FRM Part I · FRM Exam Part I · Random Variables
A portfolio's annual return R has mean 8% and standard deviation 12%. A risk manager defines a transformed variable Y = 3 - 2R, where R is in decimal form. What are E[Y] and Var(Y)?
E[Y] is 2.84 and Var(Y) is 0.0576. The mean transforms linearly: 3 minus 2 times 0.08. The constant does not affect variance, while the coefficient is squared, so variance is 4 times 0.0144, which is 0.0576.
- AE[Y] = 2.84; Var(Y) = 0.0576Correct
- BE[Y] = 2.84; Var(Y) = 0.0144
- CE[Y] = 3.16; Var(Y) = 0.0576
- DE[Y] = 2.84; Var(Y) = 0.1200
Explanation
E[Y] = 3 - 2(0.08) = 2.84. Var(Y) = (-2)^2 Var(R) = 4(0.12^2) = 4(0.0144) = 0.0576. The option with 0.0144 forgets to square the coefficient of -2, and 3.16 gets the sign wrong in the mean.
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