# Optimal Investment of DC Pension Plan under Incentive Schemes and Loss Aversion

Authors: Yinghui Dong; Wenxin Lv; Siyuan Wei; Yeyang Gong
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

We investigate the DC pension manager’s portfolio problem when the manager is remunerated through two schemes for DC pension managerial compensation under loss aversion and minimum guarantee. We apply the concavification technique and a static Lagrangian technique to solve the problem and derive the closed-form representation of the optimal wealth and portfolio processes. Theoretical and numerical results show that the incentive schemes can significantly impact the distribution of the optimal terminal wealth.


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## Body

## 1. Introduction

As special financial institutions, risk management of insurance companies is the important guarantee of business security. In the existing references, some focus on the pricing aspect, see Dong [1], Yu et al. [2], and Zhang et al. [3], while others focus on the optimal problems, see Dong et al. [4], Dong and Zheng [5, 6], Sun et al. [7], and Yu et al. [8]. The objective of the present paper is to develop optimal asset management strategies for the DC pension plan.Due to declining mortality and low interest rates, the literature on the asset allocation of defined contribution (DC) pension plans has grown up significantly in recent years. In a DC plan, the member contributes part of the salary to the pension account and upon retirement, the member can receive regular income from the pension account. The problem of optimal allocation during the accumulation phase for the DC pension plan has attracted a lot of attention, see Boulier et al. [9], Chen et al. [10], Zeng et al. [11], and Wang et al. [12].The impact of the performance of a fund with respect to a benchmark on the asset management has been extensively investigated recently. The managers are often paid by an incentive scheme that depends on the performance of the fund they manage in order to inspire them to greater efforts. Such a scheme is made up of two components: a management fee, which is proportional to the fund wealth, and an incentive fee which is composed of a combination of some options on the fund, see, for example, Basek et al. [13, 14], Carpenter [15], Basak et al. [16], and Chen and Pennacchi [17]. Incentive schemes of the manager significantly affect the investment strategy. It has been believed that the asset manager may be motivated to take risk in excess by incentive schemes. However, little focuses on the performance fee schemes in DC pension plan management, since people argue that the performance fee may trigger managers to take excessive risk, which may lead to heavy losses in DC pension plans. A typical fee structure of defined contribution (DC) pension plan is an asset-based single management fee. However, managers typically perform poorly and cannot keep up with benchmarks under a scheme with lower incentives. It is widely believed that a performance-based and option-like fee may create greater incentives for a manager to exert more efforts in terms of investment.As pointed out by He et al. [18], in the late 1990s, a performance-based fee structure was approved as a managerial incentive in DC pension plan management. However, the performance fee could induce managers to gamble at the expense of DC pension members. The purpose of a DC pension plan is to provide an adequate income for a DC member after retirement. Hence, to ensure safety of a DC pension, the performance fee scheme has not been implemented in most DC pension funds. He et al. [18] consider the optimal allocation of the DC pension fund for a manager under two fee structures: the single management fee and the mixed scheme of both management and performance fees. They compare the utilities of both fund managers and members under the two schemes and they state the win-win situation of implementing performance-based incentives in DC pension plan management. Furthermore, He et al. [18] consider the PI constraint, which can well protect the members’ benefits by keeping the optimal terminal wealth always above the minimum guarantee, see Basek and Shapiro [13].Prospect theory (PT), proposed by Kahneman and Tversky [19] and by Tversky and Kahneman [20], is one of the most notable theories in behavioral economics. With the framework of PT, investors base decisions on comparisons with a certain predefined reference point (also known as benchmark) rather than evaluating absolute level of total wealth itself. PT also suggests that individuals tend to be loss averse, that is, to be more sensitive to a loss than to a gain of the same amount. This loss aversion and risk-seeking behaviour is described in [20] by an asymmetric S-shaped utility function, convex in the domain of losses and concave in the domain of gains. In this paper, we extend the findings by He et al. [18] by incorporating loss aversion into modelling, which makes our optimization problem much more complex than that of [18]. We employ the martingale approach and the concavification technique to derive the optimal wealth and portfolio processes under a PI constraint and loss aversion. Although Dong and Zheng [5, 6] and Guan and Liang [21] also investigated the optimal allocation of the DC pension plan under loss aversion, they do not consider the remuneration schemes of DC pension.The main contribution of this paper is that we have solved an optimization problem of a DC plan under two different kinds of remuneration schemes with the framework of PT. The rest of the paper is organized as follows. In Section2, we formulate a DC investment problem under loss aversion and two different types of remuneration schemes. In Section 3, we apply the concavification technique and the martingale method to solve the optimization problem and derive closed-form expressions for the optimal wealth process and optimal control. In Section 4, we perform some numerical analysis and discuss the impacts of loss aversion and remuneration schemes on the distribution of the optimal terminal wealth. Section 5 concludes the paper.


