×

zbMATH — the first resource for mathematics

Time-consistent investment policies in Markovian markets: a case of mean-variance analysis. (English) Zbl 1402.91672
Summary: The optimal investment policy for a standard multi-period mean-variance model is not time-consistent because the variance operator is not separable in the sense of the dynamic programming principle. With a nested conditional expectation mapping, we develop an investment model with time consistency in Markovian markets. Furthermore, we examine the differences of the investment policies with a riskless asset from those without a riskless asset. Analytical solutions for time-consistent optimal investment policies and the resulting mean-variance efficient frontier are obtained. Finally, using numerical examples, we show that the optimal investment policy derived from our model is more efficient than that of the standard mean-variance model in which the trade-off is determined between the mean and variance of the terminal wealth.

MSC:
91G10 Portfolio theory
62P05 Applications of statistics to actuarial sciences and financial mathematics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Artzner, P.; Delbaen, F.; Eber, J. M.; Heath, D.; Ku, H., Coherent multi-period risk adjusted values and Bellman’s principle, Ann. Oper. Res., 152, 5-22, (2007) · Zbl 1132.91484
[2] Bielecki, T. R.; Jin, H. Q.; Pliska, S. R.; Zhou, X. Y., Continuous-time mean-variance portfolio selection with bankruptcy prohibition, Math. Financ., 15, 213-244, (2005) · Zbl 1153.91466
[3] Boda, K.; Filar, J. A., Time consistent dynamic risk measures, Math. Methods Oper. Res., 63, 169-186, (2006) · Zbl 1136.91471
[4] Çakmak, U.; Özekici, S., Portfolio optimization in stochastic markets, Math. Methods Oper. Res., 63, 151-168, (2006) · Zbl 1136.91409
[5] Çanakogˇlu, E.; Özekici, S., Portfolio selection in stochastic markets with exponential utility functions, Ann. Oper. Res., 166, 281-297, (2009) · Zbl 1163.91374
[6] Çanakogˇlu, E.; Özekici, S., HARA frontiers of optimal portfolios in stochastic markets, Eur. J. Oper. Res., 221, 129-137, (2012) · Zbl 1253.91164
[7] Çelikyurt, U.; Özekici, S., Multiperiod portfolio optimization models in stochastic markets using the mean-variance approach, Eur. J. Oper. Res., 179, 186-202, (2007) · Zbl 1163.91375
[8] Chen, Z. P.; Li, G.; Guo, J. E., Optimal investment policy in the time consistent mean-variance formulation, Insur.: Math. Econ., 52, 145-156, (2013) · Zbl 1284.91514
[9] Chen, Z. P.; Song, Z. X., Dynamic portfolio optimization under multi-factor model in stochastic markets, OR Spectr., 34, 885-919, (2012) · Zbl 1282.91297
[10] Cheridito, P.; Delbaen, F.; Kupper, M., Dynamic monetary risk measures for bounded discrete-time processes, Electron. J. Probab., 11, 57-106, (2006) · Zbl 1184.91109
[11] Costa, O.; Araujo, M., A generalized multi-period mean-variance portfolio optimization with Markov switching parameters, Automatica, 44, 2487-2497, (2008) · Zbl 1157.91356
[12] Cui, X. Y.; Li, D.; Wang, S. Y.; Zhu, S. S., Better than dynamic mean-variancetime inconsistency and free cash flow stream, Math. Financ., 22, 346-378, (2010) · Zbl 1278.91131
[13] Detlefsen, K.; Scandolo, G., Conditional and dynamic convex risk measures, Financ. Stoch., 9, 4, 539-561, (2005) · Zbl 1092.91017
[14] Dumas, B.; Luciano, E., An exact solution to a dynamic portfolio choice problem under transactions costs, J. Financ., 46, 577-596, (1991)
[15] Epstein, L. G.; Zin, S. E., Substitution, risk aversion, and the temporal behavior of consumption and asset returnsa theoretical framework, Econometrica, 57, 4, 937-969, (1989) · Zbl 0683.90012
[16] Föllmer, H.; Penner, I., Convex risk measures and the dynamics of their penalty functions, Stat. Dec., 24, 1, 61-96, (2006) · Zbl 1186.91119
[17] Guidolin, M.; Timmermann, A., International asset allocation under regime switching, skew and kurtosis preferences, Rev. Financ. Stud., 21, 889-935, (2008)
