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?
- 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.
- 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.
- 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.
- 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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