FRM: Tail risk

Explanations, applications and practice for tail risk.

• Value-at-risk aggregation counterexample

  Use a coherent joint model before comparing individual and combined loss quantiles.

  Worked examples

• Probability atoms in a loss distribution

  Identify an outcome carrying positive probability before computing a tail average.

  Concepts

• Read value at risk from a discrete loss table

  Order losses and use a declared cumulative-quantile convention.

  Lessons

• Communicating a tail-risk model result

  Attach parameters, estimation status and limitations to a threshold or tail-average number.

  Careers

• Expected shortfall and threshold conditional averages

  Compare exact selected-tail mass with strict and inclusive loss subsets.

  Comparisons

• Confidence level and selected tail probability

  Separate a percentile parameter from the mass averaged in expected shortfall.

  Concepts

• Compare tail risk across confidence levels

  Keep the loss model fixed while changing the selected quantile and tail mass.

  Lessons

• Finite-loss expected shortfall from tail mass

  Weight each loss by the portion of its probability selected in the worst tail.

  Formulas

• Expected-shortfall and boundary-mass practice

  Test exact tail probability, weighted averages and conditional subsets.

  Practice

• Expected shortfall with discrete loss outcomes

  Average exactly the selected worst-tail probability rather than an arbitrary threshold subset.

  Lessons

• Exact cumulative boundary in a loss quantile

  Apply ≥ rather than > in the declared lower-quantile rule.

  Worked examples
• 

  Historical loss record review

  Download a checklist for dates, current exposures, observation weights and quantile convention.

  Templates

• Historical-simulation value at risk: data and rank convention

  Distinguish observed outcomes, current exposures and a chosen empirical quantile.

  Lessons

• Scaling discrete tail losses by square root of time

  State a multi-period model before changing a loss horizon.

  Common mistakes

• Recording the horizon and unit of a tail-risk input

  Keep a calculator’s number tied to a meaningful loss definition.

  Tool guides
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  Loss model input record

  Download a record for scenario probabilities, loss signs and estimation status.

  Templates

• Interpreting discrete loss VaR and expected shortfall

  Read the output unit, quantile rule and selected-tail denominator together.

  Tool guides

• Joint-loss and risk-aggregation practice

  Test a stated dependence model before combining tail measures.

  Practice
• 

  Joint loss and tail allocation record

  Download the original independent-pair aggregation example and grouped tail bridge.

  Templates

• Loss-quantile jump at a cumulative boundary

  Explain a large threshold jump after a small confidence change.

  Worked examples

• Finite-model maximum loss and value at risk

  Distinguish the largest listed outcome from a confidence threshold.

  Concepts

• Confusing mean loss with expected shortfall

  Use the selected-tail denominator rather than the full model probability.

  Common mistakes

• Declared loss model and estimated loss distribution

  Separate arithmetic validation from evidence for the inputs.

  Comparisons

• Negative loss quantiles in an all-gain model

  Interpret a signed risk measure without replacing gains with zero.

  Concepts

• Partial boundary mass in expected shortfall

  Allocate only the needed fraction of a discrete boundary outcome.

  Worked examples

• Lower cumulative loss quantile

  State the loss-side percentile convention before selecting a discrete threshold.

  Formulas

• Value-at-risk and loss-quantile practice

  Test loss signs, cumulative crossings and equal-weight rank conventions.

  Practice

• Equal-weight loss ranks and tail counts

  Apply a stated empirical rank rule before selecting observations for a tail average.

  Worked examples

• Rare severe loss beyond a zero VaR

  Show why a zero threshold does not mean the loss model is harmless.

  Worked examples

• Tail-risk parameter and allocation checklist

  Review signs, probabilities, quantile convention and exact tail mass before interpreting a result.

  Revision

• Loss-quantile to tail-average study sequence

  Build a signed loss table, select the quantile and explain a boundary allocation.

  Study plans

• Same VaR with different tail severity

  Compare loss models sharing a quantile but having different severe outcomes.

  Worked examples

• Define loss signs before reading tail risk

  State what a positive or negative number means before ordering outcomes.

  Lessons

• Strict and inclusive VaR exceedance

  Keep the probability of L>q separate from L≥q when a threshold has positive mass.

  Concepts

• Averaging all losses above a VaR threshold

  Check the selected probability mass before calling an average expected shortfall.

  Common mistakes
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  Tail-mass reconciliation

  Download an allocation bridge from scenario probability to expected shortfall.

  Templates

• Checking expected shortfall with a threshold objective

  Use excess loss as an independent check on a discrete tail allocation.

  Worked examples

• Expected shortfall as a threshold objective

  Relate a loss cutoff and expected excess loss to the exact-tail average.

  Formulas

• Merging identical loss outcomes

  Check that scenario labels do not change a loss distribution.

  Worked examples
• 

  Discrete loss VaR and expected shortfall calculator

  Inspect a cumulative loss quantile and exact selected-tail allocation.

  Tools