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
April 12, 2014∗
PRELIMINARY. PLEASE DO NOT CIRCULATE.
Abstract
We study the joint determination of fund managers’ contracts and equilibrium asset prices.
Because of agency frictions, fund investors make managers’ fees more sensitive to performance
and benchmark performance against a market index. In equilibrium, managers underweight
expensive stocks that are in high demand by other traders and have endogenously high volatility
and market betas, and overweight low-demand stocks. Since benchmarking makes it risky for
managers to deviate from the index, it exacerbates cross-sectional differences in alphas and the
negative relationship between alpha on one hand, and volatility or beta on the other. Moreover,
because underweighting the expensive stocks is riskier than overweighting the cheaper ones,
benchmarking raises the price of the aggregate stock market and can increase its volatility.
Asset management contracts are not socially optimal because investors do not internalize the
feedback effect between contracts and asset prices.
∗
We thank seminar participants at Boston University, Maryland University, Central European University, Bocconi
University, the European Summer Symposium in Financial Markets (2013, Gerzensee), the FRIC Conference on
Financial Frictions (2013, Copenhagen), the AEA Annual Meeting (2014, Philadelphia), the 8th Jackson Hole
Finance Conference (2014, Jackson Hole) for helpful discussions and comments.
Address correspondence to the authors at buffa@bu.edu, d.vayanos@lse.ac.uk, and p.k.woolley@lse.ac.uk.
1
Introduction
Asset management is a large and growing industry. For example, according to French’s (2008)
presidential address to the American Finance Association, individual investors held directly only
21.5% of U.S. stocks in 2007. The remainder was held within financial institutions of various
types, run by professional managers. The risk and return of asset managers is measured against
benchmarks, and performance relative to these benchmarks affects the managers’ compensation
and the money that investors give them to manage. In this paper we study the implications of
delegated portfolio management and benchmarking on equilibrium asset prices. Unlike most of
the prior literature, we do not assume exogenous preferences or compensation contracts for asset
managers, but optimize over the contract choice.
We assume a continuous-time infinite-horizon economy with a riskless asset and multiple risky
assets (stocks). The main agents in our model are a representative fund manager and a representative fund investor. The investor cannot invest in the stocks directly and must employ the
manager. The manager receives a fee, which is an affine function of the fund’s performance and
the performance of the market index. We impose the functional form of the contract exogenously,
but optimize over the coefficients of the affine function. The manager can add value over the index because additional agents, buy-and-hold investors, hold a portfolio that differs from the index.
Hence, stock prices adjust so that holding the stocks’ effective supply, i.e., the supply net of the
holdings of buy-and-hold investors, is optimal for the manager.
We first solve the model in the case where there are no agency frictions. The fee that the
manager receives achieves optimal risk-sharing between him and the investor, and involves no
benchmarking. Stock prices depend on effective supply. Stocks in large supply are cheap and hence
offer high expected return, holding all else equal, so that the manager is induced to give them larger
weight than the index weight. Moreover, the prices of these stocks are less sensitive to shocks to
expected dividends. This is because a positive dividend shock raises not only expected dividends per
share but also the volatility of dividends per share. The compensation for the increased volatility is
larger for a stock in large effective supply, and hence the positive effect of the shock on the stock’s
price is smaller. The low price sensitivity to dividend shocks for stocks in large supply causes
them to have low idiosyncratic volatility, as well as low total return volatility and market beta in
some cases. Since these stocks have also positive alpha, our model provides a mechanism that can
generate the volatility anomaly (e.g., Haugen and Baker 1996; Ang et al. 2006, 2009) and the beta
anomaly (e.g., Black 1972; Black, Jensen and Scholes 1972; Frazzini and Pedersen 2013).
We next solve the model in the case where there are agency frictions. We model these frictions
as an unobserved action by the manager that benefits him but reduces the fund’s value. The fee that
1
the manager receives is more sensitive to the fund’s performance than under optimal risk-sharing,
and involves benchmarking. Because the fee makes the manager less willing to deviate from the
index, the effects of supply are amplified. Stocks in large supply become cheaper and less volatile,
while stocks in small supply become more expensive and more volatile. The increase in volatility
for the latter stocks can be interpreted as an amplification effect: following an increase in their
expected dividends, the manager becomes even more averse to underweighting these stocks because
they account for a larger fraction of market movements. As a consequence, he seeks to buy these
stocks, amplifying the price increase. This effect seems consistent, for example, with the behavior of
many fund managers towards technology stocks during the 1998-2000 rise in the Nasdaq. Because
the agency frictions increase cross-sectional discrepancies in expected returns and volatilities, they
amplify the volatility and beta anomalies relative to the case of no agency frictions.
Besides their effect on the prices of individual stocks, agency frictions also affect the aggregate
market. We show that the aggregate market goes up, and its expected return goes down. Therefore,
the cross-sectional effect of the frictions is asymmetric: stocks in small supply go up more than
stocks in large supply go down. The intuition is that underweighting overpriced stocks is more
costly for the manager than overweighting underpriced stocks because the former are more volatile
and hence expose the manager to greater risk of deviating from the benchmark.
In addition to the positive results, we perform a normative analysis. We show that the coefficients of the affine fee chosen by a social planner differ from those chosen by private agents. This
is because the social planner takes into account that fee choice affects equilibrium prices.
Related Literature
Our paper is closest to a literature that investigates the effects of bench-
marking and tracking error on equilibrium asset prices. Brennan (1993), Kapur and Timmermann
(2005), Arora, Ju and Ou-Yang (2006), Cuoco and Kaniel (2011), and Basak and Pavlova (2013)
assume exogenous benchmarking and study its implications for asset prices. Abstracting from
benchmarking and optimal contracts, Ross (2005) and Garcia and Vanden (2009) study mutual
funds within the standard competitive noisy rational expectations model. Garcia and Vanden
(2009) focus on the formation of mutual funds as the outcome of endogenous information acquisition. Our contribution to this literature is to derive equilibrium asset prices with endogenous
contracts for asset managers. Our asset-pricing results differ from much of this literature as well.
Optimal contracts for asset managers have mainly been analyzed in partial equilibrium settings.
Examples include Bhattacharya and Pfleiderer (1985), Starks (1987), Kihlstrom (1988), Stoughton
(1993), Heinkel and Stoughton (1994), Admati and Pfleiderer (1997), Das and Sundaram (2002),
Palomino and Prat (2003), Ou-Yang (2003), Liu (2005), Dybvig, Farnsworth and Carpenter (2010),
Cadenillas, Cvitanic and Zapatero (2007), Cvitanic, Wan and Zhang (2009) and Li and Tiwari
2
(2009).
Our analysis focuses on fund managers’ explicit incentives, deriving from their fees. A number of
papers explore instead implicit incentives, deriving from fund flows, as well as the effects that flows
have on equilibrium asset prices. Examples include Shleifer and Vishny (1997), Berk and Green
(2004), Vayanos (2004), He and Krishnamurthy (2012, 2013), Kaniel and Kondor (2013), Vayanos
and Woolley (2013). Dasgupta and Prat (2008), Dasgupta, Prat and Verardo (2011), Guerrieri and
Kondor (2012), and Malliaris and Yan (2012) emphasize implicit incentives generated by managers’
reputation concerns.
Our model relates to two important asset pricing anomalies. Black (1972) provides empirical
evidence on the beta anomaly — the negative relationship between beta and alpha. He also suggests
that this anomaly might be driven by leverage constraints: leverage-constrained investors find highbeta stocks attractive as a way to obtain leverage, and this renders those stocks overpriced. Frazzini
and Pedersen (2013) provide extensive empirical evidence for the beta anomaly across a variety of
markets, and find support for the leverage explanation. Hong and Sraer (2013) show that the beta
anomaly can be explained instead by differences in beliefs and short-sale constraints.
Ang, Hodrick, Xing and Zhang (2006, 2009) provide empirical evidence on the idiosyncratic
volatility anomaly — the negative relationship between idiosyncratic volatility and alpha. They
document strong covariation in the returns to high-idiosyncratic-risk stocks, and suggest that broad
factors that are not easily diversifiable lie behind this phenomenon. This finding is consistent with
our mechanism. Most recently, Stambaugh, Yu and Yuan (2013) consider asymmetry in arbitrage,
namely the fact that impediments to short selling stocks are larger than those to purchasing stocks,
as a possible mechanism behind the puzzle.1
Our results that agency frictions raise the price of the aggregate market, and that correcting
overpricing is costlier than correcting underpricing, relate to a literature on asset bubbles. Miller
(1977), Harrison and Kreps (1978), Scheinkman and Xiong (2003), and Hong, Scheinkman and
Xiong (2006) analyze the overvaluation generated by heterogenous beliefs and short-sales constraints. Abreu and Brunnermeier (2002, 2003) emphasize the role of synchronization risk for the
existence and persistence of bubbles. Allen and Gale (2000) investigate asset bubbles arising from
leverage constraints and agency frictions in the banking sector.
1
Wurgler (2010) and Baker, Bradley and Wurgler (2011) argue that the beta and volatility anomalies can arise
when asset managers care about their performance only relative to a market index and not in absolute terms. This
is because the risk of low-beta and high-beta stocks then becomes symmetric: low-beta stocks underperform when
the market goes up, high-beta stocks underperform when the market goes down, and managers do not care about
the direction of market movement, holding relative performance constant. The mechanism generating the anomalies
in our model is different and related to supply effects. Moreover, the investors in our model find it optimal to not
pay managers based only on relative performance.