## 2. The Model

LetΩ,ℱ,F,P be a filtered complete probability space with the filtration F≔ℱt0≤t≤T being the natural filtration generated by a standard Brownian motion Wt0≤t≤T and satisfying the usual conditions. Assume that all random variables and stochastic processes in this paper are well defined in this probability space. The DC pension fund starts at time 0, and the retirement time is T.Let the financial market consist of one riskless savings accountS0t and one risky asset S1t.The price of the risk-free assetS0tt≥0 satisfies the following differential equation:(1)dS0t=rS0tdt,where r is a riskless interest rate. The price of the risky asset S1tt≥0 is modelled by the following stochastic differential equation (SDE):(2)dS1t=S1tμdt+σdWt=S1trdt+σdWt+ϑdt,where μ>r is the expected return of the risky asset, σ>0 is the volatility of the risky asset, and ϑ=μ−rσ is the market price of risk.In a DC plan, the pension members contribute continuously to the pension plan before retirement timeT. Let ct>0 denote the aggregated amount of money contributed at time t of a cohort of fund participants. As explained in [6], individual members’ contribution rates may be random over the time, but the accumulated cash income of a large pension fund is in general deterministic and stable. We therefore, for simplicity, assume ct≡c is a constant. Assume that there are no transaction costs or taxes in the financial market. The pension account is endowed with an initial endowment x0≥0.Letπt be the total amount allocated to the risky asset at time t∈0,T. Then, the wealth process Xπt0≤t≤T with the initial value Xπ0=0 satisfies the following controlled SDE:(3)dXπt=rXπtdt+πtσdWt+ϑdt+cdt.LetHt be the discounted value at time t of total pension contribution from t to T, where(4)Ht=c∫tTe−rs−tds=cr1−e−rT−t.Traditionally, the fund manager charges a management fee as a reward, which is a fixed proportion of terminal fund wealth. More specifically, under this single management fee scheme, the payoff to the DC pension manager is(5)Ψ1XπT=k1XπT,where k1>0 is the rate of the management fee.It is well known that the performance fee may lead to a greater incentive for the fund manager. Following [18], we consider an option-like incentive scheme (also called mixed scheme) and let MT be the benchmark for the availability of the performance fee. Assume that MT is the expected accumulation of contributions until time T under the reasonable accumulation rate:(6)M≔MT=x0er¯T+cr¯er¯T−1,where r¯ is the accumulation rate. Under the mixed scheme, the payoff to the DC pension manager is(7)Ψ2XπT=k2XπT,XπT<M,k2XπT+k3XπT−M,XπT≥M,where 0<k2≤k1 is the rate of the management fee under the mixed scheme and k3>0 is the rate of the performance fee. It is obvious that the manager receives a fixed proportion of terminal wealth and a variable component only in case the DC pension fund outperforms the benchmark M at time T.The DC pension fund is supposed to meet DC members’ elementary needs. Therefore, a downside protectionM¯ should be included in DC pension fund management. We next define the set of admissible trading strategies.Definition 1.
A portfolio strategyπ=πt:t∈0,T is said to be admissible if, for all t∈0,T,πt∈ℱt, with ∫0Tπ2tdt<+∞, a.s., and XπT satisfies (3), XπT≥M¯. We denote the set of admissible portfolio strategies by A.Consider a utility function defined by(8)Ux=−Bθ−xγ1,0≤x<θ,x−θγ2,x≥θ,where θ>0 is a reference point, B>1 is called the loss aversion degree, 0<γ1<1,0<γ2<1 are the constants, γ1 characterizes the degree of loss aversion, and γ2 characterizes the degree of risk aversion, respectively.For eachi=1,2, we consider the following optimization problem with minimum guarantee:(9)maxπ∈AEUΨiXπTs.t.XπTsatisfies3withXπT≥M¯,a.s…Since the market is complete, the unique pricing kernel is as follows:(10)ξt=e−r+ϑ2/2t−ϑWt,ξ0=1,0≤t≤T,where ξt is the Arrow–Debreu value per probability unit of a security which pays out 1 at time t if the scenario ω happens, and 0 else. As this value is high in a recession and low in prosperous times, ξt directly reflects the overall state of the economy.Note that the shapes ofUΨ1⋅ and UΨ2⋅ depend on the relationship of the parameters k1,k2,M, and M¯. For simplicity, here we assume that k1M¯≤θ≤k2M. For other cases, the analysis is similar.Let(11)G1x≐UΨ1x=−Bθ−k1xγ1,0≤x<θk1,k1x−θγ2,x≥θk1,G2x≐UΨ2x=−Bθ−k2xγ1,0≤x<θk2,k2x−θγ2,θk2≤x<M,k2x+k3x−M−θγ2,x≥M.In the complete market, problem (9) can be rewritten as follows:(12)maxXπTEGiXπTs.t.EξTXπT≤x0,XπT≥M¯,a.s...


