## Optimal Dynamic Portfolio Selection: Multi-Period Mean-Variance Formulation (1998)

Venue: | Math. Finance |

Citations: | 31 - 1 self |

### BibTeX

@ARTICLE{Li98optimaldynamic,

author = {Duan Li and Wan-lung Ng},

title = {Optimal Dynamic Portfolio Selection: Multi-Period Mean-Variance Formulation},

journal = {Math. Finance},

year = {1998},

volume = {10},

pages = {387--406}

}

### OpenURL

### Abstract

The mean-variance formulation by Markowitz in 1950s and its analytical solution by Merton in 1972 paved a foundation for modern portfolio selection analysis in single period. This paper considers an analytical optimal solution to the mean-variance formulation in multiperiod portfolio selection. Specifically, analytical optimal portfolio policy and analytical expression of the mean-variance efficient frontier are derived in this paper for the multi-period mean-variance formulation. An efficient algorithm is also proposed in this paper in finding an optimal portfolio policy to maximize a utility function of the expected value and the variance of the terminal wealth. Key Words: Multi-period portfolio selection, multi-period mean-variance formulation, utility function. This research was partially supported by the Research Grants Council of Hong Kong, grant no. CUHK 4130/97E. The authors very much appreciate the constructive comments from Professor Stanley R. Pliska. y Author to whom a...

### Citations

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Citation Context ...es should be addressed. Email: dli@se.cuhk.edu.hk. 1 1 Introduction Portfolio selection is to seek a best allocation of wealth among a basket of securities. The mean-variance formulation by Markowitz =-=[13]-=- [14] provides a fundamental basis for portfolio selection in single period. An analytical expression of the mean-variance efficient frontier in single-period portfolio selection was derived by Merton... |

631 |
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Citation Context ...portfolio selection was derived by Merton [16] in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton =-=[15]-=- [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. E... |

329 |
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Citation Context ...olio selection was derived by Merton [16] in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] =-=[17]-=-, Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormo... |

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Citation Context ...as derived by Merton [16] in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson =-=[23]-=-, Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty wa... |

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Citation Context ...ingle-period portfolio selection was derived by Merton [16] in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin =-=[18]-=-, Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and P... |

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Citation Context ...vel, utility function U (E (x T ) ; V ar (x T )) is assumed to satisfy the following, @U (E (x T ) ; V ar (x T )) @E (x T ) ? 0 (77) and @U (E (x T ) ; V ar (x T )) @V ar (x T ) ! 0 (78) Knoll et al. =-=[11]-=- found that solutions via maximizing U (E (x T ) ; V ar (x T )) and via direct utility maximization are highly correlated to each other. The following multi-period portfolio selection problem is formu... |

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Citation Context ...such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano =-=[2]-=-, Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal solutions for a multi-period mean-variance formulation. The literatur... |

44 | An Analytic Derivation of the Efficient Portfolio Frontier
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Citation Context ...[14] provides a fundamental basis for portfolio selection in single period. An analytical expression of the mean-variance efficient frontier in single-period portfolio selection was derived by Merton =-=[16]-=- in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson... |

43 |
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Citation Context ...by Merton [16] in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama =-=[6]-=-, Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported... |

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(Show Context)
Citation Context ...on [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska =-=[21]-=-. Enormous difficulty was reported by Chen et al. [1] in finding optimal solutions for a multi-period mean-variance formulation. The literature in multi-period portfolio selection has been dominated b... |

36 |
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Citation Context ...ould be addressed. Email: dli@se.cuhk.edu.hk. 1 1 Introduction Portfolio selection is to seek a best allocation of wealth among a basket of securities. The mean-variance formulation by Markowitz [13] =-=[14]-=- provides a fundamental basis for portfolio selection in single period. An analytical expression of the mean-variance efficient frontier in single-period portfolio selection was derived by Merton [16]... |

22 |
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(Show Context)
Citation Context ...972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] =-=[10]-=-, Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] ... |

21 |
On Optimal Myopic Portfolio Policies with and without Serial Correlation of Yields
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(Show Context)
Citation Context ...in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson =-=[9]-=- [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al.... |