3
2
Model
2.1
Assets
Time t is continuous and goes from zero to infinity. There is a riskless asset with an exogenous
return equal to r, and N risky assets. We refer to the risky assets as stocks, but they could also be
interpreted as segments of the stock market, e.g., industry-sector portfolios or investment styles.
The price Sit of stock i = 1, .., N is determined endogenously in equilibrium. The dividend flow
Dit of stock i is given by
Dit = bi st + eit ,
(1)
where st is a component common to all stocks and eit is a component specific to stock 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 stock i to the common component
st . We set Dt ≡ (D1t , .., DN t )0 , St ≡ (S1t , .., SN t )0 , and b ≡ (b1 , .., bN )0 . We denote by dRt the
vector of stocks’ returns per share in excess of the riskless rate:
dRt ≡ Dt dt + dSt − rSt dt.
(2)
The return per dollar invested in stock i can be derived by dividing stock i’s return per share dRit
by the stock’s price Sit . Stock 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 (s̄ − st ) dt + σs st dwst ,
(3)
√
deit = κ (ēi − eit ) dt + σi eit dwit ,
(4)
where (s̄, ē1 , .., ēN , σs , σ1 , .., σN ) are positive constants, and the Brownian motions (wst , w1t , .., wN t )
are mutually independent. The square-root specification (3)-(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 both realistic and key for our analysis.
The constants (s̄, ē1 , .., ēN ) are the unconditional means of the variables (st , e1t , .., eN t ). The
2 e ) condiincrements (dst , de1t , .., deN t ) of these variables have variance rates (σs2 st , σ12 e1t , .., σN
Nt
2 ē ) unconditionally. We occasionally consider the special case of
tionally and (σs2 s̄, σ12 ē1 , .., σN
N
“scale invariance,” where the ratio of unconditional standard deviation to unconditional mean is
4
2 ) is collinear with
identical across the N + 1 processes. This occurs when the vector (σs2 , σ12 , .., σN
(s̄, ē1 , .., ēN ).
2.2
Agents
The main agents in our model are a representative investor and a representative fund manager. The
investor can invest in stocks directly by holding the index, or indirectly by employing the 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 market for individual stocks. One interpretation of this
participation friction is that the investor cannot identify stocks 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 stocks
for corporate-control or 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 stock i available to the investor and the manager is
thus θi , and represents the residual supply of stock i to them. Stocks 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. By overweighting stocks that earn high expected returns, the
manager adds value over the index. We assume that the residual supply of each stock is positive
(θi > 0 for all i). We refer to residual supply simply as supply from now on.2
The investor chooses an investment x in the index η, i.e., holds xηi shares of stock 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 stock i held by the fund. The manager can
also take a “shirking” action mt that delivers a private benefit Bmt to him but reduces the fund’s
return by 21 m2t . The manager could, for example, exert insufficient effort to reduce operating costs
or to identify a more efficient portfolio. The investor can influence the choices of zt and mt through
a compensation contract, offered at t = 0. 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 − (1/2)m2t dt
2
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.
5
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
1
dft = φ zt dRt − m2t dt − χηdRt
2
+ ψdt,
(5)
where (φ, χ, ψ) are constants. The constant φ is the fee’s sensitivity to the fund’s performance,
and the constant φχ is the sensitivity to the index performance. If χ = 0 then the manager is paid
only based on the fund’s absolute performance, while if χ 6= 0 then performance relative to the
index also matters. We assume that the manager invests his personal wealth in the riskless asset.
This is without loss of generality: since the manager is exposed to stocks through the fee, and can
adjust this exposure continuously by changing the fund’s portfolio, a personal investment in stocks
is redundant. If the manager is not employed by the investor, then he chooses a personal stock
portfolio zt , receives no fee, and has no shirking action available.
2.2.1
Manager’s Optimization Problem
The manager derives utility over intertemporal consumption. Utility is exponential:
Z
E
∞
− exp(−ρ̄c̄t − δ̄t)dt ,
(6)
0
where ρ̄ is the coefficient of absolute risk aversion, c̄t is consumption, and δ̄ is the discount rate.
We denote the manager’s wealth by W̄t .
If the manager is employed by the investor, then he chooses the fund’s portfolio zt and the
shirking action mt . His budget constraint is
dW̄t = rW̄t dt + dft + Bmt dt − c̄t dt,
(7)
where the first term is the return from the riskless asset, 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 (c̄t , zt , mt ) to maximize the expected
utility (6) subject to the budget constraint (7) and the fee (5). We denote by zt (φ, χ, ψ) and
mt (φ, χ, ψ) the manager’s optimal choices of zt and mt , and by V̄ (W̄t , st , et ) his value function,
where et ≡ (e1t , .., eN t )0 .
If the manager is not employed by the investor, then he chooses his personal portfolio zt . His
6
budget constraint is
dW̄t = rW̄t dt + zt dRt − c̄t dt,
(8)
The manager’s optimization problem is to choose controls (c̄t , zt ) to maximize (6) subject to (8).
We denote by V̄u (W̄t , st , et ) the unemployed manager’s value function.
2.2.2
Investor’s Optimization Problem
The investor derives utility over intertemporal consumption. Utility is exponential:
Z
E
∞
− exp(−ρct − δt)dt ,
(9)
0
where ρ is the coefficient of absolute risk aversion, ct is consumption, and δ is the discount rate.
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 chooses to employ 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 − (1/2)m2t dt − dft − ct dt,
(10)
where the first term is the return from the riskless asset, 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 (9) subject to the
budget constraint (10), the fee (5), the manager’s incentive compatibility (IC) constraint
zt = zt (φ, χ, ψ),
mt = mt (φ, χ, ψ),
and the manager’s individual rationality (IR) constraint at t = 0
V̄ (W̄0 , s̄, ē) > V̄u (W̄0 , s0 , e0 ).
We denote by V (Wt , st , et ) the investor’s value function.
7
If the investor chooses not to employ the manager, then her budget constraint is
dWt = rWt dt + xηdRt − ct dt.
(11)
The investor’s optimization problem is to choose controls (ct , x) to maximize (9) subject to (11). We
denote by Vu (Wt , st , et ) the investor’s value function. The investor chooses to employ the manager
if
V (W0 , s̄, ē) > Vu (W0 , s0 , e0 ).
2.3
(12)
Equilibrium Concept
We look for equilibria in which the investor employs the manager, i.e., (12) is satisfied. 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.
(ii) Given St , the investor employs 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 stock i is an affine function of st and eit :
Sit = a0i + a1i st + a2i eit ,
(13)
where (a0i , a1i , a2i ) are constants. The conjectured value functions of the manager and the investor
8
under the equilibrium prices and when the manager is employed by the investor are
"
V̄ (W̄t , st , et ) = − exp − rρ̄W̄t + q̄0 + q̄1 st +
N
X
!#
q̄2i eit
,
(14)
,
(15)
i=1
"
V (Wt , st , et ) = − exp − rρWt + q0 + q1 st +
N
X
!#
q2i eit
i=1
respectively, where (q̄0 , q̄1 , q̄2i , q0 , q1 , q2i ) are constants.
The conjectured value functions when
the manager is not employed by the investor have the same form, but for different constants
(q̄u0 , q̄u1 , q̄u2i , qu0 , qu1 , qu2i ).
3
Equilibrium without Agency Frictions
In this section we solve for equilibrium in the absence of agency frictions. We eliminate agency
frictions by setting the manager’s private-benefit parameter B to zero. When B = 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.
Proposition 1 (Equilibrium Prices and Contract without Agency Frictions). If B = 0,
then the following form an equilibrium: the price process St given by (13) with
a0i =
κ
(a1i s̄ + a2i ēi ) ,
r
(16)
bi
a1i = q
≡ bi a1 ,
ρρ̄
(r + κ)2 + 2r ρ+ρ̄
θbσs2
(17)
1
a2i = q
;
ρρ̄
(r + κ)2 + 2r ρ+ρ̄
θi σi2
(18)
the contract (φ, χ, ψ) with φ =
ρ
ρ+ρ̄ ,
χ = 0, and ψ given by (??); and the index investment x = 0.
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
9
based only on absolute performance and not on performance relative to the index. Therefore,
the manager receives a fraction φ =
ρ
ρ+ρ̄
of the return of the supply portfolio, and the investor
receives the complementary fraction 1 − φ =
ρ̄
ρ+ρ̄ .
This coincides with the standard rule for optimal
risk-sharing under exponential utility.
The coefficient a2i measures the sensitivity of stock i’s price to changes in the stock-specific
component eit of dividends. A unit increase in eit causes stock i’s dividend flow at time t to
increase by one. In the absence of risk aversion (ρρ̄ = 0), (18) implies that the price of stock i
would increase by a2i =
1
r+κ .
This is the present value, discounted at the riskless rate r, of the
increase in stock 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 stock i’s price to changes in the common component st of dividends. We normalize a1i by bi , the sensitivity of stock i’s dividend flow to changes
in st . This yields a coefficient a1 that is common to all stocks, and that measures the sensitivity
of any given stock’s price to a unit increase in the stock’s dividend flow at time t caused by an
increase in st . In the absence of risk aversion, (17) implies that the price of stock i would increase
by a1 =
1
r+κ .
Hence, a1 and a2i would be equal: an increase in a stock’s dividend flow would have
the same effect on the stock’s price regardless of whether it comes from the common or from the
stock-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 stocks rather than a specific stock.
