hard · FRM Part 2 Market Risk

A portfolio's daily loss distribution is modeled as a Student-t with low degrees of freedom (heavy tails). An analyst observes that, for this portfolio, the ratio of 99% Expected Shortfall to 99% VaR is substantially larger than the value that would obtain under a normal distribution. A colleague argues that switching the risk measure from VaR to ES eliminates the need to worry about tail-heaviness because ES 'captures the average of the tail.'

Which statement most accurately characterizes the situation?

  1. ES is more sensitive to tail-heaviness than VaR, so a fatter tail widens the ES/VaR ratio; ES does not eliminate model risk because the ES estimate itself depends on the assumed shape of the extreme tail beyond the VaR threshold.
  2. Because ES averages losses beyond the VaR threshold, it is invariant to the degrees-of-freedom parameter, so the observed ES/VaR ratio increase must instead reflect an error in the VaR estimate.
  3. The elevated ES/VaR ratio proves the loss distribution is not subadditive at the 99% level, which means ES is not a coherent risk measure for heavy-tailed data and VaR should be retained.
  4. ES and VaR converge as tails become heavier because both are dominated by the same single worst-case quantile, so the ratio should approach one, contradicting the analyst's observation.

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