securities hongkong

8/8/2019 Securities Hongkong

1/48Electronic copy available at: http://ssrn.com/abstract=951707

Optimal Portfolio Delegation with Imperfect

Information

Tao Li and Yuqing Zhou

First Draft: December 2006

This Version: April 8, 2009

Abstract

This paper investigates the contracting problems that arise in portfolio delega-

tion with imperfect information. In particular, this research studies the role of

stock indexes in the design of incentive fees in portfolio delegation. We show

that in a situation in which the fund manager possesses no superior security se-

lection skill and the portfolio choices cannot be contracted upon, stock indexes

serve as important instruments for achieving optimal risk-sharing between the

investor and the fund manager. While the current literature mainly focuses on

the implications of benchmarking either for a given class of contract forms or for

a specific type of utility functions, we characterize the optimal contracts with

general preferences and incorporate stock indexes into the analysis. Our results

indicate that stock indexes can be used to recoup or reduce the efficiency loss

caused by the uncontractibility of portfolio choices, and can provide a valuable

service even if these indexes are imperfectly constructed.

Keywords: Benchmark, Risk Sharing, Incentive, Optimal Contract, Portfolio

Delegation

JEL Classifications: D80, G11, G30, J33, M52

We would like to thank the seminar participants of Workshop in Contract Theory at CUHK (De-

cember 2006), 2007 Econometric Society North American Summer Meetings and European Meetings

for helpful comments and suggestions.Department of Economics and Finance, City University of Hong Kong, 83 Tat Chee Avenue,

Kowloon, Hong Kong, Email: [email protected] of Finance, Faculty of Business Administration, The Chinese University of Hong

Kong, Shatin, N.T., Hong Kong, Email: [email protected].


2/48Electronic copy available at: http://ssrn.com/abstract=951707

1 Introduction

The agency relationship arising in portfolio delegation and the resulting contracting

problems are quite different from those studied by the classical principal-agent the-

ory. First, the fund managers action space may include both efforts on information

collection and/or production and portfolio choices.1 As Diamond (1998) has noted in

the context of the managerial compensation design, Managers are called on to make

choices as well as to make efforts. This is especially relevant for delegated portfolio

management, in which the freedom of choices for the fund managers is much larger

than that faced by a firm manager. Thus the choice aspect is as important as, if notmore important than, the effort dimension in the design of incentive fees in the in-

vestment fund industry. Second, the wealth process of a managed portfolio has to be

endogenously determined through a well defined trading strategy that satisfies budget

constraints. Thus one cannot arbitrarily assume an exogenous relation between the

actions and payoffs in portfolio delegation as a typical principal-agent model does.2

These issues make the standard principal-agent models difficult or impossible to

apply to the delegated portfolio management. Li and Zhou (2005), which characterizes

the second-best contracts that can be written only on the final wealth or return of

portfolios, is one of the first attempts to move along this line. Built on Li and

Zhou (2005), this paper further investigates the contracting problems in portfolio

delegation with imperfect information. In particular, this research studies the role

of stock indexes in the design of incentive fees in the investment fund industry. The

model goes as follows. An investor, or a fund company, hires a fund manager to

1The action space in the terms of portfolio choice can be very rich, e.g., in a continuous-time

setting. This richness of the agents action space has important implications on the structure of theoptimal compensation schemes, see, for instance, Diamond (1998) and Mirrlees and Zhou (2005a,

2005b).2This does cause some additional technical problems that are absent in the classical principal-

agent model. This can be seen clearly from a model in a continuous-time setting, where, because of

the budget constraint, the drift rate and the diffusion rate of the underlying wealth process cannot

be controlled independently.

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manage her portfolio. The fund manager has no superior security selection skill3 in

the sense that she has no private information compared to other fund managers in the

market. We assume that the investor and fund manager share the same information

or belief about the security markets at aggregate level, which can be summarized

by assuming that both the investor and fund manager agree on the distribution of

the price density function in the case of complete markets. However, at the micro

level the fund manager does have better information than the investor about the

risk characteristics of individual securities, which partially justifies the necessity of

portfolio delegation.4 Given this, our model addresses the contracting problems in a

situation in which the fund manager does not have superior security selection skills(compared to other fund managers, of course) and her portfolio choice cannot be

contracted upon. As a result, in designing incentive fees, the investor must rely on

other variables, say stock indexes, to motivate the fund manager to make the right

choice.

We show that the first-best results are always achievable if there is a market

portfolio, which is, by definition, perfectly correlated with the underlying state price.

Hence the role of the market portfolio in portfolio delegation is to recoup the efficiency

loss due to the different preferences between the investor and her fund manager.

However, the market portfolio may not be observable hence cannot be contracted upon

in reality, thus at best some imperfect observables, e.g., a stock index, can be used

in the design of the pay schemes for fund managers. In this case, the efficiency loss

of the second-best contracts cannot be eliminated completely and the improvement

of efficiency depends on how well the index correlates with the market portfolio or

state price. We characterize the second-best contracts with imperfect information

under rather general conditions by an integral equation, which can be solved bypiece-wise ordinary differential equations (ODEs). Overall, our results shed light on

3One of the reasons why a manager who possesses no superior skill is needed is the opportunity

costs for investors to constantly monitor the markets and trade in a continuous-time setting, see

Mamaysky and Spiegel (2002) for more elaborations on this.4It is much less costly to collect market data on information at aggregate level than at micro

level. In fact, as shown in this paper, all investors need to achieve the first-best results is the market

portfolio, if it is observable hence can be contracted upon.

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and identify another role of stock indexes in the compensation of fund managers: it is

used to recoup or reduce the efficiency loss of the second-best contracts. This function

of stock indexes played in fund managers compensations has been largely ignored in

the literature.

There are several important aspects that justify our models assumptions and

confirm their empirical relevance. First, there is extensive empirical evidence showing

that, historically, the majority of fund managers fail to beat the market, even if many

of them claim to be active fund managers. Thus, the assumption that the fund

manager has no superior security selection skills has some empirical relevance in

practice. Second, in reality, the structure of a fund managers compensation mainly

depends on the past performance of the fund under management, on the funds total

return relative to a prespecified benchmark, and on bonuses that are related to the

profitability of the firm that employs her (see, e.g., Chevalier and Ellison (1997)

and Farnsworth and Taylor (2006)). While investors may place some restrictions on

the fund managers portfolio choice, the fund manager has large freedom in trading

assets in the markets and her incentive fees are seldom based on trading strategies

directly. As a result, it is reasonable to treat the fund managers portfolio choice as a

moral hazard variable, and to treat the conflicts of interest arising from the different

preferences between the investor and fund manager as a fundamental issue in the

fund management industry. Third, although our model only deals with a simple

agency relationship, that is, an investor versus a fund manager, our analysis has

general implications to the situation where investors (or fund companies) hire many

managers simultaneously.5 For example, the case of multiple fund managers can be

made precise when investors utility is exponential and the asset classes managed

by different fund managers are relatively independent. Finally, our model confirmsthe popular view held by practitioners that the use of benchmarks can be valuable,

5In reality, the agency relationship is multi-layer and complex. In general, investors (or fund

companies) allocate their capital (usually under the recommendation of agents) among many different

fund managers. However, the simple one-to-one agency relationship considered in our model will

not disappear, and investors (or fund companies) still have to face the individual fund mangers

incentive problem after the asset allocation decision has been made.

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but may be for a different reason. In our context, the stock indexes are valuable

because they can be used to recoup or reduce the efficiency loss caused by the fund

managers portfolio choice, and not to induce the fund managers efforts in collecting

information. As a direct application, our model is mostly relevant to a wide range of

passively managed funds, the major objective of which is to balance the portfolios

risk and return that align with investors preferences and the managers of which need

a right incentive to do so.

Overall, we believe that the problem arising from the conflicts of interest due to

the different preferences between the investors and fund managers is among the most

fundamental ones in the fund management industry. It has to be fully addressed in

the first place before a full-fledged theory of portfolio delegation can be developed.

