FRM Part I · FRM Exam Part I · Random Variables
A risk manager models a stock's price as lognormal, where the continuously compounded annual return is normal with mean 6% and standard deviation 20%. What is the expected value of the price ratio S(1)/S(0)? (e^0.08 = 1.0833, e^0.06 = 1.0618, e^0.04 = 1.0408)
The expected price ratio is 1.0833. For a lognormal variable the mean is exp(μ + σ²/2), here exp(0.06 + 0.02) = exp(0.08). Using only exp(0.06) gives the median, not the mean.
- A1.0833Correct
- B1.0618
- C1.0408
- D1.0600
Explanation
For a lognormal variable, E[e^X] = exp(μ + σ²/2) = exp(0.06 + 0.04/2) = exp(0.08) = 1.0833. Option B ignores the variance term (exp of the mean only, which is the median). Option C subtracts the term instead of adding it (exp(0.04)).
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