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FRM Part I · FRM Exam Part I · Common Univariate Random Variables

A risk analyst models daily returns as a mixture of two normal distributions with identical means of zero. With probability 0.90 a return is drawn from a calm regime with standard deviation 1%, and with probability 0.10 from a stressed regime with standard deviation 3%. What is the unconditional variance of daily returns (in %²)?

The unconditional variance is 1.80 (%²). With both components having zero mean, mixture variance equals the probability-weighted sum of component variances: 0.9 times 1 plus 0.1 times 9, giving 1.80.

  1. A1.80Correct
  2. B1.90
  3. C1.34
  4. D1.00

Explanation

With equal means, the mixture variance is the probability-weighted average of the component variances: 0.90 x 1 + 0.10 x 9 = 0.90 + 0.90 = 1.80. Option 1.34 comes from weighting the standard deviations first and then squaring (1.2 squared = 1.44 is also wrong). Option 1.00 ignores the stressed regime.

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