VaR is subject to a significant positive bias

Koji Inui, Masaaki Kijima, Atsushi Kitano

Research output: Contribution to journalArticlepeer-review

12 Citations (Scopus)


This article shows that value-at-risk (VaR), the most popular risk measure in financial practice, has a considerable positive bias when used for a portfolio with fat-tail distribution. The bias increases with higher confidence level, heavier tails, and smaller sample size. Also, the Harrell-Davis quantile estimator and its simulation counterpart, called the bootstrap estimator, tend to have a more significant positive bias for fat-tail distributions.

Original languageEnglish
Pages (from-to)299-311
Number of pages13
JournalStatistics and Probability Letters
Issue number4
Publication statusPublished - 15 May 2005


  • Concave ordering
  • Harrell-Davis estimator
  • Historical simulation
  • Value-at-risk


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