Summary  
This chapter explains how to compute the expected value of a variable by summing the products of each outcome and its probability, and contrasts that forward-looking estimate with the actual (realized) result to illustrate convergence over time.

General domain of usage  
Financial modeling / investment analysis

A stock's past return tells you what happened. **Expected return** tells you what a rational investor would anticipate going forward – based on probabilities, not hopes.

The formula is straightforward. You take each possible outcome, multiply it by its probability, and sum the results:

**Expected Return = Σ (Probability × Return)**

A stock with three scenarios:

That 6% isn't a guarantee. It's the probability-weighted average of outcomes – the number a rational investor uses to compare this asset against alternatives.

## Expected Return vs Realized Return

The expected return is calculated before the fact. The **realized return** is what actually happened. They are almost never the same number – and that gap is normal.

Over a single year, realized returns can swing wildly from expected. Over 20+ years, they tend to converge. This is why expected return is a long-game tool – it loses most of its meaning if you're investing for 18 months.

- **Expected return:** a forward-looking estimate based on probabilities;
- **Realized return:** the actual return over a specific period;
- **The gap between them:** normal in the short run, meaningful in the long run.

Expected Return is the probability-weighted average of all possible returns for an asset. It represents what a rational investor anticipates earning on average over many periods – not what they will earn in any single period.

Definition

Expected return is only as good as the probabilities you assign. If your scenario probabilities are wrong – or if you're missing a scenario entirely – your expected return is misleading. The math is clean; the inputs rarely are.

Note

An investor assigns the following probabilities to a stock: 50% chance of +18%, 30% chance of +4%, 20% chance of -12%. What is the expected return?

An investor calculates an expected return of +9% for a stock. After one year, the stock returns +22%. Which statement best describes this outcome?

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Expected Return

Expected Return vs Realized Return

1. An investor assigns the following probabilities to a stock: 50% chance of +18%, 30% chance of +4%, 20% chance of -12%. What is the expected return?

2. An investor calculates an expected return of +9% for a stock. After one year, the stock returns +22%. Which statement best describes this outcome?