I. INTRODUCTION

Evidence indicates that investors feel overly anxious or nervous when they invest in risky assets without assistance because investors have little financial knowledge and related information. Hence, they are willing to hire money managers or advisors to help them invest. Managers may have indispensable knowledge concerning how to diversify investments or even how to earn a premium. Additionally, money managers provide investors with peace of mind.

Many researchers are interested in similar problems. In this section, we cite the essential literature on incentives for money managers. Chevalier and Ellison, using semiparametric modeling, report that the shape of the flow-performance relationship creates incentives for fund managers to raise or reduce the riskiness of a fund and that these incentives depend on the fund's year-to-date return (see Judith and Ellison, 1997). By examining the labor market for mutual fund managers, they find that "termination" is more performance-sensitive for younger managers (see Judith and Ellison, 1999). They also identify possible implicit incentives created by the termination-performance relationship. The shape of the termination-performance relationship may give younger managers an incentive to avoid unsystematic risk. Inderst and Ottaviani focus on distorted incentives to sell financial products, distortions that arise not only from actual kickbacks but also from the difficulty of incentivizing salesmen to sell the appropriate products (see Roman and Ottaviani, 2009, 2012a,b). Guerrieri and Kondor demonstrate that performance generates a "reputational premium" that influences investors' decisions to hire or fire money managers (Veronica and Kondor, 2012). Hackethal, Inderst, and Meyer find that retail investors who report a heavy reliance on their advisors' recommendations have a substantially higher trading volume and purchase a higher fraction of investment products that their advisors were incentivized to sell (promoted products)-see Hackethal et al.

2012. Gennaioli, Shleifer and Vishny focus on the incentives of the money management organization itself when its clients' choices are mediated by trust (see Gennaioli et al., 2015).

These researches rely on Expected utility theory (EUT)(see Neumann and Morgenstern, 1944). However, substantial experimental and empirical evidence reveals that EUT is incompatible with human observed behavior in reality. Demonstrated violations of EUT are as follows.

1. EUT makes the underlying assumption that decision-makers are rational and uniformly risk averse. But, empirical evidence shows that investors' risk attitude often changes.

Specially, Kahneman and Tversky did a experiment to observe changes of personal risk attitude. The first choice is between a sure gain of $240anda25$1,000. The second choice is between a sure loss of $750anda75$1,000. For the first choice, 84% of respondents chose a sure gain of $240,whichisconsistentwithriskaversion.Forsecondchoice,87$1,000, which is consistent with risk seeking. This experiment indicates that people exhibit risk aversion in gains and exhibit risk seeking in losses.

2. EUT generally uses the level of wealth, not changes in wealth. However, in the previous experiment, we find that personal risk attitude changes in the domains of gains and losses. So, we can show that gains and losses seem to be what people care about, rather than the level of wealth.

To prove our point of view, we consider the following problem. Decision 1: Assume yourself richer by $300thanyouaretoday.Then,choosebetweenasuregainof$100 and a 50% chance to gain 200; Decision 2: Assume yourself richer by $500thanyouaretoday.Thenchoosebetweenasurelossof$100 and a 50% chance to lose $200.Inbothcases,thedecisionisbetween$400 with certainty and a prospect with a 50% chance of $500anda$300. Yet, 72% of respondents chose a sure gain of $100inDecision1and64$200 Decision 2. The choice of many indicates risk aversion for Decision 1, but risk seeking for Decision 2. This problem shows that risk attitude is not same across gains and losses, implying that it is the change in wealth, rather than the level, that matters to people. People evaluate an outcome based on the gain and loss from a reference point, usually taken to be current wealth. Notice that in this problem the two decisions assume different starting wealth positions.EUT cannot incorporate the reference point.

3. Researchers also notice that persons seem to feel a loss more strongly than a gain of equivalent absolute value. But, EUT doesn't involve this.

To more clearly explain our point of view, we consider the following problem. What value of x would make you indifferent between a sure gain of 0 and the prospect which is a 50% chance to gain x and a 50% chance to lose $25?Theaverageofresponseinthisexperimentwas$61. That is, for a fair gamble, when the loss is $25,thetypicalpersonrequiresagainof$61 to be indifferent between accepting or rejective the gamble. It is quite clear that people are quite averse to a loss. Loss aversion is the term that describes the observation that, for most people, losses loom larger than gain.

