The Principle-Agent Conflict Problem in a Continuous-Time Delegated Asset Management Model

Li, Yanan; Li, Chuanzheng

doi:https://doi.org/10.1155/2021/3770868

Mathematical Problems in Engineering

On this page

Abstract Introduction Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article Corrigendum

!

A Corrigendum for this article has been published. To view the article details, please click the ‘Corrigendum’ tab above.

Special Issue

Stochastic Optimal Control and its Applications

View this Special Issue

Research Article | Open Access

Volume 2021 | Article ID 3770868 | https://doi.org/10.1155/2021/3770868

The Principle-Agent Conflict Problem in a Continuous-Time Delegated Asset Management Model

Yanan Li¹and Chuanzheng Li¹

Academic Editor: Xiaofeng Zong

Received09 Jul 2021

Revised06 Aug 2021

Accepted17 Aug 2021

Published26 Aug 2021

Abstract

This paper considers the principle-agent conflict problem in a continuous-time delegated asset management model when the investor and the fund manager are all risk-averse with risk sensitivity coefficients , respectively. Suppose that the investor entrusts his money to the fund manager. The return of the investment is determined by the manager’s effort level and incentive strategy, but the benefit belongs to the investor. In order to encourage the manager to work hard, the investor will determine the manager’s salary according to the terminal income. This is a stochastic differential game problem, and the distribution of income between the manager and the investor is a key point to be solved in the custody model. The uncertain form of the incentive strategy implies that the problem is different from the classical stochastic optimal control problem. In this paper, we first express the investor’s incentive strategy in term of two auxiliary processes and turn this problem into a classical one. Then, we employ the dynamic programming principle to solve the problem.

1. Introduction

Since professional asset management institutions can make efficient investment decisions, save investors’ time and effort, and simplify the investment process, more and more investors now entrust their money to fund managers, securities firms, and other asset management organizations. Nowadays, scholars pay more and more attention to asset management problems. We can refer to [1–5] to name just a few.

The whole asset management process involves two parties: the investor and the manager. The return of the investment is closely related to the manager’s effort level and investment strategy, but the interests belong to the investor. So, the investor and manager’s relation poses a principal-agent conflict. An important part of discussing the asset management problem is finding the investor’s optimal incentive mode under the principle agent conflict.

There are many papers committed to solving principal-agent conflict problems. Most of the early literature studies investigate the discrete-time case (we can refer to [6–8] or a summary book [9]). The problem in continuous-time models is discussed for the first time in [10]. It points out that the investor’s optimal incentive mode is linear. See references [11–14] for further work. In recent years, the maximum principle or the martingale representation theorem is often used to solve this problem in continuous-time models. For the literature using the maximum principle, we can refer to [15, 16], and for the literature of using the martingale representation theorem, we can refer to [17, 18]. However, since this problem often needs to solve a backward stochastic differential equation (BSDE) that rarely has explicit solutions, there are few articles which give analytical solutions to this problem. In order to get explicit solutions of principal-agent conflict problems, the authors of [19] express the investor’s incentive strategy in terms of two auxiliary processes and turn the principle agent problem into a classical stochastic differential game problem.

Although there are many papers committed to solving principal-agent conflict problems in continuous-time models, the delegated asset management problems are usually investigated in discrete-time models for the sake of simplicity. Thus, there are some contributions in this paper:(i)This paper considers the delegated asset management problem in a continuous-time model(ii)Learning from [19], this paper gives explicit value functions and the optimal strategies of both sides by expressing the investor’s incentive strategy in terms of two auxiliary processes and turning the problem into a classical stochastic differential game problem(iii)In order to make the model more realistic, this paper brings in risk sensitivity coefficients to represent the subjects’ risk aversion attitudes

This paper is organized as follows. In Section 2, we establish a continuous-time model of the fund management problem. In Section 3, we discuss the manager’s optimization problem under fixed investor’s incentive strategy. By substituting the manager’s optimal strategy into the investor’s optimal problem, both the investor and the manager’s optimal strategies are obtained in Section 4.