Eqs. (17) and (18) imply that a1 < a2i if
θbσs2
=
N
X
!
θi bi
σs2 > θi σi2 .
(19)
i=1
Eq. (19) evaluates how a unit increase in stock i’s dividend flow at time t affects the covariance
between the dividend flow of stock i and of the supply portfolio. This covariance captures the
relevant risk in our model. The left-hand side of the inequality in (19) 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 (19) holds, the change in st has a larger effect on the covariance compared to the change
in eit . Therefore, it has a larger attenuating effect on the price.
10
Eq. (19) holds when the number N of stocks exceeds a threshold, which can be zero. This
is because the left-hand side increases when stocks are added, while the right-hand side remains
constant. In the special case of scale invariance, (19) takes the intuitive form
θbs̄ > θi ēi .
(20)
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 stock i. Eq. (20) obviously holds when N is large enough.
3.1
Supply Effects
We next examine how differences in supply in the cross-section of stocks are reflected into prices
and return moments. We compare two stocks i and i0 that differ only in supply and are otherwise
identical. This amounts to determining how the price and return of stock i change when the stock’s
P
supply becomes θi0 instead of θi , holding constant aggregate quantities such as θb = N
i=j θj bj .
We perform this comparison locally, setting θi0 = θi + dθi . This amounts to computing partial
derivatives with respect to θi .
We compute return moments both for returns per share and for 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 the dollar return of a stock by its share return divided
by the unconditional mean of the share price. Thus, while expected dollar return, for example, is
the expected ratio of share return to share price, we approximate it by the ratio of expected share
return to expected share price.
Proposition 2 (Price and Expected Return). An increase in the supply θi of a stock i lowers
∂E(dRit )
it
the stock’s price ( ∂S
> 0) and expected dollar
∂θi < 0), and raises its expected share return (
∂θi
return (
dR
∂E E(S it)
it
∂θi
> 0).
A stock 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 stock. Therefore, the stock’s price must be high. The stock’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.
The effect of θi on the stock price is through the coefficient a2i , which measures the price
sensitivity to changes in the stock-specific component eit of dividends. When θi is small, an increase
11
in eit is accompanied by a small increase in the covariance between the dividend flow of the stock
and of the effective-supply portfolio. 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 (Return Volatility). An increase in the supply θi of a stock i lowers the stock’s
share return variance
it )
( ∂Var(dR
∂θi
< 0). It lowers its dollar return variance (
∂Var
dRit
E(Sit )
∂θi
< 0) if and
only if
a1 bi σs2 < a2i σi2 .
(21)
Since dividend changes have a large effect on the price of a stock that is in small supply, such
a stock has high share return volatility (square root of variance). This effect is concentrated on
the part of volatility that is driven by the stock-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
(21) holds.
Since the effect of supply on volatility is concentrated on the part that is driven by the stockspecific component, (21) should hold if that part is large enough. This can be confirmed, for
example, in the case of scale invariance. Eq. (21) becomes
a1 bi s̄ < a2i ēi ,
(22)
and has the simple interpretation that the volatility driven by the stock-specific component exceeds
the volatility driven by the common component. Indeed, the conditional variance rate driven by
the common component is a21 b2i σs2 st and that driven by the stock-specific component is a22i σi2 eit .
Taking expectations, we find the unconditional variance rates a21 b2i σs2 s̄ and a22i σi2 ēi , respectively.
Under scale invariance, the latter exceeds the former if (22) holds.
We next examine how supply affects the systematic and idiosyncratic parts of volatility. When
the number N of stocks is large, these coincide, respectively, with the parts driven by the common
12
and the stock-specific component. For small N , however, the systematic part includes volatility
driven by the stock-specific component. This is especially so if a “stock” is interpreted as an
industry-sector portfolio or investment style that is a large fraction of the market. To compute the
systematic and idiosyncratic parts, we regress the return dRit of stock i on the return dRηt ≡ ηdRt
of the index:
dRit = βi dRηt + dit .
(23)
The CAPM beta of stock i is
Cov(dRit , dRηt )
,
Var(dRηt )
βi =
(24)
and measures the systematic part of volatility. The variance Var(dit ) of the regression residual
measures the idiosyncratic part. These quantities are defined in per-share terms. Their per-dollar
counterparts are
βi$ =
Cov
dRηt
dRit
E(Sit ) , E(Sηt )
Var
and Var
dit
E(Sit )
dRηt
E(Sηt )
(25)
, where Sηt ≡ ηSt denotes the price of the index.
Proposition 4 (Beta and Idiosyncratic Volatility). An increase in the supply θi of a stock i
∂Var(dit )
i
lowers the stock’s share beta ( ∂β
< 0). It
∂θi < 0) and idiosyncratic share return variance (
∂θi
∂β $
lowers the stock’s dollar beta ( ∂θii < 0) if and only if
a21 bi ηbσs2 s̄ − 2a1 bi a2i ηi σi2 s̄ − a22i ηi σi2 ēi < 0,
(26)
and lowers the stock’s idiosyncratic dollar return variance (

a31 bi ηb(ηb − 2ηi bi )a2i σs2 σi2 s̄2 ēi − a1 bi (a1 bi σs2 − a2i σi2 ) 
d
∂Var E(Sit )
it
∂θi
N
X
< 0) if and only if

a22j ηj2 σj2 ēj  s̄ēi
j=1
− (2a1 bi s̄ + a2i ēi )a32i ηi2 σi4 ē2i > 0.
(27)
The share beta and idiosyncratic volatility are large for a stock that is in small supply because
13
of the effect identified in Propositions 2 and 3: changes to the stock-specific component of dividends
have a large effect on the price of such a stock. This yields high idiosyncratic volatility. It also
yields large beta because stock-specific shocks have a large contribution to the stock’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. We next determine which effect
dominates when the number N of stocks is large.
For large N , a stock’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 a stock that is in small supply. This can be confirmed, for example, in the
case of scale invariance and symmetric stocks. We denote by (bc , ēc , ηc , θc ) the common values of
(bi , ēi , ηi , θi ) across all stocks, by a2c the common value of a2i , and by x ≡
a2c ēc
a1 bc s̄
the ratio of volatility
driven by the stock-specific component to the volatility driven by the common component. We can
write (26) as
N − 2x − x2 < 0.
(28)
As N increases, (28) is satisfied for values of x that exceed an increasingly large threshold.
A stock’s idiosyncratic volatility, for large N , is driven mainly by the shocks specific to that
stock. 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 a stock that is in small supply. For example, in the case of scale invariance and symmetric
stocks, we can write (27) as
N (N − 2) − N x + (N − 2)x2 − x3 > 0
⇔ N − 2 − x > 0.
(29)
As N increases, (29) is satisfied for values of x that are below an increasingly large threshold.
To relate the effects of supply derived in Propositions 3 and 4 to cross-sectional market anomalies, we next determine how supply affects stocks’ CAPM alphas, i.e., the expected returns that
stocks are earning in excess of the CAPM. The CAPM alpha of stock i is
αi = E(dRit ) − βi E(dRηt ),
(30)
14
in per-share terms. Its per-dollar counterpart is
αi$
=E
dRit
E(Sit )
−
βi$ E
dRηt
E(Sηt )
=
αi
.
E(Sit )
(31)
Proposition 5 (Alpha). An increase in the supply θi of a stock i raises the stock’s share alpha
∂α$
i
i
( ∂α
∂θi > 0). It raises its dollar alpha ( ∂θi < 0) if and only if
bi s̄+ēi −
[1 − (r + κ)a1 ] ηbs̄ +
PN
a21 (ηb)2 σs2 s̄ +
j=1 [1
− (r + κ)a2j ] ηj ēj
PN
2 2 2
j=1 a2j ηj σj ēj
a21 bi ηbσs2 s̄ − 2a1 bi a2i ηi σi2 s̄ − a22i ηi σi2 ēi > 0.
(32)
Eq. (32) holds when alpha is positive. It holds for all values of alpha when, for example, stocks
share the same characteristics (bi , ēi ).
A stock in small supply has low share alpha because it has low expected share return (Proposition 2) and high share beta (Proposition 4). The effect of supply on dollar alpha is unambiguously
positive when alpha is positive: an increase in supply raises dollar alpha because it raises share
alpha and lowers the price. The effect can become ambiguous, however, when alpha is negative
because an increase in share alpha corresponds to a reduction in absolute value. When stocks share
the same characteristics (bi , ēi ) (but can differ in (ηi , θi )), an increase in supply unambiguously
raises dollar alpha for both positive and negative values.
Our results have implications for the relationship between risk and return. According to the
CAPM, stocks’ expected returns in excess of the riskless rate should be linearly related in the
cross-section to the stocks’ CAPM betas. Empirically, however, the intercept of the line is positive
instead of zero, and the slope is smaller than the theoretical one. As a consequence, high-beta
stocks have negative alphas, while low-beta stocks have positive alphas. This is the beta anomaly,
documented by Black (1972), Black, Jensen and Scholes (1972), and Frazzini and Pedersen (2013).
Related to the beta anomaly is the volatility anomaly. This is that high-volatility stocks have
negative alphas, while low-volatility stocks have positive alphas. The volatility anomaly has been
documented by Haugen and Baker (1996) and Ang et al. (2006). The latter paper considers
idiosyncratic volatility in addition to (total) volatility, and shows that it is also negatively related
to alpha.
