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CN 34-1054/N

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On optimal investment strategy of pension funds with a minimum guarantee under Knightian uncertainty

  • Department of Financial Engineering, Anhui Polytechnic University, Wuhu 241000, China

Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2014.03.004
  • Received Date: 15 December 2011
  • Accepted Date: 06 March 2012
  • Rev Recd Date: 06 March 2012
  • Publish Date: 30 March 2014
    Abstract

  • Abstract

    A continuous-time stochastic control model of optimal management was proposed for a defined contribution pension fund with a minimum guarantee. A pension fund managers utility was characterized from the fund wealth on an infinite horizon by α-maxmin expected utility (α-MEU), by which he differentiates ambiguity and ambiguity attitude under Knightian uncertainty. Pension fund managers value function was derived by the stochastic control theory. The explicit expressions for both the optimal allocation strategy in feedback form and the value function which is a solution to the HJB equation were obtained.

    Abstract

    A continuous-time stochastic control model of optimal management was proposed for a defined contribution pension fund with a minimum guarantee. A pension fund managers utility was characterized from the fund wealth on an infinite horizon by α-maxmin expected utility (α-MEU), by which he differentiates ambiguity and ambiguity attitude under Knightian uncertainty. Pension fund managers value function was derived by the stochastic control theory. The explicit expressions for both the optimal allocation strategy in feedback form and the value function which is a solution to the HJB equation were obtained.
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    • References(21)
  • [1]
    Chen Z, Epstein L G. Ambiguity, risk and asset returns in continuous time [J]. Econometrica, 2002, 70: 1 403-1 443.
    [2]
    Fei W Y. Optimal consumption and portfolio choice with ambiguity and anticipation [J]. Information Sciences, 2007, 117: 5 178-5 190.
    [3]
    Fei W Y. Optimal portfolio choice based on α-MEU under ambiguity [J]. Stochastic Models, 2009, 25: 455-482.
    [4]
    Xia Dengfeng, Fei Weiyin, Liu Hongjian. On study of optimal investment with ambiguity and anticipation under fluctuated discounting rate[J]. Chinese Journal of Applied Probability and Statistics, 2010, 26(3): 270-276.
    夏登峰，费为银，刘宏建．变折现率下带含糊厌恶与预期的最优投资研究 [J].应用概率统计，2010，26(3)：270-276.
    [5]
    Xia Dengfeng，Fei Weiyin，Liang Yong. Maximization of shareholders value with ambiguity[J]. Journal of University of Science and Technology of China, 2010，40(9)：920-924.
    夏登峰，费为银，梁勇．带含糊厌恶的股东价值最大化 [J].中国科学技术大学学报，2010，40(9)：920-924.
    [6]
    Fei Weiyin, Li Shujuan. Study on optimal consumption and portfolio with inflation under Knightian uncertainty[J]. Chinese Journal of Engineering Mathematics, 2012, 29(6): 799-806.
    费为银，李淑娟．Knight不确定下带通胀的最优消费和投资模型研究[J].工程数学学报，2012，29(6)：799-806.
    [7]
    Fei Weiyin, Chen Chao, Liang Yong. Optimal consumption-portfolio and retirement problem with disutility under Knightian uncertainty[J]. Chinese Journal of Applied Probability and Statistics, 2013, 29(1): 53-63.
