Asset Management Contracts and Equilibrium Prices

Asset Management Contracts and Equilibrium Prices
ANDREA M. BUFFA

Boston University
DIMITRI VAYANOS

London School of Economics, CEPR and NBER
PAUL WOOLLEY

London School of Economics September 13, 2014∗

Abstract We study the joint determination of fund managers’ contracts and equilibrium asset prices. Because of agency frictions, investors make managers’ fees more sensitive to performance and benchmark performance against a market index. This makes managers unwilling to deviate from the index and exacerbates price distortions. Because trading against overvaluation exposes managers to greater risk of deviating from the index than trading against undervaluation, agency frictions bias the aggregate market upwards. They can also generate a negative relationship between risk and return because they raise the volatility of overvalued assets. Socially optimal contracts provide steeper performance incentives and cause larger pricing distortions than privately optimal contracts.

buffa@bu.edu, d.vayanos@lse.ac.uk, p.k.woolley@lse.ac.uk. We thank Sergey Chernenko, Chris Darnell, Peter DeMarzo, Ken French, Jeremy Grantham, Zhiguo He, Ron Kaniel, seminar participants at Bocconi, Boston University, CEU, Cheung Kong, Dartmouth, LSE, Maryland, Stanford, and conference participants at AEA, BIS, CRETE, ESSFM Gerzenzee, FRIC, Jackson Hole, and LSE PWC for helpful comments.

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Introduction

Asset management is a large and growing industry. For example, individual investors held directly 47.9% of U.S. stocks in 1980 and 21.5% in 2007, with the remainder held by ﬁnancial institutions of various types, run by professional managers (French (2008)). Asset managers’ risk and return is measured against benchmarks, and performance relative to the benchmarks determines the managers’ compensation and the funds they get to manage. In this paper we study how the delegation of asset management from investors to professional managers aﬀects equilibrium asset prices. Unlike most prior literature, we endogenize both equilibrium prices and managers’ contracts, including the extent of benchmarking. We also perform a normative analysis, comparing privately optimal contracts to socially optimal ones. We show that when agency frictions between investors and managers are more severe, managers’ compensation is more sensitive to performance, and performance is tied more closely to a benchmark. As a consequence, managers become less willing to deviate from the benchmark, and the price distortions that they are hired to exploit become more severe. While distortions are exacerbated in both directions, i.e., undervalued assets become cheaper and overvalued assets become more expensive, the positive distortions dominate, biasing the aggregate market upwards and its expected return downwards. Indeed, overvalued assets account for an increasingly large fraction of market movements relative to undervalued assets. Therefore, trading against overvaluation, by underweighting the overvalued assets, exposes managers to greater risk of under-performing their benchmark than trading against undervaluation. In addition to exacerbating price distortions, agency frictions can generate a negative relationship between risk and expected return in the cross-section. Such a negative relationship has been documented empirically, with risk being measured by return volatility or CAPM beta, and contradicts basic predictions of standard theories.1 Agency frictions can generate a negative risk-return relationship because they raise the volatility of overvalued assets, through an ampliﬁcation mechanism. Consider a positive shock to the expected cashﬂows of an overvalued asset. Because the asset then accounts for a larger fraction of market movements, managers become less willing to trade against overvaluation, and prefer instead to buy the asset. Their buying pressure ampliﬁes the price increase caused by the higher cashﬂows.
Haugen and Baker (1996) and Ang, Hodrick, Xing, and Zhang (2006) document that expected return is negatively related to volatility in the cross-section of U.S. stocks. The latter paper also documents a negative relationship between expected return and the idiosyncratic component of volatility. Since volatility is negatively related to expected return, it is also negatively related to CAPM alpha, which is expected return adjusted for beta, i.e., for systematic risk. Black (1972), Black, Jensen, and Scholes (1972), and Frazzini and Pedersen (2014) document that alpha is negatively related to beta in the cross-section of U.S. stocks. The relationship between expected return and beta is almost ﬂat during 1926-2012 (Frazzini and Pedersen (2014)), and turns negative during the second half of the sample (Baker, Bradley, and Wurgler (2011)).
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Our model, presented in Section 2, is as follows. We assume a continuous-time inﬁnite-horizon economy with multiple risky assets and an exogenous riskless rate. An investor can invest in the risky assets directly by holding a market index that includes all assets according to their supplies, or indirectly by holding a fund run by a manager. Both investor and manager are price-takers, and can be interpreted as a continuum of identical investors and managers. The manager’s contract consists of a fee, paid by the investor. We assume that the fee must be an aﬃne function of the fund’s performance and the index performance, and we optimize over the coeﬃcients of that function. The manager chooses the fund’s portfolio. We model agency frictions by assuming that the manager can additionally undertake a “shirking” action that lowers the fund’s return but delivers a private beneﬁt to him. If the investor and the manager were the only agents in the model, then they would hold the index because of market clearing. Equilibrium prices would adjust to make the index an optimal portfolio, and the investor would not employ the manager because she can hold the index directly. To ensure that the manager can add value over the index, we introduce a third set of agents, buyand-hold investors, who hold a portfolio that diﬀers from the index. The portfolio choice of these agents could be driven, for example, by corporate-control or hedging considerations.2 Assets that are in low demand by the buy-and-hold investors must earn high expected returns in equilibrium, so that the manager is induced to give them a weight larger than the index weight. Conversely, assets in high demand must earn low expected returns so that the manager underweights them. The former assets are undervalued, when measuring risk by the covariance with the market index, while the latter assets are overvalued. By overweighting the undervalued assets and underweighting the overvalued assets, the manager adds value over the index. In Section 3 we solve the model in the case where there are no agency frictions. We show that the manager’s fee does not depend on the index performance, and hence there is no benchmarking. The fee depends only on the fund’s performance, in a way that implements optimal risk-sharing between the investor and the manager, who are both risk-averse. A negative relationship between risk and return in the cross-section of assets can arise even in the absence of agency frictions (but is stronger when the frictions are present). Consider an asset that is in high demand by buy-andhold investors. This asset earns low expected return and is underweighted by the manager. The reason why its return can be highly volatile is as follows. Following a positive shock to an asset’s expected cashﬂows, the asset accounts for a larger fraction of the manager’s portfolio volatility. The increase in volatility makes the manager less willing to hold the asset, and attenuates the price increase caused by the improved fundamentals. The attenuation eﬀect is weak, however, for an asset that the manager underweights because the asset’s contribution to his portfolio volatility
Fama and French (2007) perform a similar construction in a static setting and show how the presence of investors not holding the market portfolio generates superior opportunities for other investors.
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is small. Therefore, the asset’s price is highly sensitive to the cashﬂow shock, resulting in high volatility. In Section 4 we solve the model in the case where there are agency frictions. We show that the investor makes the manager’s fee more sensitive to the fund’s performance than in the frictions’ absence. This reduces the manager’s incentive to undertake the shirking action. It also exposes him to more risk, but the manager can oﬀset the increase in his personal exposure by choosing a less risky portfolio for the fund. The investor restores the manager’s incentives to take risk by making the fee sensitive to the index performance; this encourages risk-taking because the manager’s personal exposure to market drops is reduced. Benchmarking, however, only incentivizes the manager to take risk that correlates closely with the index, and discourages deviations from that benchmark. Thus, the manager becomes less willing to overweight assets in low demand by buy-and-hold investors, and to underweight assets in high demand. The former assets become more undervalued in equilibrium, and the latter assets become more overvalued. Agency frictions exacerbate not only cross-sectional price distortions but also the negative relationship between risk and return. This is because they raise the volatility of overvalued assets. Recall that in the absence of frictions, a positive shock to an asset’s expected cashﬂows is attenuated by an increase in risk premium because the asset accounts for a larger fraction of the manager’s portfolio volatility. In the presence of frictions, the risk premium instead decreases for overvalued assets, and hence the shock is ampliﬁed. Indeed, the manager underweights overvalued assets, but becomes less willing to do so when these assets account for a larger fraction of market movements. Benchmarking amounts to a short position in the underweighted assets, which the manager seeks to reduce when volatility increases. The cross-sectional price distortions that agency frictions introduce do not cancel out in the aggregate. We show that the positive distortions are more severe than the negative ones, biasing the aggregate market upwards. This is because overvalued assets account, through the ampliﬁcation eﬀect, for an increasingly large fraction of market movements relative to undervalued assets. Therefore, trading against overvaluation exposes the manager to greater risk of under-performing the index than trading against undervaluation. Endogenizing fund managers’ contracts allows us to perform a normative analysis. In Section 5 we show that the contract chosen by a social planner provides the manager with steeper incentives than the contract chosen by private agents. The price distortions under the socially optimal contract are also larger. The ineﬃciency of private contracts can be viewed as a free-rider problem, by interpreting our price-taking investor and manager as a continuum of identical such agents. When one investor in the continuum gives steeper performance incentives to her manager, this induces less shirking. At the same time, the manager oﬀsets the increase in his personal risk exposure by 3

choosing a less risky portfolio, hence exploiting mispricings to a lesser extent. Other managers, however, remain equally willing to exploit mispricings, beneﬁting their investors. When all investors give steeper incentives to their managers, mispricings become more severe in equilibrium, and all managers remain equally willing to exploit them despite being exposed to more risk. Throughout our analysis, we assume constant absolute risk aversion (CARA) utility for the investor and the manager, and square-root processes for asset cashﬂows. Square-root processes have the property that the volatility of an asset’s cashﬂows per share increases with the cashﬂow level. This property is realistic (e.g., the risk of a ﬁrm in absolute terms, i.e., not relative to the ﬁrm’s size, increases with size) and is key for our results. We underscore its importance in Section 6, where we consider a familiar CARA-normal setting, where the volatility of cashﬂows per share is constant. We show that the risk-return relationship is then always positive and agency frictions do not aﬀect the aggregate market. The combination of CARA utility and square-root processes for cashﬂows is to our knowledge new to the literature, including in a frictionless setting. We show that it yields closed-form solutions for asset prices and can accommodate any number of risky assets. The eﬀects of asset management on equilibrium prices are the subject of a growing literature. Our paper is closest to the strand of that literature that focuses on managers’ contractual incentives.3 Brennan (1993) assumes a static setting where some investors have preferences over the return relative to a benchmark. Equilibrium expected returns are given by a two-factor model, with the factors being the market portfolio and the benchmark. Basak and Pavlova (2013) assume a dynamic setting where some investors have preferences over wealth and a benchmark. Demand by these investors raises the prices of the assets included in the benchmark and makes them more volatile and more correlated with each other. In both papers benchmarks are introduced directly into investors’ utility functions. Cuoco and Kaniel (2011) model delegation and contracts explicitly, in a dynamic setting with two risky assets. Investors delegate the management of the risky part of their portfolio to managers, whose fee is a piece-wise aﬃne function of absolute return and of the return relative to a benchmark. Managers’ demand raises the prices of the assets included in the benchmark, but the eﬀect on volatility depends on the convexity of the managers’ fee. Aﬃne fees are not optimal because investors cannot commit to an allocation in the fund when choosing the fee. Garcia and Vanden (2009), Malamud and Petrov (2014), and Qiu (2014) explore static settings where managers observe private signals about the payoﬀ of a single risky asset. They show, respectively, that competition between fund managers makes prices more informative; convexity of the managers’ fee reduces
3 Other papers in the literature focus on fund ﬂows, e.g., Shleifer and Vishny (1997), Berk and Green (2004), Vayanos (2004), He and Krishnamurthy (2012, 2013), Kaniel and Kondor (2013), and Vayanos and Woolley (2013), and on managers’ reputation concerns, e.g., Dasgupta and Prat (2008), Dasgupta, Prat, and Verardo (2011), and Guerrieri and Kondor (2012).

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volatility, and the socially optimal fee provides weaker incentives than the privately optimal fee; and relative-performance evaluation can make prices less informative. Relative to these papers, we assume symmetrically informed managers and linear contracts, but we allow for a general number of risky assets. This sharpens the analysis of cross-sectional asset pricing, and indeed we derive new implications for the risk-return relationship and the pricing of the aggregate market.4 One explanation of the negative relationship between risk and return is based on leverage (Black (1972), Frazzini and Pedersen (2014)). Stocks with high CAPM beta have the same systematic risk as a suitably levered portfolio of low-beta assets, but the latter portfolio is not available to leverage-constrained investors. The demand by these investors pushes up the prices of high-beta assets, and lowers their expected return. Another explanation is based on disagreement (Hong and Sraer (2013)). Investors’ disagreement about the future return of the aggregate market is larger for high-beta assets because they are more sensitive to market movements. Moreover, assets for which disagreement is larger are priced only by optimists and hence oﬀer low expected returns because short-sale constraints drive pessimists out of the market.5 Karceski (2002) explains the negative risk-return relationship based on fund managers’ incentives. If fund ﬂows are more sensitive to performance when the market goes up, then managers prefer high-beta assets because they outperform the market during good times. Baker, Bradley, and Wurgler (2011) suggest an explanation that is based on benchmarking. Fund managers view high- and low-beta assets as equally risky because they care about deviations from a benchmark and not about absolute returns. Therefore, the expected return of high-beta assets does not reﬂect their underlying risk and is too low. Leverage constraints are implicit in both explanations because managers cannot replace high-beta assets by a suitably levered portfolio of low-beta assets. Our explanation assumes no leverage constraints or disagreement. Moreover, unlike the previous explanations, we do not show that exogenous diﬀerences in betas yield overpricing, but rather that high investor demand yields both overpricing and high beta. We also can generate a negative relationship not only between beta and CAPM alpha (expected return adjusted for beta), but also between beta and expected return. By contrast, the negative relationship that the leverage explanation generates is only between beta and alpha: for leverage-constrained investors to prefer high-beta assets, the relationship between beta and expected return must be positive.
4 A number of papers study the choice of contracts taking prices as given. Stoughton (1993) shows that when faced with steeper performance incentives, fund managers choose less risky portfolios, and their incentives to collect information on asset payoﬀs remain unchanged. Admati and Pﬂeiderer (1997) rely on this observation to show that benchmarking distorts managers’ portfolio choice without encouraging them to collect more information. Both results are shown for aﬃne contracts. Ou-Yang (2003) shows that aﬃne contracts and benchmarking can be optimal when moral hazard pertains to other activities than information collection. Bhattacharya and Pﬂeiderer (1985), Starks (1987), Stoughton (1993), Das and Sundaram (2002), Palomino and Prat (2003), Li and Tiwari (2009), and Dybvig, Farnsworth, and Carpenter (2010) study non-aﬃne contracts and whether they can dominate aﬃne ones. 5 Cohen, Polk, and Vuolteenaho (2005) ﬁnd a negative relationship between risk and return during times of high inﬂation, and relate this result to money illusion.

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2.1

Model
Assets

Time t is continuous and goes from zero to inﬁnity. There is an exogenous riskless rate r, and N risky assets. The price Sit of asset i = 1, .., N is determined endogenously in equilibrium. The dividend ﬂow Dit of asset i is given by Dit = bi st + eit , (2.1)

where st is a component common to all assets and eit is a component speciﬁc to asset i. The variables (st , e1t , .., eN t ) are positive and mutually independent, and we specify their stochastic evolution below. The constant bi ≥ 0 measures the exposure of asset i to the common component st . We set Dt ≡ (D1t , .., DN t ) , St ≡ (S1t , .., SN t ) , and b ≡ (b1 , .., bN ) . We denote by dRt ≡ (dR1t , .., dRN t ) the vector of assets’ returns per share in excess of the riskless rate: dRt ≡ Dt dt + dSt − rSt dt. (2.2)

Dividing dRit by the price Sit of asset i yields asset i’s return per dollar in excess of the riskless
$ rate. For simplicity, we refer to dRit and dRit ≡ dRit Sit

as share return and dollar return, respectively,

omitting that they are in excess of the riskless rate. Asset i is in supply of ηi > 0 shares. We denote the market portfolio by η ≡ (η1 , .., ηN ), and refer to it as the index. The variables (st , e1t , .., eN t ) evolve according to square-root processes: √ dst = κ (¯ − st ) dt + σs st dwst , s √ deit = κ (¯i − eit ) dt + σi eit dwit , e (2.3) (2.4)

where (κ, s, e1 , .., eN , σs , σ1 , .., σN ) are positive constants, and the Brownian motions (wst , w1t , .., wN t ) ¯ ¯ ¯ are mutually independent. The square-root speciﬁcation (2.3) and (2.4) allows for closed-form solutions, while also ensuring that dividends remain positive. An additional property of this speciﬁcation is that the volatility of dividends per share (i.e., of Dit ) increases with the dividend level. This property is realistic and key for our results. The constants (¯, e1 , .., eN ) are the unconditional (long-term) means of the variables (st , e1t , .., eN t ). s ¯ ¯
2 2 2 The increments (dst , de1t , .., deN t ) of these variables have variance rates (σs st , σ1 e1t , .., σN eN t ) con2¯ 2¯ 2 ¯ ditionally and (σs s, σ1 e1 , .., σN eN ) unconditionally. We occasionally consider the special case of

“scale invariance,” where the ratio of unconditional standard deviation to unconditional mean is
2 2 2 identical across the N + 1 processes. This occurs when the vector (σs , σ1 , .., σN ) is collinear with

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(¯, e1 , .., eN ). s ¯ ¯

2.2

Agents

The main agents in our model are an investor and a fund manager. Both agents are price-takers and can be interpreted as a continuum of identical investors and managers. The investor can invest in the risky assets directly by holding the index, or indirectly by holding a fund run by a manager. Employing the manager is the only way for the investor to hold a portfolio that diﬀers from the index, and hence to “participate” in the markets for the individual risky assets. One interpretation of this participation friction is that the investor cannot identify assets that oﬀer higher returns than the index, and hence must employ the manager for non-index investing. If the investor and the manager were the only agents in the model, then the participation friction would not matter. This is because the index is the market portfolio, so equilibrium prices would adjust to make that portfolio optimal for the investor. For the participation friction to matter, the manager must add value over the index. To ensure that this can happen, we introduce a third set of agents, buy-and-hold investors, who do not hold the index. These agents could be holding assets for hedging purposes, or could be additional unmodeled fund managers. We denote their aggregate portfolio by η − θ, and assume that θ ≡ (θ1 , .., θN ) is constant over time and not proportional to η. The number of shares of asset i available to the investor and the manager is thus θi , and represents the residual supply of asset i to them. Assets in large residual supply (large θi ) must earn high expected returns in equilibrium, so that the manager is willing to give them weight larger than the index weight. Conversely, assets in small residual supply must earn low expected returns so that the manager is willing to underweight them. By overweighting high-expected-return assets and underweighting low-expected-return ones, the manager adds value over the index. We assume that the residual supply of each asset is positive (θi > 0 for all i). We refer to residual supply simply as supply from now on.6 The investor chooses an investment x in the index η, i.e., holds xηi shares of asset i. She also decides whether or not to employ the manager. Both decisions are made once and for all at t = 0. If the manager is employed by the investor, then he chooses the fund’s portfolio zt ≡ (z1t , .., zN t ) at each time t, where zit denotes the number of shares of asset i held by the fund. The manager can also undertake a “shirking” action mt ≥ 0 that delivers to him a private beneﬁt Amt −
B 2 2 mt

dt,

where 1 ≥ A ≥ 0 and B ≥ 0, and reduces the fund’s return by mt dt. A literal interpretation of mt is as cash diverted from the fund, with diversion involving a deadweight cost except when A = 1 and B = 0. Alternatively, mt could be interpreted in reduced form as insuﬃcient eﬀort to lower
An alternative interpretation of our setting is that there are no buy-and-hold investors, θ is the market portfolio, and η is an index that diﬀers from the market portfolio, e.g., does not include private equity.
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operating costs or to identify a more eﬃcient portfolio. When A = 0, the private beneﬁt is nonpositive for all values of mt and there are no agency frictions. The investor can inﬂuence the choices of zt and mt through a compensation contract that she oﬀers to the manager at t = 0. (Assuming that the manager oﬀers the contract to the investor would not change our analysis, provided that competition drives the manager’s utility to his outside option of not being employed.) The contract speciﬁes a fee that the investor pays to the manager over time. It is chosen optimally within a parametrized class, described as follows. The fee is paid as a ﬂow, and the ﬂow fee dft is an aﬃne function of the fund’s return zt dRt − mt dt and the index return ηdRt . Moreover, the coeﬃcients of this aﬃne function are chosen at t = 0 and remain constant over time. Thus, the ﬂow fee dft is given by dft = φ (zt dRt − mt dt) − χηdRt + ψdt, (2.5)

where (φ, χ, ψ) are constants. The constant φ is the fee’s sensitivity to the fund’s performance, and the constant χ is the sensitivity to the index performance. We assume that the manager invests his personal wealth in the riskless rate. This is without loss of generality: since the manager is exposed to the risky assets through the fee, and can adjust this exposure by changing the fund’s portfolio, a personal investment in those assets is redundant. If the manager is not employed by the investor, then he chooses a personal portfolio zt in the risky assets, receives no fee, and has no ¯ shirking action available.7 Our setting, in which one investor contracts with one manager, ﬁts best institutional asset management, whereby large institutions such as pension funds or sovereign-wealth funds contract with asset management ﬁrms on a target return relative to a benchmark. Yet, we abstract away from a number of real-world features. For example, fees typically depend on assets under management, but we assume that they can only depend on the return achieved by the manager and on the return of the benchmark. Moreover, fees in some cases are convex, but we restrict them to be linear. We also abstract away from implicit incentives generated by fund ﬂows that depend on past returns. Our intention is to capture in a simple manner two key features of asset management contracts: managers’ fees depend on their performance, and performance is evaluated relative to a benchmark. These features are present not only in institutional asset management, but in other forms of asset management as well, such as mutual funds oﬀered to retail investors.
Ruling out the shirking action for an unemployed manager is without loss of generality: since the manager invests his personal wealth, he would not undertake the shirking action even if that action were available.
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2.2.1

