The optimal investment strategy of a DC pension plan under deposit loan spread and the O-U process
aa r X i v : . [ q -f i n . P M ] M a y The optimal investment strategy of a DC pension planunder deposit loan spread and the O-U process
Xiao Xu a , Yonggui Kao b, ∗ a Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, PR China b Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, PR China
Abstract
This paper is devoted to invest an optimal investment strategy for adefined-contribution (DC) pension plan under the Ornstein-Uhlenbeck (O-U) process and the loan. By considering risk-free asset, a risky asset drivenby O-U process and a loan in the financial market, we firstly set up thedynamic equation and the asset market model which are instrumental inachieving the expected utility of ultimate wealth at retirement. Secondly, thecorresponding Hamilton-Jacobi-Bellman(HJB) equation is derived by meansof dynamic programming principle. The explicit expression for the optimalinvestment strategy is obtained by Legendre transform method. Finally,different parameters are selected to simulate the explicit solution and thefinancial interpretation of the optimal investment strategy is given.
Keywords:
O-U process; loan; HJB equation; Legendre transform.
Mathematics Subject Classification:
1. Introduction
With the development of the global economy and society, pension is get-ting more important for the life of the elder. Besides, nowadays the agingof population is accelerating rapidly, and pension has become a focus. Theenterprise annuity is divided into two basic modes: The defined benefit (DB)plan and the defined contribution (DC) plan. In the DC pension plan, ittransfers the longevity and financial risks from the sponsor to the memberand the DC pension plan is also playing an role in social security, which can ∗ Corresponding author: Yonggui Kao
Preprint submitted to Elsevier May 22, 2020 ot be ignored. Hence, asset allocation strategy is crucial to the distributionand deployment of DC pension funds.In recent years, many scholars have focused on the optimal investmentperformance. Markowitz and Harry (1952) put forward the optimal port-folio problem for the first time and gave a theoretical proof. Interest ratewas proposed by Duffie and Rui (1996) who described the general affine pro-cess. Boulier et al. (2001) studied the asset allocation problem of DC typeenterprise fund where the interest rate obeys the the framework of Vasicek,and obtained the analytic solution through the use of martingale method.The expected utility directly after the retirement pension was used in thepaper of Blake et al. (2003) . Through the expected utility maximization,Charupat and Milevsky (2002) found fixed and variable instantaneous an-nuities in the optimal combination of the different assumptions about mor-tality rates, and then made a comparison with the optimal situation of thecumulative period. Devolder et al. (2003) assumed that the price process ofrisky assets are networked with geometric Brownian motion(GBM), that is,the price fluctuation of risky assets is set as constant. Baev and Bondarev(2007) introduced O-U process instead of GBM. Moreover, inflation risk onthe optimal DC pension was considered in Battocchio and Menoncin (2004).Gerrard et al. (2004) concentrated on income by using the stochastic optimalcontrol technology. Afterwards, Jianwu et al. (2007) obtained an explicit so-lution by applying the CEV model. Gao (2008) examined the completefinancial market with stochastic evolution of interest rate, and used Leg-endre transform to settle the optimal asset allocation strategy of DC typeenterprise fund. Hsu et al. (2008) used the CEV model for asset pricingformulas. Gu et al. (2010) considered the optimal reinsurance and invest-ment problem of Brownian motion with risk pricing process, and the assetsare described by the constant elastic variance model. The CRRA utilitymaximization and mean-variance criteria were employed by Han and Hung(2012) to determine DC plan. Zhang and Rong (2013) further paid atten-tion to the optimal allocation of DC pension with random wage under affineinterest rate model. Guan and Liang (2016) studied the optimal allocationof DC pension under the framework of random interest rate and randomfluctuation, in which the interest rate obeys an affine interest rate structure.Teng et al. (2016) came up with the interest rate which is subject to O-Uprocess. Sun et al. (2017) proposed the expected investment goal based ondeficit and surplus. Tang et al. (2018) made an on-the-spot investigationwith two situations: random interest rate and annuity inflation. The opti-2al allocation scheme with stochastic interest rate and stochastic volatilitywas characterised by Wang et al. (2018). Bian et al. (2018) paid more at-tention