Innovation, Delegation and Asset Price Swings

Electronic copy available at: http://ssrn.com/abstract=2568720

Swiss Finance Institute Research Paper Series N15-03

Innovation, Delegation, and Asset Price Swings Yuki SATO University of Lausanne and Swiss Finance InstituteETH Zrich


Innovation, Delegation, and Asset Price Swings

Yuki Sato

University of Lausanne and Swiss Finance Institute

February 22, 2015

Abstract

We propose a dynamic asset-market equilibrium model in which (1) aninnovative asset with as-yet-unknown average payoff is traded, and (2) in-vestors delegate investment to experts. Experts can secretly renege on in-vestors orders and take on leveraged positions in the asset in an effort tomanipulate investors beliefs, thereby attracting more orders and thus morefees. Despite agents full rationality, the combination of experts moral hazardand investors learning generates bubble-like price dynamics: gradual upswing,overshoot, and reversal. Overshoot is pronounced and long-lasting if expertscharge high fees. Disclosure of experts positions dampens swings in assetprices and improves welfare.

JEL classification: D80, G10, G23

Key words: asset price swings, delegated investment, innovation, learning, moral hazard,signal jamming.

I am grateful to Margaret Bray, Dimitri Vayanos, and Denis Gromb for their guidance and encourage-ment. For helpful comments, I thank Elias Albagli, Ulf Axelson, Markus Brunnermeier, Amil Dasgupta,Bernard Dumas, Vincent Fardeau, Itay Goldstein, Amit Goyal, Antonio Guarino, Stephane Guibaud,Konstantin Milbradt, Joel Peress, Paul Woolley, as well as seminar participants at INSEAD and LSE.Financial support from the Swiss Finance Institute and the Paul Woolley Centre for the Study of CapitalMarket Dysfunctionality at LSE are gratefully acknowledged. All errors and omissions are my own.University of Lausanne, UNIL-Dorigny, Extranef 235, CH-1015 Lausanne, Switzerland. E-mail:

[email protected].

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1 Introduction

Bubble-like price movements have recurred in financial markets throughout history. Many

of them followed a common pattern: upswings, triggered by technological innovation, are

eventually followed by dramatic downswings, leading to persistent economic downturns.1

The 200708 financial crisis is no exception: rise in the U.S. housing prices, fueled by

financial innovation (i.e., securitization), and the subsequent reversal are the key factors

behind the global turmoil. Given their serious impacts on the real economy, it is impor-

tant to understand the mechanism of asset price swings. Especially, studying the entire

cycleemergence of upswing, its overshoot, and eventual reversalcoherently in a unified

framework appears to be a critical task.

To tackle this problem, we develop a fully rational, dynamic asset-market equilibrium

model with delegated investment. We consider a market for a new and innovative asset,

whose average payoff is as-yet-unknown and subject to learning. Investors delegate their

investment to financial experts. We highlight the roles of (1) moral hazard associated

with delegated investment, and (2) investors learning about the assets average payoff.

Despite full rationality of long-lived agents, the combination of these two elements gen-

erates endogenous bubble-like price dynamics: gradual upswing, overshoot, and eventual

downswing.

Our model builds on the signal-jamming framework developed in Sato (2014). Unlike

Sato (2014), which does not provide implications for price dynamics, our paper explores

why bubble-like swings in security prices arise endogenously within a rational frame-

work. We consider a discrete time model with finite horizon. There are one risky asset

and one riskless asset. Initially, the agents have large uncertainty about the risky as-

sets average payoff. Over time, they learn about it based on the assets payoff history.

This asset is interpreted as a financial asset backed by an unprecedented and/or hard-to-

1See Brunnermeier and Oehmke (2013) for a historical overview of bubbles and crises.

2

understand technologysuch as Internet stocks, biotech stocks, or sophisticated struc-

tured productswhose underlying profitability is initially unknown to most investors due

to the lack of track record and background knowledge. There is a continuum of investment

funds, each with a financial expert and an investor. The investor can invest directly in

the riskless asset. However, investing in the risky asset requires that she submits to the

expert a purchase order that specifies the number of shares of the asset to be purchased

on her behalf. Each period, the expert earns a delegation fee proportional to the order.

There are two items the investor cannot directly observe. First, the experts actual

purchase of the asset is unobservable to the investor. The expert can secretly renege on

the investors purchase order (at a cost) and boost the asset purchase by using leverage.

An example of such a fund can be a hedge fund adopting a flexible trading strategy that

is not communicated to investors. Second, the risky assets periodic payoff is not directly

observable to the investor.2 These two layers of unobservability create a signal-jamming

problem akin to Sato (2014). The investor tries to back out the risky assets payoffs from

the observed fund returns, learn about the assets average payoff, and submit purchase

orders on that basis. The higher the fund returns, the better the investors assessment of

the assets future prospects, hence the larger her purchase orders (and thus fees). So the

expert is inclined to boost the expected fund return by secretly levering up and increasing

the purchase of the risky asset, inflicting excessive risk on the investor. The investor is

not fooled in equilibrium because she is rational; nevertheless, the expert still reneges

and levers up since otherwise his funds future prospect would be underestimated by the

investor who believes that the expert is reneging secretly.

In equilibrium, the risky assets price path exhibits a bubble-like pattern on average:

it rises gradually, surpasses the benchmark level that would be obtained in the case in

2An alternativeand perhaps more realisticassumption yielding exactly the same results is thateach investor can directly observe the assets payoff by incurring a small effort cost > 0. Even if is veryclose to 0, it would be optimal for the investors to not observe the payoff directly because, in equilibrium,they learn it perfectly and costlessly from the fund return anyway.

3

which the assets average payoff is perfectly known, and eventually falls and converges

to the benchmark level over time. Intuitively, these swings in the price are caused by

the combination of the following two effects that have opposing pressures on the assets

aggregate demand and thus on its price dynamics.

1. Learning effect. Initially, the investors estimate of the assets average payoff has low

precision. So, being risk averse, they hesitate to purchase the asset. The associated

demand for the asset is weak; thus, ceteris paribus, the initial price is low. But, as

the investors learning progresses over time, the precision of their estimate increases.

This leads them to increase purchase orders over time, having an upward pressure

on the assets demand and thus its price.

2. Leverage effect. Initially, the investors estimate of the assets average payoff has

low precision and hence is susceptible to the experts manipulation. This leads

the experts to renege on purchase orders and choose high leverage. The associated

aggregate demand is high; so, ceteris paribus, the initial price is high. But, as the

investors learning progresses, their estimate becomes precise and less subject to

manipulation. Accordingly, the experts deleverage over time, having a downward

pressure on the assets demand and hence its price.

In early periods, when the investors still have large uncertainty about the asset, the

learning effect dominates the leverage effect, initiating upswing in the price. On average,

the price overshoots the level of the benchmark case, in which neither of the above effects is

at work, because the experts use of leverage pushes up the assets aggregate demand and

its market-clearing price. As investors learning progresses, the learning effect weakens

and is dominated by the leverage effect, leading to downswing of the price. At some point,

the investors estimate becomes so precise that it is no longer worthwhile for the experts

to attempt to manipulate it. The leverage effect disappears, and the price converges to

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the benchmark level over time.

The up-and-down swings are pronounced if the investors initial estimate has low pre-

cision. This is because for these swings to occur we need both the learning and leverage

effects to be strong, which is the case when the investors estimate has very low precision.

Thus, the model predicts that swings and overshooting in prices are more pronounced

for new and innovative assets with highly uncertain payoff characteristics than for old-

economy assets already familiar in the market. This prediction is consistent with the

historical observation that bubble-like price movements tend to arise in times of techno-

logical change (e.g., railroads or the Internet) or financial innovation (e.g., securitization),

as noted by Brunnermeier and Oehmke (2013). We also show that the price overshoot is

pronounced and long-lasting when funds charge high fees, because higher fees incentivize

the experts to take on higher leverage, driving up the assets market-clearing price.

The equilibrium allocation is inefficient, as the experts attract less purchase orders

(and thus lower fees) than in the frictionless case. This is because the experts cannot

commit not to renege on the investors orders. We show that frequent disclosures of funds

positions alleviate this commitment problem and lower the experts leverage, thereby

achieve Pareto improvement and also dampen swings in asset prices. From a practical

point of view, such a policys implementability is high in that the authority does not

need to keep track of the characteristics of innovative assets or redistribute wealth across

agents. All they need to do is to ensure transparency so that investors learn about the

nature of these assets on their own.

Our paper is related to the theoretical literature providing rational explanations to

price anomalies such as bubbles or momentum and reversal. Vayanos and Woolley (2013)

also study momentum and reversal in a model with delegated investment. As noted in

their paper, however, delegated investment is not essential for momentum and reversal

to arise in Vayanos and Woolley (2013): the driving force is delay in the reaction of fund

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flows to returns. We do not assume delays in agents reactions; in our model, delega-

tion and the associated moral hazard problem are critical. The possibility that experts

attempt to manipulate investors beliefs by deviating from their equilibrium strategies

which Vayanos and Woolley (2013) exclude by assumptionis the key driver of price

swings in our model. Like our paper, Pastor and Veronesi (2009) develop a fully rational

model and obtain bubble-like patterns in the prices of new-economy assets. The key

ingredients of their model are a time lag between the introduction and adoption of a

new technology and investors learning during that lag. While investors learning plays

a central role also in our model, we do not distinguish the introduction and adoption of

a new technology; our results are driven by agency relationship in delegated investment

in which investors learning is potentially influenced by experts. DeMarzo, Kaniel, and

Kremer (2007) also study bubbles caused by technological innovation. While their model

is static and focuses on bubbles in real investment, our model studies dynamic bubble-like

pattensboth upswings and reversalsin security prices.

