) can possibly change. Investors receive their endowments in the beginning of each period. Their endowments consist of units of two risky assets, called A and B, and some cash. The assets are risky because their end-of-period payoffs (in US dollars) depend on the random realization of one of three states of the world, called X, Y , and Z. Investors cannot buy or sell assets directly, thus in the beginning of each period they must assign fractions of their initial resources to different managers to trade on their behalf. The allocations from different investors to one manager constitute this manager’s initial (for that period) portfolio. The fraction of this initial portfolio that is due to a single investor is this investor’s share in the manager’s fund. The managers’ initial portfolios are the sole determinant of their payoffs: the current period payoff of a manager is equal to 40% of the (expected) value of her initial portfolio. We refer to this payoff as the manager’s fee. The allocation of assets from investors to managers constitutes the first stage of a period. Managers (only) then participate in the second stage of the period, called trading stage and lasting thirty minutes. In this stage they may trade their initial portfolio to a new, final portfolio, which generates a dividend for investors according to a random realization of the state of the world (that only becomes known after the conclusion of trading). The dividend, with the 7
management fee subtracted from it, is distributed to the investors according to their shares in the fund. If this residual dividend is negative, the end-of-period payoff of all investors in the fund is equal to $0. To facilitate borrowing, in addition to trading securities A and B, managers can trade a risk-free security called a “Bond.” The Bond pays off $1 in any state of the world and is in zero net supply. All securities are traded in parallel continuous (limit-order) markets. The third and final stage of a period is the information disclosure stage. It can also be regarded as stage “zero” of the following period. In this stage a series of performance indicators for each manager are published on the experimental webpage and the university newspaper. The details of the information disclose are presented in subsection 2.3. Periods are weekly, and except for the information about past periods and the fact that managers are always the same, they are independent events. In order to avoid any last-period effects in managers’ behavior, the design includes one “last” period. In this period the managers are paid based on the realized dividend of their final portfolio, i.e., instead of being paid 40% of the expected value of the initial portfolio, they are paid 40% of the realized dividend. The remaining 60% are distributed to the investors as before, i.e., according to their shares in each fund.3 The following subsections give more detail about payoffs and trading rules, as well as about the disclosed information. Additionally, the investors’ and managers’ instructions are available on the web at http://clef.caltech.edu/exp/dp.
2.1
Trading: Assets and Dividends
Table 1 summarizes the dividends, expressed in US Cents, of the three traded assets in each state of the world. In each period the three states, X, Y, and Z, are equally likely. This information is known to both managers and investors. 3
The initial design included eight periods followed by the “last” period. While we conducted eight periods, we report the results only of the first six due to an error that caused incorrect reporting in the information disclosure stage. Thus, in effect the “last” period’s purpose of eliminating the
8
Table 1: State-dependent asset dividends in US Cents. State X Y Z Asset A 5 80 0 Asset B 0 30 80 Bond 100 100 100 According to their initial endowment, there are two types of investors. An investor of type A holds 100 units of asset A, and $6 of cash, while investor of type B holds 70 units of asset B, and $9 of cash, respectively. Because the total number of investors as well as the fraction of investors of each type varied from one period to the other, the market portfolio also varied across periods. The exact numbers for every period are summarized in Table 2. Table 2: Number of participants by type and the corresponding market portfolio weights for each period. Participants Market Portfolio Weights A B Cash Period Type A Type B I 30 34 46.9 37.2 7.59 II 28 38 42.4 40.3 7.73 37 34 52.1 33.5 7.44 III IV 37 35 51.4 34 7.46 V 34 33 50.7 34.5 7.48 35 35 50 35 7.5 VI
Investors can only choose the number of units of the risky assets (A or B, depending on the investor’s type) in their portfolio to allocate to each manager. If a manager is allocated a fraction of an investor’s risky portfolio, the same fraction of the investor’s cash is allocated to that manager as well. Investors distribute holdings to managers using a form over the internet. Before trading starts, each manager knows her initial portfolio but not those of the other managers. During the trading stage all securities can be sold short and a bankruptcy rule is used to prevent agents from committing to trades that would imply negative cash last period effect was fulfilled by periods 7 and 8, which are now excluded from the data analysis.