## 3. Optimal Trading Strategy under Loss Aversion

In this section, we will solve the optimization problem (9) and derive the optimal wealth and portfolio processes.Denote by(13)k=G2M−G2M¯M−M¯,the slope of the straight line linking points M¯,G2M¯ and M,G2M.Lemma 1.
(1)
Fork1M¯≤θ, there exists a unique solution θ/k1<z<∞ to the equation:(14)G1x−x−M¯G1′x−G1M¯=0.(2)
Letk be defined by (13). For k≥G2′M−, there exists a unique θ/k2<z¯≤M satisfying(15)G2z¯−G2M¯−G2′z¯z¯−M¯=0.(3)
Fork≤G2′M+, there exists a unique z˜≥M satisfying(16)G2z˜−G2M¯−G2′z˜z˜−M¯=0.Proof.
We first prove the first part. Denotefx=G1x−x−M¯G1′x−G1M¯. It is easy to check that fx is a decreasing, continuous function in θ/k1,∞ and fθ/k1<0, limx⟶∞fx=∞. Therefore, there exists unique θ/k1<z<∞ such that fz=0 (see Figure 1), which concludes the proof of the first part. Similarly, we can prove the second and the third parts (see Figures 2 and 3). Here, we omit the details.Figure 1
The root to equation (14).Figure 2
The root to equation (15).Figure 3
The root to equation (16).Lemma 2.
ForG2′M−<G2′M+, there exists a unique pair z1,z2 with θ/k2<z1<M<z2 satisfying(17)G2z2−G2z1z2−z1=G2′z1=G2′z2.Proof.
Note thatG2′x is continuous and strictly decreasing on k2/θ,M and M,∞, respectively. Furthermore,(18)G2′M−<G2′M+,limx⟶θ/k2+G2′x=+∞,limx⟶+∞G2′x=0.
Therefore,G2′x on k2/θ,M and G2′x on M,∞ have strictly decreasing inverse functions and there must exist unique θ/k2<η1<M and η2>M, such that(19)G2′η1=G2′M+,G2′η2=G2′M−.
For anyy∈G2′M−,G2′M+, we define θ/k2<c1y≤M≤c2y as(20)G2′c1y=G2′c2y=y.
It is obvious thatc1y and c2y are strictly decreasing and continuous on G2′M−,G2′M+ and(21)limy⟶G2′M−c1y=M,limy⟶G2′M−c2y=η2,limy⟶G2′M+c1y=η1,limy⟶G2′M+c2y=M.
Define(22)Ky=G2c2y−G2c1yc2y−c1y.
It is easy to check thatKy is continuous on G2′M−,G2′M+. Using the concave inequalities, we have(23)G2′M+c2y−M>G2c2y−G2M>G2′c2yc2y−M≥G2′M−c2y−M,G2′M+M−c1y≥G2′c1yM−c1y>G2M−G2c1y>G2′M−M−c1y.
It follows from the above two inequalities that(24)G′M+>Ky>G′M−.
In particular,(25)KG′M−>G′M−,KG′M+<G′M+.
Then, there must existG′M−<y0<G′M+ such that Ky0=y0. Define k2/θ<z1<M,z2>M by the relation G2′z1=G2′z2=y0, which concludes the existence of z1,z2. Additionally, we can conclude that the linear interpolation between G2z1 and G2z2 has to be higher than G2⋅ on z1,z2, that is,(26)G2x<G2z2+G2′z2x−z2,z1<x<z2.
Now, we turn to the uniqueness ofz1,z2. Assume there exists another pair z^1,z^2 with θ/k2<z^1<M<z^2 satisfying(27)G2z^2−G2z^1=G2′z^1z^2−z^1=G2′z^2z^2−z^1.
Without loss of generality, we assume thatz^1>z1. From (26), we have G2z^1<G2z2+G2′z2z^1−z2. So, for z2∈z^1,z^2,(28)G2z2>G2z^1+G2′z2z2−z^1>G2z^1+G2′z^1z2−z^1.
However, the linear interpolation betweenG2z^1 and G2z^2 has to be higher than G2⋅ on z^1,z^2, that is,(29)G2z2<G2z^1+G2′z^1z2−z^1,which leads to a contradiction. □Proposition 1.