Our results indicate that a properly constructed stock index that is independent of

preferences can solve or at least mitigate these conflicts of interest. Interestingly,

some benchmarks already available in the markets exhibit some required features of

the theoretically constructed market index, and thus can be used to align the interest

of the fund manager with that of the investors to achieve optimal or near optimal

risk-sharing. As a result, our model has set up a foundation upon which further moral

hazard variables can be introduced to enrich the model.

On the technical side, we follow the method developed by Li and Zhou (2005) to

solve the optimal contract with imperfect information. Similar to Li and Zhou (2005),

we show that the problem of finding the optimal contract can be converted into the

one that solves a nonlinear second-order ordinary differential equation (ODE). As a

result, the first- and second-best contracts can be fully analyzed and compared. We

show that, for a given stock index, the first-best risk sharing can be achieved if and

only if the stock index is perfectly correlated with the state price. In particular, when

the investor and fund manager have similar preferences, an incentive fee based

on final wealth alone is sufficient to achieve the first-best risk sharing (see Li and

Zhou (2005)). In case the stock index is imperfectly constructed or a market index is

impossible to construct, numerical calculation shows that the efficiency loss is small

if the market index is highly correlated to the price density function.

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There are a number of papers that study the role of benchmarking in different

contexts. Most of them focus on the effects of benchmarking on the fund managers

trading strategy for a given class of compensation schemes (see Roll (1992) and Basak,

Pavlova, and Shapiro (2003) and the reference therein), and the predictions are mixed.

Stoughton (1993) and Admati and Pfleiderer (1997) study the adverse consequences

of benchmarking in the presence of private information for a given class of compensa-

tion schemes, and show that in general the use of benchmarks will not induce the fund

manager to exert high efforts in collecting information. Ou-Yang (2003) and Dybvig,

Farnsworth, and Carpenter (2004) treat the compensation scheme with benchmarks

endogenously but focus on specific utilities. In contrast, we solve the optimal compen-sation scheme with imperfect information and general preferences in the second-best

world. Kraft and Korn (2004) also consider the role of the market portfolio played in

the fund managers compensations. However, they mainly focus on the the first-best

cases and do not characterize the second-best contracts when the market portfolio is

unobservable and hence cannot be contracted upon.

The paper is organized as follows. The next section studies the contracting prob-

lem with imperfect information in an abstract setting. The second- and first-best

contracts conditional on some signals are characterized in Section 3. The relation

between the efficiency and the quality of information is also discussed in this section.

Section 4 offers a detailed example in a continuous-time setting, in which some fea-

tures and implications of the optimal, especially the second-best, contracts are further

studied. Section 5 concludes the paper. All proofs are provided in the appendix.

2 The Basic Framework

Consider a setting in which an investor or a fund company (the principal) wishes

to hire a fund manager (the agent) to manage her portfolio. The portfolio return is

realized over a continuum of states in a single period. Let (, F, P) be the spaceof states endowed with a probability measure and be a state. As a start,we assume that there exists a rich set of financial securities such that the financial

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markets under consideration are complete. If a market is complete, then there exists a

unique state price function p () per unit probability over . The portfolio returns are

affected by the agents actions, or individual security selections in the portfolio. Thus,

the incentive scheme designed by the principal matters in order to motivate the agent

to act in the best interest of the principal. If the agents action can be observed, or,

if the principal can costlessly distinguish the payoff characteristics of the universe of

securities in the financial markets, then the contracting problem between the principal

and the agent would be relatively straightforward; the contract would simply specify

the exact portfolio of securities to be selected by the agent and the compensation

that the principal promises to provide in return should the order be followed exactly.However, if it is too costly for the principal to distinguish the payoff characteristics of

the universe of securities, then the contract can no longer specify the agents security

selection in an effective manner. Under this circumstance, the principal must design

a compensation scheme in a way that indirectly gives the agent an incentive to select

the correct set of securities. Li and Zhou (2005) study a case in which the only way

for the principal to get the agent to select a correct portfolio is to relate her pay to

the realization of the portfolio return, which is random. In this paper, we focus on

the use of imperfect information for the efficiency improvement of the contract.

To be more specific, let w 0 be the final wealth of the selected portfolio andw () be the realization of the final wealth over , which is observable. In addition,

a signal s(), possibly vector valued, can be observed by both the principal and

the agent (common knowledge and verifiable), and can consequently be contracted

upon. A compensation scheme specifies the agents wage as a function of the observed

final wealth and the signal y(w, s). Let the principals utility function be u() and

the agents utility be v () , where u() and v() are independent of states, increasingand concave over the interval [0, ). We also assume that the utilities are twicedifferentiable,6 and their first-order derivatives satisfy u() = v() = 0. If thefinal wealth is w, the net benefit for the principal is assumed to be (w, s) y(w, s),where 0 < (w, s) w. When (w, s) = w, our model is a typical principal-agentproblem. It can be interpreted as a large investor hiring a money manager to manage

6The differentiability are not necessary but for convenience.

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his portfolio.7 In general, (w, s) is used to model a situation in which the principal

may be an institution or may act as an agent of large number of investors, and

thus only receive a portion of the realization of the final wealth either explicitly as a

management fee (e.g., a contract between a fund company and investors) or implicitly

through the flow of new funds.

The principal will delegate an initial wealth w0 for the agent to manage, and

design a pay schedule y (w, s) to induce the agent to act in her best interest. Given

a compensation scheme y (w, s) , under the budget constraint, the agent will select a

portfolio such that his own expected utility is maximized. Therefore, there exists an

explicit conflict of interests between the principal and the agent, and it is interesting

to see how the principal and the agent share the risks and what the optimal contracts

are. To formalize these ideas, let the agents action space A be defined by

A = {w () 0|

p()w()dP() w0}, (1)

where the last term in equation (1) is the budget constraint. In other words, the action

space A consists of all random variables over that satisfy the budget constraint.

In contrast to those one-dimensional (or low-dimensional) action spaces studied inthe agency models in existing literature, ours is large in the sense that it is infinite

dimensional. Let the agents reservation utility be v0. Formally, the model goes as

follows:

maxy(w,s),w()

u((w, s) y(w, s)) dP() (2)

subject to

v(y(w, s)) dP() v0 (3)

andw() arg maxw()A

v(y( w(), s)) dP(), (4)

where equations (3) and (4) are the standard participation constraint and incentive

constraint, respectively. The solution y(w, s) to this contract problem is the second

7Much work in this literature has been done under such a specification. See Dybvig, Farnsworth,

and Carpenter (2004) and the references therein.

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best, whereas the solution y(w, s) to (2) and (3) only is the first best that is Pareto

efficient.

The contracting problem above can be rewritten in terms of distribution condi-

tional on the signal s.

maxy(w,s),w()

ds fs(s)

u((w, s) y(w, s)) dP(|s) (5)

subject to ds fs(s)

v(y(w, s)) dP(|s) v0 (6)and

w() arg maxw()A

ds fs(s)

v(y( w(), s)) dP(|s), (7)where P(|s) is the conditional distribution on s and the agents action space isrewritten as

A = {w () 0|

ds fs(s)

p()w()dP(|s) w0}. (8)

For the case in which no signal s is involved in the pay schedule y, the contracting

problem (5)-(7) is studied in Li and Zhou (2005). Following Li and Zhou (2005), we

reformulate the model in terms of distributions. We note that, given our model setupin that both the principals utility and the agents utility are independent of state ,

it is well-known, in the literature of portfolio choice without agency problem, that the

states only need to be distinguished by the state price p and the signal s when the

financial markets are complete. This is also true for portfolio delegations. We will

use f and F as probability density and cumulative distribution functions, respectively

and use subscripts to distinguish different variables. Specifically, let fs and fp be the

density functions of signal s and state price p, and write the joint distribution of p

and s as fs(s)fp|s(p), where fp|s(p) is the conditional probability density of state price

p on s. Also let fw(w) be the distribution functions of wealth w (). Note that fs, fp,

hence fp|s are exogenously given, whereas fw(w) is the agents choice variable. For

simplicity, we further assume fp|s is continuous and first-order differentiable.8

8These assumptions are not crucial to solve the contracting problems, but rather for convenience.

