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Probability, Statistics and Financial Data / Probability foundations

Conditional Expectation

A factor release changes a return forecast because the forecast uses new information. Conditional probability tells you how to update probabilities; conditional expectation updates the mean. This lesson computes both, derives the tower property, and separates factor risk from residual risk. It also explains why conditional expectation is the best forecast under squared-error loss.
Learning objectives. By the end of this note you should be able to:
  • compute conditional probabilities and apply the law of total probability / Bayes' rule in a finance tree;
  • define E[X∣Y=y]\mathbb{E}[X\mid Y=y] for discrete and continuous pairs and recover E[X]\mathbb{E}[X] by averaging;
  • treat E[X∣Y]\mathbb{E}[X\mid Y] as a random variable and state/prove the tower property;
  • read a return forecast given a factor as E[R∣F]\mathbb{E}[R\mid F].

In this lesson

  1. Conditional probability
  2. Conditioning random variables
  3. Conditional expectation as a random variable
  4. Law of total variance (brief)
  5. Continuous conditioning (sketch)
  6. Best predictor under squared error

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