Assumex0≥M¯e−rT−H0. Let z,z¯,z˜ and z1,z2 be determined by (14)–(17), respectively. Then, for the optimization problem (9), the optimal terminal wealth Xπ∗,β∗T is as follows with the multiplier β∗>0 satisfying EξTXπ∗,β∗T=x0+H0.
Under the single management fee scheme,(30)Xπ∗,β∗T=M¯,ξT≥G1′zβ∗,1k1β∗ξTk1γ21/γ2−1+θ,ξT<G1′zβ∗.
Under the mixed scheme,Case A. If k≥G2′M−, then(I)
Forz1<z¯,(31)Xπ∗,β∗T=M¯,ξT≥G2′z˜β∗,1k2+k3β∗ξTγ2k2+k31/γ2−1+k3M+θ,ξT<G2′z˜β∗.(II)
Forz1≥z¯,(32)Xπ∗,β∗T=M¯,ξT≥G2′z¯β∗,1k2θ+β∗ξTk2γ21/γ2−1,G2′z1β∗≤ξT<G2′z¯β∗,1k2+k3β∗ξTγ2k2+k31/γ2−1+k3M+θ,ξT<G2′z1β∗.Case B. If k<G2′M−, then Xπ∗,β∗T is given by (31).Proof.
To solve problem (12), we define the Lagrangian of problem (12) as follows:(33)ℒiXT,β=EGiXT−βξTXT,where β>0 is a Lagrange multiplier. Note that Gi⋅ is not concave, so we first derive the concave envelope of Gi⋅.
Under the single management fee scheme, we letz be the tangent point of the line starting at M¯,G1M¯ to the curve G1x,x>θ/k1. Then, the concave envelope of G1 is given by (see Figure 4)(34)G1cx=G1M¯+G1′zx−M¯,M¯≤x<z,k1x−θγ2,x≥z.
It is easy to verify that fory>0,(35)x∗y=^argmaxx≥M¯G1cx−xy=1k1yk1γ21/γ2−1+θ,0<y<G1′z,M¯,z,y=G1′z,M¯,y>G1′z.
Under the mixed fee scheme, Lemma2 states that there exists a common tangent line with tangent point pair z1,z2 to the curve G2⋅,θ/k2<x<M and G2⋅,x>M.Case A. For k≥G2M−, we let z¯ be the tangent point of the line starting at M¯,G2M¯ to the curve G2x,θ/k2<x<M.(I)
Forz1<z¯, it can be verified that there exists a unique tangent point z˜>M of the line starting at M¯,G2M¯ to the curve G2x,x>M. Then, the concave envelope of G2 is given by (see Figure 5)(36)G2cx=G2M¯+G1′z˜x−M¯,M¯≤x<z˜,G2x,x≥z˜.Similar to deriving (35), we can obtain that(37)x∗y=^argmaxx≥M¯G2cx−xy=1k2+k3yγ2k2+k31/γ2+k3M+θ,0<y<G1′z,M¯,z˜,y=G2′z˜,M¯,y>G2′z˜.(II)
Forz1≥z¯, the concave envelope of G2 is given by (see Figure 6)(38)G2cx=G2M¯+G2′z¯x−M¯,M¯≤x<z¯,G2x,z¯≤x<z1,G2z1+G2′z1x−z1,z1≤x<z2,G2x,x≥z2.So, we have(39)x∗y=^argmaxx≥M¯G2cx−xy=1k2+k3yγ2k2+k31/γ2+k3M+θ,0<y<G2′z2,z1,z2,y=G2′z2,1k2θ+yk2γ21/γ2−1,G2′z2<y<G2′z¯,M¯,z1,y=G2′z¯,M¯,y>G2′z¯.Case B. For k<G2M−, we let z˜ be the tangent point of the line starting at M¯,G2M¯ to the curve G2x,θ/k2<x<M (see Figure 7). Then, the concave envelope of G2 and x∗y is given by (36) and (37), respectively.
According to the standard approach to portfolio selection problems, the original problem (12) is equivalent to the following problem:(40)infβ>0maxXTℒiXT,βs.t.XT≥M¯,a.s...
To solve the above problem, we first consider the following problem:(41)maxXTℒiXT,βs.t.XT≥M¯,a.s…
It can be verified that for eachβ>0, an optimal solution to (41) is(42)Xπ∗,βT=x∗βξT.
Secondly, it remains to solve an optimization problem w.r.tβ:(43)infβ>0ℒiXπ∗,βT,β.
