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

A return is a mixture of two zero-mean normal distributions: with probability 0.9 the standard deviation is 1, and with probability 0.1 it is 3. Using the fact that the fourth moment of a zero-mean normal is 3σ⁴, what is the kurtosis of the mixture?

The kurtosis is about 8.33. The variance is 1.8 and the fourth moment is 27, so kurtosis is 27 divided by 1.8 squared, or 3.24. This far exceeds 3, showing the fat tails created by mixing a small chance of a high-volatility regime.

  1. A8.33Correct
  2. B3.00
  3. C15.00
  4. D27.00

Explanation

Variance = 0.9(1) + 0.1(9) = 1.8. Fourth moment = 0.9(3·1) + 0.1(3·81) = 2.7 + 24.3 = 27. Kurtosis = 27 / 1.8² = 27 / 3.24 = 8.33. Using 27/1.8 = 15 forgets to square the variance, and 27 is the raw fourth moment. A value of 3 wrongly assumes normal components imply a normal mixture.

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