2. The Principal-Agent Conflict Model

Similar to the model in [20], let us assume that the investor employs a professional fund controller (manager) to invest and the investor will get a profit and pay the manager at the terminal moment . Since the manager’s effort level cannot be observed, the investor will determine the manager’s salary according to the terminal profit of the investment. The investor’s return is determined by the terminal investment profit and the manager’s salary. The terminal investment profit is related to the manager’s investment strategy and effort level, and the incentive mechanism largely determines the manager’s strategy. Therefore, the investor needs to find the optimal incentive mechanism (the manager’s salary) to maximize his terminal net income. Meanwhile, according to the investor’s incentive mechanism, the manager shall decide his investment strategy and the best effort level to maximize his net salary (terminal salary minus effort cost). This is a non-cooperative game problem. Next, let us build a mathematical model of this problem in probability space .

Similar to the model in [18], we suppose that the manager’s effort will affect the fund income which satisfieswhere , , and is the risk-free interest rate, is a Brownian motion on , and is the manager’s effort level. Here, for the convenience of calculation, we assume that the drift coefficient of is a linear function of the manager’s effort level. In fact, as long as the drift coefficient of has the form of for some function , the same method in this paper can be used after replacing with . For more general forms of the drift coefficient of , the existence of the time value makes it hard to obtain explicit solutions.

Considering the manager’s strategy , where represents the wealth that the manager decides to operate at moment (The manager may not want to operate all the wealth since the cost of the effort will increase with the wealth operated increases. The money left will get a risk-free return.) and represents the manager’s effort level at . By some simple calculations, we can get that the investment income under this strategy satisfies

Define the natural filtration produced by as . Now, let us give the definition of both the manager and the investor’s admissible strategies. Considering the manager’s strategy . If are bounded positive predictable stochastic processes, under the strategy , (2) has a unique solution.

We call that strategy is admissible. Denote the set of all the manager’s admissible strategies by .

Remark 1. Here, we do not consider the case when or since in that case, the model is meaningless.

Suppose that the investor’s incentive strategy is a function of the investment income at and denote it by . If , the manager’s value function under is a decreasing convex function with respect to the initial wealth, we say that is the investor’s admissible strategy. Denote the set of all the investor’s admissible strategies by .

Now, let us analyze the whole game process. Referring to [15], we know that investors play a leading role in the game. Managers need to decide their effort level and investment strategy according to the investors’ incentive strategy. Therefore, first, we need to fix and investigate the manager’s optimal problem. We can get the manager’s optimal effort and investment strategy in terms of as a byproduct. Then, by substituting the manager’s optimal strategy into the wealth process, we can solve the investor’s optimal problem by using the dynamic programming principle.

Therefore, firstly, we fix the investor’s incentive strategy and consider the manager’s optimal problem. Suppose that the manager is risk-averse and denote his risk sensitivity coefficient by . Referring to [18], we suppose that the manager needs to pay to manage units of capital in unit time under the effort level . Here, is a constant which represents the effort cost parameter. The objective of the manager is to find the optimal effort level and investment strategy to maximize his net income (salary minus effort cost), which is equivalent to minimize

Denote the manager’s optimal strategy by , then the value function is

Suppose that the investor is risk-averse too, his risk-sensitive coefficient is . Next, we consider the investor’s optimal problem.

If the manager’s salary is too high, the investor’s income will be reduced. If the manager’s salary is too low, the manager’s enthusiasm wanes, which also deduces the investor’s terminal income. Therefore, the investor needs to find a reasonable incentive strategy to maximize his net income, that is, minimizewhere is the investment income process under strategy . Thus, the investor’s value function is

Remark 2. The problem discussed above is not a standard stochastic optimal control problem since the form of is uncertain, and we cannot solve it directly by using standard stochastic optimal methods. In Section 3, we give another form of the incentive strategy and transform the game problem into a classical one. Then, we can use the dynamic programming principle to solve the problem.