By the complementary slackness condition, the optimal solutionβ∗>0 to problem (43) can be directly obtained from the following equation:(44)EξTXπ∗,β∗T=x0+H0,which yields the optimal solution Xπ∗,β∗T to problem (12).
It is easy to see that in each case,Vβ=EξTXπ∗,βT is continuous and strictly decreasing in β. Furthermore, limβ⟶0+Vβ=∞ and limβ⟶∞Vβ=M¯e−rT. Thus, there exists a unique β∗ satisfying condition (44), which implies that Xπ∗,β∗T is feasible to (12). We next show Xπ∗,β∗T is indeed the optimal terminal wealth to problem (12). Let XT be an arbitrary terminal wealth satisfying the budget constraint in problem (12). Then, we have(45)EGiXπ∗,β∗T−EGiXT≥EGiXπ∗,β∗T−β∗ξTXπ∗,β∗T+β∗x0+H0−EGiXT−β∗ξTXT+β∗x0+H0≥0,where the first equality holds since the budget constraint holds with equality for Xπ∗,β∗T while with inequality for XT and the second equality follows from the fact that Xπ∗,β∗T solves (41) for the multiplier β∗.Figure 4
Concave envelope ofG1.Figure 5
Concave envelope ofG2 for k≥G2M−,z1<z¯.Figure 6
Concave envelope ofG2 for k≥G2M−,z1≥z¯.Figure 7
Concave envelope ofG2 for k<G2M−.Remark 1.
Note that ifM¯=0, then the loss aversion states a risk-seeking preference for the loss states, and therefore the terminal wealth equals to 0, which implies in the case of no regulation there is a moral hazard problem due to loss aversion. In contrast to the unconstrained optimal terminal wealth, the optimal terminal wealth under the PI constraint in bad market scenarios is kept at the level M¯ in order to satisfy the PI constraint while the unconstrained terminal wealth is always 0. Therefore, the PI constraint well protects the DC member’s benefits in bad market scenarios.Remark 2.
Different from the derived results under the concave utility in [18], the linear envelope exists both under the single and the mixed fee schemes due to loss aversion. The optimal terminal wealth Xπ∗,β∗T is a function of the state price density at maturity ξT and the multiplier β∗. Under the single incentive scheme, the optimal terminal wealth takes a two-region form. When the state price density ξT is relatively low, the optimal terminal wealth Xπ∗,β∗T is similar to the smooth utility; when ξT increases above the boundary point of the bad-state region, G1′z/β∗, the optimal terminal wealth drops to M¯ due to the risk-seeking preference in the the domain of losses. Under the mixed incentive scheme, Xπ∗,β∗T takes a two-region form for k<G2′M− and may take a two- or three-region form for k≥G2′M− according to the the relative position of the values of z1 and z¯. In all cases, the optimal terminal wealth is determined by the utility changing points and ends with the minimum guarantee M¯ when the state price density at maturity ξT increases above the boundary point of the bad-state region.After obtaining the optimal terminal wealth, we can derive the closed-form expression for the optimal investment strategy given in the following result.Proposition 2.