4. EUT cannot explain why a person might buy a lottery ticket and insurance. In fact, overweighting of small probabilities and subjective probabilities lead to this behavior.

For example, choose between a 0.1% chance to gain $5,000andasuregainof$5. Even though the expected value of the two case are equal ($5)aswealmostcertainlyhaveobserve,manypeopleprefertheformertothelatter,consistentwithriskseeking,Suchachoiceisindicativeofriskseekinginthedomainofgains.Earlierweobservedanotherinstanceofriskseeking,butthisiswasindomainoflosses.Itseemsthatriskseekingcanalsooccurinthedomainofgainsaswell.Now,weconsideranotherchoice.Choosebetweena0.1$5,000 and a sure loss of $5. In this case many people choose the latter, consistent with risk aversion. But this implies risk aversion in the negative domain. In sum, while we normally have risk aversion in the positive domain, when there is a quite low probability of a payoff this generally shifts to risk seeking. On the other hand, while we normally have risk seeking in the negative domain, when there is a quite low probability of a loss this generally shifts to risk aversion. This is what Kahneman and Tversky characterized as the fourfold pattern of risk attitudes. This pattern suggests risk aversion for gains and risk seeking for losses when the outcome probability is high, and risk seeking for gains and risk aversion for losses when the outcome probability is low. In onw study, they found that 92%(22 out of 25) of subjects displayed the full patters.

To explain the above violations of EUT and more accurately characterize personal behaviors, Kahneman and Tversky propose prospect theory (PT) and cumulative prospect theory (CPT)-see Kahneman and Tversky, 1992a and 1992b.

When we research the interaction between investors and managers, we find investors' behavior also deviates from the predictions of EUT. So, in this article, we apply CPT-investors rather than risk averse investors in the traditional sense. That is, we adopt a cumulative prospect theory (CPT) approach to the investor's preferences. In a original way, we model the investor's trust in the manager and associated anxiety from investing in a risky asset to determine a CPT investor's optimal portfolio and a manager's optimal fees.

The novelty of our contribution relative to Gennaioli et al.(2015) and other works using classical preferences is that we rely on cumulative prospect theory rather than the classical mean-variance framework. Gennaioli, Shleifer and Vishny (2015) assume that all investors are rational. Hence, they develop a model relying on the mean-variance framework, which is certainly the most well-known investment decision rule. However, all of our investors are CPT investors, namely, these investors' behavior coincides with cumulative prospect theory. Specifically, we employ a value function for CPT investors, while Gennaioli, Shleifer and Vishny (2015) use a classical utility function. In the value function of our CPT investors, we consider the gain relative to a reference wealth level, i.e., the benchmark wealth. However, Gennaioli et al. (2015) use absolute wealth in the investors' utility function. Moreover, to accurately characterize a CPT investor's psychology, we combine probability distortions with the value functions to measure the CPT investors' satisfaction, while Gennaioli, Shleifer and Vishny (2015) use only a simple probability and the investors' utility functions to measure investor satisfaction. In other words, Gennaioli et al.(2015) only employ the expected utility functions to measure investor satisfaction. The value function and probability distortions are the key contributions of cumulative prospect theory and allow our CPT model to outperform traditional theory in real financial markets.

Another innovation is that we focus on the dynamic portfolio selection but not the static portfolio choices. Gennaioli et al. only discuss the managers' optimal fees when the investors invest in the risky asset in a single period. However, we explore the managers' optimal fees and the investor's portfolio selection under multi-period setting. Moreover, based on the subgame perfect investment strategies, we obtain the CPT investor's optimal portfolio in multi-period.

In their analysis of multiple financial products, Gennaioli et al. (2015) only consider two risky assets, while we study two or more such assets. Moreover, their research relies on the assumption that the two risky assets are uncorrelated, an assumption that we need not make.

There are two main differences between our results and those based on classical preferences. First, in our results, the optimal fees are not symmetric. In particular, the dominant managers obtain a higher fee than do subordinate managers regardless of the changes in risk of risky assets (a risky asset) and the changes in the dispersion of trust in the population. Another difference is that, in our work, these fees are not proportional to expected returns. Moreover, the optimal fees decline nonlinearly as risk of risky assets (a risky asset) decreases and the dispersion of trust in the population decreases.