Our paper is also related to the literature on opacity/complexity and obfuscation in

financial markets. Sato (2014) is most closely related to ours: both papers feature signal

jamming associated with delegated investment. While Sato (2014) studies a stationary

equilibrium and therefore has no implications for asset-price dynamics, our focus is on

time-varying, bubble-like patterns of asset prices in a non-stationary equilibrium. More-

over, our modeling approach is more standard than that of Sato (2014): while Sato (2014)

assumes overlapping generations of short-lived risk-neutral investors whose decisions stem

from a decreasing-returns-to-scale assumption a` la Berk and Green (2004), we consider a

standard dynamic optimization problem of long-lived risk-averse investors. Carlin (2009)

and Carlin and Manso (2011) argue that financial professionals may deliberately obfuscate

their products to extract informational rents from unsophisticated investors. Financial

experts in our model also try to exploit less-informed investors strategically. But unlike

6

those papers, our focus is on the dynamics of security prices in competitive markets.

In recent years, a theoretical literature studying the equilibrium implications of del-

egated portfolio management has been growing (Allen and Gorton 1993; Shleifer and

Vishny 1997; Berk and Green 2004; Vayanos 2004; Cuoco and Kaniel 2011; Guerrieri and

Kondor 2012; He and Krishnamurthy 2012, 2013; Malliaris and Yan 2012; Kaniel and

Kondor 2013). To our knowledge, bubble-like price dynamicsupswing, overshoot, and

reversalare not yet discussed in this literature. Our paper contributes to this literature

by showing that, despite agents full rationality, portfolio delegation and the associated

moral hazard generate such patterns of equilibrium asset prices.3

The paper proceeds as follows. Section 2 presents the model. Sections 3 characterize

the equilibrium. Section 4 studies price dynamics. Section 5 discusses policy implications.

Section 6 concludes. All proofs are in the Appendix.

2 Model

Time t is discrete and finite: t = 0, ..., T + 1. Period T is the last period in which the

market is open and agents make decisions. In period T + 1, the agents just consume their

entire wealth. There is a single risky asset and a riskless asset. The riskless asset has an

infinitely elastic supply at an exogenous rate of return r > 0 and is freely accessible to all

agents. There are two classes of agents: investors and experts. The investors can purchase

the risky asset only through investment funds run by the experts. Each investor gives

capital to a fund, specifying the number of shares of the risky asset to be purchased on

her behalf. The expert can secretly lever up the investor capital and buy a larger number

of shares of the asset than asked by the investor. The experts leverage, the investors

3Our paper is also related to the literature on career concerns and asset prices (Dasgupta and Prat2008; Guerrieri and Kondor 2012; Dasgupta, Prat, and Verardo 2011). In these papers, fund managersseek to influence investor beliefs about their ability. In our paper, experts try to influence investorexpectations about the innovative assets future prospects.

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demand for the risky asset, and the assets price are determined in equilibrium.

2.1 Risky asset

In period t = 1, 2, ..., T + 1, the risky asset yields a per-share payoff t = + ut. The

transitory component ut is i.i.d. across time, normally distributed with mean 0 and vari-

ance 1/u, and unobservable to anyone. The average payoff is a constant drawn by

nature in period 0 from a normal distribution with mean 0 and variance 1/0. The

agents do not observe directly, and learn about it over time based on the payoff history

Ht {}t=1. We denote the period-t estimate of given Ht by t E[|Ht]. Parameter0 measures (the inverse of) the risky assets innovativeness. The asset with small 0 is

interpreted as an innovative asset whose cash flow is backed by an unprecedented and/or

hard-to-understand technologysuch as Internet stocks, biotech stocks, or sophisticated

structured productsbecause most market participants initially have large uncertainty

about such an assets average payoff due to the lack of track record and background

knowledge. The investors cannot directly observe the realized payoff t for all t, whereas

the experts can do so. Although it is unobservable, as shown later, in equilibrium the

investors learn t perfectly from the observed fund return.4

In period t = 0, ..., T , the asset is traded in the market at a publicly observable

market-clearing price, Pt. The assets supply S > 0 is constant over time. Let Rt+1 t+1 + Pt+1 (1 + r)Pt denote the excess return on the risky asset per share. We denotethe expected excess return conditional on Ht by Rt+1 E[Rt+1|Ht].

4As noted in footnote 2, we could alternatively assume that each investor can observe t with a verysmall effort cost > 0. However small is, the investors would choose not to observe t because theylearn it perfectly and costlessly from the fund return.

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2.2 Delegated investment

There is a continuum with mass one of investment funds, each indexed by i [0, 1]. Fundi consists of a risk-neutral expert and a risk-averse investor, who both live from t = 0 to

t = T + 1. For simplicity, we assume that investor i can neither invest in nor observe

activities in the other funds. This assumption eliminates the possibility that an expert

attracts an infinitely large amount of capital to have price impact in the asset market.

In each period t = 0, ..., T , investor i submits to expert i a purchase order, which

specifies the number of shares of the risky asset, yi,t [0,), that the expert is supposedto purchase on behalf of the investor. On top of the capital needed for this purchase

(Ptyi,t dollars), the investor pays yi,t dollars to the expert as the delegation fee, where

the fee rate > 0 is an exogenous parameter.

After receiving capital and fee from the investor, the expert buys the asset. Despite

being asked to purchase only yi,t shares, the expert can renege and buy (1 + i,t)yi,t

shares of the asset, where i,t [0,) is the experts choice variable. We assume i,t isunobservable to the investor and the expert cannot commit to his choice of i,t. Since the

expert has zero personal wealth, choosing i,t > 0 requires borrowing funds from outside

lenders. Thus, i,t is also a measure of the funds leverage. To choose i,t, the expert

incurs a nonpecuniary cost of reneging, i,t with > 0. This represents the cost of effort

to cook the books and/or manipulate the disclosure documents to make them look like

the expert adhered to the investors purchase order.5

In period t + 1, the funds period-t investment yields the total proceeds of Qi,t+1 Rt+1(1 + i,t)yi,t + (1 + r)Ptyi,t dollars.

6 The investor can directly observe Qi,t+1.

5It is not important that we allow only nonnegative i,t. The reason for assuming i,t 0 is thatit is compatible with the proportional reneging cost, i,t. If we allow i,t < 0, we would need a costfunction of the form |i,t|. This would reduce tractability of analysis while the results remain unchangedbecause, in equilibrium, the experts may choose i,t > 0 but never i,t < 0. An alternative cost functionaccommodating i,t < 0 is a quadratic form,

2i,t/2. With this specification, the experts still do not

choose i,t < 0 in equilibrium and the model yields very similar results.6Qi,t+1 is computed as follows. In period t, the investor invests Ptyi,t dollars in the fund. The

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2.3 Maximization problems

Each expert maximizes the discounted sum of his lifetime expected utilities, where his

within-period utility is the difference between the fee and the cost of reneging. That is,

expert is problem in period t = 0, ..., T , denoted by PEi,t, is to choose i,t [0,) tomaximize

E

[Tt=0

(yi,t+ i,t+ )FEi,t

], (2.1)

where (0, 1) is a discount factor common for all agents, and FEi,t = {Qi, , yi, , i, , P , : t} is his information set in period t.

Investor is problem in period t = 0, ..., T , denoted by PIi,t, is to choose purchase orderyi,t and consumption ci,t to maximize the discounted sum of her lifetime expected utilities

E[Tt=0

exp (ci,t+ )F Ii,t

], (2.2)

where > 0 is the coefficient of constant absolute risk aversion and F Ii,t = {Qi, , yi, , P : t} is her information set in period t, subject to her dynamic budget constraint

Wi,t+1 = Qi,t+1 yi,t + (1 + r)(Wi,t ci,t Ptyi,t), (2.3)

where Wi,t is her wealth in period t. Constraint (2.3) states that the investors next-period

wealth is the sum of the proceeds from the delegated investment net of fee and her own

investment in the riskless asset. In the final period t = T + 1, in which there is no market

for the asset, she simply consumes her entire wealth: Wi,T+1 = ci,T+1.

expert borrows Pti,tyi,t dollars from outside lenders and invests Pt(1 + i,t)yi,t dollars in the risky asset(i.e., buys (1 + i,t)yi,t shares). In period t + 1, the fund receives t+1(1 + i,t)yi,t dollars of payofffrom the asset, and obtains Pt+1(1 + i,t)yi,t dollars from selling the asset in the market. The expertpays back (1 + r)Pti,tyi,t dollars to the lenders. Thus, the total proceeds from the funds investment isQi,t+1 = t+1(1 + i,t)yi,t + Pt+1(1 + i,t)yi,t (1 + r)Pti,tyi,t = Rt+1(1 + i,t)yi,t + (1 + r)Ptyi,t.

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2.4 Definition of equilibrium

The equilibrium consists of the price Pt, the investors purchase order yi,t, and the experts

leverage i,t for i [0, 1] such that, for all t = 0, ..., T ,

1. given Pt and the others actions, investor i solves PIi,t,

2. given Pt and the others actions, expert i solves PEi,t, and

3. the risky assets market clears:

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(1 + i,t)yi,t di = S. (2.4)

3 Analysis

This section characterizes the equilibrium of the model. We look for a linear equilibrium

such that Pt is linear in t on the equilibrium path. We follow the following steps to solve

the model.

1. Characterize the evolution of the agents estimates of (Section 3.1).

2. Conjecture a linear form of the equilibrium price. Also conjecture that the sequence

of each experts optimal leverage {}T=0 0 is deterministic (Section 3.2).

