# CHAPTER 9

# Statistical Properties and Tests of Efficient Frontier Portfolios

**C J Adcock**

Sheffield University Management School, Sheffield, UK

## INTRODUCTION

The standard theory of portfolio selection due to Markowitz (1952) makes an implicit assumption that asset returns follow a multivariate normal distribution. The original concept of minimizing portfolio variance subject to achieving a target expected return is equivalent to assuming that investors maximize the expected value of a utility function that is quadratic in portfolio returns. Stein's lemma (1981) means that Markowitz's efficient frontier arises under normally distributed returns for any suitable utility function, subject only to relatively undemanding regularity conditions. Recent developments of the lemma (e.g., Liu, 1994; Landsman and Nešlehová, 2008) show that the efficient frontier arises under multivariate probability distributions that are members of the elliptically symmetric class (Fang, Kotz and Ng, 1990). There is a direct extension to a mean-variance-skewness efficient surface for some classes of multivariate distributions that incorporate asymmetry (Adcock, 2014a).

The literature that addresses the theory and application of Markowitzian portfolio selection is very large. Much of this published work assumes that the parameters of the underlying distributions, most commonly the vector of expected returns and the covariance matrix, are known. In reality, they are not known and all practical portfolio selection ...