to a discrete-time model with mean-variance by a Markov chain.Guambe et al. (2019) further stated a investment problem, which consistsof inflation and mortality risks. Optimal investment with transaction costover an infinite horizon was developed by Blake and Sass (2002). Based onprevious work, Mudzimbabwe (2019a) investigated a unsophisticated numer-ical solution method. Chen et al. (2019) construsted the investmal strategyfor fund administers in a framework of Markov. A jump diffusion model wasdemonstrated by Mudzimbabwe (2019b) . Dong and Zheng (2020) attemptedto apply S-shaped utility. According to Zhang et al. (2020), mean-variancecriterion and the Cox-Ingersoll-Ross (CIR) model were adopted. Most of theabove literatures are involved with (CIR) model, Vasicek model, variancemodel and etc, however few of them apply O-U model. At the same time,they do not take loan into account in their financial market. We know that itis more accurate to adopt O-U process which reflects the fluctuation of assetprice. What’s more, with the upgrading and adjustment of China’s industry,capital driven by economic development will become the main thrust, thatis to say, the era of capital economy has come when loan will be a normalstate.Based on the above settings, a risky asset is depicted by the O-U processin this paper. In the framework of a discrete-time, the business adminis-trator is to make the expectation of the terminal wealth under the utilityframework before the retirement. By adopting the theory of the stochasticcontrol, the original nonlinear HJB equation is achieved which is hard todepict closed-form expressions. And then, we introduce the Legendre trans-form and separation of variables. In this case, nonlinear partial differentialequations(NPDE) are transformed into linear partial differential equations.Finally, we derive the explicit expressions of the DC scheme. In summary,this article has two innovations: (i) we describe the optimal investment prob-lem under the O-U process with CRRA utility function; (ii) Deposit and loanspreads are taken into account, and related financial explanation is presented.The rest is laid out as bellow. Section 2 characterizes the assumptions ofthe model. Section 3 shows the definition of the value function and derivesthe corresponding HJB equations and by using principle of dynamic pro-gramming. Section 4 completes the closed-form solutions for the stochasticdynamic programming problem under Legendre transform and CRRA util-ity function. In section 5, we present the numerical simulation analysis. We3ave made a summary in the final chapter.
2. The economy and model
We list the following assumptions for our model.
Assumption 1.
Consider a financial market which ignores transactionfees. We use a finite-time horizon and continuous-time model. The uncer-tainty is represented by a complete probability space (Ω , F, P ). Assumption 2.
Suppose that the financial market involves with threetradable assets: a bank account, a stock and a loan.
Assumption 3.
We denote the price of the bank account at time t by B ( t ), such that dB ( t ) = rB ( t ) dt, B (0) = B , r > , (1)where r is a constant rate of interest. Assumption 4.
Let the price of the stock at time t be S ( t ), which isdepicted by the stochastic differential equation (SDE).By comparison with GBM, the O-U process is closer to the change ofstock price. Here we use S ( t ) to express the price of risk assets at t , whichis described by O-U process dS ( t ) = k ( θ − S ( t )) dt + σdW ( t ) , S (0) = S , (2)where α > θ > σ > Assumption 5. let R denote the lending rate, where 0 < r < R < µ . V ( t ) is the pension wealth at time t . V ( t ), B ( t ) and Y ( t ) are the totalamount of money of the loan with interest, the risk-free asset and risky assetat time t , respectively. Definition 1. (Admissible strategy) If it meets the requirements as follow,investment loans are admissible.(1)( L ( t ) , B ( t ) , Y ( t )) is F t measurable on a complete probability space;(2) R T L ( t ) dt < + ∞ , R T B ( t ) dt < + ∞ , R T Y ( t ) dt < + ∞ , a.s. T < ∞ ;(3)For rational investors, with interest rates higher than the deposit rate,it’s impossible to choose between deposits and loans. That is, L ( t ) B ( t ) =0, with L ( t ) ≥ B ( t ) ≥ t ∈ [0 , T ]. Assume that the set of alladmissible investment and loan scheme (( L ( t ) , B ( t ) , Y ( t )) are expressed by π = { ( L ( t ) , B ( t ) , Y ( t )) : t ∈ [0 , T ] } . 4 ssumption 6. Define the retirement moment and the contribution rateof the enterprise annuity for T and c , separately. Where T and c are theconstants. Until retirement T , cL ( t ) is supplied to the pension fund for eachperiod. In order to simplify the model, the total salary is set as 1 dollar, andonly one insured person is studied.