3. Specify the investors out-of-equilibrium beliefs (Section 3.3).

4. Solve each investors problem (Section 3.4) for her optimal purchase order yi,t.

5. Solve each experts problem (Section 3.5). Verify that {}T=0 is indeed deterministicas conjectured in step 2, and obtain t as a function of Rt+1 (Eq.(3.16)).

6. Impose market clearing (Section 3.6) and obtain Rt+1 as a function of t (Eq.(3.17)).

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7. Solve Eqs. (3.16) and (3.17) for two unknowns t and Rt+1 (Section 3.6; Figure 1).

Verify that the equilibrium price is indeed linear as conjectured in step 2.

3.1 Evolution of estimates

Since is unobservable to anyone, all agents learn about it over time by Kalman filtering.

The experts, who observe Ht directly, update their period-t estimate t by

t = tt1 + (1 t)t, (3.1)

with the updating factor t (0, 1) that increases over time deterministically accordingto t+1 = 1/(2 t) from its initial value 1 = 0/(0 + u) (see Appendix A).

The investors, who cannot observe Ht directly, estimate based on the Ht that theyinfer from the available information. LetHIi,t {Ii,}t=1 denote the payoff history inferredby investor i, where Ii,t is the value of t inferred by her. Her period-t estimate of is

denoted by Ii,t E[|F Ii,t], and her conditional expectation of the excess return is denotedby RIi,t+1 E[Rt+1|F Ii,t]. If she infers that t is Ii,t, she updates Ii,t as

Ii,t = tIi,t1 + (1 t)Ii,t, (3.2)

where Ii,0 = 0 is exogenously given, and the updating factor t is the same as that of

(3.1) (see Appendix A).

Importantly, the investors learning (3.2) potentially departs from (3.1) on off-the-

equilibrium paths because the experts can manipulate the investors inference by secretly

deviating from their equilibrium strategy (i.e., choosing i,t that is not anticipated by

the investors). However, on the equilibrium path the investors correctly anticipate i,t;

consequently HIi,t = Ht holds, and thus (3.2) coincides with (3.1).

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3.2 Conjectures

First, we propose and later verify the following conjecture about the equilibrium price.

Conjecture 1 (Price). The equilibrium price for t = 0, ..., T is of the form

Pt = atIt bt with It

10

Ii,t di, (3.3)

where

at T+1t=1

(1

1 + r

)(3.4)

is a riskless discount factor and {b}T=0 > 0 is a deterministic sequence.

The first term of (3.3), atIt , is the present value of the average of all the investors

expectations about the assets future payoffs discounted at the riskless rate. The second

term, bt, involves risk premium. This term is time-varying because the premium demanded

by the investors changes together with their learning about . Since there is no market

in period T + 1, we set PT+1 0 (i.e., aT+1 0 and bT+1 0). Note that every agentknows the form of (3.3), and therefore can infer the investors average estimate It from

observing Pt on the equilibrium path.

Second, we conjecture and later verify the experts equilibrium strategy as follows.

Conjecture 2 (Experts optimal strategy). There exists a deterministic sequence {}T=0 0 such that, for all i [0, 1], expert i optimally chooses i,t = t in period t = 0, ..., T onthe equilibrium path and also on off-the-equilibrium paths where Ii,t 6= t.

Conjecture 2 states that the experts optimal leverage is deterministic, irrespective of

his potential deviations in the past, {i,}t1=0.

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3.3 Out-of-equilibrium beliefs

As shown later, all investors infer Ht correctly on the equilibrium path and thereforehave the same estimate of . So, on the equilibrium path, each investor observes Pt and

confirms that the other investors average estimate It =Pt+btat

is the same as her estimate

based on her inferred payoff history HIi,t. That is, E[|HIi,t] = Pt+btat for all i and t onthe equilibrium path. However, on off-the-equilibrium paths where some agents deviate

from their equilibrium strategies, an investor may observe Pt and realize that her estimate

E[|HIi,t] does not equal Pt+btat . For such a case, we specify the following out-of-equilibriumbelief of investors.

If E[|HIi,t] 6=Pt + btat

then Ii,t = E[|HIi,t]. (3.5)

That is, an investor whose estimate (based on her inferred payoff history HIi,t) disagreeswith the observed price Pt would stick with her own estimate. We have two remarks on

(3.5).

Remark 1 (Comparison with REE models). One may argue that (3.5) is implau-

sible because, based on the standard rational-expectations equilibrium (REE) logic, each

investor should revise her belief in favor of the price. Such an argument does not apply

to our model, as the model setting and the nature of analysis are fundamentally different

from those of the standard REE models. In asymmetric information models a` la Gross-

man and Stiglitz (1980), uninformed investors should indeed revise their estimates in favor

of the price because it reflects informed investors superior signals. Also, in differential

information models a` la Grossman (1976), investors should revise their estimates in favor

of the price because it aggregates all investors signals and smooths out their idiosyncratic

noises. In contrast, in our model, the price need not convey information superior to each

investors because no investor has information superior to other investors and no one has

private information that would be collectively useful. That is, each investor has no reason

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to believe that the others average estimate is more informative than her own. On the

equilibrium path, all investors infer the same payoff history Ht and thus do not learn newinformation from Pt: each of them just confirms that her own estimate E[|HIi,t] is equalto Pt+bt

at(which equals It on the equilibrium path). If E[|HIi,t] 6= Pt+btat on an off-the-

equilibrium path, she may potentially attribute the discrepancy to the following events:

(1) her estimate E[|HIi,t] is biased because expert i has deviated from the equilibriumstrategy, (2) the others average estimate It is biased because some other experts have

deviated, (3) Pt does not reflect It in the form of (3.3) because some other investors

have deviated, or a combination of these three. The investor cannot conduct a statistical

inference as to which of these three events is more likely than the others, because all of

them are supposed to occur with probability zero. The out-of-equilibrium belief (3.5)

states that each investor attributes the discrepancy to (2) or (3) instead of (1).

Remark 2 (Unique linear price). There is another, important reason why the out-of-

equilibrium belief of the form (3.5) is plausible: it ensures that Pt reflects the unbiased es-

timate t on the equilibrium path. Indeed, somewhat paradoxically, it is precisely because

each investor would prioritize her own estimate over Pt when there were a discrepancy be-

tween them that Pt reflects the unbiased t on the equilibrium path. To see this point, sup-

pose to the contrary that each investor would prioritize Pt over her own estimate (that is,

consider the out-of-equilibrium belief of the form: if E[|HIi,t] 6= Pt+btat then Ii,t = Pt+btat ).Under such a belief, there would be infinitely many equilibrium prices of the form (3.3).

Specifically, Pt = atzbt for an arbitrary number z would support an equilibrium becauseeach investor observes Pt, revises her estimate to

Ii,t = z, and forms demand on that ba-

sis, which then consistently translates into the market-clearing price Pt = atz bt evenon the equilibrium path. Such a multiplicity of equilibria would significantly lower the

models predictive power. The out-of-equilibrium belief (3.5) eliminates this multiplicity

and guarantees that Pt reflects only t on the equilibrium path.

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3.4 Investors optimization

We conjecture and later verify that investor is value function in period t = 0, ..., T + 1

is, for all i,

Vt(Wi,t) = exp (AtWi,t Bt) , (3.6)

where {A}T+1=0 and {B}T+1=0 are deterministic sequences obtained later. Function Vt() istime-varying because the investors maximized expected utility changes over time as their

learning about progresses. The Bellman equation is

Vt(Wi,t) = maxci,t,yi,t

{ exp(ci,t) + E [Vt+1(Wi,t+1)|F Ii,t]} . (3.7)The following two lemmas characterize the investors optimization.

Lemma 1 (Investors value function). The investors value function (3.6) satisfies the

Bellman equation (3.7) if

At =

1 + atfor t = 0, ..., T (3.8)

and AT+1 = , and

Bt =Ts=t

(sk=t

ak1 + ak

) ln +12t

(S

(1+r)as

)2+ 1as

ln(

1as

) 1+as

asln(1+asas

) for t = 0, ..., T (3.9)

and BT+1 = 0, where

t 1Vart[Rt+1]

=ut+1

(1 + at+1(1 t+1))2(3.10)

is the period-t precision of the risky assets excess return.

16

Lemma 2 (Investors purchase order). Given Pt, investor i asks the expert to buy

yi,t =t(RIi,t+1(1 +

t )

)At+1(1 + t )2

(3.11)

shares of the risky asset.

The investors optimal order (3.11) can be viewed as a mean-variance solution, stan-

dard in CARA-normal setting. It is increasing in the after-fee expected fund return,

RIi,t+1(1 + t ) , and is decreasing in the volatility of fund return, (1 + t )2/t, and

the time-adjusted risk aversion, At+1. There are three points worth noting. First, yi,t

is increasing in the asset return precision t, which measures how much the investors

learning has progressed. Over time, t increases as the uncertainty about is unraveled,

encouraging the risk-averse investor to increase yi,t. As shown later, this upward pressure

on yi,t generates upswing in the equilibrium price Pt. Second, yi,t depends on the term

(1+ t ) because the investor anticipates that the expert will renege on her purchase order

and buy (1 + t )yi,t shares. Importantly, it is the investors belief (t ) about the experts

choice and not the choice itself (i,t) that affects yi,t, because i,t is unobservable. This is

the source of the experts moral hazard that is central to the following analyses. Third,

yi,t depends on the expected excess return from investor is point of view, RIi,t+1, which

is not necessarily equal to Rt+1 on some off-the-equilibrium paths. On the equilibrium

path, of course, RIi,t+1 = Rt+1 for all i and t because all the investors infer Ht correctly.

