Expected value is a probability-weighted average of possible numerical outcomes. It summarizes a distribution, meaning the possible values and their probabilities. It need not be an outcome that can actually occur, and a positive expected return can coexist with a chance of losing money.
A random variable assigns a number to an uncertain outcome. For a finite list of possible values, label them x₁ through xₙ, where n is the number of values. Let pᵢ be the probability of value xᵢ. The subscript i labels an entry. Each probability is between 0 and 1, and all probabilities sum to 1. The listed cases must be mutually exclusive, meaning no overlap, and exhaustive, meaning none are missing.
Write E[X] for the expected value of random variable X: E[X] = p₁ × x₁ + … + pₙ × xₙ. Multiply each value by its probability, then add. Probability has no units, so the answer has the same units as X. Equal weighting is appropriate only when the values are equally likely.
Illustrative one-year return: assume a 60% chance of gaining 10% and a 40% chance of losing 5%, with no other outcomes. The expected return is 0.60 × 10% + 0.40 × (−5%) = 4%. Yet the realized return, meaning the return that actually occurs, can only be +10% or −5% under this model. The 4% mean is neither of them.
Expected value does not show the full range or likelihood of losses. Its usefulness depends on the outcome list and probability inputs. An exact calculation can summarize a poor model accurately. Real investments can have many more outcomes than a two-case illustration.
Expected return for one period is not geometric growth, the compounded growth rate across periods. It also does not encode risk preference: how a person values uncertain gains and losses. The Academy lesson applies the calculation to a euro balance, changes an assumption and checks these limits.
No. A weighted average can fall between the listed values. In the illustrative two-outcome model, a 4% expected return does not add a third possible return.
No. The illustrative 4% mean includes a 40% probability of losing 5%. The mean alone does not describe that downside.
No. Teaching assumptions make the calculation visible. A real application needs justified probability estimates, clearly defined outcomes and a matching time horizon.
Browse all S&P 500 tickers to see this metric applied to individual companies.
Educational research only — not investment advice.