Given The Following Probability Distribution Calculate The Expected Return

Calculating the expected return from a probability distribution is fundamental in finance and statistics. This guide explains the concept, provides a step-by-step calculation method, and includes an interactive calculator to compute the expected return from any given probability distribution.

What is Expected Return?

The expected return, also known as the expected value or mean, is a fundamental concept in probability and statistics. It represents the average outcome we would expect if an experiment or investment were repeated many times under identical conditions.

In finance, the expected return is often used to evaluate the potential profitability of an investment. It's calculated by multiplying each possible outcome by its probability of occurrence and then summing all these products.

For example, if you're considering an investment with three possible outcomes: a 20% chance of earning $100, a 50% chance of earning $50, and a 30% chance of earning $20, the expected return would be calculated as:

(0.20 × $100) + (0.50 × $50) + (0.30 × $20) = $20 + $25 + $6 = $51

How to Calculate Expected Return

Calculating the expected return involves these steps:

Identify all possible outcomes and their corresponding probabilities
Multiply each outcome by its probability
Sum all the products to get the expected return

Formula: Expected Return = Σ (Outcome × Probability)

Where Σ represents the sum of all possible outcomes and their probabilities.

The expected return is particularly useful in decision-making because it provides a single value that summarizes the entire probability distribution. However, it's important to note that the expected return doesn't account for the variability or risk associated with the outcomes.

Example Calculation

Let's consider a simple example of a probability distribution for a stock investment:

Outcome ($)	Probability
100	0.30
50	0.50
20	0.20

To calculate the expected return:

Multiply each outcome by its probability:
- 100 × 0.30 = 30
- 50 × 0.50 = 25
- 20 × 0.20 = 4
Sum the results: 30 + 25 + 4 = 59

The expected return for this investment is $59. This means, on average, you can expect to earn $59 from this investment.

Interpreting the Result

The expected return provides valuable information for decision-making, but it's important to consider it in conjunction with other factors:

Risk Assessment: While the expected return gives an average outcome, it doesn't account for the variability or risk of the outcomes. Investments with higher expected returns often come with higher risks.
Diversification: Combining investments with different probability distributions can help manage risk and potentially improve the overall expected return.
Time Horizon: The expected return is calculated for a specific time period. Longer time horizons typically result in higher expected returns but also higher risks.

In practical terms, the expected return helps investors make informed decisions by providing a benchmark against which to evaluate potential investments. However, it's essential to consider the full range of possible outcomes and the associated probabilities when making investment decisions.

Frequently Asked Questions

What is the difference between expected return and actual return?: The expected return is the average outcome calculated from a probability distribution, while the actual return is the specific outcome realized in a particular instance. The actual return may differ significantly from the expected return due to the inherent variability in the outcomes.
Can the expected return be negative?: Yes, the expected return can be negative if the weighted average of all possible outcomes is negative. This typically indicates that the investment has a higher probability of losing money than gaining money.
How does expected return relate to risk?: The expected return provides a measure of the average outcome, but it doesn't account for the variability or risk associated with the outcomes. Investments with higher expected returns often come with higher risks, as they typically involve greater uncertainty.
Is the expected return always a good measure of an investment's potential?: While the expected return is a useful measure, it's important to consider it in conjunction with other factors such as risk, diversification, and time horizon. An investment with a high expected return may not necessarily be the best choice if it comes with unacceptable levels of risk.
How can I use the expected return to make investment decisions?: The expected return provides a benchmark against which to evaluate potential investments. By comparing the expected returns of different investments, you can make more informed decisions. However, it's essential to consider the full range of possible outcomes and the associated probabilities when making investment decisions.