Expected Value

The calculation requires a complete and mutually exclusive set of outcomes: every possibility must be listed once, and the assigned probabilities must sum to one. Each outcome is multiplied by its probability and the products are added. If a stock returns 20% with probability 0.25, 8% with probability 0.50 and minus 10% with probability 0.25, the weighted sum is 5% plus 4% minus 2.5%, giving 6.5%. Omitting an outcome or letting probabilities sum to less than one silently biases the result toward whatever remains in the set.

For a continuous random variable the same logic applies with a density function replacing discrete probabilities, but at Level I the discrete formulation carries almost all the workload. The critical structural point is that the result is a single number summarising an entire distribution, and it discards all information about dispersion, skew and tail behaviour.

Expectation is a linear operator, and that linearity is what makes it usable in finance. A constant multiple passes straight through: the expected value of a scaled random variable equals the scale times the expected value. Adding a constant simply shifts the expectation by that constant. Most importantly, the expected value of a sum of random variables equals the sum of their individual expected values, whether or not those variables are correlated.

This is why the expected return on a portfolio is the weighted average of the expected returns of its constituent assets, with no correlation term anywhere in the expression. Variance behaves entirely differently, requiring covariances. The asymmetry between the two moments is one of the foundations of diversification: combining assets can reduce portfolio variance without reducing portfolio expected return. One caution: the expected value of a product of random variables equals the product of their expected values only when the variables are independent.

Expected values are rarely computed once and left alone. New information changes the probabilities, and the conditional expected value is the probability-weighted outcome given that a specified event has occurred. The total probability rule for expected value reassembles the unconditional figure: weight each conditional expectation by the probability of the scenario it is conditioned on, then sum. Analysts use this structure whenever a forecast depends on a sequenced event, such as a rate decision followed by an earnings release.

Probability trees formalise the sequence. Each branch carries a conditional probability, the path probability is the product along the branch, and the terminal nodes carry the outcomes. Working backwards from the terminal values produces the expected value at every intermediate node. This is the same machinery behind expected credit loss, where probability of default, loss given default and exposure combine multiplicatively, and behind binomial valuation of options and bonds with embedded features.

In practice, expected value underlies scenario-weighted earnings forecasts, expected loss provisioning under credit risk frameworks, capital budgeting with uncertain cash flows, and the risk-neutral expectations discounted in derivative pricing. Analysts also treat it as the input for the next step rather than the output: the deviations of outcomes from the expected value, squared and probability-weighted, define variance, so the expectation must be computed first before any dispersion measure exists.

A candidate should be able to compute an expected value from a probability table or tree, update it with conditional information, chain it into variance and standard deviation, and derive a portfolio expected return from asset weights. Equally important is interpretation: recognising that the expectation may be a value the variable can never take, that it says nothing about the likelihood of any single outcome, and that it inherits all the fragility of the subjective probabilities fed into it.