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January  2016, 12(1): 229-249. doi: 10.3934/jimo.2016.12.229

Time consistent policy of multi-period mean-variance problem in stochastic markets

Zhiping Chen 1, , Jia Liu 2, and  Gang Li 2,

1. 

Department of Computing Science, School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi, 710049
2. 

Department of Computing Science, School of Mathematics and Statistics, Xi'an Jiaotong University, 710049, Xi'an, Shaanxi, China, China

Received  April 2014 Revised  January 2015 Published  April 2015

Due to the non-separability of the variance operator, the optimal investment policy of the multi-period mean-variance model in Markovian markets doesn't satisfy the time consistency. We propose a new weak time consistency in stochastic markets and show that the pre-commitment optimal policy satisfies the weak time consistency at any intermediate period as long as the investor's wealth is no more than a specific threshold. When the investor's wealth exceeds the threshold, the weak time consistency no longer holds. In this case, by modifying the pre-commitment optimal policy, we derive a wealth interval, from which we determine a more efficient revised policy. The terminal wealth obtained under this revised policy can achieve the same mean as, but not greater variance than those of the terminal wealth obtained under the pre-commitment optimal policy; a series of superior investment policies can be obtained depending on the degree the investor wants the conditional variance to decrease. It is shown that, in the above revising process, a positive cash flow can be taken out of the market. Finally, an empirical example illustrates our theoretical results. Our results generalize existing conclusions for the multi-period mean-variance model in deterministic markets.
Keywords: policy revision., Markovian markets, Bellman's optimality principle, mean-variance, Time inconsistency.
Mathematics Subject Classification: Primary: 91G10; Secondary: 49N10, 93E2.
Citation: Zhiping Chen, Jia Liu, Gang Li. Time consistent policy of multi-period mean-variance problem in stochastic markets. Journal of Industrial and Management Optimization, 2016, 12 (1) : 229-249. doi: 10.3934/jimo.2016.12.229
References:
[1]

P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, Coherent multi-period risk adjusted values and Bellman's principle, Annals of Operations Research, 152 (2007), 5-22. doi: 10.1007/s10479-006-0132-6.

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016. doi: 10.1093/rfs/hhq028.

[3]

D. Bertsimas, G. J. Lauprete and A. Samarov, Shortfall as a risk measure: properties, optimization and app1ications, Journal of Economic Dynamics and Control, 28 (2004), 1353-1381.

[4]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, Working Paper, Stockolm School of Economics, 2009. Available from: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1694759.

[5]

T. Björk, A. Murgoci and X. Y. Zhou, Mean variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[6]

K. Boda and J. A. Filar, Time consistent dynamic risk measures, Mathematical Methods of Operations Research, 63 (2006), 169-186. doi: 10.1007/s00186-005-0045-1.

[7]

M. Britten-Jones and A. Neuberger, Option prices, implied price processes, and stochastic volatility, Journal of Finance, 55 (2000), 839-866. doi: 10.1111/0022-1082.00228.

[8]

U. Çakmak and S. Özekici, Portfolio optimization in stochastic markets, Mathematical Methods of Operations Research, 63 (2006), 151-168. doi: 10.1007/s00186-005-0020-x.

[9]

E. Çanakoğlu and S. Özekici, Portfolio selection in stochastic markets with exponential utility functions, Annals of Operations Research, 166 (2009), 281-297. doi: 10.1007/s10479-008-0406-2.

[10]

U. Çelikyurt and S. Özekici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, European Journal of Operational Research, 179 (2007), 186-202.

[11]

Z. P. Chen, G. Li and J. E. Guo, Optimal investment policy in the time consistent mean-variance formulation, Insurance: Mathematics and Economics, 52 (2013), 145-156. doi: 10.1016/j.insmatheco.2012.11.007.

[12]

Z. P. Chen, G. Li and Y. G. Zhao, Time-consistent investment policies in Markovian markets: A case of mean-variance analysis, Journal of Economic Dynamic and Control, 40 (2014), 293-316. doi: 10.1016/j.jedc.2014.01.011.