The beta and volatility anomalies are puzzling. Yet, even more puzzling is that the negative
relationship between risk and return can arise even when alpha is replaced by expected return. That
15
is, high-risk stocks earn expected returns that are not only lower than the CAPM benchmarks, but
can also be lower than the expected returns of low-risk stocks. Ang et al. (2006) document this
effect both when risk is measured by volatility and when it is measured by idiosyncratic volatility.
The same effect has also been shown for beta in some cases (e.g., Jylha 2014).
The results in this section suggest a mechanism that could help explain the anomalies, even in
the absence of agency frictions. A negative relationship between alpha or expected return on one
hand, and volatility or beta on the other, can be generated by the way that these variables depend on
supply. Stocks in small supply earn low expected dollar return (Proposition 2) and negative alpha
(Proposition 5). Under some conditions, they also have high dollar return volatility (Proposition
3), high idiosyncratic dollar return volatility (Proposition 4), and high dollar beta (Proposition
4). Under these conditions our model can explain both the negative relationship between risk and
alpha, as well as the more puzzling negative relationship between risk and expected return.
The conditions for small supply to be associated with high risk are the least restrictive in the
case of idiosyncratic volatility. When N is large, as is the case when our model is applied to stocks,
stocks in small supply have high idiosyncratic volatility for almost all of the parameter region.
Hence, our model could help explain the idiosyncratic volatility anomaly of Ang et al. (2006).
Explaining the volatility anomaly within our model requires that idiosyncratic volatility exceeds
systematic volatility. This condition is plausible when our model is applied to stocks.
The beta anomaly is the hardest to explain within our model. When N is large, stocks in small
supply have small beta for almost all of the parameter region. Hence, our model could explain the
beta anomaly only when N is small, in which case stocks must be interpreted as market segments
such as industry-sector portfolios or investment styles. Given that the anomaly holds for individual
stocks, their dividends must be having a significant segment-specific component (not present in our
model) and so must supply. The mechanism suggested by our model could be relevant in such cases.
One could argue, for example, that during the 1998-2000 run-up of technology stocks, demand by
some investors for that segment of the market pushed up prices and lowered expected returns, while
also introducing a significant segment-specific component of volatility that raised beta.
4
Equilibrium with Agency Frictions
16
In this section we characterize the equilibrium in the presence of agency frictions, i.e., positive
private benefits for the fund manager.
Proposition 6 (Equilibrium Prices and Contract). If B 6= 0, then the price process St given
by (13) and the contract (φ, χ, ψ) form an equilibrium if
a0i =
κ
(a1i s̄ + a2i ēi ) ,
r
(33)
bi
a1i = p
≡ bi a1 ,
(r + κ)2 + 2rρ̄φ(θ − χη)bσs2
(34)
1
,
a2i = q
(r + κ)2 + 2rρ̄φ(θi − χηi )σi2
(35)
(φ, χ) solve the system of equations:
"
#
N
X
(1 − φ) 2
ēi θi (a2i − ǎ2i ) = 0,
B + (r + κ) s̄θb (a1 − ǎ1 ) +
φ2
(36)
i=1
s̄ηb (a1 − ǎ1 ) +
N
X
ēi ηi (a2i − ǎ2i ) = 0,
(37)
i=1
where
1
ǎ1 = p
,
2
(r + κ) + 2rρ[θ − φ(θ − χη)]bσs2
(38)
1
ǎ2i = q
,
(r + κ)2 + 2rρ[θi − φ(θi − χηi )]σi2
(39)
and ψ is given by (A.56).
4.1
Optimal Contract
The optimal level of the shirking action that the manager undertakes is
mt = B/φ,
(40)
and is positive when the private benefit B is positive. Making the fee more sensitive to the fund’s
performance, i.e., increasing φ, lowers mt . At the same time, the manager becomes over-exposed to
risk, and hence invests too little in stocks from the investor’s viewpoint. The investor can reduce
17
the manager’s risk exposure by compensating the manager based on his performance relative to
the index, i.e., setting χ > 0. This induces the manager to increase his investment in stocks. At
the same time, the additional investment is according to the index weights.
In the special case where θ and η are collinear, the first best can be implemented by making the
manager residual claimant to the fund’s return, i.e., φ = 1, and adjusting appropriately the index
part of the compensation.
Corollary 1 (Optimal Risk Sharing with Benchmarking). If θi = kηi for i = 1, .., N and a
scalar k, then the optimal contract entails
φ = 1,
χ=
(41)
k ρ̄
,
ρ + ρ̄
(42)
and the agency friction does not affect prices and risk-sharing.
In the general case where θ and η may not be collinear, the system of equations (36) and (37)
characterizing the optimal contract can be written as:
(1 − φ) B 2
+ θS̄ − θS̄ˇ = 0,
φ2
r
(43)
η S̄ − η S̄ˇ = 0,
(44)
(r + κ)
(ǎ1 bi s̄ + ǎ2i ēi )
S̄ˇi =
r
(45)
where
is the hypothetical price of stock i if the investor were marginal in pricing stocks. Eq. (44) shows that
the investor’s valuation of the index η coincides with the market valuation, and hence the investor
and the manager share optimally index risk. This is because by changing the index component of
the fee, the investor can induce a reallocation of the index between her and the manager. Eq. (43)
also shows that the investor’s valuation of the fund’s portfolio θ is larger than that of the manager.
This is because the optimal contract subjects the manager to a disproportionate share of fund risk
to improve incentives.
Figure 1 plots the parameters φ and χ of the optimal contract as a function of the private
benefit for the numerical example of Section 3. As the agency friction, i.e., the private benefit,
18
Figure 1: Equilibrium Contract with Agency Frictions
The sensitivity of the manager’s fee to the fund’s performance (φ) and to the index performance (χ) as a
function of the manager’s private benefit. There are two groups of stocks, with two stocks in each group.
Stocks within each group have identical characteristics, and the only difference across groups is effective
supply. Parameter values are: ρ = 1, ρ̄ = 50, N = 4, ηi = 1, θ1 = θ2 = 0.8, θ3 = θ4 = 0.2, bi = 1, σi = 1.5,
σs = 1, κ = 0.1, ēi = s̄ = 1, r = 0.04, for i = 1, .., N .
BENCHMARKING c
ABSOLUTE PERFORMANCE f
0.10
0.20
0.08
0.15
0.06
0.10
0.04
0.05
0.02
0.00
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.00
0.00
0.10
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
increases, the manager’s fee becomes more sensitive to the fund’s performance and to the index
performance, i.e., φ and χ increase.
4.2
Asset Pricing Implications
The pricing coefficients in (33), (34) and (35) preserve the intuitive structure of their counterparts
in the benchmark case without frictions. Perhaps not surprisingly, the main difference is in the
risk adjustment. Indeed, in the presence of benchmarking, not only the fund portfolio θ but also
the index portfolio η matters for pricing purposes. In particular, the risk adjustment in prices is
determined by the tracking portfolio φ(θ − χη). Agency frictions give rise to amplification and,
cross-sectionally, affect risky assets in an asymmetric fashion.
In what follows, we discuss in details the underlying mechanism and the economic intuitions
behind the equilibrium pricing. To this purpose, we consider some numerical examples by assuming
an economy characterized by four symmetric risky assets, two of which are in large and two in small
effective supply. Figure 2 plots several equilibrium quantities against the severity of the agency
friction, that is the amount of private benefits B. The blue line represents one of the stocks in large
effective supply, the red line one of the stocks in small supply.
19
Asset Price Levels
Recall that, absent agency frictions, stocks in large effective supply (high θi )
are cheap, that is their equilibrium price is lower than the price of stocks in small effective supply.
Prices are low because the manager requires a high risk premium to absorb the large supply. The
opposite occurs when θi is low and, in this case, stocks are expensive.
What happens in the presence of moral-hazard? Interestingly, the dispersion in asset prices
increases: expensive stocks become even more expensive, whereas stocks that are cheap are traded
at an even lower price. The intuition for this result is as follows. In the presence of agency frictions,
benchmarking is part of the optimal contract. The manager, therefore, has an incentive to reduce
the deviations from the benchmark by selecting a portfolio that is more in line with the index, thus
reducing the risk of the tracking portfolio. Note that, without frictions, stocks in large residual
supply are overweighed in the fund portfolio compared to the index, whereas stocks in small residual
supply are underweighted. This is precisely how the fund manager can provide a superior portfolio
to the investor. As a consequence, a fund portfolio closer to the benchmark is achieved by, (i)
reducing the demand for assets in large residual supply, and (ii) increasing the demand for those in
small residual supply. In equilibrium, this reduces the price pressure in the stocks available to the
manager in large supply, and increases the price pressure in the stocks available in small supply.
The former become cheaper, the latter more expensive. Hence, the supply effects on price levels
are amplified by the agency friction. The first plot in Panel A of Figure 2 provides a graphical
representation of this result.
The effect on expected returns follows naturally from the one on asset prices. The agency
friction increases the expected returns on high-θ stocks, and it decreases the expected returns on
low-θ stocks, as shown in the thirds graph in Panel A of Figure 2.
Asset Volatility
To illustrate the amplification effect that moral-hazard has on asset volatility,
consider the following example. Suppose a stock in small effective supply receives a positive shock
to its dividend flow. Higher dividends implies an increase in dividend volatility, which in turn
makes the fund manager, who is subject to benchmarking, more eager to buy the stock in small
supply, in order to reduce the risk of deviating from the index. A higher demand reduces the asset
risk premium and amplifies the price increase initially induced by the positive dividend shock.