    费为银,陈超,梁勇.Knight不确定下考虑负效用的消费和投资问题研究[J].应用概率统计,2013,29(1):53-63.
    [8]
    Li Juan，Fei Wei-Yin，Shi Xueqin， et al. Optimal trading strategy under disordered asset return and partial information[J]. Journal of Mathematics, 2012,32(4): 693-700.
    李娟,费为银,石学芹,等.部分信息下资产收益率发生紊乱的最优投资策略[J]. 数学杂志, 2012,32(4): 693-700.
    [9]
    Li Juan, Fei Weiyin, Shi Xueqin, et al. Optimal trading strategy under disordered asset return and Knightian uncertainty[J]. Applied Mathematics A Journal of Chinese Universities, 2013, 28(1): 13-22.
    李娟,费为银，石学芹，等.奈特不确定下资产收益率发生紊乱的最优投资模型研究[J].高校应用数学学报A辑， 2013, 28(1): 13-22.
    [10]
    Fei W Y. Optimal consumption-leisure, portfolio and retirement selection based on α-maxmin expected CES utility with ambiguity[J]. Applied Mathematics A Journal of Chinese University, Series B, 2012, 27(4): 435-454.
    [11]
    Deelstra G, Grasselli M, Koehl P F. Optimal investment strategies in the presence of a minimum guarantee [J]. Insur Math Econ, 2003, 33: 189-207.
    [12]
    Deelstra G, Grasselli M, Koehl P F. Optimal design of a guarantee for defined contribution funds[J]. J Econ Dyn Control, 2004, 28: 2 239-2 260.
    [13]
    Giacinto M D, Federico S, Gozzi F. Pension funds with a minimum guarantee: A stochastic control approach [J]. Finance Stoch, 2011, 15(2): 297-342.
    [14]
    Shi Xueqin, Fei Weiyin, Li Juan, et al. Research on stochastic control of pension funds with a minimum guarantee and dividend[J]. Mathematical Theory and Application, 2011，31(3)：85-93.
    石学芹，费为银，李娟，等．带最低保障和红利的养老基金随机控制问题研究[J].数学理论与应用，2011，31(3)：85-93.
    [15]
    Chen Z J, Wang B. Infinite time interval BSDEs and the convergence of g-martingales[J]. J Austral Math Soc (Series A), 2000, 69: 187-211.
    [16]
    Merton R C. Lifetime portfolio under uncertainty: The continuous-time case [J]. Rev Econ Stat, 1969, 51: 247-257.
    [17]
    Merton R C. Optimum consumption and portfolio rules in a continuous-time model [J]. J Econ Theory, 1971, 3: 373-413.
    [18]
    刘宏建，费为银，祖纷，等．股票波动率具有模型不确定的最优消费与投资问题[J].工程数学学报，2014，31(1)：35-43.
    [19]
    Peng S. Nonlinear expectations and stochastic calculus under uncertainty[DB/OL]. arXiv:1002.4546v1, 2010.
    [20]
    Epstein L, Ji S. Ambiguity volatility, possibility and utility in continuous time[DB/OL]. arXiv:1103.1652, 2013.
    [21]
    Fei W Y, Fei C. Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion[DB/OL]. arXiv:1309.0209v1, 2013.
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    [1]
    Chen Z, Epstein L G. Ambiguity, risk and asset returns in continuous time [J]. Econometrica, 2002, 70: 1 403-1 443.
    [2]
    Fei W Y. Optimal consumption and portfolio choice with ambiguity and anticipation [J]. Information Sciences, 2007, 117: 5 178-5 190.
    [3]
    Fei W Y. Optimal portfolio choice based on α-MEU under ambiguity [J]. Stochastic Models, 2009, 25: 455-482.
    [4]
    Xia Dengfeng, Fei Weiyin, Liu Hongjian. On study of optimal investment with ambiguity and anticipation under fluctuated discounting rate[J]. Chinese Journal of Applied Probability and Statistics, 2010, 26(3): 270-276.
    夏登峰，费为银，刘宏建．变折现率下带含糊厌恶与预期的最优投资研究 [J].应用概率统计，2010，26(3)：270-276.
    [5]
