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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.

Formal definitions for an individual

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 expected return rf + ${\displaystyle \pi }$, 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

${\displaystyle u(r_{f})=\mathbb {E} [u(r_{f}+\pi +x)].}$

Here the left side is the degree of attractiveness of the risk-free asset—the known utility of its known return—and the right side is the degree of attractiveness of the risky asset—the expected utility of its risky return. 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

${\displaystyle u(C)=\mathbb {E} [u(r)];}$

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.

Suppose a game show participant may choose one of two doors, one that hides $1,000 and one that hides$0. Further suppose that the host also allows the contestant to take $500 instead of choosing a door. The two options (choosing between door 1 and door 2, or taking$500) have the same expected value of $500, so no risk premium is being offered for choosing the doors rather than the guaranteed$500.
A contestant unconcerned about risk is indifferent between these choices. A risk-averse contestant will choose no door and accept the guaranteed $500, while a risk-loving contestant will derive utility from the uncertainty and will therefore choose a door. If too many contestants are risk averse, the game show may encourage selection of the riskier choice (gambling on one of the doors) by offering a positive risk premium. If the game show offers$1,600 behind the good door, increasing to $800 the expected value of choosing between doors 1 and 2, the risk premium becomes$300 (i.e., $800 expected value minus$500 guaranteed amount). Contestants requiring a minimum risk compensation of less than $300 will choose a door instead of accepting the guaranteed$500.