9
holdings in the end of the period. Whenever a manager attempts to submit an order to buy or sell an asset, her cash holdings after dividends are computed for all states of the world, given her current asset holdings and her outstanding orders (orders awaiting trade) that are likely to trade, including the order she is trying to submit. If these hypothetical cash holdings turn out negative for some state of the world, she is not allowed to submit the order.
2.2
Payoffs
Below we formalize the payoff functions for the investors and managers in the main experiment. Let wg be the initial endowment of assets for a type g investor, g ∈ {A, B}, and denote investor ig ’s let hg be his initial cash holdings. Let mig = m1ig , . . . , m32 ig distribution of his initial asset endowment among the 32 managers, ig ∈ {1, . . . , Ig }, where Ig is the total number of investors of type g. The initial portfolio of manager j, j ∈ {1, . . . , 32}, is composed of mjA units of asset A, and mjB units of asset B, where mjA =
IA X
mjiA ,
mjB =
iA =1
IB X
mjiB .
iB =1
Manager j’s initial cash holding is IA IB X X mjiA mjiB h = hA + hB . w w A B i =1 i =1 j
A
B
The expected value of manager j’s initial portfolio determines that manager’s ¯ g denote the mean dividend of asset g. The mean-value of payoff. Specifically, let D manager j’s portfolio is ¯ A mj + D ¯ B mj + hj . V¯ j mj , hj = D A B
10
(2.1)
Manager j’s payoff is 40% of the mean-value of her initial portfolio. That is, P ay j = 0.4 × V¯ j mj , hj .
(2.2)
Investor ig ’s share in fund j is the fraction j
¯ g mj + mig hg D ig w j sig = ¯ j j gj . V (m , h )
(2.3)
Given her initial portfolio of risky assets, mj = mjA , mjB , and her cash holding, hj , manager j can trade to a final portfolio, denoted m ˜j = m ˜ jA , m ˜ jB , and final ˜ j .4 These holdings and the realized state of the world, cash plus bond holdings h x ∈ {X, Y, Z} determine manager j’s earnings, Πj , as follows: ˜ j ; x = DA (x) m ˜j, Πj m ˜ j, h ˜ jA + DB (x) m ˜ jB + h where Dg (x) denotes the dividend of asset g in state of the world x. Manager j’s earnings after the manager’s fee, to which refer as manager j’s resid ual, equal Πj − 0.4 × V¯ j . Investor ig ’s earnings from manager j equal his share in this manager’s residual, i.e., sjig Πj − 0.4 × V¯ j . Investor ig ’s payoff is the sum of his earnings from all managers, as given in the expression below:
P ayig (x) =
32 X
sjig
h i j j ˜j j j j ¯ Π m ˜ , h ; x − 0.4 × V m , h .
(2.4)
j=1
2.3
Disclosure of Information
Experimental periods were run on Tuesday every week. Indicators of managers’ performance in a given trading period were published on the following Monday. 4
The dividend on Bonds is $1. Additionally, we normalize prices so that the price of Bonds is ˜ j in all definitions we give here. $1. This allows us to merge Bonds and cash in one term, h
11
Numerical values of the indicators were published both in the university newspaper (The California Tech) and on the experimental web-page. Indicators of performance were announced for all managers and for a market index, called the Dow-tech, which was composed of one unit of asset A, one unit of asset B, and $1. For each fund and the index, we reported the following four indicators: Returns, Market Share5 , Residual, and Risky Share. In what follows we describe these indicators in detail. We first define the value of a mutual fund’s initial portfolio to be the valuation of this portfolio in terms of asset prices (as opposed to its mean-value, which is defined in terms of asset mean dividends). The value, V j , of manager j’s initial portfolio is V j mj , hj ; p = pA mjA + pB mjB + hj , where p = (pA , pB , 1) is the vector of asset prices, normalized so that the price of the bond is equal to 1 (the Bond is the numeraire). In continuous markets assets are traded at many different prices. We take p to be the average price over the last five minutes of trade in a period. The (realized) Return of mutual fund j when the realized state of the world is x and the average trading price is p, is given by
rj =
˜ j ; x − V j (mj , hj ; p) ˜ j, h Πj m V j (mj , hj ; p)
.
