When evaluating an asset’s past performance, we can make use of the historical (or realized) average return. In that sense, the historical average return is a backward-looking measure. But, in order to forecast an asset’s future performance, we need a forward-looking measure. This measure is called the expected return. In this post, we explain the expected return formula and offer an easy-to-use expected return calculator.
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Expected return formula
Let’s motivate the concept of expected return with an example. Hillary has savings worth $5,000 and is considering investing in the U.S. stock market. She believes that the probabilities that the market will be in a “good state” and “bad state” next year are 60% and 40%, respectively.
Hillary is interested in a stock that 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. She wants to calculate the expected return for this stock. 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, the expected return formula needs to take probabilities of different return outcomes into account.
Formally, the expected return formula can be written as follows:
E[R] = ∑ pi Ri
This means that an asset’s expected return E[R] is a probability-weighted average of all possible return outcomes where pi is the probability of the ith state and Ri is the return under state i.
In the example we gave above, 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, Hillary could easily calculate the expected return of the stock she 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.
Expected return calculator
You can use the expected return calculator below to calculate the expected return for any risky asset. The calculator allows up to six different states. Please note the following instructions:
- Please ensure the probabilities add up to 100% (the sum of the probabilities is given in the final row).
- If your calculation requires, say, 3 states only, simply leave 0s in the fields involving the states 4, 5, and 6.
Summary
In this post, we introduce the concept of expected return as a forward-looking performance measure. We explain the expected return formula using examples. In essence, the expected return of a risky asset depends on (i) the returns the asset yields under different states of the market, and (ii) the probabilities of those states occurring. Finally, we offer an expected return calculator, which can be used for expected return calculations involving up to six different states.
Calculating expected returns is a difficult task in reality. One approach is to rely on asset pricing models such as the CAPM, which posits a linear relationship between expected returns and beta.
Further reading on expected return calculator and formula
Black (1993), ‘Estimating expected return‘, Financial Analysts Journal, Vol. 49(5), pp. 36-38.
What is next?
This lesson is part of our free course on investments.
- Next lesson: We will discuss the risk-free rate of return.
- Previous lesson: We explained the concept of return volatility.
We hope you find our expected return calculator useful. If that is the case, feel free to share this post with others on social media or other platforms.
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