Abstract: Using expectations regarding utilities to make decisions in a risk environment hides a paradox, which is called the expected utility enigma. Moreover, the mystery has not been solved yet; an imagined utility function on the risk-return plane has been applied to establish the mean-variance model, but this hypothetical utility function not only lacks foundation, it also holds an internal contradiction. This paper studies these basic problems. Through risk preference VNM condition is proposed to solve the expected utility enigma. How can a utility function satisfy the VNM condition? This is a basic problem that is hard to deal with. Fortunately, it is found in this paper that the VNM utility function can have some concrete forms when individuals have constant relative risk aversion. Furthermore, in order to explore the basis of mean-variance utility, an MV function is founded that is based on the VNM utility function and rooted in underlying investment activities. It is shown that the MV function is just the investor’s utility function on the risk-return plane and that it has normal properties. Finally, the MV function is used to analyze the laws of investment activities in a systematic risk environment. In doing so, a tool, TRR, is used to measure risk aversion tendencies and to weigh risk and return.

Key words: VNM condition, relative risk aversion tendency, mean-variance utility, systematic risk