In addition, the state price density functions can not be arbitrarily specified; there should be no

arbitrage opportunity. Relevant restrictions are given explicitly when we work with specific examples.

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Now define feasible action spaces in the terms of state price and wealth distribution

as

Ap = {w(p,s) 0|

ds fs(s)

pw(p,s)fp|s(p) dp w0} (9)and

Aw = {fw|s(w) 0|

ds fs(s)

F1

p|s (1 Fw|s(w))wfw|s(w) dw w0}, (10)

where we have used the following

p = F1p|s (1 Fw|s(w)), (11)

to rewrite the budget constraint in terms of wealth distribution. Equation (11) implies

that that w is a nonincreasing function ofp that reduces the size of the agents action

space, and seems to be restrictive. However, the optimal choice is always within the

agents action space Aw, as shown by the following lemma.

Lemma 1 Take an arbitrary utility function G(w, s) such that

maxw()A

G(w(), s()) dP()

exists. Then

maxw()A

ds fs(s)

G(w(), s) dP(|s)

= maxw(p,s)Ap

ds fs(s)

G(w(p), s)fp|s(p) dp

= maxfw|s(w)Aw

ds fs(s)

G(w, s)fw|s(w) dw,

where Ap and Aw are defined in equations (9) and (10), respectively.

Notice that the optimizations are pointwise in the dimension of the signal s. Based

on this observation, Lemma 1 is a straightforward extension of a similar result in Li

and Zhou (2005) without a signal or information.

Now Lemma 1 enables us to reformulate the agents problem in the terms of wealth

distribution, hence the principals problem represented by equations (5)-(7) can be

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reformulated as follows:

maxy(w,s),fw|s(w)

ds fs(s)

u((w, s) y(w, s))fw|s(w)dw (12)

subject to ds fs(s)

v(y(w, s))fw|s(w) dw v0 (13)

and

fw|s(w) arg maxfw|s(w)Aw

ds fs(s)

v(y(w, s))fw|s(w) dw (14)

To solve the principal-agent problem (12)-(14), the method we are going to use

here is the basic technique in the calculus of variations. The basic idea goes as follows.

Suppose that fw|s(w) is an optimal solution, then perturb fw|s(w) by (s) (w, s),

where (s) is a small constant for each s and is an arbitrary integrable function that

satisfies

(w, s) dw = 0 for each s. Such a restriction makes sure

ds fs(s)

[fw|s(w)+

(w, s)] dw = 1. This is analogous to the finite dimensional case, in which is a di-

rectional vector. The fact that fw|s is an optimal solution implies that all directional

derivatives at fw|s for each s are zero. The details for applying this technique are

provided in Li and Zhou (2005).

To solve the principals contracting problem, we first need to solve the agents

problem, the resulting Lagrange of which is

V(fw|s, , f) =

ds fs(s)

v(y(w, s))fw|s(w) dw

ds fs(s)

fw|s(w)F

1p|s (1 Fw|s(w))w dw w0

+ ds fs(s) f(w, s)fw|s(w) dw, (15)where is positive constant and f is a nonnegative function of wealth w and sig-

nal s, which is equal to zero if fw|s(w) > 0. As indicated by the Lagrange V, themaximization over signal s can be done state-by-state or point maximization. This

observation simplifies the problem significantly. That is, for each s, the agents prob-

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lem is equivalent to choosing fw|s(w) to maximize

v(y(w, s))fw|s(w) dw +

fw|s(w)F1p|s (1 Fw|s(w))w dw w0

+

f(w, s)fw|s(w) dw.

This problem is similar to what has been studied in Li and Zhou (2005).

Lemma 2 Given a pay schedule y(w, s). Suppose the agents problem has an optimal

solution.9 Then the necessary and sufficient condition for optimality is that, for any

s, there exist constants, > 0, (s), and a density function fw|s(w) 0 such thatx(w, s)

w

F1p|s (1 Fw|s(t)) dt + (s), (16)

where x(w, s) is the concavification of the agents utility functionv(y(w, s)) and is an

arbitrary number such thatfw|s > 0. Furthermore, fw|s(w) 0 forw {t|v(y(t, s)) =x(t, s)}.

An immediate implication of Lemma 2 is that we can restrict our feasible pays

that make the agents indirect utility x(w, s) = v(y(w, s)) concave in w. All other

feasible pay schedules have an equivalent concave pay that is different only on the

set offw|s = 0. This observation enables us to further simplify the principals problem

by replacing the incentive constraint by equation (16). Lemma 2 makes it legitimate

to use the first-order approach in solving the contracting problem.

Specifically, let x(w, s) = v(y(w, s)), therefore, y(w, s) = v1(x) = h(x(w, s)) for

fixed s. Since v (y) is increasing and concave, h(x) is increasing and convex. The

principals problem is then reformulated as follows10

maxx,fw|sAw

dsfs(s)

u((w, s) h(x(w, s)))fw|s(w) dw (17)

9For example, ify(w, s) is bounded above by a linear function for all s, then x(w, s) will have an

optimal solution.10The signals in and in pay schedule y could be different. In this case, the contracting problem

can be solved by perturbing fw|s1,s2 and fs becomes a joint distribution ofs1 and s2.

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subject to

x(w, s) = w

F1p|s (1

Fw|s(t)) dt + (s), (18)

where > 0 and (s) are free variables, which are constrained byds fs(s)

x(w, s)fw|s(w)dw v0. (19)

The reformulation of the original problem leads to equations (17)-(19), in which

the principal selects x(w, s) and fw|s(w) to maximize his utility u subject to the

first-order condition constraint (18) plus the participation and budget constraints.

Indeed, x (w, s) and its concavification x(w, s) are identical in a distribution sense,

which is exactly what matters in our principal-agent problem. If x (w, s) is concave

and nondecreasing for each s, then x (w, s) = x (w, s) . Equation (18) alone also reveals

an important feature of the optimal contract. That is, the optimal contract must be

designed in such a way that the compensation is a nondecreasing function of the final

wealth. Of course, to obtain additional features of the optimal contract, we need to

solve the principals maximization problem. Lemma 2 tells us that the principal only

needs to focus on the class of nondecreasing and concave functions in the selection of

x (w, s).

Our discussions so far lead us to conclude that, on the one hand, for any smooth,

increasing and concave indirect function x(w, s) and a number > 0 there exists a

distribution function

Fw|s(w) = 1 Fp|s( 1

x(w, s)) (20)

such that equation (18) is satisfied. In other words, for any x(w, s), there is a unique

fw|s(w) that implements it. On the other hand, for any distribution function fw|s(w)and > 0, there exists a unique x(w, s) that satisfies equations (18) and (19).

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3 The Optimal Contracts

3.1 The Second-Best Contracts

In this section we characterize the optimal contracts of the agency problem discussed

in the previous section. Define

Us(fw|s, , , w, v, f) =

u ((w, s) h(x(w, s))) fw|s(w) dw

w fw|s(w)F1

p|s (1 Fw|s(w))w dw w0+ v

x(w, s)fw|s(w) dw v0

+

f(w, s)fw|s(w) dw, (21)

where > 0, w > 0, v 0, and f(w, s) 0 if fw|s(w) = 0. The Lagrangian forthe principals maximization problem is

U(fw|s, , , w, v, f) = Us(fw|s, , , w, v, f)fs(s) ds.

This means that maximizing Uis equivalent to maximizing Us for all s.

Note that, in addition to the budget and the individual rationality constraints, the

objective function Uis also dependent on two choice variables and (s). These twovariables can be handled separately from the function fw|s(w) by point maximization.

Proposition 1 For any given multipliers w and v, and a density function fw|s(w),

the objective function

Uis a concave function of (, (s)). At optimum, (, (s))

must satisfy the following first-order conditions

U

= 1

ds fs(s)

fw|s(w)[x(w, s) (s)][uh v] dw = 0, (22)

U(s)

=

fw|s(w)[uh v] dw = 0 (23)

for all s. In addition, the last equation implies v > 0, that is, the participation

constraint must be binding at optimum.

