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Estimating 95% and 99% VaR from Simulation Samples

Article Quant Q&A · Author: techie11
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Summary

The document addresses how to identify a VaR quantile in a finite set of simulated outcomes. A quoted question compares 99% VaR estimated from 1,000 replications with 95% VaR, but gives an incorrect pair of order statistics for the latter.

The accepted response corrects the tail count: at a 95% confidence level, 5% of outcomes lie in the loss tail, corresponding to 50 observations out of 1,000. The estimate is therefore around the 50th worst simulated outcome, with interpolation conventions potentially determining whether adjacent sorted values are used. The exchange offers a basic order-statistic explanation rather than a broader treatment of sampling error. It does not quantify uncertainty or address how simulation design affects the reliability of extreme quantiles.

Key ideas

  • A 95% VaR places 5% of simulated outcomes in the loss tail.
  • With 1,000 replications, that tail contains 50 observations.
  • The relevant VaR estimate is near the 50th worst outcome, subject to quantile convention.
  • Extreme quantile estimates rely on relatively few tail observations and can be uncertain.

Tags

Risk managementStatisticsBacktesting
Full text
# A simple question about VaR estimation


# A simple question about VaR estimation












"A 99% VaR using 1,000 (simulation) replications should be expected to have only 10 observations in the left tail, which is not a large number. The VaR estimate is derived from the 10th and 11th sorted numbers. In contrast, a 95% VaR is measured from the 15th and 51st sorted numbers, which is more precise"

Could anybody enlighten me why 95% VaR estimation uses the 15th data point?

## Answer by AlRacoon (score 3, accepted)

https://quant.stackexchange.com/a/58018

I think there is a mistake in your definition. It should be between "50th and 51st" sorted numbers. 95% VAR means 5% is in the tail. 5% * 1000 = 50. The 95% VAR will be the 50th worst outcome of your 1000 simulations.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.