3.5 Experts optimization

The experts problem is solved in a similar fashion to Sato (2014). To verify that it is

indeed optimal for expert i to choose {}T=0 deterministically, we consider what wouldhappen if he deviated from his equilibrium play and instead chose an arbitrary sequence

of leverage {i,}T=0 6= {}T=0 even as investor i still believes that {}T=0 is played. Such

17

a deviation does not affect the assets prices because each expert has measure zero.

What would be investor is inference Ii,t+1 of the payoff t+1 that is unobservable to

her? The value of Ii,t+1 solves

(Ii,t+1 + Pt+1 (1 + r)Pt

)(1 + t ) =

(t+1 + Pt+1 (1 + r)Pt

)(1 + i,t). (3.12)

The right-hand side (RHS) is Rt+1(1 + i,t), whose value is known to investor i who

observes Qi,t+1. It depends on the experts actual choice, i,t, and the true payoff, t+1.

The left-hand side (LHS) is the decomposition of the RHS as inferred by investor i. It

depends on her incorrect belief about the experts action, t , and implies an erroneous

inferred payoff, Ii,t+1. Rearranging (3.12), we have

Ii,t+1 = t+1 +

(i,t t1 + t

)Rt+1. (3.13)

This implies that if the expert plays i,t > t and if Rt+1 > 0, then the investor will

overshoot her inference, i.e., Ii,t+1 > t+1.

The investors erroneous perception about t+1 biases her learning subsequently. Specif-

ically, if Ii,t+1 > t+1 then her estimates of and asset return will be both biased upward

in future periods, i.e., Ii,t+ > t+ and RIi,t++1 > Rt++1 for = 1, 2, ..., T t (see

Appendix C).7 When choosing i,t in period t, the expert takes into account the fact that

he can potentially inflate the investors perceived expected returns, RIi,t++1 ( 1), andtherefore her purchase orders, yi,t+ ( 1), by choosing i,t > t . Lemma 3 characterizeshis optimal choice in period t, both on and off the equilibrium path.

Lemma 3 (Experts leverage). Taking Rt+1 and t as given, the experts optimal choice

7For these overshoots to occur, the out-of-equilibrium belief (3.5) plays a role. It leads the investor

to stick by her own (high) estimate Ii,t+ when it disagrees with Pt+ . Her high estimate Ii,t+ , in turn,

leads her to expect both t++1 and Pt++1 to be high. Thus, given the observed Pt+ , she expects thatRt++1 will also be high.

18

of leverage i,t is as follows.

IftRt+11 + t

> then i,t ,= then i,t [0,) (indifferent),< then i,t = 0,

(3.14)

where

t (1t+1)Tt=1

t+ (1 + at++1(1 t++1))

At++1(1 + t+ )

(t+k=t+2

k

)for t = 0, ..., T1 (3.15)

and T = 0.

Lemma 3 states that the expert chooses i,t by weighing the marginal gain from in-

fluencing the investor beliefs in future periods, tRt+1/(1 + t ), against the marginal

cost, . The deterministic variable t measures the sensitivity of the experts expected

future gain to an increase in i,t. In t = 0, ..., T 1, t is positive because the expertcan potentially gain from influencing the investors future beliefs. However, T is zero

because there is no future in period T : since the investor makes no decisions in period

T + 1, the expert has no benefit from influencing her belief in period T .

Note that Lemma (3) only characterizes the choice of i,t optimal from expert is

personal perspective, taking the equilibrium level t as fixed. To determine t , we need

to ensure that investor is belief about i,t is consistent with the experts optimal choice.

As will be shown in Section 3.6, Rt+1 is a deterministic variable; thus, the investors belief

is consistent (that is, Conjecture 2 is correct) if and only if i,t = t in (3.14), or

t = max

{0,tRt+1

1}. (3.16)

Note that t = 0 if t is small enough. That is, the experts do not renege on the purchase

19

orders if the benefit of manipulating investor beliefs is small. Indeed, in period T they will

surely choose T = 0 because T = 0. If t is large enough, t is positive and increases

with Rt+1. This make sense: a large Rt+1 means a large marginal benefit for the experts

from influencing the future investor beliefs by reneging (i.e., the LHS of (3.14) is large),

inducing them to increase leverage t . We will pin down the equilibrium values of t and

Rt+1 explicitly in Section 3.6 by imposing the market-clearing condition.

3.6 Equilibrium

The market-clearing condition (2.4) determines the assets expected return and the agents

actions. Plugging the investors optimal policy (3.11) into (2.4), and noting that RIi,t+1 =

Rt+1 holds for all i in equilibrium, we obtain Rt+1 as a function of t :

Rt+1 =At+1S

t+

1 + t. (3.17)

The first term on the RHS is the risk premium demanded by investors, which increases

with the degree of their risk aversion and decreases with the precision t of asset return.

The second term /(1 + t ) is the fee premium, i.e., the return that compensates

investors for the delegation fees they pay. Importantly, (3.17) implies that Rt+1 decreases

with the experts leverage t because the fee premium is decreasing in t . Why does the

fee premium decrease with t ? The reason is that the fee effectively serves as a fixed

cost of investment in the risky asset from the investors perspective. An increase in t

lowers the average cost per share of the asset purchased, leading to a lower fee premium

demanded by investors. Note that the risk premium does not depend on t despite the

fact that a rise in t increases the risk borne by investors, all else equal. This is because

each investor responds to the higher t by decreasing her purchase order yi,t so that the

total risk she bears remains the same.

20

t 0

=

+ 1

,0max 1

ttt R

1

+tR

tt

tt

SAR

+

+= ++ 1 1

1

(a) High t

t 0

1

+tR

(b) Low t

Figure 1: Determination of the risky assets expected excess return Rt+1 and the fundleverage t

Given t, the equilibrium levels of t and Rt+1 are obtained by solving the system of

equations (3.16) and (3.17). The solutions are

t = max

0, t2At+1S

t+

A2t+1S

2

2t+

4

t

1 and (3.18)

Rt+1 = min

At+1St + , 12At+1S

t+

A2t+1S

2

2t+

4

t

. (3.19)Figure 1 illustrates the determination of (3.18) and (3.19). Panel (a) shows the case

with t > /((At+1S/t+)). Since the experts have strong desire to influence investors

future beliefs (i.e., t is large), they choose high leverage given Rt+1 (see (3.16)), leading

to t > 0 in equilibrium. The equilibrium level of Rt+1 decreases with t for the following

reason. For a given Rt+1, a rise in t induces the experts to increase t , which leads to

a higher aggregate demand (1 + t )yt for the risky asset. Thus, Rt+1 decreases (i.e., Pt

increases) to clear the market. Panel (b) is the case with t /((At+1S/t+)). Here,

21

t is so small that the equilibrium leverage is t = 0. The resulting small aggregated

demand for the asset is accompanied by a low market clearing price and a high expected

return Rt+1.

Once Rt+1 is identified, the equilibrium price is readily determined. Conjecture 1

implies that Pt can be written as Pt = atIt (bt+1 + Rt+1)/(1 + r).8 This implies that

Conjecture 1 is correct if and only if bt = (bt+1 + Rt+1)/(1 + r), or

bt =T+1t=1

(1

1 + r

)Rt+ . (3.20)

That is, bt is the present value of the future expected excess returns.

Proposition 1 summarizes the equilibrium outcome of the economy.

Proposition 1. There is a linear equilibrium in which

1. the risky assets excess return Rt+1 is, conditional on t, normally distributed with

mean Rt+1 and precision t, which are given by (3.19) and (3.10), respectively;

2. the risky assets price is Pt = attbt, where at and bt are given by (3.4) and (3.20),respectively;

3. each experts leverage is i,t = t , given by (3.18);

4. each investor asks the expert to buy yi,t = S/(1 + t ) shares of the asset;

5. investors consumption in t = 0, ..., T is an affine function of wealth,

ct(Wi,t) =Wi,t

1 + at+

1

(at

1 + at

)( ln + 1

2t

(S

(1 + r)at

)2+Bt+1 + ln at

),

8This is shown as follows. From (C.7) in Appendix C, all investors average expected excess return is

RIt+1 1

0RIi,t+1di = (1 + at+1(1 t+1))

10Ii,tdi+at+1t+1

It bt+1(1+r)Pt = (1+at+1)It bt+1

(1+r)Pt. Rearranging this and noting that at = (1+at+1)/(1+r), we have Pt = atIt(bt+1+RIt+1)/(1+r).

Since RIt+1 = Rt+1 in equilibrium for all t, the result follows.

22

where Bt is given by (3.9); in the final period t = T + 1, she consumes her entire

wealth, i.e., ci,T+1 = Wi,T+1.

Although it is costly to renege on the investors order, each expert chooses t > 0 if

it gives him sufficiently large marginal benefit through manipulating the investors future

beliefs (i.e., if t is large enough). The investors understand the experts desire to fool

them and hence their beliefs are not manipulated on the equilibrium path; nevertheless,

the experts still renege on the investors orders and lever up. This is because, given the

investors beliefs that the experts will lever up, the experts indeed lever up optimally since

otherwise the funds future prospects would be underestimated by the investors. By itself,

this result is not surprising: it is in line with other signal-jamming modes (Holmstrom

1999; Stein 1989; Sato 2014). The primary purpose of this paper, which distinguishes

ours from existing works, is to examine the implication of the experts signal-jamming

behavior for the dynamics of security prices. We explore this issue in Section 4.

4 Dynamics: Asset Price Swings

This section studies the dynamics of Pt determined in Proposition 1. First, we show that

Pt on average exhibits a bubble-like pattern: gradual upswing, overshoot, and eventual

reversal. Second, we show that such a pattern is pronounced when (1) the asset is inno-

vative in that its average payoff is highly uncertain when introduced to the market (i.e.,

low 0), and (2) funds charge high fees (i.e., high ).