Manager’s Optimization Problem

The manager derives utility over intertemporal consumption. Utility is exponential:
∞

E
0

¯ − exp(−¯ct − δt)dt , ρ¯

(2.6)

¯ where ρ is the coeﬃcient of absolute risk aversion, ct is consumption, and δ is the discount rate. ¯ ¯ ¯ We denote the manager’s wealth by Wt . The manager decides at t = 0 whether or not to accept the contract oﬀered by the investor. If the manager accepts the contract and is hence employed by the investor, then he chooses at each time t the fund’s portfolio zt and the shirking action mt . His budget constraint is B ¯ ¯ ¯ dWt = rWt dt + dft + Amt − m2 dt − ct dt, 2 t (2.7)

where the ﬁrst term is the riskless return, the second term is the fee paid by the investor, the third term is the private beneﬁt from shirking, and the fourth term is consumption. The manager’s optimization problem is to choose controls (¯t , zt , mt ) to maximize the expected utility (2.6) subject to c the budget constraint (2.7) and the fee (2.5). We denote by zt (φ, χ, ψ) and mt (φ, χ, ψ) the manager’s ¯ ¯ optimal choices of zt and mt , and by V (Wt , st , et ) his value function, where et ≡ (e1t , .., eN t ) . If the manager is not employed by the investor, then he chooses his personal portfolio zt . His ¯ budget constraint is ¯ ¯ dWt = rWt dt + zt dRt − ct dt. ¯ ¯ (2.8)

The manager’s optimization problem is to choose controls (¯t , zt ) to maximize (2.6) subject to (2.8). c ¯ ¯ ¯ We denote by Vu (Wt , st , et ) his value function. The manager accepts the contract oﬀered by the investor if ¯ ¯ V (W0 , s0 , e0 ) ¯ ¯ Vu (W0 , s0 , e0 ). (2.9)

2.2.2

Investor’s Optimization Problem

The investor derives utility over intertemporal consumption. Utility is exponential:
∞

E
0

− exp(−ρct − δt)dt ,

(2.10)

where ρ is the coeﬃcient of absolute risk aversion, ct is consumption, and δ is the discount rate.

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The investor chooses an investment x in the index η, and whether or not to employ the manager. Both decisions are made at t = 0. If the investor employs the manager, then she oﬀers him a contract (φ, χ, ψ). We denote the investor’s wealth by Wt . The investor’s budget constraint is dWt = rWt dt + xηdRt + zt dRt − mt dt − dft − ct dt, (2.11)

where the ﬁrst term is the riskless return, the second term is the return from the investment in the index, the third and fourth term are the return from the fund, the ﬁfth term is the fee paid to the manager, and the sixth term is consumption. The investor’s optimization problem is to choose controls (ct , x, φ, χ, ψ) to maximize the expected utility (2.10) subject to the budget constraint (2.11), the fee (2.5), the manager’s incentive compatibility constraint zt = zt (φ, χ, ψ), mt = mt (φ, χ, ψ), and the manager’s individual rationality constraint (2.9). We denote by V (Wt , st , et ) the investor’s value function. If the investor does not employ the manager, then her budget constraint is dWt = rWt dt + xηdRt − ct dt. (2.12)

The investor’s optimization problem is to choose controls (ct , x) to maximize (2.10) subject to (2.12). We denote by Vu (Wt , st , et ) her value function. The investor employs the manager if V (W0 , s0 , e0 ) Vu (W0 , s0 , e0 ). (2.13)

2.3

Equilibrium Concept

We look for equilibria in which the investor employs the manager, i.e., oﬀers a contract that the manager accepts. These equilibria are described by a price process St , a compensation contract (φ, χ, ψ) that the investor oﬀers to the manager, and a direct investment x in the index by the investor.

Deﬁnition 1 (Equilibrium prices and contract). A price process St , a contract (φ, χ, ψ), and an index investment x, form an equilibrium if: (i) Given St and (φ, χ, ψ), zt = θ − xη solves the manager’s optimization problem. 10

(ii) Given St , the investor chooses to employ the manager, and (x, φ, χ, ψ) solve the investor’s optimization problem.

The equilibrium in Deﬁnition 1 involves a two-way feedback between prices and contracts. A contract oﬀered by the investor aﬀects the manager’s portfolio choice, and hence equilibrium prices. Equilibrium prices are determined by the market-clearing condition that the fund’s portfolio zt plus the portfolio xη that the investor holds directly add up to the supply portfolio θ. Conversely, the contract that the investor oﬀers to the manager depends on the equilibrium prices. We conjecture that the equilibrium price of asset i is an aﬃne function of st and eit : Sit = a0i + a1i st + a2i eit , where (a0i , a1i , a2i ) are constants. (2.14)

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Equilibrium without Agency Frictions

In this section we solve for equilibrium in the absence of agency frictions. We eliminate agency frictions by setting the parameter A in the manager’s private-beneﬁt function Amt − B m2 to zero. t 2 This ensures that the private beneﬁt is non-positive for all values of mt . When A = 0, the investor and the manager share risk optimally, through the contract. The equilibrium becomes one with a representative agent, whose risk tolerance is the sum of the investor’s and the manager’s. We compute prices in that equilibrium in closed form. We show that the combination of exponential utility and square-root dividend processes—which to our knowledge is new to the literature—yields a framework that is not only tractable but can also help address empirical puzzles about the riskreturn relationship.

Theorem 3.1 (Equilibrium Prices and Contract without Agency Frictions). When A = 0, the following form an equilibrium: the price process St given by (2.14) with a0i = a1i = a2i = κ (a1i s + a2i ei ) , ¯ ¯ r bi ρ¯ ρ 2 (r + κ)2 + 2r ρ+¯ θbσs ρ

(3.1) ≡ a1 bi , ; (3.2)

1 ρ¯ ρ 2 (r + κ)2 + 2r ρ+¯ θi σi ρ ρ ρ+¯ , 0, 0 ρ

(3.3)

the contract (φ, χ, ψ) =

; and the index investment x = 0. 11

Since x = 0, the investor does not invest directly in the index. Market-clearing hence implies that the fund holds the supply portfolio θ. Since, in addition, χ = 0, the manager is compensated based only on absolute performance and not on performance relative to the index. Therefore, the manager receives a fraction φ = risk-sharing under exponential utility. The coeﬃcient a2i measures the sensitivity of asset i’s price to changes in the asset-speciﬁc component eit of dividends. A unit increase in eit causes asset i’s dividend ﬂow at time t to increase by one. In the absence of risk aversion (ρ¯ = 0), (3.3) implies that the price of asset i ρ would increase by a2i =
1 r+κ . ρ ρ+¯ ρ

of the return of the supply portfolio, and the investor ρ ¯ ρ+¯ . ρ

receives the complementary fraction 1 − φ =

This coincides with the standard rule for optimal

This is the present value, discounted at the riskless rate r, of the

increase in asset i’s expected dividends from time t onwards: the dividend ﬂow at time t increases by one, and the eﬀect decays over time at rate κ. The coeﬃcient a1i measures the sensitivity of asset i’s price to changes in the common component st of dividends. We normalize a1i by bi , the sensitivity of asset i’s dividend ﬂow to changes in st . This yields a coeﬃcient a1 that is common to all assets, and that measures the sensitivity of any given asset’s price to a unit increase in the asset’s dividend ﬂow at time t caused by an increase in st . In the absence of risk aversion, (3.2) implies that the price of asset i would increase by a1 =
1 r+κ .

Hence, a1 and a2i would be equal: an increase in an asset’s dividend ﬂow would have

the same eﬀect on the asset’s price regardless of whether it comes from the common or from the asset-speciﬁc component. Risk aversion lowers a1 and a2i . This is because increases in st or eit not only raise expected dividends but also make them riskier, and risk has a negative eﬀect on prices when agents are risk averse. The eﬀect of increased risk attenuates that of higher expected dividends. One would expect the attenuation to be larger when the increased risk comes from increases in st rather than in eit . This is because agents are more averse to risk that aﬀects all assets rather than a speciﬁc asset. Equations (3.2) and (3.3) imply that a1 < a2i if
N 2 θbσs 2 2 σs > θi σi .

= i=1 θi bi

(3.4)

Equation (3.4) evaluates how a unit increase in asset i’s dividend ﬂow at time t aﬀects the covariance between the dividend ﬂow of asset i and of the supply portfolio. This covariance captures the relevant risk in our model. The left-hand side of the inequality in (3.4) is the increase in the covariance when the increase in dividend ﬂow is caused by an increase in st . The right-hand side is the increase in the same covariance when the increase in dividend ﬂow is caused by an increase in eit . When (3.4) holds, the change in st has a larger eﬀect on the covariance compared to the 12

change in eit . Therefore, it has a larger attenuating eﬀect on the price. Equation (3.4) holds when the number N of assets exceeds a threshold, which can be zero. This is because the left-hand side increases when assets are added, while the right-hand side remains constant. In the special case of scale invariance, (3.4) takes the intuitive form θb¯ > θi ei . s ¯ (3.5)

The left-hand side is the dividend ﬂow of the supply portfolio that is derived from the common component. The right-hand side is the dividend ﬂow of the same portfolio that is derived from the component speciﬁc to asset i. Equation (3.5) obviously holds when N is large enough.

3.1

Supply Eﬀects

We next examine how diﬀerences in supply in the cross-section of assets are reﬂected into prices and return moments. We compare two assets i and i that diﬀer only in supply (θi = θi ), but have otherwise identical characteristics ((bi , ei , σi , ηi ) = (bi , ei , σi , ηi )). When comparing the prices of ¯ ¯ the assets at a time t, we assume that the asset-speciﬁc components of dividends at t are also identical (eit = ei t ). We take asset i to be the one in smaller supply (θi < θi ). We compute unconditional (long-term) moments of returns, and consider both returns per share and returns per dollar invested. Moments of share returns can be computed in closed form. To compute closed-form solutions for moments of dollar returns, we approximate from now on the dollar return of an asset by its share return divided by the unconditional mean of the share price,
$ $ i.e., modify the deﬁnition of dRit to dRit ≡ dRit E(Sit ) .

For example, expected dollar return is the

expected ratio of share return to share price but we approximate it by the ratio of expected share return to expected share price.

Proposition 3.1 (Price and Expected Return). Suppose that A = 0. An asset i in smaller supply than an otherwise identical asset i has higher price at time t (Sit > Si t ), higher expected price (E(Sit ) > E(Si t )), lower expected share return (E(dRit ) < E(dRi t )), and lower expected dollar
$ $ return (E(dRit ) < E(dRi t )).

An asset i in small supply θi must oﬀer low expected share return, so that the manager is induced to hold a small number of shares of the asset. Therefore, the asset’s price must be high. The asset’s expected dollar return is low because of two eﬀects working in the same direction: low expected share return in the numerator, and high price in the denominator.

13

The eﬀect of θi on the asset price is through the coeﬃcient a2i , which measures the price sensitivity to changes in the asset-speciﬁc component eit of dividends. When θi is small, an increase in eit is accompanied by a small increase in the covariance between the dividend ﬂow of the asset and of the supply portfolio θ. Therefore, the positive eﬀect that the increase in eit has on the price through higher expected dividends is attenuated by a small negative eﬀect due to the increase in risk. As a consequence, a2i is large. Since an increase in eit away from its lower bound of zero has a large eﬀect on the price, the price is high. Note that θi does not have an eﬀect through the coeﬃcient a1i , which measures the price sensitivity to changes in the common component st of dividends. This coeﬃcient depends on θi only through the aggregate quantity θb, which is constant in cross-sectional comparisons.

Proposition 3.2 (Return Volatility). Suppose that A = 0. An asset i in smaller supply than an otherwise identical asset i has higher share return variance (Var(dRit ) > Var(dRi t )). It has
$ $ higher dollar return variance (Var(dRit ) > Var(dRi t )) if and only if 2 2 2 2 D1 ≡ a1 bi (a2i + a2i )(σi s − σs ei ) + 2(a2i a2i σi ei − a2 b2 σs s) > 0. ¯ ¯ ¯ ¯ 1 i

(3.6)

Since dividend changes have a large eﬀect on the price of an asset that is in small supply, such an asset has high share return volatility (square root of variance). This eﬀect is concentrated on the part of volatility that is driven by the asset-speciﬁc component, while there is no eﬀect on the part that is driven by the common component. Whether small supply is associated with high or low dollar return volatility depends on two eﬀects working in opposite directions: high share return volatility in the numerator, and high price in the denominator. The ﬁrst eﬀect dominates when (3.6) holds. Since the eﬀect of supply on volatility is concentrated on the part that is driven by the assetspeciﬁc component, (3.6) should hold if that part is large enough. This can be conﬁrmed, for example, in the case of scale invariance. Equation (3.6) becomes √ a2i a2i ei > a1 bi s, ¯ ¯ (3.7)

and has the simple interpretation that the volatility driven by the common component is smaller than the geometric average, across assets i and i , of the volatilities driven by the asset-speciﬁc
2 components. Indeed, the conditional variance rate driven by the common component is a2 b2 σs st 1 i 2 for both assets. The conditional variance rate driven by the asset-speciﬁc component is a2 σi eit 2i 2 for asset i and a2 σi eit for asset i . Taking expectations, we ﬁnd the unconditional variance rates 2i 2¯ 2¯ 2¯ a2 b2 σs s, a2 σi ei , and a2 σi ei , respectively. Under scale invariance, the former is smaller than the 1 i 2i 2i

14

geometric average of the latter if (3.7) holds. We next examine how supply aﬀects the systematic and idiosyncratic parts of volatility. When the number N of assets is large, these coincide, respectively, with the parts driven by the common and the asset-speciﬁc component. For small N , however, the systematic part includes volatility driven by the asset-speciﬁc component. To compute the systematic and idiosyncratic parts, we regress the return dRit of asset i on the return dRηt ≡ ηdRt of the index: dRit = βi dRηt + d it . The CAPM beta of asset i is βi = Cov(dRit , dRηt ) , Var(dRηt ) (3.9) (3.8)

and measures the systematic part of volatility. The variance Var(d it ) of the regression residual measures the idiosyncratic part. These quantities are deﬁned in per-share terms. Their per-dollar counterparts are
$ βi = $ $ Cov(dRit , dRηt ) $ Var(dRηt )

, dRηt E(Sηt ) ,

(3.10)

$ and Var(d $ ), where dRηt ≡ it

Sηt ≡ ηSt , and d

$ it

≡

d it E(Sit ) .

Proposition 3.3 (Beta and Idiosyncratic Volatility). Suppose that A = 0. An asset i in smaller supply than an otherwise identical asset i has higher share beta (βi > βi ) and idiosyncratic share return variance (Var(d it ) > Var(d i t )). $ $ It has higher dollar beta (βi > βi ) if and only if

2 2 2 D2 ≡ a1 bi (a2i + a2i )ηi σi s + a2i a2i ηi σi ei − a2 bi ηbσs s > 0, ¯ ¯ ¯ 1

(3.11)
$ i t ))

and higher idiosyncratic dollar return variance (Var(d $ ) > Var(d it 
2 a2 (ηb)2 σs s + ¯ 1 j=1 N

if and only if


2 2 a2 ηj σj ej  a1 bi sD1 ¯ ¯ 2j

−

2 2 2 2 ¯ ¯ ¯ ¯ ¯ ¯ a2 bi ηbσs s + a2 ηi σi ei (a1 bi s + a2i ei ) + a2 bi ηbσs s + a2 ηi σi ei (a1 bi s + a2i ei ) D2 > 0. ¯ ¯ 1 2i 1 2i

(3.12) The share beta and idiosyncratic volatility are large for an asset that is in small supply because of the eﬀect identiﬁed in Propositions 3.1 and 3.2: changes to the asset-speciﬁc component of dividends

15

have a large eﬀect on the price of such an asset. This yields high idiosyncratic volatility. It also yields large beta because asset-speciﬁc shocks have a large contribution to the asset’s covariance with the index. Whether small supply is associated with high or low dollar beta and idiosyncratic volatility depends on two eﬀects working in opposite directions: high share beta and idiosyncratic volatility in the numerator, and high price in the denominator. To build intuition on which eﬀect dominates, we consider the case where the number N of assets is large. For large N , an asset’s covariance with the index is driven mainly by the common shocks, whose eﬀect on price does not depend on supply. Since supply aﬀects only a small fraction of the covariance, the eﬀect of supply on price should dominate that on share beta. Hence, dollar beta should be small for an asset that is in small supply. This can be conﬁrmed, for example, in the case of scale invariance and symmetric assets with identical characteristics (bi , ei , ηi , θi ). (To ensure ¯ that assets i and i diﬀer in their supply, we assume that θi − θi is close but not equal to zero.) We denote by (bc , ec , ηc , θc ) the common values of (bi , ei , ηi , θi ) across all assets, by a2c the common ¯ ¯ value of a2i , and by y ≡ a2c ec ¯ a1 bc s ¯

the ratio of volatility driven by the asset-speciﬁc component to the

volatility driven by the common component. We can write (3.11) as 2y + y 2 − N > 0. (3.13)

As N increases, (3.13) is satisﬁed for values of y that exceed an increasingly large threshold. An asset’s idiosyncratic volatility, for large N , is driven mainly by the shocks speciﬁc to that asset. Since the eﬀect of supply is only through those shocks, while common shocks account for a potentially large fraction of the price, the eﬀect of supply on idiosyncratic share volatility should dominate the eﬀect of supply on price. Hence, idiosyncratic dollar return volatility should be large for an asset that is in small supply. For example, in the case of scale invariance and symmetric assets, we can write (3.12) as N − 2 − y > 0. (3.14)

As N increases, (3.14) is satisﬁed for values of y that are below an increasingly large threshold. We ﬁnally determine how supply aﬀects assets’ CAPM alphas, i.e., the expected returns that assets are earning in excess of the CAPM. The CAPM alpha of asset i is αi = E(dRit ) − βi E(dRηt ), (3.15)

16

in per-share terms. Its per-dollar counterpart is
$ $ $ $ αi = E(dRit ) − βi E(dRηt ) =

αi . E(Sit )

(3.16)

Proposition 3.4 (Alpha). Suppose that A = 0. An asset i in smaller supply than an otherwise
$ $ identical asset i has lower share alpha (αi < αi ) and lower dollar alpha (αi < αi ).

