# Nils Framstad: Generalizations of elliptical distributions and their portfolio separation properties

Nils Framstad, UiO, holder et seminar med tittelen: Generalizations of elliptical distributions and their portfolio separation properties

The elliptical distributions, originating with Schoenberg ('Metric spaces and completely monotone functions', Ann. of Math 1938), can with minor modifications replace the Gaussian in linear regression and in models like CAPM and the mutual fund theorem. The latter, also known as the portfolio separation property, states conditions under which a financial market can be replaced by a few (two in the prototypical model) indices ('funds') without welfare loss to the investors. Such conditions can be formulated in terms of the investors' preferences (the Cass--Stiglitz type theorems) or in terms of the returns distributions (Ross-type theorem).

In the talk I will show how the (Ross) portfolio separation property fares under various generalizations of ellipticity in a single period model. Distribution classes to be discussed, are (i) generalizations of the skew-elliptical (in the Azzalini sense), composed of conditioning some coordinates of an intracorrelated elliptical vector; (ii) the pseudo-isotropic distributions, where the ellipticity is formulated in a quasi-norm, rather than the usual weighted Euclidean norm; (iii) the class where the defining property of the pseudo-isotropic class is weakened to hold only for convex combinations; (iv) the alpha-stable class.

A trick employed by Khanna and Kulldorff ('A generalization of the mutual fund theorem', Finance Stoch. 1999) will be used to extend the portfolio separation properties of the single-period case to a the corresponding geometric Lévy process model if the distribution is infinitely divisible. The construction itself can be simplified to a level where there is hardly any stochastic analysis left, but both infinite divisibility itself and applicability of the models, will raise unsolved problems. The talk will be based on my 'Portfolio separation properties of the skew-elliptical distributions, with generalizations' (Statistics and Probability Letters, 2011, http://dx.doi.org/10.1016/j.spl.2011.07.006 ), my preprint 'Portfolio Separation with α-symmetric and Psuedo-isotropic Distributions' (http://www.sv.uio.no/econ/english/research/memorandum/pdf-files/2011/Memo-12-2011.pdf), and work in (and out of) progress.