Except for our application of CPT in money doctor, CPT is applied extensively in the contexts of optimal investment strategy and optimal insurance contracts.

Regarding optimal investment strategy, in a continuous-time setting, Jin and Zhou (2008) formulate a general behavioral portfolio selection model using Kahneman and Tversky's CPT. In a discrete-time setting, Bernard and Ghossoub (2009) consider how a CPT investor chooses his/her optimal portfolio in a single-period model with one risky and one riskless asset. In the same vein, He and Zhou (2011a,b) address and formulate the general well-posedness issue and investigate the case in which the reference point is not the risk-free return. Pirvu and Schulze (2012) extend this work to a multi-asset context. To the best of our knowledge, Shi et al. (2014) are the first to consider the CPT allocation problem in a multi-period framework. Deng and Pirvu investigate optimal portfolio selection with one risk-free asset and one risky asset in a multi-period setting under CPT (see Deng and Pirvu, 2015). Compared with Shi et al's study, their novelty is that the optimal strategies are time inconsistent due to the time-changing benchmark.

Regarding optimal insurance contracts, there are fewer results (and those that exist are less sophisticated) than in the context of optimal portfolio choice. From the insured's perspective, Dhiab investigates the demand for insurance under CPT (see Dhiab, 2015). From the insurer's perspective, Bernard et al. explore the optimal insurance policy for an insurer with a linear cost function (see Bernard et al., 2015). Based on Bernard et al.'s work, Deng identifies the optimal insurance policy for the more general insurer cost functions, addressing both linear and nonlinear cost functions (see Deng, 2015).

Although CPT is more practical than traditional models, it is not more widely used than traditional theory. As noted above, CPT is applied primarily to address problems related to optimal investment strategy and optimal insurance contracts. To the best of our knowledge, for many other other problems related to CPT, in particular the optimal fees that managers charge under CPT, there are no existing studies. This lack of research is certainly not because this problem is unimportant or uninteresting; rather, we believe, this lack of research is because addressing such problems using CPT is considerably more complicated than doing so using the traditional decision optimization.

In extant CPT research, researchers only consider the value function of a CPT investor who invests in a risky asset without assistance. Moreover, they focus solely on the optimal portfolio for the investor. However, we are interested in investors' interaction with portfolio managers and thus develop a value function for a CPT investor investing in risky assets with the assistance of a portfolio manager, not on his own. Our models accurately characterize and reflect the reduced anxiety experienced by CPT investors when bearing risk with the assistance of a trusted portfolio manager. Moreover, we determine not only the investor's optimal portfolio but also the money managers' optimal fees. However, our original models pose an enormous computational challenge, as the traditional method is not suitable for our new problem. To overcome these computational difficulties, we use rigorous derivation to obtain the implicit solutions. Then, we employ an effective software tool and obtain approximately explicit solutions.

The remainder of this paper is organized as follows. In Section 2, we discuss a basic setting. In section 3, we attain the CPT investor's optimal portfolio in multi-period under CPT. In section 4, we investigate managers' total profits. In section 5, we determine the optimal fees charged by money managers. Section 6 contains the numerical analysis. In Section 7, we compare our results with those of Gennaioli et al. and explain our results from an economic perspective. The paper concludes with an Appendix containing the proofs.

II. THE BASIC SETUP

In this paper, we analyze the dynamic optimal strategies. We consider a financial market in which the CPT-investor can invest in one risk-free asset and one risky asset. The investing horizon is $[0,T]$, where T is a finite deterministic constant. The time of investment t takes on discrete values $(t=0,1,\dots ,T-1)$. Moreover $r_t$ denotes the return of the risk-free asset from the time t to the time $t+1$, and $x_t$ denotes the return of the risky asset from the time t to the time $t+1$. We assume that a CPT-investor i has wealth $W_{i,t}$ at the time t and invest the amount $v_{i,t}$ in the risky asset and all of remaining wealth $W_{i,t}-v_{i,t}$ in the risk-free asset. The investor's wealth $W_{i,t+1}$ at time $t+1$ is given by the equation

Here $y_t=x_t-r_t$ is a random variable and represents the excess return on the risky asset over the risk-free asset from the time t to the time $t+1$. The excess return $\{y_t{\}}_{\{t=0,1,\dots ,T-1\}}$ is an adapted stochastic process defined over the probability space $(\mathrm{\Omega },{\mathcal{F}}_t,\mathcal{F},P)$. The information set at the beginning of period t is denoted as ${\mathcal{F}}_t=\sigma (y_0,y_1,\dots ,y_{t-1})$. Besides, assume the variance of the excess return is ${\sigma }^2$ and the expected value of the excess return is $\mu$.