An asset in small supply has low share alpha because it has low expected share return (Proposition 3.1) and high share beta (Proposition 3.3). The eﬀect of supply on share alpha carries through to dollar alpha, so an asset in small supply has low dollar alpha as well. Our results have implications for the relationship between risk and expected return in the cross-section of assets. Standard theories predict that this relationship should be positive: riskier assets should compensate investors with higher expected return. Empirically, however, a negative relationship has often been documented. Moreover, in those instances where a positive relationship has been documented, it has been found to be weaker than the theoretical one. The weakness of the relationship has been shown using alpha, which is expected return in excess of its theoretical value. Alpha has been shown to be negative for riskier assets and positive for less risky ones. This means that the expected returns of the former assets are not as high as predicted by theory, while the expected returns of the latter assets are not as low. The empirical ﬁndings concern both CAPM alpha, as well as alphas computed using other risk-adjustment methods such as the Fama-French three-factor model. Haugen and Baker (1996) and Ang, Hodrick, Xing, and Zhang (2006) document that U.S. stocks with high return volatility earn lower returns on average than stocks with low volatility. The latter paper also shows that the negative relationship holds not only for return volatility but also for the idiosyncratic component of that volatility. Since a negative relationship holds between volatility and expected return, it also holds between volatility and alpha: adjusting for risk can only make the negative relationship stronger. Since alpha averages to zero across stocks, high-volatility stocks earn negative alpha and low-volatility stocks earn positive alpha. The negative relationship between volatility on one hand and expected return or alpha on the other is known as the volatility anomaly. Black (1972), Black, Jensen, and Scholes (1972), and Frazzini and Pedersen (2014) document that U.S. stocks with high CAPM beta earn negative alpha while stocks with low beta earn positive alpha. The relationship between expected return and beta is almost ﬂat during 1926-2012 (Frazzini and Pedersen (2014)), and turns negative during the second half of the sample (Baker, Bradley, and Wurgler (2011)). The negative relationship between beta and alpha, as well as the weak or negative relationship between beta and expected return, is known as the beta anomaly. 17

The results in this section suggest a mechanism that could help explain the volatility and beta anomalies, even in the absence of agency frictions. A negative relationship between volatility or beta on one hand, and expected return or alpha on the other, can be generated by the way that these variables depend on supply. Assets in small supply earn low expected dollar return (Proposition 3.1) and negative alpha (Proposition 3.4). Under some conditions, they also have high dollar return volatility (Proposition 3.2), high idiosyncratic dollar return volatility (Proposition 3.3), and high dollar beta (Proposition 3.3). Under these conditions our model can generate a negative relationship both between risk and alpha, as well as between risk and expected return. We further explore the relationship between risk and return implied by our model in the next section, where we add agency frictions and quantify the eﬀects in the context of a numerical example.

4

Equilibrium with Agency Frictions

In this section we solve for equilibrium in the presence of agency frictions. We introduce agency frictions by setting the parameter A in the manager’s private-beneﬁt function Amt −
B 2 2 mt

to a

positive value. For simplicity, we set the parameter B to zero. This pins down immediately the coeﬃcient φ that characterizes how sensitive the manager’s fee is to the fund’s performance. Indeed, if φ < A, then the manager will choose an arbitrarily large shirking action mt . This forces the investor to oﬀer φ ≥ A, in which case there is no shirking, i.e., mt = 0.8 When A ≤ oﬀers φ = ρ ρ+¯ . ρ ρ ρ+¯ , ρ

the

constraint φ ≥ A is not binding, since in the equilibrium without agency frictions the investor When instead A > ρ ρ+¯ , ρ

the constraint is binding, and the investor oﬀers φ = A.

Allowing B to be positive yields a richer theory of contract determination, both on the positive and on the normative front. The asset pricing results, however, remain essentially the same. For this reason we defer the case B > 0 to Section 5, where we perform a normative analysis of contracts.

Theorem 4.1 (Equilibrium Prices and Contract with Agency Frictions). Suppose that B = 0. When A> ρ ρ+¯ , ρ ρ ρ+¯ ρ

≥ A > 0, the equilibrium in Theorem 3.1 remains an equilibrium. When

the following form an equilibrium: the price process St given by (2.14) with a0i given by

(3.1), a1i = a2i =
8

bi
2 (r + κ)2 + 2rρ(φθ − χη)bσs ¯ 1 2 (r + κ)2 + 2rρ(φθi − χηi )σi ¯

≡ a1 bi , ;

(4.1) (4.2)

For φ = A, the manager is indiﬀerent between all values of mt . We assume that he chooses mt = 0, as would be the case for any positive value of B, even arbitrarily small.

18

the contract (φ, χ, ψ) with φ = A, ψ = 0, and χ > 0 being the unique solution to
N

(a1 − a1 ) ηbˆ0 + ˇ s i=1 (a2i − a2i ) ηi ei0 = 0, ˇ ˆ

(4.3)

where a1 ≡ ˇ a2i ≡ ˇ 1 (r + κ)2
2 + 2rρ[(1 − φ)θ + χη]bσs 1

, ,

(4.4) (4.5)

2 (r + κ)2 + 2rρ[(1 − φ)θi + χηi ]σi

st ≡ st + κ s, and eit ≡ eit + κ ei ; and the index investment x = 0. ˆ ˆ r¯ r¯ When A > ρ ρ+¯ , ρ

the investor renders the manager’s fee more sensitive to the fund’s performance ρ ρ+¯ ). ρ

compared to the equilibrium without agency frictions (φ = A >

This exposes the manager

to more risk, but eliminates his incentive to undertake the shirking action mt . If the increase in φ were the only change in the contract, then the manager would respond by scaling down the fund’s holdings of the risky assets and investing more in the riskless rate. This would oﬀset the increase in his personal risk exposure caused by the larger φ. The investor restores the manager’s incentives to take risk by making the fee sensitive to the index performance (χ > 0). This induces the manager to scale up the fund’s holdings of the risky assets because his personal exposure to market drops becomes smaller. The increase in the risky-asset holdings, however, is according to the weights in the index η and not those in the supply portfolio θ. The fund’s portfolio thus changes in response to the increases in φ and χ, and becomes closer to the index. This causes equilibrium prices to change, as we show in Section 4.1. The investor does not invest directly in the index (x = 0) because she can control the fund’s index exposure by changing χ. The compensation that the manager receives for performance relative to the index is analogous to relative-performance evaluation in models of optimal contracting under moral hazard (e.g., Holmstrom 1979). The mechanism is somewhat diﬀerent, however. In typical moral-hazard models, relative-performance evaluation is used to insulate the agent from risk that he cannot control. In our model, instead, the agent can control his risk exposure through his choice of the fund’s portfolio. Compensation based on relative performance is instead used to induce the agent to take risk. Equations (4.1) and (4.2) show how the contract parameters (φ, χ) aﬀect equilibrium prices. Prices are determined by the covariance with the portfolio φθ − χη. This is the portfolio that describes the manager’s personal risk exposure: the fee is φ times the fund’s return, which in equilibrium is the return of the supply portfolio θ, minus χ times the return of the index portfolio

19

η. The covariance is multiplied by the manager’s risk aversion coeﬃcient ρ. Prices are determined ¯ by the manager’s risk aversion and risk exposure because the manager is marginal in pricing the assets. We examine the properties of prices in Sections 4.1 and 4.2. Equation (4.3), which characterizes the the contract parameter χ, can be given an intuitive interpretation. The quantity Si0 = a1 bi s0 + a2i ei0 is the price of asset i at time zero. We can ˆ ˆ ˇ also construct the counterpart Si0 ≡ a1 bi s0 + a2i ei0 of this expression for the coeﬃcients a1 and a2i ˇ ˆ ˇ ˆ ˇ ˇ deﬁned in (4.4) and (4.5). This is the hypothetical price of asset i at time zero under the assumption ˇ that the asset is priced from the investor instead of the manager. The price Si0 can be derived from Si0 by replacing the manager’s risk exposure φθ − χη by the investor’s exposure (1 − φ)θ + χη, and the manager’s risk-aversion coeﬃcient ρ by the investor’s coeﬃcient ρ. Equation (4.3) states ¯ ˇ that the investor and the manager agree on their valuation of the index: η S0 = ηS0 . This is because the investor can invest directly in the index, and hence is marginal in pricing the index. The investor and the manager can disagree on their valuation of other portfolios. In particular, and as we show in the proof of Theorem 4.1, the investor values the supply portfolio more than ˇ the manager: θS0 > θS0 . The investor could acquire more of the supply portfolio by lowering φ, but this would incentivize the manager to undertake the shirking action mt . Proposition 4.1 summarizes how the contract parameters (φ, χ) depend on agency frictions, as measured by the private-beneﬁt parameter A.

Proposition 4.1 (Eﬀect of Agency Frictions on Manager’s Contract). Suppose that A > ρ ρ+¯ ρ

and B = 0. Following an increase in the private-beneﬁt parameter A, the manager’s fee becomes

∂φ ∂χ more sensitive to the fund’s performance ( ∂A > 0) and to the index performance ( ∂A > 0).

4.1

Cross-Sectional Pricing and Ampliﬁcation

We next examine how agency frictions, as measured by the private-beneﬁt parameter A, aﬀect the cross-section of asset prices and of return moments. Following an increase in A, the manager’s fee becomes more sensitive to the fund’s performance, and performance becomes benchmarked to the index to a larger extent. This renders the manager less willing to deviate from the index. Recall that in equilibrium the manager deviates from the index by overweighting assets in large supply and underweighting assets in small supply. Hence, when A increases, the prices of large-supply assets must decrease so that the manager remains equally willing to overweight them, and the prices of small-supply assets must increase so that the manager remains equally willing to underweight them. Proposition 4.2 conﬁrms these results in two simple cases of the model. First, when dividends vary over time only because of the asset-speciﬁc component. This case can be derived by setting the volatility parameter σs of the common component to zero. Second, when supply is the only driver 20

of cross-sectional variation, i.e., the remaining characteristics (bi , ei , σi , ηi ) are common to all assets. ¯ We denote the common values of these characteristics by (bc , ec , σc , ηc ). For simplicity, we assume ¯ that in both cases and for the rest of Section 4 the time-zero values of the processes (st , e1t , .., eN t ) are equal to the processes’ unconditional means, i.e., (s0 , e10 , .., eN 0 ) = (¯, e1 , .., eN ). s ¯ ¯

Proposition 4.2 (Eﬀect of Agency Frictions on Price and Expected Return). Suppose that A > ρ ρ+¯ ρ

and B = 0. Following an increase in the private-beneﬁt parameter A, the following

results hold: (i) When σs = 0, there exists a threshold γ > 0 such that the prices of assets i for which decrease ( ∂Sit ∂A < 0), and the prices of assets i for which θi ηi θi ηi

>γ

< γ increase

( ∂Sit ∂A

> 0). Both sets

of assets are non-empty. (ii) When (bi , ei , σi , ηi ) = (bc , ec , σc , ηc ) for all i, the price of asset i = arg max ¯ ¯ creases ( ∂Sit < 0), and the expected price of asset i = arg min ∂A 0). For assets whose prices decrease, expected returns increase, both in share ( ∂E(dRit ) > 0) and dollar ∂A (
$ ∂E(dRit ) ∂A

j∈{1,..,N } θj de∂E(Sit ) > j∈{1,..,N } θj increases ( ∂A

> 0) terms. Conversely, for assets whose prices increase, expected returns decrease, both
$ ∂E(dRit ) ∂A

in share ( ∂E(dRit ) < 0) and dollar ( ∂A

< 0) terms.

When the time-variation of dividends is only asset-speciﬁc, the eﬀect of agency frictions on prices takes a simple form. Assets are ordered according to the ratio θi ηi

of the weight in the supply

portfolio θ relative to the index portfolio η. Assets for which the ratio exceeds a threshold γ, and are hence overweighted by the manager, drop in price when A increases. Conversely, underweighted assets, for which the ratio is below γ, rise. These results can also be stated in terms of the risk premium associated to the asset-speciﬁc component, i.e., the compensation that the manager requires for bearing the asset-speciﬁc risk. Agency frictions raise the risk premium for the assets in large supply, and lower it for the assets in small supply. When the time-variation of dividends has a common component in addition to the asset-speciﬁc one, the analysis becomes more complicated. This is because agency frictions aﬀect the risk premium associated to the common component and this eﬀect is the same for all assets regardless of their supply. When supply is the only driver of cross-sectional variation, agency frictions raise the risk premium associated to the common component. Since they also raise the risk premium associated to the asset-speciﬁc component for the large-supply assets, the prices of these assets decrease. For the small-supply assets instead, the two eﬀects go in opposite directions because the risk premium associated to the asset-speciﬁc component decreases. The prices of these assets 21

increase in expectation, i.e., in terms of their unconditional (long-term) means. Proposition 4.2 implies that agency frictions exacerbate price distortions caused by supply. Indeed, assets in large supply, which the manager overweights in equilibrium, trade at low prices holding else equal. Agency frictions cause their prices to become even lower. Conversely, assets in small supply, which the manager underweights, trade at high prices, and agency frictions cause their prices to rise further. Agency frictions eﬀectively raise the supply of assets whose supply is already large, and lower the supply of assets whose supply is already small. Formally, in the presence of frictions, prices are determined by the covariance with the portfolio φθ − χη that describes the manager’s personal risk exposure. Frictions raise φ and χ in such a way that φθi − χηi increases for large-θi assets and decreases for small-θi assets. Agency frictions aﬀect not only prices and expected returns, but also the volatility of returns. Indeed, because they magnify diﬀerences in supply, they also magnify the relationship between supply and volatility shown in Section 3.1. As shown in that section, supply is related to volatility through an attenuation eﬀect. Following a positive shock to an asset’s expected cashﬂows, the asset accounts for a larger fraction of the manager’s portfolio volatility. The increase in volatility makes the manager less willing to hold the asset, and attenuates the price increase caused by the improved fundamentals. The extent of attenuation depends on the asset’s supply. If supply is large, then attenuation is strong because the asset’s contribution to portfolio volatility is large. Therefore, the cashﬂow shock has a weak eﬀect on the asset’s price, resulting in low share return volatility. Conversely, if supply is small, then attenuation is weak, and share return volatility is high. The eﬀect of agency frictions on return volatility is most striking for assets in small supply. For these assets, the attenuation eﬀect described in the previous paragraph can reverse sign and become an ampliﬁcation eﬀect. Consider again a positive shock to an asset’s expected cashﬂows. Following the shock, the asset accounts for a larger fraction of the manager’s portfolio volatility. The manager, however, cares not only about the volatility of his portfolio, as is the case in the absence of agency frictions, but also about the volatility of his deviation from the index. When the shock concerns an asset in small supply, which the manager underweights, the latter volatility increases and it can be reduced by buying the asset, i.e., reducing the underweight. Buying pressure by the manager to reduce the underweight ampliﬁes the price increase caused by the improved fundamentals and results in high share return volatility. This is the ampliﬁcation eﬀect. Conversely, for assets in large supply, which the manager overweights, the volatility of the manager’s deviation from the index can be reduced by selling the asset, i.e., reducing the overweight. Hence, attenuation is stronger, resulting in lower share return volatility. Proposition 4.3 conﬁrms these results in the two special cases of the model, and examines the behavior of dollar return volatility as well.

22

Proposition 4.3 (Eﬀect of Agency Frictions on Return Volatility). Suppose that A >

ρ ρ+¯ ρ

and B = 0. Following an increase in the private-beneﬁt parameter A, the following results hold: (i) When σs = 0, the return volatility of assets i for which ( ∂Var(dRit ) < ∂A θ which ηi < γ i 0) and dollar (
$ ∂Var(dRit )

θi ηi

> γ decreases, both in share
$ ∂Var(dRit ) ∂A

∂A

< 0) terms, and the return volatility of assets i for > 0) and dollar ( > 0) terms, where

increases, both in share

( ∂Var(dRit ) ∂A

the threshold γ > 0 is as in Proposition 4.2. Both sets of assets are non-empty. (ii) When (bi , ei , σi , ηi ) = (bc , ec , σc , ηc ) for all i, the share return volatility of asset i = arg max ¯ ¯ decreases ( ∂Var(dRit ) < 0). The return volatility of asset i = arg min j∈{1,..,N } θj ∂A ∂Var(dR$ ) both in share ( ∂Var(dRit ) > 0) and dollar ( ∂A it > 0) terms, provided that ∂A
2 2 a2i σi > a1 bi σs . j∈{1,..,N } θj

increases,

(4.6)

Condition (4.6) is suﬃcient for the share return volatility, and necessary and suﬃcient for the dollar return volatility.

When the time-variation of dividends is only asset-speciﬁc, the eﬀect of agency frictions takes the same form as in Proposition 4.2: assets whose weight in the supply portfolio relative to the index portfolio exceeds a threshold γ become less volatile, and assets below the threshold become more volatile. When a time-varying common component is added, the analysis becomes more complicated but the results have a similar ﬂavor. Agency frictions lower the share return volatility of assets in large supply. They also raise the share return volatility of assets in small supply, under the suﬃcient condition (4.6). Moreover, (4.6) is necessary and suﬃcient for the dollar return volatility of small-supply assets to increase with agency frictions. Condition (4.6) parallels (3.6), which is necessary and suﬃcient for dollar return volatility to be higher for assets in small supply: under scale invariance and i = i , (3.6) and (4.6) become identical, and require that the volatility driven by the asset-speciﬁc component exceeds that driven by the common component.

4.2

Aggregate Market

We next examine how agency frictions aﬀect the valuation of the aggregate market, i.e., the index η. Recall from Proposition 4.2 that frictions cause the prices of assets in large supply to drop and the prices of assets in small supply to rise. The eﬀect on the aggregate market is thus ambiguous a priori. We show, however, that the cross-sectional diﬀerences do not cancel out, and the aggregate market goes up. Therefore, agency frictions distort the prices of small-supply assets upwards more than they distort the prices of large-supply assets downwards.

23

The intuition for the asymmetry lies in the interaction between the manager’s risk-taking incentives and the ampliﬁcation eﬀect described in Section 4.1. When agency frictions are more severe, the manager becomes less willing to deviate from the index because he is benchmarked more tightly on it. The manager’s deviations are to overweight assets in large supply, which earn high expected return, and underweight assets in small supply, which earn low expected return. The latter deviation becomes increasingly costly relative to the former because of the ampliﬁcation eﬀect: since the share return volatility of small-supply assets increases, the manager is exposed to an increased risk of deviating from the index by underweighting these assets. Therefore, when the manager is benchmarked more tightly on the index, he becomes particularly keen to reduce the underweights. As a consequence, the price of the underweights goes up more than the price of the overweights goes down. Proposition 4.4 conﬁrms these results in two simple cases of the model. These cases parallel those in Propositions 4.2 and 4.3, but are somewhat more restrictive. When dividends vary over time only because of the asset-speciﬁc component, assets are assumed symmetric in terms of some of their characteristics. And when the time-variation of dividends has also a common component, supply is assumed to take only two values.

Proposition 4.4 (Eﬀect of Agency Frictions on Aggregate Market). Suppose that A > B = 0, and that one of the following conditions holds: (i) σs = 0, and (σi , ηi ) = (σc , ηc ) for all i or (σi , θi ) = (σc , θc ) for all i. (ii) (bi , ei , σi , ηi ) = (bc , ec , σc , ηc ) for all i, and θi can take only two values. ¯ ¯

ρ ρ+¯ , ρ

Following an increase in the private-beneﬁt parameter A, the expected price of the aggregate market increases ( (
$ ∂E(dRηt )

∂E(Sηt ) ∂A

> 0), and the expected return decreases both in share (

∂E(dRηt ) ∂A

< 0) and dollar

∂A

< 0) terms.

