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FRM Part I · FRM Exam Part I · Measures of Financial Risk

A distortion risk measure uses the distortion function g(s) = s^0.5 applied to the survival function of a loss that takes value 0 with probability 0.75 and USD 100 with probability 0.25. What is the distorted expected loss, and how does it compare with the true expected loss of USD 25?

Distorting the survival probability 0.25 with the square root gives 0.5, and multiplying by the USD 100 loss gives USD 50. This exceeds the USD 25 expected loss because the concave distortion inflates tail probabilities, reflecting risk aversion.

  1. AUSD 25, equal to the expected loss because the loss has only two outcomes
  2. BUSD 50, higher than the expected loss, reflecting risk aversionCorrect
  3. CUSD 12.5, lower than the expected loss
  4. DUSD 75, because the distortion is applied to the probability of the zero outcome

Explanation

For a loss that is 0 or 100, the survival function is S(x)=0.25 for 0<=x<100 and 0 afterwards. The distorted measure is the integral of g(S(x)) dx from 0 to 100 = sqrt(0.25) x 100 = 0.5 x 100 = 50. Since g(s)=s^0.5 is above s on (0,1), tail probability is inflated, giving 50 > 25.

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