Econometrical analysis of the sample efficient frontier

Abstract

The efficient frontier is a parabola in the mean-variance space which is uniquely determined by three characteristics. Assuming that the portfolio asset returns are independent and multivariate normally distributed, we derive tests and confidence sets for all possible arrangements of these characteristics. Note that all of our results are based on the exact distributions for a finite sample size. Moreover, we determine a confidence region of the whole efficient frontier in the mean-variance space. It is shown that this set is bordered by five parabolas.asset allocation, efficient frontier, portfolio analysis, mean-variance portfolio, parameter uncertainty, interval estimation,

Construction and Inferences of the Efficient Frontier in Elliptical Models

“…Given r = r 0 it holds that the x i 's are independently distributed with x i |r = r 0 ∼ N (µ, r 2 0 Σ). Application of Lemma 1 of Bodnar and Schmid (2009) leads to the statement of the theorem. The theorem is proved.…”

“…Similar test statistic was considered by Bodnar and Schmid (2009) in the normal case. The distribution of T is derived in Theorem 2 under the null and alternative hypotheses.…”

“…The proof of Theorem 2 follows from Proposition 3 of Bodnar and Schmid (2009) by considering first the conditional distribution of X given r = r 0 and then integrating over r 0 . The theorem is proved.…”

“…Using the proof of Theorem 1 of Bodnar and Schmid (2009), we get the expression of B. (3.20) where Basak et al (2005) and Siegel and Woodgate (2007) showed that the sample efficient frontier is overly optimistic and overestimates the true location of the efficient frontier in the mean-variance space.…”

“…The first two define the location of the vertex, while the third one is the slope parameter. Assuming the asset returns to be independently and normally distributed, Jobson (1991) obtained an asymptotic confidence region of the efficient frontier, while Bodnar and Schmid (2009) derived an exact joint confidence set of the three parameters of the efficient frontier. This set is used to determine a region in the mean-variance space where all optimal portfolios lie with a given probability (1 − α).…”