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Survey on Multi-period Mean–Variance Portfolio Selection Model

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Abstract

Due to the non-separability of the variance term, the dynamic mean–variance (MV) portfolio optimization problem is inherently difficult to solve by dynamic programming. Li and Ng (Math Finance 10(3):387–406, 2000) and Zhou and Li (Appl Math Optim 42(1):19–33, 2000) develop the pre-committed optimal policy for such a problem using the embedding method. Following this line of research, researchers have extensively studied the MV portfolio selection model through the inclusion of more practical investment constraints, realistic market assumptions and various financial applications. As the principle of optimality no longer holds, the pre-committed policy suffers from the time-inconsistent issue, i.e., the optimal policy computed at the intermediate time t is not consistent with the optimal policy calculated at any time before time t. The time inconsistency of the dynamic MV model has become an important yet challenging research topic. This paper mainly focuses on the multi-period mean–variance (MMV) portfolio optimization problem, reviews the essential extensions and highlights the critical development of time-consistent policies.

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Notes

  1. The filtration \({\mathcal {F}}_t\) represents the information available at time t (see [10] for detail of discrete-time market model).

  2. If the first asset is the benchmark, the wealth dynamic becomes \(x_{t+1} = r^1_t x_t + {\hat{\varvec{r}}}_t^{\top } {\hat{\varvec{u}}}_t\), where \({\hat{\varvec{r}}}_t=(r^2_t - r^1_t, r^3_t - r^1_t,\cdots , r^n_t- r^1_t)^{\top }\) is the excess return vector and \({\hat{\varvec{u}}}_t \in {\mathbb {R}}^{n-1}\) is the truncated decision vector of \(\varvec{u}_t\).

  3. Similar model is also considered in [11].

  4. Reader may refer [5, 12] for the detailed discussion of three models.

  5. If the risk-free rate is random, then it becomes the similar model with all assets being risky.

  6. In this paper, we define the boundary of the product term as \(\prod _{k=t_1}^{t_2} a_k =1\) if \(t_1>t_2\).

  7. The detail derivation of policy (5) is given in [13]. Note that the policy (5) is different from the one given in [5], as the problem formulations are different.

  8. We re-calibrate the date given in Example 1 in [5] from yearly based statistics to the monthly based ones. Then, \(T=12\) means that the investment horizon is 12 months.

  9. We adopt the similar data for Fig. 1(a) and set \(T=12\) months. As for the static MV model, we regard the whole horizon \(T=12\) as a single period and solve the static MV portfolio selection for the buy and hold policy. We then simulate 5000 sample paths of the return vector and compute the terminal wealth for different policies.

  10. Comparing with [9], one important extension given in [22] is that it drops the nonnegative assumption \(\text {E}[\varvec{p}_t\varvec{p}_t^{\top }]^{-1}\text {E}[\varvec{p}_t]\geqslant 0\) for \(t=0,\cdots ,T-1\). The new setting fits the fact that the expected value of premium vectors may be negative in the time-varying market.

  11. Take the practice of AIS as an example. For investors with assets under management (AUM) between US$100,000 and US$250,000, the annual fee charged by AIS is 0.80% of AUM or US$1,500, whichever is greater.

  12. Note that \(T_s\) is related to the management fee, which can be fixed by the method given in Sect. 3 in [12].

  13. The time-consistent mean–variance policies in the market with all risky assets can be found in [56].

  14. A more detailed comparison between the pre-committed policy and different types of time-consistent policies can be found in [60].

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Acknowledgements

All the authors of this paper would like to thank Professor Duan Li for his guidance, help and encouragement during their Ph.D. and postdoctoral time at the Chinese University of Hong Kong.

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Authors and Affiliations

  1. School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, 200433, China

    **ang-Yu Cui

  2. School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai, 200433, China

    Jian-Jun Gao

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hong Kong, China

    Xun Li

  4. Academy of Statistics and Interdisciplinary Sciences, Faculty of Economics and Management, East China Normal University, Shanghai, 200062, China

    Yun Shi

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  1. **ang-Yu Cui
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Correspondence to Jian-Jun Gao.

Additional information

This paper is dedicated to the late Professor Duan Li in commemoration of his contributions to optimization, financial engineering, and risk management.

This work is partially supported by the National Natural Science Foundation of China (Nos. 71971132, 61573244, 71671106, 71971083 and 72171138), by the Key Program of National Natural Science Foundation of China (No. 71931004), by Shanghai Institute of International Finance and Economics, by Program for Innovative Research Team of Shanghai University of Finance and Economics and by the Open Research Fund of Key Laboratory of Advanced Theory and Application in Statistics and Data Science-MOE.

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Cui, XY., Gao, JJ., Li, X. et al. Survey on Multi-period Mean–Variance Portfolio Selection Model. J. Oper. Res. Soc. China 10, 599–622 (2022). https://doi.org/10.1007/s40305-022-00397-6

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