随机利率与随机波动率环境下的DC型养老金计划
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摘要
为实现养老金的保值增值,基金管理人将养老金投资于金融市场.假设金融市场包含一种无风险资产、一种股票和一种零息票债券,其中,利率期限结构满足随机仿射利率模型,而股票价格波动率满足Heston随机波动率模型.基金管理人希望寻找一种最优投资组合以最大化其终端财富的期望效用.假设基金管理人对风险的偏好满足幂效用或指数效用,运用随机动态规划原理和变量替换方法,得到幂效用和指数效用下最优投资策略的显式解.最后,通过数值算例分析主要模型参数对最优投资策略的影响.研究结果表明,利率风险、股市波动风险以及缴费率都对缴费确定(DC)型养老金的投资决策产生较大的影响.
In order to preserve and increase the value of pension fund, a fund manager can invest pension fund into a financial market with a risk-free asset, a stock and a zero-coupon bond. In the financial market, short rate is assumed to follow stochastic affine interest rate model, while stock price is supposed to be driven by Heston's stochastic volatility model. The fund manager expects to obtain an optimal portfolio to maximize the expected utility of terminal utility.Assuming the risk-aversion of the fund manager to satisfy power utility or exponential utility, the explicit solutions to the optimal portfolios with power utility or exponential utility are obtained by using the principle of stochastic dynamic programming and the variable change technique. Finally, a numerical example is given to analyse the impact of main market parameters on the optimal portfolios. Research results show that interest rate risk, volatility risk and contribution rate have a considerable influence on decision-making of fund manager for the defined-contribution pension found.
In order to preserve and increase the value of pension fund, a fund manager can invest pension fund into a financial market with a risk-free asset, a stock and a zero-coupon bond. In the financial market, short rate is assumed to follow stochastic affine interest rate model, while stock price is supposed to be driven by Heston's stochastic volatility model. The fund manager expects to obtain an optimal portfolio to maximize the expected utility of terminal utility.Assuming the risk-aversion of the fund manager to satisfy power utility or exponential utility, the explicit solutions to the optimal portfolios with power utility or exponential utility are obtained by using the principle of stochastic dynamic programming and the variable change technique. Finally, a numerical example is given to analyse the impact of main market parameters on the optimal portfolios. Research results show that interest rate risk, volatility risk and contribution rate have a considerable influence on decision-making of fund manager for the defined-contribution pension found.
引文
[1]谷爱玲,李仲飞,曾燕. Ornstein-Uhlenbeck模型下DC养老金计划的最优投资策略[J].应用数学学报, 2013,36(3):715-726.(Gu A L, Li Z F, Zeng Y. Optimal investment strategy under Ornstein-Uhlenbeck model for a DC pension plan[J]. Acta Mathematicae Applicatae Sinica, 2013,36(3):715-726.)
[2] Boulier J F, Huang S J, Taillard G. Optimal management under stochastic interest rates:The case of a protected defined contribution pension fund[J].Insurance:Mathematics and Economics, 2001, 28(1):173-189.
[3] Deelstra G, Grasselli M, Koehl P F. Optimal investment strategies in the presence of a minimum guarantee[J].Insurance:Mathematics and Economics, 2003, 33(1):189-207.
[4] Gao J W. Stochastic optimal control of DC pension funds[J]. Insurance:Mathematics and Economics, 2008,42(3):1159-1164.
[5]张初兵,荣喜民.仿射利率模型下确定缴费型养老金的最优投资[J].系统工程理论与实践, 2012, 32(5):1048-1056.(Zhang C B, Rong X M. Optimal investment for DC pension under the affine interest rate model[J]. Systems Engineering—Theory&Practice, 2012, 32(5):1048-1056.)
[6]殷俊,李媛媛.基于随机利率和通货膨胀的缴费确定型养老金计划最优资产配置策略[J].当代经济科学,2013, 35(2):11-20.(Yin J, Li Y Y. Stochastic interest rate and inflation based optimal asset allocation tactics for defined-contribution pension plans[J]. Modern Economic Science, 2013,35(2):11-20.)
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[8]常浩,王春峰,房振明.通胀风险下基于HARA效用的DC型养老金计划[J].运筹学学报, 2016, 20(4):39-51.(Chang H, Wang C F, Fang Z M. Defined contribution pension fund scheme with HARA preference under inflation risk[J]. Operations Research Transactions, 2016,20(4):39-51.)
[9] Heston S L. A close-form solution for options with stochastic volatility with applications to bond and currency options[J]. Review of Financial Studies, 1993,6(2):327-343.
[10] Kraft H. Optimal portfolios and Heston’s stochastic volatility model:An explicit solution for power utility[J].Quantitative Finance, 2005, 5(3):303-313.
[11] Li Z F, Zeng Y, Lai Y Z. Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model[J]. Insurance:Mathematics and Economics, 2012, 50(1):191-203.
[12] Zhao H, Rong X M, Zhao Y G. Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model[J].Insurance:Mathematics and Economics, 2013, 53(3):504-514.
