Expected Value
Expected value is the probability-weighted average of a random variable — the center of the distribution you actually face, not the mode or the ‘base case’ slide.
Definition
E[X] = Σ x p(x) or ∫ x dF(x). Linearity holds even when variables are dependent: E[aX+bY] = a E[X] + b E[Y]. It is not the outcome you will get; it is the long-run average under the measure you chose (physical or risk-neutral).
Why it matters
Every price and every VaR starts here. Mixing the physical mean with the risk-neutral mean is how people ‘prove’ free lunches.
Case
A coin that pays 100 or 0 with 50% has E = 50. A crash put with 1% physical probability can still have a high price because the risk-neutral probability is larger — different measures, different E.
How to read it
Name the measure. If the slide says ‘expected return 8%’ without a probability model, it is a hope.