Lesson 4 of 23 · All lessons in this module
  1. The central limit theorem
  2. The standard error of the sample meanVideo lesson
  3. Confidence intervals for the mean
  4. Value at risk: a one-sided confidence interval
  5. Populations, samples and sampling error
  6. Stratified and cluster sampling
  7. Good estimators, and the CLT across sampling designs
  8. Convenience and judgmental sampling, and choosing a method
  9. The four steps and stating the hypotheses
  10. Type I and Type II errors, significance and powerVideo lesson
  11. Critical values, p-values and the decision ruleVideo lesson
  12. p-values and p-hacking
  13. Testing a single mean
  14. Two means: independent vs paired samplesVideo lesson
  15. Testing variances: chi-square and FVideo lesson
  16. The t-test for a Pearson correlation
  17. Sample size, p-values and many correlations at once
  18. Parametric vs nonparametric tests
  19. The Spearman rank correlation testVideo lesson
  20. Contingency tables and the chi-square test of independenceVideo lesson
  21. Interpreting the result: standardized residuals and mosaics
  22. Choosing the test: four steps and a map of every test
  23. Integrated cases: testing whether a new strategy beats the old one

This lesson is part of the 2027 curriculum. You are following the 2026 curriculum, where it is not taught in this form. Switch if you are sitting the exam under the 2027 curriculum.

Lesson 4 of 23 · 13 min

Value at risk: a one-sided confidence interval

Value at risk turns a one-sided confidence interval into a risk number: the loss a position should not exceed over a set horizon at a chosen confidence level.

In short

  • A confidence interval for a mean is only meaningful when the data have a stable mean and a finite, stable variance. Trending price levels fail that test, so analysts work with returns instead.
  • Value at risk (VaR) is the lower bound of a one-sided confidence interval: a 99% one-month VaR of EUR 2 million means a 1% chance of losing more than EUR 2 million in a month.
  • VaR can be stated as a percentage of value or as a money amount.
  • Parametric (variance–covariance) VaR with a zero mean return: VaR=z×σ×P\text{VaR} = z \times \sigma \times P, with one-sided z of 1.645 (95%) or 2.33 (99%).
  • Other methods: historical simulation and Monte Carlo simulation.
  • Limits of parametric VaR: assumes normal returns, the same distribution at every horizon, that history is a guide, and linear exposures (a poor fit for options).

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