    Xia Dengfeng，Fei Weiyin，Liang Yong. Maximization of shareholders value with ambiguity[J]. Journal of University of Science and Technology of China, 2010，40(9)：920-924.
    夏登峰，费为银，梁勇．带含糊厌恶的股东价值最大化 [J].中国科学技术大学学报，2010，40(9)：920-924.
    [6]
    Fei Weiyin, Li Shujuan. Study on optimal consumption and portfolio with inflation under Knightian uncertainty[J]. Chinese Journal of Engineering Mathematics, 2012, 29(6): 799-806.
    费为银，李淑娟．Knight不确定下带通胀的最优消费和投资模型研究[J].工程数学学报，2012，29(6)：799-806.
    [7]
    Fei Weiyin, Chen Chao, Liang Yong. Optimal consumption-portfolio and retirement problem with disutility under Knightian uncertainty[J]. Chinese Journal of Applied Probability and Statistics, 2013, 29(1): 53-63.
    费为银,陈超,梁勇.Knight不确定下考虑负效用的消费和投资问题研究[J].应用概率统计,2013,29(1):53-63.
    [8]
    Li Juan，Fei Wei-Yin，Shi Xueqin， et al. Optimal trading strategy under disordered asset return and partial information[J]. Journal of Mathematics, 2012,32(4): 693-700.
    李娟,费为银,石学芹,等.部分信息下资产收益率发生紊乱的最优投资策略[J]. 数学杂志, 2012,32(4): 693-700.
    [9]
    Li Juan, Fei Weiyin, Shi Xueqin, et al. Optimal trading strategy under disordered asset return and Knightian uncertainty[J]. Applied Mathematics A Journal of Chinese Universities, 2013, 28(1): 13-22.
    李娟,费为银，石学芹，等.奈特不确定下资产收益率发生紊乱的最优投资模型研究[J].高校应用数学学报A辑， 2013, 28(1): 13-22.
    [10]
    Fei W Y. Optimal consumption-leisure, portfolio and retirement selection based on α-maxmin expected CES utility with ambiguity[J]. Applied Mathematics A Journal of Chinese University, Series B, 2012, 27(4): 435-454.
    [11]
    Deelstra G, Grasselli M, Koehl P F. Optimal investment strategies in the presence of a minimum guarantee [J]. Insur Math Econ, 2003, 33: 189-207.
    [12]
    Deelstra G, Grasselli M, Koehl P F. Optimal design of a guarantee for defined contribution funds[J]. J Econ Dyn Control, 2004, 28: 2 239-2 260.
    [13]
    Giacinto M D, Federico S, Gozzi F. Pension funds with a minimum guarantee: A stochastic control approach [J]. Finance Stoch, 2011, 15(2): 297-342.
    [14]
    Shi Xueqin, Fei Weiyin, Li Juan, et al. Research on stochastic control of pension funds with a minimum guarantee and dividend[J]. Mathematical Theory and Application, 2011，31(3)：85-93.
    石学芹，费为银，李娟，等．带最低保障和红利的养老基金随机控制问题研究[J].数学理论与应用，2011，31(3)：85-93.
    [15]
    Chen Z J, Wang B. Infinite time interval BSDEs and the convergence of g-martingales[J]. J Austral Math Soc (Series A), 2000, 69: 187-211.
    [16]
    Merton R C. Lifetime portfolio under uncertainty: The continuous-time case [J]. Rev Econ Stat, 1969, 51: 247-257.
    [17]
    Merton R C. Optimum consumption and portfolio rules in a continuous-time model [J]. J Econ Theory, 1971, 3: 373-413.
    [18]
    刘宏建，费为银，祖纷，等．股票波动率具有模型不确定的最优消费与投资问题[J].工程数学学报，2014，31(1)：35-43.
    [19]
    Peng S. Nonlinear expectations and stochastic calculus under uncertainty[DB/OL]. arXiv:1002.4546v1, 2010.
    [20]
    Epstein L, Ji S. Ambiguity volatility, possibility and utility in continuous time[DB/OL]. arXiv:1103.1652, 2013.
    [21]
    Fei W Y, Fei C. Optimal stochastic control and optimal consumption and portfolio with G-Brownian motion[DB/OL]. arXiv:1309.0209v1, 2013.
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