[13]

Z. P. Chen and J. Liu, Time consistent risk measure under two-level information structure and its application in dynamic portfolio selection, Working Paper, Xi'an Jiaotong university, 2014. Available from: http://t.cn/RhZ6Vrt.

[14]

Z. P. Chen and Y. Wang, Two-sided coherent risk measures and their application in realistic portfolio optimization, Journal of Banking and Finance, 32 (2008), 2667-2673. doi: 10.1016/j.jbankfin.2008.07.004.

[15]

Z. P. Chen and L. Yang, Nonlinearly weighted convex risk measure and its application, Journal of Banking and Finance, 35 (2011), 1777-1793. doi: 10.1016/j.jbankfin.2010.12.004.

[16]

P. Cheridito and M. Kupper, Composition of time-consistent dynamic monetary risk measures in discrete time, Journal of Theoretical and Applied Finance, 14 (2011), 137-162. doi: 10.1142/S0219024911006292.

[17]

X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346-378. doi: 10.1111/j.1467-9965.2010.00461.x.

[18]

C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance Stochastic, 17 (2013), 227-271. doi: 10.1007/s00780-012-0189-9.

[19]

R. J. Elliott, T. K. Siu and L. Chan, On pricing barrier options with regime switching, Journal of Computational and Applied Mathematics, 256 (2014), 196-210. doi: 10.1016/j.cam.2013.07.034.

[20]

L. G. Epstein and S. E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica, 57 (1989), 937-969. doi: 10.2307/1913778.

[21]

H. Geman and S. Ohana, Time-consistency in managing a commodity portfolio: A dynamic risk measure approach, Journal of Banking and Finance, 32 (2008), 1991-2005. doi: 10.1016/j.jbankfin.2007.05.020.

[22]

R. Korn and H. Kraft, A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40 (2002), 1250-1269. doi: 10.1137/S0363012900377791.

[23]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100.

[24]

X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504.

[25]

H. J. Lüthi and J. Doege, Convex risk measures for portfolio optimization and concepts of flexibility, Mathematical Programming, 104 (2005), 541-559. doi: 10.1007/s10107-005-0628-x.

[26]

F. Riedel, Dynamic coherent risk measures, Stochastic Processes and their Applications, 112 (2004), 185-200. doi: 10.1016/j.spa.2004.03.004.

[27]

B. Roorda and J. M. Schumacher, Time consistency conditions for acceptability measures, with an applications to Tail Value at Risk, Insurance: Mathematics and Economics, 40 (2007), 209-230. doi: 10.1016/j.insmatheco.2006.04.003.

[28]

A. Ruszczyński, Risk-averse dynamic programming for Markov decision processes, Mathematical Programming, Series B, 125 (2010), 235-261. doi: 10.1007/s10107-010-0393-3.

[29]

A. Shapiro, On a time consistency concept in risk averse multistage stochastic programming, Operations Research Letters, 37 (2009), 143-147. doi: 10.1016/j.orl.2009.02.005.

[30]

M. C. Steinbach, Markowitz revisited: Mean-variance models in financial portfolio analysis, SIAM Review, 43 (2001), 31-85. doi: 10.1137/S0036144500376650.

[31]

T. Wang, A class of dynamic risk measure, Working Paper, University of British Columbia, 1999. Available from: http://web.cenet.org.cn/upfile/57263.pdf.

[32]

J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A time-consistent strategy, European Journal of Operational Research, 209 (2011), 184-201. doi: 10.1016/j.ejor.2010.09.038.

[33]

S. Z. Wei and Z. X. Ye, Multi-period optimization portfolio with bankruptcy control in stochastic market, Applied Mathematics and Computation, 186 (2007), 414-425. doi: 10.1016/j.amc.2006.07.108.

[34]

L. Xu, R. M. Wang and D. J. Yao, Optimal stochastic investment games under Markov regime switching market, Journal of Industrial and Management Optimization, 10 (2014), 795-815. doi: 10.3934/jimo.2014.10.795.

