Expected Return Calculator (probability-weighted scenarios with standard deviation)

Understanding the expected return calculator — concept and principle

This calculator finds the expected return of an investment from scenarios you define, each with a probability and a return. It also gives the standard deviation (risk), the probability of a loss, the best and worst scenario returns and the expected gain on an amount. Use it for portfolio planning, finance courses and thinking through investment scenarios.

How it is calculated

• Expected return E[R] = Σ pᵢ × Rᵢ.
• Standard deviation σ = √Σ pᵢ (Rᵢ − E[R])².
• Probability of a loss = sum of probabilities of scenarios with negative returns.
• Expected gain = amount × E[R].

Probabilities should add up to 100%.

The principle

Expected return is a probability-weighted average: each possible outcome counts in proportion to how likely it is. It describes the average result over many repetitions, not what will happen in any single year; in one year you get one of the scenarios. The standard deviation measures how far outcomes spread around the average, a common measure of risk. Two investments with the same expected return can have very different risk, which is why both numbers matter.

Worked example

Default inputs: 25% chance of −8%, 50% of +7%, 25% of +18%; amount $10,000.

1. E[R] = 0.25 × (−8) + 0.50 × 7 + 0.25 × 18 = −2 + 3.5 + 4.5 = 6%.
2. Deviations: −14, 1, 12 points; squared × probability: 49, 0.5, 36; sum 85.5.
3. σ = √85.5 ≈ 9.25%.
4. Probability of a loss = 25%; best scenario +18%, worst −8%.
5. Expected gain on $10,000 = $600.

How to read the result

A higher expected return is attractive only if the risk is acceptable. Here there is a one-in-four chance of losing 8%, although the average is +6%. Use the standard deviation to compare risk between options, and think about whether you could live with the worst scenario.

Common mistakes

• Probabilities that do not add up to 100%.
• Treating the expected return as a guaranteed or most likely result.
• Using scenarios that are too optimistic or leave out bad cases.
• Comparing expected returns without comparing risk.

Frequently asked questions

What does expected return mean?

The probability-weighted average return over all scenarios.

Why is the standard deviation important?

It shows how widely results can vary around the average, which is a key part of risk.

Can the expected return be a value that never happens?

Yes. Here 6% is not one of the scenarios; it is the long-run average.

Sources: Probability-weighted mean and standard deviation over discrete scenarios (standard; e.g. OpenStax, Principles of Finance (2022), Ch. 15 How to Think about Investing)