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CFA Level I · CFA Level I Exam · Statistical Distributions for Financial Asset Prices and Returns

A portfolio manager has a normally distributed portfolio with an expected return of 8% and a standard deviation of 10%. The client requires that the probability of a return below 0% be at most 5%. Using a z-value of 1.645 for a 5% lower tail, the portfolio most likely:

The portfolio fails the requirement. Its safety-first ratio is 0.80, since the 0% threshold lies 8% below the mean with 10% risk, well under the 1.645 needed for a 5% shortfall probability. The actual shortfall probability is roughly 21%.

  1. Asatisfies the requirement, because the threshold lies more than 1.645 standard deviations below the mean
  2. Bfails the requirement, because the threshold lies fewer than 1.645 standard deviations below the meanCorrect
  3. Cfails the requirement, because the shortfall probability exceeds 20%

Explanation

The threshold is (8-0)/10 = 0.80 standard deviations below the mean. A 5% shortfall probability requires at least 1.645 standard deviations. Since 0.80 is less than 1.645, the shortfall probability is about 21%, above 5%. The first option reverses the comparison; the third gives a threshold that is too high, since 21% is not far above 20%, but its stated reason is that probability exceeds 20%, which is also true only marginally and is not the tested comparison.

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