[35]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33. doi: 10.1007/s002450010003.

[36]

S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457. doi: 10.1109/TAC.2004.824474.

show all references

References:
[1]

P. Artzner, F. Delbaen, J. M. Eber, D. Heath and H. Ku, Coherent multi-period risk adjusted values and Bellman's principle, Annals of Operations Research, 152 (2007), 5-22. doi: 10.1007/s10479-006-0132-6.

[2]

S. Basak and G. Chabakauri, Dynamic mean-variance asset allocation, Review of Financial Studies, 23 (2010), 2970-3016. doi: 10.1093/rfs/hhq028.

[3]

D. Bertsimas, G. J. Lauprete and A. Samarov, Shortfall as a risk measure: properties, optimization and app1ications, Journal of Economic Dynamics and Control, 28 (2004), 1353-1381.

[4]

T. Björk and A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, Working Paper, Stockolm School of Economics, 2009. Available from: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1694759.

[5]

T. Björk, A. Murgoci and X. Y. Zhou, Mean variance portfolio optimization with state dependent risk aversion, Mathematical Finance, 24 (2014), 1-24. doi: 10.1111/j.1467-9965.2011.00515.x.

[6]

K. Boda and J. A. Filar, Time consistent dynamic risk measures, Mathematical Methods of Operations Research, 63 (2006), 169-186. doi: 10.1007/s00186-005-0045-1.

[7]

M. Britten-Jones and A. Neuberger, Option prices, implied price processes, and stochastic volatility, Journal of Finance, 55 (2000), 839-866. doi: 10.1111/0022-1082.00228.

[8]

U. Çakmak and S. Özekici, Portfolio optimization in stochastic markets, Mathematical Methods of Operations Research, 63 (2006), 151-168. doi: 10.1007/s00186-005-0020-x.

[9]

E. Çanakoğlu and S. Özekici, Portfolio selection in stochastic markets with exponential utility functions, Annals of Operations Research, 166 (2009), 281-297. doi: 10.1007/s10479-008-0406-2.

[10]

U. Çelikyurt and S. Özekici, Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, European Journal of Operational Research, 179 (2007), 186-202.

[11]

Z. P. Chen, G. Li and J. E. Guo, Optimal investment policy in the time consistent mean-variance formulation, Insurance: Mathematics and Economics, 52 (2013), 145-156. doi: 10.1016/j.insmatheco.2012.11.007.

[12]

Z. P. Chen, G. Li and Y. G. Zhao, Time-consistent investment policies in Markovian markets: A case of mean-variance analysis, Journal of Economic Dynamic and Control, 40 (2014), 293-316. doi: 10.1016/j.jedc.2014.01.011.

[13]

Z. P. Chen and J. Liu, Time consistent risk measure under two-level information structure and its application in dynamic portfolio selection, Working Paper, Xi'an Jiaotong university, 2014. Available from: http://t.cn/RhZ6Vrt.

[14]

Z. P. Chen and Y. Wang, Two-sided coherent risk measures and their application in realistic portfolio optimization, Journal of Banking and Finance, 32 (2008), 2667-2673. doi: 10.1016/j.jbankfin.2008.07.004.

[15]

Z. P. Chen and L. Yang, Nonlinearly weighted convex risk measure and its application, Journal of Banking and Finance, 35 (2011), 1777-1793. doi: 10.1016/j.jbankfin.2010.12.004.

[16]

P. Cheridito and M. Kupper, Composition of time-consistent dynamic monetary risk measures in discrete time, Journal of Theoretical and Applied Finance, 14 (2011), 137-162. doi: 10.1142/S0219024911006292.

[17]

X. Y. Cui, D. Li, S. Y. Wang and S. S. Zhu, Better than dynamic mean-variance: Time inconsistency and free cash flow stream, Mathematical Finance, 22 (2012), 346-378. doi: 10.1111/j.1467-9965.2010.00461.x.

[18]

C. Czichowsky, Time-consistent mean-variance portfolio selection in discrete and continuous time, Finance Stochastic, 17 (2013), 227-271. doi: 10.1007/s00780-012-0189-9.