(2.5)
The Market Share of mutual fund j is the ratio of its mean-value with respect to the total mean-value of all mutual funds. Specifically, it is given by V¯ j (mj , hj ) . msj = P32 ¯k k k k=1 V (m , h )
(2.6)
The Residual of mutual fund j is the difference between this fund’s earnings 5
The Market Share indicator was called Volume in the published reports.
12
and the manager’s fee (as defined in Subsection 2.2), if this number is non-negative. The residual is zero otherwise. That is, j j j ˜j ¯ Π = max{Π m ˜ , h ; x − 0.4 × V¯ j mj , hj , 0}.
(2.7)
The Risky Share of mutual fund j is the fraction of the total value of mutual fund j’s final portfolio in terms of prices, that is given by holdings of risky assets. It is made specific in the following equation: νj =
pA m ˜ jA + pB m ˜ jB , ˜j pA m ˜ jA + pB m ˜ jB + h
(2.8)
where p is again taken to be the average price for the last five minutes of trade in a period. Tables 3 to 6 contain the values of these indicators, as published in the university newspaper and the experimental web-page, for every experimental period.
2.4
End Session
One week after the last period of the main session, another pseudo-period was run, that looked like other periods of the main session in all except the managers’ payoff. Investors made their distributions of assets among managers and managers participated in a 30-minutes trading period, in exactly the same way as in the main session. At the end of the trading time, each manager received a fee equal to 40% of the earnings generated by her final portfolio and the draw of the state of the world. Investors were again paid their share of the manager’s earnings after the fee, which in this case equaled their share taken from the remainder 60% of earnings. In other ˜ j ; x in state of words, manager j’s payoff in the end session equaled 0.4 × Πj m ˜ j, h i P32 j h j j ˜j the world x. Investor ig ’s payoff equaled j=1 sig 0.6 × Π m ˜ ,h ;x . The reasoning underlying the addition of the end session is the prevention of an unraveling of manager’s reputation considerations, and a consequent mistrust of 13
Table 3: Return (in %) of every mutual fund and every period of the main experimental session. The number reported in the table is 100 × rj , where rj is as defined in equation (2.5). Experimental Period (State Realization) I (Z) II (Y) III (X) IV (Y) V (Y) VI (X) 20.54 82.99 -77.86 56.99 57.00 -79.70
Dow-Tech Mutual Fund: Albite -44.18 Alexandrite 119.13 10.47 Allanite Alunite 38.82 65.85 Amazonite Amblygonite -134.56 Amosite 0.72 -21.88 Andalusite Anthophyllite 33.23 Atacamite 33.52 13.60 Barite 43.71 Bassanite Beidelite 17.53 -1.39 Bementite Bentonite -2.15 Bertrandite 115.89 10.24 Biotite 28.48 Birnessite Bloedite -98.78 Boracite 22.92 Calcite 26.90 Carnallite -100.51 Celestite 13.00 Chalcopyrite 15.59 Chlorite 5.94 Colemanite -1.19 Cornadite 23.03 Cristobalite 33.17 Cryolite 67.25 Dolomite -15.93 Dumortierite 161.05 Dunite -39.27
396.59 113.11 49.25 151.01 90.56 128.56 190.00 78.60 79.59 99.66 42.12 103.08 125.26 256.98 160.26 -4.83 99.33 -8.78 220.04 134.16 143.55 255.01 38.49 123.11 100.74 194.06 197.94 124.76 51.22 2.40 98.53 70.75
