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FRM Part I · FRM Exam Part I · Random Variables

A portfolio's one-day profit-and-loss in USD millions has a continuous CDF. An analyst states that the 5% VaR is 8, meaning F_L(8) = 0.95 for the loss L. A risk manager says the loss variable is L = −P, where P is profit, and the profit CDF is F_P. Which statement about F_P is correct?

F_P(−8) = 0.05, so the 5% quantile of profit is −8. A 95% loss quantile of 8 means there is a 95% chance the loss is at most 8, equivalently a 95% chance profit is at least −8, leaving 5% probability below −8.

  1. AF_P(−8) = 0.05, so the 5% quantile of profit is −8Correct
  2. BF_P(8) = 0.05, so the 5% quantile of profit is 8
  3. CF_P(−8) = 0.95, so the 95% quantile of profit is −8
  4. DF_P(8) = 0.95, so the 5% quantile of profit is −8

Explanation

P(L ≤ 8) = 0.95 means P(−P ≤ 8) = P(P ≥ −8) = 0.95, so P(P < −8) = 0.05 for a continuous variable, giving F_P(−8) = 0.05. The 5% profit quantile is therefore −8. The option with F_P(−8) = 0.95 reverses the tail, and the option with F_P(8) = 0.05 ignores the sign flip.

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