3. The Manager’s Optimization Problem

Define , , and . Then, can be denoted by

Using the results of Section 3.4 in [21], we know that, under the incentive strategy , the manager’s value function satisfies the HJB equation:and the boundary condition

Since is a decreasing convex function of , for , we can define the Hamiltonian function:where

Theorem 1. is the minimum point of in (10).

Proof. According to the definition, we know that is a convex function of . So, the minimum point of in (10) is the stable point under constraint conditions . By some simple calculations, we haveCombining the above two equations, we can obtain the stable point of :The proof is done.

Remark 3. In this case, the optimal investment strategy is similar to that without principal-agent relationships. The only difference is that the numerator of the optimal investment strategy is changed from into . Clearly, this is due to the existence of the agency relationship.

Apparently, the investor’s incentive strategy and the manager’s value function are one-to-one. In the following, we will use auxiliary stochastic processes to determine the manager’s value function and transform the investor’s incentive strategy into . Then, the problem in Section 2 can be translated into a classical stochastic optimal control problem.

First, let us give the space of auxiliary stochastic processes . Fix , let be -predicable processes which satisfy

Denote the set of all the processes satisfying the above conditions by .

For some and , define the -progressively measurable process on the filtration space bywhere is the investment income process. Clearly, for fixed , is only related to the investment income process and is measurable, suppose that it is an incentive strategy (we prove it in Corollary 1). In the following, we give the relationship between and the manager’s value function. First, we give the following lemma.

Lemma 1. Defineand then we have .

Proof. On the one hand, since are all predictable stochastic processes, referring to (12) and (13), we can get that are bounded positive predictable stochastic processes. On the other hand, are independent of . Taking into (2), we can get the Lipschitz continuity and linear growth of the coefficients in (2) with respect to ; then, (2) has a unique solution. The proof is done.
Denote the investment income process under by . We also have the following theorem.

Theorem 2. Denote the manager’s value function with a terminal return by . We can obtain that

Furthermore, the manager’s optimal strategy is .

Proof. , we haveUsing Ito’s formula, we haveIt follows from (16) that is a martingale. Integrating and taking expectations on both sides of (21), we can getFurthermore, by simple calculations, under , we haveUsing (23) and Ito’s formula, we can obtainWith similar methods, integrating and taking expectations on both sides of (24), we haveThis implies that is the manager’s optimal strategy andUp till now, fixing , we can get the manager’s optimal strategy and represent the manager’s value function. In Section 4, we begin to consider the investor’s optimization problem. That is, finding the optimal to maximize the investor’s net profit.

4. The Investor’s Optimization Problem

Suppose that the investor’s wealth is at . Apparently, the investor’s value function is uniquely determined by the wealth process and the manager’s value function. So, the objective of the investor is to find the optimal to minimize his value function. Define

Referring to Theorem 4.1 in [19], we know that if Assumption 3.2, Assumption 4.3, and Assumption 4.4 in [19] hold, the investor’s value function satisfies

Here, is the minimum pay in order to make sure that the manager takes the job.

Section 4.1 gives the verification of the three assumptions.

4.1. The Verification of Assumptions

Assumption 1 (Assumption 3.2 in [19]). has at least one extreme point . For any , we have .

Proof. This is the result of Theorem 1 and Lemma 1.
The Hamiltonian function can be expressed asHere,Defineand we have the following assumption.

Assumption 2 (Assumption 4.3 in [19]). has at least one extreme point ; furthermore, .

Proof. On the one hand, the right hand of is a parabola with an opening up with respect to ; so, the minimum point is attained at the axis of the parabola , that is, . On the other hand, since is predictable, we can get that is a positive predictable process. Furthermore, are independent of . This implies the Lipschitz continuity and linear growth of the coefficients in (2) with respect to the investment income process; then, (2) has a unique solution.

Assumption 3 (Assumption 4.4 in [19]). , is bounded.

Proof. We can get the result directly from .

4.2. The Investor’s Value Function

Clearly, as soon as we get , we can obtain . The following theorem gives the partial differential equation satisfied by .