3. Model Formulation
Let V ( t ) = B ( t ) + Y ( t ) − L ( t ) + ct denote the pension wealth at time t ∈ [0 , T ]. The dynamics of wealth has the following form: dV ( t ) = rB ( t ) dt + Y ( t ) dS ( t ) S ( t ) − RL ( t ) dt + cdt. (3)Based on (1) and (2), we rewrite (3) as dV ( t ) = { rX + [ k ( θ − s ) s − r ] Y + ( r − R ) L − rct + c } dt + σs Y dt. (4) Next, the goal is to maximize the expected discounted utility and ulti-mate wealth over a limited retirement period. That is to seek the optimuminvestment project Y ( t ).Applying the stochastic control theory, we define the value function as H ( t, s, v ) = max Y E [ U ( v ) | S ( t ) = s, V ( t ) = v ] , < t < T, where U ( · ) is an increasing concave utility function and satisfies the condi-tions U ′ (+ ∞ ) < U ′ (0) < + ∞ .As described in Fleming and Soner (2006), by the aid of Itˆ o ’s formula, wehavesup Y { H t +[ rx + ( k ( θ − s ) s − r ) Y + ( r − R ) L − rct + c ] H v + k ( θ − s ) s H s (5)+ 12 σ s Y H vv + 12 σ H ss + σ s Y H vs } = 0 , and it’s accompanied by a boundary condition H ( T, s, v ) = U ( v ), where H t , H s , H v , H vv , H ss and H sv represent the different partial derivatives of H ( T, s, v ).5ccording to v = V ( t ) = B ( t ) + Y ( t ) − L ( t ) + ct and 0 < r < R < µ , if v > Y ( t ) + ct , the investor will reject the loan; If v ≤ Y ( t ) + ct , the investorwill choose to load, but the total amount will not exceed Y ( t ) + ct − v , thatis, L ∗ ( t ) = Y ( t ) + ct − v = max { , Y ( t ) + ct − v } .From the above setting, the two situations are discussed as follows:(1)In the case of v ≥ Y ( t ) + ct , substituting L ∗ ( t ) = 0 back into (5), theHJB equation can be rewritten as sup { H t + [ rv + ( k ( θ − s ) s − r ) Y − rct + c ] H v + k ( σ − s ) H s + 12 σ s Y H vv + 12 σ H ss + σ s Y H vs } = 0 H ( T, s, v ) = U ( v ) . (6)(2)In the case of v ≥ Y ( t )+ ct , putting L ∗ ( t ) = 0 in (5), the correspondingHJB equation can be rewritten as sup { H t + [ rv + ( k ( θ − s ) s − r ) Y − rct + c ] H v + k ( θ − s ) H s + 12 σ s Y H vv + 12 σ H ss + σ s Y H vs } = 0 H ( T, s, v ) = U ( v ) . (7)Take the derivative of (6) with respect to Y and we have Y ∗ = − k ( θ − s ) − rsσ H v H vv − s H vs H vv . (8)Similarly, we can also get the efficient investment strategy of this prob-lem(7) Y ∗ = − k ( θ − s ) − Rsσ H v H vv − s H vs H vv . (9)Plugging Y ∗ and Y ∗ into (6) and (7), we derive respectively H t + k ( θ − s ) H s + 12 σ H ss + ( rx − rct + c ) H v − [ k ( θ − s ) − rs ] σ H v H vv (10) − σ H vs H vv − [ k ( θ − s ) − rs ] H v H vs H vv = 0 , H t + k ( θ − s ) H s + 12 σ H ss + ( Rx − Rct + c ) H v − [ k ( θ − s ) − Rs ] σ H v H vv (11) − σ H vs H vv − [ k ( θ − s ) − Rs ] H v H vs H vv = 0 . Obviously, the stochastic control problem is transformed into a NPDE.Next, we alternate the NPDE into the linear PDE based on the dual trans-formation.