Suppose now that the positive shock affects a stock in large residual supply. As in the case without
frictions, the increase in dividend volatility induces a large increase in the risk premium that strongly
attenuates the positive price effect of the shock. In addition, in the presence of benchmarking, a
higher dividend volatility makes the fund manager more eager to sell the stock in large supply,
again to be closer to the index. This raises the stock risk premium and hence reduces the price
increase even further. Therefore, agency frictions amplify also the supply effects on asset return
20
Figure 2: Pricing Implications of Agency Frictions: Assets
Equilibrium quantities as a function of the manager’s private benefit. Panel A: average asset prices S̄i , asset
return volatilities per share invested Var(dRit ) and asset expected returns per dollar invested E(dRit /S̄i ).
Panel B: asset return volatilities per dollar invested Var(dRit /S̄i ), dollar beta βi$ and the dollar alpha αi$ .
There are two groups of stocks, with two stocks in each group. Stocks within each group have identical
characteristics, and the only difference across groups is effective supply. Parameter values are: ρ = 1,
ρ̄ = 50, N = 4, ηi = 1, θ1 = θ2 = 0.8, θ3 = θ4 = 0.2, bi = 1, σi = 1.5, σs = 1, κ = 0.1, ēi = s̄ = 1, r = 0.04,
for i = 1, .., N .
Panel A
ASSET RETURN VOLATILITIES HshareL
ASSET PRICES
EXPECTED ASSET RETURNS Hdollar, %L
16
20
60
14
15
12
50
10
10
40
8
6
30
5
4
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
Panel B
ASSET RETURN VOLATILITIES Hdollar, %L
CAPM-BETA HdollarL
30
CAPM-ALPHA Hdollar, %L
1.2
10
1.0
25
5
0.8
20
0.6
0
0.4
15
-5
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
volatility: expensive assets becomes more volatile, cheap assets less volatile, as illustrated in the
second graph in Panel A of Figure 2.
Anomalies
Since the supply effects on the level and volatility of asset prices are exacerbated by
the agency friction, it may not come as a surprise that moral-hazard also amplifies the asset pricing
anomalies discussed in the previous section. The three graphs in Panel B of Figure 2 highlight
21
these results. The negative cross-sectional correlation between alpha on one hand, and volatility
or beta on the other becomes stronger as the friction gets worse. As prices and volatility of an
expensive stock (low θi ) increase, the stock becomes a bigger part of the index, contributing to a
large fraction of its variation. This raises the correlation with the index and hence the asset beta.
The opposite holds true for cheap stocks (high θi ).
Aggregate Market
The amplification of supply effects induced by the agency friction has an
impact on the aggregate market. This is the case because the amplification mechanisms, discussed
above, affect stocks in large and in small effective supply asymmetrically. The asymmetry can be
seen from the graphs in Figure 2.
The intuition for the asymmetry is as follows. Because the optimal contract involves benchmarking, the fund manager becomes less willing to deviate from the market index. Because stocks
in small effective supply are more volatile and account for larger fraction of the market movement
than stocks available in large effective supply, deviating from the index by underweighting the former is more risky than by overweighting the latter. Therefore, the fund manager is more willing
to deviate from the index by overweighting high-θ stocks in order to capture some of the “undervaluation”, but less willing to underweighting the expensive stocks to exploit the “over-valuation”.
In other words, the amplification effect is asymmetric because the fund manager does not deviate
from the index in a symmetric fashion across stocks. In the aggregate, therefore, the market index
goes up.
The asymmetry in the supply effects also contributes to the increase in market volatility. However, this can reverse for large levels of systematic risk relative to idiosyncratic risk, when the
agency friction is sufficiently low. The effects of the agency friction on the price level of the index,
on its expected return, and on its volatility, are plotted in the three graphs in Figure 3.
5
Normative Analysis
One of the advantages of analyzing optimal contracts in general equilibrium is that we can address
public policy questions. Are privately optimal contracts socially optimal? Do they generate excessive mispricings or volatility? Is there scope for regulation? In what follows we set up and solve
the problem of a regulator in our economy.
22
Figure 3: Pricing Implications of Agency Frictions: Aggregate Market
Average market prices S̄η , expected returns on the market per dollar invested E(dRmt /S̄η ) and market return
volatility per dollar invested Var(dRmt /S̄η ), as a function of the manager’s private benefit. There are two
groups of stocks, with two stocks in each group. Stocks within each group have identical characteristics, and
the only difference across groups is effective supply. Parameter values are: ρ = 1, ρ̄ = 50, N = 4, ηi = 1,
θ1 = θ2 = 0.8, θ3 = θ4 = 0.2, bi = 1, σi = 1.5, σs = 1, κ = 0.1, ēi = s̄ = 1, r = 0.04, for i = 1, .., N .
MARKET INDEX EXPECTED RETURN Hdollar, %L
MARKET INDEX PRICE
MARKET INDEX RETURN VOLATILITY Hdollar, %L
183.0
7.05
16
182.5
15
7.00
182.0
14
181.5
6.95
13
181.0
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
Regulator’s problem
0.08
0.10
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
12
0.00
SEVERITY OF AGENCY FRICTION HBL
0.02
0.04
0.06
0.08
0.10
A regulator chooses a contract, within the same contract space available
to the investor, to maximize the sum of the certainty equivalents of the investor and the manager:
1
(φ∗ , χ∗ ) ∈ arg max θS̄ +
rρ
q0 + q1 s̄ +
N
X
!
q2i ēi
i=1
1
+
rρ̄
q̄0 + q̄1 s̄ +
N
X
!
q̄2i ēi
(46)
i=1
This objective is the same as the investor’s. The difference between the regulator’s and the investor’s problems is that the regulator takes into account any effects that the contract may have
on equilibrium asset prices.
Proposition 7 (Socially Optimal Contract). The socially optimal contract (φ∗ , χ∗ ) solves the
following system of equations:
"
#
N
X
(1 − φ) 2
B + (r + κ) s̄θb (a1 − ǎ1 ) +
ēi θi (a2i − ǎ2i ) = 0,
φ3
(47)
i=1
s̄ηb (a1 − ǎ1 ) +
N
X
ēi ηi (a2i − ǎ2i ) = 0,
i=1
where where ǎ1 and ǎ2i are defined in (38) and (39), respectively.
23
(48)
In the presence of moral hazard, the socially optimal contract differs from the privately optimal
one.
Corollary 2 (Inefficiency). In the presence of agency frictions (B > 0), the privately optimal
contract is not socially optimal.
6
Concluding Remarks
In a general equilibrium model of delegated asset management, where optimal delegation contracts
and asset prices are determined jointly, we provide novel economy mechanisms that help rationalizing the long-standing volatility and beta anomalies. The presence of moral-hazard induces
amplification effects that exacerbate these asset pricing anomalies, and affect the cross-section of
asset returns asymmetrically. Our results also provide new insights on how agency frictions in the
asset management industry can impact the aggregate market and be the driving force for asset
bubbles. Optimal benchmarking plays a central role in the underlying amplification mechanism.
We also highlight a source of inefficiency. Privately optimal contracts are not socially optimal, as
the investors fails to internalize the feedback effect between delegation contracts and asset prices.
This opens the door for public-policy analysis of asset management contracts.
24
Appendix A: Proofs
Proof of Proposition 1. The proposition follows from Proposition 6 for the special case B = 0.
Proof of Proposition 2. Substituting a0i from (16), we can write the price (13) of stock i as
Sit = a1 bi
κ
κ
s̄ + st + a2i
ēi + eit .
r
r
(A.1)
Eq. (17) implies that a1 does not depend on θi , holding the aggregate quantity θb constant. Eq. (18) implies
that a2i decreases in θi . Therefore, (A.1) implies that Sit decreases in θi .
Substituting a0i from (16), we can write the share return (A.16) of stock i as
√
√
dRit = [1 − a1 (r + κ)]bi st + [1 − a2i (r + κ)]eit dt + a1 bi σs st dwst + a2i σi eit dwit .
(A.2)
The expected share return is
E(dRit ) = [1 − a1 (r + κ)]bi s̄ + [1 − a2i (r + κ)]ēi dt,
(A.3)
and increases in θi because a2i decreases in θi .
Eqs. (17) and (18) imply that the terms in square brackets in (A.3) are positive and so is E(dRit ). Since
dRit
E(dRit ) increases in θi , and Sit decreases in θi , the expected dollar return E E(S
of stock i increases in
)
it
θi .
Proof of Proposition 3. Eq. (A.2) implies that the share return variance of stock i is
2
Var(dRit ) = E (dRit )2 − [E(dRit )]
= E (dRit )2
=E
a21 b2i σs2 st + a22i σi2 eit dt
= a21 b2i σs2 s̄ + a22i σi2 ēi dt,
(A.4)
2
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 decreases in θi , (A.4) implies that Var(dRit ) decreases in θi .
Eq. (A.1) implies that the expected share price of stock i is
E(Sit ) =
κ+r
(a1 bi s̄ + a2i ēi ) .
r
(A.5)
25
Therefore, the dollar return variance of stock i is
Var
dRit
E(Sit )
=
a21 b2i σs2 s̄ + a22i σi2 ēi
dt.
