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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
Risk-free bond
A risk-free bond is a theoretical bond that repays interest and principal with absolute certainty. The rate of return would be the risk-free interest rate. In practice, government bonds are treated as risk-free bonds, as governments can raise taxes or indeed print money to repay their domestic...

, 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
Willingness to accept
Willingness to accept is the amount that а person is willing to accept to abandon a good or to put up with something negative, such as pollution. It is the minimum monetary amount required for sale of a good or acquisition of something undesirable to be accepted by an individual...

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
Expected utility hypothesis
In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different utility...

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.

## Example

Suppose a game show
Game show
A game show is a type of radio or television program in which members of the public, television personalities or celebrities, sometimes as part of a team, play a game which involves answering questions or solving puzzles usually for money and/or prizes...

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
Risk neutral
In economics and finance, risk neutral behavior is between risk aversion and risk seeking. If offered either €50 or a 50% chance of each of €100 and nothing, a risk neutral person would have no preference between the two options...

is indifferent between these choices. However, a risk averse
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....

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.

## Finance

In finance
Finance
"Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...

, the risk premium refers to the amount by which an asset's expected rate of return exceeds the risk-free interest rate
Risk-free interest rate
Risk-free interest rate is the theoretical rate of return of an investment with no risk of financial loss. The risk-free rate represents the interest that an investor would expect from an absolutely risk-free investment over a given period of time....

. When measuring risk, a common approach is to compare the risk-free return on T-bills and the risky return on other investments (using the ex post return as a proxy for the ex ante expected return). The difference between these two returns can be interpreted as a measure of the excess expected return on the risky asset. This excess expected return is known as the risk premium.
• Equity: In the equity market the risk premium is the expected return of a company stock, a group of company stocks, or a portfolio of all stock market company stocks, minus the risk-free rate. The return from equity is the sum of the dividend yield
Dividend yield
The dividend yield or the dividend-price ratio on a company stock is the company's total annual dividend payments divided by its market capitalization, or the dividend per share, divided by the price per share. It is often expressed as a percentage...

and capital gains. The risk premium for equities is also called the equity premium. Note that this is an unobservable quantity since no one knows for sure what the expected rate of return on equities is. Nonetheless, most people believe that there is a risk premium built into equities, and this is what encourages investors to place at least some of their money in equities.
• Debt: In the context of bonds
Bond (finance)
In finance, a bond is a debt security, in which the authorized issuer owes the holders a debt and, depending on the terms of the bond, is obliged to pay interest to use and/or to repay the principal at a later date, termed maturity...

, the term "risk premium" is often used imprecisely to refer to the credit spread
The financial term, credit spread is the yield spread, or difference in yield between different securities, due to different credit quality. The credit spread reflects the additional net yield an investor can earn from a security with more credit risk relative to one with less credit risk...

(the difference between the bond interest rate and the risk-free rate). To see why this is inconsistent with the given definition, imagine that the risk free rate is 3% and XYZ corporate bonds are yielding 10%. Does that mean that the expected return in excess of the risk free rate is 7%? Almost certainly not; after all, there is surely a positive probability of a default, as well as a positive probability of positive or negative capital gains due to fluctuations in the market prices of bonds. In reality, the risk premium (as defined above) is likely to be significantly less than the credit spread; it could even be negative, if the bond's default scenarios are negatively correlated with most other bonds' default scenarios. See Capital asset pricing model
Capital asset pricing model
In finance, the capital asset pricing model is used to determine a theoretically appropriate required rate of return of an asset, if that asset is to be added to an already well-diversified portfolio, given that asset's non-diversifiable risk...

.

• Interest
Interest
Interest is a fee paid by a borrower of assets to the owner as a form of compensation for the use of the assets. It is most commonly the price paid for the use of borrowed money, or money earned by deposited funds....

• Risk
Risk
Risk is the potential that a chosen action or activity will lead to a loss . The notion implies that a choice having an influence on the outcome exists . Potential losses themselves may also be called "risks"...

• Risk aversion
Risk aversion
Risk aversion is a concept in psychology, economics, and finance, based on the behavior of humans while exposed to uncertainty....

• Risk neutral
Risk neutral
In economics and finance, risk neutral behavior is between risk aversion and risk seeking. If offered either €50 or a 50% chance of each of €100 and nothing, a risk neutral person would have no preference between the two options...

• Risk-loving
Risk-loving
In economics and finance, a risk lover is a person who has a preference for risk. While most investors are considered risk averse, one could view casino goers as risk loving...

• Minimum acceptable rate of return
Minimum acceptable rate of return
In business and engineering, the minimum acceptable rate of return, often abbreviated MARR, or hurdle rate is the minimum rate of return on a project a manager or company is willing to accept before starting a project, given its risk and the opportunity cost of forgoing other projects...

• Expected utility hypothesis
Expected utility hypothesis
In economics, game theory, and decision theory the expected utility hypothesis is a theory of utility in which "betting preferences" of people with regard to uncertain outcomes are represented by a function of the payouts , the probabilities of occurrence, risk aversion, and the different utility...