Risk premium

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 riskfree asset, in order to induce an individual to hold the risky asset rather than the riskfree 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.
Contents
Formal definitions
Let an individual's increasing, concave von NeumannMorgenstern utility function be u, let r_{f} be the return on the riskfree asset, and let r be the random return on the risky asset. Write r as the sum of its hypothetical expected return r_{f} + π and its zeromean risky component x. Then the risk premium π is defined by
 u(r_{f}) = Eu(r_{f} + π + x).
Thus the risk premium is the amount by which the risky asset's expected return must in fact exceed the riskfree return in order to make the risky and riskfree 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.
Finance
In finance, the risk premium refers to the amount by which an asset's expected rate of return exceeds the riskfree interest rate. When measuring risk, a common approach is to compare the riskfree return on Tbills 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 riskfree rate. The return from equity is the sum of the dividend yield 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, the term "risk premium" is often used imprecisely to refer to the credit spread (the difference between the bond interest rate and the riskfree 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.
See also
 Interest
 Risk
 Risk aversion
 Risk neutral
 Riskloving
 Minimum acceptable rate of return
 Expected utility hypothesis
 LIBOR–OIS spread
External links
 Hussman Funds  Estimating the LongTerm Return on Stocks  June 1998
 Earnings Quality and the Equity Risk Premium: A Benchmark Model
 Ruben D. Cohen (2002) “The Relationship Between the Equity Risk Premium, Duration and Dividend Yield (download),” Wilmott Magazine, pp 8497, November issue.
 Ruben D. Cohen “The Longrun Behaviour of the S&P Composite Price Index and its Risk Premium (download).”
Categories: Actuarial science
 Financial markets
 Wagering
 Gambling terminology
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