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To determine fw|s, we use a variational technique as illustrated in the case of the

agents problem.

Proposition 2 Fixed (w, v, , (s)). The first-order necessary condition for an

optimal solution fw|s to the contract problem is that there exists a function c(s) such

that

u((w, s) h(x(w, s))) + f(w, s) +

v w

x(w, s)

+w

1

fp|s(p) a

0

(uh v)fw|s(t) dt da = c(s) (24)holds almost everywhere, where p = F1

p|s (1 Fw|s(a)).

As shown in Li and Zhou (2005), this condition can handle corner solutions quite

easily, e.g., is discontinuous and/or nondifferentiable at some points. However, to

ease the exposition, we only focus on the interior solutions in this paper.

For the region(s) in which f(w, s) = 0, a differential equation can be derived

to further investigate the property of the optimal solutions. Taking derivatives with

respect to w to the first-order condition (24) implies that

d[u + (v w/) x]dw

+

fp|s(p)

w0

[uh v]fw|s(t) dt = 0, (25)

where p = F1p|s (1 Fw|s(w)). This differential-integral equation becomes an ordinary

differential equation by taking derivatives one more time and using the fact that

fw|s(w) = 1

fp|s(x/)x.

Define

D(s) = {w| 2uh + (u [ hx] + ( v)x)fp|s

fp|s< 0}, (26)

where = 2v w/ and the prime denotes the derivatives with respect to w.

Proposition 3 Suppose u((w, s) h(x)) is a concave function of (w, x) for all s.Then f(w, s) 0 for all (w, s).

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In addition, the agents indirect utility, x = v(y), satisfies the following ordinary

differential equation (ODE)

[ 2uh] x + u [ hx]2 uh[x]2 + u

+ x (u [ hx] + ( v)x)fp|s(p)

fp|s(p)= 0, (27)

which has a unique solution in D(s), given a set of boundary conditions (x(), x()),where p = x/, where is the first-order derivative with respect to w.

The optimal pay schedule y also satisfies an ordinary differential equation (ODE)

as:

(v 2u)y + v

v(v u)[y]2 + u[ y]2 + u

+

vy + v[y]2

(u[ y] + ( v)vy)fp|s(p)

fp|s(p)= 0, (28)

where p = vy/. However, it seems to be easier to work with the agents indirect

utility x. The contracting problem can be solved by solving the ODE (27) and the

multipliers are determined by the following integrals.

The first-order conditions for (, (s)) in equations (22) and (23) can be rewritten

as follows ds fs(s)

0

fp|s(x/)xx[uh v] dw = 0, (29)

and 0

fp|s(x/)x[uh v] dw = 0, (30)

for each s. And the budget and participation constraints becomeds fs(s)

0

wfp|s(x/)xx dw = 2w0 (31)

and ds fs(s)

0

fp|s(x/)xx dw = v0. (32)

When we have solved for x(w, s), the optimal pay schedule is given by y = h(x(w, s)).

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3.2 The First-Best Contracts

It is clear that the second-best contracts are equivalent to the first-best one if the

signal s is the same as the underlying state . The first-best contract consists of a

final wealth function w(), which is equivalent to a detailed instruction of portfolios,

and a pay schedule y() such that the pair (w(), y()) maximizes the principals

expected utility under the budget constraint:

maxw()A, y()

u((w(), s) y()) dP() (33)

subject to the participation constraints:

v(y()) dP() v0. (34)

This maximization problem is straightforward when u((w, s) y) is concave for alls and y, but it becomes complicated when (w, s) is convex. Therefore, it is helpful

to reformulate the problem into the principals choosing the distribution of wealth fw

instead of w. However, we cannot directly apply Lemma 1 to transform the problem

into choosing fw because of the additional term y().

Lemma 3 The first-best pay schedule y(), which together withw() solves the max-

imization problem (33)-(34), can be written as y(w(), s).

Lemma 3 shows we can replace y() by y(w(), s) in the maximization problem

(33)-(34). Then, applying Lemma 1 to u + v implies that the first-best contracting

problem is equivalent to

maxfw|s(w)Aw, y(w,s)

ds fs(s)

u((w, s) y(w, s))fw|s(w) dw

subject to ds fs(s)

v(y(w, s))fw|s(w) dw v0.

In addition, and for the sake of comparison with the case of the second best, we use

y(w, s) = h(x) = v1(x). Then the Lagrangian of the reformulated maximization

problem is as follows:

L(fw|s, x , w, v, f) =

Ls(fw|s, x , w, v, f)fs(s) ds,

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where

Ls(fw|s, x , w, v, f) = u((w, s) h(x))fw|s(w) dw w

F1

p|s (1 Fw|s(w))wfw|s(w) dw w0

+ v

xfw|s(w) dw v0

+

f(w, s)fw|s(w) dw.

Using the variational method that perturbs fw|s by ff(w, s) and x by xx(w, s),

where f and x are two constants and

f(w, s) dw = 0 and

x(w, s)fw|s(w) dw < ,

we immediately have the following.

Proposition 4 A pair(fw|s, x(w, s)) is a first-best solution to the principals problem

if and only if it satisfies the following first-order conditions

u((w, s)h(x(w, s)))+vx(w, s)+f(w, s) ww

F1p|s (1Fw|s(t)) dt = C(s) (35)

and

[u((w, s) h(x))h(x) v] fw|s(w) = 0 (36)

almost everywhere for any such that fw|s() > 0, where C(s) is independent of w

and x.

Similar to the second-best case, the first-order condition (35) can be used to handle

corner solutions. However, the corner solutions are quite straightforward for the first-

best case, that is, replacing u + vx + f in equation (35) by the concavification of

u + vx along the dimension w. Thus f for the first-best case can be predetermined

hence the interior part of the solutions are solved by simpler conditions.

Corollary 1 If f(w, s) = 0 at (w, s), then the first-best contracts are given by

y(w(p), s) = [v]1

wp

v(w, s)

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and the final wealth is implicitly given by

(w, s) = [u]1

wp

(w, s)

+ [v]1

wp

v(w, s)

,

where (w, s) is the first-order derivative with respect to w, and w and v are de-

termined by the budget and participation constraints.

If u((w, s) h(x)) is a concave function of (w, x) for alls, thenf(w, s) 0 forall (w, s).

Corollary 1 shows that the first-best contracts have closed-form solution for many

types of utility functions, e.g., the class of power utilities. In addition, Corollary 1

also reveals another interesting implication of the model. The first-best contracts do

not depend on any signals if the gross benefit of the principal does not depend on

a signal even they are defined on a finer partition of the state space than the state

price. To achieve the first-best results, all it needs is the distribution of the state price

if it can be contracted upon. However, the second-best contracts can be improved by

conditioning on some signals even though they may not be perfect substitutes for the

state price.

3.3 Information and Efficiency

From the formulations of the contracting problems, it is clear that if the signal s

represents a finer partition on the state space then the first-best contracts are

achievable. That is the second-best contracts are the same as the first-best ones. A

formal statement of this observation is as follows.

Theorem 1 If fp|s(p) is singular, then the second-best contracts are the same as the

first-best contracts.

An important and interesting question in portfolio delegation is: what state vari-

ables can be contracted upon? In practice, one may expect that since the security

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price (or returns) processes can be observed, they can thus be contracted upon. How-

ever, casual observations tell us that practical pay schedules are hardly contracted

upon all the perceivable contingent price states. Enforceability, liquidity, and other

market imperfections prevent us from using all price information in the design of

compensation schemes. In addition, Theorem 1 shows that we do not need such full

price information to achieve the first-best results. If there exists an index of market

portfolio, which is perfectly correlated with the state price, then the first-best results

are always achievable by writing contracts based on this index. However, such an

index of the market portfolio may not exist or the market portfolio is not observable

in reality. In practice, many pay contracts use some passive indexes as a benchmarkin the design of compensation schemes because of high liquidity, easy enforcement,

and low contracting costs of these passive indexes. If these indexes are not perfect

substitutes for the state price itself, then how and in what degree are these imper-

fect signals able to improve the second-best results? To further explore this and

other contracting issues as well as to illustrate how the model can be applied, we will

go through a detailed example in a continuous-time setup, which is widely used in

modeling the dynamics of securities prices.

