medium · 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 treated as invariant to the degrees-of-freedom parameter of the Student-t, so the higher observed ES/VaR ratio must instead signal an error in the VaR estimate itself.
  3. The elevated ES/VaR ratio is taken as proof that the loss distribution violates subadditivity at the 99% confidence level, meaning ES is therefore not a coherent risk measure for heavy-tailed data and VaR should be kept instead.
  4. ES and VaR are assumed to converge as the return tails grow heavier because both risk measures become dominated by the very same single worst-case quantile outcome, so their ratio should approach one as the degrees of freedom fall further.

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