hard · FRM Part 1 Valuation and Risk Models

When computing ES from a discrete probability distribution, if the cumulative probability doesn't land exactly on (1 - α), which method is commonly used to ensure the ES remains a coherent risk measure?

1. Switch entirely to a parametric normal approximation so the model avoids discrete probability jumps at the tail boundary altogether.
2. Ignore the VaR threshold observation itself and average only the outcomes that strictly exceed that threshold value.
3. Always round up to the next worst observation to stay conservative, even when its probability exceeds the (1 - α) mass.
4. Include a fractional portion of the VaR threshold loss to ensure the total probability in the calculation exactly equals (1 - α).

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