4.3

Numerical Example

We illustrate our results with a numerical example. We set the investor’s risk-aversion coeﬃcient ρ to one. This is a normalization because we can redeﬁne the units of the consumption good. We set the manager’s risk-aversion coeﬃcient ρ to 50, meaning that the manager accounts for ¯ of aggregate risk tolerance. The ratio ρ ρ+¯ ρ ρ ρ+¯ ρ

= 2%

reﬂects the “size” of fund managers relative to fund

investors, and can be related to the size of the ﬁnancial sector. Philippon (2008) reports that the GDP share of the Finance and Insurance industry was 5.5% on average during 1960-2007 in the US. Since only part of that industry concerns asset management, 5.5% can be viewed as an upper 24

bound for

ρ ρ+¯ . ρ

We set the riskless rate r to 4%. We set the mean-reversion parameter κ to 10%, log(2) 0.1

meaning that the half-life of dividend shocks is

= 6.93 years.

If the assets in our model are interpreted as individual stocks, then supply eﬀects will be small. Indeed, since there is a large number of stocks, the speciﬁc risk associated to each stock is small relative to the aggregate risk tolerance. We interpret instead our assets as segments of the stock market, e.g., style portfolios such as value and growth, or industry-sector portfolios. Under this interpretation, asset-speciﬁc risk concerns market segments, and so does the demand by the buyand-hold investors. The volatility and beta anomalies that we show in this section are also at the segment level. We set the number N of assets to six. We assume that the six assets are divided into two groups, with three assets in each group. Assets in each group have identical characteristics, except for supply. Thus, assets are identical in terms of number of shares ηi included in the index, sensitivity bi of dividends to the dividends’ common component st , long-run mean ei of the asset-speciﬁc component, and volatility parameter ¯ σi of the asset-speciﬁc component. We assume that the index η includes one share of each asset (ηi = 1 for i = 1, .., 6). This is a normalization because we can redeﬁne one share of each asset. We set the supply of assets 1, 2, and 3, left over by the buy-and-hold investors, to 0.7 share, and the corresponding supply of assets 4, 5, and 6 to 0.3 share (θ1 = θ2 = θ3 = 0.7 and θ4 = θ5 = θ6 = 0.3). We set the dividend sensitivities to one (bi = 1 for i = 1, .., 6). This is a normalization because we can redeﬁne st . The remaining parameters are the long-run mean s of the common component of dividends, ¯ the long-run mean ei of the asset-speciﬁc component, the volatility parameter σs of the common ¯ component, and the volatility parameter σi of the asset-speciﬁc component. We determine σi as function of (¯, ei , σs ) by imposing scale invariance, i.e., s ¯
2 σs s ¯

=

2 σi ei . ¯

We determine the ratio

ei ¯ s ¯

based on

the fraction of assets’ return variance that is idiosyncratic. Finally, we determine (¯, σs ) based on s the mean and the variance of the return of the aggregate market η. We set (¯, σs ) = (0.65, 1); under s these choices the market’s expected return (in excess of the riskless rate) is 5.27% in the absence of agency frictions (A ≤ ρ ρ+¯ ), ρ

and the market’s return volatility is 16.6%. We also set ei = 0.4; under ¯

this choice idiosyncratic risk accounts for 60% of assets’ return variance in the absence of frictions. Our results are sensitive to the size of idiosyncratic risk, as we show in Propositions 3.2, 3.3, and 4.3, and emphasize again later in this section. Figure 1 plots the sensitivity φ of the manager’s fee to the fund’s performance and the sensitivity χ to the index performance as a function of the private-beneﬁt parameter A. We express the sensitivities as percentages, and allow A to vary from zero to 0.15. Thus, under the maximum value of A, the manager’s fee increases by ﬁfteen cents when the fund’s assets increase by one dollar. A sensitivity of that magnitude is typical for hedge funds; under the 2/20 contracts that 25

are common in the industry, hedge-fund managers receive 20% of proﬁts. For mutual funds, typical fees are 0.5-2% of assets under management, but the fees’ sensitivity to returns may be signiﬁcantly higher because high returns attract inﬂows (an eﬀect which is absent from our model). Figure 1 conﬁrms the result of Proposition 4.1 that φ and χ increase in A for A > ρ ρ+¯ . ρ

Figure 1: Optimal Contract
(%) (%)

( )

( )

The sensitivity φ of the manager’s fee to the fund’s performance and the sensitivity χ to the index performance as a function of the private-beneﬁt parameter A. There are two groups of assets, with three assets in each group. Assets in each group have identical characteristics, except for supply. Parameter values are: ρ = 1, ρ = 50, r = 4%, κ = 10%, N = 6, ηi = 1, θ1 = θ2 = θ3 = 0.7, θ4 = θ5 = θ6 = 0.3, bi = 1, ¯ σ2 σ2 s = 0.65, ei = 0.4, σs = 1, ss = ei , for i = 1, .., 6. ¯ ¯ ¯ ¯i

Figure 2 plots price and return moments for individual assets as a function of the private-beneﬁt parameter A. The blue solid line represents assets in large supply and the red dashed line assets in small supply. Consistent with Proposition 3.1, assets in small supply are more expensive than assets in large supply and earn lower expected return. Moreover, consistent with Proposition 4.2, the supply-driven distortions are exacerbated by agency frictions. These eﬀects are signiﬁcant quantitatively. In the absence of frictions, small-supply assets earn an expected return of 4.5% and large-supply assets earn 6%. When A = 0.15, the expected return of small-supply assets drops to 2.5% and that of large-supply assets rises to 12.5%. Thus, the expected-return diﬀerential increases from 1.5% to 10%. The results on return volatility parallel those on expected return. Assets in small supply are more volatile than assets in small supply, and the supply-driven distortions are exacerbated by agency frictions. In the absence of frictions, small-supply assets have return volatility equal to 25.8% and small-supply assets 25.4%. When A = 0.15, the volatility of small-supply assets rises to 30% and that of large-supply assets drops to 25.2%. Thus, the volatility diﬀerential increases from 26

Figure 2: Agency Frictions and Cross-Sectional Pricing
( %)
( )

( ) ( )

( ) ( %)

( )

(

%)

( )

( )

( )

Expected price E(Sit ), expected dollar return

1 $ dt E(dRit ),

share return volatility

1 dt Var(dRit ),

dollar

1 1 $ $ $ return volatility dt Var(dRit ), dollar beta βi , and dollar alpha dt αi , as a function of the private-beneﬁt parameter A. There are two groups of assets, with three assets in each group. Assets in each group have identical characteristics, except for supply. Parameter values are as in Figure 1. Assets 1, 2, and 3, in the high-supply group are represented by the blue solid line, and assets 4, 5, and 6, in the low-supply group are represented by the red dashed line.

0.4% to 4.8%, with most of the increase being driven by the assets in small supply. The results on CAPM beta parallel those on volatility except when agency frictions are small. In the absence of frictions, small-supply assets have beta equal to 0.99 and large-supply assets have a slightly larger beta equal to 1.01. The diﬀerence in beta reverses when A exceeds 0.05. When A = 0.15, the beta of small-supply assets rises to 1.1 and that of large-supply assets drops to 0.72. The results in Figure 2 are consistent with the volatility and beta anomalies. Assets in small supply have high return volatility and low expected return. Hence, volatility is negatively related to expected return in the cross-section. The negative relationship continues to hold when volatility

27

Figure 3: Agency Frictions and the Aggregate Market
( %) ( %)

( )

( )

( )

1 1 $ $ Expected price E(Sηt ), expected dollar return dt E(dRηt ), and dollar return volatility dt Var(dRηt ) of the aggregate market, as a function of the private-beneﬁt parameter A. There are two groups of assets, with three assets in each group. Assets in each group have identical characteristics, except for supply. Parameter values are as in Figure 1.

is replaced by its idiosyncratic component (not plotted in Figure 2). It also continues to hold when volatility is replaced by beta, provided that agency frictions are large enough (A > 0.05). Moreover, the negative relationship is reinforced when expected return is replaced by alpha. Indeed, alpha is expected return adjusted for beta, and assets in small supply both earn low expected return, and have high beta when frictions are large enough. For example, when A = 0.15, the alpha diﬀerential between large- and small-supply assets is 13%, which is larger than the expected-return diﬀerential of 10.5%. Propositions 3.2, 3.3, and 4.3 show that volatility or beta are not always negatively related to supply; the relationship is negative when idiosyncratic risk accounts for a large enough fraction of return variance. The same applies to the relationship between risk measures and expected return since supply and expected return are positively related. For example, under scale invariance and no agency frictions, Proposition 3.2 implies that volatility is negatively related to expected return if the asset-speciﬁc component of dividends accounts for more than half of return variance. Our numerical example meets this condition. The extent of idiosyncratic risk that is required to generate a negative risk-return relationship can be signiﬁcantly lower, however, if the assumption of scale invariance is dropped. Figure 3 plots price and return moments for the aggregate market as a function of the privatebeneﬁt parameter A. Consistent with Proposition 4.4, agency frictions cause the price of the aggregate market to increase and its expected return to drop. The eﬀect is quantitatively small,

28

however: the expected return in the absence of agency frictions is 5.27%, and it drops to 5.18% when A = 0.15. One reason why the eﬀect is small is that fund investors can react to market overvaluation by lowering their direct investment in the index or by changing the manager’s contract to induce him to hold a smaller position in the index. If investors lacked the ability or understanding to undertake these actions, the eﬀects of agency frictions on the aggregate market might be signiﬁcantly larger. Figure 3 shows additionally that agency frictions can have a non-monotonic eﬀect on the volatility of the aggregate market: volatility decreases for small frictions, but increases when frictions become large enough. This result stands in-between that in Cuoco and Kaniel (2011), who ﬁnd that market volatility decreases with the extent of benchmarking, and that in Basak and Pavlova (2013), who ﬁnd that volatility increases.

5

Social Optimality

In this section we examine whether the privately optimal contract, determined in Section 4, is socially optimal. We assume that a social planner chooses contract parameters (φ, χ, ψ) at time zero. This is the social planner’s only intervention: given the contract, the manager is free to choose the fund’s portfolio zt and the shirking action mt , and prices St must clear markets. Without loss of generality, we restrict the investor’s direct investment in the index to be zero (x = 0).9 The social planner maximizes the investor’s value function at time zero, subject to the manager’s incentive compatibility and individual rationality constraints. This optimization problem is the same as the investor’s but the social planner internalizes that a change in the contract parameters aﬀects equilibrium prices. Formally, the value functions of the investor and the manager at time ¯ ¯ zero can be written as V (W0 , s0 , e0 , φ, χ, ψ, S) and V (W0 , s0 , e0 , φ, χ, ψ, S), respectively, where S consists of the parameters (a01 , .., a0N , a11 , .., a1N , a21 , .., a2N ) that describe the price process. The investor chooses (φ, χ, ψ) taking S as given. The social planner instead internalizes the dependence of S on (φ, χ). The social planner’s optimization problem involves the utility of the investor and the manager, but not of the buy-and-hold investors. These investors, however, are neutral for our normative analysis, in the sense that the contract choice does not aﬀect their asset holdings and dividend stream. Indeed, buy-and-hold investors are endowed with the portfolio η − θ at time zero and do not trade. Therefore, the dividend stream that they receive from their portfolio does not depend
The set A of allocations that the social planner can achieve when restricting x to be zero includes the set A of allocations when the investor can choose any value of x. This is because the social planner can induce the investor to choose x = 0 through an appropriate choice of χ, without aﬀecting the allocation. Moreover, the social planner’s optimal allocation in A also belongs to A . Indeed, Proposition 5.1 shows that under the optimal allocation the investor and the manager agree on their valuation of the index ((4.3) holds). Therefore, the investor would choose x = 0 even if the restriction x = 0 were lifted.
9

29

on prices and on the contract choice. When the parameter B in the manager’s private-beneﬁt function Amt − B m2 is equal to zero, as t 2 assumed in Section 4, the social planner’s problem yields the same solution as the investor’s. Indeed, the coeﬃcient φ that characterizes the fee’s sensitivity to the fund’s performance must satisfy φ ≥ A, so that the manager does not choose an arbitrarily large shirking action mt . Moreover, any φ ≥ A yields no shirking, i.e., mt = 0. When A exceeds the value investor. The social planner chooses the same contract as the investor because φ = A is a corner solution. The diﬀerences in marginal trade-oﬀs between the social planner and the investor become apparent when instead φ is an interior solution. Interior solutions are possible when the parameter B is positive. Theorem 5.1 generalizes the equilibrium derived in the previous section to B > 0. Proposition 5.1 solves the social planner’s problem and shows that solutions for the investor and the social planner diﬀer. ρ ρ+¯ ρ

that φ takes in the absence of

agency frictions, the constraint φ ≥ A is binding. Hence, the social planner sets φ = A, as does the

Theorem 5.1 (Equilibrium Prices and Contract with General Agency Frictions). When ρ ≥ A > 0, the equilibrium in Theorem 3.1 remains an equilibrium. When A > ρ+¯ and ρ ¯ ¯ B ∈ [0, B] ∪ [B, ∞) for two constants B > B, the following form an equilibrium: the price process ρ ρ+¯ ρ

St given by (2.14), (3.1), (4.1), and (4.2); the contract (φ, χ, ψ) with A ≥ φ > solving the system of equations φ(1 − φ) + r (a1 − a1 ) θbˆ0 + r ˇ s B φ(1 − φ) + r (a1 − a1 ) θbˆ0 + r ˇ s B
N

ρ ρ+¯ ρ

and χ > 0

(a2i − a2i ) θi ei0 = 0 ˇ ˆ i=1 N

and

φ

is as follows. When B is positive but close to zero, φ = A

is a corner solution, as in the case B = 0. When B exceeds a threshold, φ = A ceases to be a ρ corner solution, and the solution becomes interior to the interval ( ρ+¯ , A). Intuitively, the investor’s ρ

beneﬁt from raising φ is that the manager has a smaller incentive to undertake the shirking action. At the same time, larger φ involves a cost to the investor because the manager becomes less willing to take risk and hence to exploit price diﬀerentials driven by supply, i.e., invest relatively more in

30

high-θ assets and less in low-θ assets. When B increases, the manager derives a smaller beneﬁt from shirking. Hence the investor’s beneﬁt from raising φ is smaller, which is why φ decreases below A when B exceeds a threshold. When B becomes large, and so the manager’s beneﬁt from shirking converges to zero, φ converges to its value ρ ρ+¯ ρ

under no agency frictions.

The system of equations (4.3), (5.2), and (5.1) that determines (φ, χ, ψ) can have multiple solutions when B > 0. This means that multiple equilibria can exist. The comparison between socially and privately optimal contract shown in Proposition 5.1 applies to the privately optimal contract in any of these equilibria. Equilibrium may fail to exist for intermediate values of B. This is because the investor may not be willing to employ the manager. Note that the investor is willing to employ the manager not ¯ only when the beneﬁt of shirking is small (B ≥ B) but also—and more surprisingly—when it is large (B ≤ B). This is because of a general-equilibrium eﬀect: when the beneﬁt from shirking is large, equilibrium prices are more distorted, making the supply portfolio an even better investment than the index portfolio.

Proposition 5.1 (Socially Optimal Contract). When tract (φ∗ , χ∗ , ψ ∗ ) is as in Theorem 3.1. When A > A≥ φ∗ > ρ ρ+¯ ρ ρ ρ+¯ , ρ

ρ ρ+¯ ρ

≥ A ≥ 0, the socially optimal con-

the socially optimal contract is as follows:

and

χ∗

> 0 are the unique solution to the system of equations
N

1−φ + r (a1 − a1 ) θbˆ0 + r ˇ s B 1−φ + r (a1 − a1 ) θbˆ0 + r ˇ s B

(a2i − a2i ) θi ei0 = 0 ˇ ˆ i=1 N

and

φ Si t ), lower expected share return (E(dRit ) < E(dRi t )), and same share return volatility (Var(dRit ) = Var(dRi t )). The eﬀects of agency frictions have a similar ﬂavor to those of supply. When frictions are more
Computing dollar returns under OU processes is made complicated by the fact that prices can be zero or even negative. We use expected prices, as under SR processes, and assume that these are positive as would be the case if the unconditional means (¯, e1 , .., eN ) are large enough. s ¯ ¯
10

33

severe, prices become more distorted but share return volatilities are unaﬀected. The aggregate market is also unaﬀected because a change in share return volatilities is key for such an eﬀect.

Proposition 6.2 (Eﬀects of Agency Frictions with OU Cashﬂow Processes). Suppose that A> ρ ρ+¯ , ρ

B = 0, and (st , e1t , .., eN t ) evolve according to (6.1) and (6.2). Following an increase

in the private-beneﬁt parameter A, prices and expected share returns are aﬀected as described in Proposition 4.2, while share return volatilities and the price of the aggregate market are unaﬀected.

7

Conclusion

In this paper we study how the delegation of asset management aﬀects equilibrium prices. Unlike most prior literature, we endogenize both equilibrium prices and fund managers’ contracts. We show that because of agency frictions, managers are compensated based on their performance relative to a benchmark. As a consequence, they become less willing to deviate from the benchmark, and the price distortions that they are hired to exploit become more severe. While distortions are exacerbated in both directions, i.e., undervalued assets become cheaper and overvalued assets become more expensive, the positive distortions dominate, biasing the aggregate market upwards. This is because overvalued assets account for an increasingly large fraction of market movements relative to undervalued assets, and hence trading against overvaluation is riskier for managers than trading against undervaluation. Agency frictions can also generate a negative relationship between risk and expected return in the cross-section, in line with empirical evidence but in contrast to standard theories. This is because they raise the volatility of overvalued assets, through an ampliﬁcation mechanism. Following a positive shock to the expected cash ﬂows of an overvalued asset, the asset accounts for a larger fraction of market movements. This makes it riskier for managers to underweight the asset, and the resulting buying pressure ampliﬁes the price increase caused by the higher cash ﬂows. Our model combines square-root processes for asset cashﬂows with constant absolute risk aversion utility. This combination is to our knowledge new to the literature, including in a frictionless setting. We show that it yields closed-form solutions for asset prices and can accommodate any number of risky assets. It also generates more realistic properties for asset cashﬂows and prices than the tractable CARA-normal alternative. The eﬀects of agency frictions in our model are reﬂected only on prices and not on fund managers’ portfolios. This is because we are assuming a representative manager, who must hold in equilibrium a ﬁxed portfolio supplied by other agents. A natural extension of our analysis is to introduce multiple managers with diﬀerent levels of agency frictions, and examine how their port34

folios diﬀer. Such an extension could provide additional points of contact with the data. For example, Christoﬀersen and Simutin (2014) ﬁnd that fund managers who face greater pressure to meet benchmarks hold a larger fraction of their portfolios in high-beta stocks and achieve lower alphas. Their ﬁndings are in the spirit of our theoretical results. Our analysis can also be extended on the normative front. One extension is to examine whether the result that socially optimal contracts cause larger pricing distortions than privately optimal contracts continues to hold when distortions aﬀect the real economy. Another extension is to allow for more degrees of freedom in contract design, e.g., the choice of benchmark which in our model is assumed to be the index, and study how privately optimal choices compare with socially optimal ones.