Assume thatx0≥M¯e−rT−H0. Then, for the optimization problem (12), the optimal wealth process and the optimal strategy are given as follows with the multiplier β∗>0 satisfying EξTXπ∗,β∗T=x0+H0.
Under the single management fee scheme, the optimal wealth process at timet is given by(46)Xπ∗,β∗t=A1ξt,t−Ht,where(47)A1ξt,t=e−rT−tM¯Φd1,tG1′z+θk11−Φd1,tG1′z+χzΦ′d1,tG1′zΦ′d2,tG1′zΦd2,tG1′z,with Φ being the standard normal cumulative distribution function and(48)d1,tx=lnx/ξtβ∗+r−ϑ2/2T−t−ϑT−t,d2,tx=−d1,tx+ϑT−t1−γ2,χx=k1x−θk1.
The percentage of wealth invested in the risky assets is given by(49)π∗,β∗t=ϑσ1−γ2Xπ∗,β∗t+Ht︸total wealth−e−rT−tΦd1,tG1′zM¯+θk11−Φd1,tG1′z︸deposit term+1−γ2Φ′d1,tG1′ze−rT−tϑT−tz−M¯︸risk seeking term.
Under the mixed scheme,Case A. If k≥G2′M−, then(I)
Forz1<z¯, the optimal wealth process at time t is given by(50)Xπ∗,β∗t=A¯1ξt,t−Ht,where(51)A¯1ξt,t=e−rT−tM¯Φd1,tG2′z˜+k3M+θk2+k31−Φd1,tG2′z˜+ηz˜Φ′d1,tG2′z˜Φ′d2,tG2′z˜Φd2,tG2′z˜,with(52)ηx=k2+k3x−k3M−θk2+k3.The percentage of wealth invested in the risky assets is given by(53)π∗,β∗t=ϑσ1−γ2Xπ∗,β∗t+Ht︸total wealth−e−rT−tΦd1,tG2′z˜M¯+k3M+θk2+k31−Φd1,tG2′z˜︸deposit term+1−γ2Φ′d1,tG2′z˜e−rT−tϑT−tz˜−M¯︸risk−seeking term.(II)
Forz1≥z¯, the optimal wealth process at time t is given by(54)Xπ∗,β∗t=A¯2ξt,t−Ht,where(55)A¯2ξt,t=e−rT−tM¯−θk2Φd1,tG2′z¯+θk2Φd1,tG2′z1+k2z¯−θk2Φ′d1,tG2′z¯Φ′d2,tG2′z¯Φd2,tG2′z¯+ψz1Φ′d1,tG2′z1Φ′d2,tG2′z1Φd2,tG2′z1,with(56)ψx=k2+k3k2γ2/1−γ2k2x−θk2.The percentage of wealth invested in the risky assets is given by(57)π∗,β∗t=ϑσ1−γ2Xπ∗,β∗t+Ht︸total wealth−e−rT−tk2M¯−θk2Φd1,tG2′z¯+θk2Φd1,tG2′z1︸deposit term+1−γ2z¯−M¯ϑT−tΦ′d1,tG2′z¯+k2ψz1−θk2ϑT−tΦ′d1,tG2′z1︸risk−seeking term.Case B. If k<G2′M−, then the optimal wealth process at time t and the percentage of wealth invested in the risky assets are given by (50) and (53), respectively.Proof.
The price ofXπ∗,β∗T at t is calculated as follows:(58)Xπ∗,β∗t=1ξtEξTXπ∗,β∗Tℱt−Ht.
Note thatlogξT is a normal distribution. By substituting the expressions for Xπ∗,β∗T given in Proposition 1 into (58), we can easily obtain the formula for Xπ∗,β∗t by some straightforward calculations.
LetXπ∗,β∗t=Aξt,t. Then,(59)dXπ∗,β∗t=It,Htdt+∂A∂ξ−ξtϑdWt,for some It,Ht, and we are only interested in the diffusion part. Comparing it with (3), we have(60)π∗,β∗t=−ϑσ∂A∂ξξt.
Some simple calculations yield the results.Remark 3.
Note that, under the concave utility in [18], a risk-seeking term only exists in the optimal policy under the mixed incentive scheme generated by the incentive of the performance fee. However, with the framework of PT, there exists a risk-seeking term both under the single and the mixed incentive schemes due to loss aversion. Therefore, the potential gambling risk-taking policy exists both under the single and the mixed incentive schemes.


