Quantitative Risk Management: A Practical Guide to Financial Risk, + Website

Appendix 8.1: Small-Sample Distribution of VaR and Standard Errors

Cramér (1974, para 28.5 and para 28.6) discusses the distribution of observed quantiles and extreme values or order statistics.

Distribution of Quantiles

Consider a sample of n observations {x_i} from a one-dimensional distribution, with the Z percent quantile q_z (for example, we might have Z = 0.01 and for a standard normal distribution the quantile q_z = −2.3263). If n*Z is not an integer and the observations are arranged in ascending order, {x₁ ≤ x₂ ≤...≤ x_n}, then there is a unique quantile equal to the observed value x_{μ +1} where μ = integer smaller than n*Z.⁴⁵ Then the density of the observed quantile (x_{μ +1}) is

(8.2)

where

F(x) = underlying distribution function

f(x) = density function

This expression can be integrated numerically to find the mean, variance, and any confidence bands desired for any quantile, given an underlying distribution F(x). But the use of (8.2) is limited because it applies only to sample quantiles when n*Z is not an integer. With 100 observations the Z = .01 quantile will be indeterminate between the first and second observations and formula (8.2) cannot be used. Either the first or the second observation could be used to estimate the 1 percent-quantile, and expression (8.4) below could be applied, but neither the first nor the second observation is ideal as an estimator of the 1 percent ...