Mean–variance–skewness efficient surfaces, Stein’s lemma and the multivariate extended skew-Student distribution

• Published in: European journal of operational research Vol. 234; no. 2; pp. 392 - 401
• Main Author: Adcock, C.J.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 16.04.2014; Elsevier Sequoia S.A
• Subjects: Computation; Finance; Investment policy; Mathematical analysis; Mathematical models; Multivariate analysis; Multivariate statistics; Operational research; Portfolio investments; Probability distribution; Quadratic programming; Random variables; Studies; Utility theory; Vectors (mathematics); Multivariate statistics; Utility theory; Finance
• ISSN: 0377-2217; 1872-6860
• DOI: 10.1016/j.ejor.2013.07.011

•This paper extends Stein’s lemma for the multivariate extended skew-t distribution.•Efficient portfolios are located on a mean–variance–skewness efficient surface.•This surface is a direct extension of Markowitz’ efficient frontier.•The multivariate models introduced by Simaan admit the same properties.•There are also mean–variance–skewness efficient hyper-surfaces. Recent advances in Stein’s lemma imply that under elliptically symmetric distributions all rational investors will select a portfolio which lies on Markowitz’ mean–variance efficient frontier. This paper describes extensions to Stein’s lemma for the case when a random vector has the multivariate extended skew-Student distribution. Under this distribution, rational investors will select a portfolio which lies on a single mean–variance–skewness efficient hyper-surface. The same hyper-surface arises under a broad class of models in which returns are defined by the convolution of a multivariate elliptically symmetric distribution and a multivariate distribution of non-negative random variables. Efficient portfolios on the efficient surface may be computed using quadratic programming.