## 4. Numerical Analysis

In this section, we do some numerical calculations to investigate the influence of incentive schemes on the optimal terminal wealth.For all numerical computations, unless specified otherwise, the benchmark data used are as follows:r=0.03,μ=0.07,σ=0.2,c=1,B=2.25,γ1=0.2,γ2=0.3,k1=0.05,k2=0.04,k3=0.1, θ=10,r¯=0.06,x0=20,M¯=50,andT=40.Figure8 presents the optimal terminal value Xπ∗,β∗T versus ξT for different θ. It is observed that there exists H∗, such that for ξT≥H∗, the optimal terminal value under the mixed incentive scheme is dominated by Xπ∗,β∗T under the single fee scheme. This is consistent with the intuition that the manger will take more risky strategy to attain the performance fee, which leads to a higher left-tail risk. Furthermore, the mixed incentive scheme also leads to an increase in the optimal terminal wealth of good economic states. We can also note that with the reference point θ increasing, the bad-state region enlarges. This is because the manager is risk seeking towards the terminal wealth. To achieve a higher reference point, the manager invests much more money in the risky asset, which results in a higher left-tail risk.Figure 8
Xπ∗,β∗T versus ξT for different θ.Figure9 shows the optimal terminal value Xπ∗,β∗T versus ξT under two incentive schemes with different minimum guarantee M¯. From it, we can see that a higher left-tail risk under the mixed incentive scheme can be improved by increasing the minimum guarantee. Therefore, we can control the left-tail risk triggered by the mixed incentive scheme via a stricter PI constraint.Figure 9
Xπ∗,β∗T versus ξT for different M¯ under mixed incentive scheme.Figure10 represents the optimal terminal value Xπ∗,β∗T versus ξT for different k3. We can see that a higher performance fee rate k3 leads to enlargement of the bad-state region and increase in the optimal terminal wealth in the good-state region. This is due to the fact that in order to attain the performance fee, the manager takes more risky strategies such that Xπ∗,εT becomes more volatile.Figure 10
Xπ∗,β∗T versus ξT for different k3.Figure11 plots the expectation of the optimal terminal value EXπ∗,β∗T versus k3. We can see that the expectation of the optimal terminal value EXπ∗,β∗T under the mixed incentive scheme is greater than that under the single incentive scheme, and EXπ∗,β∗T under the mixed incentive scheme increases with k3. As pointed out in Figure 10, to achieve the performance fee, the manager invests much more money in the risky asset, which results in a higher expectation.Figure 11
EXπ∗,β∗T versus k3.Figure12 shows the optimal terminal value Xπ∗,β∗T versus ξT for different k1. We can see that with k1 increasing, the bad-state region shrinks and the optimal terminal wealth in the good-state region decreases. This is because that as k1 increases, the manager is paid much more management fee and then the manager takes more prudent strategies such that Xπ∗,β∗T becomes less volatile. Therefore, a single incentive scheme with a higher management fee rate cannot provide incentive for a manager to increase the optimal terminal wealth.Figure 12
Xπ∗,β∗T versus ξT for different k1.Figure13 plots the optimal terminal value Xπ∗,β∗T versus ξT for different k2. It is seen that a higher fee rate k2 leads to shrinkage in the bad-state region shrinks and decrease in the optimal terminal wealth in the good-state region. As explained in Figure 12, with k2 increasing, the manager can receive much more management fee and then chooses to take more prudent strategies.Figure 13
Xπ∗,β∗T versus ξT for different k2.


## 5. Conclusions

In this paper, we investigate the optimal portfolio selection problem for a DC plan manager with an S-shaped utility under a single fee scheme and a mixed fee scheme, respectively. We use a concavification technique and a Lagrange dual method to derive closed-form optimal wealth and portfolio processes of the DC pension fund. Due to loss aversion, the potential gambling risk-taking policy exists both under the single and the mixed incentive schemes. In particular, the mixed incentive scheme can induce the manager to take much more risky strategies to achieve a higher optimal terminal value. Financial regulation such as PI constraint can be used to prevent gambling when the performance deteriorates.



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*Source: 5145848-2020-05-18.xml*