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FRM Part I · FRM Exam Part I · Multivariate Random Variables

Two asset returns X and Y are jointly bivariate normal. X has mean 4% and standard deviation 10%, Y has mean 6% and standard deviation 20%, and the correlation is 0.5. Which statement about the distribution of the portfolio return P = X + Y is correct?

The portfolio return is normal with a mean of 10% and a standard deviation of about 26.5%. Linear combinations of jointly normal variables stay normal, and the variance is 0.01 + 0.04 + 2(0.5)(0.1)(0.2) = 0.07, whose square root is about 26.46%.

  1. AP is normal with mean 10% and standard deviation 30%
  2. BP is normal with mean 10% and standard deviation about 26.5%Correct
  3. CP is normal with mean 10% and standard deviation about 22.4%
  4. DP is not normal because X and Y are correlated

Explanation

Any linear combination of jointly normal variables is normal. Variance = 0.01 + 0.04 + 2(0.5)(0.10)(0.20) = 0.07, so the standard deviation is sqrt(0.07) = 26.46%. Option A ignores diversification (assumes correlation of 1); option C assumes zero correlation (sqrt(0.05)).

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