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Continuous-time mean–variance portfolio selection with only risky assets

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  • Yao, Haixiang
  • Li, Zhongfei
  • Chen, Shumin
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    Abstract

    We investigate in this paper a continuous-time mean–variance portfolio selection problem in a general market setting with multiple assets that all can be risky. Using the Lagrange duality method and the dynamic programming approach, we derive explicit closed-form expressions for the efficient investment strategy and the mean–variance efficient frontier. We provided a necessary and sufficient condition under which the global minimum variance is zero and there exists a risk-free wealth process. Our results reveal that, even if there is no risk-free asset in the market, there can still exist a risk-free wealth process, the global minimum variance can be zero, and the efficient frontier can be a straight line in the mean–standard derivation plane. In addition, we further prove the validity of the two-fund separation theorem.

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    Bibliographic Info

    Article provided by Elsevier in its journal Economic Modelling.

    Volume (Year): 36 (2014)
    Issue (Month): C ()
    Pages: 244-251

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    Handle: RePEc:eee:ecmode:v:36:y:2014:i:c:p:244-251

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    Web page: http://www.elsevier.com/locate/inca/30411

    Related research

    Keywords: Continuous time mean–variance; Hamilton–Jacobi–Bellman equation; Portfolio selection; Dynamic programming; Two-fund separation theorem;

    References

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    1. John Y. Campbell & Luis M. Viceira, 1998. "Who Should Buy Long-Term Bonds?," NBER Working Papers 6801, National Bureau of Economic Research, Inc.
    2. Xie, Shuxiang, 2009. "Continuous-time mean-variance portfolio selection with liability and regime switching," Insurance: Mathematics and Economics, Elsevier, vol. 45(1), pages 148-155, August.
    3. Fu, Chenpeng & Lari-Lavassani, Ali & Li, Xun, 2010. "Dynamic mean-variance portfolio selection with borrowing constraint," European Journal of Operational Research, Elsevier, vol. 200(1), pages 312-319, January.
    4. Xie, Shuxiang & Li, Zhongfei & Wang, Shouyang, 2008. "Continuous-time portfolio selection with liability: Mean-variance model and stochastic LQ approach," Insurance: Mathematics and Economics, Elsevier, vol. 42(3), pages 943-953, June.
    5. Yao, Haixiang & Zeng, Yan & Chen, Shumin, 2013. "Multi-period mean–variance asset–liability management with uncontrolled cash flow and uncertain time-horizon," Economic Modelling, Elsevier, vol. 30(C), pages 492-500.
    6. Chen, Ping & Yang, Hailiang & Yin, George, 2008. "Markowitz's mean-variance asset-liability management with regime switching: A continuous-time model," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 456-465, December.
    7. Viceira, Luis M., 2012. "Bond risk, bond return volatility, and the term structure of interest rates," International Journal of Forecasting, Elsevier, vol. 28(1), pages 97-117.
    8. Jianming Xia & Jia-An Yan, 2006. "Markowitz'S Portfolio Optimization In An Incomplete Market," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 203-216.
    9. Elliott, Robert J. & Siu, Tak Kuen & Badescu, Alex, 2010. "On mean-variance portfolio selection under a hidden Markovian regime-switching model," Economic Modelling, Elsevier, vol. 27(3), pages 678-686, May.
    10. Chun Hung Chiu & Xun Yu Zhou, 2011. "The premium of dynamic trading," Quantitative Finance, Taylor & Francis Journals, vol. 11(1), pages 115-123.
    11. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, 09.
    12. Chiu, Mei Choi & Li, Duan, 2006. "Asset and liability management under a continuous-time mean-variance optimization framework," Insurance: Mathematics and Economics, Elsevier, vol. 39(3), pages 330-355, December.
    13. Chiu, Mei Choi & Wong, Hoi Ying, 2011. "Mean-variance portfolio selection of cointegrated assets," Journal of Economic Dynamics and Control, Elsevier, vol. 35(8), pages 1369-1385, August.
    14. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, 03.
    15. Duan Li & Wan-Lung Ng, 2000. "Optimal Dynamic Portfolio Selection: Multiperiod Mean-Variance Formulation," Mathematical Finance, Wiley Blackwell, vol. 10(3), pages 387-406.
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