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

A stock is priced at 100 today. Its continuously compounded one-year return is normally distributed with a mean of 8% and a standard deviation of 20%. Using Φ(0.4) = 0.6554 and Φ(0.2) = 0.5793, what is the probability that the price after one year exceeds 100?

The probability is 65.54%. The terminal price exceeds 100 when the log return is positive. With a mean of 8% and a standard deviation of 20%, the cutoff of zero is 0.4 standard deviations below the mean, so the probability is Φ(0.4) = 0.6554.

  1. A34.46%
  2. B50.00%
  3. C57.93%
  4. D65.54%Correct

Explanation

The price exceeds 100 only if the log return is above 0. z = (0 − 0.08)/0.20 = −0.4, so P(r > 0) = 1 − Φ(−0.4) = Φ(0.4) = 0.6554. The 50% option ignores the positive mean, and 34.46% is the probability of a loss.

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