-82.53 8.04 -44.77 -92.82 -65.71 -151.04 -75.85 -139.44 -119.18 -73.12 -68.75 -84.90 -88.90 -83.28 -79.58 -79.86 -65.89 -68.71 -56.92 -32.08 -36.50 -100.67 -68.88 -121.49 -90.37 42.61 -69.36 -74.29 -100.37 -100.54 -68.26 -54.42
14
124.55 95.97 183.56 37.93 140.10 169.80 46.72 133.15 45.22 11.05 -5.77 180.47 57.75 65.47 34.24 193.54 64.34 -101.90 -50.20 65.30 -7.17 206.02 58.69 -3.00 103.16 -28.34 157.00 -80.28 -22.45 69.67 49.24 57.26
66.00 -22.00 105.00 134.00 139.00 156.00 8.00 50.00 -22.00 72.00 28.00 69.00 9.00 177.00 62.00 145.00 59.00 -46.00 -99.00 66.00 -17.00 206.00 -12.00 103.00 71.00 52.00 149.00 2.00 37.00 -65.00 48.00 -3.00
-190.00 95.00 -5.00 -107.00 -125.00 -93.00 -149.00 -527.00 -145.00 -96.00 -97.00 -119.00 54.00 -13.00 -109.00 93.00 -74.00 56.00 -58.00 -120.00 -52.00 -127.00 3.00 -60.00 -125.00 -68.00 -131.00 -131.00 -121.00 -74.00 -90.00 76.00
Table 4: Market Share (in %) indicator for every mutual fund in every period. The number reported in the table is 100 × v j , where v j is as defined in equation (2.6).
I Mutual Fund: Albite Alexandrite Allanite Alunite Amazonite Amblygonite Amosite Andalusite Anthophyllite Atacamite Barite Bassanite Beidellite Bementite Bentonite Bertrandite Biotite Birnessite Bloedite Boracite Calcite Carnallite Celestite Chalcopyrite Chlorite Colemanite Cornadite Cristobalite Cryolite Dolomite Dumortierite Dunite
2.02 2.38 2.42 2.34 4.62 5.31 2.33 3.36 2.78 3.38 2.93 2.84 3.17 2.96 2.31 2.01 2.48 2.47 3.44 3.06 3.35 3.29 2.78 4.41 4.42 3.52 2.74 3.12 3.42 4.02 2.84 3.48
Experimental Period II III IV V 0.34 11.61 1.96 2.27 6.68 4.38 0.98 0.84 1.70 4.39 2.19 3.18 1.83 0.64 0.69 10.57 1.09 7.71 0.28 1.44 1.80 0.62 1.82 1.58 2.21 0.97 3.62 1.87 6.15 12.36 1.53 0.70
8.14 10.36 1.16 5.69 6.19 2.58 3.89 1.22 0.89 2.72 1.12 2.11 2.06 4.66 1.62 2.98 1.56 0.99 3.72 1.67 2.90 3.90 1.64 1.54 2.96 2.99 6.65 3.07 2.82 2.28 1.52 2.38
15
4.28 19.95 2.03 1.88 3.71 6.60 1.81 1.08 2.03 1.25 0.46 1.09 1.37 3.20 1.12 0.90 1.19 1.04 1.86 3.30 3.80 2.32 1.23 2.94 2.67 15.66 4.42 1.55 0.91 1.20 1.82 1.34
2.42 20.34 2.91 1.78 5.65 5.48 0.77 2.27 0.86 1.74 0.84 5.66 0.89 2.23 0.78 4.54 1.36 0.58 0.79 3.77 1.41 6.53 0.89 0.49 2.09 4.82 9.08 2.22 1.13 1.97 0.91 2.80
VI 2.53 8.67 2.74 2.21 6.80 11.69 1.29 0.85 0.63 0.74 0.57 2.68 0.74 5.27 0.76 6.27 1.07 0.51 0.46 2.44 0.71 10.80 0.50 2.65 1.33 6.26 13.42 0.69 0.70 1.18 1.07 1.77
Table 5: Residual (in US Dollars) for every mutual fund in every experimental period. ¯ j , as defined in equation (2.7). The number reported in the table is Π
Albite Alexandrite Allanite Alunite Amazonite Amblygonite Amosite Andalusite Anthophyllite Atacamite Barite Bassanite Beidellite Bementite Bentonite Bertrandite Biotite Birnessite Bloedite Boracite Calcite Carnallite Celestite Chalcopyrite Chlorite Colemanite Cornadite Cristobalite Cryolite Dolomite Dumortierite Dunite
I 7.41 95.01 37.85 51.37 131.63 0.00 31.70 28.41 58.54 71.76 48.50 66.23 53.88 38.20 29.67 77.38 38.99 47.79 0.00 55.80 65.03 0.00 45.34 76.05 64.75 46.27 51.21 66.14 98.90 139.73 39.79 16.50
