# Difference between uGarchRoll Value at Risk and manual calculations

I am forecasting an equity return density using the uGarchRoll commandin R studio, specifying the standardized t-distribution as error distribution. The uGarchRoll command already forecasts the Value at Risk. However I find that the values forecasted are not the same as those calculated by myself. Generally, the Value at Risk forecast should be the same number as if I would be using the qt(p, df) command? Or at least the destandardized value?

An example: uGarchRoll forecasts: mu=0.1262, sigma = 1.059, df=4.68, VaR (5%) = -1.51

By testing via the command qt(0.05,4.68) I receive -2.045887

Destandardized value would be: mu + sigma * qt = -2.045887

What am I getting wrong? The uGarchRoll command is pretty standard so the mistake cannot really lie in there can it?

For your purpose, you need a random variable with zero mean and unit variance. However, the variance of the Student-t distribution is $\frac{\nu}{\nu-2}$ for $\nu > 2$ and not one, where $\nu$ are the degrees of freedom.

You get the correct VaR by multiplying the quantile of the Student-t distribution with $\sqrt{\frac{\nu-2}{\nu}}$:

> 0.1262 + 1.059 * qt(0.05, 4.68) * sqrt((4.68-2) / 4.68)
[1] -1.51334


Alternatively, using the rugarch package which defaults to standardized distributions:

> rugarch::qdist("std", 0.05, mu=0.1262, sigma=1.059, shape=4.68)
[1] -1.51334