# Parametric VaR with Student-t distribution

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

• Hi Josh.V, welcome to Quant.SE! It's not clear to me what the issue is. What is it you want to achieve, do you want to use the formula, are you wondering whether it is suitable or something else? Note that a question such as that might still be off-topic, please check the faq to see what's on-topic. – Bob Jansen Aug 27 '15 at 16:25
• I've edited the question. Hopefully it is more clear now. I want to confirm how to implement Parametric VaR using a Student t-distribution. The formula is one that I have found but I'm not sure if its correct. Hopefully someone will be able to confirm if it is the correct formula, and if it is incorrect, what would be the correct formula to follow. – Josh.V Aug 27 '15 at 16:33
• It's better, so I reopened it, it might be too basic but it's good to have it answered. – Bob Jansen Aug 27 '15 at 16:35
• It is much clear now. I have changed the mean $X$ to $\mu$. – Gordon Aug 27 '15 at 16:45

## 1 Answer

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 $$VAR(T) = \nu/(\nu-2).$$ Thus you have to define $\sigma = sd * \sqrt{(\nu-2)/\nu}$ and a random variable $$X = \sigma T.$$ Then you will have that $$VAR(X) = VAR(T \sigma) = \sigma^2 VAR(T) = sd^2 (\nu-2)/\nu * \nu/(\nu-2) = sd^2.$$

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.