21 |
The Art of Investing
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Citation Context ...monstrate the adoption of the multi-period mean-variance formulations and the efficiency of the solution methods derived in this paper. Example 1 Consider the case study in Chapter 7 of Sharpe et al. =-=[24]-=- by assuming a stationary multi-period process with T = 4. An investor has one unit wealth in the very beginning of the planning horizon. The investor is trying to find the best allocation of his weal... |

13 |
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Citation Context ...ssin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson =-=[8]-=- and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal solutions for a multi-period mean-variance formulation. The literature in multi-period portfolio selection has ... |

7 |
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Citation Context ...rontier in single-period portfolio selection was derived by Merton [16] in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. =-=[1]-=-, Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakans... |

7 |
Investment Analysis and Management
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(Show Context)
Citation Context ...ied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis =-=[7]-=-, Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal solutions for a multi-period mean-variance for... |

5 |
A Transition Model for Portfolio Revision
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(Show Context)
Citation Context ...riance efficient frontier in single-period portfolio selection was derived by Merton [16] in 1972. The problem of multi-period portfolio selection has been studied in the literature, such as in Smith =-=[25]-=-, Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], ... |

3 |
On the optimality of some multiperiod portfolio selection criteria
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(Show Context)
Citation Context ...ti-period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber =-=[3]-=-[4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal sol... |

3 |
The multi-period consumption investment problem and single period analysis
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(Show Context)
Citation Context ...period portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3]=-=[4]-=-[5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal soluti... |

3 |
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Citation Context ...h [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark =-=[20]-=-, Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal solutions for a multi-period mean-variance formulation. The literature in multi-perio... |

2 |
Safety-first dynamic portfolio selection
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(Show Context)
Citation Context ... 2wE (x T ) js)] is proportional to [\Gammaw;s]. We must haves= 1 + 2wE (x T ) js. Q.E.D. The optimal solution of the auxiliary problem (A (; w)) can be derived analytically using dynamic programming =-=[12]-=-. The optimal portfolio policy for auxiliary problem (A (; w)) at each time period t is of the following form, u t (x t ; fl) = \GammaK t x t + v t (fl) t = 0; 1; : : : ; T \Gamma 1 (40) where fl = w ... |

2 |
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(Show Context)
Citation Context ... has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4][5], Winkler and Barry =-=[26]-=-, Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal solutions for a multi-period mean... |

1 |
Finance as a Dynamic Process. Englewood Cliffs
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(Show Context)
Citation Context ...iod portfolio selection has been studied in the literature, such as in Smith [25], Chen et al. [1], Mossin [18], Merton [15] [17], Samuelson [23], Fama [6], Hakansson [9] [10], Elton and Gruber [3][4]=-=[5]-=-, Winkler and Barry [26], Francis [7], Dumas and Luciano [2], Ostermark [20], Grauer and Hakansson [8] and Pliska [21]. Enormous difficulty was reported by Chen et al. [1] in finding optimal solutions... |

1 |
Dynamic portfolio analysis: Mean-variance formulation and iterative parametric dynamic programming," Thesis of
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(Show Context)
Citation Context ...ps all his money in the riskless asset. When setting T = 1 in our formulation, problems (P 1(oe)) and (P 2(ffl)) are reduced to the single-period mean-variance formulation [13]. It can be 15 verified =-=[19]-=- that the expressions of the efficient frontier are exactly the same as given in this paper and in Eq. (35) of Merton [16] when setting T = 1. The work of multi-period mean-variance approach presented... |

1 |
Citron (1971):"On noninferior performance index vector
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(Show Context)
Citation Context ...rst-order necessary optimality condition for optimalsis \Gammaw @E (x 2 T ( ; w)) @ + [1 + 2wE (x T ) js] @E (x T ( ; w)) @ = 0 (38) On the other side, whens2 \Pi A ( ; w), we have the following from =-=[22]-=-, \Gammaw @E (x 2 T ( ; w)) @ +s@E (x T ( ; w)) @ = 0 (39) Hence, the vector [\Gammaw; (1 + 2wE (x T ) js)] is proportional to [\Gammaw;s]. We must haves= 1 + 2wE (x T ) js. Q.E.D. The optimal solutio... |