A CPT-invest doesn't feel anxious when he invests in the risk-free asset. However, the CPT-investor feels anxious and nervous when he invests in the risky asset. The reason is that the CPT-invest should not only have professional knowledge and relevant information but also spend much time and energy on the analysis of portfolio in order to attain a satisfied return of the risky asset. But it is very difficult for an average CPT-investor to do these. Thus, the CPT-investor is willing to invest in the risky asset on a trusted and experienced wealth manager rather than on his own. In a pure mathematical sense, this assumption is not very strict, but it clearly presents the foundation and background of our research and makes the practical meanings of our result more explicit. As the viewpoint of Gennaioli, Shieifer and Vishny, a money manager plays a similar role in the financial market to a doctor. In particular, almost of investors do not know how to invest expect for the investment in a risk-free asset, so they want to seek some financial advice from a trusted and experienced manager. It is like that almost of patients have little idea of how to be treated but some simple and safe treatments, thus they prefer to seek some medical advice from an expert and trusted doctor. The present article discusses the similar problem that CPT-investors invest in a risk asset on a trusted money manager but not on their own. Actually, the relationship between the CPT-investors and the money managers in our research is also like the relationship between the patients and the doctors.

Each of the manager who the CPT-investor i can choose is denoted by $j$. In general, $j=1,2,3,\dots$. The CPT-investor delegates his wealth management to the portfolio manager who this investor most trusts. We add the element of trust to the traditional value function of a CPT-investor. Denoted by ${\tau }_{i,j}\in [0,1]$ the investor i's level of trust on the money manager j. The higher ${\tau }_{i,j}$ represents that the investor puts more trust in the manager j and the investor i suffers less anxiety when he invests his wealth in a risky asset with the manager j. The fee rate charged by money manager j is denoted by $f_j$. Let $a_{i,j}=a{\tau }_{i,j}$, where a is a nonnegative constant. Indeed, $a_{i,j}$ is the measure of the CPT-investor i's anxiety when he hires the manager j. a represents the measure of the CPT-investor i's anxiety when he hires the most trusted manager. The idea will be specially formalize in the next section.

In this paper, we only consider a simple case in which the CPT-investor hires one of two managers, A or B. So, in the section 2.3, we directly set j = A and j = B. We suppose that half of the CPT-investors trust the manager A more than the manager B. These investors are denoted by A-trusting investors. Similarly, there are half of the CPT-investors trusting the manager B more than the manager A, who are called B-trusting investors. The anxiety which an A-investor suffers when he invests in the risky asset on the manager A equals a. To similar, the anxiety suffered by a B-investor for bearing risky with the manager B's financial advice is also a. It can differ from different CPT-investors how much they trust one manager over the other. Particularly, in the population of CPT-investors ${\tau }_i$ satisfies uniformly distributed on $[1-\theta ,1]$ for both A-trusting investors and B-trusting investors, where parameter $\theta \in [0,1]$ captures the dispersion of trust in the population. The lower is $\theta$, the less investors trust one manager more than the other. When $\theta =0$, the investor trusts one manager as much as the other. This dispersion in trust level makes the money managers gain respect of the CPT-investors who trust these managers more. And, these managers can charge the optimal and positive fees even in a competitive financial market. The trust is permanent and does not depend on the change of returns.

Simply speaking, we consider the following the optimal problems. Two managers A and B decide the optimal fees at the time $t$ ( $t=0,1,\dots ,T-1$ ) in order to attract the CPT-investors and gain most total profits in competition at the time $T$. From the CPT-investors' standpoint, at the time $t$, the investors choose the optimal portfolio in order to maximize their own objective function at the time $T$.

