Portfolio Theory for 𝛼-Symmetric and Pseudoisotropic Distributions: 𝑘-Fund Separation and the CAPM
Nils Christian Framstad
Journal of Probability and Statistics Volume 2015 (2015)
The shifted pseudoisotropic multivariate distributions are shown to satisfy Ross’ stochastic dominance criterion for two-fund monetary separation in the case with risk-free investment opportunity and furthermore to admit the Capital Asset Pricing Model under an embedding in L𝛼 condition if 1<𝛼≤2, with the betas given in an explicit form. For the 𝛼-symmetric subclass, the market without risk-free investment opportunity admits 2𝑑-fund separation if 𝛼 = 1 + 1/(2𝑑 − 1), 𝑑 ∈ N, generalizing the classical elliptical case 𝑑=1, and we also give the precise number of funds needed, from which it follows that we cannot, except degenerate cases, have a CAPM without risk-free opportunity. For the symmetric stable subclass, the index of stability is only of secondary interest, and several common restrictions in terms of that index can be weakened by replacing it by the (no smaller) indices of symmetry/of embedding. Finally, dynamic models with intermediate consumption inherit the separation properties of the static models.