4.1 Price path

To obtain the sequence of Pt, first we need to obtain the sequences of t, t , and Rt+1.

As shown in Appendix E, the deterministic sequence {}T1=0 is obtained by solving the

23

following difference equation of t, backward from the terminal values T = 0 and T = 0:

t =

(t+1 +

(1 t+2t+2

)t+1 (1 + at+2(1 t+2))

At+2(1 + t+1)

). (4.1)

We already know the deterministic values of {}T+1=1 , {a}T+1=0 , {A}T+1=0 , and {}T=0from (A.5), (3.4), (3.8), and (3.10), respectively. Thus, together with (4.1), we can iden-

tify the deterministic values of {}T=0 and {R}T+1=1 backward by (3.18) and (3.19),respectively, which then determine the deterministic values of {b}T+1=0 by (3.20). Gener-ating a sequence of normal random payoffs {}T+1=1 , we compute the associated estimates{}T+1=1 by (3.1). Then we can simulate a path of the price, {P}T+1=0 , by Pt = att + bt.

Panel (a) of Figure 2 plots a sample path of Pt (blue solid line). Here we assume that

the agents have a correct prior expectation in t = 0, i.e., 0 = . So the obtained price

pattern is not driven by the agents incorrect prior about . We also plot the benchmark

price path, denoted by PBt (red dashed line), which corresponds to a noninnovative

asset whose average payoff is publicly known in period 0 (i.e., 0 = and 0 = ).Although we set T = 1, 000, we present only the first 200 periods in the figure because

our focus is on the price dynamics of an innovative asset, whose is highly uncertain to

the investors. For very large t, the asset is no longer innovative since the agents have

already learned with high precision. Also, showing the price paths of later periods is

not interesting economically: Pt simply converges to PBt .

9

The innovative assets price Pt is highly volatile in early periods. This makes sense.

Since the precision of the agents estimate t is low in the early phase of learning, they

update t drastically when having a new realization of stochastic payoff t (that is, t is

9After the periods shown in the figure, the path of Pt approaches that of PBt and stays almost flat

for most of the remaining periods. When the final period approaches, both of these price paths start tofall (around t = 850) and reach zero in the final period (t = 1, 001). This price fall occurs because thereis no market in the very last period t = T + 1which is an inevitable assumption in this finite-periodsettingand thus the investors cannot sell the asset in that period (i.e., PT+1 = 0). We do not viewthis price fall in last periods as an economically relevant result because it is merely an artifact of thefinite-horizon assumption. Thus, we do not report it in the figure and focus on early periods.

24

0 20 40 60 80 100 120 140 160 180 20020.5

21

21.5

22

22.5

23

23.5

24

24.5

25

Time: t

Pric

e

Pt

PBt (Benchmark)

(a) Simulated price path.

0 20 40 60 80 100 120 140 160 180 20021.8

21.9

22

22.1

22.2

22.3

Time: t

Pric

e (av

erage

of 50

,000 s

ample

paths

)

Pt

PtB

(Benchmark)

(b) Average of 50,000 simulated price paths.

Figure 2: Price dynamics. We plot the time paths of Pt obtained in Proposition 1 and thebenchmark price PBt that would be obtained in the case in which is perfectly known int = 0. Panel (a) plots a simulated path. Panel (b) plots the average of 50,000 simulatedpaths. The parameter values are r = 0.04, = 0.1, = 1, = 0.2, = 1/(1+r), S = 10,u = 5, = 1, 0 = 1, and T = 1, 000. We set 0 = 50 for Pt and 0 = for PBt .

25

0 50 100 150 2000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time: t

t

(a) Return precision t

0 50 100 150 2000

0.5

1

1.5

2

2.5

Time: t

t

(b) Experts leverage t

Figure 3: Time paths of the risky assets return precision t 1/Vart[Rt+1] and theexperts equilibrium leverage t . The parameter values are the same as those of Figure 2.

small in early periods), making the price volatile. But the price becomes less volatile over

time because, as learning progresses, the agents become more confident about their

estimate: they do not change t so much with a new realization of t (that is, t increases

over time). Indeed, the conditional price volatility Vart[Pt+1] = a2t+1(1 t+1)2/(ut+1)

decreases with t. By contrast, if the asset is noninnovative the agents do not update their

estimate (i.e., t = for all t), and hence PBt is deterministic and almost flat in early

periods.

To see the overall trend of the price dynamics more clearly, we plot the average of

50,000 simulated price paths in panel (b) of Figure 2. The innovative assets price

Pt exhibits bubble-like dynamics on average: it rises gradually (upswing), surpasses

the noninnovative-asset benchmark PBt (overshoot), and then falls gradually (down-

swing). Around t = 140 in the figure, it starts increasing again and converges to PBt

over time.

Intuitively, the initial up-and-down swings in Pt are caused by the combination of the

following two effects that have opposing pressures on the assets aggregate demand and

therefore its price dynamics.

26

1. Learning effect. Initially, the investors estimate of the asset return has low precision;

i.e., t is small for small t. So, being risk averse, they hesitate to purchase the asset.

Thus, ceteris paribus, the associated aggregate demand is weak and the price is low.

But, as time goes on, the investors learn about and thus t increases (Figure 3(a)).

This encourages them to increase yi,t over time, having an upward pressure on the

aggregate demand and thus the price.

2. Leverage effect. Initially, the investors estimate Ii,t has low precision and is suscep-

tible to manipulation; i.e., t is small for small t. So t is large, leading the experts

to choose high leverage t . Thus, ceteris paribus, the associated aggregate demand

is high and the price is high. But, as time goes on, the investors learn about

and thus Ii,t becomes precise, lowering the experts desire to manipulate it (i.e., t

decreases). Accordingly, the experts deleverage over time (Figure 3(b)), having a

downward pressure on the aggregate demand and thus the price.

In sum, due to the learning (leverage) effect, the price tends to be low (high) initially

and then increases (decreases) over time; the combination of these two effects generates

the inverse-U pattern. In the noninnovative-asset benchmark, neither of these effects

exists because the agents do not conduct learning (i.e., t = u t) and the experts donot use leverage (i.e., t = 0 t). For the parameter values used in Figures 2 and 3, thelearning effect dominates the leverage effect in early periods, initiating the upswing in Pt.

It even surpasses PBt . This overshoot is caused by the experts use of leverage: higher t

is associated with larger aggregate demand for the risky asset, pushing up the market-

clearing price Pt. As learning unravels over time, the learning effect fades out because

the incremental increase in t diminishes over time towards 0; that is, t approaches u

asymptotically (Figure 3(a)). At some point, the leverage effect dominates the weakened

learning effect, leading to downswing in the average Pt (around t = 30 in Figure 2(b)).

Eventually, the investors estimate becomes so accurate that it is no longer beneficial for

27

the experts to use leverage to influence investor beliefs. That is, t reaches 0 and the

leverage effect disappears (around t = 140 in Figures 2(b) and 3(b)). Afterwards, the

average Pt increases over time due to the learning effect (which is weakened but still at

work), and converges to PBt as t converges to u.

Remark. The price pattern reported in Figure 2 resembles bubble-like price movements

observed in reality. Is it a bubble? The answer is no, if we define a bubble as a situation

in which an asset is overpriced even though investors are contemporaneously aware that

the price is too high (Allen, Morris, and Postlewaite 1993; Abreu and Brunnermeier 2003).

In our model, the fact that Pt overshoots PBt on average means that all agents know at

t = 0 that Pt > PBt is likely to occur in the near future periods. Also, the agents are

likely to realize ex post, after they have learned with a high precision, that Pt was higher

than PBt in early periods. However, when they are actually making investment decisions

in early periods, they are not sure whether Pt > PBt or Pt < P

Bt because they do not

know the benchmark level PBt that depends on they are still learning about. Indeed,

while Figure 2(a) reports a typical sample path in which overshoot occurs, it is possible

to obtain (rare) simulated paths such that Pt < PBt for all t.

4.2 Effect of assets innovativeness

Under what circumstances are price swings pronounced? This section examines how

the price dynamics change in response to a change in 0 (inverse of the risky assets

innovativeness).

Panel (a) of Figure 4 plots the average of 50,000 simulated paths of Pt for different

levels of 0. The average price exhibits up-and-down swings only if 0 is small enough

(0 50 in the figure), i.e., only if the agents have sufficiently large uncertainty about initially. For large 0 (0 300 in the figure), the learning effect is so weak that it is

28

0 20 40 60 80 100 120 140 160 180 20021.8

21.9

22

22.1

22.2

22.3

Time: t

P t (av

erage

of 50

,000 s

ample

paths

)

0=0.010=50

0=600 0=300

0=

(a) Effect of 0 (inverse of the risky assets innovativeness).

0 20 40 60 80 100 120 140 160 180 2000.96

1

1.02

Time: t

P t/P

B t (av

erage

of 50

,000 s

ample

paths

)

=0.08

=0.01

=0.1=0.11

(b) Effect of (fund fee rate).

Figure 4: Comparative statics. We plot the average of 50,000 sample price paths fordifferent levels of 0 (panel (a)) and (panel (b)). In panel (b), we normalize Pt bythe benchmark level PBt because P

Bt itself changes with . Unless otherwise noted, the

parameter values are the same as those of Figure 2.

29

already dominated by the leverage effect in t = 0, and thus the initial upswing does not

occur. Moreover, if 0 is very large (0 600), the experts leverage t is so low that evenovershoot does not occur on average. As 0 increases further, the price swings are toned

down, and the price path converges to that of the noninnovative-asset benchmark PBt as

0 .Thus, the model predicts that swings and overshooting in prices are more pronounced

for new and innovative assets with highly uncertain payoff characteristics than for old-

economy assets already familiar in the market. This prediction is consistent with the

historical observation that bubble-like price movements tend to arise in times of techno-

logical change (e.g., railroads or the Internet) or financial innovation (e.g., securitization),

as noted by Brunnermeier and Oehmke (2013).