4 An Example in Continuous-Time

Our analysis in the previous sections can be carried over to a continuous-time model,

where passive indexes as contractible signals and managers dynamic portfolio selec-

tion issues can be addressed explicitly. In this section we will not seek generality in

applying our approach, but rather we will develop a specific model to highlight some

important points our approach can address. Such a model is also more relevant in

reality. In our continuous-time model, the time horizon is [0, T], the principal (in-

vestors or a fund company) hires a manager to manage her initial wealth w0 at time 0

and the final wealth under the managers management at time T, which is random, is

denoted by w = WT. In practice, a fund under management normally has an external

fund inflow or outflow in the process of portfolio selection. While introducing an

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exogenous fund flow causes no additional conceptual difficulty, we assume away the

external flow issue for simplicity. In other words, throughout this section we assume

that the managers trading strategies are self-financed.

There are N+1 securities available in the market for the manager to trade. One

is a risk-free bond, and the others are N risky securities. We assume that the bonds

price S0t follows a deterministic process and has a constant short rate r, dS0 = rS0dt.

For the N risky securities, their price process S = (S1,...,SN) is assumed to follow

a multi-dimensional geometric Brownian motion

dS

S = dt + dB, S(0) > 0, (37)

where dSS

, , and B are the transpose of

dS1

S1,..., dS

N

SN

, (1,...,N) and (B1,...,Bd)

respectively, and is a N d matrix. Note that B is a standard Brownian motionin Rd on a probability space (, F, P) , with the standard filtration denoted by Ft.We will further assume that the financial market is complete, thus set d = N without

loss of generality. Under such a condition, is a N N matrix with a full rank.

A managers trading strategy is an admissible process = 1,...,N that isprogressively measurable with respect to the filtration Ft and satisfies

E

T0

2dt

<

almost surely, where i is the fraction of total wealth held in the i-th risky security.

Given a trading strategy, then the corresponding total wealth process as follows

dWtWt

=

r + ( r1)

dt + dB; W0 = w0, (38)

where 1 is a vector with all elements 1. Note that the managers trading strategy

cannot be observed by the principal, thus cannot be contracted upon.

Given our model setup, the density process t at time T for the equivalent mar-

tingale measure is given by

T = exp

1

22T BT

, (39)

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where = 1 ( r1). The associated (deflated) state-price p at time T is p =erTT, and the corresponding state price density function fp(p) can be written as

fp (p) =1

2ppexp

1

2

lnp p

p

2, (40)

where p =

r + 122T and p = T.

Similar to the static case in the previous section, the principal designs compen-

sation schemes y (w, s) based on the final wealth and security price information s,

where s could be the information generated by a set of security (or portfolio) price

processes, and the principal and the fund manager maximize their expected utility

functions based on the final wealth at time T and the relevant information s. Ideally,

all security price processes, together with the wealth process, can be available for

contracting. In practice, however, it is very costly, if not impossible, to contract upon

all security prices due to liquidity problems or other market imperfections. Therefore,

practitioners typically design their contract based upon the final wealth and a set of

actively traded passive index funds (or portfolios of securities).

4.1 Stock Index

A passive index is formed by a subset of the N stocks. A trading strategy s such

that

is = 1 forms an index whose return follows a geometric Brownian motion by

equation (38). The gross return for this index over a period [0, T] is given by

Rs = exp

s

1

2s2

T + s BT

,

which follows a normal distribution with a mean of s

= s

1

2

s2T and a

variance ofs = sT. Since p and s is joint normal, the conditional distributionof state price p on the signal s = ln Rs is

fp|s(p) =1

2(1 2)ppexp

1

2(1 2)2p

lnp p p

s[s s]

2, (41)

where = s ps

is the correlation between p and s. Thus a contract can use the

return of such an index to improve the efficiency.

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A special case in which an index is perfectly correlated with the state price offers

an ideal solution to the inefficiency of the second-best contracts if such an index exists

and is observable. From the discussion above, we know that the trading strategy for

such an index is ms = , where m is a constant scaler. This shows the trading

strategy for this index is

s =

1( r1)

1 []1 ( r1) . (42)

Note that s is a constant, preference-free, and is determined by the price parameters

only.11 This index is also known as the market portfolio in the asset pricing literature.

The relation between the state price p and the return of the market portfolio is

Rs = R0p 1m , (43)

where

R0 = exp

s

r1

m

1

2m

1 +

1

m

2p

T

and

m = 1

1

( r1) .

The parameter m can be interpreted as the relative risk aversion coefficient of aninvestor who optimally invests all wealth in the stocks.

4.2 Market Portfolio

As a benchmark, we first study the contracting problem in the case in which there is

a market portfolio that is observable, and hence can be used in contracts. As shown

in Theorem 1 the second-best contracts are the same as the first-best ones for this

case, so, we only need to solve the first-best contracts. We also assume that the gross

payoffs to the principal is:

(w, s) = 0w + 1(w w0RseT)+ = w0[0R + 1(R RseT)+],11See, e.g., Merton (1971, 1973).

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where R = WT/W0 and Rs is the gross return of the market portfolio,12 given by

equation (43), and is a constant. The option-like pays capture some incentives or

the effects of fund flows.

Since the gross payoff of the principal is convex, we have to concavify the welfare

function u + vv for each Rs. This can be done by setting the following equations,

which are one of the direct implications of equation (35) in Proposition 4,

u[(wl, s) y(wl, s)] + vvy(wl, s)= u[(wh, s) y(wh, s)] + vvy(wh, s) (44)

= u((wh, s) y(wh, s)) + vv(y(wh, s)) u((wl, s) y(wl, s)) vv(y(wl, s))wh wl .

Here fw|s(w) 0 on the interval (wl, wh). Due to the endogeneity of the multiplierv and the pay schedule y, this concavification cannot be done without solving the

contracting problem. Coupling these equations with the first-order conditions in

Proposition 4 gives us the solution of the first-best contracting problem.

Let us consider a specific example in which both of the principal and agent have

a power utility asw1

1 .

Let p and a be the relative risk aversion coefficient of the principal and the agent,

respectively. We also use a certain equivalent pay wr to express the reservation of the

agent, which is defined as solving

w1ar1 a = v0. (45)

This seems to be a sensible way of defining agents reservation because agents are

competing with pay levels rather than utility levels in the markets for money man-

agers.

12The first-best contract for the case of an arbitrary benchmark portfolio is also straightforward

as given in Corollary 1.

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Corollary 2 Suppose that both the principal and agent have power utility and the

market portfolio is observable. Then the optimal pay schedule is given by

y(p) =

wpv0

1a

if p pwp

v(0+1)

1a

if p < p

and the final wealth is given by

w(p) =

10

wp0

1p

+

wpv0

1a

if p p

10+1 wp0+1

1p

+ 1w0Rp 1m + wpv(0+1)

1a if p < p,

where R = R0eT and p, w, and v are determined by the system of equations:

(0 + 1)1

p1

1

p1

0

p

1 p 1p

w p 1p +

a1 a

wv

1a

p1

a

=1w0R0 + 1

p1

m , (46)

e

1p

pp

1p2

p

2p

1

p1

0 [1 A (p)] + (0 + 1)1

p

1

A (p)

1p

w

+ e1aa

(p 1a2a 2p)

1

a1

0 [1 A (a)] + (0 + 1)1

a1A (a)

wv

1a

+ e1mm

(p 1m2m

2p) 1w0R0 + 1

A (m) = w0, (47)

and

e1aa

(p 1a2a

2p) 1

a1

0 [1 A (a)] + (0 + 1)1

a1A (a)

wv

1 1a

= (1 a)v0, (48)

where

A() = N

ln p p

p+

1

p

,

where N() is the cumulative distribution function of a standard normal random vari-able.

