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Multi-period mean–variance portfolio optimization with management fees

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Abstract

Due to limited capital and limited information from stock market, some individual investors prefer to construct a portfolio of funds instead of stocks. But, there will be management fees paid to the fund managers during the investment, which are in general proportional to the net asset value of the funds. Motivated by this phenomena, this paper considers multi-period mean–variance portfolio optimization problem with proportional management fees. Using stochastic dynamic programming, we derive the semi-analytical optimal portfolio policy. Our result helps clarify the benefit and cost of adopting such dynamic portfolio policy with management fees.

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Notes

  1. The main results in this paper can be readily extended to the case of correlated return vectors by adopting the technique proposed in Gao et al. (2015).

  2. Based on the proofs of Proposition 1 and Theorem 2, the key requirement of applying our technique is that the admissible set of the control variables is a cone. \(\{(u_t^i, \nu _t^i) | u_t^i\ge 0, \nu _t^i\ge 0, u_t^i \nu _t^i =0\}\) is still a cone. Thus, our technique is also applicable to the setting that the investor can only take either long position or short position on a fund.

  3. As the no bankruptcy constraint is a state constraint, our technique is not applicable to such constraint.

  4. The data of 48 industry portfolios can be found in http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html.

  5. To achieve a stable estimation, the shrinkage estimation method in Ledoit and Wolf (2003) is used in estimating the covariance matrix.

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Funding

This research was partially supported by National Natural Science Foundation of China under Grants 71671106, 71601107, by the State Key Program in the Major Research Plan of National Natural Science Foundation of China under Grant 91546202.

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Correspondence to Yun Shi.

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Cui, X., Gao, J. & Shi, Y. Multi-period mean–variance portfolio optimization with management fees. Oper Res Int J 21, 1333–1354 (2021). https://doi.org/10.1007/s12351-019-00482-4

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