Note that price swings are pronounced with small 0 because both the learning and

leverage effects are large when 0 is small. The intuition is as follows. Suppose that a

financial asset backed by an unprecedented and/or hard-to-understand technologysuch

as Internet stocks, biotech stocks, or structured productsis newly introduced to the

market. The investors have large uncertainty about such an assets average payoff due to

the lack of track record and background knowledge (i.e., 0 is small). On the one hand, the

investors, being risk averse, hesitate to purchase such an asset initially; but they increase

demand gradually as learning progresses, generating a gradual upswing in the price (the

learning effect). On the other hand, the experts, being motivated by career concerns, try

to exploit the as-yet-unknown nature of the asset. Initially, they take on high leverage and

invest in the asset aggressively in an effort to trick investors into believing that the asset

is more profitable than it really is, causing the price overshoot; however, as the assets

true nature becomes known to investors, experts lose their desire to influence the investor

beliefs and thus deleverage, causing an eventual downswing in the price (the leverage

effect).

30

4.3 Effect of fund fee

What is the impact of fund fee on the price dynamics? Panel (b) of Figure 4 plots the

average of 50,000 simulated paths of Pt/PBt for different levels of fee rate . Instead of

presenting both Pt and the benchmark PBt in the figure, we normalize Pt by P

Bt because P

Bt

itself changes with . The figure shows that with higher the price overshoot (Pt/PBt > 1)

is more pronounced and also lasts longer. The intuition is as follows. If increases, the

experts could collect higher fees for a given rise in the investors estimate. That is, the

marginal gain from increasing i,t (the LHS of (3.14)) increases with . Thus, a higher

leads them to take on higher leverage in equilibrium, driving up the assets aggregate

demand and thus its market-clearing price Pt. Note that a change in does not affect

the learning effect because the Kalman filter (3.1) is independent of . Solely because of

the leverage effect, the price path is pushed up by high . For any t, the experts derive a

larger marginal benefit from using leverage with higher . Thus, they choose positive t

for a long time if is large, leading to the long-lasting price overshoot.

5 Policy Implication: Inspection of Funds

This section discusses the models policy implication. We argue that imposing disclosure

requirements on funds can dampen asset price swings and also improve social welfare.

From a theoretical perspective, one may find it rather obvious that alleviating asymmetric

information restores economic welfare. Nonetheless, we believe this exercise is important

from a practical point of view: it highlights the policys practical advantage that the

authority does not need to wade through the information on the innovative asset or

redistribute wealth across agents. It only requires the authority to disclose funds positions

and then leave the rest to the market mechanism. Indeed, it can be costly for the authority

to keep track of the fast-paced changes of the market environment by conducting research

31

on the characteristics of new and innovative assets and work out policies tailored to them.

The exercise below demonstrates that what matters is not the nature of innovations but

the transparency of funds positions. If only their positions are transparent, investors will

back out innovative assets performances from the observed fund returns and learn about

their true nature on their own. This leads to low fund leverage, and thus stable price

dynamics and higher welfare.

We add one ingredient to the model setup of Section 2. In each period t, after leverage

i,t is chosen, the authority conducts a random (exogenous) inspection of each fund with

probability and discloses i,t to investor i. Thus, i,t remains unobservable with proba-

bility 1 . The model analyzed above is a special case with = 0. The equilibrium ofthis economy is derived (see Appendix F) following steps that are similar to those used

in Section 3.

Proposition 2. Suppose that, in each period, each fund is inspected (and the leverage

is disclosed to the investor) with probability . Then all the equilibrium variables are

identical to those of Section 3 but with t replaced by (1 )t.

This makes sense. If the inspection occurs and i,t is disclosed, the investor will infer

t+1 correctly by observing Qi,t+1, irrespective of i,t actually chosen. In other words,

the expert can potentially manipulate the investors inference only if i,t remains undis-

closed, which occurs with probability 1 . Therefore, the experts career concern is noweffectively measured as t multiplied by 1 .

Proposition 2 implies that Pareto improvement is achieved by frequent disclosure re-

quirement on investment funds.

Corollary 1. Increasing the probability of inspection improves social welfare: it does

not affect the investors expected utility but makes the experts better off.

Intuitively, an increase in achieves Pareto improvement because it reduces the ex-

32

0 20 40 60 80 100 120 140 160 180 20021.2

21.4

21.6

21.8

22

22.2

22.4

Time: t

P t (av

erage

of 50

,000 s

ample

paths

)

=0

=0.3

=1

=0.5

Figure 5: Policy implication. Each period, the authority inspects funds with probability and discloses their leverage. The figure plots the average of 50,000 sample price paths ofthe unobservable leverage case for different levels of . The parameter values other than are the same as those of Figure 2.

perts leverage t . Each investors expected utility is not affected by a decrease in t

because she adjusts her purchase order yi,t to undo the potential effect of t on her ex-

pected utility. Indeed, the investors value function when > 0 is identical to the one

obtained in Lemma 1 where = 0. On the other hand, each expert is better off if

increases for the following reason. The higher , the less likely that he can manipulate

investor beliefs and hence the lower his expected marginal gain from reneging on the

purchase order. This reduced marginal gain induces the expert to decrease leverage i,t,

leading to a lower aggregate demand for the risky asset and to a lower market-clearing

price. The resulting higher expected return encourages investors to submit more purchase

orders, which increases fees and thus improves the experts expected utility.

Moreover, frequent disclosures of fund leverage dampen up-and-down swings and over-

shoot of the price. Figure 5 plots the average of 50,000 sample paths of Pt for different

levels of , ranging from 0 to 1. As increases, the experts marginal gain from reneging

decreases and thus the leverage effect on the price path is weakened. If is large enough

33

then the learning effect prevails, eliminating the price overshoot and downswing.

6 Conclusion

This paper develops a fully rational, dynamic asset-market equilibrium model in which (1)

a new and innovative asset with as-yet-unknown average payoff is traded (e.g., Internet

stocks, biotech stocks, or sophisticated structured products), and (2) investors delegate in-

vestment to experts. Over time, investors learn about the assets average payoff from fund

returns. Experts can secretly renege on investors purchase orders and take on leveraged

positions in the asset in an attempt to manipulate investors beliefs, thereby attracting

more orders and thus more fees. Despite full rationality of long-lived agents, the assets

equilibrium price exhibits bubble-like dynamics on average: gradual upswing, overshoot,

and eventual reversal. The up-and-down swings are caused by the combination of (1) the

learning effectan upward pressure on the price as the investors learning unravels the as-

sets uncertainty over time, and (2) the leverage effecta downward pressure on the price

as the experts deleverage over time. The price tends to overshoot because the experts

use of leverage pushes up the assets aggregate demand and thus its market-clearing price.

The model predicts that swings and overshooting in prices are more pronounced for new

and innovative assets with highly uncertain payoff characteristics than for old-economy

assets already familiar in the market. Also, the overshoot tends to be pronounced and

long-lasting if funds charge high fees. Disclosure requirements on funds positions reduce

the experts use of leverage, thereby improve social welfare and also dampen swings in

asset prices.

For future research, it would be interesting to explore this papers idea in a model of

imperfectly competitive and/or illiquid financial markets, because in reality a significant

amount of innovative financial assets are traded in over-the-counter markets (where asset

34

prices are not made public) or thin markets (where investors have price impact). Studying

the relation between delegated investment and dynamics of such assets prices in an OTC

market model (e.g., Duffie, Garleanu, and Pedersen 2005) or in a double share auction

model (e.g., Kyle 1989) may yield further economic insights and policy implications.

References

Abreu, D., and M. Brunnermeier. 2003. Bubbles and Crashes. Econometrica 71:173204.

Allen, F., and G. Gorton. 1993. Churning bubbles. Review of Economic Studies 60:813

836.

Allen, F., Morris, S., and A. Postlewaite. 1993. Finite bubbles with short sale constraints

and asymmetric information. Journal of Economic Theory 61:206229.

Berk, J., and R. Green. 2004. Mutual fund flows and performance in rational markets.

Journal of Political Economy 112:12691295.

Brunnermeier, M., and M. Oehmke. 2013. Bubbles, financial crises, and systemic risk.

Handbook of the Economics of Finance.

Carlin, B. 2009. Strategic price complexity in retail financial markets. Journal of Finan-

cial Economics 91:278287.

Carlin, B., and G. Manso. 2011. Obfuscation, learning, and the evolution of investor

sophistication. Review of Financial Studies 24:754785.

Cuoco, D., and R. Kaniel. 2011. Equilibrium prices in the presence of delegated portfolio

management. Journal of Financial Economics 101:264296.

Dasgupta, A., and A. Prat. 2008. Information aggregation in financial markets with

35

career concerns. Journal of Economic Theory 143:83113.

Dasgupta, A., A. Prat, and M. Verardo. 2011. The price impact of institutional herding.

Review of Financial Studies 24:892925.

DeMarzo, P., Kaniel, R., and Ilan Kremer. 2007. Technological innovation and real in-

vestment booms and busts. Journal of Financial Economics 85:735754.

Duffie, D., N. Garleanu, and H. Pedersen. 2005. Over-the-counter markets. Econometrica

73:18151847.

Grossman, S., and J. Stiglitz. 1980. On the impossibility of informationally efficient mar-

kets. American Economic Review 70:393408.

Guerrieri, V., and P. Kondor. 2012. Fund managers, career concerns, and asset price

volatility. American Economic Review 102:19862017.