Theorem 3. is the viscosity solution ofwhere

Proof. By the definition of , we can obtain that it satisfies (33). Furthermore, according to the dynamic programming principle, we haveBy using Ito’s formula with respect to from to , we haveCombining with the above two equations, we can getThat is, satisfies (32). The proof is done.
Next, we are going to solve (32) and (33). Considering the boundary condition, we guesswhere is a function of which satisfies .
If the variables in the solution can be separated from each other, (32) can be easily solved. However, (32) contains , which is a cross term of and . To cancel the cross term, we introduce . Using Ito’s formula, we can getWe can also obtain . DefineObviously, solving is equivalent to solving . Using a similar method as the one in Theorem 3, we can get thatThe first step in solving (41) is to find its minimum point. Define , it is shown in Section 3 that and are one-to-one. Then, (41) is transformed intoNow, the problem of finding the minimum point in (41) is changed into a problem of finding the minimum point in (43).
According to (38), we suppose that . By some simple calculations, we can get thatTaking them into (43), we haveSince the right hand of (45) is continuous, the minimum point can only be attained at the stable points or the boundary points, which depends on the parameter values. Denote the minimum point of (45) by , and denote the corresponding minimum point of (41) by . It is shown from the Appendix that and are constants concerning , and . Let .

Remark 4. On the one hand, the exponential form of the objective function implies that is independent of . On the other hand, the benefit and the cost brought by the manager’s effort are only related to , so is independent of . Furthermore, in this paper, we consider the discounted benefit and cost brought by the manager’s effort; so, is independent of .

Remark 5. It is shown from figures in the Appendix that decreases with an increase in (the drift coefficient of the fund wealth process), (the effort cost coefficient), and (the manager’s risk aversion level). It increases with an increase in (the investor’s risk aversion level).

Remark 6. Define , considering (12) and (13), we can get that and are constants.

Taking the minimum point into (45) and solving it, we can get

Here,is a constant. As a consequence, we can also get the following results.

4.3. The Investor’s Excitation Mechanism

In this section, let us analyze the investor’s excitation mechanism. Denote . From the above analysis, we know that

Using Ito’s formula, we have

Furthermore, we can get that the investment income under satisfieswhich implies

Define constantand then we can obtain

So,can be deduced immediately. Since , we can get the strategy

It is a linear function of , which is the discounted profit of the investment.

Remark 7. It follows from the above results that the manager’s wages increase with the increase of the cost coefficient, the effort level and the discounted profit of the investment. Furthermore, the longer the work, the higher the salary. It is consistent with reality.

We can also get the following corollary.

Corollary 1.

This implies that is a decreasing convex function of . Thus, the assumption in Section 2 that is an incentive strategy is proved.

Appendix

Define

We know that there are three kinds of points which may be the minimum point of (45):(i)The points which satisfy (ii)The points which satisfy (iii)The points which satisfy

With parameters fixed, we can easily decide which is the minimum point of (45). In the following, we will investigate the form of those points.

The first kind of points is the solution of the following equations:

We can deduce from (A.2) that

It also follows from (A.3) that

Combining the above two equations, we can get that

Clearly, by solving (A.5) and (A.6), we can get that and are constants.

Denote the second kind of point by . Thus, satisfies (A.5) with replaced with 0 and we can get that is a constant.

Denote the third kind of points by . They satisfy (A.4). By solving it, we can get that is a constant.

Denote the minimum point of (45) by . It follows from the above analysis that and are all constants. For different , and , by calculating (A.6), (A.2), or (A.3), we can get different .

By using R, we plot the following figures which indicate the effect of , and on (Figure 1).

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (11901404).

