medium · 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 to a parametric normal approximation to avoid discrete probability jumps.
2. Ignore the VaR threshold value and only average the outcomes that strictly exceed it.
3. Always round up to the next observation to be conservative, even if it exceeds the (1 - α) probability threshold.
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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