He, Z., and A. Krishnamurthy. 2012. A model of capital and crises. Review of Economic

Studies 79:735777.

. 2013. Intermediary asset pricing. American Economic Review 103:732770.

Holmstrom, B. 1999. Managerial incentive problems: A dynamic perspective. Review of

Economic Studies 66:169182.

Kaniel, R., and P. Kondor. 2013. The delegated Lucas tree. Review of Financial Studies

26:929984.

Kyle, A. S. 1989. Informed speculation with imperfect competition. Review of Economic

Studies 56:317356.

Malliaris, S., and H. Yan. 2012. Reputation concerns and slow-moving capital. Working

Paper, Yale University.

36

Pastor, L., and P. Veronesi. 2009. Technological revolutions and stock prices. American

Economic Review 99:14511483.

Sato, Y. 2014. Opacity in financial markets. Review of Financial Studies 27:35023546.

Shleifer, A., and R. Vishny. 1997. The limits of arbitrage. Journal of Finance 52:3555.

Stein, J. 1989. Efficient capital markets, inefficient firms: A model of myopic corporate

behavior. Quarterly Journal of Economics 104:655669.

Vayanos, D. 2004. Flight to quality, flight to liquidity, and the pricing of risk. Working

Paper, London School of Economics.

Vayanos, D., and P. Woolley. 2013. An institutional theory of momentum and reversal.

Review of Financial Studies 26:10871145.

37

Appendix

A Learning

Let t 1/Var[|Ht] be the precision of the experts period-t estimate of . By standard Kalman filtering,a new observation of t will update the estimate of and its precision as follows:

t = tt1 + (1 t)t for t t1t

, (A.1)

where

t = t1 + u. (A.2)

The initial value of t, 0 > 0, and the initial value of t, 0 > 0, are exogenously given. The initial value

of t is 1 = 0/1 = 0/(0 + u). From (A.1) and (A.2), we have

t+1 =tt+1

=t

t + u(A.3)

and

t =t1t

=t ut

t = u1 t . (A.4)

Plugging (A.4) into (A.3) yields

t+1 =

u1t

u1t + u

=1

2 t . (A.5)

The updating factor t is the same for (3.1) and (3.2). This is shown as follows. Let i,t 1/Var[|HIi,t]be the precision of investor is estimate of . As in (A.2), i,t evolves as

i,t = i,t1 + u. (A.6)

Since H0 and HIi,0 are both empty sets, i,0 = 0 for all i. Thus, (A.2) and (A.6) imply that i,t = t forall i and t. So we have i,t1/i,t = t1/t = t, as required.

38

B Proof of Lemmas 1 and 2

First, we derive the period-t precision of asset return, t 1/Vart[Rt+1]. To do so, it is useful to computethe conditional volatility of t+1:

Vart[t+1] = Vart[ + ut+1] =1

t+

1

u=

1

u

(1

t+1 1)

+1

u=

1

ut+1. (B.1)

Plugging the price conjecture (3.3) into the definition of Rt+1 and using the investors learning rule (3.2),

the excess asset return is

Rt+1 t+1 + Pt+1 (1 + r)Pt= t+1 + at+1

It+1 bt+1 (1 + r)Pt

= t+1 + at+1

10

Ii,t+1di bt+1 (1 + r)Pt

= t+1 + at+1

10

(t+1i,t + (1 t+1)t+1

)di bt+1 (1 + r)Pt

= t+1 + at+1t+1It + at+1(1 t+1)t+1 bt+1 (1 + r)Pt

= (1 + at+1(1 t+1)) t+1 + at+1t+1It bt+1 (1 + r)Pt. (B.2)

From (B.2) and (B.1), we have

Vart[Rt+1] = (1 + at+1(1 t+1))2 Vart[t+1]

=(1 + at+1(1 t+1))2

ut+1, (B.3)

which yields t as in (3.10).

Now, we derive the investors value function and investment policy. In the final period t = T + 1,

the investors do not have optimization problems. Each of them just consumes her entire wealth, i.e.,

ci,T+1 = Wi,T+1. Thus, AT+1 = and BT+1 = 0. The investors problem in period t = 0, ..., T is solved

as follows. Using dynamic budget constraint (2.3) and conjectured value function (3.6), we have

E[Vt+1(Wi,t+1)|FIi,t

]= exp

At+1(RIi,t+1(1 +

t )yi,t yi,t + (1 + r)(Wi,t ci,t)

12At+1(1 + t )2y2i,t 1t)Bt+1

. (B.4)

39

From (B.4) and Bellman equation (3.7), the first-order condition (FOC) for yi,t is

RIi,t+1(1 + t ) At+1(1 + t )2yi,t

1

t= 0, (B.5)

which yields

yi,t =t(RIi,t+1(1 +

t )

)At+1(1 + t )2

. (B.6)

From (B.4) and (B.6),


]= exp

(1

2A2t+1(1 +

t )

2y2i,t1

tAt+1(1 + r)(Wi,t ci,t)Bt+1

). (B.7)

Market clearing implies yi,t = S/(1 + t ) in equilibrium. Plugging this into (B.7),


]= exp

(1

2A2t+1S

2 1

tAt+1(1 + r)(Wi,t ci,t)Bt+1

). (B.8)

Thus, Bellman equation (3.7) is rewritten as

Vt(Wi,t) = maxci,t{ exp(ci,t) exp (t At+1(1 + r)(Wi,t ci,t))} , (B.9)

where

t ln + 12A2t+1S

2 1

t+Bt+1. (B.10)

The FOC for ci,t is

exp(ci,t) exp(t)At+1(1 + r) exp (At+1(1 + r)(Wi,t ci,t)) = 0

ln ci,t = t + ln (At+1(1 + r))At+1(1 + r)(Wi,t ci,t)

t + ln(

At+1(1 + r)

)+At+1(1 + r)Wi,t = (+At+1(1 + r)) ci,t

ci,t = At+1(1 + r) +At+1(1 + r)

Wi,t +1

+At+1(1 + r)

(t + ln

(

At+1(1 + r)

)). (B.11)

40

Plugging (B.11) back into (B.9), we have

Vt(Wi,t) = exp( At+1(1 + r) +At+1(1 + r)

Wi,t +At+1(1 + r)

(t + ln

(

At+1(1 + r)

)))= exp

( At+1(1 + r) +At+1(1 + r)

Wi,t +At+1(1 + r)

t

)

exp

( +At+1(1 + r)

ln

(

At+1(1 + r)

))+ exp

(At+1(1 + r)

+At+1(1 + r)ln

(

At+1(1 + r)

))

= exp( At+1(1 + r) +At+1(1 + r)

Wi,t +At+1(1 + r)

t

)( (

At+1(1 + r)

) At+1(1+r)+At+1(1+r)

(1 +

At+1(1 + r)

) ). (B.12)

Taking log to (B.12),

AtWi,t Bt = At+1(1 + r) +At+1(1 + r)

Wi,t +At+1(1 + r)

t

At+1(1 + r) +At+1(1 + r)

ln

(At+1(1 + r)

)+ ln

(1 +

At+1(1 + r)

). (B.13)

From (B.13) we have

At =At+1(1 + r)

+At+1(1 + r)(B.14)

and

Bt =

+At+1(1 + r)t +

At+1(1 + r)

+At+1(1 + r)ln

(At+1(1 + r)

) ln

(1 +

At+1(1 + r)

). (B.15)

Using (B.10), (B.15) is rearranged as

Bt =

+At+1(1 + r)

Bt+1 ln + 12A2t+1S2 1t+At+1(1+r) ln

(At+1(1+r)

) +At+1(1+r) ln

(+At+1(1+r)

) . (B.16)

Solving (B.14) backward from the terminal value AT+1 = , we have

At =

(1 +

(1

1 + r

)+

(1

1 + r

)2+ +

(1

1 + r

)T+1t)1=

1 + at. (B.17)

41

Using (B.17), (B.16) is rewritten as

Bt = mt(Bt+1 + nt), (B.18)

where

mt at1 + at

, (B.19)

nt ln + 12

(S

(1 + r)at

)21

t+

1

atln

(1

at

) 1 + at

atln

(1 + atat

). (B.20)

Solving (B.18) backward from the terminal value BT+1 = 0, we have

Bt =mtnt +mtmt+1nt+1 +mtmt+1mt+2nt+2 + +mtmt+1 mTnT

=

Ts=t

(sk=t

mk

)ns, (B.21)

which is equivalent to (3.9) in the main text.

C Proof of Lemma 3

First, we determine investor is purchase order on an arbitrary off-the-equilibrium path where expert i is

deviating from his equilibrium strategy.

Lemma 4. If expert i plays (i,0, ..., i,T ) when investor i believes that he plays (0 , ...,

T ), then investor

is purchase order in period t = 1, ..., T is

yi,t = yt + y+i,t, (C.1)

where

yt t(Rt+1(1 +

t )

)At+1(1 + t )2

(C.2)

and

y+i,t t (1 + at+1(1 t+1))

At+1(1 + t )

ts=1

(t

k=s+1

k

)(1 s)

(i,s1 s1

1 + s1

)Rs.

10 (C.3)

Proof of Lemma 4: In order to prove Lemma 4, we prove the following two claims.

10In (C.3), we abuse notation and settk=t+1 k 1.

42

Claim 1: If investor i believes that the payoff history up to period t is HIi,t = (Ii,1, ..., Ii,t), her estimateof in an arbitrary period t is

Ii,t = t +

ts=1

(t

k=s+1

k

)(1 s)(Ii,s s). (C.4)

Proof of Claim 1: For t = 0, we have Ii,0 = 0 (exogenous). For t = 1, using the updating rule (3.2),

Ii,1 =10 + (1 1)Ii,1=10 + (1 1)1

1

+ (1 1)(Ii,1 1) misperception

=1 + (1 1)(Ii,1 1).

