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IAI Actuarial Core Principles · Economic Modelling · Mean-variance portfolio theory

An investor's one-year portfolio return is normally distributed with mean 12% and standard deviation 20%. The investor uses a 5% Value at Risk (loss threshold exceeded with 5% probability), taking the 95th percentile of the standard normal as 1.645. Measured as a return, what is the 5th percentile of the portfolio return, i.e. the return level below which the portfolio falls with 5% probability?

The 5th percentile return is the mean minus 1.645 standard deviations: 12% minus 32.9%, giving -20.9%. This is the return level that the portfolio falls below with only 5% probability under the normal assumption.

  1. A-20.9%
  2. B-17.0%
  3. C-8.9%Correct
  4. D-12.0%
  5. 32.9%

Explanation

The 5th percentile return = 12% - 1.645 x 20% = 12% - 32.9% = -20.9%. Checking the options: -20.9% is listed as the first option, so the key is -20.9%. Using 1.645 x 20% = 32.9 and adding gives 44.9%, not relevant; -8.9% would arise from subtracting 20.9 from 12 incorrectly... the correct figure is -20.9%.

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