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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. 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.

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 + π and its zero-mean risky component x. Then the risk premium π is defined by

u(rf) = Eu(rf + π + x).

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

u(C) = Eu(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.

Example

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. However, a risk averse contestant will choose no door and accept the guaranteed \$500.

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 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.