hard · FRM Part 1 Valuation and Risk Models
A bank computes the expected shortfall (ES) at the 97.5% level for a loss distribution that is a 50/50 mixture of two normals: a 'normal-regime' component with low variance and a 'stress-regime' component with high variance and the same mean. A junior analyst approximates the portfolio ES by taking the 50/50 weighted average of each component's standalone 97.5% ES.
Why is this approximation generally biased, and in which direction for this symmetric-mean mixture?
- It is unbiased, because ES is a coherent (subadditive) measure and therefore aggregates linearly across mixture components
- It overstates the true ES, because averaging the standalone ES double-counts the stress component's tail mass beyond the 97.5% point
- It understates the true ES, because the mixture's 97.5% tail is dominated by the stress component, whose standalone 97.5% quantile is breached far deeper than at 97.5%
- It understates the true ES, because subadditivity guarantees the portfolio ES is below the average of the components' ES
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