Formal Definitions
Let an individual's increasing, concave von Neumann-Morgenstern utility function be u, let rf be the return on the risk-free asset, and let r be the random return on the risky asset. Write r as the sum of its hypothetical expected return rf + and its zero-mean risky component x. Then the risk premium is defined by
Thus the risk premium is the amount by which the risky asset's expected return must in fact exceed the risk-free return in order to make the risky and risk-free assets equally attractive.
Further, the certainty equivalent C is defined by
thus the certainty equivalent is the certain value which is equally attractive as the risky asset; due to risk aversion the certainty equivalent will be less than the expected return on the risky asset.
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