Lesson 10 of 14 · All lessons in this module
  1. Random variables: PMF, PDF and CDF
  2. Expected value, variance and standard deviationVideo lesson
  3. Variance and covariance shortcuts, and the law of large numbers
  4. Discrete distributions: uniform, binomial and Poisson
  5. Continuous uniform and normal distributions, and simulating random draws
  6. Log-normal and logistic distributions, and the moments of key distributions
  7. Integrated cases: reading probabilities and moments from a return model
  8. Conditional probability and probability trees
  9. Conditional expected values and variancesVideo lesson
  10. Covariance from a joint probability function
  11. Joint, marginal and conditional distributions as information arrives
  12. Bayes' formula: updating a probability with new informationVideo lesson
  13. Bayes in practice: diffuse priors, missing inputs and screening tests
  14. Integrated cases: a tree with model forecasts and a Bayes update

This lesson is part of the 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2027 curriculum.

Lesson 10 of 14 · 14 min

Covariance from a joint probability function

When returns are forecast with scenarios, covariance is the probability-weighted sum of the cross-products of each asset's deviation from its expected return.

In short

  • A joint probability function P(X,Y)P(X, Y) gives the probability that X and Y take particular values together.
  • Step 1: expected return of each asset. Step 2: deviations in each cell. Step 3: multiply the deviations, weight by the joint probability, sum.
  • No n−1n - 1: probabilities already do the weighting.
  • Correlation needs each variance too, computed with the same probabilities.
  • X and Y are independent if and only if P(X,Y)=P(X)P(Y)P(X, Y) = P(X)P(Y) for every pair of values. Independence is stronger than zero correlation.
  • If X and Y are uncorrelated (and so if they are independent), E(XY)=E(X)E(Y)E(XY) = E(X)E(Y).

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