# Expected return – A forward-looking return measure

Historical (or realized) returns are particularly useful when evaluating an asset’s past performance. As such they are backward-looking measures. But, in order to forecast an asset’s future performance, we need a forward-looking measure. This measure is called the expected return.

Learning objectives:

• Learn how to calculate an asset’s expected return.
• Understand why we need to use probabilities in expected return calculations.

Let’s motivate the concept of expected return with an example. François has savings worth €5,000 and is considering investing in the French stock market. He believes that the probabilities that the market will be in a “good state” and “bad state” next year are 60% and 40%, respectively. According to his calculations, one of the stocks he is interested in is expected to go up by 10% when the market is in the good state and go down by —5% when the market is in the bad state. Clearly, the 10% outcome is more likely than the —5% outcome as the market is more likely to be in the good state (60%) than in the bad state (40%). Therefore, we need a measure that weighs each possible outcome by its probability, and the expected return measure just does that.

## Expected return is a probability-weighted average return

Formally, we can define an asset’s expected return E[R] as a probability-weighted average of all possible return outcomes:

E[R] = ∑ pi Ri

where pi is the probability of the ith state and Ri is the return under state i. We should highlight that the symbol μ is commonly used to denote an asset’s expected return.

In our example, there are only two possible states: the good state and the bad state. And, as a result, there are only two possible return outcomes: 10% (under the good state) and —5% (under the bad state). So, François could easily calculate the expected return of the stock he is interested in as follows:

E[R] = 60% * (+10%) + 40% * (—5%) = 4%

In reality, there are not just two but many possible return outcomes, and it is difficult to know the probability associated with each outcome. In later lessons, we will cover asset pricing models that will help us estimate the expected returns of risky assets.

##### What is next?

This post is part of the series on investments. The next post in the series is the risk-free rate of return. The previous post was the return volatility.