4. Model solution
Definition 2.
Let f : R n → R be a convex function. Legendre transformcan be defined as follows: L ( z ) = sup x { f ( x ) − zx } , < t < T. (12)Then the function L ( z ) is called Legendre dual function of Legendre .With reference to Jose et al. (2006) , a specific definition is proposed byˆ H ( t, s, z ) = sup v> { H ( t, s, v ) − zx | < v < ∞} , < t < T, where z > v .The value of v where this optimum is denoted by g ( t, s, z ), so that, g ( t, s, z ) = inf v> { v | H ( t, s, v ) ≥ zx + ˆ H ( t, s, v ) } , < t < T. From the above equation, we can getˆ H ( t, s, z ) = H ( t, s, g ) − zg, (13)Where g ( t, s, z ) = v and H v = z .The function ˆ H is related to g by g = − ˆ H z . (14)7y differentiating (14), we achieve H t = ˆ H t , H s = ˆ H s , H vv = − H zz , H ss = ˆ H ss − ˆ H sz ˆ H zz , H sv = − ˆ H sz ˆ H zz . (15)At the terminal time T , we defineˆ U ( z ) = sup v> { U ( z ) − zv | < v < ∞} , (16) G ( z ) = sup v> { U ( z ) − zv | < v < ∞} . In addition, there exists g ( T, s, z ) = ( U ′ ) − , which is a boundary condi-tion.Plugging (15) into (10) and (11), we deriveˆ H t + k ( θ − s ) ˆ H s + 12 σ ˆ H ss + ( rx − rct + c ) z (17)+ [ k ( θ − s ) − rs ] z ˆ H zz σ − [ k ( θ − s ) − rs ] z ˆ H sz = 0 . Differentiating both sides of (17) with respect to z , we obtainˆ H tz +[ k ( θ − s ) − rs ] ˆ H sz + 12 σ ˆ H ssz + ( rx − rct + c ) + rzg z + [ k ( θ − s ) − rs ] z ˆ H zz σ (18)+ [ k ( θ − s ) − rs ] z ˆ H zzz σ − [ k ( θ − s ) − rs ] ˆ H sz − [ k ( θ − s ) − rs ] z ˆ H szz = 0 . Due to (14), we get v = g = − ˆ H z , ˆ H tz = − g t , ˆ H sz = − g s , ˆ H zz = − g z , (19)ˆ H ssz = − g ss , ˆ H szz = − g sz , ˆ H zzz = − g zz . We recover (18) by using (19), and then obtain the following partialdifferential equation g t +[ k ( θ − s ) − rs ] g s + 12 σ g ss − ( rg − rct + c ) − rzg z + [ k ( θ − s ) − rs ] zg z σ (20)+ [ k ( θ − s ) − rs ] z g zz σ − [ k ( θ − s ) − rs ] g s − [ k ( θ − s ) − rs ] zg sz = 0 . Y ∗ Y ∗ = − [ k ( θ − s ) − rs ] szσ g z + sg s . (21) Theorem 4.2.1.
If the price of the risk-free asset, the price of the risk assetand the wealth process follow (1)-(3) respectively, the optimal portfolio ofthe enterprise annuity is specified by according to (22)-(30) Y ∗ ( t ) = [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ , v ≤ ct + [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ v − ct, [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ + ct < v < [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ + ct [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ , v ≥ ct + [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ . (22) Proof.
In the light of the logarithmic utility function, a definition is alsoprovided U ( x ) = lnx, x > , Depending on the form of logarithmic utility function, we have g ( T, s, z ) = 1 z .