[19]

R. J. Elliott, T. K. Siu and L. Chan, On pricing barrier options with regime switching, Journal of Computational and Applied Mathematics, 256 (2014), 196-210. doi: 10.1016/j.cam.2013.07.034.

[20]

L. G. Epstein and S. E. Zin, Substitution, risk aversion, and the temporal behavior of consumption and asset returns: A theoretical framework, Econometrica, 57 (1989), 937-969. doi: 10.2307/1913778.

[21]

H. Geman and S. Ohana, Time-consistency in managing a commodity portfolio: A dynamic risk measure approach, Journal of Banking and Finance, 32 (2008), 1991-2005. doi: 10.1016/j.jbankfin.2007.05.020.

[22]

R. Korn and H. Kraft, A stochastic control approach to portfolio problems with stochastic interest rates, SIAM Journal on Control and Optimization, 40 (2002), 1250-1269. doi: 10.1137/S0363012900377791.

[23]

D. Li and W. L. Ng, Optimal dynamic portfolio selection: Multi-period mean-variance formulation, Mathematical Finance, 10 (2000), 387-406. doi: 10.1111/1467-9965.00100.

[24]

X. Li, X. Y. Zhou and A. E. B. Lim, Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40 (2002), 1540-1555. doi: 10.1137/S0363012900378504.

[25]

H. J. Lüthi and J. Doege, Convex risk measures for portfolio optimization and concepts of flexibility, Mathematical Programming, 104 (2005), 541-559. doi: 10.1007/s10107-005-0628-x.

[26]

F. Riedel, Dynamic coherent risk measures, Stochastic Processes and their Applications, 112 (2004), 185-200. doi: 10.1016/j.spa.2004.03.004.

[27]

B. Roorda and J. M. Schumacher, Time consistency conditions for acceptability measures, with an applications to Tail Value at Risk, Insurance: Mathematics and Economics, 40 (2007), 209-230. doi: 10.1016/j.insmatheco.2006.04.003.

[28]

A. Ruszczyński, Risk-averse dynamic programming for Markov decision processes, Mathematical Programming, Series B, 125 (2010), 235-261. doi: 10.1007/s10107-010-0393-3.

[29]

A. Shapiro, On a time consistency concept in risk averse multistage stochastic programming, Operations Research Letters, 37 (2009), 143-147. doi: 10.1016/j.orl.2009.02.005.

[30]

M. C. Steinbach, Markowitz revisited: Mean-variance models in financial portfolio analysis, SIAM Review, 43 (2001), 31-85. doi: 10.1137/S0036144500376650.

[31]

T. Wang, A class of dynamic risk measure, Working Paper, University of British Columbia, 1999. Available from: http://web.cenet.org.cn/upfile/57263.pdf.

[32]

J. Wang and P. A. Forsyth, Continuous time mean variance asset allocation: A time-consistent strategy, European Journal of Operational Research, 209 (2011), 184-201. doi: 10.1016/j.ejor.2010.09.038.

[33]

S. Z. Wei and Z. X. Ye, Multi-period optimization portfolio with bankruptcy control in stochastic market, Applied Mathematics and Computation, 186 (2007), 414-425. doi: 10.1016/j.amc.2006.07.108.

[34]

L. Xu, R. M. Wang and D. J. Yao, Optimal stochastic investment games under Markov regime switching market, Journal of Industrial and Management Optimization, 10 (2014), 795-815. doi: 10.3934/jimo.2014.10.795.

[35]

X. Y. Zhou and D. Li, Continuous-time mean-variance portfolio selection: A stochastic LQ framework, Applied Mathematics and Optimization, 42 (2000), 19-33. doi: 10.1007/s002450010003.

[36]

S. S. Zhu, D. Li and S. Y. Wang, Risk control over bankruptcy in dynamic portfolio selection: A generalized mean-variance formulation, IEEE Transactions on Automatic Control, 49 (2004), 447-457. doi: 10.1109/TAC.2004.824474.

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