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

A stock is priced at 50 today. The continuously compounded one-year return ln(S1/S0) is normal with mean 0.04 and variance 0.09. Which value is closest to the expected stock price in one year, E[S1]? (e^0.04 = 1.0408; e^0.085 = 1.0887; e^0.13 = 1.1388; e^−0.005 = 0.9950)

The expected price is about 54.44. For a lognormal price, the mean equals S0 times exp of the log-return mean plus half the variance: 50 × exp(0.04 + 0.045) = 50 × 1.0887. Using only exp(0.04) gives the median of 52.04, not the mean.

  1. A49.75
  2. B52.04
  3. C54.44Correct
  4. D56.94

Explanation

For a lognormal variable, E[S1] = S0·exp(μ + σ²/2) = 50·exp(0.04 + 0.045) = 50·1.0887 = 54.44. The 52.04 option is the median, 50·e^0.04, which omits the σ²/2 term. The 49.75 option subtracts σ²/2, and 56.94 adds the full σ².

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