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Work out expected values and variances conditional on given information

CFA® Level I · Quantitative Methods · Statistical Distributions for Financial Asset Prices and Returns · QM.15.a

The idea

A conditional expected value is the expected outcome given that a particular scenario occurs. The total probability rule links these to the unconditional expectation: E(X) = E(X | S1) x P(S1) + E(X | S2) x P(S2) + ..., provided the scenarios are mutually exclusive and exhaustive. A probability tree lays this out, with scenario probabilities on the first branches and conditional outcomes beyond them. Conditional variance measures dispersion within a scenario around its conditional mean, weighting each squared deviation by its conditional probability. New information changes the scenario probabilities, and updating them updates the expected value; this is how an analyst revises a forecast when evidence arrives.

A practice question

From the question bank, with the reason each wrong option is wrong.

A stock's expected return is 12% if the economy expands and 2% if it contracts. The probability of expansion is 0.6. The unconditional expected return is:

  1. 7.0%Why not: Took a simple average of the two conditional returns instead of weighting by probability.
  2. 8.0% — correct
  3. 6.0%Why not: Attached the 0.6 probability to the contraction scenario, swapping the weights.

By the total probability rule, E(R) = 0.6 x 12% + 0.4 x 2% = 7.2% + 0.8% = 8.0%.

A harder one

Given an expansion, a stock returns either 15% or 9%, each with probability 0.5. Its conditional variance of return, given expansion, in squared percent, is:

  1. 9.0 — correct
  2. 3.0Why not: Reported the conditional standard deviation, the square root, instead of the variance.
  3. 18.0Why not: Summed the squared deviations without weighting each by its 0.5 probability.

The conditional mean is 12%. Variance = 0.5 x (15 - 12)^2 + 0.5 x (9 - 12)^2 = 4.5 + 4.5 = 9.0.

The hardest tier

A portfolio returns 16%, 8% or -4% in three scenarios with probabilities 0.25, 0.50 and 0.25. Its standard deviation of return is closest to:

  1. 10.07%Why not: Treated the three outcomes as an equally weighted sample and ignored the scenario probabilities.
  2. 5.50%Why not: Weighted the absolute deviations from the mean rather than the squared deviations.
  3. 7.14% — correct

E(R) = 4 + 4 - 1 = 7%. Variance = 0.25 x 81 + 0.50 x 1 + 0.25 x 121 = 51.0, so the standard deviation is sqrt(51) = 7.14%.

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