In response to (20), we construct its corresponding solution g ( t, s, z ) = 1 z f ( s t ) + ϕ ( t ) . (23)In the meantime, we quote the boundary conditions by f ( s T ) = 1 and ϕ ( T ) = 0.Suppose it is a convex function and we can attain df ( x ) f ( x ) − z = 0 . Assume x is the optimum point, and there is L ( z ) = f ( x ) − zx . If f ( x ) = lnx , we have x = z .As a result, L ( z ) = f ( 1 z ) − ln z − − lnz − g t = ϕ t , g s = 1 z f + s, g z = − z f (24) g ss = 1 z f ss , g sz = − z f s , g zz = 2 z f. Substituting (23) back into (20), we obtain ϕ t + k ( θ − s ) 1 z f s + σ z f ss + rct − c − rϕ = 0 . (25)By observation, (24) can be decomposed into two equations, which issupplied to eliminate the dependence on s . Furthermore, since the boundaryconditions are f ( s T ) = 1 and ϕ ( T ) = 0, we have k ( θ − s ) 1 z f s + σ z f ss = 0 f ( s T ) = 1 , (26)and ( ϕ t − rϕ + rct − c = 0 ϕ ( T ) = 0 . (27)By integrating the two equations, we derive the solution to (25) f ( s t ) = 1 . The corresponding solution of (26) is given by ϕ ( t ) = ct − cT e r ( t − T ) , consequently, g = 1 z + ct − cT e r ( t − T ) . (28)Due to g ( t, s, v ) = v , we derive1 z = v − ct + cT e r ( t − T ) . (29)Finally, the optimal strategy Y ∗ can be rewritten as10 ∗ = − [ k ( θ − s ) − rs ] szσ g z + sg s (30)= [ k ( θ − s ) − rs ] sσ = [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ . From the equivalence of r and R , we can get another optimal investmentstrategy Y ∗ Y ∗ = [ k ( θ − s ) − Rs ][ v − ct + cT e r ( t − T ) ] sσ . (31)The above results are discussed as follows.(1) If v ≤ [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ + ct , then Y ∗ = [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ . (32)(2) If v ≤ ( [ k ( θ − s ) − Rs ][ v − ct + cT e r ( t − T ) ] sσ + ct ) , then Y ∗ = [ k ( θ − s ) − Rs ][ v − ct + cT e r ( t − T ) ] sσ . (33)(3) If [ k ( θ − s ) − Rs ][ v − ct + cT e r ( t − T ) ] sσ + ct < v < [ k ( θ − s ) − rs ][ v − ct + cT e R ( t − T ) ] sσ + ct , wewill proceed with two cases.(i)With Y ( t ) + ct ∈ [ [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ + ct, v ], since L ∗ ( t ) = Y ( t ) + ct − v = max { , Y ( t ) + ct − v } , then L ∗ ( t ) = 0. It means that the investmentrefuses to lend in this case. Let the left bracket of (6) be φ ( Y ). Because φ ( Y ) is increasing with respect to Y ≤ [ [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ , v − ct ], φ ( Y ) attains its maximum at Y ∗ ( t ) = v − ct . (ii)With Y ( t ) + ct ∈ [ v, [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ + ct ], since L ∗ ( t ) = Y ( t ) + ct − v = max { , Y ( t ) + ct − v } , then L ∗ ( t ) = v − ct . Denote (7) the left bracket by φ ( Y ). Consid-ering φ ( Y ) decreases of Y in the interval [ v − ct, [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ ],we have that φ ( Y ) reaches the maximum at Y ∗ ( t ) = v − ct . Hence, in theinterval [ [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ , [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ ], Y ∗ ( t ) = v − ct .11n overall, the optimal investment strategy Y ∗ ( T )can be expressed as Y ∗ ( t ) = [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ , v ≤ ct + [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ v − ct, [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ + ct < v < [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ + ct [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ , v ≥ ct + [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ . (34) Theorem 4.2.2.