35

Appendix
Proof of Theorem 3.1. The theorem follows from the proof of Theorem 4.1, which covers the case A = 0. Proof of Proposition 3.1. Substituting a0i from (3.1), we can write the price (2.14) of asset i as Sit = a1 bi st + a2i eit , ˆ ˆ and asset i’s expected price as E(Sit ) = r+κ (a1 bi s + a2i ei ) , ¯ ¯ r (A.2) (A.1)

ˆ where st ≡ st + κ s and eit ≡ eit + κ ei . Since asset i diﬀers from asset i only in its supply (θi = θi ), its ˆ r¯ r¯ price diﬀers from the price of asset i only because a2i = a2i . Since, in addition, θi < θi , (3.3) implies that a2i > a2i . Therefore, (A.1) implies that Sit > Si t , and (A.2) implies that E(Sit ) > E(Si t ). Substituting a0i from (3.1), we can write the share return (A.23) of asset i as √ √ dRit = [1 − (r + κ)a1 ]bi st + [1 − (r + κ)a2i ]eit dt + a1 bi σs st dwst + a2i σi eit dwit . The expected share return is E(dRit ) = {[1 − (r + κ)a1 ]bi s + [1 − (r + κ)a2i ]¯i } dt ¯ e = [bi s + ei − rE(Sit )] dt, ¯ ¯ (A.4) (A.3)

where the second step follows from (A.2). Since E(Sit ) > E(Si t ), (A.4) implies that E(dRit ) < E(dRi t ). The expected dollar return is
$ E(dRit ) = E

dRit E(Sit )

=

E(dRit ) bi s + ei ¯ ¯ = − r dt, E(Sit ) E(Sit )

(A.5)

where the third step follows from (A.4). Since bi s + ei > 0 and E(Sit ) > E(Si t ), (A.5) implies that ¯ ¯
$ $ E(dRit ) < E(dRi t ).

Proof of Proposition 3.2. Equation (A.3) implies that the share return variance of asset i is Var(dRit ) = E (dRit )2 − [E(dRit )] = E (dRit )2 =E
2 2 a2 b2 σs st + a2 σi eit dt 1 i 2i 2

2 2 = a2 b2 σs s + a2 σi ei dt, ¯ ¯ 1 i 2i 2

(A.6)

where the second step follows because the term E (dRit )2 is of order dt and the term [E(dRit )] is of order (dt)2 . Since a2i > a2i , (A.6) implies that Var(dRit ) > Var(dRi t ).

36

Equations (A.2) and (A.6) imply that the dollar return variance of asset i is
$ Var(dRit ) = Var

dRit E(Sit )

=

2 2 a2 b2 σs s + a2 σi ei ¯ ¯ 1 i 2i dt. 2 (r+κ)2 (a1 bi s + a2i ei ) ¯ ¯ r2

(A.7)

$ $ Since asset i diﬀers from asset i only in its supply, (A.7) implies that Var(dRit ) > Var(dRi t ) if and only if 2 2 a2 b2 σs s + a2 σi ei ¯ ¯ 1 i 2i 2 2 a2 b2 σs s + a2 σi ei ¯ ¯ 1 i 2i

(a1 bi s + a2i ei ) ¯ ¯

2

>

(a1 bi s + a2i ei ) ¯ ¯

2 2 2

2 2 2 2 ⇔ a2 b2 σs s + a2 σi ei (a1 bi s + a2i ei ) − a2 b2 σs s + a2 σi ei (a1 bi s + a2i ei ) > 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 i 2i 1 i 2i 2 2 2 2 ⇔ (a2i − a2i ) a1 bi (a2i + a2i )(σi s − σs ei ) + 2(a2i a2i σi ei − a2 b2 σs s) > 0. ¯ ¯ ¯ ¯ 1 i

(A.8) (A.9)

Since a2i > a2i , (A.9) is equivalent to (3.6). Proof of Proposition 3.3. Equation (A.3) implies that the share return of the index is
N

dRηt = [1 − (r + κ)a1 ]ηbst + j=1 √ [1 − (r + κ)a2i ]ηj ejt dt + a1 ηbσs st dwst +

N

√ a2j ηj σj ejt dwjt . (A.10)

j=1

Equations (3.9), (A.3) and (A.10) imply that the share beta of asset i is βi =
2 2 ¯ ¯ E (dRit dRηt ) a2 bi ηbσs s + a2 ηi σi ei Cov(dRit , dRηt ) 1 2i = = . N 2] 2 (ηb)2 σ 2 s + 2 η2 σ2 e Var(dRηt ) E [(dRηt ) a1 s¯ j=1 a2j j j ¯j

(A.11)

Since a2i > a2i , (A.11) implies that βi > βi . Equation (A.2) implies that the expected share price of the index is  E(ηSt ) = r+κ a1 ηb¯ + s a2j ηj ej  . ¯ r j=1
N

 (A.12)

Equations (3.10), (A.2), (A.11), and (A.12) imply that the dollar beta of asset i is
$ βi = βi 2 2 a1 ηb¯ + j=1 a2j ηj ej s ¯ ¯ ¯ E(Sηt ) a2 bi ηbσs s + a2 ηi σi ei 1 2i = . N 2 2¯ 2¯ E(Sit ) a1 bi s + a2i ei ¯ ¯ a2 (ηb)2 σs s + j=1 a2 ηj σj ej 1 2j N

(A.13)

$ $ Equation (A.13) implies that βi > βi if and only if 2 2 2 2 a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ 1 2i 2i > 1 a1 bi s + a2i ei ¯ ¯ a1 bi s + a2i ei ¯ ¯ 2 2 2 2 ⇔ a2 bi ηbσs s + a2 ηi σi ei (a1 bi s + a2i ei ) − a2 bi ηbσs s + a2 ηi σi ei (a1 bi s + a2i ei ) > 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2i 1 2i

(A.14) (A.15)

⇔ (a2i − a2i ) a1 bi (a2i +

2 a2i )ηi σi s + ¯

2 a2i a2i ηi σi ei ¯

−

2 a2 bi ηbσs s ¯ 1

> 0.

Since a2i > a2i , (A.15) is equivalent to (3.11).

37

Equations (A.6), (A.10), and (A.11) imply that the idiosyncratic share return variance of asset i is Var(d it ) 2 = Var(dRit ) − βi Var(dRηt ) 2 2 = a2 b2 σs s + a2 σi ei − ¯ ¯ 1 i 2i 2 2 a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ 1 2i 2¯ a2 (ηb)2 σs s + 1 i t) N j=1 2

2 2¯ a2 ηj σj ej 2j

dt.

(A.16)

Equation (A.16) implies that Var(d
2 a2 σi ei ¯ 2i

it )

> Var(d
2

if and only if
2 σi ei ¯ 2 2 ¯ a2 bi ηbσs s + a2 ηi σi ei ¯ 1 2i 2¯ a2 (ηb)2 σs s + 1 N j=1 2

−

2 2 ¯ ¯ a2 bi ηbσs s + a2 ηi σi ei 1 2i 2¯ a2 (ηb)2 σs s + 1  N j=1

2 2¯ a2 ηj σj ej 2j N

>

a2 2i

−

2 2¯ a2 ηj σj ej 2j 2 2 2 − a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ 1 2i 2


2 2 a2 ηj σj ej  − ¯ 2j 2 2 a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ 1 2i

2 2 ¯ ¯ ⇔ (a2 − a2 )σi ei a2 (ηb)2 σs s + 1 2i 2i j=1

>0


2 ⇔ (a2 − a2 ) a2 (ηb)2 σs s + ¯ 2i 2i 1

N


2 2 2 2 a2 ηj σj ej − ηi 2a2 bi ηbσs s + (a2 + a2 )ηi σi ei  > 0. ¯ ¯ ¯ 2j 1 2i 2i

(A.17)

j=1

Eq. (A.17) holds because a2i > a2i , (ηb)2 ≥ (ηi bi + ηi bi )(ηb) = 2ηi bi ηb,
N 2 2 2 2 2 2 2 2 a2 ηj σj ej ≥ a2 ηi σi ei + a2 ηi σi ei = (a2 + a2 )ηi σi ei . ¯ ¯ ¯ ¯ 2j 2i 2i 2i 2i j=1

Equations (A.2) and (A.16) imply that the idiosyncratic dollar return variance of asset i is
2 2 ¯ ¯ a2 b2 σs s + a2 σi ei − 1 i 2i
2 2 2 a2 bi ηbσs s+a2 ηi σi ei ¯ 2i ¯ 1 N 2 (ηb)2 σ 2 s+ 2 η2 σ2 e a1 s¯ j=1 a2j j j ¯j

(

)

Var(d

$ it )

= Var

d it E(Sit )

=

(r+κ)2 r2 $ it )

(a1 bi s + a2i ei ) ¯ ¯

2

dt.

(A.18)

Equation (A.18) implies that Var(d
2 2 a2 b2 σs s + a2 σi ei − ¯ ¯ 1 i 2i

> Var(d

$ i t)
2

if and only if
2 2 a2 b2 σs s + a2 σi ei − ¯ ¯ 1 i 2i
2 ¯ 2i (a2 bi ηbσs s+a2 1 2¯ a2 (ηb)2 σs s+ 1 N j=1 2 ηi σi ei ) ¯ 2

2 2 ¯ 2i ¯ (a2 bi ηbσs s+a2 ηi σi ei ) 1 2¯ a2 (ηb)2 σs s+ 1 N j=1

2 2¯ a2 ηj σj ej 2j

(a1 bi s + a2i ei ) ¯ ¯ 
2 ⇔ a2 (ηb)2 σs s + ¯ 1 j=1 N

2

>

2 2¯ a 2 ηj σ j ej 2j

(a1 bi s + a2i ei ) ¯ ¯
2

2


2 2 a2 ηj σj ej  ¯ 2j 2 2 2 2 a2 b2 σs s + a2 σi ei (a1 bi s + a2i ei ) − a2 b2 σs s + a2 σi ei (a1 bi s + a2i ei ) ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 i 2i 1 i 2i 2 2 2 2

−

2 a2 bi ηbσs s + ¯ 1

2 2 a2 ηi σi ei ¯ 2i

2 2 (a1 bi s + a2i ei ) − a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ ¯ ¯ 1 2i

(a1 bi s + a2i ei ) ¯ ¯

> 0. (A.19)

Using the same calculations as when deriving (A.9) from (A.8) and (A.15) from (A.14), we can write the right-hand side of (A.19) as (a2i − a2i )¯i times the right-hand side of (3.12). Since a2i > a2i , (A.19) is e equivalent to (3.12). Proof of Proposition 3.4. Since E(dRit ) < E(dRi t ) and βi > βi , (3.15) implies that αi < αi . Equations

38

(3.16), (A.2), (A.4), (A.12), and (A.13) imply that the dollar alpha of asset i is
2 2 r[1 − (r + κ)a1 ]ηb¯ + r j=1 [1 − (r + κ)a2j ]ηj ej s ¯ a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ r(bi s + ei ) ¯ ¯ 1 2i −r− . = N 2 η2 σ2 e 2 (ηb)2 σ 2 s + (r + κ)(a1 bi s + a2i ei ) ¯ ¯ (r + κ)(a1 bi s + a2i ei ) ¯ ¯ a1 s¯ j=1 a2j j j ¯j N

$ αi

(A.20)
$ $ Equation (A.20) implies that αi < αi if and only if 2 2 [1 − (r + κ)a1 ]ηb¯ + j=1 [1 − (r + κ)a2j ]ηj ej s ¯ bi s + e i ¯ ¯ a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ 1 2i − N 2 η2 σ2 e 2 (ηb)2 σ 2 s + a1 bi s + a2i ei ¯ ¯ a1 bi s + a2i ei ¯ ¯ a1 s¯ j=1 a2j j j ¯j N

<

2 2 [1 − (r + κ)a1 ]ηb¯ + j=1 [1 − (r + κ)a2j ]ηj ej s ¯ bi s + e i ¯ ¯ a2 bi ηbσs s + a2 ηi σi ei ¯ ¯ 1 2i − N 2 η2 σ2 e 2 (ηb)2 σ 2 s + a1 bi s + a2i ei ¯ a1 bi s + a2i ei ¯ ¯ ¯ a1 s¯ j=1 a2j j j ¯j

N

⇔ (a2i − a2i )¯i (bi s + ei ) + e ¯ ¯ ×

[1 − (r + κ)a1 ]ηb¯ + s
2¯ a2 (ηb)2 σs s + 1

N ¯ j=1 [1 − (r + κ)a2j ]ηj ej N 2 η2 σ2 e j=1 a2j j j ¯j

2 2 2 2 a2 bi ηbσs s + a2 ηi σi ei (a1 bi s + a2i ei ) − a2 bi ηbσs s + a2 ηi σi ei (a1 bi s + a2i ei ) > 0 ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ 1 2i 1 2i

⇔ (a2i − a2i )¯i bi s + ei + e ¯ ¯

[1 − (r + κ)a1 ] ηb¯ + s
2¯ a2 (ηb)2 σs s + 1

N ¯ j=1 [1 − (r + κ)a2j ] ηj ej N 2 2 2¯ j=1 a2j ηj σj ej

2 2 2 × a1 bi (a2i + a2i )ηi σi s + a2i a2i ηi σi ei − a2 bi ηbσs s ¯ ¯ ¯ 1

> 0.

(A.21)

Since a2i > a2i , (A.21) is equivalent to [1 − (r + κ)a1 ] ηb¯ + s
2¯ a2 (ηb)2 σs s + 1 N ¯ j=1 [1 − (r + κ)a2j ] ηj ej N 2 2 2¯ j=1 a2j ηj σj ej

bi s+¯i + ¯ e

2 2 2 a1 bi (a2i + a2i )ηi σi s + a2i a2i ηi σi ei − a2 bi ηbσs s > 0. ¯ ¯ ¯ 1

(A.22) Since
N j=1 $ ηj αj = 0, there exists i∗ such that αi∗ ≥ 0 and αi∗ ≥ 0. Equation (A.20) written for i∗ implies

that the left-hand side of (A.22) is larger than [1 − (r + κ)a1 ] ηb¯ + s
2¯ a2 (ηb)2 σs s + 1 N ¯ j=1 [1 − (r + κ)a2j ] ηj ej N 2 η2 σ2 e j=1 a2j j i ¯i

(r + κ)(a1 bi s + a2i∗ ei ) + ¯ ¯

2 2 2 2 2 × a2 bi ηbσs s + a2 ∗ ηi σi ei + a1 bi (a2i + a2i )ηi σi s + a2i a2i ηi σi ei − a2 bi ηbσs s ¯ ¯ ¯ ¯ ¯ 1 2i 1

= (r + κ)(a1 bi s + a2i∗ ei ) + ¯ ¯

[1 − (r + κ)a1 ] ηb¯ + s
2¯ a2 (ηb)2 σs s + 1

N ¯ j=1 [1 − (r + κ)a2j ] ηj ej N 2 2 2¯ j=1 a2j ηj σi ei

2 2 × a1 bi (a2i + a2i )ηi σi s + (a2 ∗ + a2i a2i )ηi σi ei > 0. ¯ ¯ 2i

Therefore, (A.22) holds, and so does (A.21). Proof of Theorem 4.1. We allow A to be zero so that the proof can also cover Theorem 3.1. The proof assumes B = 0, but when A = 0 the proof carries through unchanged to B > 0. We proceed in two steps:

39

• Step 1: We ﬁx a contract (φ, χ, ψ) with φ ≥ A, and show that for the price function (2.14) and the coeﬃcients (a0i , a1i , a2i ) given by (3.1), (4.1), and (4.2), zt = θ solves the optimization problem of an employed manager. Hence, markets clear provided that the manager accepts the contract (φ, χ, ψ) and the investor invests x = 0 in the index. • Step 2: We ﬁx prices given by (2.14), (3.1), (4.1), and (4.2), and show that the investor decides to employ the manager, i.e., oﬀer a contract that the manager accepts, and that the contract (φ, χ, ψ) given in Theorems 3.1 and 4.1, and the index investment x = 0, solve the investor’s optimization problem. Hence, an equilibrium exists, and is as in the theorems. Step 1. Substituting Sit from (2.14) into (2.2), and using (2.1), (2.3), and (2.4), we can write the excess return dRit of asset i as √ √ dRit = µit dt + a1i σs st dwst + a2i σi eit dwit , where µit ≡ (κa1i s + κa2i ei − ra0i ) + [bi − (r + κ)a1i ]st + [1 − (r + κ)a2i ]eit . ¯ ¯ We set µt ≡ (µ1t , .., µN t ) . We conjecture that the value function of an employed manager takes the form
N

(A.23)

(A.24)

¯ ¯ V (Wt , st , et ) = − exp − rρWt + q0 + q1 st + ¯¯ ¯ ¯ i=1 q2i eit ¯

,

(A.25)

where (¯0 , q1 , q21 , .., q2N ) are constants. The manager’s Bellman equation is q ¯ ¯ ¯ max ¯¯ ¯ − exp(−¯ct ) + DVt − δ Vt = 0, ρ¯ (A.26)

ct ,zt ,mt ¯

¯ ¯ where DVt is the drift of Vt . ¯ Using (2.3), (2.4), (2.5), (2.7), and (A.23), we ﬁnd that the dynamics of Jt ≡ rρWt + q0 + q1 st + ¯¯ ¯ ¯ are
N N ¯ i=1 q2i eit

¯ ¯ ¯ dJt = Gt dt + Ht dwst + i=1 ¯ Kit dwit ,

(A.27)

where
N

¯ Gt ≡ rρ rWt + (φzt − χη)µt + ψ + (A − φ)mt − ct + κ q1 (¯ − st ) + ¯ ¯ ¯ ¯ s i=1 N

q2i (¯i − eit ) , ¯ e

¯ Ht ≡ r ρ ¯ i=1 √ (φzit − χηi )a1i + q1 σs st , ¯

√ ¯ Kit ≡ [rρ(φzit − χηi )a2i + q2i ] σi eit . ¯ ¯

40

¯ ¯ ¯ ¯ ¯ Using V (Wt , st , et ) = − exp(Jt ), (A.27), and Ito’s lemma, we ﬁnd that the drift DVt of Vt is ¯ ¯ DVt = −Vt 1 ¯2 1 ¯ G t − Ht − 2 2
N

¯2 Kit i=1 .

(A.28)

Substituting into (A.26), we can write the manager’s Bellman equation as ¯ − exp(−¯ct ) − Vt ρ¯ 1 ¯2 1 ¯ G t − Ht − 2 2
N

ct ,zt ,mt ¯

max

¯2 Kit i=1 ¯¯ − δ Vt = 0.

(A.29)

The ﬁrst-order condition with respect to ct is ¯ ρ exp(−¯ct ) + rρVt = 0. ¯ ρ¯ ¯¯ ¯ Using (A.25) to substitute for Vt , and solving for ct , we ﬁnd ¯ 1 ¯ ct = rWt + ¯ ρ ¯
N

q0 − log(r) + q1 st + ¯ ¯ i=1 q2i eit ¯

.

