Optimal portfolio for a defined-contribution pension plan under a constant elasticity of variance model with exponential
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Optimal portfolio for a defined-contribution pension plan under a constant elasticity of variance model with exponential utility Xiaoqian SUN1 , Xuelin YONG1 , Jianwei GAO2 1 School of Mathematical Sciences and Physics, North China Electric Power University, Beijing 102206, China 2 School of Economics and Management, North China Electric Power University, Beijing 102206, China
c Higher Education Press 2020
Abstract Based on the Lie symmetry method, we derive the explicit optimal invest strategy for an investor who seeks to maximize the expected exponential (CARA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model. We examine the point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem. The symmetries compatible with the terminal condition enable us to transform the (2 + 1)-dimensional HJB equation into a (1 + 1)dimensional nonlinear equation which is linearized by its infinite-parameter Lie group of point transformations. Finally, the ansatz technique based on variables separation is applied to solve the linear equation and the optimal strategy is obtained. The algorithmic procedure of the Lie symmetry analysis method adopted here is quite general compared with conjectures used in the literature. Keywords Lie symmetry, portfolio, defined-contribution (DC) pension plan, constant elasticity of variance (CEV) model, exponential utility MSC2020 22E70, 58D19, 91B02 1
Introduction
The defined contribution (DC) pension plan, where contributions are fixed and benefits depend on the returns on the funds portfolio, has become popular in global pension market. The key to a DC plan is that the employee accepts the investment risk and is responsible for ensuring that there are enough funds in the plan to meet their needs upon retirement [8,10]. Received August 31, 2020; accepted October 16, 2020 Corresponding author: Xuelin YONG, E-mail: [email protected]
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In order to find the optimal investment strategy for the investors in a DC pension scheme, Gao [9] supposed that the market structure consists of two financial assets, a riskless asset Bt which evolves as dBt = rBt dt, and a single risky asset St which is described by the constant elasticity of variance (CEV) model dSt = µdt + kStβ dWt , St where r is a constant rate of interest, µ > r is an expected instantaneous rate of return, kStβ is the instantaneous volatility, β < 0 is the elasticity parameter, and Wt is a standard Brownian motion. Furthermore, the contribution rate is assumed as a fixed constant c ∈ (0, 1) compared with the one unit wage. Then the wealth process of pension Vt ∈ [0, T ] is given by dVt = πt Vt
dBt dSt + (1 − πt )Vt + cdt St Bt
= [πt (µ − r)Vt + rVt + c]dt + πt Vt kStβ dWt ,
(1)
where πt and 1 − πt denote the proportion of the pension fund invested in the risky and risk-free assets, respectively, and T is the horizon for the fund investment and can be interpreted as the retirement date in a DC
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