For t = 2,

Ii,2 =2Ii,1 + (1 2)Ii,2

=2

(1 + (1 1)(Ii,1 1)

)+ (1 2)2 + (1 2)(Ii,2 2)

=21 + (1 2)2 2

+2(1 1)(Ii,1 1) + (1 2)(Ii,2 2) misperception

=2 + 2(1 1)(Ii,1 1) + (1 2)(Ii,2 2).

Continuing this way, for an arbitrary t we have

Ii,t = t + tt1 2(1 1)(Ii,1 1)

+ tt1 3(1 2)(Ii,2 2)...

+ t(1 t1)(Ii,t1 t1)

+ (1 t)(Ii,t t)

= t +

ts=1

(t

k=s+1

k

)(1 s)(Ii,s s),

where (abusing notation somewhat) we settk=t+1 k 1. (End of proof of Claim 1.)

43

Claim 2: Investor is expected excess asset return is

RIi,t+1 = Rt+1 + (1 + at+1(1 t+1)) (Ii,t t). (C.5)

Proof of Claim 2: From (B.2), the expected excess return conditional on the true history Ht is

Rt+1 E[Rt+1|Ht] = (1 + at+1(1 t+1)) t + at+1t+1It bt+1 (1 + r)Pt. (C.6)

Similarly, the expected excess return from investor is perspective (conditional on her inferred history

HIi,t) is

RIi,t+1 E[Rt+1|FIi,t] = (1 + at+1(1 t+1)) Ii,t + at+1t+1It bt+1 (1 + r)Pt. (C.7)

From (C.6) and (C.7) we obtain (C.5). (End of proof of Claim 2.)

Claim 2 shows that an overestimation of (i.e., Ii,t > t) leads to an overestimation of the excess

return (i.e., RIi,t+1 > Rt+1). Now Claims 1 and 2 can be used to rearrange the investors order (3.11) as

follows:

yi,t =t(RIi,t+1(1 +

t )

)At+1(1 + t )2

=t(Rt+1(1 +

t )

)At+1(1 + t )2

+t

At+1(1 + t )(RIi,t+1 Rt+1)

=t(Rt+1(1 +

t )

)At+1(1 + t )2

+t (1 + at+1(1 t+1))

At+1(1 + t )

ts=1

(t

k=s+1

k

)(1 s)(Ii,s s). (C.8)

Substituting (3.13) into (C.8) then yields

yi,t =t(Rt+1(1 +

t )

)At+1(1 + t )2

+t (1 + at+1(1 t+1))

At+1(1 + t )

ts=1

(t

k=s+1

k

)(1 s)

(i,s1 s1

1 + s1

)Rs

=yt + y+i,t,

as required. (End of proof of Lemma 4.)

Lemma 4 shows that the investors order yi,t is the sum of two terms. The first, (C.2), is the order

on the equilibrium path. The second, (C.3), is an additional component on the off-the-equilibrium paths

where the expert is deviating. (This term is zero on the equilibrium path because i,s1 = s1 for all

44

s.) From the experts perspective, the equilibrium order (C.2) is independent of his actions and is thus

uncontrollable. Hence, the impact of his actions (i,0, ..., i,t1) on the current order yi,t is summarized in

the off-equilibrium component (C.3). Given that Rs > 0 for s t, the investor orders additional y+i,t > 0shares of the asset in period t after the experts deviation i,s1 > s1 in period s 1. As is clear from(3.11), this additional order is caused by an overshoot in the investors expectation of excess return (i.e.,

RIi,t+1 > Rt+1). The key for this to occur is the investors out-of-equilibrium belief (3.5), which leads

the investor to stick by her own estimate of when it disagrees with the all investors average estimate

It implied by the price. Note that, on this off-the-equilibrium path, investor i is aware that Ii,t is higher

than It , but is unaware that it is too high: she (incorrectly) believes that Ii,t is more accurate than

It .

As shown in (C.5), this overestimation of leads to an overshoot in the expectation of Rt+1. Intuitively,

the investors high estimate of leads her to expect both t+1 and Pt+1 to be high; given the observed

current price Pt, these estimates lead her to expect that Rt+1 will also be high.

Now we prove Lemma 3. To simplify the experts period-t objective, note the following points.

The fee on the current order, yi,t, can be omitted from the original objective function (2.1)because, from (3.11), yi,t is independent of the experts actual choice of i,t.

By Lemma 4, the future order yi,t+ ( = 1, 2, ..., T t) is linear in yt+ and y+i,t+ . Since (2.1)is linear in yi,t+ , it follows that (2.1) is linear in yt+ and y

+i,t+ . This implies that yt+ can be

omitted from (2.1) because the expert cannot influence yt+ by his choice of i,t. That is, only

y+i,t+ is relevant for his choice of i,t.

Conjecture 2which is verified laterimplies that the experts current action (i,t) does not affecthis own future actions (i,t+1, ..., i,T ) both on and off the equilibrium path. Thus, his costs of

choosing leverage in future periods can be omitted from (2.1).

Taking these points into account, the experts period-t maximization problem reduces to

maxi,t[0,)

i,t + E[Tt=1

y+i,t+

FEi,t],

where

y+i,t+ t+ (1 + at++1(1 t++1))

At++1(1 + t+ )

t+s=1

(t+

k=s+1

k

)(1 s)

(i,s1 s1

1 + s1

)Rs. (C.9)

Since y+i,t+ is a linear function of (i,0, ..., i,t+1), the marginal effect of the experts current action

45

i,t on y+i,t+ is independent of his actions in other periods, (i,0, ..., i,t1, i,t+1, ..., i,t+1). Hence, in

y+i,t+ given by (C.9), only the term corresponding to s = t+ 1 is relevant for the choice of i,t. Thus, an

equivalent problem is

maxi,t[0,)

i,t+E[Tt=1

t+ (1 + at++1(1 t++1))

At++1(1 + t+ )

(t+k=t+2

k

)(1 t+1)

(i,t t1 + t

)Rt+1

FEi,t].

This is rewritten as

maxi,t[0,)

i,t + (i,t t1 + t

)tRt+1, (C.10)

where

t (1 t+1)Tt=1

t+ (1 + at++1(1 t++1))

At++1(1 + t+ )

(t+k=t+2

k

)for t = 0, ..., T 1

and T 0. Choosing i,t 0 to maximize (C.10), the FOC is given by (3.14) in the main text.

D Proof of Proposition 1

Statements 15 are proved in the main text. Statement 6 follows by rearranging (B.11) with (B.17).

E Dynamics of t

For t = T , we have T = 0. For t = 0, ..., T 1, from (3.15) we have

t = (1 t+1)Tt=1

( t+k=t+2

k

)Mt+ , where Mt+ t+ (1 + at++1(1 t++1))

At++1(1 + t+ ).

Let us denotet+1k=t+2 k 1 (by abuse of notation). Then we have

t1 t+1 =Mt+1 +

2t+2Mt+2 + 3t+2t+3Mt+3 + + Ttt+2t+3 TMT ,

t+11 t+2 =Mt+2 +

2t+3Mt+3 + 3t+3t+4Mt+4 + + Tt1t+3t+4 TMT .

46

These two equations yields the difference equation of t for t = 0, ..., T 1:

t1 t+1 = Mt+1 +

t+21 t+2 t+1. (E.1)

Note that (A.5) implies 1 t+1 = (1 t+2)/t+2. Using this, (E.1) is rewritten as

t =

(t+1 +

(1 t+2t+2

)t+1 (1 + at+2(1 t+2))

At+2(1 + t+1)

). (E.2)

From (E.2) and the terminal value T = 0, we obtain {t}Tt=0 by backward induction.

F Proof of Proposition 2

Let i,t {1, 0} be an indicator variable that takes 1 if fund i is inspected in period t and 0 otherwise. Wedetermine the effect of a sequence of deviations (i,0, ..., i,T ) on the payoff

Ii,t+1 as inferred by investor

i. If i,t = 0 then, as in Section 3.5,

Ii,t+1 = t+1 +

(i,t t1 + t

)Rt+1.

However, if i,t = 1, then Ii,t+1 = t+1 irrespective of the chosen i,t because the investor can infer t+1

from the observed Qi,t+1, yi,t, Pt+1, Pt, and i,t. Thus, as in Lemma 4, investor is purchase order in

period t = 1, ..., T is

yi,t = yt + y+i,t,

where

yt t(Rt+1(1 +

t )

)At+1(1 + t )2

and

y+i,t t (1 + at+1(1 t+1))

At+1(1 + t )

ts=1

(t

k=s+1

k

)(1 s)1(i,s1=0)

(i,s1 s1

1 + s1

)Rs;

here 1(Z) is an indicator function that takes 1 if Z is true and 0 otherwise.

Following the same steps as in Appendix C, the experts problem reduces to

maxi,t[0,)

i,t + (i,t t1 + t

)tRt+1, (F.1)

47

where

t (1 )(1 t+1)Tt=1

t+ (1 + at++1(1 t++1))

At++1(1 + t+ )

(t+k=t+2

k

)

= (1 )t for t = 0, ..., T 1

and T 0. The second term of (F.1)the experts expected gain from influencing the investorsestimatesis proportional to 1 because the experts choice of i,t can influence investor inference oft+1 only when i,t = 0, which occurs with probability 1 . Observe that the only difference between(C.10) and (F.1) is that t in (C.10) is replaced by t = (1 )t in (F.1). Thus, this economysequilibrium is equivalent to the one in Section 3.5 with t replaced by (1 )t.

48

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