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Under the first-best contracts, both pay schedule and final wealth are monotone

functions of state price. Hence there is a unique relation between pay schedule and

final wealth. This is especially useful in comparisons between the first- and second-

best contracts. Without confusion, we also call this relation the first-best contracts.

The following graphs illustrate this relations for two numerical examples.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.5 1 1.5 2 2.5

(w, s) = 0.02 w + 0.05 (w - w0 Rs)+

(w, s) = 0.02 w(w, s) = w

Figure 1: First-best contracts against final wealth with the same reservation. Both

final wealth and pay schedules are normalized by the initial wealth. Preferences are

power utilities and the relative risk aversion coefficients of the principal and agent

are p = 3 and a = 0.3, respectively. The parameters are: p = 0.5, s = 10%,

s = 30%, r = 5%. The gap in the graph represents the region of {w|fw|s(w) = 0}

caused by the covexity of (w, s).

Figure 1 plots the first-best contracts in the case in which the principal (a fund

company or an individual investor) is more risk averse than the agent. This plot

shows that the agent takes more risk by offering a convex pay schedule. This is the

result of the optimal risk sharing between principal and agent. Intuitively, the first-

best pay schedule becomes concave if the agent is more risk averse than the principal

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as illustrated in Figure 2.

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0 1 2 3 4 5 6

(w, s) = 0.02 w + 0.05 (w - w0 Rs)+

(w, s) = 0.02 w(w, s) = w

Figure 2: First-best contracts against final wealth with the same reservation. Both

final wealth and pay schedules are normalized by the initial wealth. Preferences are

power utilities and the relative risk aversion coefficients of the principal and agent

are p = 0.3 and a = 3, respectively. The parameters are: p = 0.5, s = 10%,

s = 30%, r = 5%. The gap in the graph represents the region of {w|fw|s(w) = 0}caused by the covexity of (w, s).

Although there is no incentive concern in the first-best contracts, the first-best

pay schedule may be very sensitive to the performance of a delegated portfolio. Such

sensitivities are solely due to the optimal risk sharing and becomes a constant if both

principal and agent share the same preferences. If the first-best model presented

here is a good approximation for some of the delegated portfolio management, e.g.,

existing a good proxy for the market portfolio, then different pay contracts across fund

companies are simply the results of different risk attitudes among fund companies and

managers.

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4.3 Imperfect Signals

In practice, the market portfolio is not observable due to various reasons, hence an

index with a subset of stock is used in the contracting problem. In this case, we have

to seek the second-best solutions. Due to the numerical complexity, we only consider

a simple case in which

(w, s) = 0w.

By equation (41), we have

fp|s(p)

fp|s(p) = 1

(1 2)2pp

lnp ps s p|s

, (49)

where s = ln Rs and p|s = p ps s (1 2)2p is a constant. Substituting thisinto the ODE (27) and using p = x/ yields, for each s,

2uh u [0 hx] + ( v)x

(1 2)2px

ln x ps

s p|s ln

x

= u [0 hx]2 + uh[x]2. (50)

Because the righthand side of the ODE is positive and x

0, the solution has tosatisfy the constraint

2uh u [0 hx] + ( v)x

(1 2)2px

ln x ps

s p|s ln

< 0. (51)

The second-order ODE (50) can be solved numerically by transforming it into a

system of first-order ODEs given a set of boundary conditions. Because the ODE

contains parameters that are determined endogenously, it is not obvious to choose

starting values. However, the asymptotic behavior of the ODE can be obtained bycombining it with the differential-integral equation (25).

Lemma 4 Given(, w, v). Ifx is a second-best solution to the contracting problem,

then

u[0 hx] + ( v) x 0as w goes to .

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This lemma sets additional constraints on the feasible solutions to the ODE. Such

constraints make the searching a numerical solution much easier.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.5 1 1.5 2 2.5 3

First-best

P = 1%

P = 99%

s = -50%

s = -38%

s = -26%

s = -14%

s = -02%

s = 10%

s = 22%

s = 34%

s = 46%

s = 58%

s = 70%

Figure 3: Optimal contracts against final wealth. Both final wealth and pay schedules

are normalized by the initial wealth. Label s represents the second-best conditional

on returns of a benchmark portfolio and P represents the conditional cumulative

probability distributions of final wealth. Preferences are power utilities and the rel-

ative risk aversion coefficients of the principal and agent are p = 3 and a = 0.3,

respectively. The gross benefit of the principal (w) = 0.02w. The parameters are:

p = 0.5, s = 10%, s = 30%, r = 5%, and = 0.8.

We present two numerical examples in the following. In the first example, theprincipal is assumed to collect a 2% fee on the final wealth from the fund investors.

The model parameters are stated in the figures captions. Figure 3 plots the optimal

contracts, both the first- and second-best, against the final wealth, where the second-

best pay schedules are conditional on the returns of the benchmark index. In addition,

the cumulative distribution conditional on the benchmark returns for 1% and 99%

is also ploted in the graph. One of the striking features of the second-best contracts

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is the flatness conditional on the benchmark returns within a reasonable range of

portfolio returns. This indicates managers will not get large rewards for dazzling

performance relative to the benchmark. However, they do get hefty rewards when

the returns of the benchmark are high even the return of the managed portfolio is

below the benchmark.

0.002

0.00205

0.0021

0.00215

0.0022

1 1.5 2 2.5 3 3.5

s = 9.94%

s = 10.06%

s = 10.18%

P = 1.00%

P = 99.00%

Figure 4: Second-best contracts against final wealth. Both final wealth and pay

schedules are normalized by the initial wealth. Label s represents the second-best

conditional on returns of a benchmark portfolio and P represents the conditional

cumulative probability distributions of final wealth. Preferences are power utilities

and the relative risk aversion coefficients of the principal and agent are p = 3 and

a = 0.3, respectively. The gross benefit of the principal (w) = 0.02w. The param-eters are: p = 0.5, s = 10%, s = 30%, r = 5%, and = 0.8.

Another interesting feature to notice is the similarities between the contour curves

for fixed conditional cumulative probabilities, e.g., the curve with P = 1% in Figure 3,

and the first-best pay schedule. This illustrates the role of the benchmark in designing

the second-best contracts. The benchmark makes the second-best contracts more

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first-best like, and thses two kinds of contracts become exactly the same when the

return of benchmark is perfectly correlated with the state price. This is of course an

indication of efficiency improvement by incorporating the benchmark in designing the

second-best contracts. Such efficiency gains are also evident by the clear separations

of the conditional pay schedules. The usefulness of the benchmark is determined by

the correlation between the benchmark and state price; the conditional pay schedules

get closer with a lower correlation.

Of course the flatness of the second-best contracts is only a relative term. For

example, the pay should converge to 0 with the gross return. The plots truncate the

pay schedule to illustrate its insensitivity to returns in a reasonable range. Figure 4

further illustrates the detailed second-best contract conditional on particular levels of

the benchmark returns. It is obvious that the conditional pay schedules are concave

but the absolute variations are quite small. This is why they look very flat in Figure

3.

Figures 5 and 6 replicate the previous two plots but with different preferences and

gross benefit of the principal (e.g., a wealthy individual investor). At this time, the

principal is less risk averse. Although, as indicated by Figures 1 and 2, the shapesof the first-best contracts are quite distinguished by the relative risk aversion be-

tween the principal and agent, the conditional second-best contracts are quite similar

in terms of flatness and concavity, especially in the reasonable range of portfolio re-

turns. All qualitative observations we have discussed for the previous case are exactly

applied here, too. The less sensitivity to the benchmark is due the less divergence of

preferences between the principal and agent.