2
(κ+r)2
(a1 bi s̄ + a2i ēi )
r2
Differentiating with respect to a2i , we find that Var
(A.6)
dRit
E(Sit )
increases in a2i and hence decreases in θi if and
only if (21) holds.
Proof of Proposition 4. Eq. (A.2) implies that the share return of the index is
dRηt = [1 − a1 (r + κ)]ηbst +
N
X
√
[1 − a2i (r + κ)]ηj ejt dt + a1 ηbσs st dwst +
N
X
√
a2j ηj σj ejt dwjt . (A.7)
j=1
j=1
Eqs. (24), (A.2) and (A.7) imply that the share beta of stock i is
βi =
Cov(dRit , dRηt )
E (dRit , dRηt )
a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi
=
=
,
PN
Var(dRηt )
E [(dRηt )2 ]
a21 (ηb)2 σs2 s̄ + j=1 a22j ηj2 σj2 ēj
and decreases in θi holding aggregate quantities such as
PN
j=1
(A.8)
a22j ηj2 σj2 ēj constant because a2i decreases in
θi .
Eq. (A.5) implies that the expected share price of the index is


N
X
κ+r 
a1 ηbs̄ +
a2j ηj ēj  .
E(ηSt ) =
r
j=1
(A.9)
Eqs. (25), (A.5), (A.8), and (A.9) imply that the dollar beta of stock i is
βi$
PN
a1 ηbs̄ + j=1 a2j ηj ēj
a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi
=
.
PN
a1 bi s̄ + a2i ēi
a21 (ηb)2 σs2 s̄ + j=1 a22j ηj2 σj2 ēj
The right-hand side of (A.10) increases in a2i and hence decreases in θi if and only if the function
a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi
a1 bi s̄ + a2i ēi
increases in a2i . This happens if and only if (26) holds.
26
(A.10)
Eqs. (A.4), (A.7), and (A.8) imply that the idiosyncratic share return variance of stock i is
Var(dit ) = Var(dRit ) − βi2 Var(dRηt )
"
=
a21 b2i σs2 s̄
+
a22i σi2 ēi
2 #
a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi
dt.
−
PN
a21 (ηb)2 σs2 s̄ + j=1 a22j ηj2 σj2 ēj
(A.11)
Since a2i decreases in θi , Var(dit ) decreases in θi holding aggregate quantities such as
PN
j=1
a22j ηj2 σj2 ēj
constant if and only if

I(a2i ) ≡ (a21 b2i σs2 s̄ + a22i σi2 ēi ) a21 (ηb)2 σs2 s̄ +
N
X

a22j ηj2 σj2 ēj  − a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi
2
j=1
increases in a2i holding the same quantities constant. The latter happens if and only if


N
X
∂I(a2i )
= σi2 ēi a21 (ηb)2 σs2 s̄ +
a22j ηj2 σj2 ēj − 2ηi a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi  > 0.
∂a22i
j=1
(A.12)
Since asset i0 6= i satisfies (bi0 , σi0 , ēi0 , ηi0 , θi0 ) = (bi , σi , ēi , ηi , θi + dθi ), the following inequalities hold:
(ηb)2 ≥ (ηi bi + ηi0 bi0 )(ηb) = 2ηi bi ηb,
N
X
a22j ηj2 σj2 ēj ≥ a22i ηi2 σi2 ēi + a22i0 ηi20 σi20 ēi0 = 2ηi a22i ηi σi2 ēi ,
j=1
implying that (A.12) also holds.
Eqs. (A.5) and (A.11) imply that the idiosyncratic dollar return variance of stock i is
Var
dit
E(Sit )
a21 b2i σs2 s̄ + a22i σi2 ēi −
=
(κ+r)2
r2
2
ηi σi2 ēi )
(a21 bi ηbσs2 s̄+a
P 2i
a21 (ηb)2 σs2 s̄+
(a1 bi s̄ + a2i ēi )
N
j=1
2
a22j ηj2 σj2 ēj
2
dt,
(A.13)
and decreases in θi holding aggregate quantities constant if and only if
I1 (a2i ) ≡
h
i
2
PN
(a21 b2i σs2 s̄ + a22i σi2 ēi ) a21 (ηb)2 σs2 s̄ + j=1 a22j ηj2 σj2 ēj − a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi
2
(a1 bi s̄ + a2i ēi )
increases in a2i holding the same quantities constant. The latter happens if and only if (27) holds, as can
be seen by computing the partial derivative of I1 (a2i ) with respect to a2i .
Proof of Proposition 5. Since the expected share return of stock i increases in θi and the stock’s share
beta decreases in θi , (30) implies that the stock’s share alpha increases in θi . Eqs. (31), (A.3), and (A.10)
27
imply that the dollar alpha of stock i is
αi$
PN
[1 − a1 (r + κ)]ηbs̄ + j=1 [1 − a2j (r + κ)]ēj
bi s̄ + ēi
a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi
=
− (r + κ) −
.
PN
a1 bi s̄ + a2i ēi
a1 bi s̄ + a2i ēi
a21 (ηb)2 σs2 s̄ + j=1 a22j ηj2 σj2 ēj
(A.14)
Differentiating with respect to a2i holding aggregate quantities constant, we find that αi$ decreases in a2i
and hence increases in θi if and only if (32) holds. When αi$ > 0, the left-hand side of (32) is larger than
(r + κ)(a1 bi s̄ + a2i ēi ) +
[1 − (r + κ)a1 ] ηbs̄ +
PN
j=1
[1 − (r + κ)a2j ] ηj ēj
PN
a21 (ηb)2 σs2 s̄ +
j=1
a22j ηj2 σi2 ēi
× a21 bi ηbσs2 s̄ + a22i ηi σi2 ēi − a21 bi ηbσs2 s̄ − 2a1 bi a2i ηi σi2 s̄ − a22i ηi σi2 ēi
(r + κ)(a1 bi s̄ + a2i ēi ) + 2
PN
[1 − (r + κ)a1 ] ηbs̄ +
[1 − (r + κ)a2j ] ηj ēj 2
2a2i ηi σi2 ēi + a1 bi a2i ηi σi2 s̄ > 0,
2 2 2
j=1 a2j ηj σi ēi
j=1
a21 (ηb)2 σs2 s̄ +
PN
and hence (32) holds. When (bi , ēi ) = (bc , ēc ) for all i, (32) becomes
bc s̄+ ēc −
[1 − (r + κ)a1 ] bc η1s̄ +
a21 b2c (η1)2 σs2 s̄
PN
j=1
+
[1 − (r + κ)a2j ] ηj ēc
PN
2 2 2
j=1 a2j ηj σj ēc
a21 b2c η1σs2 s̄ − 2a1 bc a2i ηi σi2 s̄ − a22i ηi σi2 ēc > 0,
(A.15)
where 1 ≡ (1, .., 1)0 . Eq. (A.15) holds under the sufficient condition
bc s̄ + ēc −
⇔bc s̄ + ēc −
bc η1s̄ +
a21 b2c (η1)2 σs2 s̄
PN
+
j=1
PN
ηj ēc
2 2 2
j=1 a2j ηj σj ēc
a21 b2c η1σs2 s̄ > 0
bc η1s̄ + η1ēc
a21 b2c η1σs2 s̄ > 0,
PN
+ j=1 a22j ηj2 σj2 ēc
a21 b2c (η1)2 σs2 s̄
which holds.
Proof of Proposition 6. We proceed with the following steps:
• Step 1: We fix a contract (φ, χ, ψ), and for the conjectured pricing function and fund manager’s value
function in (13) and (14) respectively, we solve for the coefficients (a0i , a1i , a2i , q̄0 , q̄1 , q̄2i ) for i = 1, .., N
that form an equilibrium (i.e., that satisfy condition (i) of Definition 1).
• Step 2: For a given contract (φ, χ, ψ) and equilibrium asset prices, we solve for the coefficients
(q0 , q1 , q2i ) for i = 1, .., N that characterize the conjectured investor’s value function in (15).
• Step 3: Given the value functions obtained and the equilibrium asset prices, we solve for the optimal
contract (i.e., that satisfy condition (ii) of Definition 1).
28
The excess return dRit defined in (2), given the pricing function in (13) becomes equal to
dRit = (a1i κs̄ + a2i κēi − a0i r) + [bi − a1i (r + κ)]st + [1 − a2i (r + κ)]eit dt
√
√
+ a1i σs st dwst + a2i σi eit dwit .
(A.16)
Step 1. The Bellman equation for the manager’s optimization problem is given by
max
c̄t ,zt ,mt
− exp(−ρ̄c̄t ) + DV̄t − δ̄ V̄t = 0,
(A.17)
PN
where DV̄t is the drift of the diffusion process dV̄t . Define J¯t ≡ (rρ̄W̄t + q̄0 + q̄1 st + i q̄2i eit ). Given the
conjectured value function in (14), it follows that
1 ¯ 2
¯
dV̄t = −V̄t dJt − (dJt ) ,
2
(A.18)
where the diffusion process dJ¯t can be expressed in terms of (Ḡt , H̄t , K̄it ) as
dJ¯t = Ḡt dt + H̄t dwst +
N
X
K̄it dwit .
(A.19)
i
We also conjecture (and later verify) that the pricing coefficient a1i take the form: a1i = bi a1 , where a1 is
constant and not asset-dependent. Using Ito’s lemma:
N
X
Et (dRt ) φ 2
Ḡt ≡ rρ̄ rW̄t − c̄t + φ(zt − χη)
− mt + Bmt + ψ + q̄1 κ(s̄ − st ) +
q̄2j κ(ēj − ejt ),
dt
2
j
√
H̄t ≡ q̄1 + rρ̄φ(zt − χη)ba1 σs st ,
√
K̄it ≡ q̄2i + rρ̄φ(zit − χηi )a2i σi eit .