Experimental Period II III IV V 27.75 0.00 198.71 71.12 398.53 145.59 787.06 177.48 42.66 2.83 124.41 110.61 92.75 0.00 46.76 78.94 197.72 0.00 187.20 258.46 178.10 0.00 386.10 276.77 45.57 0.00 49.10 11.98 23.16 0.00 52.91 57.78 44.82 0.00 54.55 7.64 142.55 0.00 22.37 52.83 45.34 0.00 6.37 16.88 100.13 0.00 65.96 169.05 65.45 0.00 40.82 14.38 36.56 0.00 102.01 122.24 27.67 0.00 27.04 22.03 109.82 0.00 57.66 215.50 33.03 0.00 37.33 37.52 73.34 0.00 0.00 1.84 15.66 0.00 4.84 -7.15 51.65 8.82 104.29 109.79 67.26 12.62 51.06 14.14 38.52 0.00 155.34 403.01 35.75 0.00 37.25 9.66 54.24 0.00 42.34 18.37 67.77 0.00 109.17 63.70 49.33 66.15 126.92 125.06 172.12 0.00 241.04 435.21 67.02 0.00 0.00 31.73 134.87 0.00 8.88 25.18 134.47 0.00 39.79 -2.42 46.54 0.00 50.08 22.95 15.99 1.12 39.85 37.17
16
VI 0.00 319.36 35.66 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 20.00 58.24 0.00 227.22 0.00 13.95 0.17 0.00 1.27 0.00 7.37 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 57.17
Table 6: Risky Share (in %) for every mutual fund in every experimental period. The number reported in the table is 100 × ν j , where ν j is as defined in equation (2.8).
Dow-Tech Albite Alexandrite Allanite Alunite Amazonite Amblygonite Amosite Andalusite Anthophyllite Atacamite Barite Bassanite Beidellite Bementite Bentonite Bertrandite Biotite Birnessite Bloedite Boracite Calcite Carnallite Celestite Chalcopyrite Chlorite Colemanite Cornadite Cristobalite Cryolite Dolomite Dumortierite Dunite
I 86.61 78.88 100.49 110.26 99.75 101.77 99.77 65.11 78.9 101.64 75.97 76.8 92.39 99.87 77.55 83 99.84 77.81 13.33 57.99 99.62 70.9 117.61 45.65 80.66 37.15 0 108.77 87.53 99.97 71.69 76.44 20.89
II 84.75 242.13 100 74.32 100 96.11 99.99 112.09 120.55 99.97 106.65 62.05 99.84 111.98 106.61 98.58 69.59 75.01 11.6 88.08 99.41 61.12 122.34 22.71 99.04 99.55 101.46 120.29 82.17 63.35 31.31 96.19 65.04
Experimental Period III IV V 85.24 86.92 86.92 93.6 47.98 113.42 -10.92 109.86 82.65 57.29 99.36 26.77 100.76 99.98 99.33 72.49 114.23 76.35 172.01 89.47 82.67 86.3 90.28 113.63 150.07 104.64 120.57 127.92 101.73 104.15 80.78 8.47 60.86 85.09 29.38 39.22 99.75 99.93 96.06 99.68 55.49 11.08 92.29 11.92 54.4 90.21 77.15 90.66 98.77 115.29 70.92 77.77 99.6 93.41 75.28 -4.27 7.45 65.02 -73.78 -174.07 34.69 89.15 66.86 40.1 4.79 -42.74 123.86 119.12 119.08 75.19 98.78 56.31 135.73 -23.95 -0.49 99.56 90.74 96.46 -39.2 99.4 77.81 74.92 71.86 114.19 81.65 30.96 83.43 109.82 101.35 99.74 109.33 -17.45 35.92 74.98 61.67 93.61 58.14 99.38 -2.33
17
VI 86.6 197.28 -117.53 -7.2 119.57 121.52 102.85 145.98 475.63 159.8 101.55 92.17 121.64 -63.61 12.15 119.3 -152.87 79.08 -91.59 67.24 128.35 30.56 155.1 -18.67 75.45 120.65 97.51 145.84 132.77 122.84 78.45 95.43 -110.85
investors. In the absence of the end session, managers have no reputation reason to perform in the investors’ best interest in the last period. Investors’ knowledge of this could have lead to an equilibrium where investors do not use any performance indicators to allocate their assets and managers do not exert any effort to satisfy investors. Thus, by design, the incentives of the managers were fully aligned with those of investors. Moreover, conditional on having identical relative holdings of assets and cash, a manager endowed with a larger initial portfolio at the beginning of the end session was better off than one receiving a smaller allocation from investors. This means that the possibility of reputation considerations on the side of the managers was left intact for the main session.