[13]张初兵,荣喜民,侯如靖,等. Heston模型下确定缴费型养老金的投资组合优化[J].系统工程, 2012, 30(12):39-44.(Zhang C B, Rong X M, Hou R J, et al. Investment portfolio optimization for defined-contribution pension under a Heston model[J]. Systems Engineering, 2012,30(12):39-44.)
[14] Liu J. Portfolio selection in stochastic environments[J].The Review of Financial Studies, 2007, 20(1):1-39.
[15] Li J Z, Wu R. Optimal investment problem with stochastic interest rate and stochastic volatility:Maximizing a power utility[J]. Applied Stochastic Models in Business and Industry, 2009, 25(3):407-420.
[16] Noh E J, Kim J H. An optimal portfolio model with stochastic volatility and stochastic interest rate[J]. J of Mathematical Analysis and Applications, 2011, 375(2):510-522.
[17] Chang H, Rong X M. An investment and consumption problem with CIR interest rate and stochastic volatility[J].Abstract and Applied Analysis, 2013(1):1-12.
[18] Guan G H, Liang Z X. Optimal management of DC pension plan in a stochastic interest rates and stochastic volatility framework[J]. Insurance:Mathematics and Economics, 2014, 57(1):58-66.
[2] Boulier J F, Huang S J, Taillard G. Optimal management under stochastic interest rates:The case of a protected defined contribution pension fund[J].Insurance:Mathematics and Economics, 2001, 28(1):173-189.
[3] Deelstra G, Grasselli M, Koehl P F. Optimal investment strategies in the presence of a minimum guarantee[J].Insurance:Mathematics and Economics, 2003, 33(1):189-207.
[4] Gao J W. Stochastic optimal control of DC pension funds[J]. Insurance:Mathematics and Economics, 2008,42(3):1159-1164.
[5]张初兵,荣喜民.仿射利率模型下确定缴费型养老金的最优投资[J].系统工程理论与实践, 2012, 32(5):1048-1056.(Zhang C B, Rong X M. Optimal investment for DC pension under the affine interest rate model[J]. Systems Engineering—Theory&Practice, 2012, 32(5):1048-1056.)
[6]殷俊,李媛媛.基于随机利率和通货膨胀的缴费确定型养老金计划最优资产配置策略[J].当代经济科学,2013, 35(2):11-20.(Yin J, Li Y Y. Stochastic interest rate and inflation based optimal asset allocation tactics for defined-contribution pension plans[J]. Modern Economic Science, 2013,35(2):11-20.)
[7]伍慧玲,董洪斌.带有通胀风险和随机收入的确定缴费养老计划[J].系统工程理论与实践, 2016, 36(3):545-558.(Wu H L, Dong H B. Multi-period mean-variance defined contribution pension management with inflation and stochastic income[J]. Systems Engineering—Theory&Practice, 2016, 36(3):545-558.)
[8]常浩,王春峰,房振明.通胀风险下基于HARA效用的DC型养老金计划[J].运筹学学报, 2016, 20(4):39-51.(Chang H, Wang C F, Fang Z M. Defined contribution pension fund scheme with HARA preference under inflation risk[J]. Operations Research Transactions, 2016,20(4):39-51.)
[9] Heston S L. A close-form solution for options with stochastic volatility with applications to bond and currency options[J]. Review of Financial Studies, 1993,6(2):327-343.
[10] Kraft H. Optimal portfolios and Heston’s stochastic volatility model:An explicit solution for power utility[J].Quantitative Finance, 2005, 5(3):303-313.
[11] Li Z F, Zeng Y, Lai Y Z. Optimal time-consistent investment and reinsurance strategies for insurers under Heston’s SV model[J]. Insurance:Mathematics and Economics, 2012, 50(1):191-203.
[12] Zhao H, Rong X M, Zhao Y G. Optimal excess-of-loss reinsurance and investment problem for an insurer with jump-diffusion risk process under the Heston model[J].Insurance:Mathematics and Economics, 2013, 53(3):504-514.
[13]张初兵,荣喜民,侯如靖,等. Heston模型下确定缴费型养老金的投资组合优化[J].系统工程, 2012, 30(12):39-44.(Zhang C B, Rong X M, Hou R J, et al. Investment portfolio optimization for defined-contribution pension under a Heston model[J]. Systems Engineering, 2012,30(12):39-44.)
[14] Liu J. Portfolio selection in stochastic environments[J].The Review of Financial Studies, 2007, 20(1):1-39.
[15] Li J Z, Wu R. Optimal investment problem with stochastic interest rate and stochastic volatility:Maximizing a power utility[J]. Applied Stochastic Models in Business and Industry, 2009, 25(3):407-420.
[16] Noh E J, Kim J H. An optimal portfolio model with stochastic volatility and stochastic interest rate[J]. J of Mathematical Analysis and Applications, 2011, 375(2):510-522.
[17] Chang H, Rong X M. An investment and consumption problem with CIR interest rate and stochastic volatility[J].Abstract and Applied Analysis, 2013(1):1-12.
[18] Guan G H, Liang Z X. Optimal management of DC pension plan in a stochastic interest rates and stochastic volatility framework[J]. Insurance:Mathematics and Economics, 2014, 57(1):58-66.