Stochastic Dominance, Almost Stochastic Dominance, Moment, and Test Statistics

We first plan to extend Wong (2007) and others to study the expected-utility maximization, hierarchy, and diversification properties of stochastic dominance (SD) for risk-averse and risk-loving investors to any order. We will also study the conditions such that the preferences of risk-averse and risk-loving investors could be the same or reversed. We will construct examples to demonstrate the usefulness of our approach.

We then plan to extend Egozcue and Wong (2010) and others to investigate the diversification preferences of risk-averse and risk-loving investors and other types of investors including investors with S-shaped and reversed S-shaped utility functions for the independent and identical distributed variables. In particular, we plan to examine whether these investors would prefer to invest in an individual asset, any convex combination portfolio, or the equally weighted portfolio. Thereafter, we will extend the theory to include the dependent cases.

Thereafter, we plan to extend Levy and Levy (2002) and others to first develop a theory of almost SD (ASD) for risk-loving investors. We will study the expected-utility maximization, hierarchy, and diversification relationships in ASD for risk-lovers and risk averters to any order. We will also study the conditions such that different orders of almost SD for risk-averse and risk-loving investors could be the same or reversed. In addition, we will study the relationship between any order of ASD for risk averters and risk lovers with the corresponding moment.

We will then extend Bai, Li, Liu, and Wong (2011) and others to derive the limiting process of n-order SD and ASD statistics for risk averters and risk lovers. We will take into consideration that the underlying processes are dependent as well as independent. We will also develop a bootstrap method to compute the critical values for the tests. In addition, we will include testing other conditions in the SD and ASD testing procedures to reveal the true SD and ASD relationships among the prospects being compared. For example, we will test whether the means of the assets are equal for the third order SD. We will also conduct simulation to compare the performance of the SD and ASD tests we plan to develop.

At last, we readdress the issue analyzed by Levy (2009) to study the preferences of risk- averse and risk-loving investors on stocks and bonds by using the new SD and ASD theory and tests we plan to develop in this project.