The Bellman equation in (A.17) can be written as
"
max
c̄t ,mt zt
N
− exp(−ρ̄c̄t ) − V̄t
1X 2
1
K̄it
Ḡt − H̄t2 −
2
2 i
!
#
− δ̄ V̄t = 0.
(A.20)
The FOC with respect to c̄t is
ρ̄ exp(−ρ̄c̄t ) + rρ̄V̄t = 0
⇒
1
c̄t = rW̄t +
ρ̄
q̄0 − ln r + q̄1 st +
N
X
i
29
!
q̄2i eit
.
(A.21)
The FOC with respect to mt is
B − φmt = 0
⇒
mt =
B
.
φ
(A.22)
The FOC with respect to zit is
rρ̄φ
Et (dRit )
− rρ̄φbi a1 q̄1 + rρ̄φ(zt − χη)ba1 σs2 st − rρ̄φa2i q̄2i + rρ̄φ(zit − χηi )a2i σi2 eit = 0. (A.23)
dt
Substituting (??) into (A.23) and imposing market clearing zt = θ we obtain
(bi a1 κs̄ + a2i κēi − ra0i ) + bi [1 − a1 (r + κ)]st + [1 − a2i (r + κ)]eit
−bi a1 [q̄1 + rρ̄φ(θ − χη)ba1 ]σs2 st − a2i [q̄2i + rρ̄φ(θi − χηi )a2i ]σi2 eit = 0.
(A.24)
The LHS of (A.24) is affine in st and eit . Collecting terms in st and eit , we can write (A.24) as
A0i + A1 bi st + A2i eit = 0,
(A.25)
A0i ≡ κ(a1 bi s̄ + a2i ēi ) − ra0i ,
(A.26)
A1 ≡ 1 − a1 (r + κ) − a1 [q̄1 + rρ̄φ(θ − χη)ba1 ] σs2 ,
(A.27)
A2i ≡ 1 − a2i (r + κ) − a2i [q̄2i + rρ̄φ(θi − χηi )a2i ] σi2 .
(A.28)
where
Substituting the three FOCs in (A.21), (A.22), and (A.24) into (A.20), the fund manager’s Bellman equation
can be written as affine in st and eit ,
Q̄0 + Q̄1 st +
N
X
Q̄2i eit = 0,
(A.29)
i
where
Q̄0 ≡ δ̄ − r + r ln r + rρ̄
!
N
X
B2
+ ψ − rq̄0 + κ q̄1 s̄ +
q̄2i ēi ,
2φ
i
(A.30)
1
1
2
Q̄1 ≡ −q̄1 (r + κ) − q̄12 σs2 + [rρ̄φ(θ − χη)ba1 ] σs2 ,
2
2
(A.31)
1 2 2 1
2
σi + [rρ̄φ(θi − χηi )a2i ] σi2 .
Q̄2i ≡ −q̄2i (r + κ) − q̄2i
2
2
(A.32)
Asset prices are such zt = θ, and the value function in (14) satisfies the Bellman equation in (A.17) for those
30
prices, if the scalars (q̄0 , q̄1 , {a0i , a1 , a2i , q̄2i }i ) solve the system of equations
{A0i = 0, A1 = 0, A2i = 0, Q̄0 = 0, Q̄1 = 0, Q̄2i = 0}i ,
for i = 1, .., N . The equilibrium is obtained as follows. Solving the quadratic equation Q̄1 = 0 for q̄1 (and
considering the positive root) we obtain
q̄1 = −
(r + κ) −
q
2
(r + κ)2 + [rρ̄φ(θ − χη)ba1 ] σs4
σs2
.
(A.33)
Substituting (A.33) into A1 = 0 we obtain
1 − a21 rρ̄φ(θ − χη)bσs2 = a1
q
2
(r + κ)2 + [rρ̄φ(θ − χη)ba1 ] σs4 .
(A.34)
Squaring both terms and simplifying yields
1 − a21 (r + κ)2 + 2rρ̄φ(θ − χη)bσs2 = 0,
(A.35)
which can be solved explicitly for a1 and gives
a1 = p
1
(r +
κ)2
(A.36)
+ 2rρ̄φ(θ − χη)bσs2
Therefore, we just verified that a1 is the same for any asset i. Equation (34) obtains. Substituting (A.36)
into (A.33) yields
q̄1 = − p
rρ̄φ(θ − χη)b
(r + κ)2 + 2rρ̄φ(θ − χη)bσs2
−
(r + κ) −
p
(r + κ)2 + 2rρ̄φ(θ − χη)bσs2
.
σs2
(A.37)
Solving the quadratic equation Q̄2i = 0 for q̄2i (and considering the positive root) we obtain
q̄2i = −
(r + κ) −
q
2
(r + κ)2 + [rρ̄φ(θi − χηi )a2i ] σi4
σi2
.
(A.38)
Substituting (A.38) into A2i = 0 we obtain
1 − a22i rρ̄φ(θi − χηi )σi2 = a2i
q
2
(r + κ)2 + [rρ̄φ(θi − χηi )a2i ] σi4 .
(A.39)
Squaring both terms and simplifying yields
1 − a22i (r + κ)2 + 2rρ̄φ(θi − χηi )σi2 = 0,
(A.40)
31
which can be solved for a2i , and gives (35). Substituting (35) into (A.38) yields
q̄2i
= −p
rρ̄φ(θi − χηi )
(r + κ)2 + 2rρ̄φ(θi − χηi )σi2
−
(r + κ) −
p
(r + κ)2 + 2rρ̄φ(θi − χηi )σi2
,
σi2
(A.41)
for i = 1, .., N . The coefficient a0i of the pricing function in (33) is obtained by substituting (34) and (35)
into the linear equation A0i = 0. The coefficient q̄0 of the fund manager’s value function is obtained by
substituting (A.37) and (A.41) into the linear equation Q̄0 = 0:
q̄0 =
2
B
κ
δ̄
+ ln r − 1 + ρ̄
+ψ +
r
2φ
r
q̄1 s̄ +
N
X
!
q̄2i ēi
.
(A.42)
i
Step 2. The Bellman equation for the investor’s optimization problem is given by
max [− exp(−ρct ) + DVt − δVt ] = 0,
(A.43)
ct
where DVt is the drift of the diffusion process dVt . Define Jt ≡ (rρWt + q0 + q1 st +
PN
i
q2i eit ). Given the
conjectured value function in (15), it follows that
1
dVt = −Vt dJt − (dJt )2 ,
2
(A.44)
where the diffusion process dJt can be expressed in terms of (Gt , Ht , Kit ) as
dJt = Gt dt + Ht dwst +
N
X
Kit dwit .
(A.45)
i
Using Ito’s lemma:
Et (dRt ) (1 − φ) 2
Gt ≡ rρ rWt − ct + [zt − φ(zt − χη)]
−
mt − ψ
dt
2
+ q1 κ(s̄ − st ) +
N
X
q2j κ(ēj − ejt ),
j
√
Ht ≡ {q1 + rρ[zt − φ(zt − χη)]ba1 } σs st ,
√
Kit ≡ {q2i + rρ[zit − φ(zit − χηi )]a2i } σi eit .
The Bellman equation in (A.43) can be written as
"
max
ct
− exp(−ρct ) − Vt
N
1
1X 2
Gt − Ht2 −
Kit
2
2 i
!
32
#
− δVt = 0.
(A.46)
The FOC with respect to ct is
ρ exp(−ρct ) + rρVt = 0
⇒
1
ct = rWt +
ρ
q0 − ln r + q1 st +
N
X
!
q2i eit
.
(A.47)
i
Substituting (A.47) into (A.46), imposing market clearing zt = θ, and considering the FOCs of the fund
manager in (A.22) and (A.23), the investor’s Bellman equation can be written as affine in st and eit ,
Q0 + Q1 st +
N
X
Q2i eit = 0,
(A.48)
i
where
Q0 ≡ (δ − r + r ln r) − rρ
!
N
X
1−φ 2
B + ψ − rq0 + κ q1 s̄ +
q2i ēi ,
2φ2
i
1
2
Q1 ≡ −q1 (r + κ) − q12 σs2 + rρ [θ − φ(θ − χη)] ba1 {rρ̄φ(θ−χη)ba1 − rρ
2 [θ−φ(θ−χη)]ba1 +(q̄1 −q1 )} σs ,
2
(A.49)
(A.50)
1 2 2
2
Q2i ≡ −q2i (r + κ) − q2i
σi + rρ [θi − φ(θi − χηi )] a2i {rρ̄φ(θi −χηi )a2i − rρ
2 [θi −φ(θi −χηi )]a2i +(q̄2i −q2i )} σi .