3
Results in the Main Experiment
3.1
Prices and Final Holdings
Table 7 presents the market portfolio and the corresponding state-dependent aggregate wealth for each of the six periods. As can be seen the market portfolio changed from one period to the next, with changes primarily due to changes in the aggregate supply of security A. Table 7: Market portfolio and state-dependent aggregate endowments (in US dollars) in each period. Asset / Period I II III IV V VI Asset A 3000 2800 3700 3700 3400 3500 Asset B 2380 2660 2380 2450 2310 2450 State State State State
/ Period X Y Z
I 150 3114 1904
II 140 3038 2128
III 185 3674 1904
IV 185 3695 1960
V 170 3413 1848
VI 175 3535 1960
Table 8 presents the trading volume and the turnover for each asset in the six trading periods. As apparent from the table, the trading activity in each of the secu18
rities increased as the periods advanced, with the increase being more pronounced in security B, which was the security with more stable aggregate supply across periods. Table 8: Trading volume and asset turnover by period. Trading Volumea Asset / Period I II III IV V VI Asset A 1114 1484 1652 1739 1648 1988 Asset B 942 928 1257 1383 1601 1919 Bond 212 135 217 279 349 446
Asset / Period Asset A Asset B
Asset Turnoverb I II III IV 0.37 0.53 0.45 0.47 0.40 0.35 0.53 0.56
V 0.48 0.69
VI 0.57 0.78
a
Trading volume is the number of assets that traded during a period. Asset turnover is calculated by dividing the trading volume over a period by the total number of securities outstanding (as presented in Table 7). b
We next turn to evaluating the pricing “quality” in the markets across periods. The Arrow-Debreu securities pricing model makes predictions about state prices. Because we have complete markets, we can invert the prices of the traded securities for the state prices, and compute the ratio of the state prices and the state probabilities. According to the Arrow-Debreu model, these state price probability ratios should be inversely related to aggregate wealth in the corresponding state. The state when aggregate wealth was lowest should be most expensive relative to its probability. In this experiment (as evident from Table 7), the state with the lowest aggregate payoff was X. The state where aggregate wealth was highest should carry the lowest state price probability ratio. In all periods the highest payoff state was Y . Thus, the Arrow-Debreu asset pricing model predicts that in all periods the state price probability ratio for state X should be the highest, followed by that for state Z, and the ratio should be the lowest for state Y . The ranking of state-price probabilities evolved in time in each session. Figure 1 presents the evolution of state price probabilities in each of the periods. The state prices are computed every time a transaction occurs. 19
Figure 1: State-price probabilities in time for each session. SPP at every trading time, session 061030. The endowments in each state are ranked: X
