# Elliptically Contoured Models in Statistics and Portfolio Theory

For that reason many authors assume equal means for the portfolio asset returns or, in other words, the global minimum variance portfolio GMVP. The GMVP has the lowest risk of any feasible portfolio.

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The subject of our paper are the weights of the GMVP portfolio. Results about the distribution of the estimated optimal weights are of great importance for evaluating the efficiency of the underlying portfolio Barberis ; Fleming et al. Jobson and Korkie studied the weights result- ing from the Sharpe ratio approach under the assumption that the returns are independent and normally distributed.

They derive approximations for the mean and the variance of the estimated weights, together with the asymptotic covariance matrix.

In Jobson and Korkie a test for the mean—variance efficiency is derived. A performance measure is introduced which is related to the Sharpe ratio. Britten-Jones analyzed tests for the efficiency of the mean—variance optimal weights under the normality assumption of the returns. Using a regression procedure the exact distribution of the normalized weights is derived.

Okhrin and Schmid proved several distributional properties for various optimal portfolio weights like, e. They considered the case of finite sample size and of infinite sample size as well. The aim of the present paper is to derive a test for the general linear hypoth- esis of the GMVP weights.

Multifactor Models

This hypothesis is treated in great detail within the theory of linear models e. It covers a large number of relevant and important testing problems. Our test statistic is derived in a similar way. Contrary to linear models its distribution under the alterna- tive hypothesis is not a non-central F-distribution. This shows that our results cannot be obtained in a straightforward way from the theory of linear mod- els. A great advantage of the approach suggested in this paper is that the assumptions on the distribution of the returns are very weak. In all of the above cited papers the stock returns are demanded to be independent and normally distributed.

The assumption of normality is found appropriate due to positive theoretical features, e. Fama found that monthly stock returns can be well described by a normal approach. However, in the case of daily returns the assumption of normality and independence might not be appropriate since it is very likely that the underlying distributions have heavy tails Osborne ; Fama , ; Markowitz ; Rachev and Mittnik For such a case the application of the multivariate t-distribution has been suggested by Zellner and Sutradhar Moreover, the assumption of independent returns turns out to be questionable, too.

Numerous studies demonstrated that frequently stock returns are uncorrelated but not independent Engle ; Bollerslev ; Nelson Here we assume that the matrix of returns fol- lows a matrix elliptically contoured distribution Fang and Zhang ; Fang and Zhang ; Gupta and Varga As shown in Bodnar and Schmid this family turns out to be very suitable to describe stock returns because the returns are neither assumed to be independent nor to be normally dis- tributed.

## Elliptically Contoured Models in Statistics and Portfolio Theory

Furthermore, it is in line with the results of Andersen et al. The family covers a wide class of distributions like, e. Elliptically contoured distributions have been already discussed in financial literature. While Chamberlain showed that elliptical distributions imply mean—variance utility functions, Berk argued that one of the necessary conditions for the capital asset pricing model CAPM is an elliptical distribution for the asset returns. Furthermore, Zhou extended findings of Gibbons et al. The first paper dealing with the application of matrix elliptically contoured distributions in finance, however, seems to be Bodnar and Schmid They introduced a test for the global minimum var- iance.

It is analyzed whether the lowest risk is larger than a given benchmark value or not.

## Multivariate elliptically contoured stable distributions: theory and estimation

In this paper a test for the weights of the GMVP is proposed. In Sect. It turns out that the distribution of the test sta- tistic is independent of the type of elliptical symmetry. The proposed testing procedure has financial and statistical interpretations even if the distribution of the returns is heavy-tailed. Furthermore, we find that the density of the estimator of the GMVP weights has a quite large tail index. Hence, its higher moments do exist even if the second moment does not exist for the distribution of the returns. Final remarks are presented in Sect.

The proofs of all results are given in the Sect. The k-dimensional vector of returns of these assets at time n is denoted by Xn. The weight of the ith asset in the portfolio is denoted by wi. Here 1 denotes the vector of ones. Suppose that for Xn the second moments ex- ist. Another approach consists in maximizing the Sharpe ratio of a portfolio without a risk free asset.

The Sharpe ratio is still one of the most popular measures for the evaluation of a portfolio and the asset performance Cochrane ; MacKinley and Pastor The portfolio with maximum Sharpe ratio can be equivalently presented as a global tangency portfolio in a classical quadratic optimization problem. This portfolio is known as GMVP. It will be subject of the following analysis. Given the sample X1 ,. The multivariate t-distribution belongs to the family of elliptically contoured distributions.

Here we consider linear combinations of the GMVP weights. The problem of testing the efficiency of a portfolio has been recently discussed in a large number of studies. Schmid efficiency of a portfolio. Jobson and Korkie and Gibbons et al. More re- cently, Britten-Jones has given the exact F-statistics for testing the efficiency of a portfolio with respect to portfolio weights which is based on a single linear regression. In this section we introduce a test of the general linear hypothesis for the GMVP weights. First, in Sect. M and r are assumed to be known.

This means that the investor is interested to know whether the weights of the GMVP fulfill p linear restrictions or not. This is a very general testing problem and it includes many important special cases cf. Greene , pp. Because the distribution of the underlying quantities is different than in the case of a linear model we cannot apply these well-known results. Now let Fi,j denote the F-distribution with degrees i and j. Its density is writ- ten as fi,j.

In the following we make also use of the hypergeometric function cf. Abramowitz and Stegun , chap. This fact simpli- fies the power study of the test. In Fig. Note that the T-statistic under H1 does not possess the non-central F-distribution which is obtained in the theory of linear models. The number of observations n is equal to and k is equal to 7. The figure illustrates the good performance of the test.

Moreover, it can be seen that its power decreases if p increases. Theorem 2 has many important applications. If, e.