If the price of the risk-free asset, the price of the risk assetand the wealth process follow (1)-(3) respectively, the expected maximumutility of the enterprise annuity for problem (10) and (11) is(1)In the case of v ≥ Y ( t ) + ct , H = ln ( v − ct + cT e r ( t − T ) ) . (2)In the case of v < Y ( t ) + ct , H = ln ( v − ct + cT e R ( t − T ) ) . Proof.
We first prove the first case. Combining(27) with − ˆ H z ( t, s, v ) = g ,we have ˆ H z = − z − ct + cT e r ( t − T ) . (35)From (34), integrating yieldsˆ H = − lnz + − ctz + czT e r ( t − T ) + m, Where m is a constant.Taking into account ˆ H = H − zg and the terminal condition m = −
1, weobtain H = ln ( v − ct + cT e r ( t − T ) ) . By the same token, we derive H = ln ( v − ct + cT e R ( t − T ) ) .
5. Numerical analysis
Based on these simulation results, we provide some economic explanationsand discuss the behavioral features related to loss aversion, and contributionrate. We take the initial time t = 5, and the investor will retire at T = 20.12n the financial market, other parameters are r = 0 . R = 0 . σ = 0 . c = 0 . σ on the optimal investment strategy Y ∗ ( t ) is taken into account. Assume that the wealth value is 500, 1200at time t, respectively, and the volatility σ varies at [1 , . v = 500,then v ≤ ct + [ k ( θ − s ) − Rs ][ v − ct + cT e R ( t − T ) ] sσ . If σ varies in the range [1 , .
4] and v = 1200, then v ≥ ct + [ k ( θ − s ) − rs ][ v − ct + cT e r ( t − T ) ] sσ . Y * Fig. 1. Effect of parameter σ on Y ∗ when v = 500 Y * Fig.2. Effect of parameter σ on Y ∗ when v = 120013ig.2 presents the impacts of σ on Y ∗ invested in the risky asset. We cansee, from Fig.1, Y ∗ reduces while σ grows. In an economic sense, the moredrastic the stock price varies, the more uncertain the market is. Investorsare afraid to take risks and will engage in conservative risk-averse behaviors,that is, they will increase their investment in risk-free assets. v Y * Fig.3. Effect of parameter v on Y ∗ when R > r
Fig. 3 shows v on Y ∗ invested in the risky asset. We adopt the instan-taneous volatility σ = 0 .
2. From Fig. 3, we may find that Y ∗ increaseswith the initial wealth v . This can be explained by the fact that employeesbecome richer, they become more capable of taking risks. Therefore, thepension manager tend to spend money on risky assets to get more return. Y * =0.2=0.15=0.1 Fig. 4. Effects of parameters σ and µ on Y ∗ σ and µ on the robust optimal investment strategy. Asshown in Fig. 4, Y ∗ decreases with regard to σ . A higher σ leads toa larger expected drop in volatility and an increased probability of a largeadverse movement in the risky assets price. In addition, under the elasticitycoefficient σ is fixed, when µ is raising, Y ∗ also rises. This is because thatan increase in the expected instantaneous rate makes the member improvehis ability to resist risk, and hence she invests more in the stock. R Y * Fig. 5. Effect of parameter R on Y ∗ Fig. 5 displays that as the lending rate R increases, the proportion ofwealth invested in the stock becomes larger. With the increase of R , thereis more risk in the market and it takes a lot of time to invest. As a result,the manager will reduce the amount of money on risk assets which is also inline with the economic market.
6. Conclusion
Optimal portfolio has always been the core of financial market research.We do the research about problem with the the O-U process under
CRRA utility. With the help of dynamic programming principle and dual transformmethod, the closed-form of optimal asset allocation strategy is obtained.Finally, MATLAB software is used for programming. More importantly, wedo an analysis of the volatility of the stocks, the initial wealth, the elasticitycoefficient and the lending rate on the investment behavior.
