# Risk premium

For an individual, a risk premium is the minimum amount of money by which the expected return on a risky asset (such as stock) must exceed the known return on a risk-free asset (such as a Treasury bond) 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 + $\pi$ , necessary for indifference between the risky and risk-free assets, and its zero-mean risky component x. Then the risk premium $\pi$  is defined by

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

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

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

thus the certainty equivalent is the certain value that 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 of observed risk premium

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.

## Finance

Risk premiums are commonly used in finance. This is because investors generally will not willingly take on risk unless they expect some compensation for doing so. This makes risk premiums quite an important topic in finance, since they are crucial to being able to provide a value for financial assets.

In finance, a common approach for measuring risk premia 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 stock 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 and capital gains. The risk premium for equities is also called the equity premium. This risk premium is an unobservable quantity since it is not known what the expected rate of return on equities is for the average market participant (even though each individual participant knows their own expectation). 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 to refer to the credit spread (the difference between the bond interest rate and the risk-free rate).

Using the risk premium to produce valuations

As aforementioned, one of the biggest reasons for ascertaining a risk premiums is to estimate the value of financial assets. There are a number of models used in finance in order to provide this the most widely used is the Capital Asset Pricing Model or CAPM. CAPM uses investment risk and what return on investment an investor should expect to help estimate a value for the investment. In Finance, CAPM is generally used to estimate the required rate of return for an equity. This required rate of return can then be used to estimate a price for the stock which can be done via a number of methods. The formula for CAPM is:

The Risk Free Rate + The Beta of the Security * The Market Risk Premium 

In this model, we use the risk premium of the market and multiply this with the beta of the security. The beta of a security is the measure of a stocks relative volatility in comparison to the market, that is it is a measure of how closely the share price of an equity move when compared to the market. If the beta of a stock is 1 then a 10% increase in the market will translate to a 10% increase in stock price. If the Beta of a stock is 1.5 then a 10% increase in the market will translate to a 15% increase in the stock and if the beta of a stock is 0.5 a 10% market increase will translate to a 5% stock increase. This beta is generally found via statistical analysis of the share price history of a stock. Therefore CAPM aims to provide a simple model in order to estimate the required return of an investment which uses the theory of risk premiums. This helps to provide investors with a simple means of determining what return an investment should be giving relative to its risk.