What Is Expected Return?

The expected return is the profit or loss that an investor anticipates on an investment that has known historical rates of return (RoR). It is calculated by multiplying potential outcomes by the chances of them occurring and then totaling these results.

Understanding Expected Return

Expected return calculations are a key piece of both business operations and financial theory, including in the well-known models of the modern portfolio theory (MPT) or the Black-Scholes options pricing model. For example, if an investment has a 50% chance of gaining 20% and a 50% chance of losing 10%, the expected return would be 5% = (50% x 20% + 50% x -10% = 5%).

The expected return is a tool used to determine whether an investment has a positive or negative average net outcome. The sum is calculated as the expected value (EV) of an investment given its ;potential returns in different scenarios, as illustrated by the following formula:

Expected Return = Σ (Return_i x Probability_i)

where "i" indicates each known return and its respective probability in the series

The expected return is usually based on historical data and is therefore not guaranteed into the future; however, it does often set reasonable expectations. Therefore, the expected return figure can be thought of as a long-term weighted average of historical returns.

In the formulation above, for instance, the 5% expected return may never be realized in the future, as the investment is inherently subject to systematic and unsystematic risks. Systematic risk is the danger to a market sector or the entire market, whereas unsystematic risk applies to a specific company or industry.

Calculating Expected Return

When considering individual investments or portfolios, a more formal equation for the expected return of a financial investment is:

where:

• r_a = expected return;
• r_f = the risk-free rate of return;
• β = the investment's beta; and
• r_m =the expected market return

In essence, this formula states that the expected return in excess of the risk-free rate of return depends on the investment's beta, or relative volatility compared to the broader market.

The expected return and standard deviation are two statistical measures that can be used to analyze a portfolio. The expected return of a portfolio is the anticipated amount of returns that a portfolio may generate, making it the mean (average) of the portfolio's possible return distribution. The standard deviation of a portfolio, on the other hand, measures the amount that the returns deviate from its mean, making it a proxy for the portfolio's risk.

Limitations of the Expected Return

To make investment decisions solely on expected return calculations can be quite naïve and dangerous. Before making any investment decisions, one should always review the risk characteristics of investment opportunities to determine if the investments align with their portfolio goals.

For example, assume two hypothetical investments exist. Their annual performance results for the last five years are:

• Investment A: 12%, 2%, 25%, -9%, and 10%
• Investment B: 7%, 6%, 9%, 12%, and 6%

Both of these investments have expected returns of exactly 8%. However, when analyzing the risk of each, as defined by the standard deviation, investment A is approximately five times riskier than investment B. That is, investment A has a standard deviation of 11.26% and investment B has a standard deviation of 2.28%. Standard deviation is a common statistical metric used by analysts to measure an investment's historical volatility, or risk.

In addition to expected returns, investors should also consider the likelihood of that return. After all, one can find instances where certain lotteries offer a positive expected return, despite the very low chances of realizing that return.

Expected Return Example

The expected return does not just apply to a single security or asset. It can also be expanded to analyze a portfolio containing many investments. If the expected return for each investment is known, the portfolio's overall expected return is a weighted average of the expected returns of its components.

For example, let's assume we have an investor interested in the tech sector. Their portfolio contains the following stocks:

• Alphabet Inc., (GOOG): $500,000 invested and an expected return of 15%
• Apple Inc. (AAPL): $200,000 invested and an expected return of 6%
• Amazon.com Inc. (AMZN): $300,000 invested and an expected return of 9%

With a total portfolio value of $1 million the weights of Alphabet, Apple, and Amazon in the portfolio are 50%, 20%, and 30%, respectively. Thus, the expected return of the total portfolio is:

• (50% x 15%) + (20% x 6%) + (30% x 9%) = 11.4%

The Bottom Line

Expected return is an estimate of the average return that an investment or portfolio investments should generate over a certain period of time. In general, riskier assets or securities demand a higher expected return to compensate for the additional risk. Expected return is not a guarantee, but rather a prediction based on historical data and other relevant factors. It can be used by investors to compare different investment options and make informed decisions about their portfolios, and is a key input for various financial models such as modern portfolio theory (MPT) and the capital asset pricing model (CAPM).