Classical mean-variance model revisited: pseudo efficiency

Abstract

Investigating the inverse problem of the classical Markowitz mean-variance formulation: Given a mean-variance pair, find initial investment levels and their corresponding portfolio policies such that the given mean-variance pair can be realized, we reveal that any mean-variance pair inside the reachable region can be achieved by multiple portfolio policies associated with different initial investment levels. Therefore, in the mean-variance world for a market of all risky assets, the common belief of monotonicity: ‘The larger you invest, the larger expected future wealth you can expect for a given risk (variance) level’ does not hold, which stimulates us to extend the classical two-objective mean-variance framework to an expanded three-objective framework: to maximize the mean and minimize the variance of the final wealth as well as to minimize the initial investment level. As a result, we eliminate from the policy candidate list the set of pseudo efficient policies that are efficient in the original mean-variance space, but inefficient in this newly introduced three-dimensional objective space.

Keywords:

• mean-variance portfolio selection
• minimum cost policy
• binding budget spending
• optimal wealth management

Acknowledgements

This paper is dedicated in memory of Professor Douglas J. White. The second author is in debt to Professor Douglas White for his invaluable advices during the second author’s career development. The research was partially supported by Research Grants Council, Hong Kong (CUHK414808 and CUHK414513), National Natural Science Foundation of China (No. 71201094 and No. 11371350), Shanghai Pujiang Program 12PJC051, and Key Laboratory of Random Complex Structures and Data Science, CAS (No. 2008DP173182). The authors are thankful to Professor Harry M. Markowitz for pointing out to the second author the literature on long-short portfolio literature. The second author also appreciates the support from the Patrick Huen Wing Ming Chair Professorship of Systems Engineering and Engineering Management.