Acknowledgements
References
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The optimal investment strategy for dc pension plan witha dynamic investment target. Xitong Gongcheng Lilun Yu Shijian/system EngineeringTheory & Practice 37 (9), 2209–2221.Tang, M.-L., Chen, S.-N., Lai, G. C., Wu, T.-P., 2018. Asset allocation for a dc pen-sion fund under stochastic interest rates and inflation-protected guarantee. InsuranceMathematics & Economics 78, 87–104.Teng, L., Ehrhardt, M., Gnther, M., 2016. On the heston model with stochastic correlation.International Journal of Theoretical & Applied Finance 19 (06), 1–42.Wang, Pei, Li, Zhongfei, 2018. Robust optimal investment strategy for an aam of dc pen-sion plans with stochastic interest rate and stochastic volatility. Insurance Mathematics& Economics.Zhang, C., Rong, X., 2013. Optimal investment for dc pension with stochastic salary undera cev model. Discrete Dynamics in Nature and Society 000 (1), 1–9.Zhang, L., Li, D., Lai, Y., 2020. Equilibrium investment strategy for a defined contri-bution pension plan under stochastic interest rate and stochastic volatility. Journal ofComputational and Applied Mathematics 368, 112536.u, M., Yang, Y., Li, S., Zhang, J., 2010. Constant elasticity of variance model for pro-portional reinsurance and investment strategies. Insurance: Mathematics & Economics46 (3), 580 – 587.Guambe, C., Kufakunesu, R., van Zyl, G., Beyers, C., 2019. Optimal asset allocation fora dc plan with partial information under inflation and mortality risks, 67–83.Guan, G., Liang, Z., 2016. Optimal management of dc pension plan under loss aversionand value-at-risk constraints. Insurance Mathematics & Economics 69 (jul.), 224–237.Han, N. W., Hung, M. W., 2012. Optimal asset allocation for dc pension plans underinflation. Insurance Mathematics & Economics 51 (1), 172–181.Hsu, Y., Lin, T., Lee, C., 2008. Constant elasticity of variance (cev) option pricing model:Integration and detailed derivation. Mathematics and Computers in Simulation (MAT-COM) 79.Jianwu, Xiao, , , Zhai, Hong, , , Chenglin, Qin, 2007. The constant elasticity of variance(cev) model and the legendre transformcdual solution for annuity contracts. InsuranceMathematics & Economics 40 (2), p.302–310.Jose, Luis, Menaldi, 2006. Controlled markov processes and viscosity solutions 25.Markowitz, Harry, 1952. Portfolio selection*. Journal of Finance 7 (1), 77–91.Mudzimbabwe, W., 2019a. A simple numerical solution for an optimal investment strategyfor a dc pension plan in a jump diffusion model. Journal of Computational and AppliedMathematics, 55–61.Mudzimbabwe, W., 2019b. A simple numerical solution for an optimal investment strategyfor a dc pension plan in a jump diffusion model. Journal of Computational and AppliedMathematics 360, 55–61.Sun, J., Li, Z., Li, Y., 2017. The optimal investment strategy for dc pension plan witha dynamic investment target. Xitong Gongcheng Lilun Yu Shijian/system EngineeringTheory & Practice 37 (9), 2209–2221.Tang, M.-L., Chen, S.-N., Lai, G. C., Wu, T.-P., 2018. Asset allocation for a dc pen-sion fund under stochastic interest rates and inflation-protected guarantee. InsuranceMathematics & Economics 78, 87–104.Teng, L., Ehrhardt, M., Gnther, M., 2016. On the heston model with stochastic correlation.International Journal of Theoretical & Applied Finance 19 (06), 1–42.Wang, Pei, Li, Zhongfei, 2018. Robust optimal investment strategy for an aam of dc pen-sion plans with stochastic interest rate and stochastic volatility. Insurance Mathematics& Economics.Zhang, C., Rong, X., 2013. Optimal investment for dc pension with stochastic salary undera cev model. Discrete Dynamics in Nature and Society 000 (1), 1–9.Zhang, L., Li, D., Lai, Y., 2020. Equilibrium investment strategy for a defined contri-bution pension plan under stochastic interest rate and stochastic volatility. Journal ofComputational and Applied Mathematics 368, 112536.