III. THE MODEL

a) The Benchmarked Wealth

Let $R_t^k=\prod _{j=t}^{k-1}(1+r_j)(0\le t\le T-1,1\le k\le T)$, with $k\ge t$ be the value of 1 dollar (in the portfolio at time $t$ ) at time $k$. If the initial time is $t$, we let the benchmark be $R_t^k{W}_{i,t}$ at the time $k$ (this is the amount at time $k$ of $W_t$ invested in the risk free asset at time $t$ ). The benchmarked wealth at time $t+1$, given initial time $t$ is

Given the initial time t the benchmarked wealth at time $t+2$ is

Generally, we let the initial time be $t(t=0,1,2,\dots ,T-1)$. We can characterize the benchmarked wealth at the end of the investment horizon by the following Proposition.

Proposition 3.1. (See Deng and Pirvu (2015)) If the initial time is t (t = 0, 1, 2,..., T - 1), then the benchmarked wealth at T is:

b) The CPT Risk Criterion

Before we define the value function and the objective function of a CPT-investor, we introduce two indispensable and classical definitions.

Definition 3.1. (see Tversky and Kahneman (1992a) and Tversky and Kahneman (1992b)) The value function u is defined as follows:

where $u^{+}:{\bar{\mathbb{R}}}^{+}\to {\bar{\mathbb{R}}}^{+}$ and $u^{-}:{\bar{\mathbb{R}}}^{+}\to {\bar{\mathbb{R}}}^{+}$ satisfy:

(i) $u(0)=u^{+}(0)=u^{-}(0)=0;$

(ii) $u^{+}(+\mathrm{\infty })=u^{-}(+\mathrm{\infty })=+\mathrm{\infty };$

(iii) $u^{+}(x)=x^{\alpha }$, with and $x\ge 0;$

(iv) $u^{-}(x)=\beta x^{\alpha }$, with and $x\ge 0$.

Definition 3.2. Let $F_W(\cdot )$ be the cumulative distribution function (CDF) of a random variable $W$. We define the two probability weight functions (distortions) $T^{+}:[0,1]\to [0,1]$ and $T^{-}:[0,1]\to [0,1]$ as follows:

When the CPT investor is risk averse, he feels less anxious and nervous about bearing the risk when he has the assistance of a trusted and experienced money manager. The CPT investor is only risk averse when the benchmark wealth is nonnegative, i.e., $x\ge 0$. When x < 0, the CPT investor is risk seeking. Thus, we only need to modify value function $u^{+}$ and can leave value function $u^{-}$ unchanged.

Definition 3.3. When investor i chooses manager j to invest wealth in the risky asset, if ${\tau }_i$ is uniformly distributed on $[1-\theta ,1]$, we definite the value function of a CPT investor as follows:

It is worth stressing that $a_{i,j}$ has a different meaning from that used by Gennaioli et al. In our model, $a_{i,j}$ is the measure of CPT investor i's trust when he hires manager j. In particular, $a$ represents a measure of CPT investor i's anxiety when he hires his most trusted manager. However, in Gennaioli et al.'s model, $a_{i,j}$ is the measure of CPT investor i's anxiety when he hires manager j:

This different meaning is attributable to different reference levels. In our model, $a_{i,j}$, as a measure of trust, makes the value function of a risk-averse investor without a manager's help, $W_i^{\alpha }$, increase to that of a risk-averse investor with a manager's help, $(W_i-f_j+\frac{a{\tau }_{i,j}}{{\sigma }^2}{)}^{\alpha }$. In other words, we take the value function of a risk-averse investor without a manager's help, $W_i^{\alpha }$, as a reference level. In Gennaioli et al.'s model, $a_{i,j}$, as a measure of anxiety, makes the value function of a risk-neutral investor $E(c)$ decrease to that of a risk-averse investor, $E(c)-\frac{a_{i,j}}{2}Var(c)$. In other words, they take the value function of a risk-neutral investor, $E(c)$, as a reference level. If we examine the value function of a risk-averse investor without a manager's help, $E(c)-Var(c)$, as a reference in Gennaioli et al.'s model, we find that $1-\frac{a_{i,j}}{2}$ has a similar meaning to that of our $a_{i,j}$. That is, in Gennaioli et al.'s model, the measure of trust is $1-\frac{a_{i,j}}{2}$.