Bootstrapowe metody estymacji wartości zagrożonej ryzykiem

Interval bootstrap methods can be used to estimate Value at Risk, defined as a quantile of fixed order of random variable being the value of losses from investments. These methods are applied when there is no information about the distribution class of the variable considered, which  is the advantage of bootstrap methods compared with parametric methods. Semiparametric estimation procedures are of particular importance. They can be used in the estimation of high-order quantiles. They guarantee the occurrence of large values in the generated bootstrap samples. The paper presents nonparametric and semiparametric bootstrap estimation methods and the results of simulation studies for higher-order quantiles of a heavy-tailed distribution. The application of the methods analysed provide confidence intervals with greater accuracy compared to the nonparametric classical method. The procedures under discussion are used in VaR estimation of daily returns of selected shares at Warsaw Stock Exchange.