There are two opposite forces working against each other in the second-best con-tracting problem; one is the risk-sharing, the other is the similarity. The latter means

designing a pay schedule such that effective preferences for the agent, v(y), is similar

to the principals. As shown in the case of first-best contracting problem, optimal

risk-sharing is to let the less risk-averse party take more risk. However, the similarity

condition requires that the less risk-averse party taking more risk, hence the effective

preferences after splitting the gross returns are similar. These two opposite condi-

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0.0005

0.0006

0.0007

0.0008

0.0009

0.001

0.0011

0.0012

0 0.5 1 1.5 2 2.5 3 3.5

s = 70%

s = 60%

s = 50%

s = 40%

s = 30%

s = 20%

s = 10%

s = 00%

s = -10%

s = -20%

s = -30%s = -40%

s = -50%

First-best

P = 01%

P = 99%

Figure 5: Optimal contracts against final wealth. Both final wealth and pay schedules

are normalized by the initial wealth. Label s represents the second-best conditional

on returns of a benchmark portfolio and P represents the conditional cumulative

probability distributions of final wealth. Preferences are power utilities and the rel-

ative risk aversion coefficients of the principal and agent are p = 1.5 and a = 3,

respectively. The gross benefit of the principal (w) = w. The parameters are:

p = 0.5, s = 10%, s = 30%, r = 5%, and = 0.8.

tions make the second-best contracts flat, especially in comparison to the first-best

contracts. This qualitative feature of the second-best contracts also holds for the ones

without any signals. As shown in Li and Zhou (2005), the flatness is also prevalent in

the case in which the state price follows a uniform distribution, and it also depends

on the level of reservation, which affects the relative strength between risk-sharing

and preference similarity.

Although the second-best contracts are concave in the reasonable range of portfolio

returns for the two examples, we cannot conclude that they are always concave,

especially for extreme returns. Again, as shown in Li and Zhou (2005), the second-

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0.00079237

0.00079238

0.00079239

0.0007924

0.00079241

0.00079242

0 1 2 3 4 5 6 7 8

s = 10%

P = 1%

P = 99%

Figure 6: Second-best contracts against final wealth. Both final wealth and pay

schedules are normalized by the initial wealth. Label s represents the second-best

conditional on returns of a benchmark portfolio and P represents the conditional

cumulative probability distributions of final wealth. Preferences are power utilities

and the relative risk coefficients of the principal and agent are p = 1.5 and a = 3,

respectively. The gross benefit of the principal (w) = w. The parameters are:

p = 0.5, s = 10%, s = 30%, r = 5%, and = 0.8.

best contracts can be a combination of locally concave and convex curves. A more

detailed study of the second-best contracts is interesting. However, such an inquiry,

which calls for a rigorous analytical treatment of the ODE and a tailor-made numerical

method, is outside the scope of this paper.

As a final note, we notice that the shape of the second-best contract in the clas-

sical principal-agent models with moral hazard is steeper than that of the first-best

contract. This is consistent with our intuition, since the principal needs to motivate

the agent to exert a high effort. In contrast, the shape of the second-best contract in

our context is flatter than that of the first-best contract, as the investor does not have

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to motivate a high managers effort. This different prediction of the optimal contract

shows the key difference between the model with effort and the model with choice.

5 Conclusion

The stock indexes are created for many different purposes. In this paper we show

that one of their side effects is socially valuable, that is, they can help to reduce, or

eliminate when properly constructed, agency costs when the fund mangers portfolio

choice cannot be contracted upon. The condition for a stock index to achieve the

first-best risk sharing is strikingly simple: it only needs to be a market portfolio,

which is by definition perfectly correlated with the state price. In this situation, the

funds total return plus the benchmark return can implement the first-best contract.

In general, however, the market portfolio may not be observable. Then, a close

substitute, e.g., a stock index, can be used in the contract design. In this case the

magnitude of efficiency improvement depends on how close the stock index is to the

market portfolio.

The next natural step to enrich our model would be to introduce the fund man-

agers private information or other information structures into the analysis. Doing so

would shed light on the nature of contracts for the managers of actively managed

funds, which would be a complement to what we have done in this paper. This is

a largely unexplored area. As mentioned earlier, Stoughton (1993), and Admati and

Pfleiderer (1997) point out some problems of the popular linear contract and bench-

marks in the presence of private information. However, their conclusion is based on

a given class of compensation schemes. The true role of benchmarks in the design ofoptimal compensation schemes in this rich environment is still not well understood,

and a lot of work is left to be done along this line.

33


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Appendix: Proofs

Proof of Lemma 1

See Lemma 1 in Li and Zhou (2005).

Proof of Lemma 2

We need the following result first to proceed the proof. This result is also used in

solving the contracting problems later.

Lemma A.1 For any given distribution function fw|s, letE[pw] be the present value

of the wealth under a perturbed density function fw|s = fw|s + , where is a constant

and satisfies

(t, s) dt = 0. Then,

dE[pw]

d=

ds fs(s)

0

(w, s)

w

F1p|s (1 Fw|s(t)) dt

dw

and

d2E[pw]

d2=

ds fs(s)

0

1

fp|s(F1

p|s (1 Fw|s(w)))

w0

(t, s) dt

2dw,

where is an arbitrary number such that fw|s() > 0 or F1

p|s (1 Fw|s()) is finite.

Proof. Perturb the density function fw|s by (w, s), where is a small constant and

(w, s) is an arbitrary piece wise smooth function that satisfies

0

(w, s) dw = 0.

Let Fw|s denote the cumulative distribution function conditional on the signal s. Note

that

1 Fw|s(w) = 1 w

0

[fw|s(t) + (t, s)] dt =

w

[fw|s(t) + (t, s)] dt.

Let

Es,[w] =

0

fw|s(w)F1

p|s (1 Fw|s(w))wdw.

34


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Then, the first derivative of the perturbed expected wealth is

dEs,[w]

d =

0(w, s)wF

1p|s (1 F

w|s(w)) dw

0

wfw|s(w)

fp|sw

0(t, s)dtdw.

Applying the integration by parts to the second integral of the right hand side yields0

wfw|s(w)

fp|s

w0

(t)dtdw

=

w

tfw|s(t)

fpdt

w0

(t) dt

0

0

(w)

w

tfw|s(t)

fp|sdtdw

=

0

(w)wF1

p|s (1 Fw|s(w)) F1p|s (1 Fw|s())

w


dw,

where is a positive number. This shows that

dEs,[w]

d=

0

(w)

w


dw.

Then, taking derivatives to the above equation again and integrating by parts shows

that

d2Es,[w]

d2 =

0(w, s)

w

1

fp|st

0(a, s) dadt

fp|s

0(a, s) da

dw

=

0

1

fp|s(F1

p|s (1 Fw|s(w)))

w0

(t, s) dt

2dw,

where the term with equals zero. Finally, the relation between E and Es, leads to

the lemma.

Let x(w, s) = v(y(w, s)). To maximize the Lagrangian given by (15) is to maximize

the following:

Vs =

0

x(w, s)fw|s(w) dw

0

fw|s(w)[F1

p|s (1 Fw|s(w))]w dw w0

+

0

f(w, s)fw|s(w) dw,

where is a positive constant and f(w, s) is a nonnegative function, which equals

zero when fw|s > 0 and is nonnegative when fw|s = 0. Then, following the proce-

dure as described in the proof of Lemma 1, let fw|s(w) = fw|s(w) + (w, s), where

35


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0

(w, s) dw = 0 and fw|s is the optimal solution. Then, define the perturbed La-

grangian as

Vs, =

0

[x(w, s) + f(w, s)] fw|s(w) dw

0

fw|s(w)[F1

p|s (1 Fw|s(w))]w dw

.

Using Lemma 1, the first two derivatives of the Lagrangian with respect to are

dVs,d

=

0

(w, s)

x(w, s) + f(w, s)

w

F1

p|s (1 F

w|s(t)) dt + F1

p|s (1 F

w|s())

dw,

andd2Vs,

d2=

0

1

fp|s(F1

p|s (1 Fw|s(t)))

w0

(t, s) dt

2dw < 0.

This shows that the Lagrangian Vs always has an interior maximum and the first-order condition dV

s,

d|=0 = 0 is both sufficient and necessary when we restrict fw|s 0.