[18] Hamilton, J., A new approach to the economic analysis of nonstationary time series and the business cycle, Econometrica, 57, 357-384, (1989) · Zbl 0685.62092
[19] Honda, T., Optimal portfolio choice for unobservable and regime-switching mean returns, J. Econ. Dyn. Control, 28, 45-78, (2003) · Zbl 1179.91233
[20] Jobert, L.; Rogers, L. C.G., Valuations and dynamic convex risk measures, Math. Financ., 18, 1-22, (2008) · Zbl 1138.91501
[21] Koopmans, T. C., Stationary ordinal utility and impatience, Econometrica, 28, 2, 287-309, (1960) · Zbl 0149.38401
[22] Kovacevic, R.; Pflug, G. Ch., Time consistency and information monotonicity of multiperiod acceptability functionals, Radon Ser. Comput. Appl. Math., 8, 347-370, (2009) · Zbl 1185.91094
[23] Kreps, M. K.; Porteus, E. L., Temporal resolution of uncertainty and dynamic choice theory, Econometrica, 46, 1, 185-200, (1978) · Zbl 0382.90006
[24] Li, D.; Ng, W. L., Optimal dynamic portfolio selectionmulti-period mean-variance formulation, Math. Financ., 10, 387-406, (2000) · Zbl 0997.91027
[25] Li, X.; Zhou, X. Y.; Lim, A. E.B., Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM J. Control Optim., 40, 1540-1555, (2001) · Zbl 1027.91040
[26] Liu, P.; Xu, K.; Zhao, Y., Market regimes, sectorial investments, and time-varying risk premiums, Int. J. Manag. Financ., 7, 107-133, (2011)
[27] Ma, Y.; MacLean, L.; Xu, K.; Zhao, Y., A portfolio optimization model with regime-switching risk factors for sector exchange traded funds, Pac. J. Optim., 7, 2, 281-296, (2011) · Zbl 1226.91068
[28] Markowitz, H., Portfolio selection, J. Financ., 7, 77-91, (1952)
[29] Merton, R. C., Lifetime portfolio selection under uncertaintythe continuous time case, Rev. Econ. Stat., 51, 247-257, (1969)
[30] Riedel, F., Dynamic coherent risk measures, Stoch. Process. Appl., 112, 185-200, (2004) · Zbl 1114.91055
[31] Roorda, B.; Schumacher, J. M.; Engwerda, J. C., Coherent acceptability measures in multiperiod models, Math. Financ., 15, 589-612, (2005) · Zbl 1107.91059
[32] Roorda, B.; Schumacher, J. M., Time consistency conditions for acceptability measures, with an applications to tail value at risk, Insur.: Math. Econ., 40, 2, 209-230, (2007) · Zbl 1141.91547
[33] Ruszczyński, A., Risk-averse dynamic programming for Markov decision processes, Math. Program., Ser. B, 125, 235-261, (2010) · Zbl 1207.49032
[34] Sheldon, R., A first course in probability, (2010), Pearson Education, Inc. Pearson Prentice Hall · Zbl 1307.60002
[35] Sotomayor, L.; Cadenillas, A., Explicit solutions of consumption-investment problems in financial markets with regime switching, Math. Financ., 19, 215-236, (2009)
[36] Tobin, J., 1965. The theory of portfolio selection. In: The Theory of Interest Rates, Macmillan, London.
[37] Wang, J.; Forsyth, P. A., Continuous time mean variance asset allocationa time-consistent strategy, Eur. J. Oper. Res., 209, 184-201, (2011) · Zbl 1208.91139
[38] Weber, S., Distribution-invariant dynamic risk measures, information and dynamic consistency, Math. Financ., 16, 419-441, (2006) · Zbl 1145.91037
[39] Wei, S. Z.; Ye, Z. X., Multi-period optimization portfolio with bankruptcy control in stochastic market, Appl. Math. Comput., 186, 414-425, (2007) · Zbl 1185.91168
[40] Yin, G.; Zhou, X. Y., Markowitz’s mean-variance portfolio selection with regime switchingfrom discrete-time models to their continuous-time limits, IEEE Trans. Autom. Control, 49, 3, 349-360, (2003) · Zbl 1366.91148
[41] Zhou, X. Y.; Yin, G., Markowitz’s mean-variance portfolio selection with regime switchinga continuous-time model, SIAM J. Control Optim., 42, 4, 1466-1482, (2003) · Zbl 1175.91169
[42] Zhu, S. S.; Li, D.; Wang, S. Y., Risk control over bankruptcy in dynamic portfolio selectiona generalized mean-variance formulation, IEEE Trans. Autom. Control, 49, 447-457, (2004) · Zbl 1366.91150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.