Schur Convex Functions and its Applications

Abstract

This master's thesis focuses on a brief review of the Schur convex function's notions on majorization between vectors and doubly stochastic matrix. We provide an application of majorization in risk management via portfolio diversification. Firstly, we review the classification of a wide class of stable distributions which are defined using the characteristic function, into four classes based on the stability parameter $\alpha$. The main results in this thesis are derived from the original work of Ibragimov who classified and analyzed Value at Risk in classes of heavy-tailed risk distribution. In this thesis, the two main results, which are consistent with the empirical findings on the Value at Risk measure are presented based on majorization theory. The first main result affirms the coherence of Value at Risk as a risk measure in log-concave risk distributions (which are light-tailed) and moderately heavy-tailed symmetric stable risk distributions. The second main result shows that for extremely heavy-tailed symmetric stable risk distribution, Value at Risk is not coherence risk and that diversification of the portfolio only increases it.