2
(A.51)
The value function in (15) satisfies the Bellman equation in (A.43) for the equilibrium asset prices, if
the scalars (q0 , q1 , {q2i }i ) solve the system of equations
{Q0 = 0, Q1 = 0, Q2i = 0}i ,
for i = 1, .., N . Substituting (34), (35), (A.37), and (A.41) into (A.50) and (A.51), we can explicitly solve
for q1 and q2i , obtaining
p
(r + κ)2 + 2rρ[θ − φ(θ − χη)]bσs2
q1 = − p
−
,
σs2
(r + κ)2 + 2rρ̄φ(θ − χη)bσs2
p
(r + κ) − (r + κ)2 + 2rρ[θi − φ(θi − χηi )]σi2
rρ[θi − φ(θi − χηi )]
q2i = − p
−
,
σi2
(r + κ)2 + 2rρ̄φ(θi − χηi )σi2
rρ[θ − φ(θ − χη)]b
(r + κ) −
(A.52)
(A.53)
for i = 1, .., N , respectively. The coefficient q0 is obtained by substituting (A.52) and (A.53) into the linear
equation Q0 = 0:
q0 =
δ
1−φ 2
κ
+ ln r − 1 − ρ
B
+
ψ
+
r
2φ2
r
q1 s̄ +
N
X
i
33
!
q2i ēi
,
(A.54)
Step 3. The flat fee ψ is pinned down by the fund manager’s participation constraint:
"
− exp − rρ̄W̄0 + q̄0 + q̄1 s̄ +
N
X
!#
= − exp − rρ̄W̄0 + Π .
q̄2i ēi
(A.55)
i=1
Substituting (A.42) into (A.55), and solving for ψ, we obtain
1
ψ=
ρ̄
"
#
N
X
δ̄
r+κ
B2
Π − − ln r + 1 −
q̄1 (φ, χ)s̄ +
.
q̄2i (φ, χ)ēi −
r
rρ̄
2φ
i
(A.56)
To determine the optimality conditions for an equilibrium contract, we proceed as follows. Suppose the
contract (φ, χ, ψ) and asset prices defined by (a0i , a1i , a2i ), for i = 1, .., N , form an equilibrium. Definition 1
implies that equilibrium asset prices are linked to the equilibrium contract through the fund manager’s FOC
(with respect to the fund portfolio). Note, however, that being price-taker, the investor does not internalize
this link when determining the optimal contract. Indeed, suppose the investor offers an alternative contract
(φ̃, χ̃, ψ̃) to the fund manager. Holding constant the equilibrium prices, the optimal portfolio of the fund
manager zt must satisfy the following condition:
φ̃(zit − χ̃ηi ) = φ(θi − χηi )
(A.57)
which implies that
zit = χ̃ηi + (φ/φ̃)(θi − χηi )
(A.58)
Equation (A.57) is an optimality condition for the portfolio zt , as it guarantees that zt satisfies the fund
manager’s FOC for the equilibrium prices (a0i , a1i , a2i ). Condition (A.57) also implies that the coefficients
(q̄1 , q̄2 ) of the fund manager’s value function remain unchanged under the alternative contract (φ̃, χ̃, ψ̃). The
coefficients of the investor’s value function are instead given by
q̃0 =
δ
+ ln r − 1 − ρ
r
1 − φ̃ 2
B + ψ̃
2φ̃2
!
κ
+
r
q̃1 s̄ +
N
X
!
q̃2i ēi
,
h
i
(r + κ)
q̃1 = −rρ χ̃η + (φ/φ̃)(θ − χη) − φ(θ − χη) ba1 −
σs2
r
h
i
(r + κ)2 + 2rρ χ̃η + (φ/φ̃)(θ − χη) − φ(θ − χη) bσs2
+
,
σs2
h
i
(r + κ)
q̃2i = −rρ χ̃ηi + (φ/φ̃)(θi − χηi ) − φ(θi − χηi ) a2i −
σi2
r
h
i
(r + κ)2 + 2rρ χ̃ηi + (φ/φ̃)(θi − χηi ) − φ(θi − χηi ) σi2
+
,
σi2
34
(A.59)
i
(A.60)
(A.61)
for i = 1, .., N , where the flat payment
1
ψ̃ =
ρ̄
δ̄
r+κ
Π − − ln r + 1 −
r
rρ̄
q̄1 s̄ +
N
X
!
−
q̄2i ēi
i
B2
2φ̃
(A.62)
can be substituted in (A.59).
The contract (φ̃, χ̃) is optimal if it maximizes the investor’s ex-ante value function:3
"
(φ̃, χ̃) ∈ arg max − exp − rρW0 + q̃0 + q̃1 s̄ +
N
X
!#
q̃2i ēi
i=1
= arg max q̃0 + q̃1 s̄ +
N
X
q̃2i ēi
i=1
B2
= arg max
2
1 − φ̃
1
−
φ̃
φ̃2
!
r+κ
+
rρ
q̃1 s̄ +
N
X
!
q̃2i ēi
.
(A.63)
i=1
Moreover, the contract (φ̃, χ̃) is an equilibrium contract if, by construction, it is equal to (φ, χ). Therefore,
the conditions for an equilibrium contract are given by the FOCs of (A.63) with respect to (φ̃, χ̃), evaluated
at (φ, χ):
(1 − φ) 2 r + κ
B +
φ3
rρ
N
X
∂ q̃1
∂ q̃2i
(φ, χ) +
ēi
(φ, χ)
s̄
∂ φ̃
∂ φ̃
i=1
s̄
!
= 0,
(A.64)
N
X
∂ q̃1
∂ q̃2i
(φ, χ) +
ēi
(φ, χ) = 0,
∂ χ̃
∂ χ̃
i=1
(A.65)
where
∂ q̃1
1
(φ, χ) = rρ(θ − χη)b (a1 − ǎ1 ) ,
φ
∂ φ̃
(A.66)
∂ q̃2i
1
(φ, χ) = rρ(θi − χηi ) (a2i − ǎ2i ) ,
φ
∂ φ̃
(A.67)
∂ q̃1
(φ, χ) = −rρηb (a1 − ǎ1 ) ,
∂ χ̃
(A.68)
∂ q̃2i
(φ, χ) = −rρηi (a2i − ǎ2i ) ,
∂ χ̃
(A.69)
for i = 1, .., N , and the coefficients (a1 , a2i , ǎ1 , ǎ2i ) are defined in (A.36), (35), (38) and (39). After simplifying
and subtracting (A.65) from (A.64), we obtain the system of two non-linear equations defined by (36) and
(37). Substituting the equilibrium (φ, χ) into (A.56), we obtain the equilibrium flat payment ψ.
3
Note that the problem in (A.63) coincides with maximizing the sum of the fund manager’s and investor’s certainty
equivalent.
35
Proof of Corollary 1. Suppose θi = kηi for i = 1, .., N and a scalar k. The coefficients (a1 , ǎ1 , a2i , ǎ2i )
become equal to
a1 = p
ǎ1 = p
a2i = p
ǎ2i = p
1
(r +
κ)2
(r +
κ)2
+ 2rρ̄φ(k − χ)ηbσs2
,
1
+ 2rρ[k − φ(k − χ)]ηbσs2
1
(r + κ)2 + 2rρ̄φ(k − χ)ηi σi2
(r +
,
(A.71)
,
1
κ)2
(A.70)
+ 2rρ[k − φ(k − χ)]ηi σi2
(A.72)
.
(A.73)
Optimal risk sharing (first best) can be achieved if a1 = ǎ1 and a2i = ǎ2i . This entails imposing the following
restriction on the contract space
ρ̄φ(k − χ) = ρ[k − φ(k − χ)].
(A.74)
Given (A.74), Eq. (37) is always satisfied, whereas Eq. (36) is satisfied for φ = 1. Substituting φ = 1 into
(A.74), and solving for χ, we obtain
χ=
k ρ̄
,
ρ + ρ̄
(A.75)
Proof of Proposition 7. Let us fix a contract (φ, χ, ψ). The equilibrium prices and the agents’ value functions are characterized by the coefficients (a0i , a1i , a2i , q̄1 , q̄2i , q̄0 , q1 , q2i , q0 ) in (33), (34), (35), (A.37), (A.41),
(A.42), (A.52), (A.53), and (A.54), respectively. Substituting these coefficients into (46), and simplifying,
the regulator’s problem can be written as
B2
(φ∗ , χ∗ ) ∈ arg max
2
B2
= arg max
2
2φ − 1
φ2
2φ − 1
φ2
" X
#
N
q1
q̄1
q2i
q̄2i
+ (r + κ) s̄ θa1 +
+
+
ēi θi a2i +
+
rρ rρ̄
rρ
rρ̄
i=1
(A.76)
" X
#
N
r + κ s̄
1
1
ēi
1
1
+
+
+
+
r
σs2 ρ̄a1
ρǎ1
σ 2 ρ̄a2i
ρǎ2i
i=1 i
(A.77)
where (A.77) is obtained by substituting (q1 , q2i , q̄1 , q̄2i ) into (A.76). Simplifying terms, the FOCs of (A.76)
with respect to (φ, χ) are given by
"
#
N
X
(1 − φ) 2
B + (r + κ) s̄(θ − χη)b (a1 − ǎ1 ) +
ēi (θi − χηi ) (a2i − ǎ2i ) = 0,
φ3
i=1
+s̄ηb (a1 − ǎ1 ) +
N
X
i=1
36
ēi ηi (a2i − ǎ2i ) = 0,
(A.78)
(A.79)
respectively. After subtracting (A.79) from (A.78), we obtain the system of two non-linear equations defined
by (47) and (48).
Proof of Corollary 2. Suppose the contract (φ, χ) is privately optimal, that is it solves equations (36) and
(37). Evaluating the LHS of (47) at the privately optimal contract (φ, χ), we can conclude that it is positive,
implying that φ∗ 6= φ.
37
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