Lesson 5 of 7 · 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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Covariance from a joint probability function · Portfolio Mathematics