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Quantitative Risk Management / Loss Distributions, VaR, and Coherent Risk Measures

Expected Shortfall and Other Tail Risk Measures

VaR sets a threshold and counts whether losses cross it, but it is blind to what happens once that threshold is crossed. A portfolio whose losses are tightly clustered just above its 99% VaR level and one whose losses extend to catastrophic magnitudes in that 1% region have identical VaR figures — yet they are profoundly different risks.

Expected Shortfall (ES) — also called CVaR or tail conditional expectation in the literature — corrects this by averaging all the losses above the VaR threshold. It is the dominant tail risk measure in modern regulation: the Fundamental Review of the Trading Book (FRTB, Basel III finalised rules) replaced VaR with ES at 97.5% as the primary market-risk capital measure for internal models, effective from 2025 implementation.

This note: defines ES formally as an integral of quantiles; derives closed-form ES for the normal and Student tt distributions; introduces upper partial moments and the stop-loss transform; compares VaR and ES numerically to show ES's superior sensitivity to tail shape; and closes with an informal preview of the coherence axioms that give ES its axiomatic justification (fully developed in the next lesson).
Learning objectives. By the end of this note you should be able to:
  • state the integral definition of ES and explain its relationship to VaR;
  • prove the conditional-expectation representation ESα=E[L∣L≥VaRα]\text{ES}_\alpha = \mathbb{E}[L \mid L \ge \text{VaR}_\alpha] for continuous FLF_L;
  • compute ES for a normal loss distribution using the standard normal density;
  • compute ES for a Student tt loss distribution using the tt density and tail quantile;
  • explain why ES is more sensitive to tail heaviness than VaR is, using the asymptotic ratio ESα/VaRα\text{ES}_\alpha / \text{VaR}_\alpha as α→1\alpha \to 1;
  • describe upper partial moments and their role as alternative tail risk measures.

In this lesson

    1. Expected Shortfall: definition
    1. The conditional expectation representation
    1. ES in closed form
    1. Asymptotic behaviour: the ES/VaR ratio
    1. Other tail risk measures
    1. ES and coherence: a preview

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