Different stocks offer different levels of expected return. What causes stock A’s expected return to be higher than stock B’s expected return? How does the expected return of a risky asset relate to the risk-free rate of return? In this post, we answer both questions by introducing the concept of risk premium.

__Learning objectives__

- Define the risk premium for an asset and interpret how it relates to that asset’s expected return and the risk-free rate of return.
- Understand why risk premiums vary across both between and within asset classes.

## The formal definition of risk premium

The formal definition of risk premium *π* on a risky asset is easy enough to understand. It is simply the difference between that asset’s expected return *E[R]* and the risk-free rate of return *R _{f}*:

*π = E[R] − R _{f}*

The same equation can be rewritten as:

*E[R] = R _{f} + π *

This tells us that a risky asset’s expected return is the risk-free rate PLUS a risk premium. It implies that we would invest in a risky asset only if it offers a return higher than the risk-free rate. That markup constitutes the risk premium. This is a direct consequence of investors’ aversion towards risk: Risk-averse investors would bear risk only if they are rewarded for doing so. And, the risk premium is the reward.

## The risk-return tradeoff

In general, the riskier the asset, the higher the risk premium should be. So, if asset A is riskier than asset B, we have *π _{A} > π_{B}*. If that is the case, we can deduce that asset A’s expected return, which is

*E[R*, should be higher than asset B’s expected return, which is

_{A}] = R_{f}+ π_{A}*E[R*, as well:

_{B}] = R_{f}+ π_{B}*E[R*since

_{A}] > E[R_{B}]*π*. This answers the question we posed at the start of this post: The difference between the expected returns of stock A and stock B is down to the difference between their

_{A}> π_{B}*π*s. That is, the riskier stock will command a higher risk premium and, thus, will have a higher expected return.

The fact that risk and (expected) return goes hand-in-hand is known as the **risk-return tradeoff**, which is a fundamental principle in finance. An investor can expect to earn a higher level of expected return only if the investor agrees to bear a higher degree of risk.

Note that the risk premium of the risk-free asset must be zero by definition. Therefore, the expected return on the risk-free asset is *E[R] = R _{f} + π = R_{f}*, which is of course the risk-free rate of return. This makes intuitive sense. While the risk-free asset has

*π*= 0, risk-averse investors would not invest in risky assets if they didn’t offer a positive risk premium.

Let’s wrap up with a simple, numerical example. Suppose that the risk premium for the Tesla stock is 8% per year and the risk-free rate of return is 2% per year. This would imply that the expected return on Tesla shares is 8% + 2% = 10%.

##### What is next?

This post is part of the series on investments. The next post in the series explains how to calculate the return of a portfolio of assets. The previous explained the investors’ appetite for risk by distinguishing between risk-averse, risk-neutral, and risk-seeking behavior.

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