(A.30)

The ﬁrst-order condition with respect to mt is mt = 0 (A.31)

because φ ≥ A. (For φ > A, the manager has a strict preference for mt = 0. For φ = A, the manager is indiﬀerent between all values of mt , and we assume that he chooses mt = 0.) The ﬁrst-order condition with respect to zit is
N

rρφµit − rρφa1i rρ ¯ ¯ ¯ i=1 2 2 (φzit − χηi )a1i + q1 σs st − rρφa2i [rρ(φzit − χηi )a2i + q2i ] σi eit = 0. (A.32) ¯ ¯ ¯ ¯

The portfolio zt = θ solves the manager’s optimization problem if (A.32) holds for zt = θ and for all values of (st , e1t , .., eN t ). Substituting µit from (A.24), and dividing by rρφ throughout, we can write (A.32) for ¯ zt = θ as A0i + A1i st + A2i eit = 0, where A0i ≡ κ(a1i s + a2i ei ) − ra0i , ¯ ¯
N

(A.33)

A1i ≡ bi − (r + κ)a1i − a1i rρ ¯ i=1 2 (φθi − χηi )a1i + q1 σs , ¯

2 A2i ≡ 1 − (r + κ)a2i − a2i [rρ(φθi − χηi )a2i + q2i ] σi . ¯ ¯

The left-hand side of (A.33) is an aﬃne function of (st , eit ). Therefore, (A.32) holds for zt = θ and for all

41

values of (st , e1t , .., eN t ) if A0i = A1i = A2i = 0. Before linking these equations to the coeﬃcients (a0i , a1i , a2i ) given in the proposition, we determine a set of additional equations that follow from the requirement that the manager’s Bellman equation (A.29) holds. Using (A.30), (A.31), and (A.32) to substitute ct , mt , and ¯ µit , we can write (A.29) for zt = θ as
N

¯ ¯ Q0 + Q1 st + i=1 ¯ Q2i eit = 0,

(A.34)

where
N

¯ Q0 ≡ rq0 − rρψ − κ q1 s + ¯ ¯ ¯¯ i=1 q2i ei ¯ ¯
N

¯ + r − δ − r log(r),
2 2 σs ,

1 2 2 1 ¯ ¯ rρ ¯ (φθi − χηi )a1i Q1 ≡ (r + κ)¯1 + q1 σs − q 2 2 i=1 1 2 2 1 2 2 ¯ ¯ ¯ Q2i ≡ (r + κ)¯2i + q2i σi − [rρ(φθi − χηi )a2i ] σi . q 2 2

The left-hand side of (A.34) is an aﬃne function of (st , e1t , .., eN t ). Therefore, (A.34) holds for zt = θ and ¯ ¯ ¯ ¯ for all values of (st , e1t , .., eN t ) if Q0 = Q1 = Q21 = .. = Q2N = 0. ¯ ¯ ¯ We next show that equations A0i = A1i = A2i = 0 and Q0 = Q1 = Q2i = 0 determine the coeﬃcients (a0i , a1i , a2i , q0 , q1 , q2i ) uniquely, with (a0i , a1i , a2i ) being as in the proposition. This will imply that zt = θ ¯ ¯ ¯ solves the manager’s optimization problem given the prices in the proposition. Equation A1i = 0 implies that a1i = a1 bi , with a1 being independent of i. Hence, A1i = 0 can be replaced by A1 = 0 with A1 ≡ 1 − (r + κ)a1 − a1 [rρ(φθ − χη)ba1 + q1 ] σs . ¯ ¯ 2 ¯ Moreover, Q1 can be written as 1 2 2 1 2 2 ¯ Q1 = (r + κ)¯1 + q1 σs − [rρ(φθ − χη)ba1 ] σs . q ¯ ¯ 2 2 ¯ The quadratic equation Q1 = 0 has the unique positive root11
4 (r + κ)2 + [rρ(φθ − χη)ba1 ] σs − (r + κ) ¯ 2 σs 2

q1 = ¯
11

.

(A.35)

¯ Holding wealth Wt constant, the manager is better oﬀ the larger st is. This is because with larger st , dividends are more volatile, and the manager must earn higher compensation in equilibrium for investing in the risky assets. (In the extreme case where volatility is zero, risky assets earn the riskless return r, and the manager derives no beneﬁt from investing in them.) Because the manager’s utility increases in st , the coeﬃcient q1 must be positive. ¯ The coeﬃcients (¯21 , .., q2N ), and the counterparts of (¯1 , q21 , .., q2N ) in the investor’s value function, must be positive q ¯ q ¯ ¯ for the same reason.

42

Substituting (A.35) into A1 = 0, we ﬁnd
2 1 − a2 rρ(φθ − χη)bσs = a1 1 ¯ 4 (r + κ)2 + [rρ(φθ − χη)ba1 ] σs ¯ 2

2 ⇒ 1 − a2 (r + κ)2 + 2rρ(φθ − χη)bσs = 0 ¯ 1

⇒ a1 =

1
2 (r + κ)2 + 2rρ(φθ − χη)bσs ¯

(A.36)

where the second equation follows from the ﬁrst by squaring both sides and simplifying. Eqs. a1i = bi a1 and (A.36) coincide with (4.1). Substituting (A.36) into (A.35) we can determine q1 : ¯ q1 = ¯
2 (r + κ)2 + rρ(φθ − χη)bσs ¯ 2 σs 2 (r + κ)2 + 2rρ(φθ − χη)bσs ¯

−

r+κ . 2 σs

(A.37)

¯ Following the same procedure to solve the system of A2i = Q2i = 0, we ﬁnd (4.2) and q2i = ¯
2 (r + κ)2 + rρ(φθi − χηi )σi ¯ 2 σi

(r +

κ)2

+ 2rρ(φθi − ¯

2 χηi )bσi

−

r+κ 2 . σi

(A.38)

¯ Finally, A0i = 0 implies (3.1), and Q0 = 0 implies κ r
N

q0 = ρψ + ¯ ¯

q1 s + ¯¯ i=1 q2i ei ¯ ¯

−1+

¯ δ + log(r). r

(A.39)

Step 2. We conjecture that the value function of the investor when he employs the manager, oﬀers contract ˜ ˜ ˜ ˜ (φ, χ, ψ) that satisﬁes φ ≥ A and can diﬀer from the equilibrium contract (φ, χ, ψ), and invests x in the index, takes the form
N

V (Wt , st , et ) = − exp − rρWt + q0 + q1 st + i=1 q2i eit

,

(A.40)

where (q0 , q1 , q21 , .., q2N ) are constants. The investor’s Bellman equation is max [− exp(−ρct ) + DVt − δVt ] = 0, ct (A.41)

where DVt is the drift of Vt . When the investor oﬀers the equilibrium contract (φ, χ, ψ), the manager’s ﬁrst-order condition (A.32) is ˜ ˜ ˜ ˜ satisﬁed for zt = θ, as shown in Step 1. When the investor oﬀers contract (φ, χ, ψ) with φ ≥ A, (A.32) is satisﬁed for zt given by ˜ φzit − χηi = φθi − χηi ˜ ⇒ zit = φθi + (χ − χ)ηi ˜ . ˜ φ (A.42)

˜ ˜ ˜ ˜ ˜ ˜ ˜ This is because (A.32) depends on (φ, χ, ψ) only through the quantity φzit − χηi : if (φ, χ, ψ) changes, then ˜ ˜ zit also changes in a way that φzit − χηi is kept constant. The economic intuition is that the manager chooses ˜

43

the fund’s portfolio zt to “undo” a change in contract: his personal risk exposure, through the fee, is the ˜ ˜ ˜ same under (φ, χ, ψ) and (φ, χ, ψ). The manager’s personal risk exposure arises through the fee’s variable ˜ ˜ ˜ ˜ component, which is (φzt − χη)dRt under (φ, χ, ψ), and (φθ − χη)dRt under (φ, χ, ψ). Eq. (A.42) relies on ˜ the assumption that the investor and the manager take asset prices as given and independent of the contract. ˜ ˜ ˜ Formally, the contract (φ, χ, ψ) in (A.32) does not aﬀect the price coeﬃcients (ai0 , a1i , a2i ). We drop the time subscript from the portfolio zt in (A.42) because that portfolio is constant over time. ˜ Using (2.3), (2.4), (2.5), (2.11), (A.23), mt = 0 (which holds because φ ≥ A), and (A.42), we ﬁnd that the dynamics of Jt ≡ rρWt + q0 + q1 st +
N N i=1 q2i eit

are

dJt = Gt dt + Ht dwst + i=1 Kit dwit ,

(A.43)

where
N

˜ Gt ≡ rρ rWt + (xη + z − φθ + χη)µt − ψ − ct + κ q1 (¯ − st ) + s i=1 q2i (¯i − eit ) , e

√ Ht ≡ [rρ(xη + z − φθ + χη)ba1 + q1 ] σs st , √ Kit ≡ [rρ(xηi + z − φθi + χηi )a2i + q2i ] σi eit . Proceeding as in Step 1, we can write the investor’s Bellman equation (A.41) as 1 2 1 Gt − Ht − 2 2
N 2 Kit i=1

max − exp(−ρct ) − Vt cbt − δVt = 0.

(A.44)

The ﬁrst-order condition with respect to ct is ρ exp(−ρct ) + rρVt = 0, and yields 1 ρ
N

ct = rWt +

q0 − log(r) + q1 st + i=1 q2i eit

.

(A.45)

Using (A.45) to substitute ct , we can write (A.44) as
N

Q0 + Q1 st + i=1 Q2i eit = 0,

(A.46)

44

where
N

˜ Q0 ≡ rq0 + rρψ − κ q1 s + ¯ i=1 q2i ei ¯

¯ + r − δ − r log(r),

1 2 2 Q1 ≡ (r + κ)q1 + q1 σs − rρ(xη + z − φθ + χη)ba1 2 1 2 × rρ(φθ − χη)ba1 − rρ(xη + z − φθ + χη)ba1 + q1 − q1 σs , ¯ ¯ 2 1 2 2 Q2i ≡ (r + κ)q2i + q2i σi − rρ(xηi + zi − φθi + χηi )a2i 2 1 2 × rρ(φθi − χηi )a2i − rρ(xηi + zi − φθi + χηi )a2i + q2i − q2i σi . ¯ ¯ 2

The left-hand side of (A.34) is an aﬃne function of (st , e1t , .., eN t ). Therefore, (A.34) holds for all values of (st , e1t , .., eN t ) if Q0 = Q1 = Q21 = .. = Q2N = 0. Using A1 = 0 we can simplify Q1 to 1 2 2 Q1 = (r + κ)q1 + q1 σs + rρ(xη + z − φθ + χη)ba1 2 1 1 2 2 × rρ(xη + z − φθ + χη)ba1 σs − q1 σs + r + κ − , 2 a1 and using A2i = 0 we can simplify Q2i to 1 2 2 Q2i = (r + κ)q2i + q2i σi + rρ(xηi + zi − φθi + χηi )a2i 2 1 1 2 2 × rρ(xηi + zi − φθi + χηi )a2i σi − q2i σi + r + κ − . 2 a2i Using the simpliﬁed expressions, we ﬁnd that the positive root of Q1 = 0 is q1 =
2 (r + κ)2 + 2rρ(xη + z − φθ + χη)bσs − (r + κ) − rρ(xη + z − φθ + χη)ba1 , 2 σs

(A.47)

and the positive root of Q2i = 0 is q2i =
2 (r + κ)2 + 2rρ(xηi + zi − φθi + χηi )σi − (r + κ) − rρ(xηi + zi − φθi + χηi )a2i . 2 σi

(A.48)

Moreover, Q0 = 0 implies ˜ κ q0 = −ρψ + r
N

q1 s + ¯ i=1 q2i ei ¯

−1+

δ + log(r). r

(A.49)

˜ ˜ ˜ If the investor decides to employ the manager, then she chooses a contract (φ, χ, ψ) and index investment x to maximize her time-zero value function V (W0 , s0 , e0 ). This objective is equivalent to q0 + q1 s0 +

45

N i=1 q2i ei0

because of (A.40), and the latter objective is equivalent to
N

˜ −ρψ + q1 s0 + ˆ i=1 q2i ei0 ˆ

(A.50)

because of (A.49). The maximization is subject to the manager’s individual rationality (IR) constraint (2.9). ˜ ˜ ˜ ¯ ¯ To derive the time-zero value function V (W0 , s0 , e0 ) of an employed manager under a contract (φ, χ, ψ), we ˜ recall from (A.42) that the contract does not aﬀect the manager’s personal risk exposure (φzt − χη = φθ−χη). ˜ ˜ ≥ A. Hence, the The contract also does not aﬀect the manager’s shirking action mt , which is zero because φ value function is as in Step 1, i.e., as under the equilibrium contract (φ, χ, ψ), with (¯1 , q21 , .., q2N ) given by q ¯ ¯ (A.37) and (A.38), and q0 given by ¯ κ q0 = ρψ + ¯ ¯˜ r
N

q1 s + ¯¯ i=1 q2i ei ¯ ¯

−1+

¯ δ + log(r) r

¯ ¯ instead of (A.39). The time-zero value function Vu (W0 , s0 , e0 ) of an unemployed manager follows by the same ˜ ˜ ˜ argument. An unemployed manager can be viewed as an employed one with contract (φ, χ, ψ) = (1, 0, 0) and shirking action mt = 0. Hence, the value function is as in Step 1, with (¯1 , q21 , .., q2N ) given by (A.37) q ¯ ¯ and (A.38), and q0 given by ¯ κ q0 = ¯ r
N

q1 s + ¯¯ i=1 q2i ei ¯ ¯

−1+

¯ δ + log(r). r

The manager’s IR constraint (2.9) thus reduces to ˜ ψ ≥ 0. ˜ ˜ The investor chooses ψ that meets this constraint with equality: ψ = 0. Substituting into (A.50), we can write the investor’s optimization problem as
N ˜ ˜ φ,χ,x

max q1 s0 + ˆ i=1 q2i ei0 ˆ

,

˜ subject to the constraint φ ≥ A. Because this problem is concave, the ﬁrst-order conditions characterize an ˜ ˜ optimum. To conﬁrm that (φ, χ, x) = (φ, χ, 0) is an optimum, we thus need to check that the ﬁrst-order conditions are satisﬁed for (φ, χ, 0). Equation (A.42) implies that ∂z ˜ ∂φ ∂z ∂χ ˜ θ =− , φ =
˜ ˜ (φ,χ)=(φ,χ)

(A.51) (A.52)

˜ ˜ (φ,χ)=(φ,χ)

η . φ

46

Using (A.47), (A.48), (A.51), and (A.52), we ﬁnd ∂ ˜ ∂φ ∂ ˜ ∂φ
N

q1 s0 + ˆ i=1 N

q2i ei0 ˆ
˜ ˜ (φ,χ,x)=(φ,χ,0)

=

rρ rρ (a1 − a1 ) θbˆ0 + ˇ s Φ, (a2i − a2i ) θi ei0 ≡ ˇ ˆ φ φ i=1 rρ rρ (a1 − a1 ) ηbˆ0 + ˇ s (a2i − a2i ) ηi ei0 ≡ − X, ˇ ˆ φ φ i=1
N

N

(A.53)

q1 s0 + ˆ i=1 q2i ei0 ˆ
˜ ˜ (φ,χ,x)=(φ,χ,0)

=−

(A.54) ∂ ∂x
N

q1 s0 + ˆ i=1 q2i ei0 ˆ
˜ ˜ (φ,χ,x)=(φ,χ,0)

= −rρX.

(A.55)

The ﬁrst-order conditions with respect to χ and x require that X = 0, which is equivalent to (4.3). The ˜ ˜ ﬁrst-order condition with respect to φ requires that Φ is non-positive if φ = A and is equal to zero if φ > A. To show that the values of (φ, χ) implied by these conditions are as in Theorems 3.1 and 4.1, we ﬁrst characterize the solution χ of (4.3) and then determine the sign of Φ. Given φ ∈ [0, 1], X is increasing in χ because a1 is increasing in χ from (4.1), a2i is increasing in χ from (4.2), a1 is decreasing in χ from (4.4), and a2i is decreasing in χ from (4.5). It converges to ∞ when χ goes ˇ ˇ to χ ≡ min ¯ (r + κ)2 1 (r + κ)2 1 + φθb , min 2 + φθi η 2 2rρσs ¯ ηb i=1..,N 2rρσi ¯ i ,

and to −∞ when χ goes to χ ≡ − min (r + κ)2 1 (r + κ)2 1 + (1 − φ)θb , min + (1 − φ)θi 2 2rρσs ¯ ηb i=1..,N 2rρσi ¯ 2 ηi .

Therefore, (4.3) has a unique solution χ(φ). Moreover, X is decreasing in φ because a1 is decreasing in φ from (4.1), a2i is decreasing in φ from (4.2), a1 is increasing in φ from (4.4), and a2i is increasing in φ from ˇ ˇ ρ (4.5). Therefore, χ(φ) is increasing in φ. Since X = 0 for (φ, χ) = ( ρ+ρ , 0), χ(φ) has the same sign as ¯

φ−

ρ ρ+ρ . ¯

We next substitute χ(φ) into Φ, and show property (P): Φ has the same sign as

ρ ρ+ρ ¯

− φ. Property (P)

ρ will imply that the values of (φ, χ) are as in Theorems 3.1 and 4.1. Indeed, when A ≤ ρ+ρ , Φ cannot be ¯ ˜ negative: the ﬁrst-order condition with respect to φ would then imply that φ = A ≤ ρ , and property (P)

would imply that Φ has to be non-negative. Therefore, Φ = 0, which implies φ = instead A > χ(φ) > 0. ρ ρ+ρ , ¯

χ(φ) = 0. When ˜ φ cannot be strictly larger than A: the ﬁrst-order condition with respect to φ would then ρ ρ+ρ ¯

ρ+ρ ¯ ρ and ρ+ρ ¯

imply that Φ = 0, and property (P) would imply that φ =

< A. Therefore, φ = A >

ρ ρ+ρ , ¯

which implies

47

Setting ∆ ≡ a1 − a1 and ∆i ≡ a2i − a2i , we can write Φ and (4.3) as ˇ ˇ
N

Φ = ∆θbˆ0 + s i=1 N

∆i θi ei0 , ˆ

(A.56)

∆ηbˆ0 + s i=1 ∆i ηi ei0 = 0, ˆ

(A.57)

Eqs. (4.1) and (4.4) imply that ∆ has the same sign as [ρ − (ρ + ρ)φ]θb + (ρ + ρ)χηb. ¯ ¯ Likewise, (4.2) and (4.5) imply that ∆i has the same sign as [ρ − (ρ + ρ)φ]θi + (ρ + ρ)χηi . ¯ ¯ For φ = ρ ρ+ρ , ¯

χ(φ) = 0, and hence ∆ = ∆i = 0 and Φ = 0. For φ <
N

ρ ρ+ρ , ¯

Φ = ∆θbˆ0 + s i=1 ∆i θi ei0 ˆ
N

>− = 0,

(ρ + ρ)χ(φ) ¯ ρ − (ρ + ρ)φ ¯

∆ηbˆ0 + s i=1 ∆i ηi ei0 ˆ

where the second step follows by distinguishing cases according to the signs of ∆ and ∆i , and the third step follows from (A.57). The inequality in the second step is strict. This is because θ is not proportional to η, and hence the components of the vector [ρ − (ρ + ρ)φ]θ + (ρ + ρ)χη cannot all be zero. For φ > ¯ ¯ ρ ρ+ρ , ¯

the

same reasoning implies that Φ < 0. Therefore, property (P) holds. Note that property (P) implies that ˇ when A > ρ , the investor values the supply portfolio more than the manager: θS0 > θS0 . This is because ρ+ρ ¯

ˇ θS0 − θS0 has the same sign as −Φ, which is positive when φ >

ρ ρ+ρ . ¯

˜ ˜ ˜ Setting (φ, χ, ψ) = (φ, χ, ψ) in (A.47), (A.48), and (A.49), and using (A.42), we ﬁnd that the coeﬃcients q1 , q2i , and q0 when the investor oﬀers the equilibrium contract (φ, χ, ψ) are q1 = q2i =
2 (r + κ)2 + 2rρ((1 − φ)θ + χη)bσs − (r + κ) − rρ((1 − φ)θ + χη)ba1 , 2 σs 2 (r + κ)2 + 2rρ((1 − φ)θi + χηi )σi − (r + κ) − rρ((1 − φ)θi + χηi )a2i , 2 σi N

(A.58) (A.59) (A.60)

κ q0 = −ρψ + r

q1 s + ¯ i=1 q2i ei ¯

−1+

δ + log(r). r

The investor decides to employ the manager if (2.13) is satisﬁed. To derive the time-zero value function Vu (W0 , s0 , e0 ) of the investor when he does not employ the manager, we can follow the same steps as when she employs the manager, but with two modiﬁcations. First, we replace xη + z − φθ + χη by xη since the

48

investor’s only exposure to the risky assets when he does not employ the manager is through the investment ˜ x in the index. Second, we replace ψ by zero because the investor does not oﬀer a contract. The value function is given by (A.40), with q1u ≡ q2iu = q0u = κ r
2 (r + κ)2 + 2rρxηbσs − (r + κ) − rρxηba1 , 2 σs 2 (r + κ)2 + 2rρxηi σi − (r + κ) − rρxηi a2i , 2 σi N

(A.61) (A.62) (A.63)

q1u s + ¯ i=1 q2iu ei ¯

−1+

δ + log(r), r

instead of q1 , q2i , and q0 , respectively. The investor’s optimization problem is
N

max q1u s0 + ˆ x i=1

q2iu ei0 . ˆ

The investor decides to employ the manager if
N ˜ ˜ φ,χ,x N

ˆ max q1 s0 + i=1 q2i ei0 ˆ

> max q1u s0 + ˆ x i=1

q2iu ei0 ˆ

.

