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

A portfolio has five equally likely loss outcomes (in $ millions): 0, 2, 4, 6 and 10. A spectral risk measure applies weights of 0.1, 0.1, 0.2, 0.2 and 0.4 to the losses sorted from smallest to largest. What is the value of the measure?

The measure is $6.2 million, found by multiplying each sorted loss by its weight and summing: 0.2 + 0.8 + 1.2 + 4.0. Equal weighting would give only $4.4 million, so the increasing weights add a risk-aversion premium for the larger losses.

  1. A$4.4 million
  2. B$6.2 millionCorrect
  3. C$2.8 million
  4. D$10.0 million

Explanation

Weighted sum = 0(0.1) + 2(0.1) + 4(0.2) + 6(0.2) + 10(0.4) = 0 + 0.2 + 0.8 + 1.2 + 4.0 = 6.2. The expected loss of 4.4 comes from equal weights of 0.2 and ignores risk aversion. Applying the weights in reverse order gives 2.8, which is wrong because larger weights must go to larger losses.

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