Parametric VaR with Student-t distribution

Question

Im using VaR to estimate parametric VaR. I have been able to do this using a Normal Distribution, however I want to also do this using a Student t-distribution and I'm unsure how to implement that in Matlab.

I have a dataset of portfolio values, I have log returns and returns as well as mean and standard deviation. The only suggested method I was able to find elsewhere was the following:

$\mu - t_\alpha * (sd/\sqrt{n})$

Where $\mu$ is the mean, sd is the standard deviation, t is the t stat at the alpha level and n is the number of returns. This would give the lower level of the alpha confidence interval.

Can anyone confirm if this is the correct or incorrect method to implement parametric VaR using a t-distribution. If it is incorrect, how would I be able to implement it?

Thanks

Answer 1

You got some things wrong:

1. You don't have to devide sd by $\sqrt{n}$, the division is already part of the definition of $sd$.
2. The $t$ distribution has a parameter $\nu$, the degrees of freedom.
3. The variance of a standard $t$ distributed random variable $T$ is $$ VAR(T) = \frac{\nu}{(\nu-2)}. $$ Thus you have to define $$\sigma = sd * \sqrt{\frac{(\nu-2)}{\nu}}$$ and a random variable $$ X = \sigma T. $$ Then you will have that $$ VAR(X) = VAR(T \sigma) = \sigma^2 VAR(T) = sd^2 \frac{(\nu-2)}{\nu} \times \frac{\nu}{(\nu-2)} = sd^2. $$

For VaR you estimate $\nu$ and $sd$ and look at $$ \mu - \sigma t_{\alpha}, $$ where $\sigma$ is defined above and $t_{\alpha}$ is the quantile of a t-distribution with $\nu$ degrees of freedom.