(A.64)

˜ To show that (A.64) holds, we show that it holds when setting (φ, x) = (φ, 0) in the left-hand side. Using (A.58), (A.59), (A.61), (A.62), and setting f1 (y) ≡ f2i (y) ≡
2 (r + κ)2 + 2rρyσs − (r + κ) − rρya1 , 2 σs 2 (r + κ)2 + 2rρyσi − (r + κ) − rρya2i , 2 σi

for a scalar y, we can write the latter condition as χ−χ ˜ + χ ηb s0 + ˆ f2i (1 − φ)θi + φ i=1
N N

max f1 (1 − φ)θb + χ ˜

χ−χ ˜ + χ ηi ei0 ˆ φ (A.65)

> max f1 (xηb)ˆ0 + s x i=1

f2i (xηi )ˆi0 . e

The function f1 (y) is concave and maximized for y given by 1
2 (r + κ)2 + 2rρyσs ρ ¯ θb, ⇔y= ρ+ρ ¯

− a1 = 0

where the second step follows from (3.2). Likewise, the function f2 (y) is concave and maximized for y given

49

by 1
2 + 2rρyσs ρ ¯ ⇔y= θi , ρ+ρ ¯

(r +

κ)2

− a2i = 0

where the second step follows from (3.3). For any given x, we can write (1 − φ)θ + χ−χ ˜ ρ ¯ +χ η =λ θ + (1 − λ)xη φ ρ+ρ ¯

by deﬁning (λ, χ) by ˜ ρ ¯ ≡ 1 − φ, ρ+ρ ¯ χ−χ ˜ + χ. (1 − λ)x ≡ φ λ Since 1 ≥ φ ≥ ρ ρ+ρ , ¯

λ ∈ [0, 1]. Therefore, the arguments of f1 (y) and f2 (y) in the left-hand side of (A.65) are

convex combinations of the corresponding arguments in the right-hand side and of the maximands of f1 (y) and f2 (y). Concavity of f1 (y) and f2 (y) then implies that the values of f1 (y) and f2 (y) in the left-hand side of (A.65) exceed the corresponding values in the right-hand side. Moreover, at least one of the inequalities is strict. This is because θ is not proportional to η and hence the arguments of f1 (y) and f2 (y) in the right-hand side of (A.65) cannot all coincide with the maximands of f1 (y) and f2 (y). Therefore, (A.65) holds. Proof of Proposition 4.1. When A > increasing in φ, χ is also increasing in A. Proof of Proposition 4.2. We ﬁrst compute the derivatives of (a1 , a2i , a1 , a2i , Sit , E(Sit )) with respect to ˇ ˇ A. Diﬀerentiating (4.1), (4.2), (4.4), and (4.5), and using φ = A, we ﬁnd ∂a1 ∂A ∂a2i ∂A ∂ˇ1 a ∂A ∂ˇ2i a ∂A ∂χ 2 η bσs , ∂A ∂χ 2 = −rρa3 θi − ¯ 2i η i σi , ∂A ∂χ 2 = rρˇ3 θ − a1 η bσs , ∂A ∂χ 2 = rρˇ3 θi − a2i ηi σi . ∂A = −rρa3 θ − ¯ 1 (A.66) (A.67) (A.68) (A.69) ρ ρ+ρ , ¯

φ is equal to A and hence is increasing in A. Since χ(φ) is

Diﬀerentiating (4.3) with respect to A, and using (A.66)-(A.69) and (s0 , e10 , .., eN 0 ) = (¯, e1 , .., eN ), we ﬁnd s ¯ ¯ − (r + κ) ρa3 + ρˇ3 ¯ 1 a1 θ− ∂χ 2 η bσs ηb¯ − (r + κ) s ρa3 + ρˇ3 ¯ 2i a2i ∂A i=1
N 2 ¯ 3 a3 ¯ i=1 ρa2i + ρˇ2i ηi θi σi ei . N 3 + ρˇ3 ) η 2 σ 2 e ρ a2i i i ¯i i=1 (¯a2i N

θi −

∂χ 2 ηi σi ηi ei = 0 ¯ ∂A (A.70)

2 ρa3 + ρˇ3 ηbθbσs s + ¯ 1 a1 ¯ ∂χ ⇒ = 3 + ρˇ3 ) (ηb)2 σ 2 s + ∂A (¯a1 ρ a1 s¯

50

Substituting

∂χ ∂A

from (A.70) into (A.66) and (A.67), we ﬁnd
N

2 2 rρa3 j=1 ρa3 + ρˇ3 ηj (ηbθj − θbηj )σs σj ej ¯ 1 ¯ 2j a2j ¯ ∂a1 , = N 2¯ ∂A (¯a3 + ρˇ3 ) (ηb)2 σs s + j=1 ρa3 + ρˇ3 ηj σj ej ρ 1 a1 a2j 2 2 ¯ ¯ 2j N

(A.71)

2 2 2 2 rρ ρa3 + ρˇ3 a3 ηb(ηi θb − θi ηb)σs σi s + rρa3 ¯ ¯ 1 a1 2i ¯ ¯ 2i j=1 ρa3 + ρˇ3 ηj (ηi θj − θi ηj )σi σj ej ¯ 2j a2j ¯ ∂a2i . = N 3 + ρˇ3 ) (ηb)2 σ 2 s + 3 + ρˇ3 η 2 σ 2 e ∂A (¯a1 ρ a1 a2j j j ¯j ¯ s¯ j=1 ρa2j

(A.72) Diﬀerentiating (A.1) with respect to A, we ﬁnd that ∂a1 ∂a2i bi s t + ˆ eit , ˆ ∂A ∂A and
∂E(Sit ) ∂A ∂Sit ∂A

has the same sign as (A.73)

has the same sign as (A.74)
∂Sit ∂A

∂a2i ∂a1 bi s + ¯ ei . ¯ ∂A ∂A When σs = 0, (A.71)-(A.73) imply that θi > ηi
N j=1 N j=1 2 ρa3 + ρˇ3 ηj θj σj ej ¯ 2j a2j ¯

is negative if

ρa3 + ρˇ3 ηj σj ej ¯ 2j a2j 2 2 ¯

,

(A.75)

and is positive if (A.75) holds in the opposite direction. This establishes the threshold result in the proposition, with γ equal to the right-hand side of (A.75). Since γ is a weighted average of with the weights ρa3 + ρˇ3 ηj σj ej ¯ 2j a2j 2 2 ¯
N j =1 θj ηj

over j ∈ {1, .., N }

ρa3 + ρˇ3 ¯ 2j a2j

2 2 ¯ η j σj ej θj j∈{1,..,N } ηj ,

that are positive and sum to one, (A.75) holds for i = arg max for i = arg min θj j∈{1,..,N } ηj .

and the opposite inequality holds

Therefore, each inequality holds for a non-empty set of assets.

When (bi , ei , σi , ηi ) = (bc , ec , σc , ηc ) for all i, we can write (A.71) and (A.72) as ¯ ¯ ¯ 2j a2j N θj − j =1 θj rρa3 bc ηc σs σc ec j=1 ρa3 + ρˇ3 ¯ 1 2 2 2¯ ∂a1 = , N 2¯ ∂A (¯a3 + ρˇ3 ) b2 ηc N 2 σs s + j=1 ρa3 + ρˇ3 ηc σc ec ρ 1 a1 c 2 ¯ 2j a2j 2 2 ¯
N N N N

(A.76)

2 2 rρ ρa3 + ρˇ3 a3 b2 ηc N σs σc s ¯ ¯ 1 a1 2i c 2 ¯ ¯ 3 2 4¯ ¯ 3 a3 j=1 θj − N θi + r ρa2i ηc σc ec j=1 ρa2j + ρˇ2j (θj − θi ) ∂a2i = , N 2¯ ∂A (¯a3 + ρˇ3 ) b2 ηc N 2 σs s + j=1 ρa3 + ρˇ3 ηc σc ec ρ 1 a1 c 2 ¯ 2j a2j 2 2 ¯

(A.77) respectively. We next show some properties of (a1 , a1 ). The expression ρa3 + ρˇ3 decreases in θj because ˇ ¯ 2j a2j ρ ¯ ¯ 2j a2j of (4.2), (4.5), and φ = A ∈ ( ρ+ρ , 1]. Using this observation and denoting by a2 the value of ρa3 + ρˇ3 for ¯

51

¯ θ≡

N j =1

θj

N N

, we ﬁnd 
N

 θj 

ρa3 + ρˇ3 N θj − ¯ 2j a2j j=1 j =1

=
¯ θj ≤θ

¯ ρa3 + ρˇ3 N (θj − θ) + ¯ 2j a2j
¯ θj >θ

¯ ρa3 + ρˇ3 N (θj − θ) ¯ 2j a2j

<
¯ θ j ≤θ N

¯ a2 N (θj − θ) + ¯
¯ θ j >θ

¯ a2 N (θj − θ) ¯

= j=1 ¯ a2 N (θj − θ) = 0. ¯

(A.78)

Equations (A.76) and (A.78) imply that opposite signs, i = arg max have a2i > i = arg min
∂ˇ1 a ∂A

∂a1 ∂A

< 0. Since (A.66) and (A.68) imply that ρ ρ+ρ ¯ j∈{1,..,N } θj

∂a1 ∂A

and

∂ˇ1 a ∂A

have We

> 0. Since, in addition, a1 = a1 for A = ˇ (A.77) implies that
∂a2i ∂A

(no frictions), a1 < a1 for A ∈ ˇ and i = arg min a2i <

ρ ( ρ+ρ , 1]. ¯

next show analogous properties of (a2i , a2i ) for i = arg max ˇ j∈{1,..,N } θj , a opposite signs, ∂ˇ2i ∂A

< 0. Since (A.67) and (A.69) ρ ρ+ρ , ¯

> 0. Since, in addition, a2i = a2i for A = ˇ
∂a2i ∂A

j∈{1,..,N } θj . For a imply that ∂a2i and ∂ˇ2i ∂A ∂A ρ a2i for A ∈ ( ρ+ρ , 1]. For ˇ ¯

j∈{1,..,N } θj , (A.77) implies that ρ a2i for A ∈ ( ρ+ρ , 1]. Since, in addition, ˇ ¯ ∂a1 j∈{1,..,N } θj , ∂A

> 0. Therefore, repeating the previous argument, we ﬁnd

a2j and a2j decrease in θj , a2i = max{a2j , a2j }j=1,..,N . ˇ ˇ < 0, and (A.73) imply that
∂Sit ∂A

For i = arg max

< 0,

∂a2i ∂A

< 0. For i = arg min

j∈{1,..,N } θj ,

the second term in the numerator in (A.77) is positive. Denoting the numerator in (A.76) by N1 , and the ﬁrst term in the numerator in (A.77) by N2 , the term bc sN1 + ec N2 is positive because ¯ ¯
N 2 2 2 a3 b2 ηc σs σc sec ¯¯ 1 c j=1 N

 ρa3 + ρˇ3 N θj − ¯ 2j a2j

N





N

 θj − N θi 

2 2 θj  + ρa3 + ρˇ3 a3 b2 ηc N σs σc sec  ¯ 1 a1 2i c 2 ¯¯ j =1 j=1 N

 ρa3 ¯ 2j + ρˇ3 a2j N θj −

 θj  + (¯ + ρ
2 2 2 ρ)a3 a3 b2 ηc N σs σc sec ¯¯ 1 2i c

 

N

 θj − N θi 

>

2 2 2 a3 b2 ηc σs σc sec ¯¯ 1 c j=1 N

j =1

j=1

 ρa3 + ρˇ3 N θj − ¯ 2j a2j

N


2 2 2 θj  + a3 b2 ηc σs σc sec ¯¯ 1 c

N

 ρa3 + ρˇ3  ¯ 2j a2j

N

 θj − N θi 

2 2 2 > a3 b2 ηc σs σc sec ¯¯ 1 c j=1 N 2 2 2 = a3 b2 ηc σs σc sec ¯¯ 1 c j=1 N 2 2 2 = a3 b2 ηc σs σc sec N ¯¯ 1 c

j =1

j=1 N

j =1

 ρa3 + ρˇ3 N θj − ¯ 2j a2j

N

 θj − N θi 

θj + j =1 j =1

ρa3 + ρˇ3 (θj − θi ) > 0, ¯ 2j a2j j=1 where the ﬁrst step follows from a1 < a1 and the second from a2i = max{a2j , a2j }j=1,..,N . Therefore, (A.74) ˇ ˇ implies that
∂E(Sit ) ∂A

> 0.

The results on expected returns in both cases of the proposition follow by combining the results on prices (which translate to expected prices) with (A.4) and (A.5). Proof of Proposition 4.3. Diﬀerentiating (A.6) with respect to A, we ﬁnd that
∂Var(dRit ) ∂A

has the same

52

sign as a1 ∂a1 2 2 ∂a2i 2 b σ s + a2i ¯ σ ei . ¯ ∂A i s ∂A i
$ ∂Var(dRit ) ∂A

(A.79) has the same sign as ∂a2i ∂a1 bi s + ¯ ei ¯ ∂A ∂A (A.80)

Likewise, diﬀerentiating (A.7), we ﬁnd that a1

∂a1 2 2 ∂a2i 2 2 2 bi σs s + a2i ¯ σ ei (a1 bi s + a2i ei ) − a2 b2 σs s + a2 σi ei ¯ ¯ ¯ ¯ ¯ 1 i 2i ∂A ∂A i ∂a2i ∂a1 2 2 = a1 − a2i a2i σi − a1 bi σs bi sei . ¯¯ ∂A ∂A When σs = 0, (A.72) and (A.79) imply that
∂Var(dRit ) ∂A

is negative if (A.75) holds, and is positive if
$ ∂Var(dRit ) ∂A

(A.75) holds in the opposite direction. Likewise, (A.71), (A.72), and (A.80) imply that

is negative

if (A.75) holds, and is positive if (A.75) holds in the opposite direction. These observations establish the threshold results in the proposition. The same argument as in the proof of Proposition 4.2 implies that each inequality holds for a non-empty set of assets. Consider next the case (bi , ei , σi , ηi ) = (bc , ec , σc , ηc ) for all i. For i = arg max ¯ ¯
∂a2i ∂A ∂a1 j∈{1,..,N } θj , ∂A

< 0,

< 0, and (A.79) imply that

∂Var(dRit ) ∂A

< 0. For i = arg min
2 a1 b2 σs sN1 ¯ c

j∈{1,..,N } θj ,

the second term in the

numerator in (A.77) is positive. The term

+

2 a2i σi ec N2 ¯

(where N1 denotes the numerator in

(A.76), and N2 the ﬁrst term in the numerator in (A.77)) is positive if (4.6) holds. This follows from N2 > 0 and because bc sN1 + ec N2 is positive as shown in the proof of Proposition 4.2. Therefore, (A.79) implies ¯ ¯
∂Var(dRit ) ∂A because of ∂a1 ∂A

that

> 0 if (4.6) holds. Moreover, (A.80) implies that < 0 and
∂a2i ∂A

$ ∂Var(dRit ) ∂A

> 0 if and only if (4.6) holds,

> 0.
∂E(Sηt ) ∂A

Proof of Proposition 4.4. Diﬀerentiating (A.12) with respect to A, we ﬁnd that sign as ∂a2i ∂a1 ηb¯ + s η i ei . ¯ ∂A ∂A i=1 Equations (A.71) and (A.72) imply that (A.81) has the same sign as
N 2 a3 ηbσs s ¯ 1 i=1 N N 2 2 a3 ρa3 + ρˇ3 ηi ηj (ηi θj − θi ηj )σi σj ei ej a2j ¯¯ 2i ¯ 2j i=1 j=1 N N N 2 2 ρa3 + ρˇ3 ηi (ηbθi − θbηi )σi ei + ρa3 + ρˇ3 ηbσs s ¯ 2i a2i ¯ ¯ 1 a1 ¯ i=1 N N

has the same

(A.81)

2 a3 ηi (ηi θb − θi ηb)σi ei ¯ 2j

+

=

2 ρηbσs s ¯ i=1

a3 a3 1 ˇ2i

−

a3 a3 ˇ1 2i

ηi (ηbθi −

2 θbηi )σi ei ¯

+ρ i=1 j=1

2 2 a3 a3 ηi ηj (ηi θj − θi ηj )σi σj ei ej . ¯¯ 2i ˇ2j

(A.82)

53

When σs = 0, the ﬁrst term in (A.82) is zero. The second term can be written as
N N 2 2 a3 ηi σi ei ¯ 2i i=1 i=1 2 a3 ηi θi σi ei ˇ2i ¯ N N 2 a3 ηi θi σi ei ¯ 2i i=1 i=1

ρ

−ρ

a3 ηi σi ei ˇ2i 2 2 ¯

,

and has the same sign as
N 2 ¯ ˇ3 i=1 a2i ηi θi σi ei N 3 η2 σ2 e ˇ i=1 a2i i i ¯i

−

N 3 2 ¯ i=1 a2i ηi θi σi ei . N 3 η2 σ2 e i=1 a2i i i ¯i θi ηi

(A.83)

Both fractions in (A.83) are weighted averages of sum to one. The weights are wi ≡ ˇ a3 ηi σi ei ˇ2i 2 2 ¯ N ˇ3 2 2 ¯ j=1 a2j ηj σj ej

over i ∈ {1, .., N } with weights that are positive and

for the ﬁrst fraction, and wi ≡
2 2 a3 ηi σi ei ¯ 2i N j=1 2 2¯ a3 ηj σj ej 2j

for the second fraction. When (σi , ηi ) = (σc , ηc ) for all i, the ratio a3 wi = 2i wi ˇ a3 ˇ2i
N j=1 N j=1 2 2 a3 ηj σj ej ¯ 2j

a3 ηj σj ej ˇ2j 2 2 ¯

ρ of weights depends on i only through θi . It also decreases in θi because (4.2), (4.5), and φ ∈ ( ρ+ρ , 1] imply ¯

that wi wi ˇ

a3 2i a3 ˇ2i

decreases in θi . Denote by i∗ the asset that maximizes θi within the set of assets for which wi∗ ≥ wi∗ . ˇ
N i=1

(That set is non-empty: wi < wi for all i ∈ {1, .., N } is not possible since ˇ we ﬁnd
N

wi =

N i=1

wi = 1.) Since ˇ

decreases in θi , it is larger than one for θi < θi∗ , and smaller than one for θi > θi∗ . Using this property,

(wi − wi ) ˇ i=1 θi = ηc =

(wi − wi ) ˇ θi ≤θi∗

θi + ηc

(wi − wi ) ˇ θi >θi∗

θi ηc wi wi ˇ wi wi ˇ θi ηc θi∗ ηc

wi 1 − ˇ θi ≤θi∗

wi wi ˇ

θi + ηc

wi 1 − ˇ θi >θi∗

> θi ≤θi∗

wi ˇ

wi 1− wi ˇ

θ i∗ + ηc

wi 1 − ˇ θi >θi∗

= θi ≤θi∗ N

θ i∗ (wi − wi ) ˇ + ηc θi∗ = 0. ηc

(wi − wi ) ˇ θi >θi∗

θ i∗ ηc (A.84)

= i=1 (wi − wi ) ˇ

Therefore, (A.83) is positive and so is (A.82). When (σi , θi ) = (σc , θc ) for all i, we can follow the same steps to show that (A.83) is again positive. The modiﬁcations are that wi wi ˇ

and

θi ηi

depend on i only through ηi

54

rather than only through θi , increases in θi .

wi wi ˇ

increases in ηi rather than decreases in θi , and

θi ηi

decreases in ηi rather than

When (bi , ei , σi , ηi ) = (bc , ec , σc , ηc ) for all i, the second term in (A.82) is positive because of the previous ¯ ¯ argument. The ﬁrst term can be written as
N 3 2 2 ρb2 ηc N σs σc sec ¯¯ c i=1

 a3 a3 − a3 a3 N θi − ˇ1 2i 1 ˇ2i

N

 θj  ,

j=1 N


3 2 2 = ρb2 ηc N σs σc sec a3 ¯¯ c 1

 a3 − a3 N θi − ˇ2i 2i

N

 θj  + (a3 − a3 ) ˇ1 1

N

 a3 N θi − 2i

N

 θj  . (A.85)

i=1

j=1

i=1

j=1

The same argument as for equation (A.78) establishes that
N

 a3 N θi − 2i

N

 θj  < 0.

i=1

j=1

Since, in addition, a1 < a1 , the second term in the bracket in (A.85) is positive. When θi can take only two ˇ values, a2i > a2i for the smaller value, and a2i < a2i for the larger value, as shown in the proof of Proposition ˇ ˇ 4.2. Therefore, the ﬁrst term in the bracket in (A.85) is positive, and so (A.82) is positive. The results on expected returns in both cases of the proposition follow by combining the result on expected prices with (A.4) and (A.5). Proof of Theorem 5.1. We proceed in two steps, as in the proof of Theorem 4.1. ¯ Step 1. Same as for Theorem 4.1, except that we do not impose the restriction φ ≥ A, and we replace Gt , (A.31), and (A.39) by B ¯ Gt ≡ rρ rWt + (φzt − χη)µt + ψ + (A − φ)mt − m2 − ct + κ q1 (¯ − st ) + ¯ ¯ ¯ ¯ s q2i (¯i − eit ) , ¯ e 2 t i=1 mt = A−φ 1{φ≤A} , B (A − φ)2 1{φ≤A} 2B + κ r
N N

q0 = ρ ψ + ¯ ¯

q1 s + ¯¯ i=1 q2i ei ¯ ¯

−1+

¯ δ + log(r), r

(A.86)

respectively, where 1S is the indicator function of the set S.

