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# Expected Return: Formula, How It Works, Limitations, Example

## 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.

### Key Takeaways

• The expected return is the amount of profit or loss an investor can anticipate receiving on an investment.
• An expected return is calculated by multiplying potential outcomes by the odds of them occurring and then totaling these results.
• Expected returns cannot be guaranteed.
• The expected return for a portfolio containing multiple investments is the weighted average of the expected return of each of the investments.
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## 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 (Returni x Probabilityi)

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.

When considering individual investments or portfolios, a more formal equation for the expected return of a financial investment is: Expected return = risk free premium + Beta (expected market return - risk free premium).  Investopedia

where:

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.

The expected return is not absolute, as it is a projection and not a realized return.

## 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.

Pros
• Gauges the performance of an asset

• Weighs different scenarios

Cons
• Doesn't take risk into account

• Based largely on historic data

## 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%

## How Is Expected Return Used in Finance?

Expected return calculations are a key piece of both business operations and financial theory, including in the well-known models of modern portfolio theory (MPT) or the Black-Scholes options pricing model. It is a tool used to determine whether an investment has a positive or negative average net outcome. The calculation is usually based on historical data and therefore cannot be guaranteed for future results, however, it can set reasonable expectations.

## What Are Historical Returns?

Historical returns are the past performance of a security or index, such as the S&P 500. Analysts review historical return data when trying to predict future returns or to estimate how a security might react to a particular economic situation, such as a drop in consumer spending. Historical returns can also be useful when estimating where future points of data may fall in terms of standard deviations.

## How Does Expected Return Differ From Standard Deviation?

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. 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.

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