Certainty equivalent

For an individual, a risk premium is the minimum amount of money by which the expected return on a risky asset must exceed the known return on a risk-free asset in order to induce an individual to hold the risky asset rather than the risk-free asset. It is positive if the person is risk averse. Thus it is the minimum willingness to accept compensation for the risk.

The certainty equivalent, a related concept, is the guaranteed amount of money that an individual would view as equally desirable as a risky asset.

For market outcomes, a risk premium is the actual excess of the expected return on a risky asset over the known return on the risk-free asset.

Let an individual's increasing, concave von Neumann-Morgenstern utility function be u, let r_f 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 expected return r_f + ${\displaystyle \pi }$, necessary for indifference between the risky and risk-free assets, and its zero-mean risky component x. Then the risk premium ${\displaystyle \pi }$ is defined by

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