55

Step 2. Same as for Theorem 4.1, with the following changes. We replace Gt , (A.49), and (A.60) by
N

˜ ˜ Gt ≡ rρ rWt + (xη + z − φθ + χη)µt − ψ − (1 − φ)mt − ct + κ q1 (¯ − st ) + s i=1 q2i (¯i − eit ) , e

˜ ˜ ˜ (1 − φ)(A − φ) 1 ˜ q0 = −ρ ψ + {φ≤A} B (1 − φ)(A − φ) q0 = −ρ ψ + 1{φ≤A} B

+

κ r

N

q1 s + ¯ i=1 N

q2i ei ¯

−1+

δ + log(r), r δ + log(r), r

(A.87)

κ + r

q1 s + ¯ i=1 q2i ei ¯

−1+

(A.88)

respectively. The manager’s individual rationality constraint becomes ˜2 ˜ (A − φ) 1 ˜ ψ+ {φ≤A} ≥ 0, 2B
˜ ˜ and the investor chooses ψ = − (A−φ) 1{φ≤A} . The investor’s optimization problem becomes ˜ 2B
2

˜ ˜ φ,χ,x

max −ρ

N ˜ ˜ (A − φ)(2 − A − φ) 1{φ≤A} + q1 s0 + ˆ q2i ei0 ˆ ˜ 2B i=1

,

˜ without the constraint φ ≥ A. The ﬁrst-order conditions with respect to χ and x are equivalent to (4.3). ˜ ˜ is Using (A.53), we ﬁnd that the ﬁrst-order condition with respect to φ r 1−φ + Φ = 0 if φ < A, B φ 1−φ r + Φ ≥ 0 and Φ ≤ 0 B φ Φ = 0 if φ > A. Equations (A.89)-(A.91) rule out that Φ is positive. When A ≤ then imply that φ ≤ A ≤ Φ = 0, which implies φ = be smaller Therefore, ρ ρ+ρ , and property ¯ ρ ρ+ρ and χ(φ) = 0. ¯ ρ ρ+ρ , ¯

(A.89) if φ = A, (A.90) (A.91) Φ cannot be negative: (A.91) would φ cannot be strictly larger than A: ρ ρ+ρ ¯

(P) would imply that Φ has to be non-negative. Therefore, When instead A > ρ ρ+ρ , ¯

(A.91) would then imply that Φ = 0, and property (P) would imply that φ =

< A. Moreover, φ cannot imply that φ > ρ ρ+ρ . ¯

ρ than ρ+ρ : (A.89) would then imply that Φ < 0, and property (P) would ¯ ρ φ ∈ ( ρ+ρ , A], which implies χ(φ) > 0. Equations (A.89) and (A.90) yield ¯

(5.1) and (5.2). The

condition that the investor decides to employ the manager becomes
N ˜ ˜ (A − φ)(2 − A − φ) 1{φ≤A} + q1 s0 + ˆ q2i ei0 ˆ ˜ 2B i=1 N

˜ ˜ φ,χ,x

max −ρ

> max q1u s0 + ˆ x i=1

q2iu ei0 ˆ

(A.92)

instead of (A.64). Equation (A.92) is satisﬁed for B = 0, as shown in Theorem 4.1. It is also satisﬁed for B = ∞ because the left-hand side of (A.92) becomes identical to that of (A.64). By continuity, it is satisﬁed for B close to zero and to inﬁnity. Proof of Proposition 5.1. The social planner maximizes the investor’s value function V (W0 , s0 , e0 ) at

56

time zero, subject to the manager’s incentive compatibility (IC) and individual rationality (IR) constraints. The IC constraint is that the manager’s choices of (zt , mt ) are optimal given the contract. The IR constraint ¯ ¯ ¯ ¯ is that the manager’s value function V (W0 , s0 , e0 ) exceeds the value function Vu (W0 , s0 , e0 ) from being unemployed. From Steps 1 and 2 of the proof of Theorem 5.1, the social planner’s problem reduces to maximizing
N

rρW0 + q0 + q1 s0 + i=1 q2i ei0

subject to
N N

rρW0 + q0 + q1 s0 + ¯¯ ¯ ¯ i=1 q2i ei0 ≥ q0u + q1u s0 + ¯ ¯ ¯ i=1 q2iu ei0 , ¯

where (q0 , q1 , q21 , .., q2N ) are given by (A.58), (A.59), and (A.88), (¯0 , q1 , q21 , .., q2N ) are given by (A.37), q ¯ ¯ ¯ (A.38), and (A.86), and (¯0u , q1u , q21u , .., q2N u ) are the counterparts of (¯0 , q1 , q21 , .., q2N ) for an unemployed q ¯ ¯ ¯ q ¯ ¯ ¯ manager. The values of (¯0u , q1u , q21u , .., q2N u ) computed in Theorem 5.1 depend on (φ, χ). (In particular, q ¯ ¯ ¯ (¯1u , q21u , .., q2N u ) = (¯1 , q21 , .., q2N ).) This is because the manager computes his value function when unq ¯ ¯ q ¯ ¯ employed under the equilibrium prices, which depend on the contract. The values of (¯0u , q1u , q21u , .., q2N u ) q ¯ ¯ ¯ computed by the social planner, however, do not depend on (φ, χ). This is because the social planner internalizes that when the manager is unemployed, prices change and do not depend on the contract. Using (A.86) and (A.88), we can write the social planner’s problem as 1 (A − φ)(2 − A − φ) ¯ 1{φ≤A} + max r(W0 + W0 ) − φ,χ 2B ρ
N

q 1 s0 + ˆ i=1 q2i ei0 ˆ

+

1 ρ ¯

N

q 1 s0 + ¯ˆ i=1 q2i ei0 ¯ ˆ

. (A.93)

Since the investor and the manager are endowed collectively with the portfolio θ at time zero, the problem (A.93) is equivalent to (A − φ)(2 − A − φ) 1 1{φ≤A} + 2B ρ
N

max rθS0 − φ,χ q 1 s0 + ˆ i=1 q2i ei0 ˆ

+

1 ρ ¯

N

q 1 s0 + ¯ˆ i=1 q2i ei0 ¯ ˆ

.

(A.94)

Using (A.58), (A.59), q1 = ¯
2 (r + κ)2 + 2rρ(φθ − χη)bσs − (r + κ) ¯ − rρ(φθ − χη)ba1 , ¯ 2 σs

(A.95)

which follows from (4.1) and (A.37), q2i = ¯
2 (r + κ)2 + 2rρ(φθi − χηi )σi − (r + κ) ¯ − rρ(φθi − χηi )a2i , ¯ 2 σi

(A.96)

which follows from (4.2) and (A.38), and Si0 = a1 bi s0 + a2i ei0 , ˆ ˆ

57

which follows from (A.1), we can write (A.94) as max − φ,χ (A − φ)(2 − A − φ) 1{φ≤A} 2B
2 (r + κ)2 + 2rρ((1 − φ)θ + χη)bσs s0 + ˆ 2 σs i=1 2 (r + κ)2 + 2rρ(φθ − χη)bσs ¯ s0 + ˆ 2 σs i=1 N N 2 (r + κ)2 + 2rρ((1 − φ)θi + χηi )σi ei0 ˆ 2 σi

+

1 ρ 1 ρ ¯

+

2 (r + κ)2 + 2rρ(φθi − χηi )σi ¯ ei0 ˆ 2 σi

.

(A.97)

The ﬁrst-order condition with respect to χ is (4.3). The ﬁrst-order condition with respect to φ is 1−φ + rΦ = 0 if φ < A, B 1−φ + rΦ ≥ 0 and Φ ≤ 0 B Φ = 0 if φ > A. (A.98) if φ = A, (A.99) (A.100)

Since (A.97) is strictly concave, the ﬁrst-order conditions characterize a unique maximum (φ∗ , χ∗ ). Using the same arguments as in the proof of Theorem 5.1, we ﬁnd that (φ∗ , χ∗ , ψ ∗ ) are as in the theorem. We ﬁnally show that φ∗ ≥ φ and χ∗ ≥ χ for the privately optimal (φ, χ), with the inequalities being strict when φ < A. Since χ∗ solves (4.3), it is equal to χ(φ∗ ) for the function χ(φ) deﬁned in the proof of Theorem 4.1. Since χ(φ) is increasing in φ, it suﬃces to show the inequalities for φ∗ . When φ < A, φ is determined by (A.89). Using (A.89), we can write the left-hand side of (A.98) as (1 − φ)2 1 − φ φ(1 − φ) − = > 0. B B B Since the derivative of the social planner’s objective with respect to φ is positive at the privately optimal φ, φ∗ > φ. When φ = A, φ satisﬁes (A.90). Using (A.90), we ﬁnd that the left-hand side of (A.98) is larger than
(1−φ)2 B

> 0. Since the derivative of the social planner’s objective with respect to φ is positive at the

privately optimal φ = A, φ∗ = φ = A. Proof of Theorem 6.1. We follow the same steps as in the proof of Theorem 4.1. Step 1. Equation (A.23) is replaced by dRit = µit dt + a1i σs dwst + a2i σi dwit , (A.101)

which can be derived by substituting Sit from (2.14) into (2.2), and using (2.1), (6.1), and (6.2). The ¯ ¯ manager’s Bellman equation (A.29) remains the same, except that (Ht , Kit ) are replaced by
N

¯ Ht ≡ r ρ ¯ i=1 (φzit − χηi )a1i + q1 σs , ¯

¯ Kit ≡ [rρ(φzit − χηi )a2i + q2i ] σi . ¯ ¯

58

The ﬁrst-order condition (A.32) is replaced by
N 2 2 (φzit − χηi )a1i + q1 σs − rρφa2i [rρ(φzit − χηi )a2i + q2i ] σi = 0. ¯ ¯ ¯ ¯ i=1

rρφµit − rρφa1i rρ ¯ ¯ ¯

(A.102)

¯ ¯ ¯ The terms (A0i , A1i , A2i , Q0 , Q1 , Q2i ) are replaced by
N N 2 (φθi − χηi )a1i + q1 σs − ¯ i=1 i=1 2 a2i [rρ(φθi − χηi )a2i + q2i ] σi , ¯ ¯

A0i ≡ κ(a1i s + a2i ei ) − ra0i − a1i rρ ¯ ¯ ¯ A1i ≡ bi − (r + κ)a1i , A2i ≡ 1 − (r + κ)a2i ,
N

¯ Q0 ≡ rq0 − rρψ − κ q1 s + ¯ ¯ ¯¯ i=1 q2i ei ¯ ¯
2

¯ + r − δ − r log(r)
N

−

1 rρ ¯ (φθi − χηi )a1i 2 i=1

N

2 σs − i=1

1 2 2 [rρ(φθi − χηi )a2i ] σi , ¯ 2

1 2 2 ¯ Q1 ≡ (r + κ)¯1 + q1 σs , q ¯ 2 1 2 2 ¯ Q2i ≡ (r + κ)¯2i + q2i σi . q ¯ 2 ¯ ¯ ¯ Setting Q1 = Q2i = 0 yields q1 = q2i = 0. Setting A0i = A1i = A2i = 0 yields (6.3)-(6.5). Setting Q0 = 0, ¯ ¯ and using (6.4) and (6.5), yields
2 2 rρ2 [(φθ − χη)b] σs + i=1 (φθi − χηi )σi ¯ ¯ δ q0 = ρψ − 1 + + log(r) + ¯ ¯ . r 2(r + κ)2 2 N

(A.103)

Step 2. The investor’s Bellman equation (A.44) remains the same, except that (Ht , Kit ) are replaced by Ht ≡ [rρ(xη + z − φθ + χη)ba1 + q1 ] σs , Kit ≡ [rρ(xηi + z − φθi + χηi )a2i + q2i ] σi . The terms (Q0 , Q1 , Q2i ) are replaced by
N

˜ Q0 ≡ rq0 + rρψ − κ q1 s + ¯ i=1 q2i ei ¯

+ r − δ − r log(r)

1 2 − rρ(xη + z − φθ + χη)ba1 rρ(φθ − χη)ba1 − rρ(xη + z − φθ + χη)ba1 + q1 − q1 σs ¯ ¯ 2 − 1 2 rρ(xηi + zi − φθi + χηi )a2i rρ(φθi − χηi )a2i − rρ(xηi + zi − φθi + χηi )a2i + q2i − q2i σi , ¯ ¯ 2 i=1
N

1 2 2 Q1 ≡ (r + κ)q1 + q1 σs , 2 1 2 2 Q2i ≡ (r + κ)q2i + q2i σi . 2

59

Setting Q1 = Q2i = 0 yields q1 = q2i = 0. Setting Q0 = 0, and using (6.4) and (6.5), yields δ ˜ q0 = −ρψ − 1 + + log(r) r 2 rρ(xη + z − φθ + χη)b ρ(φθ − χη)b − 1 ρ(xη + z − φθ + χη)b σs ¯ 2 + (r + κ)2 +
N i=1 1 2 rρ(xηi + zi − φθi + χηi ) ρ(φθi − χηi ) − 2 ρ(xηi + zi − φθi + χηi ) σi ¯ . (r + κ)2

(A.104)

˜ ˜ ˜ If the investor decides to employ the manager, then she chooses a contract (φ, χ, ψ) and index investment x to maximize q0 + q1 s0 + i=1 q2i ei0 . Using q1 = q2i = 0 and (A.104), and noting that the investor chooses ˜ ˜ ψ = 0 to meet the manager’s IR constraint ψ ≥ 0 with equality, we can write the investor’s optimization problem as 1 2 ¯ max (xη + z − φθ + χη)b ρ(φθ − χη)b − ρ(xη + z − φθ + χη)b σs 2 + 1 2 rρ(xηi + zi − φθi + χηi ) ρ(φθi − χηi ) − ρ(xηi + zi − φθi + χηi ) σi ¯ 2 i=1
N N

˜ ˜ φ,χ,x

,

˜ ˜ ˜ subject to the constraint φ ≥ A. The partial derivatives of the investor’s objective with respect to (φ, χ, x) ˜ ˜ at (φ, χ, x) = (φ, χ, 0) are
2 [ρ − (ρ + ρ)φ] (θb)2 σs + ¯ N 2 2 i=1 θi σi 2 + (ρ + ρ)χ ηbθbσs + ¯ N i=1 2 η i θ i σi

φ −
2 [ρ − (ρ + ρ)φ] ηbθbσs + ¯ N i=1 2 2 ηi θi σi + (ρ + ρ)χ (ηb)2 σs + ¯ N i=1

,

(A.105)

2 2 ηi σi

φ
N N 2 2 ηi θi σi − (ρ + ρ)χ (ηb)2 σs + ¯ i=1 i=1 2 2 ηi σi ,

,

(A.106) (A.107)

2 − [ρ − (ρ + ρ)φ] ηbθbσs + ¯

˜ ˜ respectively. Because the investor’s optimization problem is concave, (φ, χ, x) = (φ, χ, 0) is an optimum if (A.106) and (A.107) are equal to zero, and if (A.105) is non-positive when φ = A and is equal to zero when φ > A. Setting (A.106) and (A.107) to zero yields (6.7). Using (6.7), we can write (A.105) as
2 (ηb)2 σs + N i=1 2 2 ηi σi 2 (θb)2 σs + N 2 2 i=1 θi σi N i=1 2 − ηbθbσs + N i=1 2 ηi θi σi 2

[ρ − (ρ + ρ)φ] ¯

.

(A.108)

2 φ (ηb)2 σs +

2 2 η i σi

Since θ is not proportional to η, the Cauchy-Schwarz inequality implies that the denominator in (A.108) is positive. Therefore, (A.108) has the same sign as ρ − (ρ + ρ)φ. Property (P) in the proof of Theorem 4.1 is ¯ thus satisﬁed, and the same argument as in that proof implies that φ is given by (6.6). Proof of Proposition 6.1. Equation (6.3) implies that a0i > a0i . Since, in addition, (6.4) and (6.5) imply that (a1i , a2i ) = (a1i , a2i ), Sit > Si t . Equations (A.23), which remains valid under OU processes,

60

and E(Sit ) > E(Si t ) imply that E(dRit ) < E(dRi t ). Proceeding as in the derivation of (A.6), we ﬁnd
2 2 Var(dRit ) = a2 b2 σs + a2 σi dt = 1 i 2i 2 2 b2 σ s + σ i i dt, 2 (r + κ)

(A.109)

where the second step follows from (6.4) and (6.5). Therefore, Var(dRit ) = Var(dRi t ). Proof of Proposition 6.2. Equations (6.3)-(6.5), (6.7), and φ = A imply that ρ bi ¯ ∂Sit = ∂A
N j=1 2 2 2 2 ηj (ηbθj − θbηj )σs σj + ηb(ηi θb − θi ηb)σs σi + 2 (r + κ)2 (ηb)2 σs + N i=1 2 2 η i σi N j=1 2 2 ηj (ηi θj − ηj θi )σi σj

. (A.110)

When σs = 0, the result follows as in the proof of Proposition 4.2. When (bi , ei , σi , ηi ) = (bc , ec , σc , ηc ) ¯ ¯ for all i, the ﬁrst term in the numerator in (A.110) is zero. The second and third terms are negative for i = arg max j∈{1,..,N } θj ,

and positive for i = arg min

j∈{1,..,N } θj .

The results on expected returns in both

cases follow by combining the results on prices with (A.4). The share return variance of asset i is given by (A.109) and hence does not depend on A. Multiplying (A.110) by ηi and summing across i, we ﬁnd ρ ηb ¯ ∂Sηt = ∂A
N i=1 2 2 ηi (ηbθi − θbηi )σs σi + ηb N i=1 2 2 ηi (ηi θb − θi ηb)σs σi + N i=1 2 2 ηi σi N i=1 N j=1 2 2 ηi ηj (ηi θj − ηj θi )σi σj

.

2 (r + κ)2 (ηb)2 σs +

(A.111) Since the ﬁrst two terms in (A.111) cancel and the third term is zero,
∂Sηt ∂A

= 0.

61

References
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Asset Management Contracts and Equilibrium Prices

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