Yet this is true if and only if there exists a function c(s) which does not depend on

w, such that

x(w, s)

w

F1p|s (1 Fw|s(t)) dt F1p|s (1 Fw|s()) + f(w, s) c(s) (52)

holds almost everywhere. As x is bounded above by v((w, s)), there exists a function

Fw|s or fw|s such that equation (52) holds.

This equality and Lemma 1 imply that

x(, s) F1p (1 Fw()) = c(s)

for all {t| > fw|s(t) > 0}.Let (s) = x(, s) and define

x(w, s) =

w

F1p|s (1 Fw|s(t)) dt + (s).

As x(w, s) x(w, s) and x(w, s) is nondecreasing and concave, x(w, s) is the smallestconcavification of x(w, s) for each s. Otherwise, it contradicts the first-order condi-

tion.

36


38/48

Proof of Proposition 1

The first-order derivatives ofUwith respect to and are straightforward by notingthat

x(w, s) =

w

F1p|s (1 Fw|s(t)) dt + (s).

Then the second order derivatives are given by

2U2

= 12

ds fs(s)

0

fw|s(w)[x(w, s) (s)]2 [uh]

xdw < 0;

2Ud(s)

= 1 ds fs(s)

0

fw|s(w)[x(w, s) (s)] [uh]

xdw;

2U2 (s)

=

ds fs(s)

0

fw|s(w)[uh]

xdw < 0,

where the negativeness of the second derivatives is due to the fact that

[uh]

x= u[h]2 + uh > 0,

because both u and v are concave functions. A direct calculation shows

2U2

2U2

2U(s)

2> 0.

Since this holds for any (, (s)), the solution to the first-order condition maximizes

the objective globally and is unique.


Let Us,

denote the perturbed Us

, that is

Us, =

0

[u + vx + f(w)]f

w|s dw wEs,[pw] + ww0 vv0.

Then the first-order derivative of U isdUs,

d=

0

(w, s)[u + vx + f(w)] dw

+

0

fw|sd[u + vx

]

ddw w dE

s,[pw]

d. (53)

37


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First note that, using (18) or Lemma 2, we have

dx

(w)d = w

1fp|s

t0

(a, s) da dt.

Then, 0

fw|s(w)d[u + vx

]

ddw =

0

[uh v]fw|sdx(w)

ddw

=

0

[uh v]fw|s(w)w

1

fp|s

t0

(a, s) dadt

dw,

where we use the fact that the derivatives of and do not depend on w. Note that

the last two terms are equal to zero due to the first-order conditions in Proposition

1. Therefore, integration by parts shows0

fw|s(w)d[u + vx

]

ddw

=

0

[uh v]fw|s(w)w

1

fp|s

t0

(a, s) dadt

dw

= w

0

[uh v]fw|s(t) dtw

1

fp|s

t0

(a, s) dadt

0

0

1fp|s

w

0

(t, s) dt

w

0

[uh v]fw|s dt

dw

=

0

(w, s)

w0

1

fp|s

t0

[uh v]fw|s dadt

dw,

where we have use the first-order condition for , equation (23). Using this equation

and Lemma 1, the first derivative of U with respect to is:dUs,

d=

0

(w, s) u + vx + f +

w

0

1

fp|s t

0

[uh v]fw|s dadt dw w

0

(w, s)

w


dw, (54)

38


40/48

where we have used Lemma 1. However, when = 0, we have x = x and

dUs,

d

=0=

0

(w, s)

u + vx + f + w

1fp|s

t0

[uh v]fw|s dadt dw w

0

(w, s)

w


dw

=

0

(w, s)

u + vx + f

+

w

1

fp|s

t0

[uh v]fw|s dadt w

x +w

dw = 0,

where the second last equality is obtained by using the constraints (18) or Lemma 2.

The above integral equals zero for any arbitrary such that0

(w, s) dw = 0

if and only if there exists a c(s) that is independent of w such that

u((w, s) h(x(w, s))) + vx(w, s) w

x(w, s) + f(w, s)

+ w

1

fp|s t

0

[uh v]fw|s(a) dadt = c(s) (55)

holds almost everywhere, where we ignore the constant terms. Specifically, we have

u((w, s) h(x(w, s))) +

v w

x(w, s)

+

w

1

fp|s

t0

[uh v]fw|s(a) dadt = c(s)

when fw|s(w) > 0.


The derivation of the second-order ODE is straightforward. See Li and Zhou (2005)

for the other results.

39


41/48

Proof of Lemma 3

Let F(w, y|s) is the joint cumulative distribution function for given w() A andy() conditional on the signal s, we then have

u((w(), s) y()) P(d)

=

ds fs(s)

u((w(), s) y()) P(d|s)

=

ds fs(s)

u((w, s) y) dF(w, y|s)

=

ds fs(s)

dFw|s(w)

u((w, s) y) dF(w, y|s, w)

ds fs(s)

u((w, s)) E[y|s, w]) dFw|s(w).

The last inequality is due to Jensens inequality and becomes equality if

y() = y(w, s) for all {|(w() = w, s() = s}.

This means the optimal pay takes the form of y(w, s).


Because x or (y(w)) is a free choice function, the maximization problem is quite

straightforward. It is similar to the case of agents problem for fw. The second order

is automatically satisfied by the budget constraint. And for the case of x, the second

order is due to that fact that u((w) h(x)) is a concave function of x for any fixedw. We skip the details of the calculations.

Proof of Corollary 1

Iff(w, s) = 0, fw|s > 0. Then the two first-order conditions in Proposition 4 become

u = wp and u = vv

. (56)

40


42/48

Then it is straightforward to check that these two equations imply the pay schedule

and the final wealth in the corollary.

Having recognized that u + vx is concave in w if the condition is satisfied, the

second part follows immediately.

Proof of Theorem 1

It is trivial to examine the contracting problems or the first-order conditions. If fp|s

is singular at s, then fw|s(w) is also singular. Therefore, the first-order conditions

(22) and (23) hold only ifuh = v, then the third first-order condition (24) becomes

equation (35) in light of equation (18).

Proof of Corollary 2

By Proposition 4, for any w / (wl, wh), where wl and wh satisfy equations (44), wehave u = wp and u

= vv. Substituting these equation into equations (16) leads

to

u((wh, s) y(wh, s)) + vv(y(wh, s)) u((wl, s) y(wl, s)) vv(y(wl, s))wh wl

= wp, (57)

where s = Rs is as defined by equation (43). Also, by Corollary 1, we have the pay

schedule as given by

y(w(p), s(p)) =

wp

v0

1

aif w

wl

wpv(0+1)

1a

if w wh

and the final wealth is given by

w(p) =

10

wp0

1p

+

wpv0

1a

if w wl

10+1

wp

0+1

1p

+ 1w0Rp 1m +

wp

v(0+1)

1a

if w wh.

41


43/48

Then, substituting these into (57) yields equation (46).

Given p, the pay schedule and final wealth can be rewritten as in the the theorem.Then, the budget constraint isp

0

w(p)pfp(p) dp +

p

w(p)pfp(p) dp = w0.

Using the identities, for any ,p0

pfp(p) dp =1

2p

ln p

ex e 1

22p(xp)2

dx

= exp p 2p

2N ln p p +

2p

p ,

and p

pfp(p) dp = exp

p

2p2

1 N

ln p p + 2p

p

gives us equation (47). Similarly, equation (48) is also obtained from the participation

constraint.

Proof of Lemma 4

Since both fp|s andw

0[uh v]fw|s(t) dt converge to 0 as w goes to , LHopitals

rule implies

fp|s(p)

w0

[uh v]fw|s(t) dt [uh v]fp|s(p)

fp|s(p)

as w goes to . Then, equation (25) shows

u[0

hx] + (

v)x

[uh

v]

fp|s(p)

fp|s(p).

Suppose this converges to a constant b. That is

u[0 hx] + ( v)x [uh v]fp|s(p)

fp|s(p)

b. (58)

As w goes to , x = p converges to 0 and fp|s(p)fp|s

(p)converges to 0. In this case, at

best, uh 1p

by equation (58) andfp|s(p)

fp|s

(p) p

lnp, hence b has to equal 0.

42


44/48

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