References

A. Almazan, K. C. Brown, M. Carlson, and D. A. Chapman, “Why constrain your mutual fund manager?” Journal of Financial Economics, vol. 73, no. 2, pp. 289–321, 2004.
View at: Publisher Site | Google Scholar
M. K. Brunnermeier and L. H. Pedersen, “Market liquidity and funding liquidity,” Review of Financial Studies, vol. 22, no. 6, pp. 2201–2238, 2009.
View at: Publisher Site | Google Scholar
P. H. Dybvig, H. K. Farnsworth, and J. N. Carpenter, “Portfolio performance and agency,” Review of Financial Studies, vol. 23, no. 1, pp. 1–23, 2010.
View at: Publisher Site | Google Scholar
S. Gervais, A. W. Lynch, and D. K. Musto, “Fund families as delegated monitors of money managers,” Review of Financial Studies, vol. 18, no. 4, pp. 1139–1169, 2005.
View at: Publisher Site | Google Scholar
V. Guerrieri and P. Kondor, “Fund managers, career concerns, and asset price volatility,” The American Economic Review, vol. 102, no. 5, pp. 1986–2017, 2012.
View at: Publisher Site | Google Scholar
S. Ross, “The economic theory of agency: the principal’s problem,” The American Economic Review, vol. 63, pp. 134–139, 1973.
View at: Google Scholar
J. A. Mirrlees, “The optimal structure of incentives and authority within an organization,” The Bell Journal of Economics, vol. 7, no. 1, pp. 105–131, 1976.
View at: Publisher Site | Google Scholar
B. Holmstrom, “Moral hazard and observability,” The Bell Journal of Economics, vol. 10, no. 1, pp. 74–91, 1979.
View at: Publisher Site | Google Scholar
P. Bolton and M. Dewatripont, Contract Theory, MIT Press, Cambridge, UK, 2005.
B. Holmstrom and P. Milgrom, “Aggregation and linearity in the provision of intertemporal incentives,” Econometrica, vol. 55, no. 2, pp. 303–328, 1987.
View at: Publisher Site | Google Scholar
H. Schättler and J. Sung, “The first-order approach to the continuous-time principal-agent problem with exponential utility,” Journal of Economic Theory, vol. 61, no. 2, pp. 331–371, 1993.
View at: Publisher Site | Google Scholar
H. Schättler and J. Sung, “On optimal sharing rules in discrete-and continuous-time principal-agent problems with exponential utility,” Journal of Economic Dynamics and Control, vol. 21, no. 2-3, pp. 551–574, 1997.
View at: Publisher Site | Google Scholar
H. M. Müller, “The first-best sharing rule in the continuous-time principal-agent problem with exponential utility,” Journal of Economic Theory, vol. 79, no. 2, pp. 276–280, 1998.
View at: Publisher Site | Google Scholar
H. M. Müller, “Asymptotic efficiency in dynamic principal-agent problems,” Journal of Economic Theory, vol. 91, no. 2, pp. 292–301, 2000.
View at: Publisher Site | Google Scholar
J. Cvitanic, X. Wan, and J. Zhang, “Optimal compensation with hidden action and lump-sum payment in a continuous-time model,” Applied Mathematics and Optimization, vol. 59, pp. 99–146, 2009.
View at: Google Scholar
J. Cvitanic and J. Zhang, Contract Theory in Continuous Time Models, Springer-Verlag, Berlin, Germany, 2013.
Y. Sannikov, “A continuous-time version of the principal-agent problem,” The Review of Economic Studies, vol. 75, no. 3, pp. 957–984, 2008.
View at: Publisher Site | Google Scholar
Z. He, “A Model of dynamic compensation and capital structure,” Journal of Financial Economics, vol. 100, no. 2, pp. 351–366, 2011.
View at: Publisher Site | Google Scholar
J. Cvitanic, D. Possamai, and N. Touzi, “Dynamic programming approach to principal-agent problems,” Finance and Stochastics, vol. 22, pp. 1–37, 2018.
View at: Google Scholar
Z. He and W. Xiong, “Delegated asset management, investment mandates, and capital immobility,” Journal of Financial Economics, vol. 107, no. 2, pp. 239–258, 2013.
View at: Publisher Site | Google Scholar
H. Pham, Continuous-time Stochastic Control and Optimization with Financial Applications, Springer-Verlag, New York, NY, USA, 2009.

Copyright

Copyright © 2021 Yanan Li and Chuanzheng Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies